{"nodes":[{"data_url":"../dependencies.json","id":"roadmap","kind":"scope","result":false,"route":"../roadmap/index.html","scope":"roadmap","status":"not_ready","summary":"Trough maps Lean formalization libraries: what each one formalizes and how its results depend on one another, not what the library may add. Each library gets the article unit that suits it, such as curated concepts for Mathlib, the declarations its roadmaps name and the results its docstrings name for Tau Ceti, blueprint nodes for the research projects, headline results for Formal Frontier and project cards for\u2026","title":"Trough"},{"data_url":"chapters/mathlib.json","id":"mathlib","kind":"scope","parent":0,"result":false,"route":"../roadmap/mathlib/index.html","scope":"mathlib","status":"not_ready","summary":"Mathlib's maintainers keep four curated lists of what the library contains: an overview of its main concepts, an undergraduate curriculum, Freek Wiedijk's 100 theorems, and the theorems of the 1000+ theorems project. This chapter has one article per concept on those lists that Mathlib has, read at Mathlib commit 9bac2d06 (2026-11-09): 887 articles, 1 of them stated in Mathlib without a proof. Concepts on the lists\u2026","title":"Mathlib"},{"data_url":"scopes/mathlib/overview.json","id":"mathlib/overview","kind":"scope","parent":1,"result":false,"route":"../roadmap/mathlib/overview/index.html","scope":"mathlib/overview","status":"mathlib","summary":"Mathlib's own overview of what it contains, area by area (docs/overview.yaml).","title":"Mathlib overview"},{"data_url":"scopes/mathlib/overview/general-algebra.json","id":"mathlib/overview/general-algebra","kind":"scope","parent":2,"result":false,"route":"../roadmap/mathlib/overview/general-algebra/index.html","scope":"mathlib/overview/general-algebra","status":"mathlib","title":"General algebra"},{"data_url":"scopes/mathlib/overview/general-algebra/category-theory.json","id":"mathlib/overview/general-algebra/category-theory","kind":"scope","parent":3,"result":false,"route":"../roadmap/mathlib/overview/general-algebra/category-theory/index.html","scope":"mathlib/overview/general-algebra/category-theory","status":"mathlib","title":"Category theory"},{"id":"mathlib/overview/general-algebra/category-theory/category","kind":"node","parent":4,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/category-theory/index.html#category","scope":"mathlib/overview/general-algebra/category-theory","status":"mathlib","summary":"A category consists of objects and of arrows, called morphisms, between them, together with a rule for composing arrows end to end that is associative and has an identity arrow at every object. Sets with functions, groups with homomorphisms, and topological spaces with continuous maps are all categories. The notion lets constructions and arguments that recur across algebra, topology and logic be stated once, in\u2026","title":"Category"},{"id":"mathlib/overview/general-algebra/category-theory/small-category","kind":"node","parent":4,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/category-theory/index.html#small-category","scope":"mathlib/overview/general-algebra/category-theory","status":"mathlib","summary":"A small category is a category whose objects form a set and whose morphisms form a set, rather than a proper class. Smallness is the size condition under which a category can be handled like any other set-sized mathematical object: diagrams are indexed by small categories, functors from a small category to a locally small category form a locally small category, and categories of presheaves on a small category are\u2026","title":"Small category"},{"id":"mathlib/overview/general-algebra/category-theory/functor","kind":"node","parent":4,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/category-theory/index.html#functor","scope":"mathlib/overview/general-algebra/category-theory","status":"mathlib","summary":"A functor is a structure-preserving map between two categories: it sends objects to objects and morphisms to morphisms, respecting identities and composition. Functors are the basic way of comparing categories, and many constructions of algebra and topology, such as taking the fundamental groupoid of a space, the homology of a chain complex or the underlying set of a group, are functors. Saying that a construction\u2026","title":"Functor"},{"id":"mathlib/overview/general-algebra/category-theory/natural-transformation","kind":"node","parent":4,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/category-theory/index.html#natural-transformation","scope":"mathlib/overview/general-algebra/category-theory","status":"mathlib","summary":"A natural transformation is a morphism between two functors with the same source and target categories. It assigns to each object of the source a morphism in the target, and these morphisms are compatible with every morphism of the source. The notion makes precise the informal idea of a construction that is \"natural\", one defined uniformly without arbitrary choices, and it is the basic ingredient of functor\u2026","title":"Natural transformation"},{"id":"mathlib/overview/general-algebra/category-theory/yoneda-embedding","kind":"node","parent":4,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/category-theory/index.html#yoneda-embedding","scope":"mathlib/overview/general-algebra/category-theory","status":"mathlib","summary":"The Yoneda embedding of a category $\\mathcal{C}$ sends each object $X$ to the presheaf of morphisms into $X$, the contravariant functor $Y \\mapsto \\mathrm{Hom}(Y, X)$ to sets. It is fully faithful, so $\\mathcal{C}$ sits inside the category of presheaves on $\\mathcal{C}$ without losing or gaining any morphisms. The underlying result, the Yoneda lemma, identifies the natural transformations from the presheaf of\u2026","title":"Yoneda embedding"},{"id":"mathlib/overview/general-algebra/category-theory/isomorphism","kind":"node","parent":4,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/category-theory/index.html#isomorphism","scope":"mathlib/overview/general-algebra/category-theory","status":"mathlib","summary":"An isomorphism in a category is a morphism that can be undone: a morphism $f\\colon X \\to Y$ for which there is a morphism back from $Y$ to $X$ whose composites with $f$, in either order, are identities. Two objects joined by an isomorphism are indistinguishable by anything the category can express, so isomorphism, rather than equality, is the natural notion of sameness for objects. It specializes to bijections of\u2026","title":"Isomorphism"},{"id":"mathlib/overview/general-algebra/category-theory/adjunction","kind":"node","parent":4,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/category-theory/index.html#adjunction","scope":"mathlib/overview/general-algebra/category-theory","status":"mathlib","summary":"An adjunction is a relationship between two functors running in opposite directions between two categories, $F \\colon \\mathcal{C} \\to \\mathcal{D}$ and $G \\colon \\mathcal{D} \\to \\mathcal{C}$, in which morphisms $F(X) \\to Y$ in $\\mathcal{D}$ correspond exactly, naturally in $X$ and $Y$, to morphisms $X \\to G(Y)$ in $\\mathcal{C}$. The functor $F$ is then called the left adjoint and $G$ the right adjoint, and one\u2026","title":"Adjunction"},{"id":"mathlib/overview/general-algebra/category-theory/monad","kind":"node","parent":4,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/category-theory/index.html#monad","scope":"mathlib/overview/general-algebra/category-theory","status":"mathlib","summary":"A monad on a category $\\mathcal{C}$ is an endofunctor $T$ of $\\mathcal{C}$ together with two natural transformations, a unit $\\eta \\colon 1_{\\mathcal{C}} \\to T$ and a multiplication $\\mu \\colon T T \\to T$, subject to associativity and unit laws that mirror those of a monoid. Monads arise from every adjunction, and conversely every monad arises from one. They give a uniform language for algebraic structure: groups,\u2026","title":"Monad"},{"id":"mathlib/overview/general-algebra/category-theory/comma-category","kind":"node","parent":4,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/category-theory/index.html#comma-category","scope":"mathlib/overview/general-algebra/category-theory","status":"mathlib","summary":"A comma category is a category built from two functors with a common target: its objects are morphisms from the image of the first functor to the image of the second, and its morphisms are commutative squares relating them. The construction packages many familiar categories in one definition, among them slice and coslice categories, arrow categories and categories of structured arrows, and it gives a uniform\u2026","title":"Comma category"},{"id":"mathlib/overview/general-algebra/category-theory/limits","kind":"node","parent":4,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/category-theory/index.html#limits","scope":"mathlib/overview/general-algebra/category-theory","status":"mathlib","summary":"A limit of a diagram in a category is an object equipped with morphisms to every object of the diagram, compatible with the arrows of the diagram, through which every other such compatible family factors in exactly one way. Products, equalizers, pullbacks, terminal objects and inverse limits are all special cases, so the single notion lets one state and prove facts about all of them at once. The dual notion,\u2026","title":"Limits"},{"id":"mathlib/overview/general-algebra/category-theory/presheafed-space","kind":"node","parent":4,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/category-theory/index.html#presheafed-space","scope":"mathlib/overview/general-algebra/category-theory","status":"mathlib","summary":"A presheafed space is a topological space together with a presheaf on it, an assignment of an object of a fixed category $\\mathcal{C}$ to each open set, with restriction maps between them. It is a very general setting for \"a space with functions on it\", with the functions replaced by objects of an arbitrary category, and the structures of geometry are built from it by adding conditions: a sheafed space asks the\u2026","title":"Presheafed space"},{"id":"mathlib/overview/general-algebra/category-theory/sheafed-space","kind":"node","parent":4,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/category-theory/index.html#sheafed-space","scope":"mathlib/overview/general-algebra/category-theory","status":"mathlib","summary":"A sheafed space is a topological space together with a sheaf on it with values in a fixed category $\\mathcal{C}$: an object of $\\mathcal{C}$ for each open set, with restriction maps, such that data given locally on an open cover and agreeing on overlaps glue uniquely. It is the common frame for spaces described by their local functions. Smooth manifolds with their smooth functions, and spectra of commutative rings\u2026","title":"Sheafed space"},{"id":"mathlib/overview/general-algebra/category-theory/monoidal-category","kind":"node","parent":4,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/category-theory/index.html#monoidal-category","scope":"mathlib/overview/general-algebra/category-theory","status":"mathlib","summary":"A monoidal category is a category equipped with a way of multiplying objects, a tensor product $X \\otimes Y$, together with a unit object $I$, such that the product is associative and unital up to specified, well-behaved isomorphisms. The notion abstracts the cartesian product of sets, the tensor product of vector spaces and the composition of functors, and it is the setting in which one can speak of monoids and\u2026","title":"Monoidal category"},{"id":"mathlib/overview/general-algebra/category-theory/cartesian-closed","kind":"node","parent":4,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/category-theory/index.html#cartesian-closed","scope":"mathlib/overview/general-algebra/category-theory","status":"mathlib","summary":"A cartesian closed category is a category with finite products in which, for any two objects $A$ and $B$, there is an object $B^A$ of \"morphisms from $A$ to $B$\" that behaves like a function space: morphisms $A \\times Y \\to B$ correspond exactly to morphisms $Y \\to B^A$. The category of sets is the basic example, with $B^A$ the set of all functions. Cartesian closed categories are a natural setting for currying\u2026","title":"Cartesian closed"},{"id":"mathlib/overview/general-algebra/category-theory/abelian-category","kind":"node","parent":4,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/category-theory/index.html#abelian-category","scope":"mathlib/overview/general-algebra/category-theory","status":"mathlib","summary":"An abelian category is a category whose morphisms can be added and in which kernels, cokernels, images and exact sequences behave as they do for modules over a ring. The notion isolates the formal properties of abelian groups and modules that homological algebra actually uses, so that constructions such as chain complexes, homology and derived functors can be carried out once, uniformly, for modules, sheaves,\u2026","title":"Abelian category"},{"id":"mathlib/overview/general-algebra/category-theory/grothendieck-topology","kind":"node","parent":4,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/category-theory/index.html#grothendieck-topology","scope":"mathlib/overview/general-algebra/category-theory","status":"mathlib","summary":"A Grothendieck topology on a category specifies, for each object, which collections of morphisms into it count as coverings of that object. It abstracts the way the open subsets of a topological space cover larger open sets, keeping only the formal properties that the notion of a sheaf needs. A category equipped with a Grothendieck topology is called a site, and sheaves on sites are the setting for \u00e9tale\u2026","title":"Grothendieck topology"},{"data_url":"scopes/mathlib/overview/general-algebra/numbers.json","id":"mathlib/overview/general-algebra/numbers","kind":"scope","parent":3,"result":false,"route":"../roadmap/mathlib/overview/general-algebra/numbers/index.html","scope":"mathlib/overview/general-algebra/numbers","status":"mathlib","title":"Numbers"},{"id":"mathlib/overview/general-algebra/numbers/natural-number","kind":"node","parent":21,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/numbers/index.html#natural-number","scope":"mathlib/overview/general-algebra/numbers","status":"mathlib","summary":"The natural numbers are the numbers $0, 1, 2, 3, \\dots$ used to count the elements of finite collections and to number the positions in a sequence. Their collection is written $\\mathbb{N}$. They are where arithmetic begins: the integers, the rational numbers and the real numbers are all constructed from them, and proof by induction, one of the basic methods of mathematical argument, is a property of $\\mathbb{N}$.\u2026","title":"Natural number"},{"id":"mathlib/overview/general-algebra/numbers/integer","kind":"node","parent":21,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/numbers/index.html#integer","scope":"mathlib/overview/general-algebra/numbers","status":"mathlib","summary":"The integers are the whole numbers $\\dots, -2, -1, 0, 1, 2, \\dots$: the natural numbers, their negatives, and zero. Together they form a set written $\\mathbb{Z}$. They extend the natural numbers so that subtraction is always possible, and they are the basic example of a commutative ring. Divisibility, primes and congruences are studied in $\\mathbb{Z}$, and every ring receives exactly one structure-preserving map\u2026","title":"Integer"},{"id":"mathlib/overview/general-algebra/numbers/rational-number","kind":"node","parent":21,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/numbers/index.html#rational-number","scope":"mathlib/overview/general-algebra/numbers","status":"mathlib","summary":"A rational number is a number that can be written as a quotient $a/b$ of two integers with $b \\neq 0$. The rational numbers form the set $\\mathbb{Q}$, the smallest number system that contains the integers and in which division by any nonzero number is always possible. They are the most familiar example of a field, they sit between the integers and the real numbers, and both the real and the $p$-adic numbers are\u2026","title":"Rational number"},{"id":"mathlib/overview/general-algebra/numbers/continued-fraction","kind":"node","parent":21,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/numbers/index.html#continued-fraction","scope":"mathlib/overview/general-algebra/numbers","status":"mathlib","summary":"A continued fraction is an expression of the form","title":"Continued fraction"},{"id":"mathlib/overview/general-algebra/numbers/real-number","kind":"node","parent":21,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/numbers/index.html#real-number","scope":"mathlib/overview/general-algebra/numbers","status":"mathlib","summary":"The real numbers are the numbers used to measure continuous quantities such as length, time and mass; together they make up the real line $\\mathbb{R}$. They extend the rational numbers by filling the gaps between them, so that quantities such as $\\sqrt 2$ and $\\pi$ have a place on the line. Algebraically they form an ordered field, and what sets them apart is completeness: every nonempty set of real numbers that is\u2026","title":"Real number"},{"id":"mathlib/overview/general-algebra/numbers/extended-real-number","kind":"node","parent":21,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/numbers/index.html#extended-real-number","scope":"mathlib/overview/general-algebra/numbers","status":"mathlib","summary":"The extended real numbers are the real numbers together with two further elements, $+\\infty$ and $-\\infty$, placed above and below every real number. The resulting set, written $\\overline{\\mathbb{R}}$ or $[-\\infty, +\\infty]$, is the set of extended real numbers. Adjoining the two infinities makes the order complete: every subset acquires a supremum and an infimum. This makes the extended reals a natural home for\u2026","title":"Extended real number"},{"id":"mathlib/overview/general-algebra/numbers/complex-number","kind":"node","parent":21,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/numbers/index.html#complex-number","scope":"mathlib/overview/general-algebra/numbers","status":"mathlib","summary":"A complex number is an expression $a + bi$ in which $a$ and $b$ are real numbers and $i$ is a number whose square is $-1$. The complex numbers form a field, written $\\mathbb{C}$, that contains the real numbers and in which every nonconstant polynomial equation has a solution. This makes $\\mathbb{C}$ the natural setting for algebra, for the theory of analytic functions, and for much of number theory, geometry and\u2026","title":"Complex number"},{"id":"mathlib/overview/general-algebra/numbers/p-adic-number","kind":"node","parent":21,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-adic number"]],"source":"$p$-adic number"},"result":true,"route":"../roadmap/mathlib/overview/general-algebra/numbers/index.html#p-adic-number","scope":"mathlib/overview/general-algebra/numbers","status":"mathlib","summary":"For a prime number $p$, the $p$-adic numbers form a field $\\mathbb{Q}_p$ that contains the rational numbers and is complete for a notion of size in which a number is small when it is divisible by a high power of $p$. Just as the real numbers arise by filling in the gaps of $\\mathbb{Q}$ for the ordinary absolute value, $\\mathbb{Q}_p$ arises by filling them in for the $p$-adic absolute value. The $p$-adic numbers let\u2026","title":"$p$-adic number"},{"id":"mathlib/overview/general-algebra/numbers/p-adic-integer","kind":"node","parent":21,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-adic integer"]],"source":"$p$-adic integer"},"result":true,"route":"../roadmap/mathlib/overview/general-algebra/numbers/index.html#p-adic-integer","scope":"mathlib/overview/general-algebra/numbers","status":"mathlib","summary":"For a prime number $p$, the $p$-adic integers form the ring $\\mathbb{Z}_p$ of $p$-adic numbers whose $p$-adic absolute value is at most $1$. They play for the field $\\mathbb{Q}_p$ the role that the integers play for the rational numbers, and they encode, in a single ring, all the congruences modulo $p, p^2, p^3, \\dots$ at once. This makes $\\mathbb{Z}_p$ a basic tool of number theory: a problem about solving a\u2026","title":"$p$-adic integer"},{"id":"mathlib/overview/general-algebra/numbers/hyper-real-number","kind":"node","parent":21,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/numbers/index.html#hyper-real-number","scope":"mathlib/overview/general-algebra/numbers","status":"mathlib","summary":"The hyper-real numbers, or hyperreals, form an ordered field $^*\\mathbb R$ that contains the real numbers together with positive infinitesimals, numbers greater than $0$ but smaller than every positive real, and infinite elements, numbers larger than every real. They are the number system of nonstandard analysis. In $^*\\mathbb R$ a derivative can be computed from a quotient of infinitesimal changes and a limit can\u2026","title":"Hyper-real number"},{"data_url":"scopes/mathlib/overview/general-algebra/group-theory.json","id":"mathlib/overview/general-algebra/group-theory","kind":"scope","parent":3,"result":false,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/index.html","scope":"mathlib/overview/general-algebra/group-theory","status":"mathlib","summary":"Group theory is the study of groups: sets with an associative multiplication, an identity element and inverses. Groups record the symmetries of an object, whether the rotations of a polyhedron, the permutations of the roots of a polynomial or the automorphisms of a vector space, and they turn up in nearly every part of mathematics. The subject asks how groups are built from smaller ones, how they act on other\u2026","title":"Group theory"},{"data_url":"scopes/mathlib/overview/general-algebra/group-theory/part-01.json","id":"mathlib/overview/general-algebra/group-theory/part-01","kind":"scope","navigation":"transparent","parent":32,"result":false,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-01/index.html","scope":"mathlib/overview/general-algebra/group-theory/part-01","status":"mathlib","title":"Group to Nilpotent group"},{"id":"mathlib/overview/general-algebra/group-theory/part-01/group","kind":"node","parent":33,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-01/index.html#group","scope":"mathlib/overview/general-algebra/group-theory/part-01","status":"mathlib","summary":"A group is a set equipped with an associative binary operation that has an identity element and in which every element has an inverse. Groups are the algebraic language of symmetry: the symmetries of any object, composed one after another, form a group. They are also the common skeleton of many other structures, since the integers under addition, the nonzero real numbers under multiplication and the invertible\u2026","title":"Group"},{"id":"mathlib/overview/general-algebra/group-theory/part-01/group-morphism","kind":"node","parent":33,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-01/index.html#group-morphism","scope":"mathlib/overview/general-algebra/group-theory/part-01","status":"mathlib","summary":"A group morphism, or group homomorphism, is a map between two groups that respects the group operation. Morphisms are how groups are compared: they describe when one group is a copy of another, when one is a quotient of another, and how a group can be represented by permutations or matrices. Much of group theory, from normal subgroups to the first, second and third isomorphism theorems, is organized around them.","title":"Group morphism"},{"id":"mathlib/overview/general-algebra/group-theory/part-01/group-action","kind":"node","parent":33,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-01/index.html#group-action","scope":"mathlib/overview/general-algebra/group-theory/part-01","status":"mathlib","summary":"A group action of a group $G$ on a set $X$ lets each element of $G$ move the points of $X$, in a way that respects the group law: acting by a product is the same as acting by the factors one after the other. Groups usually arise as groups of symmetries of something, and an action records exactly that relationship. Actions let one study a group through the sets it permutes, and a set through its symmetries, and they\u2026","title":"Group action"},{"id":"mathlib/overview/general-algebra/group-theory/part-01/class-formula","kind":"node","parent":33,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-01/index.html#class-formula","scope":"mathlib/overview/general-algebra/group-theory/part-01","status":"mathlib","summary":"The class formula describes how a set on which a group acts breaks up into pieces. The set is the disjoint union of its orbits, and each orbit is a copy of the set of cosets of the stabilizer of any of its points. For a finite group acting on a finite set this turns into a counting rule: the size of the set is a sum of indices of subgroups, and each term divides the order of the group. Applied to the action of a\u2026","title":"Class formula"},{"id":"mathlib/overview/general-algebra/group-theory/part-01/burnside-lemma","kind":"node","parent":33,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-01/index.html#burnside-lemma","scope":"mathlib/overview/general-algebra/group-theory/part-01","status":"mathlib","summary":"Burnside's lemma, also called the Cauchy\u2013Frobenius lemma or the orbit-counting lemma, says that when a finite group acts on a finite set, the number of orbits equals the average number of points fixed by an element of the group. It is the basic tool for counting objects up to symmetry: colourings of a necklace up to rotation, of a cube up to rigid motion, or graphs up to relabelling of their vertices. It is also\u2026","title":"Burnside lemma"},{"id":"mathlib/overview/general-algebra/group-theory/part-01/subgroup","kind":"node","parent":33,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-01/index.html#subgroup","scope":"mathlib/overview/general-algebra/group-theory/part-01","status":"mathlib","summary":"A subgroup of a group $G$ is a subset of $G$ that is itself a group under the same operation. Subgroups are the basic way of locating smaller groups inside a larger one: much of the structure theory of groups, from Lagrange's theorem to the Sylow theorems, is a study of which subgroups a group has and how they sit inside it.","title":"Subgroup"},{"id":"mathlib/overview/general-algebra/group-theory/part-01/index-of-a-subgroup","kind":"node","parent":33,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-01/index.html#index-of-a-subgroup","scope":"mathlib/overview/general-algebra/group-theory/part-01","status":"mathlib","summary":"The index of a subgroup $H$ of a group $G$ is the number of cosets of $H$ in $G$, written $[G : H]$. It measures how large $G$ is relative to $H$, and it stays meaningful for infinite groups, where orders cannot be compared directly: the integers and the even integers are both infinite, yet the even integers have index $2$ in the integers, since they and the odd integers are the only two cosets. The index is the\u2026","title":"Index of a subgroup"},{"id":"mathlib/overview/general-algebra/group-theory/part-01/normal-subgroup","kind":"node","parent":33,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-01/index.html#normal-subgroup","scope":"mathlib/overview/general-algebra/group-theory/part-01","status":"mathlib","summary":"A normal subgroup of a group is a subgroup that is carried into itself by every conjugation. Normal subgroups are exactly the subgroups by which one can divide: the cosets of a normal subgroup form a quotient group, and they are exactly the kernels of group homomorphisms. Much of the structure theory of groups, from the isomorphism theorems to the study of simple groups, is phrased in terms of them.","title":"Normal subgroup"},{"id":"mathlib/overview/general-algebra/group-theory/part-01/simple-group","kind":"node","parent":33,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-01/index.html#simple-group","scope":"mathlib/overview/general-algebra/group-theory/part-01","status":"mathlib","summary":"A simple group is a nontrivial group with no normal subgroups other than the trivial subgroup and the group itself. Normal subgroups are exactly the kernels of homomorphisms, so a simple group has no quotient other than itself and the trivial group: every homomorphism out of it is either injective or sends everything to the identity. Simple groups play the role for finite groups that prime numbers play for\u2026","title":"Simple group"},{"id":"mathlib/overview/general-algebra/group-theory/part-01/subgroup-generated-by-a-subset","kind":"node","parent":33,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-01/index.html#subgroup-generated-by-a-subset","scope":"mathlib/overview/general-algebra/group-theory/part-01","status":"mathlib","summary":"The subgroup generated by a subset $S$ of a group $G$ is the smallest subgroup of $G$ that contains $S$. It is written $\\langle S \\rangle$, and its elements are exactly the finite products of elements of $S$ and their inverses. Groups are often described by a few generators: a rotation and a reflection generate a dihedral group, for example, and many questions about the whole group, such as whether two\u2026","title":"Subgroup generated by a subset"},{"id":"mathlib/overview/general-algebra/group-theory/part-01/quotient-group","kind":"node","parent":33,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-01/index.html#quotient-group","scope":"mathlib/overview/general-algebra/group-theory/part-01","status":"mathlib","summary":"A quotient group is the group obtained from a group $G$ by collapsing a normal subgroup $N$ to the identity: its elements are the cosets of $N$, multiplied by multiplying representatives. The construction is the standard way to discard information in a controlled manner, as when the integers are reduced modulo $n$. Every surjective homomorphism of groups is, up to isomorphism, the projection onto a quotient group,\u2026","title":"Quotient group"},{"id":"mathlib/overview/general-algebra/group-theory/part-01/first-isomorphism-theorem","kind":"node","parent":33,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-01/index.html#first-isomorphism-theorem","scope":"mathlib/overview/general-algebra/group-theory/part-01","status":"mathlib","summary":"The first isomorphism theorem says that a group homomorphism $\\varphi \\colon G \\to H$ identifies the quotient of $G$ by the kernel of $\\varphi$ with the image of $\\varphi$. Every homomorphism therefore factors as a surjection onto a quotient followed by an isomorphism and an inclusion, and the question of which groups are homomorphic images of $G$ becomes the question of which normal subgroups $G$ has. The theorem\u2026","title":"First isomorphism theorem"},{"id":"mathlib/overview/general-algebra/group-theory/part-01/second-isomorphism-theorem","kind":"node","parent":33,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-01/index.html#second-isomorphism-theorem","scope":"mathlib/overview/general-algebra/group-theory/part-01","status":"mathlib","summary":"The second isomorphism theorem describes what happens when a subgroup $H$ of a group $G$ meets a normal subgroup $N$: the product $HN$ is a subgroup, and the quotient $HN/N$ is isomorphic to $H/(H \\cap N)$. It lets one compute a quotient of a large subgroup by passing to a quotient of a smaller one, and it is a basic tool in the study of composition series. Because $H \\cap N$ lies in both $H$ and $N$, which both\u2026","title":"Second isomorphism theorem"},{"id":"mathlib/overview/general-algebra/group-theory/part-01/third-isomorphism-theorem","kind":"node","parent":33,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-01/index.html#third-isomorphism-theorem","scope":"mathlib/overview/general-algebra/group-theory/part-01","status":"mathlib","summary":"The third isomorphism theorem says that taking a quotient of a quotient group gives nothing new: if $N \\le M$ are normal subgroups of a group $G$, then $M/N$ is a normal subgroup of $G/N$, and dividing $G/N$ by it gives $G/M$. In symbols, $(G/N)/(M/N) \\cong G/M$, so the quotient by $N$ can be removed from both sides. Together with the correspondence between subgroups of $G/N$ and subgroups of $G$ containing $N$, it\u2026","title":"Third isomorphism theorem"},{"id":"mathlib/overview/general-algebra/group-theory/part-01/abelianization","kind":"node","parent":33,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-01/index.html#abelianization","scope":"mathlib/overview/general-algebra/group-theory/part-01","status":"mathlib","summary":"The abelianization of a group $G$ is the largest abelian quotient of $G$: the quotient obtained by imposing the relations $gh = hg$ for all elements $g, h$ and no others. It is written $G^{\\mathrm{ab}}$. Every homomorphism from $G$ to an abelian group passes through it, so it carries exactly the information about $G$ that is detected by homomorphisms to abelian groups. It appears throughout algebra and topology,\u2026","title":"Abelianization"},{"id":"mathlib/overview/general-algebra/group-theory/part-01/free-group","kind":"node","parent":33,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-01/index.html#free-group","scope":"mathlib/overview/general-algebra/group-theory/part-01","status":"mathlib","summary":"The free group on a set $S$ is the group whose elements are the words in the letters of $S$ and their formal inverses, with no relations imposed beyond those that every group must satisfy. It is the most general group generated by a copy of $S$: every map from $S$ into a group extends to a homomorphism in exactly one way. Because of this, every group is a quotient of a free group, and describing a group by\u2026","title":"Free group"},{"id":"mathlib/overview/general-algebra/group-theory/part-01/presented-group","kind":"node","parent":33,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-01/index.html#presented-group","scope":"mathlib/overview/general-algebra/group-theory/part-01","status":"mathlib","summary":"A presented group is a group described by a set of generators together with a set of relations that those generators must satisfy, and by nothing else. Writing $\\langle S \\mid R \\rangle$ for the group with generators $S$ and relations $R$ gives a compact, finite description of many infinite groups, and it is the standard way groups arise in combinatorial group theory and topology, for instance as fundamental groups\u2026","title":"Presented group"},{"id":"mathlib/overview/general-algebra/group-theory/part-01/schreiers-lemma","kind":"node","parent":33,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-01/index.html#schreiers-lemma","scope":"mathlib/overview/general-algebra/group-theory/part-01","status":"mathlib","summary":"Schreier's lemma gives an explicit set of generators for a subgroup of a group, built from a set of generators of the whole group and one chosen representative from each coset of the subgroup. Its best-known consequence is that a subgroup of finite index in a finitely generated group is itself finitely generated. The generators it produces, often called Schreier generators, are used in the study of subgroups of\u2026","title":"Schreier's lemma"},{"id":"mathlib/overview/general-algebra/group-theory/part-01/cyclic-group","kind":"node","parent":33,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-01/index.html#cyclic-group","scope":"mathlib/overview/general-algebra/group-theory/part-01","status":"mathlib","summary":"A cyclic group is a group in which every element is a power of one fixed element. Cyclic groups are among the most elementary groups: up to isomorphism there is exactly one finite cyclic group with $n$ elements for each $n \\ge 1$, namely the integers modulo $n$, and one infinite one, the integers under addition. They are the building blocks of the finitely generated abelian groups, and they turn up throughout\u2026","title":"Cyclic group"},{"id":"mathlib/overview/general-algebra/group-theory/part-01/nilpotent-group","kind":"node","parent":33,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-01/index.html#nilpotent-group","scope":"mathlib/overview/general-algebra/group-theory/part-01","status":"mathlib","summary":"A nilpotent group is a group that can be built from abelian pieces by repeatedly taking central extensions (extensions in which the kernel lies in the center), in finitely many steps. Every abelian group is nilpotent, and so is every finite $p$-group. Nilpotency sits between commutativity and solvability: nilpotent groups keep much of the arithmetic control that abelian groups enjoy, which makes them a basic class\u2026","title":"Nilpotent group"},{"data_url":"scopes/mathlib/overview/general-algebra/group-theory/part-02.json","id":"mathlib/overview/general-algebra/group-theory/part-02","kind":"scope","navigation":"transparent","parent":32,"result":false,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-02/index.html","scope":"mathlib/overview/general-algebra/group-theory/part-02","status":"mathlib","title":"Permutation group of a type to Third Sylow theorem"},{"id":"mathlib/overview/general-algebra/group-theory/part-02/permutation-group-of-a-type","kind":"node","parent":54,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-02/index.html#permutation-group-of-a-type","scope":"mathlib/overview/general-algebra/group-theory/part-02","status":"mathlib","summary":"The permutation group of a set $X$, also called the symmetric group on $X$, is the set of all bijections from $X$ to itself, with composition of maps as the group operation. It is the group of all symmetries of a set with no further structure. By Cayley's theorem, every group is isomorphic to a subgroup of some permutation group, so permutation groups are as general as groups. For a finite set with $n$ elements the\u2026","title":"Permutation group of a type"},{"id":"mathlib/overview/general-algebra/group-theory/part-02/cayleys-theorem","kind":"node","parent":54,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-02/index.html#cayleys-theorem","scope":"mathlib/overview/general-algebra/group-theory/part-02","status":"mathlib","summary":"Cayley's theorem says that every group is isomorphic to a group of permutations: a group $G$ can be realized inside the permutation group of the set $G$ itself, by letting each element act on the others through multiplication. The theorem shows that the abstract axioms of a group describe nothing more general than groups of symmetries of a set. It also reduces the study of finite groups, in principle, to subgroups\u2026","title":"Cayley's theorem"},{"id":"mathlib/overview/general-algebra/group-theory/part-02/alternating-group","kind":"node","parent":54,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-02/index.html#alternating-group","scope":"mathlib/overview/general-algebra/group-theory/part-02","status":"mathlib","summary":"The alternating group $A_n$ is the group of even permutations of $n$ letters, a subgroup of index two in the symmetric group $S_n$ when $n \\ge 2$. The alternating groups $A_n$ with $n \\ge 5$ form the most elementary infinite family of nonabelian simple groups: each has no normal subgroups other than the trivial group and itself. This fact is the group-theoretic reason why the general polynomial equation of degree\u2026","title":"Alternating group"},{"id":"mathlib/overview/general-algebra/group-theory/part-02/alternating-groups-on-5-letters-or-more-are-simp","kind":"node","parent":54,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-02/index.html#alternating-groups-on-5-letters-or-more-are-simp","scope":"mathlib/overview/general-algebra/group-theory/part-02","status":"mathlib","summary":"The simplicity of the alternating groups is the theorem that for $n \\ge 5$ the alternating group $A_n$, the group of even permutations of $n$ letters, has no normal subgroups other than the trivial subgroup and itself. It gives an infinite family of simple groups that are not abelian, it is the group-theoretic reason why the general equation of degree five or more cannot be solved by radicals, and the alternating\u2026","title":"Alternating groups on 5 letters or more are simple"},{"id":"mathlib/overview/general-algebra/group-theory/part-02/classification-of-abelian-simple-groups","kind":"node","parent":54,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-02/index.html#classification-of-abelian-simple-groups","scope":"mathlib/overview/general-algebra/group-theory/part-02","status":"mathlib","summary":"The classification of abelian simple groups says that an abelian group is simple exactly when its order is a prime number, so that up to isomorphism the abelian simple groups are the cyclic groups $\\mathbb{Z}/p\\mathbb{Z}$, one for each prime $p$. Finite simple groups are the building blocks from which every finite group is assembled by successive extensions, as the factors of a composition series, and this theorem\u2026","title":"Classification of abelian simple groups"},{"id":"mathlib/overview/general-algebra/group-theory/part-02/structure-of-finitely-generated-abelian-groups","kind":"node","parent":54,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-02/index.html#structure-of-finitely-generated-abelian-groups","scope":"mathlib/overview/general-algebra/group-theory/part-02","status":"mathlib","summary":"The structure theorem for finitely generated abelian groups says that every abelian group generated by finitely many elements is a direct product of finitely many cyclic groups: some copies of the integers and some finite cyclic groups of prime-power order. It classifies these groups completely up to isomorphism. Questions about such a group, such as how many elements of a given order it has or which groups occur\u2026","title":"Structure of finitely generated abelian groups"},{"id":"mathlib/overview/general-algebra/group-theory/part-02/lagranges-theorem","kind":"node","parent":54,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-02/index.html#lagranges-theorem","scope":"mathlib/overview/general-algebra/group-theory/part-02","status":"mathlib","summary":"Lagrange's theorem says that in a finite group the order (number of elements) of every subgroup divides the order of the group. It is a basic structural result of group theory: it places an arithmetic restriction on the possible orders of subgroups, and several classical facts, Fermat's little theorem among them, follow from it in a line.","title":"Lagrange's theorem"},{"id":"mathlib/overview/general-algebra/group-theory/part-02/first-sylow-theorem","kind":"node","parent":54,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-02/index.html#first-sylow-theorem","scope":"mathlib/overview/general-algebra/group-theory/part-02","status":"mathlib","summary":"The first Sylow theorem says that a finite group has a subgroup of every prime-power order that divides the order of the group. It is a partial converse to Lagrange's theorem, which says that the order of a subgroup divides the order of the group. A general divisor of $|G|$ need not be the order of any subgroup, but a divisor that is a power of a single prime always is. Together with the second and third Sylow\u2026","title":"First Sylow theorem"},{"id":"mathlib/overview/general-algebra/group-theory/part-02/second-sylow-theorem","kind":"node","parent":54,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-02/index.html#second-sylow-theorem","scope":"mathlib/overview/general-algebra/group-theory/part-02","status":"mathlib","summary":"The second Sylow theorem says that in a finite group, for each prime $p$, any two Sylow $p$-subgroups are conjugate. The first Sylow theorem guarantees that Sylow subgroups exist; the second says that, up to conjugation, there is only one of them. Together with the third Sylow theorem, which constrains how many there are, it is the main tool for analysing finite groups of a given order, and in particular for\u2026","title":"Second Sylow theorem"},{"id":"mathlib/overview/general-algebra/group-theory/part-02/third-sylow-theorem","kind":"node","parent":54,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/group-theory/part-02/index.html#third-sylow-theorem","scope":"mathlib/overview/general-algebra/group-theory/part-02","status":"mathlib","summary":"The third Sylow theorem says that in a finite group $G$ the number of Sylow $p$-subgroups, for a prime $p$, is congruent to $1$ modulo $p$ and divides the order of $G$. Together with the first and second Sylow theorems it is the basic counting tool of finite group theory: the two arithmetic conditions often leave only one possible value for the number of Sylow subgroups, and when that value is $1$ the unique Sylow\u2026","title":"Third Sylow theorem"},{"data_url":"scopes/mathlib/overview/general-algebra/rings.json","id":"mathlib/overview/general-algebra/rings","kind":"scope","parent":3,"result":false,"route":"../roadmap/mathlib/overview/general-algebra/rings/index.html","scope":"mathlib/overview/general-algebra/rings","status":"mathlib","title":"Rings"},{"id":"mathlib/overview/general-algebra/rings/ring","kind":"node","parent":65,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/rings/index.html#ring","scope":"mathlib/overview/general-algebra/rings","status":"mathlib","summary":"A ring is a set with two binary operations, addition and multiplication, such that the set is an abelian group under addition, the multiplication is associative and has an identity element, and multiplication distributes over addition. Rings capture the arithmetic shared by the integers, the integers modulo $n$, polynomials and square matrices: one can add, subtract and multiply, but not necessarily divide.\u2026","title":"Ring"},{"id":"mathlib/overview/general-algebra/rings/ring-morphism","kind":"node","parent":65,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/rings/index.html#ring-morphism","scope":"mathlib/overview/general-algebra/rings","status":"mathlib","summary":"A ring morphism, or ring homomorphism, is a map between two rings that respects addition, multiplication and the multiplicative identity. Ring morphisms are the means of comparing rings: reduction of integers modulo $n$, evaluation of polynomials at a point and passage to a quotient ring are all ring morphisms, and the notions of ideal, quotient and isomorphism of rings are organized around them.","title":"Ring morphism"},{"id":"mathlib/overview/general-algebra/rings/the-category-of-rings","kind":"node","parent":65,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/rings/index.html#the-category-of-rings","scope":"mathlib/overview/general-algebra/rings","status":"mathlib","summary":"The category of rings, usually written $\\mathbf{Ring}$, is the category whose objects are rings and whose morphisms are ring morphisms. Viewing rings this way puts constructions such as products, quotients and polynomial rings in a common language of universal properties, and lets one compare rings with other algebraic structures through functors, for instance the functor that forgets the multiplication of a ring\u2026","title":"The category of rings"},{"id":"mathlib/overview/general-algebra/rings/subring","kind":"node","parent":65,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/rings/index.html#subring","scope":"mathlib/overview/general-algebra/rings","status":"mathlib","summary":"A subring of a ring $R$ is a subset of $R$ that contains the identity $1$ and is closed under addition, negation and multiplication, so that it is a ring in its own right under the operations of $R$. Subrings are the substructures of ring theory, as subgroups are for groups: the integers inside the rational numbers, the Gaussian integers inside the complex numbers, and the image of any ring homomorphism are all\u2026","title":"Subring"},{"id":"mathlib/overview/general-algebra/rings/localization","kind":"node","parent":65,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/rings/index.html#localization","scope":"mathlib/overview/general-algebra/rings","status":"mathlib","summary":"Localization is the construction that adjoins inverses to a chosen set of elements of a commutative ring. Given a multiplicatively closed subset $S$ of a ring $R$, it produces a ring $S^{-1}R$ of fractions $r/s$ with $r \\in R$ and $s \\in S$, in which every element of $S$ becomes invertible, and it does so in the most economical way. The passage from the integers to the rational numbers is the model case.\u2026","title":"Localization"},{"id":"mathlib/overview/general-algebra/rings/local-ring","kind":"node","parent":65,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/rings/index.html#local-ring","scope":"mathlib/overview/general-algebra/rings","status":"mathlib","summary":"A local ring is a commutative ring with exactly one maximal ideal. Equivalently, it is a nonzero ring in which the elements that are not units are closed under addition. Local rings are the algebraic counterpart of looking at a geometric object near a single point: the ring of germs of functions at a point (functions defined near it, identified when they agree near it), with the germs vanishing at the point as the\u2026","title":"Local ring"},{"id":"mathlib/overview/general-algebra/rings/noetherian-ring","kind":"node","parent":65,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/rings/index.html#noetherian-ring","scope":"mathlib/overview/general-algebra/rings","status":"mathlib","summary":"A Noetherian ring is a ring in which every ideal is finitely generated; equivalently, every increasing chain of ideals eventually stops growing. The condition is a finiteness hypothesis that covers most rings met in number theory and algebraic geometry, including the integers, fields, and polynomial rings in finitely many variables over them. It is preserved by the usual constructions of commutative algebra, and it\u2026","title":"Noetherian ring"},{"id":"mathlib/overview/general-algebra/rings/ordered-ring","kind":"node","parent":65,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/rings/index.html#ordered-ring","scope":"mathlib/overview/general-algebra/rings","status":"mathlib","summary":"An ordered ring is a ring equipped with an order that is compatible with its addition and multiplication: adding the same element to both sides of an inequality preserves it, and so does multiplying both sides by a nonnegative element. The integers, the rational numbers and the real numbers are the basic examples. The notion isolates the algebraic rules used whenever one manipulates inequalities, so that facts such\u2026","title":"Ordered ring"},{"data_url":"scopes/mathlib/overview/general-algebra/ideals-and-quotients.json","id":"mathlib/overview/general-algebra/ideals-and-quotients","kind":"scope","parent":3,"result":false,"route":"../roadmap/mathlib/overview/general-algebra/ideals-and-quotients/index.html","scope":"mathlib/overview/general-algebra/ideals-and-quotients","status":"mathlib","title":"Ideals and quotients"},{"id":"mathlib/overview/general-algebra/ideals-and-quotients/ideal-of-a-commutative-ring","kind":"node","parent":74,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/ideals-and-quotients/index.html#ideal-of-a-commutative-ring","scope":"mathlib/overview/general-algebra/ideals-and-quotients","status":"mathlib","summary":"An ideal of a commutative ring $R$ is a nonempty subset of $R$ that is closed under addition and under multiplication by arbitrary elements of $R$. Ideals play for rings the part that normal subgroups play for groups: they are exactly the kernels of ring morphisms, and they are what one divides by to form quotient rings. They also carry the arithmetic of divisibility over from elements to sets, which is how\u2026","title":"Ideal of a commutative ring"},{"id":"mathlib/overview/general-algebra/ideals-and-quotients/quotient-ring","kind":"node","parent":74,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/ideals-and-quotients/index.html#quotient-ring","scope":"mathlib/overview/general-algebra/ideals-and-quotients","status":"mathlib","summary":"A quotient ring is the ring obtained from a ring $R$ by declaring the elements of a two-sided ideal $I$ (any ideal, when $R$ is commutative) to be zero. Its elements are the cosets $a + I$, added and multiplied through representatives. Quotient rings are the ring-theoretic counterpart of quotient groups: they turn congruence modulo an ideal into equality, they describe the image of every ring morphism, and they are\u2026","title":"Quotient ring"},{"id":"mathlib/overview/general-algebra/ideals-and-quotients/prime-ideal","kind":"node","parent":74,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/ideals-and-quotients/index.html#prime-ideal","scope":"mathlib/overview/general-algebra/ideals-and-quotients","status":"mathlib","summary":"A prime ideal of a commutative ring is a proper ideal with the property that whenever a product of two elements lies in it, one of the two factors already does. The definition transfers a characteristic property of a prime number (Euclid's lemma: it divides a product only if it divides one of the factors) from elements to ideals. Prime ideals are exactly the ideals whose quotient ring is an integral domain, a\u2026","title":"Prime ideal"},{"id":"mathlib/overview/general-algebra/ideals-and-quotients/maximal-ideal","kind":"node","parent":74,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/ideals-and-quotients/index.html#maximal-ideal","scope":"mathlib/overview/general-algebra/ideals-and-quotients","status":"mathlib","summary":"A maximal ideal of a commutative ring is a proper ideal that is not contained in any larger proper ideal. Maximal ideals are exactly the ideals whose quotient ring is a field, which makes them the standard way to produce fields from rings: the integers modulo a prime, finite fields, and fields obtained by adjoining a root of a polynomial all arise this way. They correspond to the closed points of the prime\u2026","title":"Maximal ideal"},{"id":"mathlib/overview/general-algebra/ideals-and-quotients/chinese-remainder-theorem","kind":"node","parent":74,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/ideals-and-quotients/index.html#chinese-remainder-theorem","scope":"mathlib/overview/general-algebra/ideals-and-quotients","status":"mathlib","summary":"The Chinese remainder theorem says that a finite system of congruences with pairwise coprime moduli always has a solution, and that the solution is unique modulo the product of the moduli. In the language of rings it says that the quotient of a commutative ring by the intersection of finitely many pairwise coprime ideals splits as a product of the separate quotients. It is the standard tool for reducing a question\u2026","title":"Chinese remainder theorem"},{"id":"mathlib/overview/general-algebra/ideals-and-quotients/fractional-ideal","kind":"node","parent":74,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/ideals-and-quotients/index.html#fractional-ideal","scope":"mathlib/overview/general-algebra/ideals-and-quotients","status":"mathlib","summary":"A fractional ideal of an integral domain $R$ is an $R$-submodule of the field of fractions $K$ of $R$ whose denominators can be cleared by a single nonzero element of $R$. Fractional ideals are to ideals what fractions are to integers: they allow ideals to be divided as well as multiplied. In the rings of algebraic number theory, the Dedekind domains, the nonzero fractional ideals form a group under multiplication,\u2026","title":"Fractional ideal"},{"id":"mathlib/overview/general-algebra/ideals-and-quotients/first-isomorphism-theorem-for-commutative-rings","kind":"node","parent":74,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/ideals-and-quotients/index.html#first-isomorphism-theorem-for-commutative-rings","scope":"mathlib/overview/general-algebra/ideals-and-quotients","status":"mathlib","summary":"The first isomorphism theorem for commutative rings says that a surjective ring morphism $f \\colon R \\to S$ identifies the quotient ring of $R$ by the kernel of $f$ with $S$. It turns the question of which rings are images of $R$ into the question of which ideals $R$ has, and it is the usual way to recognize a quotient ring: to show that $R/I$ is isomorphic to a known ring $S$, one finds a surjective morphism from\u2026","title":"First isomorphism theorem for commutative rings"},{"data_url":"scopes/mathlib/overview/general-algebra/divisibility-in-integral-domains.json","id":"mathlib/overview/general-algebra/divisibility-in-integral-domains","kind":"scope","parent":3,"result":false,"route":"../roadmap/mathlib/overview/general-algebra/divisibility-in-integral-domains/index.html","scope":"mathlib/overview/general-algebra/divisibility-in-integral-domains","status":"mathlib","title":"Divisibility in integral domains"},{"id":"mathlib/overview/general-algebra/divisibility-in-integral-domains/irreducible-element","kind":"node","parent":82,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/divisibility-in-integral-domains/index.html#irreducible-element","scope":"mathlib/overview/general-algebra/divisibility-in-integral-domains","status":"mathlib","summary":"An irreducible element of a commutative ring or monoid is a non-unit that cannot be split as a product of two non-units: whenever it is written as a product, one of the factors is a unit. Irreducible elements play the role that prime numbers play among the natural numbers, as the smallest building blocks of multiplication. They are the elements that appear in factorizations, and the question of when such\u2026","title":"Irreducible element"},{"id":"mathlib/overview/general-algebra/divisibility-in-integral-domains/coprime-element","kind":"node","parent":82,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/divisibility-in-integral-domains/index.html#coprime-element","scope":"mathlib/overview/general-algebra/divisibility-in-integral-domains","status":"mathlib","summary":"Two elements $x$ and $y$ of a commutative ring are coprime when $1$ can be written as a combination $ax + by$ with coefficients $a, b$ in the ring. For the integers this is the familiar condition that the greatest common divisor is $1$, by B\u00e9zout's identity. In a general ring it is a stronger condition than having no common non-unit factor, and it is the form of coprimality that appears in results such as the\u2026","title":"Coprime element"},{"id":"mathlib/overview/general-algebra/divisibility-in-integral-domains/unique-factorisation-domain","kind":"node","parent":82,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/divisibility-in-integral-domains/index.html#unique-factorisation-domain","scope":"mathlib/overview/general-algebra/divisibility-in-integral-domains","status":"mathlib","summary":"A unique factorisation domain (UFD) is an integral domain in which every nonzero non-unit is a product of irreducible elements, and this product is unique up to the order of the factors and multiplication of the factors by units. It is the setting in which the fundamental theorem of arithmetic generalises: the integers, polynomial rings over fields, and polynomial rings over the integers all have unique\u2026","title":"Unique factorisation domain"},{"id":"mathlib/overview/general-algebra/divisibility-in-integral-domains/greatest-common-divisor","kind":"node","parent":82,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/divisibility-in-integral-domains/index.html#greatest-common-divisor","scope":"mathlib/overview/general-algebra/divisibility-in-integral-domains","status":"mathlib","summary":"The greatest common divisor of two elements $a$ and $b$ is a common divisor of $a$ and $b$ that every other common divisor divides. For integers not both zero, the largest integer dividing both is a greatest common divisor, so that $\\gcd(12, 18) = 6$. It governs when fractions can be reduced, when linear equations $ax + by = c$ have integer solutions, and when two elements share no common factor. The same notion\u2026","title":"Greatest common divisor"},{"id":"mathlib/overview/general-algebra/divisibility-in-integral-domains/least-common-multiple","kind":"node","parent":82,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/divisibility-in-integral-domains/index.html#least-common-multiple","scope":"mathlib/overview/general-algebra/divisibility-in-integral-domains","status":"mathlib","summary":"The least common multiple of two elements $a$ and $b$ is a common multiple of both that divides every other common multiple. For nonzero natural numbers it is the smallest positive integer divisible by both, written $\\operatorname{lcm}(a, b)$, while $\\operatorname{lcm}(a, 0) = 0$; the same idea makes sense in any commutative ring, where it is defined purely in terms of divisibility. It is the counterpart of the\u2026","title":"Least common multiple"},{"id":"mathlib/overview/general-algebra/divisibility-in-integral-domains/principal-ideal-domain","kind":"node","parent":82,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/divisibility-in-integral-domains/index.html#principal-ideal-domain","scope":"mathlib/overview/general-algebra/divisibility-in-integral-domains","status":"mathlib","summary":"A principal ideal domain, or PID, is an integral domain in which every ideal is generated by a single element. The integers and the polynomials in one variable over a field are the standard examples. In a PID the arithmetic of the integers carries over almost unchanged: greatest common divisors exist and satisfy B\u00e9zout's identity, irreducible elements are prime, and every nonzero nonunit factors into irreducibles,\u2026","title":"Principal ideal domain"},{"id":"mathlib/overview/general-algebra/divisibility-in-integral-domains/euclidean-domain","kind":"node","parent":82,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/divisibility-in-integral-domains/index.html#euclidean-domain","scope":"mathlib/overview/general-algebra/divisibility-in-integral-domains","status":"mathlib","summary":"A Euclidean domain is an integral domain in which one can divide with remainder, the remainder being smaller than the divisor for some measure of size. The integers, every field and the polynomials over a field are the main examples. Division with remainder is exactly what the Euclidean algorithm needs, so in a Euclidean domain greatest common divisors can be computed, every ideal is principal, and factorization\u2026","title":"Euclidean domain"},{"id":"mathlib/overview/general-algebra/divisibility-in-integral-domains/euclids-algorithm","kind":"node","parent":82,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/divisibility-in-integral-domains/index.html#euclids-algorithm","scope":"mathlib/overview/general-algebra/divisibility-in-integral-domains","status":"mathlib","summary":"Euclid's algorithm computes the greatest common divisor of two natural numbers by repeated division with remainder. Its refinement, the extended Euclidean algorithm, also produces integers $x$ and $y$ with $\\gcd(a, b) = ax + by$. The algorithm is fast, needs nothing beyond division with remainder, and the coefficients it returns give B\u00e9zout's identity in explicit form, inverses in modular arithmetic, and solutions\u2026","title":"Euclid's' algorithm"},{"id":"mathlib/overview/general-algebra/divisibility-in-integral-domains/eulers-totient-function-varphi","kind":"node","parent":82,"reading_title":{"format":"inline-parts-v1","parts":[["text","Euler's totient function ("],["math","\\(\\varphi\\)"],["text",")"]],"source":"Euler's totient function ($\\varphi$)"},"result":true,"route":"../roadmap/mathlib/overview/general-algebra/divisibility-in-integral-domains/index.html#eulers-totient-function-varphi","scope":"mathlib/overview/general-algebra/divisibility-in-integral-domains","status":"mathlib","summary":"Euler's totient function, written $\\varphi$, assigns to each positive natural number $n$ the number of integers between $1$ and $n$ that are coprime to $n$, that is, share no common factor with $n$ other than $1$. It is one of the basic functions of elementary number theory. It counts the invertible residues modulo $n$, so it is the exponent in Euler's theorem, and it reappears in the theory of cyclic groups, roots\u2026","title":"Euler's totient function ($\\varphi$)"},{"id":"mathlib/overview/general-algebra/divisibility-in-integral-domains/lucas-lehmer-primality-test","kind":"node","parent":82,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/divisibility-in-integral-domains/index.html#lucas-lehmer-primality-test","scope":"mathlib/overview/general-algebra/divisibility-in-integral-domains","status":"mathlib","summary":"The Lucas-Lehmer primality test decides whether a Mersenne number $2^p - 1$ is prime by iterating the map $x \\mapsto x^2 - 2$ modulo that number. It needs only $p - 2$ squarings of numbers smaller than $2^p - 1$, far fewer operations than trial division, so that primality is far cheaper to decide for Mersenne numbers than for general numbers of the same size. Mersenne primes are also of independent interest: by the\u2026","title":"Lucas-Lehmer primality test"},{"data_url":"scopes/mathlib/overview/general-algebra/polynomials-and-power-series.json","id":"mathlib/overview/general-algebra/polynomials-and-power-series","kind":"scope","parent":3,"result":false,"route":"../roadmap/mathlib/overview/general-algebra/polynomials-and-power-series/index.html","scope":"mathlib/overview/general-algebra/polynomials-and-power-series","status":"mathlib","title":"Polynomials and power series"},{"id":"mathlib/overview/general-algebra/polynomials-and-power-series/polynomial-in-one-indeterminate","kind":"node","parent":93,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/polynomials-and-power-series/index.html#polynomial-in-one-indeterminate","scope":"mathlib/overview/general-algebra/polynomials-and-power-series","status":"mathlib","summary":"A polynomial in one indeterminate over a ring $R$ is a formal expression $a_0 + a_1 X + \\dots + a_n X^n$ with coefficients $a_i$ in $R$, added and multiplied by the familiar rules of high-school algebra. The polynomials over $R$ form a ring $R[X]$, the polynomial ring. It is the simplest way to enlarge a ring by one new element, and it underlies much of algebra: the theory of equations, field extensions, algebraic\u2026","title":"Polynomial in one indeterminate"},{"id":"mathlib/overview/general-algebra/polynomials-and-power-series/roots-of-a-polynomial","kind":"node","parent":93,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/polynomials-and-power-series/index.html#roots-of-a-polynomial","scope":"mathlib/overview/general-algebra/polynomials-and-power-series","status":"mathlib","summary":"The roots of a polynomial $p$ are the values $a$ at which the polynomial vanishes, that is, $p(a) = 0$. Over an integral domain, the roots of a nonzero polynomial, each counted as often as its multiplicity, form a finite multiset whose size is at most the degree of $p$. Roots tie the evaluation of a polynomial to its factorization: finding the roots of $p$ is the same as finding its factors of the form $X - a$, and\u2026","title":"Roots of a polynomial"},{"id":"mathlib/overview/general-algebra/polynomials-and-power-series/multiplicity","kind":"node","parent":93,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/polynomials-and-power-series/index.html#multiplicity","scope":"mathlib/overview/general-algebra/polynomials-and-power-series","status":"mathlib","summary":"The multiplicity of a root $a$ of a polynomial $p$ is the number of times the linear factor $X - a$ occurs in $p$: the largest $m$ such that $(X - a)^m$ divides $p$. It refines the yes-or-no question \"is $a$ a root?\" into a count, distinguishing simple roots from double and higher roots. Counting roots with multiplicity is what makes statements such as \"a polynomial of degree $n$ over the complex numbers has\u2026","title":"Multiplicity"},{"id":"mathlib/overview/general-algebra/polynomials-and-power-series/separable-polynomial","kind":"node","parent":93,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/polynomials-and-power-series/index.html#separable-polynomial","scope":"mathlib/overview/general-algebra/polynomials-and-power-series","status":"mathlib","summary":"A separable polynomial over a field is a nonzero polynomial without repeated roots: in any field over which it factors into linear factors, its roots are pairwise distinct. Separability can be decided inside the base field, without finding a single root, by checking that the polynomial and its formal derivative have no common factor of positive degree. The notion is basic to field theory: separable extensions,\u2026","title":"Separable polynomial"},{"id":"mathlib/overview/general-algebra/polynomials-and-power-series/k-x-is-euclidean","kind":"node","parent":93,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(K[X]\\)"],["text"," is Euclidean"]],"source":"$K[X]$ is Euclidean"},"result":true,"route":"../roadmap/mathlib/overview/general-algebra/polynomials-and-power-series/index.html#k-x-is-euclidean","scope":"mathlib/overview/general-algebra/polynomials-and-power-series","status":"mathlib","summary":"For a field $K$, the ring $K[X]$ of polynomials in one indeterminate over $K$ is a Euclidean domain: any polynomial can be divided by a nonzero polynomial, leaving a remainder of smaller degree. This fact, $K[X]$ is Euclidean, makes the arithmetic of polynomials over a field closely parallel to the arithmetic of the integers. It yields a Euclidean algorithm for greatest common divisors, shows that every ideal of\u2026","title":"$K[X]$ is Euclidean"},{"id":"mathlib/overview/general-algebra/polynomials-and-power-series/hilbert-basis-theorem","kind":"node","parent":93,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/polynomials-and-power-series/index.html#hilbert-basis-theorem","scope":"mathlib/overview/general-algebra/polynomials-and-power-series","status":"mathlib","summary":"The Hilbert basis theorem says that if $R$ is a commutative Noetherian ring, then so is the polynomial ring $R[x]$. Applied repeatedly, it shows that every ideal of a polynomial ring in finitely many variables over a field, or over the integers, is generated by finitely many polynomials. It is one of the basic finiteness results of commutative algebra and algebraic geometry: it guarantees that any system of\u2026","title":"Hilbert basis theorem"},{"id":"mathlib/overview/general-algebra/polynomials-and-power-series/a-x-has-gcd-and-lcm-if-a-does","kind":"node","parent":93,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(A[X]\\)"],["text"," has gcd and lcm if "],["math","\\(A\\)"],["text"," does"]],"source":"$A[X]$ has gcd and lcm if $A$ does"},"result":true,"route":"../roadmap/mathlib/overview/general-algebra/polynomials-and-power-series/index.html#a-x-has-gcd-and-lcm-if-a-does","scope":"mathlib/overview/general-algebra/polynomials-and-power-series","status":"mathlib","summary":"The statement that $A[X]$ has gcd and lcm if $A$ does says that if every two elements of an integral domain $A$ have a greatest common divisor and a least common multiple, then the same holds in the polynomial ring $A[X]$. In other words, the polynomial ring over a GCD domain is again a GCD domain. The result explains why polynomials over the integers, or over any unique factorisation domain, admit gcds even though\u2026","title":"$A[X]$ has gcd and lcm if $A$ does"},{"id":"mathlib/overview/general-algebra/polynomials-and-power-series/a-x-i-is-a-ufd-when-a-is-a-ufd","kind":"node","parent":93,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(A[{X_i}]\\)"],["text"," is a UFD when "],["math","\\(A\\)"],["text"," is a UFD"]],"source":"$A[{X_i}]$ is a UFD when $A$ is a UFD"},"result":true,"route":"../roadmap/mathlib/overview/general-algebra/polynomials-and-power-series/index.html#a-x-i-is-a-ufd-when-a-is-a-ufd","scope":"mathlib/overview/general-algebra/polynomials-and-power-series","status":"mathlib","summary":"Polynomial rings over a unique factorisation domain are unique factorisation domains: if $A$ is a unique factorisation domain (UFD), then so is the polynomial ring $A[X_i : i \\in I]$ in any family of indeterminates, finite or infinite. Together with the fact that every field and the ring of integers are UFDs, it shows that the rings $k[X_1, \\dots, X_n]$ and $\\mathbb{Z}[X_1, \\dots, X_n]$ have unique factorisation,\u2026","title":"$A[{X_i}]$ is a UFD when $A$ is a UFD"},{"id":"mathlib/overview/general-algebra/polynomials-and-power-series/eisensteins-criterion","kind":"node","parent":93,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/polynomials-and-power-series/index.html#eisensteins-criterion","scope":"mathlib/overview/general-algebra/polynomials-and-power-series","status":"mathlib","summary":"Eisenstein's criterion is a sufficient condition for a polynomial to be irreducible, read off from its coefficients alone. In its best-known form it says that an integer polynomial whose leading coefficient is not divisible by a prime $p$, whose other coefficients are all divisible by $p$, and whose constant term is not divisible by $p^2$, cannot be factored into polynomials of lower degree with rational\u2026","title":"Eisenstein's criterion"},{"id":"mathlib/overview/general-algebra/polynomials-and-power-series/polynomial-in-several-indeterminates","kind":"node","parent":93,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/polynomials-and-power-series/index.html#polynomial-in-several-indeterminates","scope":"mathlib/overview/general-algebra/polynomials-and-power-series","status":"mathlib","summary":"A polynomial in several indeterminates over a commutative ring $R$ is a finite sum of terms $c\\,X_1^{a_1}X_2^{a_2}\\cdots X_n^{a_n}$ with coefficients $c$ in $R$, such as $x^2 + y^2 - 1$ or $xyz - 2$. These polynomials form a commutative ring, written $R[X_1, \\dots, X_n]$ or $R[X_i : i \\in I]$, which generalizes the polynomials in one indeterminate. It is a basic object of commutative algebra and algebraic geometry:\u2026","title":"Polynomial in several indeterminates"},{"id":"mathlib/overview/general-algebra/polynomials-and-power-series/power-series","kind":"node","parent":93,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/polynomials-and-power-series/index.html#power-series","scope":"mathlib/overview/general-algebra/polynomials-and-power-series","status":"mathlib","summary":"A formal power series in one indeterminate $X$ over a ring $R$ is an expression $\\sum_{n \\ge 0} a_n X^n$ with coefficients $a_n$ in $R$, added and multiplied as if it were a polynomial that never ends. Nothing is evaluated and no question of convergence arises: a power series is just its sequence of coefficients, and the set $R[[X]]$ of all such series is made into a ring by a multiplication that extends that of\u2026","title":"Power series"},{"data_url":"scopes/mathlib/overview/general-algebra/algebras-over-a-ring.json","id":"mathlib/overview/general-algebra/algebras-over-a-ring","kind":"scope","parent":3,"result":false,"route":"../roadmap/mathlib/overview/general-algebra/algebras-over-a-ring/index.html","scope":"mathlib/overview/general-algebra/algebras-over-a-ring","status":"mathlib","title":"Algebras over a ring"},{"id":"mathlib/overview/general-algebra/algebras-over-a-ring/associative-algebra-over-a-commutative-ring","kind":"node","parent":105,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/algebras-over-a-ring/index.html#associative-algebra-over-a-commutative-ring","scope":"mathlib/overview/general-algebra/algebras-over-a-ring","status":"mathlib","summary":"An associative algebra over a commutative ring $R$, or $R$-algebra, is a ring $A$ whose elements can also be multiplied by scalars from $R$, in a way compatible with the multiplication of $A$. Polynomial rings, matrix rings, rings of endomorphisms of a module and field extensions are all algebras over a suitable base, and the notion lets one study such rings together with their linear structure over $R$. Most of\u2026","title":"Associative algebra over a commutative ring"},{"id":"mathlib/overview/general-algebra/algebras-over-a-ring/the-category-of-algebras-over-a-ring","kind":"node","parent":105,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/algebras-over-a-ring/index.html#the-category-of-algebras-over-a-ring","scope":"mathlib/overview/general-algebra/algebras-over-a-ring","status":"mathlib","summary":"For a commutative ring $R$, the category of algebras over $R$ has as its objects the associative unital algebras over $R$ and as its morphisms the homomorphisms of $R$-algebras. It is the natural home for the constructions of commutative and noncommutative algebra that are characterized by universal properties, such as polynomial rings, free algebras, tensor algebras and tensor products, and it sits between the\u2026","title":"The category of algebras over a ring"},{"id":"mathlib/overview/general-algebra/algebras-over-a-ring/free-algebra-of-a-commutative-ring","kind":"node","parent":105,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/algebras-over-a-ring/index.html#free-algebra-of-a-commutative-ring","scope":"mathlib/overview/general-algebra/algebras-over-a-ring","status":"mathlib","summary":"The free algebra on a set $X$ over a commutative ring $R$, written $R\\langle X\\rangle$, is the associative algebra with identity generated by the elements of $X$ subject to no relations other than those every associative $R$-algebra must satisfy. Its elements are the noncommutative polynomials in variables from $X$: finite $R$-linear combinations of words in those variables, in which $xy$ and $yx$ are different\u2026","title":"Free algebra of a commutative ring"},{"id":"mathlib/overview/general-algebra/algebras-over-a-ring/tensor-product-of-algebras","kind":"node","parent":105,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/algebras-over-a-ring/index.html#tensor-product-of-algebras","scope":"mathlib/overview/general-algebra/algebras-over-a-ring","status":"mathlib","summary":"The tensor product of algebras $A \\otimes_R B$ of two associative algebras $A$ and $B$ over a commutative ring $R$ is the tensor product of their underlying modules, made into an $R$-algebra by multiplying factor by factor. It is the standard way to combine two algebras into one in which each acts on its own side: it changes the base ring of an algebra (base change), it builds polynomial and matrix algebras over an\u2026","title":"Tensor product of algebras"},{"id":"mathlib/overview/general-algebra/algebras-over-a-ring/tensor-algebra-of-a-commutative-ring","kind":"node","parent":105,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/algebras-over-a-ring/index.html#tensor-algebra-of-a-commutative-ring","scope":"mathlib/overview/general-algebra/algebras-over-a-ring","status":"mathlib","summary":"The tensor algebra of a module $M$ over a commutative ring $R$, written $T(M)$, is the associative algebra with identity obtained by putting together all tensor powers $M \\otimes \\cdots \\otimes M$ and multiplying them by juxtaposition. It is the most general associative algebra into which $M$ maps linearly: every linear map from $M$ to an associative algebra extends uniquely to an algebra homomorphism out of\u2026","title":"Tensor algebra of a commutative ring"},{"id":"mathlib/overview/general-algebra/algebras-over-a-ring/lie-algebra","kind":"node","parent":105,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/algebras-over-a-ring/index.html#lie-algebra","scope":"mathlib/overview/general-algebra/algebras-over-a-ring","status":"mathlib","summary":"A Lie algebra is a module over a commutative ring, typically a vector space over a field, equipped with a bilinear product, the bracket, which vanishes on equal arguments and satisfies the Jacobi identity. The bracket measures how far two operations are from commuting. Lie algebras arise as the infinitesimal versions of Lie groups, as algebras of derivations and vector fields, and from any associative algebra\u2026","title":"Lie algebra"},{"id":"mathlib/overview/general-algebra/algebras-over-a-ring/exterior-algebra","kind":"node","parent":105,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/algebras-over-a-ring/index.html#exterior-algebra","scope":"mathlib/overview/general-algebra/algebras-over-a-ring","status":"mathlib","summary":"The exterior algebra of a module $M$ over a commutative ring $R$, written $\\Lambda(M)$, is the associative algebra generated by $M$ subject only to the rule that every element of $M$ squares to zero. Its product, the wedge product $\\wedge$, is the algebraic form of oriented area and volume: in an $n$-dimensional space with basis $e_1, \\dots, e_n$, the wedge of $n$ vectors is their determinant in that basis times\u2026","title":"Exterior algebra"},{"id":"mathlib/overview/general-algebra/algebras-over-a-ring/clifford-algebra","kind":"node","parent":105,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/algebras-over-a-ring/index.html#clifford-algebra","scope":"mathlib/overview/general-algebra/algebras-over-a-ring","status":"mathlib","summary":"The Clifford algebra of a module $M$ over a commutative ring $R$, equipped with a quadratic form $Q$, is the associative $R$-algebra generated by $M$ subject only to the rule that every vector $v$ squares to the scalar $Q(v)$: it is the universal associative algebra in which each vector squares to the value of the quadratic form. The complex numbers, the quaternions, the dual numbers and the exterior algebra all\u2026","title":"Clifford algebra"},{"data_url":"scopes/mathlib/overview/general-algebra/field-theory.json","id":"mathlib/overview/general-algebra/field-theory","kind":"scope","parent":3,"result":false,"route":"../roadmap/mathlib/overview/general-algebra/field-theory/index.html","scope":"mathlib/overview/general-algebra/field-theory","status":"mathlib","title":"Field theory"},{"id":"mathlib/overview/general-algebra/field-theory/field","kind":"node","parent":114,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/field-theory/index.html#field","scope":"mathlib/overview/general-algebra/field-theory","status":"mathlib","summary":"A field is a set with an addition and a multiplication in which the usual rules of arithmetic hold and division by any nonzero element is possible. The rational, real and complex numbers are the familiar examples. Fields are where linear algebra takes place, since a vector space is a module over a field, and they are the setting for the theory of polynomial equations, from splitting fields to the Galois\u2026","title":"Field"},{"id":"mathlib/overview/general-algebra/field-theory/subfield","kind":"node","parent":114,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/field-theory/index.html#subfield","scope":"mathlib/overview/general-algebra/field-theory","status":"mathlib","summary":"A subfield of a field $K$ is a subset of $K$ that contains $0$ and $1$ and is closed under addition, negation, multiplication and taking inverses of nonzero elements, so that it is itself a field under the operations of $K$. Subfields are central to field theory: a field extension $L/K$ can be viewed as a field $L$ together with a subfield isomorphic to $K$, and for a finite Galois extension the Galois\u2026","title":"Subfield"},{"id":"mathlib/overview/general-algebra/field-theory/characteristic-of-a-ring","kind":"node","parent":114,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/field-theory/index.html#characteristic-of-a-ring","scope":"mathlib/overview/general-algebra/field-theory","status":"mathlib","summary":"The characteristic of a ring is the least positive integer $n$ for which adding the identity element $1$ to itself $n$ times gives $0$, or $0$ if no such integer exists. It measures how the integers sit inside the ring, and it separates rings of characteristic zero, such as the rational and real numbers, from rings of positive characteristic, such as the integers modulo $n$ and the finite fields, where arithmetic\u2026","title":"Characteristic of a ring"},{"id":"mathlib/overview/general-algebra/field-theory/characteristic-zero","kind":"node","parent":114,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/field-theory/index.html#characteristic-zero","scope":"mathlib/overview/general-algebra/field-theory","status":"mathlib","summary":"A ring has characteristic zero when no positive multiple of its identity element vanishes: $1 + 1 + \\cdots + 1 \\neq 0$ however many terms are added. The rational, real and complex numbers all have this property, while the integers modulo $n$ for $n \\geq 1$ and every finite field do not. Characteristic zero is the setting in which the familiar arithmetic of the integers and the rationals survives inside a ring, so\u2026","title":"Characteristic zero"},{"id":"mathlib/overview/general-algebra/field-theory/characteristic-p","kind":"node","parent":114,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/field-theory/index.html#characteristic-p","scope":"mathlib/overview/general-algebra/field-theory","status":"mathlib","summary":"A ring $R$ has characteristic $p$ when adding the identity element $1$ to itself $n$ times gives $0$ exactly for the multiples $n$ of $p$. The characteristic measures how the integers sit inside a ring: in characteristic $0$ they embed faithfully, while in a positive characteristic $p$ the ring arithmetic of whole numbers wraps around modulo $p$. Commutative rings and fields of prime characteristic have an algebra\u2026","title":"Characteristic p"},{"id":"mathlib/overview/general-algebra/field-theory/frobenius-morphism","kind":"node","parent":114,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/field-theory/index.html#frobenius-morphism","scope":"mathlib/overview/general-algebra/field-theory","status":"mathlib","summary":"The Frobenius morphism (or Frobenius endomorphism) of a commutative ring of prime characteristic $p$ is the map that raises every element to its $p$-th power. In characteristic $p$ this map preserves sums as well as products, so it is a ring homomorphism from the ring to itself. In characteristic zero no power map $x \\mapsto x^n$ with $n > 1$ is additive, since $(1+1)^n = 2^n$ differs from $1^n + 1^n = 2$. It is\u2026","title":"Frobenius morphism"},{"id":"mathlib/overview/general-algebra/field-theory/algebraically-closed-field","kind":"node","parent":114,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/field-theory/index.html#algebraically-closed-field","scope":"mathlib/overview/general-algebra/field-theory","status":"mathlib","summary":"An algebraically closed field is a field in which every nonconstant polynomial in one variable has a root. In such a field every polynomial breaks up completely into linear factors, so questions about polynomial equations reduce to questions about their roots. The complex numbers are the most familiar example. Algebraically closed fields are the natural setting for linear algebra over a field (every square matrix\u2026","title":"Algebraically closed field"},{"id":"mathlib/overview/general-algebra/field-theory/existence-of-algebraic-closure-of-a-field","kind":"node","parent":114,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/field-theory/index.html#existence-of-algebraic-closure-of-a-field","scope":"mathlib/overview/general-algebra/field-theory","status":"mathlib","summary":"The existence of an algebraic closure is the theorem that every field $k$ has an algebraic extension $\\bar k$ that is algebraically closed. Such a field $\\bar k$ is called an algebraic closure of $k$; it is unique up to isomorphism over $k$. The theorem supplies, for every field, one ambient field in which every polynomial over $k$ splits into linear factors, so that roots, splitting fields and algebraic extensions\u2026","title":"Existence of algebraic closure of a field"},{"id":"mathlib/overview/general-algebra/field-theory/c-is-algebraically-closed","kind":"node","parent":114,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\C\\)"],["text"," is algebraically closed"]],"source":"$\\C$ is algebraically closed"},"result":true,"route":"../roadmap/mathlib/overview/general-algebra/field-theory/index.html#c-is-algebraically-closed","scope":"mathlib/overview/general-algebra/field-theory","status":"mathlib","summary":"The statement that $\\mathbb{C}$ is algebraically closed, usually called the fundamental theorem of algebra, says that every nonconstant polynomial with complex coefficients has a complex root. It means that the complex numbers need no further enlargement to solve polynomial equations: adjoining a square root of $-1$ to the real numbers already produces a field in which every polynomial equation of positive degree\u2026","title":"$\\C$ is algebraically closed"},{"id":"mathlib/overview/general-algebra/field-theory/field-of-fractions-of-an-integral-domain","kind":"node","parent":114,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/field-theory/index.html#field-of-fractions-of-an-integral-domain","scope":"mathlib/overview/general-algebra/field-theory","status":"mathlib","summary":"The field of fractions of an integral domain $A$ is the smallest field in which $A$ sits as a subring: its elements are quotients $a/b$ of elements of $A$ with $b \\neq 0$, added and multiplied by the rules of school arithmetic. The passage from the integers to the rational numbers is the model case, and the passage from polynomials to rational functions is another. The construction lets questions about a domain be\u2026","title":"Field of fractions of an integral domain"},{"id":"mathlib/overview/general-algebra/field-theory/algebraic-extension","kind":"node","parent":114,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/field-theory/index.html#algebraic-extension","scope":"mathlib/overview/general-algebra/field-theory","status":"mathlib","summary":"An algebraic extension is a field extension $L/K$ in which every element of $L$ is a root of some nonzero polynomial with coefficients in $K$. Algebraic extensions are the setting of most of classical field theory: finite extensions, splitting fields and Galois extensions are all algebraic, and every algebraic extension of a field embeds into its algebraic closure, which is thus the largest algebraic extension up\u2026","title":"Algebraic extension"},{"id":"mathlib/overview/general-algebra/field-theory/rupture-field","kind":"node","parent":114,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/field-theory/index.html#rupture-field","scope":"mathlib/overview/general-algebra/field-theory","status":"mathlib","summary":"A rupture field of an irreducible polynomial $f$ over a field $K$ is an extension of $K$ generated by a single root of $f$. The standard model is the quotient ring $K[X]/(f)$, in which the class of $X$ is a root of $f$ by construction. Applied to an irreducible factor, this quotient construction is how one proves that every nonconstant polynomial over a field has a root in some extension, and iterating it produces\u2026","title":"Rupture field"},{"id":"mathlib/overview/general-algebra/field-theory/splitting-field","kind":"node","parent":114,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/field-theory/index.html#splitting-field","scope":"mathlib/overview/general-algebra/field-theory","status":"mathlib","summary":"A splitting field of a polynomial $f$ over a field $K$ is a field extension of $K$ in which $f$ factors completely into linear factors and which is generated over $K$ by the roots of $f$. Every polynomial has one, and any two are isomorphic over $K$, so one may speak of \"the\" splitting field. Splitting fields are the extensions in which the roots of a polynomial can all be handled at once, which makes them the\u2026","title":"Splitting field"},{"id":"mathlib/overview/general-algebra/field-theory/perfect-closure","kind":"node","parent":114,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/field-theory/index.html#perfect-closure","scope":"mathlib/overview/general-algebra/field-theory","status":"mathlib","summary":"The perfect closure of a commutative ring $K$ of prime characteristic $p$ is the ring obtained from $K$ by formally adjoining a $p$-th root, a $p^2$-th root, and so on, of every element; along the way every nilpotent element of $K$ is sent to $0$. It is the most economical way to make the Frobenius morphism $x \\mapsto x^p$ bijective: it is a perfect ring that receives a map from $K$, and every ring homomorphism\u2026","title":"Perfect closure"},{"id":"mathlib/overview/general-algebra/field-theory/galois-correspondence","kind":"node","parent":114,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/field-theory/index.html#galois-correspondence","scope":"mathlib/overview/general-algebra/field-theory","status":"mathlib","summary":"The Galois correspondence, also called the fundamental theorem of Galois theory, says that for a finite Galois extension of fields $E/F$ the intermediate fields of the extension are in one-to-one, inclusion-reversing correspondence with the subgroups of its Galois group. It turns questions about fields, which can be infinite and hard to enumerate, into questions about a finite group, and it is the basis of the\u2026","title":"Galois correspondence"},{"id":"mathlib/overview/general-algebra/field-theory/abel-ruffini-theorem-one-direction","kind":"node","parent":114,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/field-theory/index.html#abel-ruffini-theorem-one-direction","scope":"mathlib/overview/general-algebra/field-theory","status":"mathlib","summary":"The Abel\u2013Ruffini theorem, in the direction treated here, says that if an element of a field extension can be built from the base field by the field operations and the extraction of $n$-th roots, then the Galois group of its minimal polynomial is a solvable group. It is the half of Galois's criterion for solvability by radicals that serves as an obstruction: to show that a polynomial has a root that no formula in\u2026","title":"Abel-Ruffini theorem (one direction)"},{"data_url":"scopes/mathlib/overview/general-algebra/homological-algebra.json","id":"mathlib/overview/general-algebra/homological-algebra","kind":"scope","parent":3,"result":false,"route":"../roadmap/mathlib/overview/general-algebra/homological-algebra/index.html","scope":"mathlib/overview/general-algebra/homological-algebra","status":"mathlib","title":"Homological algebra"},{"id":"mathlib/overview/general-algebra/homological-algebra/chain-complex","kind":"node","parent":131,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/homological-algebra/index.html#chain-complex","scope":"mathlib/overview/general-algebra/homological-algebra","status":"mathlib","summary":"A chain complex is a sequence of objects, such as abelian groups or modules over a ring, joined by maps that lower the degree by one, in which any two consecutive maps compose to zero. It is the basic object of homological algebra. The vanishing of these composites says that the image of each map lies inside the kernel of the next, and the failure of the two to coincide is measured by the homology of the complex.\u2026","title":"Chain complex"},{"id":"mathlib/overview/general-algebra/homological-algebra/functorial-homology","kind":"node","parent":131,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/homological-algebra/index.html#functorial-homology","scope":"mathlib/overview/general-algebra/homological-algebra","status":"mathlib","summary":"Functorial homology is the construction that assigns to a complex $K$ and a degree $i$ the homology object $H_i(K)$, the cycles in degree $i$ modulo the boundaries, together with the morphism $H_i(K) \\to H_i(L)$ that every chain map $K \\to L$ induces. The assignment respects identities and composition, so homology in each degree is a functor from complexes to the underlying category. It measures how far a complex\u2026","title":"Functorial homology"},{"data_url":"scopes/mathlib/overview/general-algebra/number-theory.json","id":"mathlib/overview/general-algebra/number-theory","kind":"scope","parent":3,"result":false,"route":"../roadmap/mathlib/overview/general-algebra/number-theory/index.html","scope":"mathlib/overview/general-algebra/number-theory","status":"mathlib","title":"Number theory"},{"id":"mathlib/overview/general-algebra/number-theory/sum-of-two-squares","kind":"node","parent":134,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/number-theory/index.html#sum-of-two-squares","scope":"mathlib/overview/general-algebra/number-theory","status":"mathlib","summary":"Fermat's theorem on sums of two squares says that an odd prime number $p$ can be written as $p = a^2 + b^2$ with integers $a$ and $b$ if and only if $p \\equiv 1 \\pmod 4$. Together with $2 = 1^2 + 1^2$, this settles which primes are sums of two squares, and because such sums are closed under multiplication it leads to a description of every sum of two squares through its prime factorization. The theorem is a basic\u2026","title":"Sum of two squares"},{"id":"mathlib/overview/general-algebra/number-theory/sum-of-four-squares","kind":"node","parent":134,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/number-theory/index.html#sum-of-four-squares","scope":"mathlib/overview/general-algebra/number-theory","status":"mathlib","summary":"Lagrange's four-square theorem says that every natural number is a sum of four squares of natural numbers. Four is the least number that works for all natural numbers, since $7$ is not a sum of three squares, so the theorem settles completely which count of squares suffices. It is a model case of an additive problem in number theory, and its proof combines an algebraic identity with a descent argument on prime\u2026","title":"Sum of four squares"},{"id":"mathlib/overview/general-algebra/number-theory/quadratic-reciprocity","kind":"node","parent":134,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/number-theory/index.html#quadratic-reciprocity","scope":"mathlib/overview/general-algebra/number-theory","status":"mathlib","summary":"Quadratic reciprocity is a theorem of elementary number theory relating two questions that look unrelated: whether an odd prime $q$ is a square modulo a different odd prime $p$, and whether $p$ is a square modulo $q$. It says the two answers agree unless both primes are congruent to $3$ modulo $4$, in which case they are opposite. Together with two supplementary laws it gives an efficient way, especially in its\u2026","title":"Quadratic reciprocity"},{"id":"mathlib/overview/general-algebra/number-theory/solutions-to-pells-equation","kind":"node","parent":134,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/number-theory/index.html#solutions-to-pells-equation","scope":"mathlib/overview/general-algebra/number-theory","status":"mathlib","summary":"The solutions to Pell's equation $x^2 - dy^2 = 1$ in the special case $d = a^2 - 1$, for an integer $a > 1$, can be written down completely: the solutions in natural numbers form a single sequence of pairs, the Pell sequence, generated from the obvious solution $(a, 1)$ by a linear recursion, and the integer solutions are these up to sign. In this case the fundamental solution is evident, and the case serves as a\u2026","title":"Solutions to Pell's equation (Number theory)"},{"id":"mathlib/overview/general-algebra/number-theory/matiyasevics-theorem","kind":"node","parent":134,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/number-theory/index.html#matiyasevics-theorem","scope":"mathlib/overview/general-algebra/number-theory","status":"mathlib","summary":"Matiyasevi\u010d's theorem, also called the MRDP theorem (after Matiyasevi\u010d, Robinson, Davis and Putnam), says that every recursively enumerable set of natural numbers (one whose members some algorithm can list) is Diophantine: it is the set of parameter values for which some polynomial equation with integer coefficients has a solution in natural numbers. Since some recursively enumerable sets cannot be decided by any\u2026","title":"Matiyasevi\u010d's theorem"},{"id":"mathlib/overview/general-algebra/number-theory/arithmetic-functions","kind":"node","parent":134,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/number-theory/index.html#arithmetic-functions","scope":"mathlib/overview/general-algebra/number-theory","status":"mathlib","summary":"An arithmetic function is a function defined on the positive integers, usually with integer, real or complex values, that records some arithmetic property of its argument: the number of divisors of $n$, the sum of those divisors, or the count of integers up to $n$ coprime to $n$. Arithmetic functions are among the basic objects of elementary and analytic number theory. A product on them, Dirichlet convolution,\u2026","title":"Arithmetic functions"},{"id":"mathlib/overview/general-algebra/number-theory/bernoulli-numbers","kind":"node","parent":134,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/number-theory/index.html#bernoulli-numbers","scope":"mathlib/overview/general-algebra/number-theory","status":"mathlib","summary":"The Bernoulli numbers $B_0, B_1, B_2, \\dots$ are a sequence of rational numbers defined by a power series identity, beginning $1, -\\tfrac12, \\tfrac16, 0, -\\tfrac1{30}, 0, \\tfrac1{42}, \\dots$. They appear throughout number theory and analysis: they give closed formulas for sums of powers of consecutive integers, up to explicit factors they are the values of the Riemann zeta function at the negative integers and at\u2026","title":"Bernoulli numbers"},{"id":"mathlib/overview/general-algebra/number-theory/chevalley-warning-theorem","kind":"node","parent":134,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/number-theory/index.html#chevalley-warning-theorem","scope":"mathlib/overview/general-algebra/number-theory","status":"mathlib","summary":"The Chevalley\u2013Warning theorem says that a polynomial equation over a finite field whose degree is smaller than its number of variables has a number of solutions divisible by the characteristic of the field. It turns a bound on degrees into a congruence on a count, and so yields existence results directly: if such an equation has one obvious solution, the count cannot be $1$, so it has another. This is the standard\u2026","title":"Chevalley-Warning theorem (Number theory)"},{"id":"mathlib/overview/general-algebra/number-theory/hensels-lemma-for-z-p","kind":"node","parent":134,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hensel's lemma (for "],["math","\\(\\mathbb{Z}_p\\)"],["text",")"]],"source":"Hensel's lemma (for $\\mathbb{Z}_p$)"},"result":true,"route":"../roadmap/mathlib/overview/general-algebra/number-theory/index.html#hensels-lemma-for-z-p","scope":"mathlib/overview/general-algebra/number-theory","status":"mathlib","summary":"Hensel's lemma for the $p$-adic integers $\\mathbb{Z}_p$ says that an approximate root of a polynomial can be refined to an exact root, provided the approximation is good compared with the size of the derivative at that point. In its most used form it says that a root modulo a prime $p$ at which the derivative is nonzero modulo $p$ lifts to a unique root in $\\mathbb{Z}_p$ congruent to it modulo $p$, and so to a root\u2026","title":"Hensel's lemma (for $\\mathbb{Z}_p$)"},{"id":"mathlib/overview/general-algebra/number-theory/ring-of-witt-vectors","kind":"node","parent":134,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/number-theory/index.html#ring-of-witt-vectors","scope":"mathlib/overview/general-algebra/number-theory","status":"mathlib","summary":"For a prime number $p$ and a commutative ring $R$, the ring of ($p$-typical) Witt vectors $W(R)$ is a ring whose elements are infinite sequences $(a_0, a_1, a_2, \\dots)$ of elements of $R$, added and multiplied by universal polynomial rules that imitate carrying digits. Its basic use is to produce rings of characteristic zero out of rings of characteristic $p$: applied to the field with $p$ elements it gives the\u2026","title":"Ring of Witt vectors"},{"id":"mathlib/overview/general-algebra/number-theory/perfection-of-a-ring","kind":"node","parent":134,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/number-theory/index.html#perfection-of-a-ring","scope":"mathlib/overview/general-algebra/number-theory","status":"mathlib","summary":"The perfection of a commutative ring $R$ of prime characteristic $p$ is the ring of sequences $(x_0, x_1, x_2, \\dots)$ of elements of $R$ in which each term is a $p$-th root of the one before it. On this ring the Frobenius map $x \\mapsto x^p$ is bijective, and it is the universal perfect commutative ring of characteristic $p$ mapping to $R$: every homomorphism to $R$ from a perfect commutative ring of\u2026","title":"Perfection of a ring"},{"data_url":"scopes/mathlib/overview/general-algebra/transcendental-numbers.json","id":"mathlib/overview/general-algebra/transcendental-numbers","kind":"scope","parent":3,"result":false,"route":"../roadmap/mathlib/overview/general-algebra/transcendental-numbers/index.html","scope":"mathlib/overview/general-algebra/transcendental-numbers","status":"mathlib","title":"Transcendental numbers"},{"id":"mathlib/overview/general-algebra/transcendental-numbers/liouvilles-theorem-on-existence-of-transcendenta","kind":"node","parent":146,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/transcendental-numbers/index.html#liouvilles-theorem-on-existence-of-transcendenta","scope":"mathlib/overview/general-algebra/transcendental-numbers","status":"mathlib","summary":"Liouville's theorem on existence of transcendental numbers says that explicit real numbers such as are transcendental: they are not roots of any nonzero polynomial with integer coefficients. The theorem names specific transcendental numbers instead of only proving that some exist. The method behind it is a lower bound on how well an algebraic number can be approximated by fractions: a number that has rational\u2026","title":"Liouville's theorem on existence of transcendental numbers"},{"data_url":"scopes/mathlib/overview/general-algebra/algebraic-number-theory.json","id":"mathlib/overview/general-algebra/algebraic-number-theory","kind":"scope","parent":3,"result":false,"route":"../roadmap/mathlib/overview/general-algebra/algebraic-number-theory/index.html","scope":"mathlib/overview/general-algebra/algebraic-number-theory","status":"mathlib","title":"Algebraic Number Theory"},{"id":"mathlib/overview/general-algebra/algebraic-number-theory/finiteness-of-the-class-number","kind":"node","parent":148,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/algebraic-number-theory/index.html#finiteness-of-the-class-number","scope":"mathlib/overview/general-algebra/algebraic-number-theory","status":"mathlib","summary":"Finiteness of the class number is the theorem that the ideal class group of the ring of integers of a number field is finite. Its order, the class number, measures how far that ring is from being a principal ideal domain, and in particular whether factorization of elements into irreducibles in it is unique. Together with the Dirichlet unit theorem it is one of the two basic finiteness results of algebraic number\u2026","title":"Finiteness of the class number"},{"id":"mathlib/overview/general-algebra/algebraic-number-theory/dirichlet-unit-theorem","kind":"node","parent":148,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/algebraic-number-theory/index.html#dirichlet-unit-theorem","scope":"mathlib/overview/general-algebra/algebraic-number-theory","status":"mathlib","summary":"The Dirichlet unit theorem describes the group of units of the ring of integers of a number field: it is the product of the finite cyclic group of roots of unity in the field and a free abelian group whose rank is determined by how the field embeds into the complex numbers. It is one of the two basic finiteness results of algebraic number theory, alongside the finiteness of the class number, and the size of the\u2026","title":"Dirichlet unit theorem"},{"id":"mathlib/overview/general-algebra/algebraic-number-theory/dirichlet-class-number-formula","kind":"node","parent":148,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/algebraic-number-theory/index.html#dirichlet-class-number-formula","scope":"mathlib/overview/general-algebra/algebraic-number-theory","status":"mathlib","summary":"The Dirichlet class number formula, also called the analytic class number formula, expresses the behaviour of the Dedekind zeta function of a number field near $s = 1$ in terms of the field's basic arithmetic invariants: its class number, regulator, number of roots of unity, discriminant and numbers of real and complex places. It ties an analytic quantity to algebraic ones, and is the main route by which the class\u2026","title":"Dirichlet class number formula"},{"data_url":"scopes/mathlib/overview/general-algebra/representation-theory.json","id":"mathlib/overview/general-algebra/representation-theory","kind":"scope","parent":3,"result":false,"route":"../roadmap/mathlib/overview/general-algebra/representation-theory/index.html","scope":"mathlib/overview/general-algebra/representation-theory","status":"mathlib","title":"Representation theory"},{"id":"mathlib/overview/general-algebra/representation-theory/representation","kind":"node","parent":152,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/representation-theory/index.html#representation","scope":"mathlib/overview/general-algebra/representation-theory","status":"mathlib","summary":"A representation of a group $G$ is a way of letting $G$ act on a vector space by linear maps, compatibly with the group multiplication. It turns questions about an abstract group into questions of linear algebra, where matrices, traces and eigenvalues are available. Representations are a main tool for studying finite groups and for the symmetry arguments of number theory, geometry and physics.","title":"Representation"},{"id":"mathlib/overview/general-algebra/representation-theory/category-of-finite-dimensional-representations","kind":"node","parent":152,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/representation-theory/index.html#category-of-finite-dimensional-representations","scope":"mathlib/overview/general-algebra/representation-theory","status":"mathlib","summary":"The category of finite-dimensional representations of a group $G$ over a field $k$ has as objects the finite-dimensional representations of $G$ over $k$, and as morphisms the linear maps that commute with the action of $G$. Treating representations as the objects of a category, rather than one at a time, is the natural setting for the main tools of representation theory: Schur's lemma is a statement about its\u2026","title":"Category of finite-dimensional representations"},{"id":"mathlib/overview/general-algebra/representation-theory/character","kind":"node","parent":152,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/representation-theory/index.html#character","scope":"mathlib/overview/general-algebra/representation-theory","status":"mathlib","summary":"The character of a finite-dimensional representation of a group is the function that assigns to each group element the trace of the linear map by which it acts. It compresses a representation, a family of matrices, into a single scalar-valued function on the group, yet for finite groups over the complex numbers it loses nothing: two representations with the same character are isomorphic. Characters turn many\u2026","title":"Character"},{"id":"mathlib/overview/general-algebra/representation-theory/orthogonality-of-characters","kind":"node","parent":152,"result":true,"route":"../roadmap/mathlib/overview/general-algebra/representation-theory/index.html#orthogonality-of-characters","scope":"mathlib/overview/general-algebra/representation-theory","status":"mathlib","summary":"The orthogonality of characters is the theorem that the characters of the irreducible representations of a finite group over the complex numbers (more generally, over an algebraically closed field whose characteristic does not divide the order of the group) form an orthonormal family for a natural pairing of functions on the group. It is the basic computational tool of character theory: it lets one decide whether a\u2026","title":"Orthogonality of characters"},{"data_url":"scopes/mathlib/overview/linear-algebra.json","id":"mathlib/overview/linear-algebra","kind":"scope","parent":2,"result":false,"route":"../roadmap/mathlib/overview/linear-algebra/index.html","scope":"mathlib/overview/linear-algebra","status":"mathlib","title":"Linear algebra"},{"data_url":"scopes/mathlib/overview/linear-algebra/fundamentals.json","id":"mathlib/overview/linear-algebra/fundamentals","kind":"scope","parent":157,"result":false,"route":"../roadmap/mathlib/overview/linear-algebra/fundamentals/index.html","scope":"mathlib/overview/linear-algebra/fundamentals","status":"mathlib","title":"Fundamentals"},{"id":"mathlib/overview/linear-algebra/fundamentals/module","kind":"node","parent":158,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/fundamentals/index.html#module","scope":"mathlib/overview/linear-algebra/fundamentals","status":"mathlib","summary":"A module is a generalization of vector spaces to a scalar semiring. It consists of a scalar semiring `R` and an additive monoid of \"vectors\" `M`, connected by a \"scalar multiplication\" operation `r \u2022 x : M` (where `r : R` and `x : M`) with some natural associativity and distributivity axioms similar to those on a ring.","title":"Module"},{"id":"mathlib/overview/linear-algebra/fundamentals/linear-map","kind":"node","parent":158,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/fundamentals/index.html#linear-map","scope":"mathlib/overview/linear-algebra/fundamentals","status":"mathlib","summary":"A map `f` between an `R`-module and an `S`-module over a ring homomorphism `\u03c3 : R \u2192+* S` is semilinear if it satisfies the two properties `f (x + y) = f x + f y` and `f (c \u2022 x) = (\u03c3 c) \u2022 f x`. Elements of `LinearMap \u03c3 M M\u2082` (available under the notation `M \u2192\u209b\u2097[\u03c3] M\u2082`) are bundled versions of such maps. For plain linear maps (i.e. for which `\u03c3 = RingHom.id R`), the notation `M \u2192\u2097[R] M\u2082` is available. An unbundled\u2026","title":"Linear map"},{"id":"mathlib/overview/linear-algebra/fundamentals/the-category-of-modules-over-a-ring","kind":"node","parent":158,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/fundamentals/index.html#the-category-of-modules-over-a-ring","scope":"mathlib/overview/linear-algebra/fundamentals","status":"mathlib","summary":"The category of R-modules and their morphisms.","title":"The category of modules over a ring"},{"id":"mathlib/overview/linear-algebra/fundamentals/quotient-space","kind":"node","parent":158,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/fundamentals/index.html#quotient-space","scope":"mathlib/overview/linear-algebra/fundamentals","status":"mathlib","summary":"The quotient of a module `M` by a submodule `p \u2286 M`.","title":"Quotient space (Fundamentals)"},{"id":"mathlib/overview/linear-algebra/fundamentals/tensor-product","kind":"node","parent":158,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/fundamentals/index.html#tensor-product","scope":"mathlib/overview/linear-algebra/fundamentals","status":"mathlib","summary":"The tensor product of two modules `M` and `N` over the same commutative semiring `R`. The localized notations are `M \u2297 N` and `M \u2297[R] N`, accessed by `open scoped TensorProduct`.","title":"Tensor product"},{"id":"mathlib/overview/linear-algebra/fundamentals/noetherian-module","kind":"node","parent":158,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/fundamentals/index.html#noetherian-module","scope":"mathlib/overview/linear-algebra/fundamentals","status":"mathlib","summary":"`IsNoetherian R M` is the proposition that `M` is a Noetherian `R`-module, implemented as the predicate that all `R`-submodules of `M` are finitely generated.","title":"Noetherian module"},{"id":"mathlib/overview/linear-algebra/fundamentals/basis","kind":"node","parent":158,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/fundamentals/index.html#basis","scope":"mathlib/overview/linear-algebra/fundamentals","status":"mathlib","summary":"A `Basis \u03b9 R M` for a module `M` is the type of `\u03b9`-indexed `R`-bases of `M`.","title":"Basis"},{"id":"mathlib/overview/linear-algebra/fundamentals/multilinear-map","kind":"node","parent":158,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/fundamentals/index.html#multilinear-map","scope":"mathlib/overview/linear-algebra/fundamentals","status":"mathlib","summary":"Multilinear maps from the `R`-module `\u03a0 i, M i` to the `S`-module `N` over a ring homomorphism `\u03c3 : R \u2192+* S`.","title":"Multilinear map"},{"id":"mathlib/overview/linear-algebra/fundamentals/alternating-map","kind":"node","parent":158,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/fundamentals/index.html#alternating-map","scope":"mathlib/overview/linear-algebra/fundamentals","status":"mathlib","summary":"An alternating map from `\u03b9 \u2192 M` to `N`, denoted `M [\u22c0^\u03b9]\u2192\u2097[R] N`, is a multilinear map that vanishes when two of its arguments are equal.","title":"Alternating map"},{"id":"mathlib/overview/linear-algebra/fundamentals/general-linear-group","kind":"node","parent":158,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/fundamentals/index.html#general-linear-group","scope":"mathlib/overview/linear-algebra/fundamentals","status":"mathlib","summary":"The group of invertible linear maps from `M` to itself","title":"General linear group (Fundamentals)"},{"data_url":"scopes/mathlib/overview/linear-algebra/duality.json","id":"mathlib/overview/linear-algebra/duality","kind":"scope","parent":157,"result":false,"route":"../roadmap/mathlib/overview/linear-algebra/duality/index.html","scope":"mathlib/overview/linear-algebra/duality","status":"mathlib","title":"Duality"},{"id":"mathlib/overview/linear-algebra/duality/dual-vector-space","kind":"node","parent":169,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/duality/index.html#dual-vector-space","scope":"mathlib/overview/linear-algebra/duality","status":"mathlib","summary":"The left dual space of an R-module M is the R-module of linear maps `M \u2192 R`.","title":"Dual vector space"},{"id":"mathlib/overview/linear-algebra/duality/dual-basis","kind":"node","parent":169,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/duality/index.html#dual-basis","scope":"mathlib/overview/linear-algebra/duality","status":"mathlib","summary":"Maps a basis for `V` to a basis for the dual space.","title":"Dual basis"},{"data_url":"scopes/mathlib/overview/linear-algebra/finite-dimensional-vector-spaces.json","id":"mathlib/overview/linear-algebra/finite-dimensional-vector-spaces","kind":"scope","parent":157,"result":false,"route":"../roadmap/mathlib/overview/linear-algebra/finite-dimensional-vector-spaces/index.html","scope":"mathlib/overview/linear-algebra/finite-dimensional-vector-spaces","status":"mathlib","title":"Finite-dimensional vector spaces"},{"id":"mathlib/overview/linear-algebra/finite-dimensional-vector-spaces/finite-dimensionality","kind":"node","parent":172,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/finite-dimensional-vector-spaces/index.html#finite-dimensionality","scope":"mathlib/overview/linear-algebra/finite-dimensional-vector-spaces","status":"mathlib","summary":"`FiniteDimensional` vector spaces are defined to be finite modules. Use `Module.Basis.finiteDimensional_of_finite` to prove finite dimension from another definition.","title":"Finite-dimensionality"},{"id":"mathlib/overview/linear-algebra/finite-dimensional-vector-spaces/isomorphism-with-k-n","kind":"node","parent":172,"reading_title":{"format":"inline-parts-v1","parts":[["text","Isomorphism with "],["math","\\(K^n\\)"]],"source":"Isomorphism with $K^n$"},"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/finite-dimensional-vector-spaces/index.html#isomorphism-with-k-n","scope":"mathlib/overview/linear-algebra/finite-dimensional-vector-spaces","status":"mathlib","summary":"A module over `R` with a finite basis is linearly equivalent to functions from its basis to `R`.","title":"Isomorphism with $K^n$"},{"id":"mathlib/overview/linear-algebra/finite-dimensional-vector-spaces/isomorphism-with-bidual","kind":"node","parent":172,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/finite-dimensional-vector-spaces/index.html#isomorphism-with-bidual","scope":"mathlib/overview/linear-algebra/finite-dimensional-vector-spaces","status":"mathlib","summary":"The bijection between a reflexive module and its double dual, bundled as a `LinearEquiv`.","title":"Isomorphism with bidual"},{"data_url":"scopes/mathlib/overview/linear-algebra/finitely-generated-modules-over-a-pid.json","id":"mathlib/overview/linear-algebra/finitely-generated-modules-over-a-pid","kind":"scope","parent":157,"result":false,"route":"../roadmap/mathlib/overview/linear-algebra/finitely-generated-modules-over-a-pid/index.html","scope":"mathlib/overview/linear-algebra/finitely-generated-modules-over-a-pid","status":"mathlib","title":"Finitely generated modules over a PID"},{"id":"mathlib/overview/linear-algebra/finitely-generated-modules-over-a-pid/structure-theorem","kind":"node","parent":176,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/finitely-generated-modules-over-a-pid/index.html#structure-theorem","scope":"mathlib/overview/linear-algebra/finitely-generated-modules-over-a-pid","status":"mathlib","summary":"Structure theorem of finitely generated modules over a PID : A finitely generated module over a PID is isomorphic to the product of a free module and a direct sum of some `R \u29f8 R \u2219 (p i ^ e i)` where the `p i ^ e i` are prime powers.","title":"Structure theorem"},{"data_url":"scopes/mathlib/overview/linear-algebra/matrices.json","id":"mathlib/overview/linear-algebra/matrices","kind":"scope","parent":157,"result":false,"route":"../roadmap/mathlib/overview/linear-algebra/matrices/index.html","scope":"mathlib/overview/linear-algebra/matrices","status":"mathlib","title":"Matrices"},{"id":"mathlib/overview/linear-algebra/matrices/ring-valued-matrix","kind":"node","parent":178,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/matrices/index.html#ring-valued-matrix","scope":"mathlib/overview/linear-algebra/matrices","status":"mathlib","summary":"`Matrix m n R` is the type of matrices with entries in `R`, whose rows are indexed by `m` and whose columns are indexed by `n`.","title":"Ring-valued matrix"},{"id":"mathlib/overview/linear-algebra/matrices/matrix-representation-of-a-linear-map","kind":"node","parent":178,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/matrices/index.html#matrix-representation-of-a-linear-map","scope":"mathlib/overview/linear-algebra/matrices","status":"mathlib","summary":"Given bases of two modules `M\u2081` and `M\u2082` over a commutative ring `R`, we get a linear equivalence between linear maps `M\u2081 \u2192\u2097 M\u2082` and matrices over `R` indexed by the bases.","title":"Matrix representation of a linear map"},{"id":"mathlib/overview/linear-algebra/matrices/determinant","kind":"node","parent":178,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/matrices/index.html#determinant","scope":"mathlib/overview/linear-algebra/matrices","status":"mathlib","summary":"The determinant of a matrix given by the Leibniz formula.","title":"Determinant"},{"id":"mathlib/overview/linear-algebra/matrices/invertibility","kind":"node","parent":178,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/matrices/index.html#invertibility","scope":"mathlib/overview/linear-algebra/matrices","status":"mathlib","summary":"The inverse of a square matrix, when it is invertible (and zero otherwise).","title":"Invertibility"},{"data_url":"scopes/mathlib/overview/linear-algebra/endomorphism-polynomials.json","id":"mathlib/overview/linear-algebra/endomorphism-polynomials","kind":"scope","parent":157,"result":false,"route":"../roadmap/mathlib/overview/linear-algebra/endomorphism-polynomials/index.html","scope":"mathlib/overview/linear-algebra/endomorphism-polynomials","status":"mathlib","title":"Endomorphism polynomials"},{"id":"mathlib/overview/linear-algebra/endomorphism-polynomials/minimal-polynomial","kind":"node","parent":183,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/endomorphism-polynomials/index.html#minimal-polynomial","scope":"mathlib/overview/linear-algebra/endomorphism-polynomials","status":"mathlib","summary":"Suppose `x : B`, where `B` is an `A`-algebra.","title":"Minimal polynomial"},{"id":"mathlib/overview/linear-algebra/endomorphism-polynomials/characteristic-polynomial","kind":"node","parent":183,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/endomorphism-polynomials/index.html#characteristic-polynomial","scope":"mathlib/overview/linear-algebra/endomorphism-polynomials","status":"mathlib","summary":"The characteristic polynomial of a matrix `M` is given by $\\det (t I - M)$.","title":"Characteristic polynomial"},{"id":"mathlib/overview/linear-algebra/endomorphism-polynomials/cayley-hamilton-theorem","kind":"node","parent":183,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/endomorphism-polynomials/index.html#cayley-hamilton-theorem","scope":"mathlib/overview/linear-algebra/endomorphism-polynomials","status":"mathlib","summary":"The Cayley-Hamilton Theorem, that the characteristic polynomial of a matrix, applied to the matrix itself, is zero.","title":"Cayley-Hamilton theorem"},{"data_url":"scopes/mathlib/overview/linear-algebra/structure-theory-of-endomorphisms.json","id":"mathlib/overview/linear-algebra/structure-theory-of-endomorphisms","kind":"scope","parent":157,"result":false,"route":"../roadmap/mathlib/overview/linear-algebra/structure-theory-of-endomorphisms/index.html","scope":"mathlib/overview/linear-algebra/structure-theory-of-endomorphisms","status":"mathlib","title":"Structure theory of endomorphisms"},{"id":"mathlib/overview/linear-algebra/structure-theory-of-endomorphisms/eigenvalue","kind":"node","parent":187,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/structure-theory-of-endomorphisms/index.html#eigenvalue","scope":"mathlib/overview/linear-algebra/structure-theory-of-endomorphisms","status":"mathlib","summary":"A scalar `\u03bc` is an eigenvalue for a linear map `f` if there are nonzero vectors `x` such that `f x = \u03bc \u2022 x`. (Def 5.5 of axler2024).","title":"Eigenvalue"},{"id":"mathlib/overview/linear-algebra/structure-theory-of-endomorphisms/eigenvector","kind":"node","parent":187,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/structure-theory-of-endomorphisms/index.html#eigenvector","scope":"mathlib/overview/linear-algebra/structure-theory-of-endomorphisms","status":"mathlib","summary":"A nonzero element of an eigenspace is an eigenvector. (Def 5.8 of axler2024)","title":"Eigenvector"},{"id":"mathlib/overview/linear-algebra/structure-theory-of-endomorphisms/existence-of-an-eigenvalue","kind":"node","parent":187,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/structure-theory-of-endomorphisms/index.html#existence-of-an-eigenvalue","scope":"mathlib/overview/linear-algebra/structure-theory-of-endomorphisms","status":"mathlib","summary":"In finite dimensions, over an algebraically closed field, every linear endomorphism has an eigenvalue.","title":"Existence of an eigenvalue"},{"data_url":"scopes/mathlib/overview/linear-algebra/bilinear-and-quadratic-forms.json","id":"mathlib/overview/linear-algebra/bilinear-and-quadratic-forms","kind":"scope","parent":157,"result":false,"route":"../roadmap/mathlib/overview/linear-algebra/bilinear-and-quadratic-forms/index.html","scope":"mathlib/overview/linear-algebra/bilinear-and-quadratic-forms","status":"mathlib","title":"Bilinear and quadratic forms"},{"id":"mathlib/overview/linear-algebra/bilinear-and-quadratic-forms/bilinear-form","kind":"node","parent":191,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/bilinear-and-quadratic-forms/index.html#bilinear-form","scope":"mathlib/overview/linear-algebra/bilinear-and-quadratic-forms","status":"mathlib","summary":"For convenience, a shorthand for the type of bilinear forms from `M` to `R`.","title":"Bilinear form"},{"id":"mathlib/overview/linear-algebra/bilinear-and-quadratic-forms/alternating-bilinear-form","kind":"node","parent":191,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/bilinear-and-quadratic-forms/index.html#alternating-bilinear-form","scope":"mathlib/overview/linear-algebra/bilinear-and-quadratic-forms","status":"mathlib","summary":"The proposition that a bilinear form is alternating","title":"Alternating bilinear form"},{"id":"mathlib/overview/linear-algebra/bilinear-and-quadratic-forms/symmetric-bilinear-form","kind":"node","parent":191,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/bilinear-and-quadratic-forms/index.html#symmetric-bilinear-form","scope":"mathlib/overview/linear-algebra/bilinear-and-quadratic-forms","status":"mathlib","summary":"The proposition that a bilinear form is symmetric","title":"Symmetric bilinear form"},{"id":"mathlib/overview/linear-algebra/bilinear-and-quadratic-forms/matrix-representation","kind":"node","parent":191,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/bilinear-and-quadratic-forms/index.html#matrix-representation","scope":"mathlib/overview/linear-algebra/bilinear-and-quadratic-forms","status":"mathlib","summary":"`BilinForm.toMatrix b` is the equivalence between `R`-bilinear forms on `M` and `n`-by-`n` matrices with entries in `R`, if `b` is an `R`-basis for `M`.","title":"Matrix representation"},{"id":"mathlib/overview/linear-algebra/bilinear-and-quadratic-forms/quadratic-form","kind":"node","parent":191,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/bilinear-and-quadratic-forms/index.html#quadratic-form","scope":"mathlib/overview/linear-algebra/bilinear-and-quadratic-forms","status":"mathlib","summary":"A quadratic form on a module.","title":"Quadratic form"},{"id":"mathlib/overview/linear-algebra/bilinear-and-quadratic-forms/polar-form-of-a-quadratic","kind":"node","parent":191,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/bilinear-and-quadratic-forms/index.html#polar-form-of-a-quadratic","scope":"mathlib/overview/linear-algebra/bilinear-and-quadratic-forms","status":"mathlib","summary":"Up to a factor 2, `Q.polar` is the associated bilinear map for a quadratic map `Q`.","title":"Polar form of a quadratic"},{"data_url":"scopes/mathlib/overview/linear-algebra/finite-dimensional-inner-product-spaces-see-also.json","id":"mathlib/overview/linear-algebra/finite-dimensional-inner-product-spaces-see-also","kind":"scope","parent":157,"result":false,"route":"../roadmap/mathlib/overview/linear-algebra/finite-dimensional-inner-product-spaces-see-also/index.html","scope":"mathlib/overview/linear-algebra/finite-dimensional-inner-product-spaces-see-also","status":"mathlib","title":"Finite-dimensional inner product spaces (see also Hilbert spaces, below)"},{"id":"mathlib/overview/linear-algebra/finite-dimensional-inner-product-spaces-see-also/existence-of-orthonormal-basis","kind":"node","parent":198,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/finite-dimensional-inner-product-spaces-see-also/index.html#existence-of-orthonormal-basis","scope":"mathlib/overview/linear-algebra/finite-dimensional-inner-product-spaces-see-also","status":"mathlib","summary":"An orthonormal basis on E is an identification of `E` with its dimensional-matching `EuclideanSpace \ud835\udd5c \u03b9`.","title":"Existence of orthonormal basis"},{"id":"mathlib/overview/linear-algebra/finite-dimensional-inner-product-spaces-see-also/diagonalization-of-self-adjoint-endomorphisms","kind":"node","parent":198,"result":true,"route":"../roadmap/mathlib/overview/linear-algebra/finite-dimensional-inner-product-spaces-see-also/index.html#diagonalization-of-self-adjoint-endomorphisms","scope":"mathlib/overview/linear-algebra/finite-dimensional-inner-product-spaces-see-also","status":"mathlib","summary":"*Diagonalization theorem*, *spectral theorem*; version 2: A self-adjoint operator `T` on a finite-dimensional inner product space `E` acts diagonally on the identification of `E` with Euclidean space induced by an orthonormal basis of eigenvectors of `T`.","title":"Diagonalization of self-adjoint endomorphisms"},{"data_url":"scopes/mathlib/overview/topology.json","id":"mathlib/overview/topology","kind":"scope","parent":2,"result":false,"route":"../roadmap/mathlib/overview/topology/index.html","scope":"mathlib/overview/topology","status":"mathlib","title":"Topology"},{"data_url":"scopes/mathlib/overview/topology/general-topology.json","id":"mathlib/overview/topology/general-topology","kind":"scope","parent":201,"result":false,"route":"../roadmap/mathlib/overview/topology/general-topology/index.html","scope":"mathlib/overview/topology/general-topology","status":"mathlib","title":"General topology"},{"data_url":"scopes/mathlib/overview/topology/general-topology/part-01.json","id":"mathlib/overview/topology/general-topology/part-01","kind":"scope","navigation":"transparent","parent":202,"result":false,"route":"../roadmap/mathlib/overview/topology/general-topology/part-01/index.html","scope":"mathlib/overview/topology/general-topology/part-01","status":"mathlib","title":"Filter to Topological vector bundle"},{"id":"mathlib/overview/topology/general-topology/part-01/filter","kind":"node","parent":203,"result":true,"route":"../roadmap/mathlib/overview/topology/general-topology/part-01/index.html#filter","scope":"mathlib/overview/topology/general-topology/part-01","status":"mathlib","summary":"A filter `F` on a type `\u03b1` is a collection of sets of `\u03b1` which contains the whole `\u03b1`, is upwards-closed, and is stable under intersection. We do not forbid this collection to be all sets of `\u03b1`.","title":"Filter"},{"id":"mathlib/overview/topology/general-topology/part-01/limit-of-a-map-with-respect-to-filters","kind":"node","parent":203,"result":true,"route":"../roadmap/mathlib/overview/topology/general-topology/part-01/index.html#limit-of-a-map-with-respect-to-filters","scope":"mathlib/overview/topology/general-topology/part-01","status":"mathlib","summary":"`Filter.Tendsto` is the generic \"limit of a function\" predicate. `Tendsto f l\u2081 l\u2082` asserts that for every `l\u2082` neighborhood `a`, the `f`-preimage of `a` is an `l\u2081` neighborhood.","title":"Limit of a map with respect to filters"},{"id":"mathlib/overview/topology/general-topology/part-01/topological-space","kind":"node","parent":203,"result":true,"route":"../roadmap/mathlib/overview/topology/general-topology/part-01/index.html#topological-space","scope":"mathlib/overview/topology/general-topology/part-01","status":"mathlib","summary":"A topology on `X`.","title":"Topological space"},{"id":"mathlib/overview/topology/general-topology/part-01/continuous-function","kind":"node","parent":203,"result":true,"route":"../roadmap/mathlib/overview/topology/general-topology/part-01/index.html#continuous-function","scope":"mathlib/overview/topology/general-topology/part-01","status":"mathlib","summary":"A function between topological spaces is continuous if the preimage of every open set is open. Registered as a structure to make sure it is not unfolded by Lean.","title":"Continuous function"},{"id":"mathlib/overview/topology/general-topology/part-01/the-category-of-topological-spaces","kind":"node","parent":203,"result":true,"route":"../roadmap/mathlib/overview/topology/general-topology/part-01/index.html#the-category-of-topological-spaces","scope":"mathlib/overview/topology/general-topology/part-01","status":"mathlib","summary":"The category of topological spaces.","title":"The category of topological spaces"},{"id":"mathlib/overview/topology/general-topology/part-01/induced-topology","kind":"node","parent":203,"result":true,"route":"../roadmap/mathlib/overview/topology/general-topology/part-01/index.html#induced-topology","scope":"mathlib/overview/topology/general-topology/part-01","status":"mathlib","summary":"Given `f : X \u2192 Y` and a topology on `Y`, the induced topology on `X` is the collection of sets that are preimages of some open set in `Y`. This is the coarsest topology that makes `f` continuous.","title":"Induced topology"},{"id":"mathlib/overview/topology/general-topology/part-01/open-map","kind":"node","parent":203,"result":true,"route":"../roadmap/mathlib/overview/topology/general-topology/part-01/index.html#open-map","scope":"mathlib/overview/topology/general-topology/part-01","status":"mathlib","summary":"A map `f : X \u2192 Y` is said to be an *open map*, if the image of any open `U : Set X` is open in `Y`.","title":"Open map"},{"id":"mathlib/overview/topology/general-topology/part-01/closed-map","kind":"node","parent":203,"result":true,"route":"../roadmap/mathlib/overview/topology/general-topology/part-01/index.html#closed-map","scope":"mathlib/overview/topology/general-topology/part-01","status":"mathlib","summary":"A map `f : X \u2192 Y` is said to be a *closed map*, if the image of any closed `U : Set X` is closed in `Y`.","title":"Closed map"},{"id":"mathlib/overview/topology/general-topology/part-01/closure","kind":"node","parent":203,"result":true,"route":"../roadmap/mathlib/overview/topology/general-topology/part-01/index.html#closure","scope":"mathlib/overview/topology/general-topology/part-01","status":"mathlib","summary":"The closure of `s` is the smallest closed set containing `s`.","title":"Closure"},{"id":"mathlib/overview/topology/general-topology/part-01/cluster-point","kind":"node","parent":203,"result":true,"route":"../roadmap/mathlib/overview/topology/general-topology/part-01/index.html#cluster-point","scope":"mathlib/overview/topology/general-topology/part-01","status":"mathlib","summary":"A point `x` is a cluster point of a filter `F` if `\ud835\udcdd x \u2293 F \u2260 \u22a5`. Also known as an accumulation point or a limit point, but beware that terminology varies. This is *not* the same as asking `\ud835\udcdd[\u2260] x \u2293 F \u2260 \u22a5`, which is called `AccPt` in Mathlib. See `mem_closure_iff_clusterPt` in particular.","title":"Cluster point"},{"id":"mathlib/overview/topology/general-topology/part-01/hausdorff-space","kind":"node","parent":203,"result":true,"route":"../roadmap/mathlib/overview/topology/general-topology/part-01/index.html#hausdorff-space","scope":"mathlib/overview/topology/general-topology/part-01","status":"mathlib","summary":"A T\u2082 space, also known as a Hausdorff space, is one in which for every `x \u2260 y` there exists disjoint open sets around `x` and `y`. This is the most widely used of the separation axioms.","title":"Hausdorff space"},{"id":"mathlib/overview/topology/general-topology/part-01/sequential-space","kind":"node","parent":203,"result":true,"route":"../roadmap/mathlib/overview/topology/general-topology/part-01/index.html#sequential-space","scope":"mathlib/overview/topology/general-topology/part-01","status":"mathlib","summary":"A topological space is said to be a *sequential space* if any sequentially closed set in this space is closed. This condition is weaker than being a Fr\u00e9chet-Urysohn space.","title":"Sequential space"},{"id":"mathlib/overview/topology/general-topology/part-01/extension-by-continuity","kind":"node","parent":203,"result":true,"route":"../roadmap/mathlib/overview/topology/general-topology/part-01/index.html#extension-by-continuity","scope":"mathlib/overview/topology/general-topology/part-01","status":"mathlib","summary":"If `i : \u03b1 \u2192 \u03b2` is a dense inducing, then any function `f : \u03b1 \u2192 \u03b3` \"extends\" to a function `g = IsDenseInducing.extend di f : \u03b2 \u2192 \u03b3`. If `\u03b3` is Hausdorff and `f` has a continuous extension, then `g` is the unique such extension. In general, `g` might not be continuous or even extend `f`.","title":"Extension by continuity"},{"id":"mathlib/overview/topology/general-topology/part-01/compactness-in-terms-of-filters","kind":"node","parent":203,"result":true,"route":"../roadmap/mathlib/overview/topology/general-topology/part-01/index.html#compactness-in-terms-of-filters","scope":"mathlib/overview/topology/general-topology/part-01","status":"mathlib","summary":"A set `s` is compact if for every nontrivial filter `f` that contains `s`, there exists `a \u2208 s` such that every set of `f` meets every neighborhood of `a`.","title":"Compactness in terms of filters"},{"id":"mathlib/overview/topology/general-topology/part-01/compactness-in-terms-of-open-covers-borel-lebesg","kind":"node","parent":203,"result":true,"route":"../roadmap/mathlib/overview/topology/general-topology/part-01/index.html#compactness-in-terms-of-open-covers-borel-lebesg","scope":"mathlib/overview/topology/general-topology/part-01","status":"mathlib","summary":"A set `s` is compact if and only if for every open cover of `s`, there exists a finite subcover.","title":"Compactness in terms of open covers (Borel-Lebesgue)"},{"id":"mathlib/overview/topology/general-topology/part-01/connectedness","kind":"node","parent":203,"result":true,"route":"../roadmap/mathlib/overview/topology/general-topology/part-01/index.html#connectedness","scope":"mathlib/overview/topology/general-topology/part-01","status":"mathlib","summary":"A connected space is a nonempty one where there is no non-trivial open partition.","title":"Connectedness"},{"id":"mathlib/overview/topology/general-topology/part-01/compact-open-topology","kind":"node","parent":203,"result":true,"route":"../roadmap/mathlib/overview/topology/general-topology/part-01/index.html#compact-open-topology","scope":"mathlib/overview/topology/general-topology/part-01","status":"mathlib","summary":"The compact-open topology on the space of continuous maps `C(X, Y)`.","title":"Compact open topology"},{"id":"mathlib/overview/topology/general-topology/part-01/stone-cech-compactification","kind":"node","parent":203,"result":true,"route":"../roadmap/mathlib/overview/topology/general-topology/part-01/index.html#stone-cech-compactification","scope":"mathlib/overview/topology/general-topology/part-01","status":"mathlib","summary":"The Stone-\u010cech compactification of a topological space.","title":"Stone-Cech compactification"},{"id":"mathlib/overview/topology/general-topology/part-01/topological-fiber-bundle","kind":"node","parent":203,"result":true,"route":"../roadmap/mathlib/overview/topology/general-topology/part-01/index.html#topological-fiber-bundle","scope":"mathlib/overview/topology/general-topology/part-01","status":"mathlib","summary":"A (topological) fiber bundle with fiber `F` over a base `B` is a space projecting on `B` for which the fibers are all homeomorphic to `F`, such that the local situation around each point is a direct product.","title":"Topological fiber bundle"},{"id":"mathlib/overview/topology/general-topology/part-01/topological-vector-bundle","kind":"node","parent":203,"result":true,"route":"../roadmap/mathlib/overview/topology/general-topology/part-01/index.html#topological-vector-bundle","scope":"mathlib/overview/topology/general-topology/part-01","status":"mathlib","summary":"The space `Bundle.TotalSpace F E` (for `E : B \u2192 Type*` such that each `E x` is a topological vector space) has a topological vector space structure with fiber `F` (denoted with `VectorBundle R F E`) if around every point there is a fiber bundle trivialization which is linear in the fibers.","title":"Topological vector bundle"},{"data_url":"scopes/mathlib/overview/topology/general-topology/part-02.json","id":"mathlib/overview/topology/general-topology/part-02","kind":"scope","navigation":"transparent","parent":202,"result":false,"route":"../roadmap/mathlib/overview/topology/general-topology/part-02/index.html","scope":"mathlib/overview/topology/general-topology/part-02","status":"mathlib","title":"Urysohn's lemma to Stone-Weierstrass theorem"},{"id":"mathlib/overview/topology/general-topology/part-02/urysohns-lemma","kind":"node","parent":224,"result":true,"route":"../roadmap/mathlib/overview/topology/general-topology/part-02/index.html#urysohns-lemma","scope":"mathlib/overview/topology/general-topology/part-02","status":"mathlib","summary":"Urysohn's lemma: if `s` and `t` are two disjoint closed sets in a normal topological space `X`, then there exists a continuous function `f : X \u2192 \u211d` such that","title":"Urysohn's lemma"},{"id":"mathlib/overview/topology/general-topology/part-02/stone-weierstrass-theorem","kind":"node","parent":224,"result":true,"route":"../roadmap/mathlib/overview/topology/general-topology/part-02/index.html#stone-weierstrass-theorem","scope":"mathlib/overview/topology/general-topology/part-02","status":"mathlib","summary":"The Stone-Weierstrass Approximation Theorem, that a subalgebra `A` of `C(X, \u211d)`, where `X` is a compact topological space, is dense if it separates points.","title":"Stone-Weierstrass theorem"},{"data_url":"scopes/mathlib/overview/topology/uniform-notions.json","id":"mathlib/overview/topology/uniform-notions","kind":"scope","parent":201,"result":false,"route":"../roadmap/mathlib/overview/topology/uniform-notions/index.html","scope":"mathlib/overview/topology/uniform-notions","status":"mathlib","title":"Uniform notions"},{"id":"mathlib/overview/topology/uniform-notions/uniform-space","kind":"node","parent":227,"result":true,"route":"../roadmap/mathlib/overview/topology/uniform-notions/index.html#uniform-space","scope":"mathlib/overview/topology/uniform-notions","status":"mathlib","summary":"A uniform space is a generalization of the \"uniform\" topological aspects of a metric space. It consists of a filter on `\u03b1 \u00d7 \u03b1` called the \"uniformity\", which satisfies properties analogous to the reflexivity, symmetry, and triangle properties of a metric.","title":"Uniform space"},{"id":"mathlib/overview/topology/uniform-notions/uniformly-continuous-function","kind":"node","parent":227,"result":true,"route":"../roadmap/mathlib/overview/topology/uniform-notions/index.html#uniformly-continuous-function","scope":"mathlib/overview/topology/uniform-notions","status":"mathlib","summary":"Mathlib states this as `Metric.uniformContinuous_iff`.","title":"Uniformly continuous function"},{"id":"mathlib/overview/topology/uniform-notions/uniform-convergence","kind":"node","parent":227,"result":true,"route":"../roadmap/mathlib/overview/topology/uniform-notions/index.html#uniform-convergence","scope":"mathlib/overview/topology/uniform-notions","status":"mathlib","summary":"A sequence of functions `F\u2099` converges uniformly to a limiting function `f` with respect to a filter `p` if, for any entourage of the diagonal `u`, one has `p`-eventually `(f x, F\u2099 x) \u2208 u` for all `x`.","title":"Uniform convergence"},{"id":"mathlib/overview/topology/uniform-notions/cauchy-filter","kind":"node","parent":227,"result":true,"route":"../roadmap/mathlib/overview/topology/uniform-notions/index.html#cauchy-filter","scope":"mathlib/overview/topology/uniform-notions","status":"mathlib","summary":"A filter `f` is Cauchy if for every entourage `r`, there exists an `s \u2208 f` such that `s \u00d7 s \u2286 r`. This is a generalization of Cauchy sequences, because if `a : \u2115 \u2192 \u03b1` then the filter of sets containing cofinitely many of the `a n` is Cauchy iff `a` is a Cauchy sequence.","title":"Cauchy filter"},{"id":"mathlib/overview/topology/uniform-notions/cauchy-sequence","kind":"node","parent":227,"result":true,"route":"../roadmap/mathlib/overview/topology/uniform-notions/index.html#cauchy-sequence","scope":"mathlib/overview/topology/uniform-notions","status":"mathlib","summary":"Cauchy sequences. Usually defined on \u2115, but often it is also useful to say that a function defined on \u211d is Cauchy at +\u221e to deduce convergence. Therefore, we define it in a type class that is general enough to cover both \u2115 and \u211d, which are the main motivating examples.","title":"Cauchy sequence"},{"id":"mathlib/overview/topology/uniform-notions/completeness","kind":"node","parent":227,"result":true,"route":"../roadmap/mathlib/overview/topology/uniform-notions/index.html#completeness","scope":"mathlib/overview/topology/uniform-notions","status":"mathlib","summary":"A complete space is defined here using uniformities. A uniform space is complete if every Cauchy filter converges.","title":"Completeness"},{"id":"mathlib/overview/topology/uniform-notions/completion","kind":"node","parent":227,"result":true,"route":"../roadmap/mathlib/overview/topology/uniform-notions/index.html#completion","scope":"mathlib/overview/topology/uniform-notions","status":"mathlib","summary":"Hausdorff completion of `\u03b1`","title":"Completion"},{"id":"mathlib/overview/topology/uniform-notions/heine-cantor-theorem","kind":"node","parent":227,"result":true,"route":"../roadmap/mathlib/overview/topology/uniform-notions/index.html#heine-cantor-theorem","scope":"mathlib/overview/topology/uniform-notions","status":"mathlib","summary":"Heine-Cantor: a continuous function on a compact uniform space is uniformly continuous.","title":"Heine-Cantor theorem"},{"data_url":"scopes/mathlib/overview/topology/topological-algebra.json","id":"mathlib/overview/topology/topological-algebra","kind":"scope","parent":201,"result":false,"route":"../roadmap/mathlib/overview/topology/topological-algebra/index.html","scope":"mathlib/overview/topology/topological-algebra","status":"mathlib","title":"Topological algebra"},{"id":"mathlib/overview/topology/topological-algebra/order-topology","kind":"node","parent":236,"result":true,"route":"../roadmap/mathlib/overview/topology/topological-algebra/index.html#order-topology","scope":"mathlib/overview/topology/topological-algebra","status":"mathlib","summary":"The order topology on an ordered type is the topology generated by open intervals. We register it on a preorder, but it is mostly interesting in linear orders, where it is also order-closed. We define it as a mixin. If you want to introduce the order topology on a preorder, use `Preorder.topology`.","title":"Order topology"},{"id":"mathlib/overview/topology/topological-algebra/intermediate-value-theorem","kind":"node","parent":236,"result":true,"route":"../roadmap/mathlib/overview/topology/topological-algebra/index.html#intermediate-value-theorem","scope":"mathlib/overview/topology/topological-algebra","status":"mathlib","summary":"Intermediate Value Theorem for continuous functions on closed intervals, case `f a \u2264 t \u2264 f b`.","title":"Intermediate value theorem"},{"id":"mathlib/overview/topology/topological-algebra/extreme-value-theorem","kind":"node","parent":236,"result":true,"route":"../roadmap/mathlib/overview/topology/topological-algebra/index.html#extreme-value-theorem","scope":"mathlib/overview/topology/topological-algebra","status":"mathlib","summary":"The extreme value theorem: a continuous function realizes its minimum on a compact set.","title":"Extreme value theorem"},{"id":"mathlib/overview/topology/topological-algebra/limit-infimum-and-supremum","kind":"node","parent":236,"result":true,"route":"../roadmap/mathlib/overview/topology/topological-algebra/index.html#limit-infimum-and-supremum","scope":"mathlib/overview/topology/topological-algebra","status":"mathlib","summary":"Defines the liminf/limsup of a function taking values in a conditionally complete lattice, with respect to an arbitrary filter.","title":"Limit infimum and supremum"},{"id":"mathlib/overview/topology/topological-algebra/topological-group","kind":"node","parent":236,"result":true,"route":"../roadmap/mathlib/overview/topology/topological-algebra/index.html#topological-group","scope":"mathlib/overview/topology/topological-algebra","status":"mathlib","summary":"A topological group is a group in which the multiplication and inversion operations are continuous.","title":"Topological group"},{"id":"mathlib/overview/topology/topological-algebra/completion-of-an-abelian-topological-group","kind":"node","parent":236,"result":true,"route":"../roadmap/mathlib/overview/topology/topological-algebra/index.html#completion-of-an-abelian-topological-group","scope":"mathlib/overview/topology/topological-algebra","status":"mathlib","summary":"Mathlib states this as `UniformSpace.Completion.instAddCommGroup`.","title":"Completion of an abelian topological group"},{"id":"mathlib/overview/topology/topological-algebra/infinite-sum","kind":"node","parent":236,"result":true,"route":"../roadmap/mathlib/overview/topology/topological-algebra/index.html#infinite-sum","scope":"mathlib/overview/topology/topological-algebra","status":"mathlib","summary":"`HasSum f a L` means that the (potentially infinite) sum of the `f b` for `b : \u03b2` converges to `a` along the SummationFilter `L`.","title":"Infinite sum"},{"id":"mathlib/overview/topology/topological-algebra/topological-ring","kind":"node","parent":236,"result":true,"route":"../roadmap/mathlib/overview/topology/topological-algebra/index.html#topological-ring","scope":"mathlib/overview/topology/topological-algebra","status":"mathlib","summary":"A topological ring is a ring `R` where addition, multiplication and negation are continuous.","title":"Topological ring"},{"id":"mathlib/overview/topology/topological-algebra/completion-of-a-topological-ring","kind":"node","parent":236,"result":true,"route":"../roadmap/mathlib/overview/topology/topological-algebra/index.html#completion-of-a-topological-ring","scope":"mathlib/overview/topology/topological-algebra","status":"mathlib","summary":"Mathlib states this as `UniformSpace.Completion.ring`.","title":"Completion of a topological ring"},{"id":"mathlib/overview/topology/topological-algebra/topological-module","kind":"node","parent":236,"result":true,"route":"../roadmap/mathlib/overview/topology/topological-algebra/index.html#topological-module","scope":"mathlib/overview/topology/topological-algebra","status":"mathlib","summary":"Class `ContinuousSMul M X` says that the scalar multiplication `(\u2022) : M \u2192 X \u2192 X` is continuous in both arguments. We use the same class for all kinds of multiplicative actions, including (semi)modules and algebras.","title":"Topological module"},{"id":"mathlib/overview/topology/topological-algebra/continuous-linear-map","kind":"node","parent":236,"result":true,"route":"../roadmap/mathlib/overview/topology/topological-algebra/index.html#continuous-linear-map","scope":"mathlib/overview/topology/topological-algebra","status":"mathlib","summary":"Continuous linear maps between modules. We only put the type classes that are necessary for the definition, although in applications `M` and `M\u2082` will be topological modules over the topological ring `R`.","title":"Continuous linear map"},{"id":"mathlib/overview/topology/topological-algebra/haar-measure-on-a-locally-compact-hausdorff-grou","kind":"node","parent":236,"result":true,"route":"../roadmap/mathlib/overview/topology/topological-algebra/index.html#haar-measure-on-a-locally-compact-hausdorff-grou","scope":"mathlib/overview/topology/topological-algebra","status":"mathlib","summary":"The Haar measure on the locally compact group `G`, scaled so that `haarMeasure K\u2080 K\u2080 = 1`.","title":"Haar measure on a locally compact Hausdorff group"},{"data_url":"scopes/mathlib/overview/topology/metric-spaces.json","id":"mathlib/overview/topology/metric-spaces","kind":"scope","parent":201,"result":false,"route":"../roadmap/mathlib/overview/topology/metric-spaces/index.html","scope":"mathlib/overview/topology/metric-spaces","status":"mathlib","title":"Metric spaces"},{"id":"mathlib/overview/topology/metric-spaces/metric-space","kind":"node","parent":249,"result":true,"route":"../roadmap/mathlib/overview/topology/metric-spaces/index.html#metric-space","scope":"mathlib/overview/topology/metric-spaces","status":"mathlib","summary":"A metric space is a type endowed with a `\u211d`-valued distance `dist` satisfying `dist x y = 0 \u2194 x = y`, commutativity `dist x y = dist y x`, and the triangle inequality `dist x z \u2264 dist x y + dist y z`.","title":"Metric space"},{"id":"mathlib/overview/topology/metric-spaces/ball","kind":"node","parent":249,"result":true,"route":"../roadmap/mathlib/overview/topology/metric-spaces/index.html#ball","scope":"mathlib/overview/topology/metric-spaces","status":"mathlib","summary":"`ball x \u03b5` is the set of all points `y` with `dist y x < \u03b5`","title":"Ball"},{"id":"mathlib/overview/topology/metric-spaces/sequential-compactness-is-equivalent-to-compactn","kind":"node","parent":249,"result":true,"route":"../roadmap/mathlib/overview/topology/metric-spaces/index.html#sequential-compactness-is-equivalent-to-compactn","scope":"mathlib/overview/topology/metric-spaces","status":"mathlib","summary":"A version of Bolzano-Weierstrass: in a (pseudo)metrizable space, a set is compact if and only if it is sequentially compact.","title":"Sequential compactness is equivalent to compactness (Bolzano-Weierstrass)"},{"id":"mathlib/overview/topology/metric-spaces/heine-borel-theorem-proper-metric-space-version","kind":"node","parent":249,"result":true,"route":"../roadmap/mathlib/overview/topology/metric-spaces/index.html#heine-borel-theorem-proper-metric-space-version","scope":"mathlib/overview/topology/metric-spaces","status":"mathlib","summary":"The Heine\u2013Borel theorem: In a proper metric space, a set is compact if and only if it is closed and bounded.","title":"Heine-Borel theorem (proper metric space version)"},{"id":"mathlib/overview/topology/metric-spaces/lipschitz-continuity","kind":"node","parent":249,"result":true,"route":"../roadmap/mathlib/overview/topology/metric-spaces/index.html#lipschitz-continuity","scope":"mathlib/overview/topology/metric-spaces","status":"mathlib","summary":"A function `f` is Lipschitz continuous with constant `K \u2265 0` if for all `x, y` we have `dist (f x) (f y) \u2264 K * dist x y`.","title":"Lipschitz continuity"},{"id":"mathlib/overview/topology/metric-spaces/holder-continuity","kind":"node","parent":249,"result":true,"route":"../roadmap/mathlib/overview/topology/metric-spaces/index.html#holder-continuity","scope":"mathlib/overview/topology/metric-spaces","status":"mathlib","summary":"A function `f : X \u2192 Y` between two `PseudoEMetricSpace`s is H\u00f6lder continuous with constant `C : \u211d\u22650` and exponent `r : \u211d\u22650`, if `edist (f x) (f y) \u2264 C * edist x y ^ r` for all `x y : X`.","title":"H\u00f6lder continuity"},{"id":"mathlib/overview/topology/metric-spaces/contraction-mapping-theorem","kind":"node","parent":249,"result":true,"route":"../roadmap/mathlib/overview/topology/metric-spaces/index.html#contraction-mapping-theorem","scope":"mathlib/overview/topology/metric-spaces","status":"mathlib","summary":"Banach fixed-point theorem, contraction mapping theorem, `EMetricSpace` version. A contracting map on a complete metric space has a fixed point. We include more conclusions in this theorem to avoid proving them again later.","title":"Contraction mapping theorem"},{"id":"mathlib/overview/topology/metric-spaces/baire-theorem","kind":"node","parent":249,"result":true,"route":"../roadmap/mathlib/overview/topology/metric-spaces/index.html#baire-theorem","scope":"mathlib/overview/topology/metric-spaces","status":"mathlib","summary":"First Baire theorem: a completely pseudometrizable topological space has Baire property.","title":"Baire theorem"},{"id":"mathlib/overview/topology/metric-spaces/arzela-ascoli-theorem","kind":"node","parent":249,"result":true,"route":"../roadmap/mathlib/overview/topology/metric-spaces/index.html#arzela-ascoli-theorem","scope":"mathlib/overview/topology/metric-spaces","status":"mathlib","summary":"Third \\(main\\) version, with pointwise equicontinuity and range in a compact subset, but without closedness. The closure is then compact.","title":"Arzela-Ascoli theorem"},{"id":"mathlib/overview/topology/metric-spaces/hausdorff-distance","kind":"node","parent":249,"result":true,"route":"../roadmap/mathlib/overview/topology/metric-spaces/index.html#hausdorff-distance","scope":"mathlib/overview/topology/metric-spaces","status":"mathlib","summary":"The Hausdorff edistance between two sets is the smallest `r` such that each set is contained in the `r`-neighborhood of the other one","title":"Hausdorff distance"},{"id":"mathlib/overview/topology/metric-spaces/gromov-hausdorff-space","kind":"node","parent":249,"result":true,"route":"../roadmap/mathlib/overview/topology/metric-spaces/index.html#gromov-hausdorff-space","scope":"mathlib/overview/topology/metric-spaces","status":"mathlib","summary":"The Gromov-Hausdorff space","title":"Gromov-Hausdorff space"},{"data_url":"scopes/mathlib/overview/analysis.json","id":"mathlib/overview/analysis","kind":"scope","parent":2,"result":false,"route":"../roadmap/mathlib/overview/analysis/index.html","scope":"mathlib/overview/analysis","status":"mathlib","title":"Analysis"},{"data_url":"scopes/mathlib/overview/analysis/topological-vector-spaces.json","id":"mathlib/overview/analysis/topological-vector-spaces","kind":"scope","parent":261,"result":false,"route":"../roadmap/mathlib/overview/analysis/topological-vector-spaces/index.html","scope":"mathlib/overview/analysis/topological-vector-spaces","status":"mathlib","title":"Topological vector spaces"},{"id":"mathlib/overview/analysis/topological-vector-spaces/local-convexity","kind":"node","parent":262,"result":true,"route":"../roadmap/mathlib/overview/analysis/topological-vector-spaces/index.html#local-convexity","scope":"mathlib/overview/analysis/topological-vector-spaces","status":"mathlib","summary":"A `LocallyConvexSpace` is a topological semimodule over an ordered semiring in which convex neighborhoods of a point form a neighborhood basis at that point.","title":"Local convexity"},{"id":"mathlib/overview/analysis/topological-vector-spaces/bornology","kind":"node","parent":262,"result":true,"route":"../roadmap/mathlib/overview/analysis/topological-vector-spaces/index.html#bornology","scope":"mathlib/overview/analysis/topological-vector-spaces","status":"mathlib","summary":"A bornology on a type `\u03b1` is a filter of cobounded sets which contains the cofinite filter. Such spaces are equivalently specified by their bounded sets, see `Bornology.ofBounded` and `Bornology.ext_iff_isBounded`","title":"Bornology"},{"id":"mathlib/overview/analysis/topological-vector-spaces/weak-topology-for-dualities","kind":"node","parent":262,"result":true,"route":"../roadmap/mathlib/overview/analysis/topological-vector-spaces/index.html#weak-topology-for-dualities","scope":"mathlib/overview/analysis/topological-vector-spaces","status":"mathlib","summary":"Mathlib states this as `WeakBilin.instTopologicalSpace`.","title":"Weak-* topology for dualities"},{"data_url":"scopes/mathlib/overview/analysis/normed-vector-spaces-banach-spaces.json","id":"mathlib/overview/analysis/normed-vector-spaces-banach-spaces","kind":"scope","parent":261,"result":false,"route":"../roadmap/mathlib/overview/analysis/normed-vector-spaces-banach-spaces/index.html","scope":"mathlib/overview/analysis/normed-vector-spaces-banach-spaces","status":"mathlib","title":"Normed vector spaces/Banach spaces"},{"id":"mathlib/overview/analysis/normed-vector-spaces-banach-spaces/normed-vector-space-over-a-normed-field","kind":"node","parent":266,"result":true,"route":"../roadmap/mathlib/overview/analysis/normed-vector-spaces-banach-spaces/index.html#normed-vector-space-over-a-normed-field","scope":"mathlib/overview/analysis/normed-vector-spaces-banach-spaces","status":"mathlib","summary":"A normed space over a normed field is a vector space endowed with a norm which satisfies the equality `\u2016c \u2022 x\u2016 = \u2016c\u2016 \u2016x\u2016`. We require only `\u2016c \u2022 x\u2016 \u2264 \u2016c\u2016 \u2016x\u2016` in the definition, then prove `\u2016c \u2022 x\u2016 = \u2016c\u2016 \u2016x\u2016` in `norm_smul`.","title":"Normed vector space over a normed field"},{"id":"mathlib/overview/analysis/normed-vector-spaces-banach-spaces/topology-on-a-normed-vector-space","kind":"node","parent":266,"result":true,"route":"../roadmap/mathlib/overview/analysis/normed-vector-spaces-banach-spaces/index.html#topology-on-a-normed-vector-space","scope":"mathlib/overview/analysis/normed-vector-spaces-banach-spaces","status":"mathlib","summary":"Mathlib states this as `NormSMulClass.toIsBoundedSMul`.","title":"Topology on a normed vector space"},{"id":"mathlib/overview/analysis/normed-vector-spaces-banach-spaces/equivalence-of-norms-in-finite-dimension","kind":"node","parent":266,"result":true,"route":"../roadmap/mathlib/overview/analysis/normed-vector-spaces-banach-spaces/index.html#equivalence-of-norms-in-finite-dimension","scope":"mathlib/overview/analysis/normed-vector-spaces-banach-spaces","status":"mathlib","summary":"The continuous linear equivalence induced by a linear equivalence on a finite-dimensional space.","title":"Equivalence of norms in finite dimension"},{"id":"mathlib/overview/analysis/normed-vector-spaces-banach-spaces/finite-dimensional-normed-spaces-over-complete-n","kind":"node","parent":266,"result":true,"route":"../roadmap/mathlib/overview/analysis/normed-vector-spaces-banach-spaces/index.html#finite-dimensional-normed-spaces-over-complete-n","scope":"mathlib/overview/analysis/normed-vector-spaces-banach-spaces","status":"mathlib","summary":"A finite-dimensional subspace is complete.","title":"Finite-dimensional normed spaces over complete normed fields are complete"},{"id":"mathlib/overview/analysis/normed-vector-spaces-banach-spaces/heine-borel-theorem-finite-dimensional-normed-sp","kind":"node","parent":266,"result":true,"route":"../roadmap/mathlib/overview/analysis/normed-vector-spaces-banach-spaces/index.html#heine-borel-theorem-finite-dimensional-normed-sp","scope":"mathlib/overview/analysis/normed-vector-spaces-banach-spaces","status":"mathlib","summary":"Any finite-dimensional vector space over a locally compact field is proper. We do not register this as an instance to avoid an instance loop when trying to prove the properness of `\ud835\udd5c`, and the search for `\ud835\udd5c` as an unknown metavariable. Declare the instance explicitly when needed.","title":"Heine-Borel theorem (finite-dimensional normed spaces are proper)"},{"id":"mathlib/overview/analysis/normed-vector-spaces-banach-spaces/norm-of-a-continuous-linear-map","kind":"node","parent":266,"result":true,"route":"../roadmap/mathlib/overview/analysis/normed-vector-spaces-banach-spaces/index.html#norm-of-a-continuous-linear-map","scope":"mathlib/overview/analysis/normed-vector-spaces-banach-spaces","status":"mathlib","summary":"Construct a continuous linear map from a linear map and a bound on this linear map. The fact that the norm of the continuous linear map is then controlled is given in `LinearMap.mkContinuous_norm_le`.","title":"Norm of a continuous linear map"},{"id":"mathlib/overview/analysis/normed-vector-spaces-banach-spaces/banach-steinhaus-theorem","kind":"node","parent":266,"result":true,"route":"../roadmap/mathlib/overview/analysis/normed-vector-spaces-banach-spaces/index.html#banach-steinhaus-theorem","scope":"mathlib/overview/analysis/normed-vector-spaces-banach-spaces","status":"mathlib","summary":"This is the standard Banach-Steinhaus theorem, or Uniform Boundedness Principle. If a family of continuous linear maps from a Banach space into a normed space is pointwise bounded, then the norms of these linear maps are uniformly bounded.","title":"Banach-Steinhaus theorem"},{"id":"mathlib/overview/analysis/normed-vector-spaces-banach-spaces/banach-open-mapping-theorem","kind":"node","parent":266,"result":true,"route":"../roadmap/mathlib/overview/analysis/normed-vector-spaces-banach-spaces/index.html#banach-open-mapping-theorem","scope":"mathlib/overview/analysis/normed-vector-spaces-banach-spaces","status":"mathlib","summary":"The Banach open mapping theorem: a surjective bounded linear map between Banach spaces is open.","title":"Banach open mapping theorem"},{"id":"mathlib/overview/analysis/normed-vector-spaces-banach-spaces/absolutely-convergent-series-in-banach-spaces","kind":"node","parent":266,"result":true,"route":"../roadmap/mathlib/overview/analysis/normed-vector-spaces-banach-spaces/index.html#absolutely-convergent-series-in-banach-spaces","scope":"mathlib/overview/analysis/normed-vector-spaces-banach-spaces","status":"mathlib","summary":"Mathlib states this as `Summable.of_norm`.","title":"Absolutely convergent series in Banach spaces"},{"id":"mathlib/overview/analysis/normed-vector-spaces-banach-spaces/hahn-banach-theorem","kind":"node","parent":266,"result":true,"route":"../roadmap/mathlib/overview/analysis/normed-vector-spaces-banach-spaces/index.html#hahn-banach-theorem","scope":"mathlib/overview/analysis/normed-vector-spaces-banach-spaces","status":"mathlib","summary":"Hahn-Banach theorem for continuous linear functions over `\ud835\udd5c` satisfying `IsRCLikeNormedField \ud835\udd5c`.","title":"Hahn-Banach theorem"},{"id":"mathlib/overview/analysis/normed-vector-spaces-banach-spaces/dual-of-a-normed-space","kind":"node","parent":266,"result":true,"route":"../roadmap/mathlib/overview/analysis/normed-vector-spaces-banach-spaces/index.html#dual-of-a-normed-space","scope":"mathlib/overview/analysis/normed-vector-spaces-banach-spaces","status":"mathlib","summary":"The *strong dual* of a topological vector space `M` over a ring `R`. This is the space of continuous linear functionals and is equipped with the topology of uniform convergence on bounded subsets. `StrongDual R M` is an abbreviation for `M \u2192L[R] R`.","title":"Dual of a normed space"},{"id":"mathlib/overview/analysis/normed-vector-spaces-banach-spaces/isometric-inclusion-in-double-dual","kind":"node","parent":266,"result":true,"route":"../roadmap/mathlib/overview/analysis/normed-vector-spaces-banach-spaces/index.html#isometric-inclusion-in-double-dual","scope":"mathlib/overview/analysis/normed-vector-spaces-banach-spaces","status":"mathlib","summary":"The inclusion of a normed space in its double strong dual is an isometry onto its image.","title":"Isometric inclusion in double dual"},{"id":"mathlib/overview/analysis/normed-vector-spaces-banach-spaces/completeness-of-spaces-of-bounded-continuous-fun","kind":"node","parent":266,"result":true,"route":"../roadmap/mathlib/overview/analysis/normed-vector-spaces-banach-spaces/index.html#completeness-of-spaces-of-bounded-continuous-fun","scope":"mathlib/overview/analysis/normed-vector-spaces-banach-spaces","status":"mathlib","summary":"Bounded continuous functions taking values in a complete space form a complete space.","title":"Completeness of spaces of bounded continuous functions"},{"data_url":"scopes/mathlib/overview/analysis/hilbert-spaces.json","id":"mathlib/overview/analysis/hilbert-spaces","kind":"scope","parent":261,"result":false,"route":"../roadmap/mathlib/overview/analysis/hilbert-spaces/index.html","scope":"mathlib/overview/analysis/hilbert-spaces","status":"mathlib","title":"Hilbert spaces"},{"id":"mathlib/overview/analysis/hilbert-spaces/inner-product-space-over-r-or-c","kind":"node","parent":280,"reading_title":{"format":"inline-parts-v1","parts":[["text","Inner product space, over "],["math","\\(R\\)"],["text"," or "],["math","\\(C\\)"]],"source":"Inner product space, over $R$ or $C$"},"result":true,"route":"../roadmap/mathlib/overview/analysis/hilbert-spaces/index.html#inner-product-space-over-r-or-c","scope":"mathlib/overview/analysis/hilbert-spaces","status":"mathlib","summary":"A (pre) inner product space is a vector space with an additional operation called inner product. The (semi)norm could be derived from the inner product, instead we require the existence of a seminorm and the fact that `\u2016x\u2016^2 = re \u27eax, x\u27eb` to be able to put instances on `\ud835\udd42` or product spaces.","title":"Inner product space, over $R$ or $C$"},{"id":"mathlib/overview/analysis/hilbert-spaces/cauchy-schwarz-inequality","kind":"node","parent":280,"result":true,"route":"../roadmap/mathlib/overview/analysis/hilbert-spaces/index.html#cauchy-schwarz-inequality","scope":"mathlib/overview/analysis/hilbert-spaces","status":"mathlib","summary":"Cauchy\u2013Schwarz inequality.","title":"Cauchy-Schwarz inequality"},{"id":"mathlib/overview/analysis/hilbert-spaces/adjoint-operator","kind":"node","parent":280,"result":true,"route":"../roadmap/mathlib/overview/analysis/hilbert-spaces/index.html#adjoint-operator","scope":"mathlib/overview/analysis/hilbert-spaces","status":"mathlib","summary":"The adjoint operator as a partially defined linear operator, denoted as `T\u2020`.","title":"Adjoint operator"},{"id":"mathlib/overview/analysis/hilbert-spaces/self-adjoint-operator","kind":"node","parent":280,"result":true,"route":"../roadmap/mathlib/overview/analysis/hilbert-spaces/index.html#self-adjoint-operator","scope":"mathlib/overview/analysis/hilbert-spaces","status":"mathlib","summary":"An element is self-adjoint if it is equal to its star.","title":"Self-adjoint operator"},{"id":"mathlib/overview/analysis/hilbert-spaces/orthogonal-projection","kind":"node","parent":280,"result":true,"route":"../roadmap/mathlib/overview/analysis/hilbert-spaces/index.html#orthogonal-projection","scope":"mathlib/overview/analysis/hilbert-spaces","status":"mathlib","summary":"The orthogonal projection onto a subspace.","title":"Orthogonal projection"},{"id":"mathlib/overview/analysis/hilbert-spaces/reflection","kind":"node","parent":280,"result":true,"route":"../roadmap/mathlib/overview/analysis/hilbert-spaces/index.html#reflection","scope":"mathlib/overview/analysis/hilbert-spaces","status":"mathlib","summary":"Reflection in a complete subspace of an inner product space. The word \"reflection\" is sometimes understood to mean specifically reflection in a codimension-one subspace, and sometimes more generally to cover operations such as reflection in a point. The definition here, of reflection in a subspace, is a more general sense of the word that includes both those common cases.","title":"Reflection"},{"id":"mathlib/overview/analysis/hilbert-spaces/orthogonal-complement","kind":"node","parent":280,"result":true,"route":"../roadmap/mathlib/overview/analysis/hilbert-spaces/index.html#orthogonal-complement","scope":"mathlib/overview/analysis/hilbert-spaces","status":"mathlib","summary":"The subspace of vectors orthogonal to a given subspace, denoted `K\u15ee`.","title":"Orthogonal complement"},{"id":"mathlib/overview/analysis/hilbert-spaces/existence-of-hilbert-basis","kind":"node","parent":280,"result":true,"route":"../roadmap/mathlib/overview/analysis/hilbert-spaces/index.html#existence-of-hilbert-basis","scope":"mathlib/overview/analysis/hilbert-spaces","status":"mathlib","summary":"A Hilbert space admits a Hilbert basis.","title":"Existence of Hilbert basis"},{"id":"mathlib/overview/analysis/hilbert-spaces/eigenvalues-from-rayleigh-quotient","kind":"node","parent":280,"result":true,"route":"../roadmap/mathlib/overview/analysis/hilbert-spaces/index.html#eigenvalues-from-rayleigh-quotient","scope":"mathlib/overview/analysis/hilbert-spaces","status":"mathlib","summary":"For a self-adjoint operator `T`, a local extremum of the Rayleigh quotient of `T` on a sphere centred at the origin is an eigenvector of `T`.","title":"Eigenvalues from Rayleigh quotient"},{"id":"mathlib/overview/analysis/hilbert-spaces/frechet-riesz-representation-of-the-dual-of-a-hi","kind":"node","parent":280,"result":true,"route":"../roadmap/mathlib/overview/analysis/hilbert-spaces/index.html#frechet-riesz-representation-of-the-dual-of-a-hi","scope":"mathlib/overview/analysis/hilbert-spaces","status":"mathlib","summary":"Fr\u00e9chet-Riesz representation: any `\u2113` in the dual of a Hilbert space `E` is of the form `fun u => \u27eay, u\u27eb` for some `y : E`, i.e. `toDualMap` is surjective.","title":"Fr\u00e9chet-Riesz representation of the dual of a Hilbert space"},{"id":"mathlib/overview/analysis/hilbert-spaces/lax-milgram-theorem","kind":"node","parent":280,"result":true,"route":"../roadmap/mathlib/overview/analysis/hilbert-spaces/index.html#lax-milgram-theorem","scope":"mathlib/overview/analysis/hilbert-spaces","status":"mathlib","summary":"The Lax-Milgram equivalence of a coercive bounded bilinear operator: for all `v : V`, `continuousLinearEquivOfBilin B v` is the unique element `V` such that `continuousLinearEquivOfBilin B v, w\u27eb = B v w`. The Lax-Milgram theorem states that this is a continuous equivalence.","title":"Lax-Milgram theorem"},{"data_url":"scopes/mathlib/overview/analysis/differentiability.json","id":"mathlib/overview/analysis/differentiability","kind":"scope","parent":261,"result":false,"route":"../roadmap/mathlib/overview/analysis/differentiability/index.html","scope":"mathlib/overview/analysis/differentiability","status":"mathlib","title":"Differentiability"},{"id":"mathlib/overview/analysis/differentiability/differentiable-function-between-normed-vector-sp","kind":"node","parent":292,"result":true,"route":"../roadmap/mathlib/overview/analysis/differentiability/index.html#differentiable-function-between-normed-vector-sp","scope":"mathlib/overview/analysis/differentiability","status":"mathlib","summary":"A function `f` has the continuous linear map `f'` as derivative at `x` if `f x' = f x + f' (x' - x) + o (x' - x)` when `x'` tends to `x`.","title":"Differentiable function between normed vector spaces"},{"id":"mathlib/overview/analysis/differentiability/derivative-of-a-composition-of-functions","kind":"node","parent":292,"result":true,"route":"../roadmap/mathlib/overview/analysis/differentiability/index.html#derivative-of-a-composition-of-functions","scope":"mathlib/overview/analysis/differentiability","status":"mathlib","summary":"The chain rule.","title":"Derivative of a composition of functions (Analysis)"},{"id":"mathlib/overview/analysis/differentiability/derivative-of-the-inverse-of-a-function","kind":"node","parent":292,"result":true,"route":"../roadmap/mathlib/overview/analysis/differentiability/index.html#derivative-of-the-inverse-of-a-function","scope":"mathlib/overview/analysis/differentiability","status":"mathlib","summary":"If `f (g x) = x` for `x` in some neighborhood of `a`, `g` is continuous at `a`, and `f` has an invertible derivative `f'` at `g a`, then `g` has the derivative `f'\u207b\u00b9` at `a`.","title":"Derivative of the inverse of a function (Analysis)"},{"id":"mathlib/overview/analysis/differentiability/rolles-theorem","kind":"node","parent":292,"result":true,"route":"../roadmap/mathlib/overview/analysis/differentiability/index.html#rolles-theorem","scope":"mathlib/overview/analysis/differentiability","status":"mathlib","summary":"Rolle's Theorem `deriv` version","title":"Rolle's theorem"},{"id":"mathlib/overview/analysis/differentiability/mean-value-theorem","kind":"node","parent":292,"result":true,"route":"../roadmap/mathlib/overview/analysis/differentiability/index.html#mean-value-theorem","scope":"mathlib/overview/analysis/differentiability","status":"mathlib","summary":"Cauchy's Mean Value Theorem, `deriv` version.","title":"Mean value theorem (Differentiability)"},{"id":"mathlib/overview/analysis/differentiability/taylors-theorem","kind":"node","parent":292,"result":true,"route":"../roadmap/mathlib/overview/analysis/differentiability/index.html#taylors-theorem","scope":"mathlib/overview/analysis/differentiability","status":"mathlib","summary":"This file defines the Taylor polynomial of a real function `f : \u211d \u2192 E`, where `E` is a normed vector space over `\u211d` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term.","title":"Taylor's theorem"},{"id":"mathlib/overview/analysis/differentiability/c-k-function","kind":"node","parent":292,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(C^k\\)"],["text"," function"]],"source":"$C^k$ function"},"result":true,"route":"../roadmap/mathlib/overview/analysis/differentiability/index.html#c-k-function","scope":"mathlib/overview/analysis/differentiability","status":"mathlib","summary":"A function is continuously differentiable up to `n` if it admits derivatives up to order `n`, which are continuous. Contrary to the case of definitions in domains (where derivatives might not be unique) we do not need to localize the definition in space or time. The parameter `n` belongs to `\u2115\u221e\u03c9` (accessible in the `ContDiff` scope), i.e. it can be a natural number, `\u221e`, or `\u03c9`.","title":"$C^k$ function"},{"id":"mathlib/overview/analysis/differentiability/leibniz-formula","kind":"node","parent":292,"result":true,"route":"../roadmap/mathlib/overview/analysis/differentiability/index.html#leibniz-formula","scope":"mathlib/overview/analysis/differentiability","status":"mathlib","summary":"Mathlib states this as `fderiv_mul`.","title":"Leibniz formula (Analysis)"},{"id":"mathlib/overview/analysis/differentiability/local-extrema","kind":"node","parent":292,"result":true,"route":"../roadmap/mathlib/overview/analysis/differentiability/index.html#local-extrema","scope":"mathlib/overview/analysis/differentiability","status":"mathlib","summary":"Fermat's Theorem: the derivative of a function at a local minimum equals zero.","title":"Local extrema"},{"id":"mathlib/overview/analysis/differentiability/inverse-function-theorem","kind":"node","parent":292,"result":true,"route":"../roadmap/mathlib/overview/analysis/differentiability/index.html#inverse-function-theorem","scope":"mathlib/overview/analysis/differentiability","status":"mathlib","summary":"If `f` has an invertible derivative `f'` at `a` in the sense of strict differentiability `(hf)`, then the inverse function `hf.localInverse f` has derivative `f'.symm` at `f a`.","title":"Inverse function theorem (Differentiability)"},{"id":"mathlib/overview/analysis/differentiability/implicit-function-theorem","kind":"node","parent":292,"result":true,"route":"../roadmap/mathlib/overview/analysis/differentiability/index.html#implicit-function-theorem","scope":"mathlib/overview/analysis/differentiability","status":"mathlib","summary":"Implicit function theorem. If `f : E \u2192 F` and `g : E \u2192 G` are two maps strictly differentiable at `a`, their derivatives `f'`, `g'` are surjective, and the kernels of these derivatives are complementary subspaces of `E`, then `implicitFunction` is the unique (germ of a) map `\u03c6 : F \u2192 G \u2192 E` such that `f (\u03c6 y z) = y` and `g (\u03c6 y z) = z`.","title":"Implicit function theorem"},{"id":"mathlib/overview/analysis/differentiability/analytic-function","kind":"node","parent":292,"result":true,"route":"../roadmap/mathlib/overview/analysis/differentiability/index.html#analytic-function","scope":"mathlib/overview/analysis/differentiability","status":"mathlib","summary":"Given a function `f : E \u2192 F`, we say that `f` is analytic at `x` if it admits a convergent power series expansion around `x`.","title":"Analytic function"},{"data_url":"scopes/mathlib/overview/analysis/convexity.json","id":"mathlib/overview/analysis/convexity","kind":"scope","parent":261,"result":false,"route":"../roadmap/mathlib/overview/analysis/convexity/index.html","scope":"mathlib/overview/analysis/convexity","status":"mathlib","title":"Convexity"},{"id":"mathlib/overview/analysis/convexity/convex-function","kind":"node","parent":305,"result":true,"route":"../roadmap/mathlib/overview/analysis/convexity/index.html#convex-function","scope":"mathlib/overview/analysis/convexity","status":"mathlib","summary":"Convexity of functions","title":"Convex function"},{"id":"mathlib/overview/analysis/convexity/characterization-of-convexity","kind":"node","parent":305,"result":true,"route":"../roadmap/mathlib/overview/analysis/convexity/index.html#characterization-of-convexity","scope":"mathlib/overview/analysis/convexity","status":"mathlib","summary":"If a function `f` is continuous on a convex set `D \u2286 \u211d`, is twice differentiable on its interior, and `f''` is nonnegative on the interior, then `f` is convex on `D`.","title":"Characterization of convexity"},{"id":"mathlib/overview/analysis/convexity/jensens-inequality-finite-sum-version","kind":"node","parent":305,"result":true,"route":"../roadmap/mathlib/overview/analysis/convexity/index.html#jensens-inequality-finite-sum-version","scope":"mathlib/overview/analysis/convexity","status":"mathlib","summary":"Convex Jensen's inequality, `Finset.sum` version.","title":"Jensen's inequality (finite sum version)"},{"id":"mathlib/overview/analysis/convexity/jensens-inequality-integral-version","kind":"node","parent":305,"result":true,"route":"../roadmap/mathlib/overview/analysis/convexity/index.html#jensens-inequality-integral-version","scope":"mathlib/overview/analysis/convexity","status":"mathlib","summary":"Jensen's inequality: if a function `g : E \u2192 \u211d` is convex and continuous on a convex closed set `s`, `\u03bc` is a probability measure on `\u03b1`, and `f : \u03b1 \u2192 E` is a function sending `\u03bc`-a.e. points to `s`, then the value of `g` at the expected value of `f` is less than or equal to the expected value of `g \u2218 f` provided that both `f` and `g \u2218 f` are integrable. See also `ConvexOn.map_centerMass_le` for a finite sum version\u2026","title":"Jensen's inequality (integral version)"},{"id":"mathlib/overview/analysis/convexity/convexity-inequalities","kind":"node","parent":305,"result":true,"route":"../roadmap/mathlib/overview/analysis/convexity/index.html#convexity-inequalities","scope":"mathlib/overview/analysis/convexity","status":"mathlib","summary":"In this file we prove several inequalities for finite sums, including AM-GM inequality, HM-GM inequality, Young's inequality, H\u00f6lder inequality, and Minkowski inequality. Versions for integrals of some of these inequalities are available in `Mathlib/MeasureTheory/Integral/MeanInequalities.lean`.","title":"Convexity inequalities"},{"id":"mathlib/overview/analysis/convexity/caratheodorys-theorem","kind":"node","parent":305,"result":true,"route":"../roadmap/mathlib/overview/analysis/convexity/index.html#caratheodorys-theorem","scope":"mathlib/overview/analysis/convexity","status":"mathlib","summary":"Carath\u00e9odory's convexity theorem","title":"Carath\u00e9odory's theorem"},{"data_url":"scopes/mathlib/overview/analysis/special-functions.json","id":"mathlib/overview/analysis/special-functions","kind":"scope","parent":261,"result":false,"route":"../roadmap/mathlib/overview/analysis/special-functions/index.html","scope":"mathlib/overview/analysis/special-functions","status":"mathlib","title":"Special functions"},{"id":"mathlib/overview/analysis/special-functions/logarithm","kind":"node","parent":312,"result":true,"route":"../roadmap/mathlib/overview/analysis/special-functions/index.html#logarithm","scope":"mathlib/overview/analysis/special-functions","status":"mathlib","summary":"The real logarithm function, equal to the inverse of the exponential for `x > 0`, to `log |x|` for `x < 0`, and to `0` for `0`. We use this unconventional extension to `(-\u221e, 0]` as it gives the formula `log (x * y) = log x + log y` for all nonzero `x` and `y`, and the derivative of `log` is `1/x` away from `0`.","title":"Logarithm"},{"id":"mathlib/overview/analysis/special-functions/exponential","kind":"node","parent":312,"result":true,"route":"../roadmap/mathlib/overview/analysis/special-functions/index.html#exponential","scope":"mathlib/overview/analysis/special-functions","status":"mathlib","summary":"The real exponential function, defined as the real part of the complex exponential","title":"Exponential"},{"id":"mathlib/overview/analysis/special-functions/trigonometric-functions","kind":"node","parent":312,"result":true,"route":"../roadmap/mathlib/overview/analysis/special-functions/index.html#trigonometric-functions","scope":"mathlib/overview/analysis/special-functions","status":"mathlib","summary":"The real sine function, defined as the real part of the complex sine","title":"Trigonometric functions"},{"id":"mathlib/overview/analysis/special-functions/inverse-trigonometric-functions","kind":"node","parent":312,"result":true,"route":"../roadmap/mathlib/overview/analysis/special-functions/index.html#inverse-trigonometric-functions","scope":"mathlib/overview/analysis/special-functions","status":"mathlib","summary":"Inverse of the `sin` function, returns values in the range `-\u03c0 / 2 \u2264 arcsin x \u2264 \u03c0 / 2`. It defaults to `-\u03c0 / 2` on `(-\u221e, -1)` and to `\u03c0 / 2` to `(1, \u221e)`.","title":"Inverse trigonometric functions"},{"id":"mathlib/overview/analysis/special-functions/hyperbolic-trigonometric-functions","kind":"node","parent":312,"result":true,"route":"../roadmap/mathlib/overview/analysis/special-functions/index.html#hyperbolic-trigonometric-functions","scope":"mathlib/overview/analysis/special-functions","status":"mathlib","summary":"The real hyperbolic sine function, defined as the real part of the complex hyperbolic sine","title":"Hyperbolic trigonometric functions"},{"id":"mathlib/overview/analysis/special-functions/inverse-hyperbolic-trigonometric-functions","kind":"node","parent":312,"result":true,"route":"../roadmap/mathlib/overview/analysis/special-functions/index.html#inverse-hyperbolic-trigonometric-functions","scope":"mathlib/overview/analysis/special-functions","status":"mathlib","summary":"`arsinh` is defined using a logarithm, `arsinh x = log (x + \u221a(1 + x^2))`.","title":"Inverse hyperbolic trigonometric functions"},{"data_url":"scopes/mathlib/overview/analysis/measures-and-integral-calculus.json","id":"mathlib/overview/analysis/measures-and-integral-calculus","kind":"scope","parent":261,"result":false,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/index.html","scope":"mathlib/overview/analysis/measures-and-integral-calculus","status":"mathlib","title":"Measures and integral calculus"},{"data_url":"scopes/mathlib/overview/analysis/measures-and-integral-calculus/part-01.json","id":"mathlib/overview/analysis/measures-and-integral-calculus/part-01","kind":"scope","navigation":"transparent","parent":319,"result":false,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/part-01/index.html","scope":"mathlib/overview/analysis/measures-and-integral-calculus/part-01","status":"mathlib","title":"Sigma-algebra to Fundamental theorem of\u2026"},{"id":"mathlib/overview/analysis/measures-and-integral-calculus/part-01/sigma-algebra","kind":"node","parent":320,"result":true,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/part-01/index.html#sigma-algebra","scope":"mathlib/overview/analysis/measures-and-integral-calculus/part-01","status":"mathlib","summary":"A measurable space is a space equipped with a \u03c3-algebra.","title":"Sigma-algebra"},{"id":"mathlib/overview/analysis/measures-and-integral-calculus/part-01/measurable-function","kind":"node","parent":320,"result":true,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/part-01/index.html#measurable-function","scope":"mathlib/overview/analysis/measures-and-integral-calculus/part-01","status":"mathlib","summary":"A function `f` between measurable spaces is measurable if the preimage of every measurable set is measurable.","title":"Measurable function"},{"id":"mathlib/overview/analysis/measures-and-integral-calculus/part-01/the-category-of-measurable-spaces","kind":"node","parent":320,"result":true,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/part-01/index.html#the-category-of-measurable-spaces","scope":"mathlib/overview/analysis/measures-and-integral-calculus/part-01","status":"mathlib","summary":"The category of measurable spaces and measurable functions.","title":"The category of measurable spaces"},{"id":"mathlib/overview/analysis/measures-and-integral-calculus/part-01/borel-sigma-algebra","kind":"node","parent":320,"result":true,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/part-01/index.html#borel-sigma-algebra","scope":"mathlib/overview/analysis/measures-and-integral-calculus/part-01","status":"mathlib","summary":"A space with `MeasurableSpace` and `TopologicalSpace` structures such that the `\u03c3`-algebra of measurable sets is exactly the `\u03c3`-algebra generated by open sets.","title":"Borel sigma-algebra"},{"id":"mathlib/overview/analysis/measures-and-integral-calculus/part-01/positive-measure","kind":"node","parent":320,"result":true,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/part-01/index.html#positive-measure","scope":"mathlib/overview/analysis/measures-and-integral-calculus/part-01","status":"mathlib","summary":"A measure is defined to be an outer measure that is countably additive on measurable sets, with the additional assumption that the outer measure is the canonical extension of the restricted measure.","title":"Positive measure"},{"id":"mathlib/overview/analysis/measures-and-integral-calculus/part-01/stieltjes-measure","kind":"node","parent":320,"result":true,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/part-01/index.html#stieltjes-measure","scope":"mathlib/overview/analysis/measures-and-integral-calculus/part-01","status":"mathlib","summary":"The measure associated to a Stieltjes function, giving mass `f b - f a` to the interval `(a, b]`. If there is a bot element, it gives zero mass to it.","title":"Stieltjes measure"},{"id":"mathlib/overview/analysis/measures-and-integral-calculus/part-01/lebesgue-measure","kind":"node","parent":320,"result":true,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/part-01/index.html#lebesgue-measure","scope":"mathlib/overview/analysis/measures-and-integral-calculus/part-01","status":"mathlib","summary":"Mathlib states this as `Real.measureSpace`.","title":"Lebesgue measure (Measures and integral calculus)"},{"id":"mathlib/overview/analysis/measures-and-integral-calculus/part-01/hausdorff-measure","kind":"node","parent":320,"result":true,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/part-01/index.html#hausdorff-measure","scope":"mathlib/overview/analysis/measures-and-integral-calculus/part-01","status":"mathlib","summary":"Hausdorff measure on an (e)metric space.","title":"Hausdorff measure"},{"id":"mathlib/overview/analysis/measures-and-integral-calculus/part-01/hausdorff-dimension","kind":"node","parent":320,"result":true,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/part-01/index.html#hausdorff-dimension","scope":"mathlib/overview/analysis/measures-and-integral-calculus/part-01","status":"mathlib","summary":"Hausdorff dimension of a set in an (e)metric space.","title":"Hausdorff dimension"},{"id":"mathlib/overview/analysis/measures-and-integral-calculus/part-01/giry-monad","kind":"node","parent":320,"result":true,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/part-01/index.html#giry-monad","scope":"mathlib/overview/analysis/measures-and-integral-calculus/part-01","status":"mathlib","summary":"Measurability structure on `Measure`: Measures are measurable w.r.t. all projections","title":"Giry monad"},{"id":"mathlib/overview/analysis/measures-and-integral-calculus/part-01/integral-of-positive-measurable-functions","kind":"node","parent":320,"result":true,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/part-01/index.html#integral-of-positive-measurable-functions","scope":"mathlib/overview/analysis/measures-and-integral-calculus/part-01","status":"mathlib","summary":"The lower Lebesgue integral of a function `f` with respect to a measure `\u03bc`.","title":"Integral of positive measurable functions"},{"id":"mathlib/overview/analysis/measures-and-integral-calculus/part-01/monotone-convergence-theorem","kind":"node","parent":320,"result":true,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/part-01/index.html#monotone-convergence-theorem","scope":"mathlib/overview/analysis/measures-and-integral-calculus/part-01","status":"mathlib","summary":"Monotone convergence theorem for nonincreasing sequences of functions.","title":"Monotone convergence theorem (Measures and integral calculus)"},{"id":"mathlib/overview/analysis/measures-and-integral-calculus/part-01/fatous-lemma","kind":"node","parent":320,"result":true,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/part-01/index.html#fatous-lemma","scope":"mathlib/overview/analysis/measures-and-integral-calculus/part-01","status":"mathlib","summary":"Fatou's lemma, version with `Measurable` functions.","title":"Fatou's lemma"},{"id":"mathlib/overview/analysis/measures-and-integral-calculus/part-01/vector-valued-integrable-function-bochner-integr","kind":"node","parent":320,"result":true,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/part-01/index.html#vector-valued-integrable-function-bochner-integr","scope":"mathlib/overview/analysis/measures-and-integral-calculus/part-01","status":"mathlib","summary":"`Integrable f \u03bc` means that `f` is measurable and that the integral `\u222b\u207b a, \u2016f a\u2016 \u2202\u03bc` is finite. `Integrable f` means `Integrable f volume`.","title":"Vector-valued integrable function (Bochner integral)"},{"id":"mathlib/overview/analysis/measures-and-integral-calculus/part-01/uniform-integrability","kind":"node","parent":320,"result":true,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/part-01/index.html#uniform-integrability","scope":"mathlib/overview/analysis/measures-and-integral-calculus/part-01","status":"mathlib","summary":"In probability theory, a family of measurable functions is uniformly integrable if it is uniformly integrable in the measure theory sense and is uniformly bounded.","title":"Uniform integrability"},{"id":"mathlib/overview/analysis/measures-and-integral-calculus/part-01/l-p-space","kind":"node","parent":320,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(L^p\\)"],["text"," space"]],"source":"$L^p$ space"},"result":true,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/part-01/index.html#l-p-space","scope":"mathlib/overview/analysis/measures-and-integral-calculus/part-01","status":"mathlib","summary":"Lp space","title":"$L^p$ space"},{"id":"mathlib/overview/analysis/measures-and-integral-calculus/part-01/bochner-integral","kind":"node","parent":320,"result":true,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/part-01/index.html#bochner-integral","scope":"mathlib/overview/analysis/measures-and-integral-calculus/part-01","status":"mathlib","summary":"The Bochner integral","title":"Bochner integral"},{"id":"mathlib/overview/analysis/measures-and-integral-calculus/part-01/dominated-convergence-theorem","kind":"node","parent":320,"result":true,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/part-01/index.html#dominated-convergence-theorem","scope":"mathlib/overview/analysis/measures-and-integral-calculus/part-01","status":"mathlib","summary":"Lebesgue dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies the convergence of their integrals. We could weaken the condition `bound_integrable` to require `HasFiniteIntegral bound \u03bc` instead (i.e. not requiring that `bound` is measurable), but in all applications proving integrability is easier.","title":"Dominated convergence theorem"},{"id":"mathlib/overview/analysis/measures-and-integral-calculus/part-01/fundamental-theorem-of-calculus-part-1","kind":"node","parent":320,"result":true,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/part-01/index.html#fundamental-theorem-of-calculus-part-1","scope":"mathlib/overview/analysis/measures-and-integral-calculus/part-01","status":"mathlib","summary":"Fundamental theorem of calculus-1: if `f : \u211d \u2192 E` is integrable on `a..b` and `f x` has finite limits `ca` and `cb` almost surely as `x` tends to `a` and `b`, respectively, then `(u, v) \u21a6 \u222b x in u..v, f x` has derivative `(u, v) \u21a6 v \u2022 cb - u \u2022 ca` at `(a, b)`.","title":"Fundamental theorem of calculus, part 1"},{"id":"mathlib/overview/analysis/measures-and-integral-calculus/part-01/fundamental-theorem-of-calculus-part-2","kind":"node","parent":320,"result":true,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/part-01/index.html#fundamental-theorem-of-calculus-part-2","scope":"mathlib/overview/analysis/measures-and-integral-calculus/part-01","status":"mathlib","summary":"Fundamental theorem of calculus-2: If `f : \u211d \u2192 E` is continuous on `[a, b]` (where `a \u2264 b`) and has a right derivative at `f' x` for all `x` in `(a, b)`, and `f'` is integrable on `[a, b]`, then `\u222b y in a..b, f' y` equals `f b - f a`.","title":"Fundamental theorem of calculus, part 2"},{"data_url":"scopes/mathlib/overview/analysis/measures-and-integral-calculus/part-02.json","id":"mathlib/overview/analysis/measures-and-integral-calculus/part-02","kind":"scope","navigation":"transparent","parent":319,"result":false,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/part-02/index.html","scope":"mathlib/overview/analysis/measures-and-integral-calculus/part-02","status":"mathlib","title":"Fubini's theorem to Divergence theorem"},{"id":"mathlib/overview/analysis/measures-and-integral-calculus/part-02/fubinis-theorem","kind":"node","parent":341,"result":true,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/part-02/index.html#fubinis-theorem","scope":"mathlib/overview/analysis/measures-and-integral-calculus/part-02","status":"mathlib","summary":"Fubini's Theorem: For integrable functions on `\u03b1 \u00d7 \u03b2`, the Bochner integral of `f` is equal to the iterated Bochner integral. `integrable_prod_iff` can be useful to show that the function in question in integrable. `MeasureTheory.Integrable.integral_prod_right` is useful to show that the inner integral of the right-hand side is integrable.","title":"Fubini's theorem"},{"id":"mathlib/overview/analysis/measures-and-integral-calculus/part-02/product-of-finitely-many-measures","kind":"node","parent":341,"result":true,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/part-02/index.html#product-of-finitely-many-measures","scope":"mathlib/overview/analysis/measures-and-integral-calculus/part-02","status":"mathlib","summary":"`Measure.pi \u03bc` is the finite product of the measures `{\u03bc i | i : \u03b9}`. It is defined to be measure corresponding to `MeasureTheory.OuterMeasure.pi`.","title":"Product of finitely many measures"},{"id":"mathlib/overview/analysis/measures-and-integral-calculus/part-02/convolution","kind":"node","parent":341,"result":true,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/part-02/index.html#convolution","scope":"mathlib/overview/analysis/measures-and-integral-calculus/part-02","status":"mathlib","summary":"The convolution of two functions `f` and `g` with respect to a continuous bilinear map `L` and measure `\u03bc`. It is defined to be `(f \u22c6[L, \u03bc] g) x = \u222b t, L (f t) (g (x - t)) \u2202\u03bc`.","title":"Convolution"},{"id":"mathlib/overview/analysis/measures-and-integral-calculus/part-02/approximation-by-convolution","kind":"node","parent":341,"result":true,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/part-02/index.html#approximation-by-convolution","scope":"mathlib/overview/analysis/measures-and-integral-calculus/part-02","status":"mathlib","summary":"`(\u03c6 i \u22c6 g i) (k i)` tends to `z\u2080` as `i` tends to some filter `l` if such that `(\u03c6 i).rOut` tends to `0` as `i` tends to `l`;","title":"Approximation by convolution"},{"id":"mathlib/overview/analysis/measures-and-integral-calculus/part-02/regularization-by-convolution","kind":"node","parent":341,"result":true,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/part-02/index.html#regularization-by-convolution","scope":"mathlib/overview/analysis/measures-and-integral-calculus/part-02","status":"mathlib","summary":"Mathlib states this as `HasCompactSupport.contDiff_convolution_left`.","title":"Regularization by convolution"},{"id":"mathlib/overview/analysis/measures-and-integral-calculus/part-02/change-of-variables-formula","kind":"node","parent":341,"result":true,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/part-02/index.html#change-of-variables-formula","scope":"mathlib/overview/analysis/measures-and-integral-calculus/part-02","status":"mathlib","summary":"Change of variable formula for differentiable functions: if a function `f` is injective and differentiable on a measurable set `s`, then the Bochner integral of a function `g : E \u2192 F` on `f '' s` coincides with the integral of `|(f' x).det| \u2022 g \u2218 f` on `s`.","title":"Change of variables formula"},{"id":"mathlib/overview/analysis/measures-and-integral-calculus/part-02/divergence-theorem","kind":"node","parent":341,"result":true,"route":"../roadmap/mathlib/overview/analysis/measures-and-integral-calculus/part-02/index.html#divergence-theorem","scope":"mathlib/overview/analysis/measures-and-integral-calculus/part-02","status":"mathlib","summary":"Divergence theorem for Bochner integral. If `f : \u211d\u207f\u207a\u00b9 \u2192 E\u207f\u207a\u00b9` is continuous on a rectangular box `[a, b] : Set \u211d\u207f\u207a\u00b9`, `a \u2264 b`, is differentiable on its interior with derivative `f' : \u211d\u207f\u207a\u00b9 \u2192 \u211d\u207f\u207a\u00b9 \u2192L[\u211d] E\u207f\u207a\u00b9` and the divergence `fun x \u21a6 \u2211 i, f' x e\u1d62 i` is integrable on `[a, b]`, where `e\u1d62 = Pi.single i 1` is the `i`-th basis vector, then its integral is equal to the sum of integrals of `f` over the faces of `[a, b]`,\u2026","title":"Divergence theorem"},{"data_url":"scopes/mathlib/overview/analysis/complex-analysis.json","id":"mathlib/overview/analysis/complex-analysis","kind":"scope","parent":261,"result":false,"route":"../roadmap/mathlib/overview/analysis/complex-analysis/index.html","scope":"mathlib/overview/analysis/complex-analysis","status":"mathlib","title":"Complex analysis"},{"id":"mathlib/overview/analysis/complex-analysis/cauchy-integral-formula","kind":"node","parent":349,"result":true,"route":"../roadmap/mathlib/overview/analysis/complex-analysis/index.html#cauchy-integral-formula","scope":"mathlib/overview/analysis/complex-analysis","status":"mathlib","summary":"Cauchy integral formula: if `f : \u2102 \u2192 E` is complex differentiable on an open disc and is continuous on its closure, then for any `w` in this open ball we have $\\oint_{|z-c|=R}(z-w)^{-1}f(z)\\,dz=2\u03c0if(w)$.","title":"Cauchy integral formula"},{"id":"mathlib/overview/analysis/complex-analysis/liouville-theorem","kind":"node","parent":349,"result":true,"route":"../roadmap/mathlib/overview/analysis/complex-analysis/index.html#liouville-theorem","scope":"mathlib/overview/analysis/complex-analysis","status":"mathlib","summary":"Liouville's theorem: a complex differentiable bounded function `f : E \u2192 F` is a constant.","title":"Liouville theorem"},{"id":"mathlib/overview/analysis/complex-analysis/maximum-modulus-principle","kind":"node","parent":349,"result":true,"route":"../roadmap/mathlib/overview/analysis/complex-analysis/index.html#maximum-modulus-principle","scope":"mathlib/overview/analysis/complex-analysis","status":"mathlib","summary":"Maximum modulus principle: if `f : E \u2192 F` is complex differentiable in a neighborhood of `c` and the norm `\u2016f z\u2016` has a local maximum at `c`, then `f` is locally constant in a neighborhood of `c`.","title":"Maximum modulus principle"},{"id":"mathlib/overview/analysis/complex-analysis/principle-of-isolated-zeros","kind":"node","parent":349,"result":true,"route":"../roadmap/mathlib/overview/analysis/complex-analysis/index.html#principle-of-isolated-zeros","scope":"mathlib/overview/analysis/complex-analysis","status":"mathlib","summary":"The *principle of isolated zeros* for an analytic function, local version: if a function is analytic at `z\u2080`, then either it is identically zero in a neighborhood of `z\u2080`, or it does not vanish in a punctured neighborhood of `z\u2080`.","title":"Principle of isolated zeros"},{"id":"mathlib/overview/analysis/complex-analysis/principle-of-analytic-continuation","kind":"node","parent":349,"result":true,"route":"../roadmap/mathlib/overview/analysis/complex-analysis/index.html#principle-of-analytic-continuation","scope":"mathlib/overview/analysis/complex-analysis","status":"mathlib","summary":"The *identity principle* for analytic functions, global version: if two functions are analytic on a connected set `U` and coincide at points which accumulate to a point `z\u2080 \u2208 U`, then they coincide globally in `U`. For higher-dimensional versions requiring that the functions coincide in a neighborhood of `z\u2080`, see `AnalyticOnNhd.eqOn_of_preconnected_of_eventuallyEq`.","title":"Principle of analytic continuation"},{"id":"mathlib/overview/analysis/complex-analysis/analyticity-of-holomorphic-functions","kind":"node","parent":349,"result":true,"route":"../roadmap/mathlib/overview/analysis/complex-analysis/index.html#analyticity-of-holomorphic-functions","scope":"mathlib/overview/analysis/complex-analysis","status":"mathlib","summary":"If `f : \u2102 \u2192 E` is complex differentiable on some set `s`, then it is analytic at any point `z` such that `s \u2208 \ud835\udcdd z` (equivalently, `z \u2208 interior s`).","title":"Analyticity of holomorphic functions"},{"id":"mathlib/overview/analysis/complex-analysis/schwarz-lemma","kind":"node","parent":349,"result":true,"route":"../roadmap/mathlib/overview/analysis/complex-analysis/index.html#schwarz-lemma","scope":"mathlib/overview/analysis/complex-analysis","status":"mathlib","summary":"The Schwarz Lemma. Let `f : E \u2192 F` be a complex analytic on an open ball with center at the origin. If `f` sends this ball to the closed ball with center `0` of the same radius and `f 0 = 0`, then for any point `z` of this disk we have `\u2016f z\u2016 \u2264 \u2016z\u2016`.","title":"Schwarz lemma"},{"id":"mathlib/overview/analysis/complex-analysis/removable-singularity","kind":"node","parent":349,"result":true,"route":"../roadmap/mathlib/overview/analysis/complex-analysis/index.html#removable-singularity","scope":"mathlib/overview/analysis/complex-analysis","status":"mathlib","summary":"Removable singularity theorem: if `s` is a punctured neighborhood of `c : \u2102`, a function `f : \u2102 \u2192 E` is complex differentiable on `s`, and $f(z) - f(c)=o((z-c)^{-1})$, then `f` redefined to be equal to `limUnder (\ud835\udcdd[\u2260] c) f` at `c` is complex differentiable on `{c} \u222a s`.","title":"Removable singularity"},{"id":"mathlib/overview/analysis/complex-analysis/phragmen-lindelof-principle","kind":"node","parent":349,"result":true,"route":"../roadmap/mathlib/overview/analysis/complex-analysis/index.html#phragmen-lindelof-principle","scope":"mathlib/overview/analysis/complex-analysis","status":"mathlib","summary":"Phragmen-Lindel\u00f6f principle in a strip `U = {z : \u2102 | a < im z < b}`. Let `f : \u2102 \u2192 E` be a function such that","title":"Phragmen-Lindel\u00f6f principle"},{"id":"mathlib/overview/analysis/complex-analysis/fundamental-theorem-of-algebra","kind":"node","parent":349,"result":true,"route":"../roadmap/mathlib/overview/analysis/complex-analysis/index.html#fundamental-theorem-of-algebra","scope":"mathlib/overview/analysis/complex-analysis","status":"mathlib","summary":"Fundamental theorem of algebra: the field `\u2102` of complex numbers is algebraically closed.","title":"Fundamental theorem of algebra"},{"id":"mathlib/overview/analysis/complex-analysis/first-main-theorem-of-value-distribution-theory","kind":"node","parent":349,"result":true,"route":"../roadmap/mathlib/overview/analysis/complex-analysis/index.html#first-main-theorem-of-value-distribution-theory","scope":"mathlib/overview/analysis/complex-analysis","status":"mathlib","summary":"First part of the First Main Theorem, quantitative version: If `f` is meromorphic on the complex plane, then the difference between the characteristic functions of `f` and `f\u207b\u00b9` is bounded by an explicit constant.","title":"First Main Theorem of Value Distribution Theory"},{"data_url":"scopes/mathlib/overview/analysis/distribution-theory.json","id":"mathlib/overview/analysis/distribution-theory","kind":"scope","parent":261,"result":false,"route":"../roadmap/mathlib/overview/analysis/distribution-theory/index.html","scope":"mathlib/overview/analysis/distribution-theory","status":"mathlib","title":"Distribution theory"},{"id":"mathlib/overview/analysis/distribution-theory/test-functions","kind":"node","parent":361,"result":true,"route":"../roadmap/mathlib/overview/analysis/distribution-theory/index.html#test-functions","scope":"mathlib/overview/analysis/distribution-theory","status":"mathlib","summary":"The type of bundled `n`-times continuously differentiable maps with compact support","title":"Test functions"},{"id":"mathlib/overview/analysis/distribution-theory/schwartz-space","kind":"node","parent":361,"result":true,"route":"../roadmap/mathlib/overview/analysis/distribution-theory/index.html#schwartz-space","scope":"mathlib/overview/analysis/distribution-theory","status":"mathlib","summary":"A function is a Schwartz function if it is smooth and all derivatives decay faster than any power of `\u2016x\u2016`.","title":"Schwartz space"},{"id":"mathlib/overview/analysis/distribution-theory/distributions","kind":"node","parent":361,"result":true,"route":"../roadmap/mathlib/overview/analysis/distribution-theory/index.html#distributions","scope":"mathlib/overview/analysis/distribution-theory","status":"mathlib","summary":"`\ud835\udcd3'^{n}(\u03a9, F) = Distribution \u03a9 F n` is the space of `F`-valued distributions on `\u03a9` with order at most `n`.","title":"Distributions"},{"id":"mathlib/overview/analysis/distribution-theory/tempered-distributions","kind":"node","parent":361,"result":true,"route":"../roadmap/mathlib/overview/analysis/distribution-theory/index.html#tempered-distributions","scope":"mathlib/overview/analysis/distribution-theory","status":"mathlib","summary":"The space of tempered distribution is the space of continuous linear maps from the Schwartz to a normed space, equipped with the topology of pointwise convergence.","title":"Tempered distributions"},{"data_url":"scopes/mathlib/overview/analysis/fourier-analysis.json","id":"mathlib/overview/analysis/fourier-analysis","kind":"scope","parent":261,"result":false,"route":"../roadmap/mathlib/overview/analysis/fourier-analysis/index.html","scope":"mathlib/overview/analysis/fourier-analysis","status":"mathlib","title":"Fourier analysis"},{"id":"mathlib/overview/analysis/fourier-analysis/fourier-transform-as-an-integral","kind":"node","parent":366,"result":true,"route":"../roadmap/mathlib/overview/analysis/fourier-analysis/index.html#fourier-transform-as-an-integral","scope":"mathlib/overview/analysis/fourier-analysis","status":"mathlib","summary":"Mathlib states this as `Real.fourier_eq`.","title":"Fourier transform as an integral"},{"id":"mathlib/overview/analysis/fourier-analysis/inversion-formula","kind":"node","parent":366,"result":true,"route":"../roadmap/mathlib/overview/analysis/fourier-analysis/index.html#inversion-formula","scope":"mathlib/overview/analysis/fourier-analysis","status":"mathlib","summary":"Fourier inversion formula: If a function `f` on a finite-dimensional real inner product space is continuous, integrable, and its Fourier transform `\ud835\udcd5 f` is also integrable, then `\ud835\udcd5\u207b (\ud835\udcd5 f) = f`.","title":"Inversion formula"},{"id":"mathlib/overview/analysis/fourier-analysis/riemann-lebesgue-lemma","kind":"node","parent":366,"result":true,"route":"../roadmap/mathlib/overview/analysis/fourier-analysis/index.html#riemann-lebesgue-lemma","scope":"mathlib/overview/analysis/fourier-analysis","status":"mathlib","summary":"Riemann-Lebesgue lemma for functions on a real inner-product space: the integral `\u222b v, exp (-2 * \u03c0 * \u27eaw, v\u27eb * I) \u2022 f v` tends to 0 as `w \u2192 \u221e`.","title":"Riemann-Lebesgue lemma (Analysis)"},{"id":"mathlib/overview/analysis/fourier-analysis/parseval-formula-for-fourier-series","kind":"node","parent":366,"result":true,"route":"../roadmap/mathlib/overview/analysis/fourier-analysis/index.html#parseval-formula-for-fourier-series","scope":"mathlib/overview/analysis/fourier-analysis","status":"mathlib","summary":"Mathlib states this as `tsum_sq_fourierCoeff`.","title":"Parseval formula for Fourier series"},{"data_url":"scopes/mathlib/overview/probability-theory.json","id":"mathlib/overview/probability-theory","kind":"scope","parent":2,"result":false,"route":"../roadmap/mathlib/overview/probability-theory/index.html","scope":"mathlib/overview/probability-theory","status":"mathlib","title":"Probability Theory"},{"data_url":"scopes/mathlib/overview/probability-theory/definitions-in-probability-theory.json","id":"mathlib/overview/probability-theory/definitions-in-probability-theory","kind":"scope","parent":371,"result":false,"route":"../roadmap/mathlib/overview/probability-theory/definitions-in-probability-theory/index.html","scope":"mathlib/overview/probability-theory/definitions-in-probability-theory","status":"mathlib","title":"Definitions in probability theory"},{"id":"mathlib/overview/probability-theory/definitions-in-probability-theory/probability-measure","kind":"node","parent":372,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/definitions-in-probability-theory/index.html#probability-measure","scope":"mathlib/overview/probability-theory/definitions-in-probability-theory","status":"mathlib","summary":"A measure `\u03bc` is called a probability measure if `\u03bc univ = 1`.","title":"Probability measure"},{"id":"mathlib/overview/probability-theory/definitions-in-probability-theory/independent-events","kind":"node","parent":372,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/definitions-in-probability-theory/index.html#independent-events","scope":"mathlib/overview/probability-theory/definitions-in-probability-theory","status":"mathlib","summary":"A family of sets is independent if the family of measurable space structures they generate is independent. For a set `s`, the generated measurable space has measurable sets `\u2205, s, s\u1d9c, univ`.","title":"Independent events"},{"id":"mathlib/overview/probability-theory/definitions-in-probability-theory/independent-sigma-algebras","kind":"node","parent":372,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/definitions-in-probability-theory/index.html#independent-sigma-algebras","scope":"mathlib/overview/probability-theory/definitions-in-probability-theory","status":"mathlib","summary":"A family of measurable space structures (i.e. of \u03c3-algebras) is independent with respect to a measure `\u03bc` (typically defined on a finer \u03c3-algebra) if the family of sets of measurable sets they define is independent. `m : \u03b9 \u2192 MeasurableSpace \u03a9` is independent with respect to measure `\u03bc` if for any finite set of indices `s = {i_1, ..., i_n}`, for any sets `f i_1 \u2208 m i_1, ..., f i_n \u2208 m i_n`, then\u2026","title":"Independent sigma-algebras"},{"id":"mathlib/overview/probability-theory/definitions-in-probability-theory/conditional-probability","kind":"node","parent":372,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/definitions-in-probability-theory/index.html#conditional-probability","scope":"mathlib/overview/probability-theory/definitions-in-probability-theory","status":"mathlib","summary":"The conditional probability measure of measure `\u03bc` on set `s` is `\u03bc` restricted to `s` and scaled by the inverse of `\u03bc s` (to make it a probability measure): `(\u03bc s)\u207b\u00b9 \u2022 \u03bc.restrict s`.","title":"Conditional probability"},{"id":"mathlib/overview/probability-theory/definitions-in-probability-theory/conditional-expectation","kind":"node","parent":372,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/definitions-in-probability-theory/index.html#conditional-expectation","scope":"mathlib/overview/probability-theory/definitions-in-probability-theory","status":"mathlib","summary":"Conditional expectation of a function, with notation `\u03bc[f | m]`.","title":"Conditional expectation"},{"id":"mathlib/overview/probability-theory/definitions-in-probability-theory/characteristic-function","kind":"node","parent":372,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/definitions-in-probability-theory/index.html#characteristic-function","scope":"mathlib/overview/probability-theory/definitions-in-probability-theory","status":"mathlib","summary":"The characteristic function of a measure in an inner product space.","title":"Characteristic function"},{"data_url":"scopes/mathlib/overview/probability-theory/random-variables-and-their-laws.json","id":"mathlib/overview/probability-theory/random-variables-and-their-laws","kind":"scope","parent":371,"result":false,"route":"../roadmap/mathlib/overview/probability-theory/random-variables-and-their-laws/index.html","scope":"mathlib/overview/probability-theory/random-variables-and-their-laws","status":"mathlib","title":"Random variables and their laws"},{"id":"mathlib/overview/probability-theory/random-variables-and-their-laws/discrete-law","kind":"node","parent":379,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/random-variables-and-their-laws/index.html#discrete-law","scope":"mathlib/overview/probability-theory/random-variables-and-their-laws","status":"mathlib","summary":"A probability mass function, or discrete probability measures is a function `\u03b1 \u2192 \u211d\u22650\u221e` such that the values have (infinite) sum `1`.","title":"Discrete law"},{"id":"mathlib/overview/probability-theory/random-variables-and-their-laws/probability-density-function","kind":"node","parent":379,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/random-variables-and-their-laws/index.html#probability-density-function","scope":"mathlib/overview/probability-theory/random-variables-and-their-laws","status":"mathlib","summary":"A random variable `X : \u03a9 \u2192 E` is said to have a probability density function (`HasPDF`) with respect to the measure `\u2119` on `\u03a9` and `\u03bc` on `E` if the push-forward measure of `\u2119` along `X` is absolutely continuous with respect to `\u03bc` and they have a Lebesgue decomposition (`HaveLebesgueDecomposition`).","title":"Probability density function"},{"id":"mathlib/overview/probability-theory/random-variables-and-their-laws/variance-of-a-real-valued-random-variable","kind":"node","parent":379,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/random-variables-and-their-laws/index.html#variance-of-a-real-valued-random-variable","scope":"mathlib/overview/probability-theory/random-variables-and-their-laws","status":"mathlib","summary":"The `\u211d`-valued variance of a real-valued random variable defined by applying `ENNReal.toReal` to `evariance`.","title":"Variance of a real-valued random variable"},{"id":"mathlib/overview/probability-theory/random-variables-and-their-laws/independence-of-random-variables","kind":"node","parent":379,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/random-variables-and-their-laws/index.html#independence-of-random-variables","scope":"mathlib/overview/probability-theory/random-variables-and-their-laws","status":"mathlib","summary":"A family of functions defined on the same space `\u03a9` and taking values in possibly different spaces, each with a measurable space structure, is independent if the family of measurable space structures they generate on `\u03a9` is independent. For a function `g` with codomain having measurable space structure `m`, the generated measurable space structure is `MeasurableSpace.comap g m`.","title":"Independence of random variables"},{"id":"mathlib/overview/probability-theory/random-variables-and-their-laws/kolmogorovs-0-1-law","kind":"node","parent":379,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kolmogorov's "],["math","\\(0\\)"],["text","-"],["math","\\(1\\)"],["text"," law"]],"source":"Kolmogorov's $0$-$1$ law"},"result":true,"route":"../roadmap/mathlib/overview/probability-theory/random-variables-and-their-laws/index.html#kolmogorovs-0-1-law","scope":"mathlib/overview/probability-theory/random-variables-and-their-laws","status":"mathlib","summary":"Kolmogorov's 0-1 law : any event in the tail \u03c3-algebra of an independent sequence of sub-\u03c3-algebras has probability 0 or 1. The tail \u03c3-algebra `limsup s atTop` is the same as `\u22c2 n, \u22c3 i \u2265 n, s i`.","title":"Kolmogorov's $0$-$1$ law"},{"id":"mathlib/overview/probability-theory/random-variables-and-their-laws/mean-of-product-of-independent-random-variables","kind":"node","parent":379,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/random-variables-and-their-laws/index.html#mean-of-product-of-independent-random-variables","scope":"mathlib/overview/probability-theory/random-variables-and-their-laws","status":"mathlib","summary":"Mathlib states this as `ProbabilityTheory.IndepFun.integral_mul_eq_mul_integral`.","title":"Mean of product of independent random variables"},{"id":"mathlib/overview/probability-theory/random-variables-and-their-laws/moment-of-a-random-variable","kind":"node","parent":379,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/random-variables-and-their-laws/index.html#moment-of-a-random-variable","scope":"mathlib/overview/probability-theory/random-variables-and-their-laws","status":"mathlib","summary":"Moment of a real random variable, `\u03bc[X ^ p]`.","title":"Moment of a random variable"},{"id":"mathlib/overview/probability-theory/random-variables-and-their-laws/bernoulli-law","kind":"node","parent":379,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/random-variables-and-their-laws/index.html#bernoulli-law","scope":"mathlib/overview/probability-theory/random-variables-and-their-laws","status":"mathlib","summary":"The Bernoulli distribution over an arbitrary measurable space `X`. Given `x y : X` and `p : I` (`I` is the `unitInterval`), it is the measure which gives mass `p` to `{x}` and `1 - p` to `{y}`.","title":"Bernoulli law"},{"id":"mathlib/overview/probability-theory/random-variables-and-their-laws/binomial-law","kind":"node","parent":379,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/random-variables-and-their-laws/index.html#binomial-law","scope":"mathlib/overview/probability-theory/random-variables-and-their-laws","status":"mathlib","summary":"The binomial probability distribution with parameter `p`.","title":"Binomial law"},{"id":"mathlib/overview/probability-theory/random-variables-and-their-laws/geometric-law","kind":"node","parent":379,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/random-variables-and-their-laws/index.html#geometric-law","scope":"mathlib/overview/probability-theory/random-variables-and-their-laws","status":"mathlib","summary":"The geometric measure with success probability `p` as a measure over `\u2115`.","title":"Geometric law"},{"id":"mathlib/overview/probability-theory/random-variables-and-their-laws/poisson-law","kind":"node","parent":379,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/random-variables-and-their-laws/index.html#poisson-law","scope":"mathlib/overview/probability-theory/random-variables-and-their-laws","status":"mathlib","summary":"The poisson measure with rate `r : \u211d\u22650` as a measure over `\u2115`.","title":"Poisson law"},{"id":"mathlib/overview/probability-theory/random-variables-and-their-laws/exponential-law","kind":"node","parent":379,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/random-variables-and-their-laws/index.html#exponential-law","scope":"mathlib/overview/probability-theory/random-variables-and-their-laws","status":"mathlib","summary":"Measure defined by the exponential distribution","title":"Exponential law"},{"id":"mathlib/overview/probability-theory/random-variables-and-their-laws/gaussian-law","kind":"node","parent":379,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/random-variables-and-their-laws/index.html#gaussian-law","scope":"mathlib/overview/probability-theory/random-variables-and-their-laws","status":"mathlib","summary":"A Gaussian distribution on `\u211d` with mean `\u03bc` and variance `v`.","title":"Gaussian law"},{"id":"mathlib/overview/probability-theory/random-variables-and-their-laws/gaussian-measures","kind":"node","parent":379,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/random-variables-and-their-laws/index.html#gaussian-measures","scope":"mathlib/overview/probability-theory/random-variables-and-their-laws","status":"mathlib","summary":"A measure is Gaussian if its map by every continuous linear form is a real Gaussian measure.","title":"Gaussian measures"},{"data_url":"scopes/mathlib/overview/probability-theory/convergence-of-a-sequence-of-random-variables.json","id":"mathlib/overview/probability-theory/convergence-of-a-sequence-of-random-variables","kind":"scope","parent":371,"result":false,"route":"../roadmap/mathlib/overview/probability-theory/convergence-of-a-sequence-of-random-variables/index.html","scope":"mathlib/overview/probability-theory/convergence-of-a-sequence-of-random-variables","status":"mathlib","title":"Convergence of a sequence of random variables"},{"id":"mathlib/overview/probability-theory/convergence-of-a-sequence-of-random-variables/convergence-in-probability","kind":"node","parent":394,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/convergence-of-a-sequence-of-random-variables/index.html#convergence-in-probability","scope":"mathlib/overview/probability-theory/convergence-of-a-sequence-of-random-variables","status":"mathlib","summary":"A sequence of functions `f` is said to converge in measure to some function `g` if for all `\u03b5 > 0`, the measure of the set `{x | \u03b5 \u2264 dist (f i x) (g x)}` tends to 0 as `i` converges along some given filter `l`.","title":"Convergence in probability"},{"id":"mathlib/overview/probability-theory/convergence-of-a-sequence-of-random-variables/almost-sure-convergence","kind":"node","parent":394,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/convergence-of-a-sequence-of-random-variables/index.html#almost-sure-convergence","scope":"mathlib/overview/probability-theory/convergence-of-a-sequence-of-random-variables","status":"mathlib","summary":"The \u201calmost everywhere\u201d filter of co-null sets.","title":"Almost sure convergence"},{"id":"mathlib/overview/probability-theory/convergence-of-a-sequence-of-random-variables/convergence-in-distribution","kind":"node","parent":394,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/convergence-of-a-sequence-of-random-variables/index.html#convergence-in-distribution","scope":"mathlib/overview/probability-theory/convergence-of-a-sequence-of-random-variables","status":"mathlib","summary":"The characterization of weak convergence of probability measures by the usual (defining) condition that the integrals of every continuous bounded function converge to the integral of the function against the limit measure.","title":"Convergence in distribution"},{"id":"mathlib/overview/probability-theory/convergence-of-a-sequence-of-random-variables/markov-inequality","kind":"node","parent":394,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/convergence-of-a-sequence-of-random-variables/index.html#markov-inequality","scope":"mathlib/overview/probability-theory/convergence-of-a-sequence-of-random-variables","status":"mathlib","summary":"Markov's inequality also known as Chebyshev's first inequality. For a version assuming `AEMeasurable`, see `mul_meas_ge_le_lintegral\u2080`.","title":"Markov inequality"},{"id":"mathlib/overview/probability-theory/convergence-of-a-sequence-of-random-variables/chebyshev-inequality","kind":"node","parent":394,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/convergence-of-a-sequence-of-random-variables/index.html#chebyshev-inequality","scope":"mathlib/overview/probability-theory/convergence-of-a-sequence-of-random-variables","status":"mathlib","summary":"Chebyshev's inequality: one can control the deviation probability of a real random variable from its expectation in terms of the variance.","title":"Chebyshev inequality"},{"id":"mathlib/overview/probability-theory/convergence-of-a-sequence-of-random-variables/strong-law-of-large-numbers","kind":"node","parent":394,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/convergence-of-a-sequence-of-random-variables/index.html#strong-law-of-large-numbers","scope":"mathlib/overview/probability-theory/convergence-of-a-sequence-of-random-variables","status":"mathlib","summary":"Strong law of large numbers, almost sure version: if `X n` is a sequence of independent identically distributed integrable random variables taking values in a Banach space, then `n\u207b\u00b9 \u2022 \u2211 i \u2208 range n, X i` converges almost surely to `\ud835\udd3c[X 0]`. We give here the strong version, due to Etemadi, that only requires pairwise independence.","title":"Strong law of large numbers"},{"id":"mathlib/overview/probability-theory/convergence-of-a-sequence-of-random-variables/central-limit-theorem","kind":"node","parent":394,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/convergence-of-a-sequence-of-random-variables/index.html#central-limit-theorem","scope":"mathlib/overview/probability-theory/convergence-of-a-sequence-of-random-variables","status":"mathlib","summary":"Central Limit Theorem: Given a sequence of random variables `X : \u2115 \u2192 \u03a9 \u2192 \u211d` that are independent, identically distributed with mean `\u03bc` and variance `v`, and a random variable `Y : \u03a9' \u2192 \u211d` following `gaussianReal 0 v`, the sequence `n \u21a6 (\u221an)\u207b\u00b9 * (\u2211 k \u2208 Finset.range n, X k \u03c9 - n * \u03bc)` converges to `Y` in distribution.","title":"Central limit theorem"},{"data_url":"scopes/mathlib/overview/probability-theory/stochastic-processes.json","id":"mathlib/overview/probability-theory/stochastic-processes","kind":"scope","parent":371,"result":false,"route":"../roadmap/mathlib/overview/probability-theory/stochastic-processes/index.html","scope":"mathlib/overview/probability-theory/stochastic-processes","status":"mathlib","title":"Stochastic Processes"},{"id":"mathlib/overview/probability-theory/stochastic-processes/martingale","kind":"node","parent":402,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/stochastic-processes/index.html#martingale","scope":"mathlib/overview/probability-theory/stochastic-processes","status":"mathlib","summary":"A family of functions `f : \u03b9 \u2192 \u03a9 \u2192 E` is a martingale with respect to a filtration `\u2131` if `f` is strongly adapted with respect to `\u2131` and for all `i \u2264 j`, `\u03bc[f j | \u2131 i] =\u1d50[\u03bc] f i`.","title":"Martingale"},{"id":"mathlib/overview/probability-theory/stochastic-processes/optional-stopping-theorem","kind":"node","parent":402,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/stochastic-processes/index.html#optional-stopping-theorem","scope":"mathlib/overview/probability-theory/stochastic-processes","status":"mathlib","summary":"The optional stopping theorem (fair game theorem): a strongly adapted integrable process `f` is a submartingale if and only if for all bounded stopping times `\u03c4` and `\u03c0` such that `\u03c4 \u2264 \u03c0`, the stopped value of `f` at `\u03c4` has expectation smaller than its stopped value at `\u03c0`.","title":"Optional stopping theorem"},{"id":"mathlib/overview/probability-theory/stochastic-processes/stopping-time","kind":"node","parent":402,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/stochastic-processes/index.html#stopping-time","scope":"mathlib/overview/probability-theory/stochastic-processes","status":"mathlib","summary":"A stopping time with respect to some filtration `f` is a function `\u03c4` such that for all `i`, the preimage of `{j | j \u2264 i}` along `\u03c4` is measurable with respect to `f i`.","title":"Stopping time"},{"id":"mathlib/overview/probability-theory/stochastic-processes/hitting-time","kind":"node","parent":402,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/stochastic-processes/index.html#hitting-time","scope":"mathlib/overview/probability-theory/stochastic-processes","status":"mathlib","summary":"Hitting time: given a stochastic process `u` and a set `s`, `hittingBtwn u s n m` is the first time `u` is in `s` after time `n` and before time `m` (if `u` does not hit `s` after time `n` and before `m` then the hitting time is simply `m`).","title":"Hitting time"},{"id":"mathlib/overview/probability-theory/stochastic-processes/gaussian-processes","kind":"node","parent":402,"result":true,"route":"../roadmap/mathlib/overview/probability-theory/stochastic-processes/index.html#gaussian-processes","scope":"mathlib/overview/probability-theory/stochastic-processes","status":"mathlib","summary":"A stochastic process is a Gaussian process if all its finite dimensional distributions are Gaussian.","title":"Gaussian processes"},{"data_url":"scopes/mathlib/overview/geometry.json","id":"mathlib/overview/geometry","kind":"scope","parent":2,"result":false,"route":"../roadmap/mathlib/overview/geometry/index.html","scope":"mathlib/overview/geometry","status":"mathlib","title":"Geometry"},{"data_url":"scopes/mathlib/overview/geometry/affine-and-euclidean-geometry.json","id":"mathlib/overview/geometry/affine-and-euclidean-geometry","kind":"scope","parent":408,"result":false,"route":"../roadmap/mathlib/overview/geometry/affine-and-euclidean-geometry/index.html","scope":"mathlib/overview/geometry/affine-and-euclidean-geometry","status":"mathlib","title":"Affine and Euclidean geometry"},{"id":"mathlib/overview/geometry/affine-and-euclidean-geometry/affine-space","kind":"node","parent":409,"result":true,"route":"../roadmap/mathlib/overview/geometry/affine-and-euclidean-geometry/index.html#affine-space","scope":"mathlib/overview/geometry/affine-and-euclidean-geometry","status":"mathlib","summary":"An `AddTorsor G P` gives a structure to the nonempty type `P`, acted on by an `AddGroup G` with a transitive and free action given by the `+\u1d65` operation and a corresponding subtraction given by the `-\u1d65` operation. In the case of a vector space, it is an affine space.","title":"Affine space"},{"id":"mathlib/overview/geometry/affine-and-euclidean-geometry/affine-function","kind":"node","parent":409,"result":true,"route":"../roadmap/mathlib/overview/geometry/affine-and-euclidean-geometry/index.html#affine-function","scope":"mathlib/overview/geometry/affine-and-euclidean-geometry","status":"mathlib","summary":"An `AffineMap k P1 P2` (notation: `P1 \u2192\u1d43[k] P2`) is a map from `P1` to `P2` that induces a corresponding linear map from `V1` to `V2`.","title":"Affine function"},{"id":"mathlib/overview/geometry/affine-and-euclidean-geometry/affine-subspace","kind":"node","parent":409,"result":true,"route":"../roadmap/mathlib/overview/geometry/affine-and-euclidean-geometry/index.html#affine-subspace","scope":"mathlib/overview/geometry/affine-and-euclidean-geometry","status":"mathlib","summary":"An `AffineSubspace k P` is a subset of an `AffineSpace V P` that, if not empty, has an affine space structure induced by a corresponding subspace of the `Module k V`.","title":"Affine subspace"},{"id":"mathlib/overview/geometry/affine-and-euclidean-geometry/barycenter","kind":"node","parent":409,"result":true,"route":"../roadmap/mathlib/overview/geometry/affine-and-euclidean-geometry/index.html#barycenter","scope":"mathlib/overview/geometry/affine-and-euclidean-geometry","status":"mathlib","summary":"A weighted sum of the results of subtracting a default base point from the given points, added to that base point, as an affine map on the weights. This is intended to be used when the sum of the weights is 1, in which case it is an affine combination (barycenter) of the points with the given weights; that condition is specified as a hypothesis on those lemmas that require it.","title":"Barycenter"},{"id":"mathlib/overview/geometry/affine-and-euclidean-geometry/affine-span","kind":"node","parent":409,"result":true,"route":"../roadmap/mathlib/overview/geometry/affine-and-euclidean-geometry/index.html#affine-span","scope":"mathlib/overview/geometry/affine-and-euclidean-geometry","status":"mathlib","summary":"The affine span of a set of points is the smallest affine subspace containing those points. (Actually defined here in terms of spans in modules.)","title":"Affine span"},{"id":"mathlib/overview/geometry/affine-and-euclidean-geometry/euclidean-affine-space","kind":"node","parent":409,"result":true,"route":"../roadmap/mathlib/overview/geometry/affine-and-euclidean-geometry/index.html#euclidean-affine-space","scope":"mathlib/overview/geometry/affine-and-euclidean-geometry","status":"mathlib","summary":"A `NormedAddTorsor V P` is a torsor of an additive seminormed group action by a `SeminormedAddCommGroup V` on points `P`. We bundle the pseudometric space structure and require the distance to be the same as results from the norm (which in fact implies the distance yields a pseudometric space, but bundling just the distance and using an instance for the pseudometric space results in type class problems).","title":"Euclidean affine space"},{"id":"mathlib/overview/geometry/affine-and-euclidean-geometry/angle","kind":"node","parent":409,"result":true,"route":"../roadmap/mathlib/overview/geometry/affine-and-euclidean-geometry/index.html#angle","scope":"mathlib/overview/geometry/affine-and-euclidean-geometry","status":"mathlib","summary":"The undirected angle between two vectors. If either vector is 0, this is \u03c0/2. See `Orientation.oangle` for the corresponding oriented angle definition.","title":"Angle"},{"data_url":"scopes/mathlib/overview/geometry/differentiable-manifolds.json","id":"mathlib/overview/geometry/differentiable-manifolds","kind":"scope","parent":408,"result":false,"route":"../roadmap/mathlib/overview/geometry/differentiable-manifolds/index.html","scope":"mathlib/overview/geometry/differentiable-manifolds","status":"mathlib","title":"Differentiable manifolds"},{"data_url":"scopes/mathlib/overview/geometry/differentiable-manifolds/part-01.json","id":"mathlib/overview/geometry/differentiable-manifolds/part-01","kind":"scope","navigation":"transparent","parent":417,"result":false,"route":"../roadmap/mathlib/overview/geometry/differentiable-manifolds/part-01/index.html","scope":"mathlib/overview/geometry/differentiable-manifolds/part-01","status":"mathlib","title":"Smooth or analytic manifold\u2026 to Sphere"},{"id":"mathlib/overview/geometry/differentiable-manifolds/part-01/smooth-or-analytic-manifold-with-boundary-and-co","kind":"node","parent":418,"result":true,"route":"../roadmap/mathlib/overview/geometry/differentiable-manifolds/part-01/index.html#smooth-or-analytic-manifold-with-boundary-and-co","scope":"mathlib/overview/geometry/differentiable-manifolds/part-01","status":"mathlib","summary":"Typeclass defining manifolds with respect to a model with corners, over a field `\ud835\udd5c`. This definition includes the model with corners `I` (which might allow boundary, corners, or not, so this class covers both manifolds with boundary and manifolds without boundary), and a smoothness parameter `n : \u2115\u221e\u03c9` (where `n = 0` means topological manifold, `n = \u221e` means smooth manifold and `n = \u03c9` means analytic manifold).","title":"Smooth or analytic manifold (with boundary and corners)"},{"id":"mathlib/overview/geometry/differentiable-manifolds/part-01/smooth-map-between-manifolds","kind":"node","parent":418,"result":true,"route":"../roadmap/mathlib/overview/geometry/differentiable-manifolds/part-01/index.html#smooth-map-between-manifolds","scope":"mathlib/overview/geometry/differentiable-manifolds/part-01","status":"mathlib","summary":"`ContMDiff I I' n f` indicates that the function `f : M \u2192 M'` between manifolds is `n` times continuously differentiable.","title":"Smooth map between manifolds"},{"id":"mathlib/overview/geometry/differentiable-manifolds/part-01/smooth-immersion","kind":"node","parent":418,"result":true,"route":"../roadmap/mathlib/overview/geometry/differentiable-manifolds/part-01/index.html#smooth-immersion","scope":"mathlib/overview/geometry/differentiable-manifolds/part-01","status":"mathlib","summary":"`f : M \u2192 N` is a `C^n` immersion if around each point `x \u2208 M`, there are charts `\u03c6` and `\u03c8` of `M` and `N` around `x` and `f x`, respectively such that in these charts, `f` looks like `u \u21a6 (u, 0)`.","title":"Smooth immersion"},{"id":"mathlib/overview/geometry/differentiable-manifolds/part-01/smooth-embedding","kind":"node","parent":418,"result":true,"route":"../roadmap/mathlib/overview/geometry/differentiable-manifolds/part-01/index.html#smooth-embedding","scope":"mathlib/overview/geometry/differentiable-manifolds/part-01","status":"mathlib","summary":"A `C^n` map `f : M \u2192 M'` is a smooth `C^n` embedding if it is a topological embedding and a `C^n` immersion.","title":"Smooth embedding"},{"id":"mathlib/overview/geometry/differentiable-manifolds/part-01/product-of-manifolds","kind":"node","parent":418,"result":true,"route":"../roadmap/mathlib/overview/geometry/differentiable-manifolds/part-01/index.html#product-of-manifolds","scope":"mathlib/overview/geometry/differentiable-manifolds/part-01","status":"mathlib","summary":"The product of two `C^n` manifolds is naturally a `C^n` manifold.","title":"Product of manifolds"},{"id":"mathlib/overview/geometry/differentiable-manifolds/part-01/disjoint-union-of-manifolds","kind":"node","parent":418,"result":true,"route":"../roadmap/mathlib/overview/geometry/differentiable-manifolds/part-01/index.html#disjoint-union-of-manifolds","scope":"mathlib/overview/geometry/differentiable-manifolds/part-01","status":"mathlib","summary":"The disjoint union of two `C^n` manifolds modelled on `(E, H)` is a `C^n` manifold modelled on `(E, H)`.","title":"Disjoint union of manifolds"},{"id":"mathlib/overview/geometry/differentiable-manifolds/part-01/interior-of-a-manifold","kind":"node","parent":418,"result":true,"route":"../roadmap/mathlib/overview/geometry/differentiable-manifolds/part-01/index.html#interior-of-a-manifold","scope":"mathlib/overview/geometry/differentiable-manifolds/part-01","status":"mathlib","summary":"The interior of a manifold `M` is the set of its interior points.","title":"Interior of a manifold"},{"id":"mathlib/overview/geometry/differentiable-manifolds/part-01/boundary-of-a-manifold","kind":"node","parent":418,"result":true,"route":"../roadmap/mathlib/overview/geometry/differentiable-manifolds/part-01/index.html#boundary-of-a-manifold","scope":"mathlib/overview/geometry/differentiable-manifolds/part-01","status":"mathlib","summary":"The boundary of a manifold `M` is the set of its boundary points.","title":"Boundary of a manifold"},{"id":"mathlib/overview/geometry/differentiable-manifolds/part-01/tangent-bundle","kind":"node","parent":418,"result":true,"route":"../roadmap/mathlib/overview/geometry/differentiable-manifolds/part-01/index.html#tangent-bundle","scope":"mathlib/overview/geometry/differentiable-manifolds/part-01","status":"mathlib","summary":"The tangent bundle to a manifold, as a Sigma type. Defined in terms of `Bundle.TotalSpace` to be able to put a suitable topology on it.","title":"Tangent bundle"},{"id":"mathlib/overview/geometry/differentiable-manifolds/part-01/tangent-map","kind":"node","parent":418,"result":true,"route":"../roadmap/mathlib/overview/geometry/differentiable-manifolds/part-01/index.html#tangent-map","scope":"mathlib/overview/geometry/differentiable-manifolds/part-01","status":"mathlib","summary":"`tangentMap I I' f` is the derivative of `f : M \u2192 M'` as a map between the tangent bundles `TM` and `TM'`.","title":"Tangent map"},{"id":"mathlib/overview/geometry/differentiable-manifolds/part-01/pullback-of-vector-fields","kind":"node","parent":418,"result":true,"route":"../roadmap/mathlib/overview/geometry/differentiable-manifolds/part-01/index.html#pullback-of-vector-fields","scope":"mathlib/overview/geometry/differentiable-manifolds/part-01","status":"mathlib","summary":"The pullback of a vector field under a map between manifolds. If the derivative of the map is not invertible, then pullback is given the junk value zero.","title":"Pullback of vector fields"},{"id":"mathlib/overview/geometry/differentiable-manifolds/part-01/lie-bracket-of-vector-field","kind":"node","parent":418,"result":true,"route":"../roadmap/mathlib/overview/geometry/differentiable-manifolds/part-01/index.html#lie-bracket-of-vector-field","scope":"mathlib/overview/geometry/differentiable-manifolds/part-01","status":"mathlib","summary":"The Lie bracket of two vector fields in a manifold.","title":"Lie bracket of vector field"},{"id":"mathlib/overview/geometry/differentiable-manifolds/part-01/jacobi-leibniz-identity","kind":"node","parent":418,"result":true,"route":"../roadmap/mathlib/overview/geometry/differentiable-manifolds/part-01/index.html#jacobi-leibniz-identity","scope":"mathlib/overview/geometry/differentiable-manifolds/part-01","status":"mathlib","summary":"The Lie bracket of vector fields in manifolds satisfies the Leibniz identity `[U, [V, W]] = [[U, V], W] + [V, [U, W]]` (also called Jacobi identity).","title":"Jacobi/Leibniz identity"},{"id":"mathlib/overview/geometry/differentiable-manifolds/part-01/integral-curves","kind":"node","parent":418,"result":true,"route":"../roadmap/mathlib/overview/geometry/differentiable-manifolds/part-01/index.html#integral-curves","scope":"mathlib/overview/geometry/differentiable-manifolds/part-01","status":"mathlib","summary":"If `v : M \u2192 TM` is a vector field on `M`, `IsMIntegralCurve \u03b3 v` means `\u03b3 : \u211d \u2192 M` is a global integral curve of `v`. That is, `\u03b3 t` is tangent to `v (\u03b3 t)` for all `t : \u211d`.","title":"Integral curves"},{"id":"mathlib/overview/geometry/differentiable-manifolds/part-01/local-existence-of-integral-curves","kind":"node","parent":418,"result":true,"route":"../roadmap/mathlib/overview/geometry/differentiable-manifolds/part-01/index.html#local-existence-of-integral-curves","scope":"mathlib/overview/geometry/differentiable-manifolds/part-01","status":"mathlib","summary":"Existence of local integral curves for a $C^1$ vector field on a `C^1` manifold without boundary.","title":"Local existence of integral curves"},{"id":"mathlib/overview/geometry/differentiable-manifolds/part-01/uniform-time-existence","kind":"node","parent":418,"result":true,"route":"../roadmap/mathlib/overview/geometry/differentiable-manifolds/part-01/index.html#uniform-time-existence","scope":"mathlib/overview/geometry/differentiable-manifolds/part-01","status":"mathlib","summary":"If there exists `\u03b5 > 0` such that the local integral curve at each point `x : M` is defined at least on an open interval `Ioo (-\u03b5) \u03b5`, then every point on `M` has a global integral curve passing through it.","title":"Uniform time existence"},{"id":"mathlib/overview/geometry/differentiable-manifolds/part-01/uniqueness-of-integral-curves","kind":"node","parent":418,"result":true,"route":"../roadmap/mathlib/overview/geometry/differentiable-manifolds/part-01/index.html#uniqueness-of-integral-curves","scope":"mathlib/overview/geometry/differentiable-manifolds/part-01","status":"mathlib","summary":"Mathlib states this as `isMIntegralCurveOn_Ioo_eqOn_of_contMDiff_boundaryless`.","title":"Uniqueness of integral curves"},{"id":"mathlib/overview/geometry/differentiable-manifolds/part-01/lie-group","kind":"node","parent":418,"result":true,"route":"../roadmap/mathlib/overview/geometry/differentiable-manifolds/part-01/index.html#lie-group","scope":"mathlib/overview/geometry/differentiable-manifolds/part-01","status":"mathlib","summary":"A (multiplicative) Lie group is a group and a `C^n` manifold at the same time in which the multiplication and inverse operations are `C^n`.","title":"Lie group"},{"id":"mathlib/overview/geometry/differentiable-manifolds/part-01/lie-algebra-of-a-lie-group","kind":"node","parent":418,"result":true,"route":"../roadmap/mathlib/overview/geometry/differentiable-manifolds/part-01/index.html#lie-algebra-of-a-lie-group","scope":"mathlib/overview/geometry/differentiable-manifolds/part-01","status":"mathlib","summary":"The Lie algebra of a Lie group, i.e., its tangent space at the identity. We use the word `GroupLieAlgebra` instead of `LieAlgebra` as the latter is taken as a generic class.","title":"Lie algebra of a Lie group"},{"id":"mathlib/overview/geometry/differentiable-manifolds/part-01/sphere","kind":"node","parent":418,"result":true,"route":"../roadmap/mathlib/overview/geometry/differentiable-manifolds/part-01/index.html#sphere","scope":"mathlib/overview/geometry/differentiable-manifolds/part-01","status":"mathlib","summary":"The unit sphere in an `n + 1`-dimensional inner product space `E` is an analytic manifold, modelled on the Euclidean space of dimension `n`.","title":"Sphere"},{"data_url":"scopes/mathlib/overview/geometry/differentiable-manifolds/part-02.json","id":"mathlib/overview/geometry/differentiable-manifolds/part-02","kind":"scope","navigation":"transparent","parent":417,"result":false,"route":"../roadmap/mathlib/overview/geometry/differentiable-manifolds/part-02/index.html","scope":"mathlib/overview/geometry/differentiable-manifolds/part-02","status":"mathlib","title":"Closed interval to 0-manifolds are discrete spaces"},{"id":"mathlib/overview/geometry/differentiable-manifolds/part-02/closed-interval","kind":"node","parent":439,"result":true,"route":"../roadmap/mathlib/overview/geometry/differentiable-manifolds/part-02/index.html#closed-interval","scope":"mathlib/overview/geometry/differentiable-manifolds/part-02","status":"mathlib","summary":"The manifold structure on `[x, y]` is smooth.","title":"Closed interval"},{"id":"mathlib/overview/geometry/differentiable-manifolds/part-02/general-linear-group-as-a-lie-group","kind":"node","parent":439,"result":true,"route":"../roadmap/mathlib/overview/geometry/differentiable-manifolds/part-02/index.html#general-linear-group-as-a-lie-group","scope":"mathlib/overview/geometry/differentiable-manifolds/part-02","status":"mathlib","summary":"The units of a complete normed ring form a Lie group.","title":"General linear group as a Lie group"},{"id":"mathlib/overview/geometry/differentiable-manifolds/part-02/discrete-spaces-are-0-manifolds","kind":"node","parent":439,"result":true,"route":"../roadmap/mathlib/overview/geometry/differentiable-manifolds/part-02/index.html#discrete-spaces-are-0-manifolds","scope":"mathlib/overview/geometry/differentiable-manifolds/part-02","status":"mathlib","summary":"A discrete space `M` is a smooth manifold over the trivial model on a trivial normed space.","title":"Discrete spaces are 0-manifolds"},{"id":"mathlib/overview/geometry/differentiable-manifolds/part-02/0-manifolds-are-discrete-spaces","kind":"node","parent":439,"result":true,"route":"../roadmap/mathlib/overview/geometry/differentiable-manifolds/part-02/index.html#0-manifolds-are-discrete-spaces","scope":"mathlib/overview/geometry/differentiable-manifolds/part-02","status":"mathlib","summary":"A charted space over a discrete space is discrete.","title":"0-manifolds are discrete spaces"},{"data_url":"scopes/mathlib/overview/geometry/smooth-vector-and-fiber-bundles.json","id":"mathlib/overview/geometry/smooth-vector-and-fiber-bundles","kind":"scope","parent":408,"result":false,"route":"../roadmap/mathlib/overview/geometry/smooth-vector-and-fiber-bundles/index.html","scope":"mathlib/overview/geometry/smooth-vector-and-fiber-bundles","status":"mathlib","title":"Smooth vector and fiber bundles"},{"id":"mathlib/overview/geometry/smooth-vector-and-fiber-bundles/smooth-vector-bundle","kind":"node","parent":444,"result":true,"route":"../roadmap/mathlib/overview/geometry/smooth-vector-and-fiber-bundles/index.html#smooth-vector-bundle","scope":"mathlib/overview/geometry/smooth-vector-and-fiber-bundles","status":"mathlib","summary":"When `B` is a manifold with respect to a model `IB` and `E` is a topological vector bundle over `B` with fibers isomorphic to `F`, then `ContMDiffVectorBundle n F E IB` registers that the bundle is `C^n`, in the sense of having `C^n` transition functions. This is a mixin, not carrying any new data.","title":"(smooth) vector bundle"},{"id":"mathlib/overview/geometry/smooth-vector-and-fiber-bundles/smooth-trivial-vector-bundle","kind":"node","parent":444,"result":true,"route":"../roadmap/mathlib/overview/geometry/smooth-vector-and-fiber-bundles/index.html#smooth-trivial-vector-bundle","scope":"mathlib/overview/geometry/smooth-vector-and-fiber-bundles","status":"mathlib","summary":"A trivial vector bundle over a manifold is a `C^n` vector bundle.","title":"(smooth) trivial vector bundle"},{"id":"mathlib/overview/geometry/smooth-vector-and-fiber-bundles/smooth-direct-sum-of-vector-bundles","kind":"node","parent":444,"result":true,"route":"../roadmap/mathlib/overview/geometry/smooth-vector-and-fiber-bundles/index.html#smooth-direct-sum-of-vector-bundles","scope":"mathlib/overview/geometry/smooth-vector-and-fiber-bundles","status":"mathlib","summary":"The direct sum of two `C^n` vector bundles over the same base is a `C^n` vector bundle.","title":"(smooth) direct sum of vector bundles"},{"id":"mathlib/overview/geometry/smooth-vector-and-fiber-bundles/smooth-pullback-vector-bundle","kind":"node","parent":444,"result":true,"route":"../roadmap/mathlib/overview/geometry/smooth-vector-and-fiber-bundles/index.html#smooth-pullback-vector-bundle","scope":"mathlib/overview/geometry/smooth-vector-and-fiber-bundles","status":"mathlib","summary":"For a `C^n` vector bundle `E` over a manifold `B` and a `C^n` map `f : B' \u2192 B`, the pullback vector bundle `f *\u1d56 E` is a `C^n` vector bundle.","title":"(smooth) pullback vector bundle"},{"id":"mathlib/overview/geometry/smooth-vector-and-fiber-bundles/smooth-hom-bundle","kind":"node","parent":444,"result":true,"route":"../roadmap/mathlib/overview/geometry/smooth-vector-and-fiber-bundles/index.html#smooth-hom-bundle","scope":"mathlib/overview/geometry/smooth-vector-and-fiber-bundles","status":"mathlib","summary":"Mathlib states this as `ContMDiffVectorBundle.continuousLinearMap`.","title":"(smooth) Hom bundle"},{"id":"mathlib/overview/geometry/smooth-vector-and-fiber-bundles/covariant-derivatives","kind":"node","parent":444,"result":true,"route":"../roadmap/mathlib/overview/geometry/smooth-vector-and-fiber-bundles/index.html#covariant-derivatives","scope":"mathlib/overview/geometry/smooth-vector-and-fiber-bundles","status":"mathlib","summary":"Bundled global covariant derivative on a vector bundle. Caution, the argument order is nonstandard: `cov \u03c3 x (X x)` corresponds to `\u2207_X \u03c3 x` on paper.","title":"Covariant derivatives"},{"data_url":"scopes/mathlib/overview/geometry/riemannian-manifolds.json","id":"mathlib/overview/geometry/riemannian-manifolds","kind":"scope","parent":408,"result":false,"route":"../roadmap/mathlib/overview/geometry/riemannian-manifolds/index.html","scope":"mathlib/overview/geometry/riemannian-manifolds","status":"mathlib","title":"Riemannian manifolds"},{"id":"mathlib/overview/geometry/riemannian-manifolds/riemannian-bundle-metric","kind":"node","parent":451,"result":true,"route":"../roadmap/mathlib/overview/geometry/riemannian-manifolds/index.html#riemannian-bundle-metric","scope":"mathlib/overview/geometry/riemannian-manifolds","status":"mathlib","summary":"A family of inner product space structures on the fibers of a fiber bundle, defining the same topology as the already existing one. This family is not assumed to be continuous or smooth: to guarantee continuity, resp. smoothness, of the inner product as a function of the base point, use `ContinuousRiemannianMetric` or `ContMDiffRiemannianMetric`.","title":"Riemannian/bundle metric"},{"id":"mathlib/overview/geometry/riemannian-manifolds/smooth-bundle-metric","kind":"node","parent":451,"result":true,"route":"../roadmap/mathlib/overview/geometry/riemannian-manifolds/index.html#smooth-bundle-metric","scope":"mathlib/overview/geometry/riemannian-manifolds","status":"mathlib","summary":"A family of inner product space structures on the fibers of a fiber bundle, defining the same topology as the already existing one, and varying continuously with the base point. See also `ContinuousRiemannianMetric` for a continuous version.","title":"Smooth bundle metric"},{"id":"mathlib/overview/geometry/riemannian-manifolds/riemannian-manifold","kind":"node","parent":451,"result":true,"route":"../roadmap/mathlib/overview/geometry/riemannian-manifolds/index.html#riemannian-manifold","scope":"mathlib/overview/geometry/riemannian-manifolds","status":"mathlib","summary":"Consider a manifold in which the tangent spaces are already endowed with an inner product, and the space is already endowed with an extended distance. We say that this is a Riemannian manifold if the distance is given by the infimum of the lengths of `C^1` paths, measured using the norm in the tangent spaces.","title":"Riemannian manifold"},{"data_url":"scopes/mathlib/overview/geometry/algebraic-geometry.json","id":"mathlib/overview/geometry/algebraic-geometry","kind":"scope","parent":408,"result":false,"route":"../roadmap/mathlib/overview/geometry/algebraic-geometry/index.html","scope":"mathlib/overview/geometry/algebraic-geometry","status":"mathlib","title":"Algebraic geometry"},{"id":"mathlib/overview/geometry/algebraic-geometry/prime-spectrum","kind":"node","parent":455,"result":true,"route":"../roadmap/mathlib/overview/geometry/algebraic-geometry/index.html#prime-spectrum","scope":"mathlib/overview/geometry/algebraic-geometry","status":"mathlib","summary":"The prime spectrum of a commutative (semi)ring `R` is the type of all prime ideals of `R`.","title":"Prime spectrum"},{"id":"mathlib/overview/geometry/algebraic-geometry/zariski-topology","kind":"node","parent":455,"result":true,"route":"../roadmap/mathlib/overview/geometry/algebraic-geometry/index.html#zariski-topology","scope":"mathlib/overview/geometry/algebraic-geometry","status":"mathlib","summary":"The Zariski topology on the prime spectrum of a commutative (semi)ring is defined via the closed sets of the topology: they are exactly those sets that are the zero locus of a subset of the ring.","title":"Zariski topology"},{"id":"mathlib/overview/geometry/algebraic-geometry/locally-ringed-space","kind":"node","parent":455,"result":true,"route":"../roadmap/mathlib/overview/geometry/algebraic-geometry/index.html#locally-ringed-space","scope":"mathlib/overview/geometry/algebraic-geometry","status":"mathlib","summary":"A `LocallyRingedSpace` is a topological space equipped with a sheaf of commutative rings such that all the stalks are local rings.","title":"Locally ringed space"},{"id":"mathlib/overview/geometry/algebraic-geometry/scheme","kind":"node","parent":455,"result":true,"route":"../roadmap/mathlib/overview/geometry/algebraic-geometry/index.html#scheme","scope":"mathlib/overview/geometry/algebraic-geometry","status":"mathlib","summary":"We define `Scheme` as an `X : LocallyRingedSpace`, along with a proof that every point has an open neighbourhood `U` so that the restriction of `X` to `U` is isomorphic, as a locally ringed space, to `Spec.toLocallyRingedSpace.obj (op R)` for some `R : CommRingCat`.","title":"Scheme"},{"id":"mathlib/overview/geometry/algebraic-geometry/nullstellensatz","kind":"node","parent":455,"result":true,"route":"../roadmap/mathlib/overview/geometry/algebraic-geometry/index.html#nullstellensatz","scope":"mathlib/overview/geometry/algebraic-geometry","status":"mathlib","summary":"Main statement of the Nullstellensatz","title":"Nullstellensatz"},{"data_url":"scopes/mathlib/overview/geometry/convex-geometry.json","id":"mathlib/overview/geometry/convex-geometry","kind":"scope","parent":408,"result":false,"route":"../roadmap/mathlib/overview/geometry/convex-geometry/index.html","scope":"mathlib/overview/geometry/convex-geometry","status":"mathlib","title":"Convex geometry"},{"id":"mathlib/overview/geometry/convex-geometry/convex-cone","kind":"node","parent":461,"result":true,"route":"../roadmap/mathlib/overview/geometry/convex-geometry/index.html#convex-cone","scope":"mathlib/overview/geometry/convex-geometry","status":"mathlib","summary":"A convex cone is a subset `s` of an `R`-module such that `a \u2022 x + b \u2022 y \u2208 s` whenever `a, b > 0` and `x, y \u2208 s`.","title":"Convex cone"},{"id":"mathlib/overview/geometry/convex-geometry/salient-cone","kind":"node","parent":461,"result":true,"route":"../roadmap/mathlib/overview/geometry/convex-geometry/index.html#salient-cone","scope":"mathlib/overview/geometry/convex-geometry","status":"mathlib","summary":"A convex cone is salient if it doesn't include `x` and `-x` for any nonzero `x`.","title":"Salient cone"},{"id":"mathlib/overview/geometry/convex-geometry/positive-cone","kind":"node","parent":461,"result":true,"route":"../roadmap/mathlib/overview/geometry/convex-geometry/index.html#positive-cone","scope":"mathlib/overview/geometry/convex-geometry","status":"mathlib","summary":"The positive cone is the convex cone formed by the set of nonnegative elements in an ordered module.","title":"Positive cone"},{"id":"mathlib/overview/geometry/convex-geometry/pointed-cone","kind":"node","parent":461,"result":true,"route":"../roadmap/mathlib/overview/geometry/convex-geometry/index.html#pointed-cone","scope":"mathlib/overview/geometry/convex-geometry","status":"mathlib","summary":"A pointed cone is a submodule of a module with scalars restricted to being nonnegative.","title":"Pointed cone"},{"id":"mathlib/overview/geometry/convex-geometry/dual-cone","kind":"node","parent":461,"result":true,"route":"../roadmap/mathlib/overview/geometry/convex-geometry/index.html#dual-cone","scope":"mathlib/overview/geometry/convex-geometry","status":"mathlib","summary":"The dual cone of a set `s` with respect to a bilinear pairing `p` is the cone consisting of all points `y` such that for all points `x \u2208 s` we have `0 \u2264 p x y`.","title":"Dual cone"},{"id":"mathlib/overview/geometry/convex-geometry/minimal-tensor-product","kind":"node","parent":461,"result":true,"route":"../roadmap/mathlib/overview/geometry/convex-geometry/index.html#minimal-tensor-product","scope":"mathlib/overview/geometry/convex-geometry","status":"mathlib","summary":"The minimal tensor product of two cones is given by all conical combinations of elementary tensor products `x \u2297\u209c y` with `x \u2208 C\u2081` and `y \u2208 C\u2082`.","title":"Minimal tensor product"},{"id":"mathlib/overview/geometry/convex-geometry/maximal-tensor-product","kind":"node","parent":461,"result":true,"route":"../roadmap/mathlib/overview/geometry/convex-geometry/index.html#maximal-tensor-product","scope":"mathlib/overview/geometry/convex-geometry","status":"mathlib","summary":"The maximal tensor product of two cones is the dual (pointed cone) of the minimal tensor product of the dual cones.","title":"Maximal tensor product"},{"data_url":"scopes/mathlib/overview/combinatorics.json","id":"mathlib/overview/combinatorics","kind":"scope","parent":2,"result":false,"route":"../roadmap/mathlib/overview/combinatorics/index.html","scope":"mathlib/overview/combinatorics","status":"mathlib","title":"Combinatorics"},{"data_url":"scopes/mathlib/overview/combinatorics/pigeonhole-principles.json","id":"mathlib/overview/combinatorics/pigeonhole-principles","kind":"scope","parent":469,"result":false,"route":"../roadmap/mathlib/overview/combinatorics/pigeonhole-principles/index.html","scope":"mathlib/overview/combinatorics/pigeonhole-principles","status":"mathlib","title":"Pigeonhole principles"},{"id":"mathlib/overview/combinatorics/pigeonhole-principles/finite","kind":"node","parent":470,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/pigeonhole-principles/index.html#finite","scope":"mathlib/overview/combinatorics/pigeonhole-principles","status":"mathlib","summary":"The pigeonhole principle for finitely many pigeons and pigeonholes. This is the `Fintype` version of `Finset.exists_ne_map_eq_of_card_lt_of_maps_to`.","title":"Finite"},{"id":"mathlib/overview/combinatorics/pigeonhole-principles/infinite","kind":"node","parent":470,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/pigeonhole-principles/index.html#infinite","scope":"mathlib/overview/combinatorics/pigeonhole-principles","status":"mathlib","summary":"The strong pigeonhole principle for infinitely many pigeons in finitely many pigeonholes. If there are infinitely many pigeons in finitely many pigeonholes, then there is a pigeonhole with infinitely many pigeons.","title":"Infinite"},{"id":"mathlib/overview/combinatorics/pigeonhole-principles/strong-pigeonhole-principle","kind":"node","parent":470,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/pigeonhole-principles/index.html#strong-pigeonhole-principle","scope":"mathlib/overview/combinatorics/pigeonhole-principles","status":"mathlib","summary":"The strong pigeonhole principle for finitely many pigeons and pigeonholes. There is a pigeonhole with at least as many pigeons as the ceiling of the average number of pigeons across all pigeonholes. (\"The maximum is at least the mean\" specialized to integers.)","title":"Strong pigeonhole principle"},{"data_url":"scopes/mathlib/overview/combinatorics/transversals.json","id":"mathlib/overview/combinatorics/transversals","kind":"scope","parent":469,"result":false,"route":"../roadmap/mathlib/overview/combinatorics/transversals/index.html","scope":"mathlib/overview/combinatorics/transversals","status":"mathlib","title":"Transversals"},{"id":"mathlib/overview/combinatorics/transversals/halls-marriage-theorem","kind":"node","parent":474,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/transversals/index.html#halls-marriage-theorem","scope":"mathlib/overview/combinatorics/transversals","status":"mathlib","summary":"This is the version of Hall's Marriage Theorem in terms of indexed families of finite sets `t : \u03b9 \u2192 Finset \u03b1`. It states that there is a set of distinct representatives if and only if every union of `k` of the sets has at least `k` elements.","title":"Hall's marriage theorem"},{"data_url":"scopes/mathlib/overview/combinatorics/enumerative-combinatorics.json","id":"mathlib/overview/combinatorics/enumerative-combinatorics","kind":"scope","parent":469,"result":false,"route":"../roadmap/mathlib/overview/combinatorics/enumerative-combinatorics/index.html","scope":"mathlib/overview/combinatorics/enumerative-combinatorics","status":"mathlib","title":"Enumerative combinatorics"},{"id":"mathlib/overview/combinatorics/enumerative-combinatorics/catalan-numbers","kind":"node","parent":476,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/enumerative-combinatorics/index.html#catalan-numbers","scope":"mathlib/overview/combinatorics/enumerative-combinatorics","status":"mathlib","summary":"The recursive definition of the sequence of Catalan numbers: `catalan (n + 1) = \u2211 i : Fin n.succ, catalan i * catalan (n - i)`","title":"Catalan numbers"},{"id":"mathlib/overview/combinatorics/enumerative-combinatorics/bell-numbers","kind":"node","parent":476,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/enumerative-combinatorics/index.html#bell-numbers","scope":"mathlib/overview/combinatorics/enumerative-combinatorics","status":"mathlib","summary":"Number of partitions of a set of cardinality `m.sum` whose parts have cardinalities given by `m`","title":"Bell numbers"},{"id":"mathlib/overview/combinatorics/enumerative-combinatorics/dyck-words","kind":"node","parent":476,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/enumerative-combinatorics/index.html#dyck-words","scope":"mathlib/overview/combinatorics/enumerative-combinatorics","status":"mathlib","summary":"A Dyck word is a list of `DyckStep`s with as many `U`s as `D`s and with every prefix having at least as many `U`s as `D`s.","title":"Dyck words"},{"data_url":"scopes/mathlib/overview/combinatorics/set-families.json","id":"mathlib/overview/combinatorics/set-families","kind":"scope","parent":469,"result":false,"route":"../roadmap/mathlib/overview/combinatorics/set-families/index.html","scope":"mathlib/overview/combinatorics/set-families","status":"mathlib","title":"Set families"},{"id":"mathlib/overview/combinatorics/set-families/four-functions-theorem","kind":"node","parent":480,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/set-families/index.html#four-functions-theorem","scope":"mathlib/overview/combinatorics/set-families","status":"mathlib","summary":"The Four Functions Theorem, aka Ahlswede-Daykin Inequality.","title":"Four functions theorem"},{"id":"mathlib/overview/combinatorics/set-families/harris-kleitman-inequality","kind":"node","parent":480,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/set-families/index.html#harris-kleitman-inequality","scope":"mathlib/overview/combinatorics/set-families","status":"mathlib","summary":"Harris-Kleitman inequality: Any two lower sets of finsets correlate.","title":"Harris-Kleitman inequality"},{"id":"mathlib/overview/combinatorics/set-families/ahlswede-zhang-identity","kind":"node","parent":480,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/set-families/index.html#ahlswede-zhang-identity","scope":"mathlib/overview/combinatorics/set-families","status":"mathlib","summary":"The Ahlswede-Zhang Identity.","title":"Ahlswede-Zhang identity"},{"id":"mathlib/overview/combinatorics/set-families/lym-inequality","kind":"node","parent":480,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/set-families/index.html#lym-inequality","scope":"mathlib/overview/combinatorics/set-families","status":"mathlib","summary":"The Lubell-Yamamoto-Meshalkin inequality, also known as the LYM inequality.","title":"LYM inequality"},{"id":"mathlib/overview/combinatorics/set-families/sperners-theorem","kind":"node","parent":480,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/set-families/index.html#sperners-theorem","scope":"mathlib/overview/combinatorics/set-families","status":"mathlib","summary":"Sperner's theorem. The size of an antichain in `Finset \u03b1` is bounded by the size of the maximal layer in `Finset \u03b1`. This precisely means that `Finset \u03b1` is a Sperner order.","title":"Sperner's theorem"},{"id":"mathlib/overview/combinatorics/set-families/maximum-size-of-an-intersecting-family","kind":"node","parent":480,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/set-families/index.html#maximum-size-of-an-intersecting-family","scope":"mathlib/overview/combinatorics/set-families","status":"mathlib","summary":"Mathlib states this as `Set.Intersecting.is_max_iff_card_eq`.","title":"Maximum size of an intersecting family"},{"id":"mathlib/overview/combinatorics/set-families/kleitmans-theorem","kind":"node","parent":480,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/set-families/index.html#kleitmans-theorem","scope":"mathlib/overview/combinatorics/set-families","status":"mathlib","summary":"Kleitman's theorem. An intersecting family on `n` elements contains at most `2\u207f\u207b\u00b9` sets, and each further intersecting family takes at most half of the sets that are in no previous family.","title":"Kleitman's theorem"},{"id":"mathlib/overview/combinatorics/set-families/vc-dimension","kind":"node","parent":480,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/set-families/index.html#vc-dimension","scope":"mathlib/overview/combinatorics/set-families","status":"mathlib","summary":"The Vapnik-Chervonenkis dimension of a set family is the maximal size of a set it shatters.","title":"VC dimension"},{"id":"mathlib/overview/combinatorics/set-families/sauer-shelah-lemma","kind":"node","parent":480,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/set-families/index.html#sauer-shelah-lemma","scope":"mathlib/overview/combinatorics/set-families","status":"mathlib","summary":"The Sauer-Shelah lemma.","title":"Sauer-Shelah lemma"},{"id":"mathlib/overview/combinatorics/set-families/kruskal-katona-theorem","kind":"node","parent":480,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/set-families/index.html#kruskal-katona-theorem","scope":"mathlib/overview/combinatorics/set-families","status":"mathlib","summary":"The Kruskal-Katona theorem.","title":"Kruskal-Katona theorem"},{"data_url":"scopes/mathlib/overview/combinatorics/graph-theory.json","id":"mathlib/overview/combinatorics/graph-theory","kind":"scope","parent":469,"result":false,"route":"../roadmap/mathlib/overview/combinatorics/graph-theory/index.html","scope":"mathlib/overview/combinatorics/graph-theory","status":"mathlib","title":"Graph theory"},{"id":"mathlib/overview/combinatorics/graph-theory/simple-graph","kind":"node","parent":491,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/graph-theory/index.html#simple-graph","scope":"mathlib/overview/combinatorics/graph-theory","status":"mathlib","summary":"A simple graph is an irreflexive symmetric relation `Adj` on a vertex type `V`. The relation describes which pairs of vertices are adjacent. There is exactly one edge for every pair of adjacent vertices; see `SimpleGraph.edgeSet` for the corresponding edge set.","title":"Simple graph"},{"id":"mathlib/overview/combinatorics/graph-theory/degree-sum-formula","kind":"node","parent":491,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/graph-theory/index.html#degree-sum-formula","scope":"mathlib/overview/combinatorics/graph-theory","status":"mathlib","summary":"The degree-sum formula. This is also known as the handshaking lemma, which might more specifically refer to `SimpleGraph.even_card_odd_degree_vertices`.","title":"Degree-sum formula"},{"id":"mathlib/overview/combinatorics/graph-theory/matching","kind":"node","parent":491,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/graph-theory/index.html#matching","scope":"mathlib/overview/combinatorics/graph-theory","status":"mathlib","summary":"The subgraph `M` of `G` is a matching if every vertex of `M` is incident to exactly one edge in `M`. We say that the vertices in `M.support` are *matched* or *saturated*.","title":"Matching"},{"id":"mathlib/overview/combinatorics/graph-theory/adjacency-matrix","kind":"node","parent":491,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/graph-theory/index.html#adjacency-matrix","scope":"mathlib/overview/combinatorics/graph-theory","status":"mathlib","summary":"`adjMatrix G \u03b1` is the matrix `A` such that `A i j = (1 : \u03b1)` if `i` and `j` are adjacent in the simple graph `G`, and otherwise `A i j = 0`.","title":"Adjacency matrix"},{"id":"mathlib/overview/combinatorics/graph-theory/strongly-regular-graphs","kind":"node","parent":491,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/graph-theory/index.html#strongly-regular-graphs","scope":"mathlib/overview/combinatorics/graph-theory","status":"mathlib","summary":"A graph is strongly regular with parameters `n k \u2113 \u03bc` if","title":"Strongly regular graphs"},{"id":"mathlib/overview/combinatorics/graph-theory/turans-theorem","kind":"node","parent":491,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/graph-theory/index.html#turans-theorem","scope":"mathlib/overview/combinatorics/graph-theory","status":"mathlib","summary":"Tur\u00e1n's theorem. `turanGraph n r` is, up to isomorphism, the unique `r + 1`-cliquefree Tur\u00e1n-maximal graph on `n` vertices.","title":"Tur\u00e1n's theorem"},{"id":"mathlib/overview/combinatorics/graph-theory/regularity-lemma","kind":"node","parent":491,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/graph-theory/index.html#regularity-lemma","scope":"mathlib/overview/combinatorics/graph-theory","status":"mathlib","summary":"Effective Szemer\u00e9di Regularity Lemma: For any sufficiently large graph, there is an `\u03b5`-uniform equipartition of bounded size (where the bound does not depend on the graph).","title":"Regularity lemma"},{"id":"mathlib/overview/combinatorics/graph-theory/triangle-counting-lemma","kind":"node","parent":491,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/graph-theory/index.html#triangle-counting-lemma","scope":"mathlib/overview/combinatorics/graph-theory","status":"mathlib","summary":"The Triangle Counting Lemma. If `G` is a graph and `s`, `t`, `u` are disjoint sets of vertices such that each pair is `\u03b5`-uniform and `2 * \u03b5`-dense, then `G` contains at least `(1 - 2 * \u03b5) * \u03b5 ^ 3 * |s| * |t| * |u|` triangles.","title":"Triangle counting lemma"},{"id":"mathlib/overview/combinatorics/graph-theory/triangle-removal-lemma","kind":"node","parent":491,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/graph-theory/index.html#triangle-removal-lemma","scope":"mathlib/overview/combinatorics/graph-theory","status":"mathlib","summary":"Triangle Removal Lemma. If there are not too many triangles (on the order of `(card \u03b1)^3`) then they can all be removed by removing a few edges (on the order of `(card \u03b1)^2`).","title":"Triangle removal lemma"},{"data_url":"scopes/mathlib/overview/combinatorics/ramsey-theory.json","id":"mathlib/overview/combinatorics/ramsey-theory","kind":"scope","parent":469,"result":false,"route":"../roadmap/mathlib/overview/combinatorics/ramsey-theory/index.html","scope":"mathlib/overview/combinatorics/ramsey-theory","status":"mathlib","title":"Ramsey theory"},{"id":"mathlib/overview/combinatorics/ramsey-theory/van-der-waerden-theorem","kind":"node","parent":501,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/ramsey-theory/index.html#van-der-waerden-theorem","scope":"mathlib/overview/combinatorics/ramsey-theory","status":"mathlib","summary":"A generalization of Van der Waerden's theorem: if `M` is a finitely colored commutative monoid, and `S` is a finite subset, then there exists a monochromatic homothetic copy of `S`.","title":"Van der Waerden theorem"},{"id":"mathlib/overview/combinatorics/ramsey-theory/hales-jewett-theorem","kind":"node","parent":501,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/ramsey-theory/index.html#hales-jewett-theorem","scope":"mathlib/overview/combinatorics/ramsey-theory","status":"mathlib","summary":"The Hales-Jewett theorem: For any finite types `\u03b1` and `\u03ba`, there exists a finite type `\u03b9` such that whenever the hypercube `\u03b9 \u2192 \u03b1` is `\u03ba`-colored, there is a monochromatic combinatorial line.","title":"Hales-Jewett theorem"},{"id":"mathlib/overview/combinatorics/ramsey-theory/hindmans-theorem","kind":"node","parent":501,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/ramsey-theory/index.html#hindmans-theorem","scope":"mathlib/overview/combinatorics/ramsey-theory","status":"mathlib","summary":"The strong form of Hindman's theorem: in any finite cover of an FS-set, one the parts contains an FS-set.","title":"Hindman's theorem (Ramsey theory)"},{"data_url":"scopes/mathlib/overview/combinatorics/additive-combinatorics.json","id":"mathlib/overview/combinatorics/additive-combinatorics","kind":"scope","parent":469,"result":false,"route":"../roadmap/mathlib/overview/combinatorics/additive-combinatorics/index.html","scope":"mathlib/overview/combinatorics/additive-combinatorics","status":"mathlib","title":"Additive combinatorics"},{"id":"mathlib/overview/combinatorics/additive-combinatorics/sets-without-arithmetic-progressions","kind":"node","parent":505,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/additive-combinatorics/index.html#sets-without-arithmetic-progressions","scope":"mathlib/overview/combinatorics/additive-combinatorics","status":"mathlib","summary":"A set is 3AP-free if it does not contain any non-trivial arithmetic progression of length three.","title":"Sets without arithmetic progressions"},{"id":"mathlib/overview/combinatorics/additive-combinatorics/roths-theorem","kind":"node","parent":505,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/additive-combinatorics/index.html#roths-theorem","scope":"mathlib/overview/combinatorics/additive-combinatorics","status":"mathlib","summary":"Roth's theorem for `\u2115`.","title":"Roth's theorem"},{"id":"mathlib/overview/combinatorics/additive-combinatorics/corners-theorem","kind":"node","parent":505,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/additive-combinatorics/index.html#corners-theorem","scope":"mathlib/overview/combinatorics/additive-combinatorics","status":"mathlib","summary":"The corners theorem for `\u2115`.","title":"Corners theorem"},{"id":"mathlib/overview/combinatorics/additive-combinatorics/behrends-construction","kind":"node","parent":505,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/additive-combinatorics/index.html#behrends-construction","scope":"mathlib/overview/combinatorics/additive-combinatorics","status":"mathlib","summary":"Mathlib states this as `Behrend.roth_lower_bound`.","title":"Behrend's construction"},{"id":"mathlib/overview/combinatorics/additive-combinatorics/doubling-constant","kind":"node","parent":505,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/additive-combinatorics/index.html#doubling-constant","scope":"mathlib/overview/combinatorics/additive-combinatorics","status":"mathlib","summary":"The doubling constant `\u03c3[A, B]` of two finsets `A` and `B` in a group is `|A + B| / |A|`.","title":"Doubling constant"},{"id":"mathlib/overview/combinatorics/additive-combinatorics/ruzsa-covering-lemma","kind":"node","parent":505,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/additive-combinatorics/index.html#ruzsa-covering-lemma","scope":"mathlib/overview/combinatorics/additive-combinatorics","status":"mathlib","summary":"Ruzsa's covering lemma for sets. See also `Finset.ruzsa_covering_add`.","title":"Ruzsa covering lemma"},{"id":"mathlib/overview/combinatorics/additive-combinatorics/ruzsa-triangle-inequality","kind":"node","parent":505,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/additive-combinatorics/index.html#ruzsa-triangle-inequality","scope":"mathlib/overview/combinatorics/additive-combinatorics","status":"mathlib","summary":"Ruzsa's triangle inequality. Subtraction version.","title":"Ruzsa triangle inequality"},{"id":"mathlib/overview/combinatorics/additive-combinatorics/plunnecke-petridis-inequality","kind":"node","parent":505,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/additive-combinatorics/index.html#plunnecke-petridis-inequality","scope":"mathlib/overview/combinatorics/additive-combinatorics","status":"mathlib","summary":"Mathlib states this as `Finset.pluennecke_petridis_inequality_add`.","title":"Pl\u00fcnnecke-Petridis inequality"},{"id":"mathlib/overview/combinatorics/additive-combinatorics/small-tripling-implies-small-powers","kind":"node","parent":505,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/additive-combinatorics/index.html#small-tripling-implies-small-powers","scope":"mathlib/overview/combinatorics/additive-combinatorics","status":"mathlib","summary":"If `A` is symmetric \\(`A\u207b\u00b9 = A`\\) and has small tripling, then `A` has small powers, in the sense that `|A ^ m|` is at most `|A|` times a constant exponential in `m`.","title":"Small tripling implies small powers"},{"id":"mathlib/overview/combinatorics/additive-combinatorics/dyson-transform","kind":"node","parent":505,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/additive-combinatorics/index.html#dyson-transform","scope":"mathlib/overview/combinatorics/additive-combinatorics","status":"mathlib","summary":"The Dyson e-transform. Turns `(s, t)` into `(s \u222a e +\u1d65 t, t \u2229 -e +\u1d65 s)`. This reduces the sum of the two sets.","title":"Dyson transform"},{"id":"mathlib/overview/combinatorics/additive-combinatorics/cauchy-davenport-theorem","kind":"node","parent":505,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/additive-combinatorics/index.html#cauchy-davenport-theorem","scope":"mathlib/overview/combinatorics/additive-combinatorics","status":"mathlib","summary":"The Cauchy-Davenport Theorem. If `s`, `t` are nonempty sets in `\u2124/p\u2124`, then the size of `s + t` is lower-bounded by `|s| + |t| - 1`, unless this quantity is greater than `p`.","title":"Cauchy-Davenport theorem"},{"id":"mathlib/overview/combinatorics/additive-combinatorics/erdos-ginzburg-ziv-theorem","kind":"node","parent":505,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/additive-combinatorics/index.html#erdos-ginzburg-ziv-theorem","scope":"mathlib/overview/combinatorics/additive-combinatorics","status":"mathlib","summary":"The Erd\u0151s\u2013Ginzburg\u2013Ziv theorem for `\u2124/n\u2124`.","title":"Erd\u0151s\u2013Ginzburg\u2013Ziv theorem"},{"id":"mathlib/overview/combinatorics/additive-combinatorics/additive-energy","kind":"node","parent":505,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/additive-combinatorics/index.html#additive-energy","scope":"mathlib/overview/combinatorics/additive-combinatorics","status":"mathlib","summary":"The additive energy `E[s, t]` of two finsets `s` and `t` in a group is the number of quadruples `(a\u2081, a\u2082, b\u2081, b\u2082) \u2208 s \u00d7 s \u00d7 t \u00d7 t` such that `a\u2081 + b\u2081 = a\u2082 + b\u2082`.","title":"Additive energy"},{"id":"mathlib/overview/combinatorics/additive-combinatorics/freiman-homomorphisms","kind":"node","parent":505,"result":true,"route":"../roadmap/mathlib/overview/combinatorics/additive-combinatorics/index.html#freiman-homomorphisms","scope":"mathlib/overview/combinatorics/additive-combinatorics","status":"mathlib","summary":"An additive `n`-Freiman homomorphism from a set `A` to a set `B` is a map which preserves sums of `n` elements.","title":"Freiman homomorphisms"},{"data_url":"scopes/mathlib/overview/dynamics.json","id":"mathlib/overview/dynamics","kind":"scope","parent":2,"result":false,"route":"../roadmap/mathlib/overview/dynamics/index.html","scope":"mathlib/overview/dynamics","status":"mathlib","title":"Dynamics"},{"data_url":"scopes/mathlib/overview/dynamics/circle-dynamics.json","id":"mathlib/overview/dynamics/circle-dynamics","kind":"scope","parent":520,"result":false,"route":"../roadmap/mathlib/overview/dynamics/circle-dynamics/index.html","scope":"mathlib/overview/dynamics/circle-dynamics","status":"mathlib","title":"Circle dynamics"},{"id":"mathlib/overview/dynamics/circle-dynamics/translation-number","kind":"node","parent":521,"result":true,"route":"../roadmap/mathlib/overview/dynamics/circle-dynamics/index.html#translation-number","scope":"mathlib/overview/dynamics/circle-dynamics","status":"mathlib","summary":"The translation number of a `CircleDeg1Lift`, $\u03c4(f)=\\lim_{n\u2192\u221e}\\frac{f^n(x)-x}{n}$. We use an auxiliary sequence `\\frac{f^{2^n}(0)}{2^n}` to define `\u03c4(f)` because some proofs are simpler this way.","title":"Translation number"},{"id":"mathlib/overview/dynamics/circle-dynamics/translation-numbers-define-a-group-action-up-to","kind":"node","parent":521,"result":true,"route":"../roadmap/mathlib/overview/dynamics/circle-dynamics/index.html#translation-numbers-define-a-group-action-up-to","scope":"mathlib/overview/dynamics/circle-dynamics","status":"mathlib","summary":"Consider two actions `f\u2081 f\u2082 : G \u2192* CircleDeg1Lift` of a group on the real line by lifts of orientation-preserving circle homeomorphisms. Suppose that for each `g : G` the homeomorphisms `f\u2081 g` and `f\u2082 g` have equal rotation numbers. Then there exists `F : CircleDeg1Lift` such that `F * f\u2081 g = f\u2082 g * F` for all `g : G`.","title":"Translation numbers define a group action up to semiconjugacy"},{"id":"mathlib/overview/dynamics/circle-dynamics/translation-number-defines-a-homeomorphism-up-to","kind":"node","parent":521,"result":true,"route":"../roadmap/mathlib/overview/dynamics/circle-dynamics/index.html#translation-number-defines-a-homeomorphism-up-to","scope":"mathlib/overview/dynamics/circle-dynamics","status":"mathlib","summary":"If two lifts of circle homeomorphisms have the same translation number, then they are semiconjugate by a `CircleDeg1Lift`. This version uses assumptions `bijective f\u2081` and `bijective f\u2082` to assume that `f\u2081` and `f\u2082` are homeomorphisms.","title":"Translation number defines a homeomorphism up to semiconjugacy"},{"data_url":"scopes/mathlib/overview/dynamics/general-theory.json","id":"mathlib/overview/dynamics/general-theory","kind":"scope","parent":520,"result":false,"route":"../roadmap/mathlib/overview/dynamics/general-theory/index.html","scope":"mathlib/overview/dynamics/general-theory","status":"mathlib","title":"General theory"},{"id":"mathlib/overview/dynamics/general-theory/omega-limit-sets","kind":"node","parent":525,"result":true,"route":"../roadmap/mathlib/overview/dynamics/general-theory/index.html#omega-limit-sets","scope":"mathlib/overview/dynamics/general-theory","status":"mathlib","summary":"The \u03c9-limit of a set `s` under `\u03d5` with respect to a filter `f` is `\u22c2 u \u2208 f, cl (\u03d5 u s)`.","title":"Omega-limit sets"},{"id":"mathlib/overview/dynamics/general-theory/fixed-points","kind":"node","parent":525,"result":true,"route":"../roadmap/mathlib/overview/dynamics/general-theory/index.html#fixed-points","scope":"mathlib/overview/dynamics/general-theory","status":"mathlib","summary":"The set of fixed points of a map `f : \u03b1 \u2192 \u03b1`.","title":"Fixed points"},{"id":"mathlib/overview/dynamics/general-theory/periodic-points","kind":"node","parent":525,"result":true,"route":"../roadmap/mathlib/overview/dynamics/general-theory/index.html#periodic-points","scope":"mathlib/overview/dynamics/general-theory","status":"mathlib","summary":"The set of periodic points of a map `f : \u03b1 \u2192 \u03b1`.","title":"Periodic points"},{"data_url":"scopes/mathlib/overview/data-structures.json","id":"mathlib/overview/data-structures","kind":"scope","parent":2,"result":false,"route":"../roadmap/mathlib/overview/data-structures/index.html","scope":"mathlib/overview/data-structures","status":"mathlib","title":"Data structures"},{"data_url":"scopes/mathlib/overview/data-structures/list-like-structures.json","id":"mathlib/overview/data-structures/list-like-structures","kind":"scope","parent":529,"result":false,"route":"../roadmap/mathlib/overview/data-structures/list-like-structures/index.html","scope":"mathlib/overview/data-structures/list-like-structures","status":"mathlib","title":"List-like structures"},{"id":"mathlib/overview/data-structures/list-like-structures/list","kind":"node","parent":530,"result":true,"route":"../roadmap/mathlib/overview/data-structures/list-like-structures/index.html#list","scope":"mathlib/overview/data-structures/list-like-structures","status":"mathlib","summary":"Linked lists: ordered lists, in which each element has a reference to the next element.","title":"List"},{"id":"mathlib/overview/data-structures/list-like-structures/array","kind":"node","parent":530,"result":true,"route":"../roadmap/mathlib/overview/data-structures/list-like-structures/index.html#array","scope":"mathlib/overview/data-structures/list-like-structures","status":"mathlib","summary":"`Array \u03b1` is the type of dynamic arrays with elements from `\u03b1`. This type has special support in the runtime.","title":"Array"},{"id":"mathlib/overview/data-structures/list-like-structures/difference-list","kind":"node","parent":530,"result":true,"route":"../roadmap/mathlib/overview/data-structures/list-like-structures/index.html#difference-list","scope":"mathlib/overview/data-structures/list-like-structures","status":"mathlib","summary":"A difference List is a Function that, given a List, returns the original contents of the difference List prepended to the given List. This structure supports `O(1)` `append` and `push` operations on lists, making it useful for append-heavy uses such as logging and pretty printing.","title":"Difference list"},{"id":"mathlib/overview/data-structures/list-like-structures/lazy-list","kind":"node","parent":530,"result":true,"route":"../roadmap/mathlib/overview/data-structures/list-like-structures/index.html#lazy-list","scope":"mathlib/overview/data-structures/list-like-structures","status":"mathlib","summary":"A monadic lazy list, controlled by an arbitrary monad.","title":"Lazy list"},{"id":"mathlib/overview/data-structures/list-like-structures/stream","kind":"node","parent":530,"result":true,"route":"../roadmap/mathlib/overview/data-structures/list-like-structures/index.html#stream","scope":"mathlib/overview/data-structures/list-like-structures","status":"mathlib","summary":"A stream `Stream' \u03b1` is an infinite sequence of elements of `\u03b1`.","title":"Stream"},{"id":"mathlib/overview/data-structures/list-like-structures/sequence","kind":"node","parent":530,"result":true,"route":"../roadmap/mathlib/overview/data-structures/list-like-structures/index.html#sequence","scope":"mathlib/overview/data-structures/list-like-structures","status":"mathlib","summary":"`Seq \u03b1` is the type of possibly infinite lists (referred here as sequences). It is encoded as an infinite stream of options such that if `f n = none`, then `f m = none` for all `m \u2265 n`.","title":"Sequence"},{"id":"mathlib/overview/data-structures/list-like-structures/weak-sequence","kind":"node","parent":530,"result":true,"route":"../roadmap/mathlib/overview/data-structures/list-like-structures/index.html#weak-sequence","scope":"mathlib/overview/data-structures/list-like-structures","status":"mathlib","summary":"Weak sequences.","title":"Weak sequence"},{"data_url":"scopes/mathlib/overview/data-structures/sets.json","id":"mathlib/overview/data-structures/sets","kind":"scope","parent":529,"result":false,"route":"../roadmap/mathlib/overview/data-structures/sets/index.html","scope":"mathlib/overview/data-structures/sets","status":"mathlib","title":"Sets"},{"id":"mathlib/overview/data-structures/sets/set","kind":"node","parent":538,"result":true,"route":"../roadmap/mathlib/overview/data-structures/sets/index.html#set","scope":"mathlib/overview/data-structures/sets","status":"mathlib","summary":"A set is a collection of elements of some type `\u03b1`.","title":"Set"},{"id":"mathlib/overview/data-structures/sets/finite-set","kind":"node","parent":538,"result":true,"route":"../roadmap/mathlib/overview/data-structures/sets/index.html#finite-set","scope":"mathlib/overview/data-structures/sets","status":"mathlib","summary":"`Finset \u03b1` is the type of finite sets of elements of `\u03b1`. It is implemented as a multiset (a list up to permutation) which has no duplicate elements.","title":"Finite set"},{"id":"mathlib/overview/data-structures/sets/multiset","kind":"node","parent":538,"result":true,"route":"../roadmap/mathlib/overview/data-structures/sets/index.html#multiset","scope":"mathlib/overview/data-structures/sets","status":"mathlib","summary":"`Multiset \u03b1` is the quotient of `List \u03b1` by list permutation. The result is a type of finite sets with duplicates allowed.","title":"Multiset"},{"id":"mathlib/overview/data-structures/sets/ordered-set","kind":"node","parent":538,"result":true,"route":"../roadmap/mathlib/overview/data-structures/sets/index.html#ordered-set","scope":"mathlib/overview/data-structures/sets","status":"mathlib","summary":"An `Ordset \u03b1` is a finite set of values, represented as a tree. The operations on this type maintain that the tree is balanced and correctly stores subtree sizes at each level. The correctness property of the tree is baked into the type, so all operations on this type are correct by construction.","title":"Ordered set"},{"data_url":"scopes/mathlib/overview/data-structures/maps.json","id":"mathlib/overview/data-structures/maps","kind":"scope","parent":529,"result":false,"route":"../roadmap/mathlib/overview/data-structures/maps/index.html","scope":"mathlib/overview/data-structures/maps","status":"mathlib","title":"Maps"},{"id":"mathlib/overview/data-structures/maps/key-value-map","kind":"node","parent":543,"result":true,"route":"../roadmap/mathlib/overview/data-structures/maps/index.html#key-value-map","scope":"mathlib/overview/data-structures/maps","status":"mathlib","summary":"`AList \u03b2` is a key-value map stored as a `List` (i.e. a linked list). It is a wrapper around certain `List` functions with the added constraint that the list have unique keys.","title":"Key-value map"},{"id":"mathlib/overview/data-structures/maps/hash-map","kind":"node","parent":543,"result":true,"route":"../roadmap/mathlib/overview/data-structures/maps/index.html#hash-map","scope":"mathlib/overview/data-structures/maps","status":"mathlib","summary":"Hash maps.","title":"Hash map"},{"id":"mathlib/overview/data-structures/maps/finitely-supported-function","kind":"node","parent":543,"result":true,"route":"../roadmap/mathlib/overview/data-structures/maps/index.html#finitely-supported-function","scope":"mathlib/overview/data-structures/maps","status":"mathlib","summary":"`Finsupp \u03b1 M`, denoted `\u03b1 \u2192\u2080 M`, is the type of functions `f : \u03b1 \u2192 M` such that `f x = 0` for all but finitely many `x`.","title":"Finitely supported function"},{"id":"mathlib/overview/data-structures/maps/finite-map","kind":"node","parent":543,"result":true,"route":"../roadmap/mathlib/overview/data-structures/maps/index.html#finite-map","scope":"mathlib/overview/data-structures/maps","status":"mathlib","summary":"`Finmap \u03b2` is the type of finite maps over a multiset. It is effectively a quotient of `AList \u03b2` by permutation of the underlying list.","title":"Finite map"},{"data_url":"scopes/mathlib/overview/data-structures/trees.json","id":"mathlib/overview/data-structures/trees","kind":"scope","parent":529,"result":false,"route":"../roadmap/mathlib/overview/data-structures/trees/index.html","scope":"mathlib/overview/data-structures/trees","status":"mathlib","title":"Trees"},{"id":"mathlib/overview/data-structures/trees/tree","kind":"node","parent":548,"result":true,"route":"../roadmap/mathlib/overview/data-structures/trees/index.html#tree","scope":"mathlib/overview/data-structures/trees","status":"mathlib","summary":"A binary tree with values stored in non-leaf nodes.","title":"Tree"},{"id":"mathlib/overview/data-structures/trees/size-balanced-binary-search-tree","kind":"node","parent":548,"result":true,"route":"../roadmap/mathlib/overview/data-structures/trees/index.html#size-balanced-binary-search-tree","scope":"mathlib/overview/data-structures/trees","status":"mathlib","summary":"An `Ordnode \u03b1` is a finite set of values, represented as a tree. The operations on this type maintain that the tree is balanced and correctly stores subtree sizes at each level.","title":"Size-balanced binary search tree"},{"id":"mathlib/overview/data-structures/trees/w-type","kind":"node","parent":548,"result":true,"route":"../roadmap/mathlib/overview/data-structures/trees/index.html#w-type","scope":"mathlib/overview/data-structures/trees","status":"mathlib","summary":"Given `\u03b2 : \u03b1 \u2192 Type*`, `WType \u03b2` is the type of finitely branching trees where nodes are labeled by elements of `\u03b1` and the children of a node labeled `a` are indexed by elements of `\u03b2 a`.","title":"W type"},{"data_url":"scopes/mathlib/overview/logic-and-computation.json","id":"mathlib/overview/logic-and-computation","kind":"scope","parent":2,"result":false,"route":"../roadmap/mathlib/overview/logic-and-computation/index.html","scope":"mathlib/overview/logic-and-computation","status":"mathlib","title":"Logic and computation"},{"data_url":"scopes/mathlib/overview/logic-and-computation/computability.json","id":"mathlib/overview/logic-and-computation/computability","kind":"scope","parent":552,"result":false,"route":"../roadmap/mathlib/overview/logic-and-computation/computability/index.html","scope":"mathlib/overview/logic-and-computation/computability","status":"mathlib","title":"Computability"},{"id":"mathlib/overview/logic-and-computation/computability/primitive-recursive-function","kind":"node","parent":553,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/computability/index.html#primitive-recursive-function","scope":"mathlib/overview/logic-and-computation/computability","status":"mathlib","summary":"`Primrec f` means `f` is primitive recursive (after encoding its input and output as natural numbers).","title":"Primitive recursive function"},{"id":"mathlib/overview/logic-and-computation/computability/ackermann-function","kind":"node","parent":553,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/computability/index.html#ackermann-function","scope":"mathlib/overview/logic-and-computation/computability","status":"mathlib","summary":"The two-argument Ackermann function, defined so that","title":"Ackermann function"},{"id":"mathlib/overview/logic-and-computation/computability/turing-machine","kind":"node","parent":553,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/computability/index.html#turing-machine","scope":"mathlib/overview/logic-and-computation/computability","status":"mathlib","summary":"A Post-Turing machine with symbol type `\u0393` and label type `\u039b` is a function which, given the current state `q : \u039b` and the tape head `a : \u0393`, either halts (returns `none`) or returns a new state `q' : \u039b` and a `Stmt` describing what to do, either a move left or right, or a write command.","title":"Turing machine"},{"id":"mathlib/overview/logic-and-computation/computability/halting-problem","kind":"node","parent":553,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/computability/index.html#halting-problem","scope":"mathlib/overview/logic-and-computation/computability","status":"mathlib","summary":"The Halting problem is not computable","title":"Halting problem"},{"id":"mathlib/overview/logic-and-computation/computability/rice-theorem","kind":"node","parent":553,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/computability/index.html#rice-theorem","scope":"mathlib/overview/logic-and-computation/computability","status":"mathlib","summary":"Rice's Theorem","title":"Rice theorem"},{"id":"mathlib/overview/logic-and-computation/computability/turing-degrees","kind":"node","parent":553,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/computability/index.html#turing-degrees","scope":"mathlib/overview/logic-and-computation/computability","status":"mathlib","summary":"Turing degrees are the equivalence classes of partial functions under Turing equivalence.","title":"Turing degrees"},{"data_url":"scopes/mathlib/overview/logic-and-computation/computability/computable-function.json","id":"mathlib/overview/logic-and-computation/computability/computable-function","kind":"scope","parent":553,"result":false,"route":"../roadmap/mathlib/overview/logic-and-computation/computability/computable-function/index.html","scope":"mathlib/overview/logic-and-computation/computability/computable-function","status":"mathlib","title":"Computable function"},{"id":"mathlib/overview/logic-and-computation/computability/computable-function/in-terms-of-general-recursive-functions","kind":"node","parent":560,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/computability/computable-function/index.html#in-terms-of-general-recursive-functions","scope":"mathlib/overview/logic-and-computation/computability/computable-function","status":"mathlib","summary":"Computable functions `\u03b1 \u2192 \u03c3` between `Primcodable` types: a function is computable if and only if it is partially recursive (as a partial function)","title":"In terms of general recursive functions"},{"id":"mathlib/overview/logic-and-computation/computability/computable-function/in-terms-of-turing-machines","kind":"node","parent":560,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/computability/computable-function/index.html#in-terms-of-turing-machines","scope":"mathlib/overview/logic-and-computation/computability/computable-function","status":"mathlib","summary":"A Turing machine + a proof it outputs `f`.","title":"In terms of Turing machines"},{"data_url":"scopes/mathlib/overview/logic-and-computation/complexity.json","id":"mathlib/overview/logic-and-computation/complexity","kind":"scope","parent":552,"result":false,"route":"../roadmap/mathlib/overview/logic-and-computation/complexity/index.html","scope":"mathlib/overview/logic-and-computation/complexity","status":"mathlib","title":"Complexity"},{"id":"mathlib/overview/logic-and-computation/complexity/polynomial-time-computation","kind":"node","parent":563,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/complexity/index.html#polynomial-time-computation","scope":"mathlib/overview/logic-and-computation/complexity","status":"mathlib","summary":"A Turing machine + a polynomial time function + a proof it outputs `f` in at most `time(input.length)` steps.","title":"Polynomial-time computation"},{"data_url":"scopes/mathlib/overview/logic-and-computation/formal-languages.json","id":"mathlib/overview/logic-and-computation/formal-languages","kind":"scope","parent":552,"result":false,"route":"../roadmap/mathlib/overview/logic-and-computation/formal-languages/index.html","scope":"mathlib/overview/logic-and-computation/formal-languages","status":"mathlib","title":"Formal languages"},{"id":"mathlib/overview/logic-and-computation/formal-languages/language","kind":"node","parent":565,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/formal-languages/index.html#language","scope":"mathlib/overview/logic-and-computation/formal-languages","status":"mathlib","summary":"A language is a set of strings over an alphabet.","title":"Language"},{"id":"mathlib/overview/logic-and-computation/formal-languages/ardens-lemma","kind":"node","parent":565,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/formal-languages/index.html#ardens-lemma","scope":"mathlib/overview/logic-and-computation/formal-languages","status":"mathlib","summary":"Arden's lemma","title":"Arden's lemma"},{"id":"mathlib/overview/logic-and-computation/formal-languages/regular-language","kind":"node","parent":565,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/formal-languages/index.html#regular-language","scope":"mathlib/overview/logic-and-computation/formal-languages","status":"mathlib","summary":"A regular language is a language that is defined by a DFA with finite states.","title":"Regular language"},{"id":"mathlib/overview/logic-and-computation/formal-languages/dfa","kind":"node","parent":565,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/formal-languages/index.html#dfa","scope":"mathlib/overview/logic-and-computation/formal-languages","status":"mathlib","summary":"A DFA is a set of states (`\u03c3`), a transition function from state to state labelled by the alphabet (`step`), a starting state (`start`) and a set of acceptance states (`accept`).","title":"DFA"},{"id":"mathlib/overview/logic-and-computation/formal-languages/nfa","kind":"node","parent":565,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/formal-languages/index.html#nfa","scope":"mathlib/overview/logic-and-computation/formal-languages","status":"mathlib","summary":"An NFA is a set of states (`\u03c3`), a transition function from state to state labelled by the alphabet (`step`), a set of starting states (`start`) and a set of acceptance states (`accept`). Note the transition function sends a state to a `Set` of states. These are the states that it may be sent to.","title":"NFA"},{"id":"mathlib/overview/logic-and-computation/formal-languages/varepsilon-nfa","kind":"node","parent":565,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\varepsilon\\)"],["text","-NFA"]],"source":"$\\varepsilon$-NFA"},"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/formal-languages/index.html#varepsilon-nfa","scope":"mathlib/overview/logic-and-computation/formal-languages","status":"mathlib","summary":"An `\u03b5NFA` is a set of states (`\u03c3`), a transition function from state to state labelled by the alphabet (`step`), a starting state (`start`) and a set of acceptance states (`accept`). Note the transition function sends a state to a `Set` of states and can make \u03b5-transitions by inputting `none`. Since this definition allows for Automata with infinite states, a `Fintype` instance must be supplied for true `\u03b5NFA`'s.","title":"$\\varepsilon$-NFA"},{"id":"mathlib/overview/logic-and-computation/formal-languages/regular-expression","kind":"node","parent":565,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/formal-languages/index.html#regular-expression","scope":"mathlib/overview/logic-and-computation/formal-languages","status":"mathlib","summary":"This is the definition of regular expressions. The names used here are meant to mirror the definition of a Kleene algebra.","title":"Regular expression"},{"id":"mathlib/overview/logic-and-computation/formal-languages/context-free-language","kind":"node","parent":565,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/formal-languages/index.html#context-free-language","scope":"mathlib/overview/logic-and-computation/formal-languages","status":"mathlib","summary":"Context-free languages are defined by context-free grammars.","title":"Context-free language"},{"id":"mathlib/overview/logic-and-computation/formal-languages/context-free-grammar","kind":"node","parent":565,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/formal-languages/index.html#context-free-grammar","scope":"mathlib/overview/logic-and-computation/formal-languages","status":"mathlib","summary":"Context-free grammar that generates words over the alphabet `T` (a type of terminals).","title":"Context-free grammar"},{"data_url":"scopes/mathlib/overview/logic-and-computation/set-theory.json","id":"mathlib/overview/logic-and-computation/set-theory","kind":"scope","parent":552,"result":false,"route":"../roadmap/mathlib/overview/logic-and-computation/set-theory/index.html","scope":"mathlib/overview/logic-and-computation/set-theory","status":"mathlib","title":"Set theory"},{"id":"mathlib/overview/logic-and-computation/set-theory/model-of-zfc","kind":"node","parent":575,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/set-theory/index.html#model-of-zfc","scope":"mathlib/overview/logic-and-computation/set-theory","status":"mathlib","summary":"The ZFC universe of sets consists of the type of pre-sets, quotiented by extensional equivalence.","title":"Model of ZFC"},{"data_url":"scopes/mathlib/overview/logic-and-computation/set-theory/ordinals.json","id":"mathlib/overview/logic-and-computation/set-theory/ordinals","kind":"scope","parent":575,"result":false,"route":"../roadmap/mathlib/overview/logic-and-computation/set-theory/ordinals/index.html","scope":"mathlib/overview/logic-and-computation/set-theory/ordinals","status":"mathlib","title":"Ordinals"},{"id":"mathlib/overview/logic-and-computation/set-theory/ordinals/ordinal","kind":"node","parent":577,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/set-theory/ordinals/index.html#ordinal","scope":"mathlib/overview/logic-and-computation/set-theory/ordinals","status":"mathlib","summary":"`Ordinal.{u}` is the type of well orders in `Type u`, up to order isomorphism.","title":"Ordinal"},{"id":"mathlib/overview/logic-and-computation/set-theory/ordinals/cantor-normal-form","kind":"node","parent":577,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/set-theory/ordinals/index.html#cantor-normal-form","scope":"mathlib/overview/logic-and-computation/set-theory/ordinals","status":"mathlib","summary":"The Cantor normal form of an ordinal `o` is the list of coefficients and exponents in the base-`b` expansion of `o`.","title":"Cantor normal form"},{"id":"mathlib/overview/logic-and-computation/set-theory/ordinals/fundamental-sequences","kind":"node","parent":577,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/set-theory/ordinals/index.html#fundamental-sequences","scope":"mathlib/overview/logic-and-computation/set-theory/ordinals","status":"mathlib","summary":"A fundamental sequence for `o` is a strictly monotonic function `Iio o.cof.ord \u2192 Iio o` with cofinal range. We provide `a = o.cof.ord` explicitly to avoid type rewrites.","title":"Fundamental sequences"},{"id":"mathlib/overview/logic-and-computation/set-theory/ordinals/epsilon-numbers","kind":"node","parent":577,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/set-theory/ordinals/index.html#epsilon-numbers","scope":"mathlib/overview/logic-and-computation/set-theory/ordinals","status":"mathlib","summary":"The epsilon function enumerates the fixed points of `\u03c9 ^ \u2b1d`. This is an abbreviation for `veblen 1`.","title":"Epsilon numbers"},{"id":"mathlib/overview/logic-and-computation/set-theory/ordinals/gamma-numbers","kind":"node","parent":577,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/set-theory/ordinals/index.html#gamma-numbers","scope":"mathlib/overview/logic-and-computation/set-theory/ordinals","status":"mathlib","summary":"The gamma function enumerates the fixed points of `veblen \u00b7 0`.","title":"Gamma numbers"},{"data_url":"scopes/mathlib/overview/logic-and-computation/set-theory/cardinals.json","id":"mathlib/overview/logic-and-computation/set-theory/cardinals","kind":"scope","parent":575,"result":false,"route":"../roadmap/mathlib/overview/logic-and-computation/set-theory/cardinals/index.html","scope":"mathlib/overview/logic-and-computation/set-theory/cardinals","status":"mathlib","title":"Cardinals"},{"id":"mathlib/overview/logic-and-computation/set-theory/cardinals/cardinal","kind":"node","parent":583,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/set-theory/cardinals/index.html#cardinal","scope":"mathlib/overview/logic-and-computation/set-theory/cardinals","status":"mathlib","summary":"`Cardinal.{u}` is the type of cardinal numbers in `Type u`, defined as the quotient of `Type u` by existence of an equivalence (a bijection with explicit inverse).","title":"Cardinal"},{"id":"mathlib/overview/logic-and-computation/set-theory/cardinals/aleph-numbers","kind":"node","parent":583,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/set-theory/cardinals/index.html#aleph-numbers","scope":"mathlib/overview/logic-and-computation/set-theory/cardinals","status":"mathlib","summary":"The `aleph` function gives the infinite cardinals listed by their ordinal index. `aleph 0 = \u2135\u2080`, `aleph 1 = succ \u2135\u2080` is the first uncountable cardinal, and so on.","title":"Aleph numbers"},{"id":"mathlib/overview/logic-and-computation/set-theory/cardinals/beth-numbers","kind":"node","parent":583,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/set-theory/cardinals/index.html#beth-numbers","scope":"mathlib/overview/logic-and-computation/set-theory/cardinals","status":"mathlib","summary":"The Beth function is defined so that `beth 0 = \u2135\u2080`, `beth (succ o) = 2 ^ beth o`, and that for a limit ordinal `o`, `beth o` is the supremum of `beth a` for `a < o`.","title":"Beth numbers"},{"id":"mathlib/overview/logic-and-computation/set-theory/cardinals/cofinality","kind":"node","parent":583,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/set-theory/cardinals/index.html#cofinality","scope":"mathlib/overview/logic-and-computation/set-theory/cardinals","status":"mathlib","summary":"The cofinality of a preorder is the smallest cardinality of a cofinal subset.","title":"Cofinality"},{"data_url":"scopes/mathlib/overview/logic-and-computation/model-theory.json","id":"mathlib/overview/logic-and-computation/model-theory","kind":"scope","parent":552,"result":false,"route":"../roadmap/mathlib/overview/logic-and-computation/model-theory/index.html","scope":"mathlib/overview/logic-and-computation/model-theory","status":"mathlib","title":"Model theory"},{"id":"mathlib/overview/logic-and-computation/model-theory/first-order-structure","kind":"node","parent":588,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/model-theory/index.html#first-order-structure","scope":"mathlib/overview/logic-and-computation/model-theory","status":"mathlib","summary":"A first-order structure on a type `M` consists of interpretations of all the symbols in a given language. Each function of arity `n` is interpreted as a function sending tuples of length `n` (modeled as `(Fin n \u2192 M)`) to `M`, and a relation of arity `n` is a function from tuples of length `n` to `Prop`.","title":"First-order structure"},{"id":"mathlib/overview/logic-and-computation/model-theory/first-order-formula","kind":"node","parent":588,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/model-theory/index.html#first-order-formula","scope":"mathlib/overview/logic-and-computation/model-theory","status":"mathlib","summary":"`Formula \u03b1` is the type of formulas with free variables indexed by `\u03b1` and no bound variables in scope.","title":"First-order formula"},{"id":"mathlib/overview/logic-and-computation/model-theory/satisfiability","kind":"node","parent":588,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/model-theory/index.html#satisfiability","scope":"mathlib/overview/logic-and-computation/model-theory","status":"mathlib","summary":"A theory is satisfiable if a structure models it.","title":"Satisfiability"},{"id":"mathlib/overview/logic-and-computation/model-theory/substructure","kind":"node","parent":588,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/model-theory/index.html#substructure","scope":"mathlib/overview/logic-and-computation/model-theory","status":"mathlib","summary":"A substructure of a structure `M` is a set closed under application of function symbols.","title":"Substructure"},{"id":"mathlib/overview/logic-and-computation/model-theory/definable-set","kind":"node","parent":588,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/model-theory/index.html#definable-set","scope":"mathlib/overview/logic-and-computation/model-theory","status":"mathlib","summary":"A subset of a finite Cartesian product of a structure is definable over a set `A` when membership in the set is given by a first-order formula with parameters from `A`.","title":"Definable set"},{"id":"mathlib/overview/logic-and-computation/model-theory/elementary-embedding","kind":"node","parent":588,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/model-theory/index.html#elementary-embedding","scope":"mathlib/overview/logic-and-computation/model-theory","status":"mathlib","summary":"An elementary embedding of first-order structures is an embedding that commutes with the realizations of formulas.","title":"Elementary embedding"},{"id":"mathlib/overview/logic-and-computation/model-theory/compactness-theorem","kind":"node","parent":588,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/model-theory/index.html#compactness-theorem","scope":"mathlib/overview/logic-and-computation/model-theory","status":"mathlib","summary":"The Compactness Theorem of first-order logic: A theory is satisfiable if and only if it is finitely satisfiable.","title":"Compactness theorem"},{"id":"mathlib/overview/logic-and-computation/model-theory/lowenheim-skolem","kind":"node","parent":588,"result":true,"route":"../roadmap/mathlib/overview/logic-and-computation/model-theory/index.html#lowenheim-skolem","scope":"mathlib/overview/logic-and-computation/model-theory","status":"mathlib","summary":"The L\u00f6wenheim\u2013Skolem Theorem: If `\u03ba` is a cardinal greater than the cardinalities of `L` and an infinite `L`-structure `M`, then there is an elementary embedding in the appropriate direction between then `M` and a structure of cardinality `\u03ba`.","title":"L\u00f6wenheim-Skolem"},{"data_url":"scopes/mathlib/undergraduate.json","id":"mathlib/undergraduate","kind":"scope","parent":1,"result":false,"route":"../roadmap/mathlib/undergraduate/index.html","scope":"mathlib/undergraduate","status":"mathlib","summary":"The undergraduate curriculum Mathlib tracks (docs/undergrad.yaml). Mathlib does not have 167 of its 566 items; they have no article here.","title":"Undergraduate mathematics"},{"data_url":"scopes/mathlib/undergraduate/linear-algebra.json","id":"mathlib/undergraduate/linear-algebra","kind":"scope","parent":597,"result":false,"route":"../roadmap/mathlib/undergraduate/linear-algebra/index.html","scope":"mathlib/undergraduate/linear-algebra","status":"mathlib","title":"Linear algebra"},{"data_url":"scopes/mathlib/undergraduate/linear-algebra/fundamentals.json","id":"mathlib/undergraduate/linear-algebra/fundamentals","kind":"scope","parent":598,"result":false,"route":"../roadmap/mathlib/undergraduate/linear-algebra/fundamentals/index.html","scope":"mathlib/undergraduate/linear-algebra/fundamentals","status":"mathlib","title":"Fundamentals"},{"id":"mathlib/undergraduate/linear-algebra/fundamentals/product-of-vector-spaces","kind":"node","parent":599,"result":true,"route":"../roadmap/mathlib/undergraduate/linear-algebra/fundamentals/index.html#product-of-vector-spaces","scope":"mathlib/undergraduate/linear-algebra/fundamentals","status":"mathlib","summary":"Mathlib states this as `Prod.instModule`.","title":"Product of vector spaces"},{"id":"mathlib/undergraduate/linear-algebra/fundamentals/vector-subspace","kind":"node","parent":599,"result":true,"route":"../roadmap/mathlib/undergraduate/linear-algebra/fundamentals/index.html#vector-subspace","scope":"mathlib/undergraduate/linear-algebra/fundamentals","status":"mathlib","summary":"Subspace of a vector space. Defined to equal `Submodule`.","title":"Vector subspace"},{"id":"mathlib/undergraduate/linear-algebra/fundamentals/sum-of-subspaces","kind":"node","parent":599,"result":true,"route":"../roadmap/mathlib/undergraduate/linear-algebra/fundamentals/index.html#sum-of-subspaces","scope":"mathlib/undergraduate/linear-algebra/fundamentals","status":"mathlib","summary":"Mathlib states this as `Submodule.completeLattice`.","title":"Sum of subspaces"},{"id":"mathlib/undergraduate/linear-algebra/fundamentals/direct-sum","kind":"node","parent":599,"result":true,"route":"../roadmap/mathlib/undergraduate/linear-algebra/fundamentals/index.html#direct-sum","scope":"mathlib/undergraduate/linear-algebra/fundamentals","status":"mathlib","summary":"The `DirectSum` formed by a collection of additive submonoids (or subgroups, or submodules) of `M` is said to be internal if the canonical map `(\u2a01 i, A i) \u2192+ M` is bijective.","title":"Direct sum"},{"id":"mathlib/undergraduate/linear-algebra/fundamentals/complementary-subspaces","kind":"node","parent":599,"result":true,"route":"../roadmap/mathlib/undergraduate/linear-algebra/fundamentals/index.html#complementary-subspaces","scope":"mathlib/undergraduate/linear-algebra/fundamentals","status":"mathlib","summary":"Mathlib states this as `Submodule.exists_isCompl`.","title":"Complementary subspaces"},{"id":"mathlib/undergraduate/linear-algebra/fundamentals/linear-independence","kind":"node","parent":599,"result":true,"route":"../roadmap/mathlib/undergraduate/linear-algebra/fundamentals/index.html#linear-independence","scope":"mathlib/undergraduate/linear-algebra/fundamentals","status":"mathlib","summary":"`LinearIndependent R v` states the family of vectors `v` is linearly independent over `R`.","title":"Linear independence"},{"id":"mathlib/undergraduate/linear-algebra/fundamentals/generating-sets","kind":"node","parent":599,"result":true,"route":"../roadmap/mathlib/undergraduate/linear-algebra/fundamentals/index.html#generating-sets","scope":"mathlib/undergraduate/linear-algebra/fundamentals","status":"mathlib","summary":"The span of a set `s \u2286 M` is the smallest submodule of M that contains `s`.","title":"Generating sets"},{"id":"mathlib/undergraduate/linear-algebra/fundamentals/existence-of-bases","kind":"node","parent":599,"result":true,"route":"../roadmap/mathlib/undergraduate/linear-algebra/fundamentals/index.html#existence-of-bases","scope":"mathlib/undergraduate/linear-algebra/fundamentals","status":"mathlib","summary":"Each vector space has a basis.","title":"Existence of bases"},{"id":"mathlib/undergraduate/linear-algebra/fundamentals/range-of-a-linear-map","kind":"node","parent":599,"result":true,"route":"../roadmap/mathlib/undergraduate/linear-algebra/fundamentals/index.html#range-of-a-linear-map","scope":"mathlib/undergraduate/linear-algebra/fundamentals","status":"mathlib","summary":"The range of a linear map `f : M \u2192 M\u2082` is a submodule of `M\u2082`. See Note [range copy pattern].","title":"Range of a linear map"},{"id":"mathlib/undergraduate/linear-algebra/fundamentals/kernel-of-a-linear-map","kind":"node","parent":599,"result":true,"route":"../roadmap/mathlib/undergraduate/linear-algebra/fundamentals/index.html#kernel-of-a-linear-map","scope":"mathlib/undergraduate/linear-algebra/fundamentals","status":"mathlib","summary":"The kernel of a linear map `f : M \u2192 M\u2082` is defined to be `comap f \u22a5`. This is equivalent to the set of `x : M` such that `f x = 0`. The kernel is a submodule of `M`.","title":"Kernel of a linear map"},{"id":"mathlib/undergraduate/linear-algebra/fundamentals/algebra-of-endomorphisms-of-a-vector-space","kind":"node","parent":599,"result":true,"route":"../roadmap/mathlib/undergraduate/linear-algebra/fundamentals/index.html#algebra-of-endomorphisms-of-a-vector-space","scope":"mathlib/undergraduate/linear-algebra/fundamentals","status":"mathlib","summary":"Mathlib states this as `Module.End.instAlgebra`.","title":"Algebra of endomorphisms of a vector space"},{"data_url":"scopes/mathlib/undergraduate/linear-algebra/duality.json","id":"mathlib/undergraduate/linear-algebra/duality","kind":"scope","parent":598,"result":false,"route":"../roadmap/mathlib/undergraduate/linear-algebra/duality/index.html","scope":"mathlib/undergraduate/linear-algebra/duality","status":"mathlib","title":"Duality"},{"id":"mathlib/undergraduate/linear-algebra/duality/transpose-of-a-linear-map","kind":"node","parent":611,"result":true,"route":"../roadmap/mathlib/undergraduate/linear-algebra/duality/index.html#transpose-of-a-linear-map","scope":"mathlib/undergraduate/linear-algebra/duality","status":"mathlib","summary":"The transposition of linear maps, as a linear map from `M \u2192\u2097[R] M'` to `Dual R M' \u2192\u2097[R] Dual R M`.","title":"Transpose of a linear map"},{"id":"mathlib/undergraduate/linear-algebra/duality/orthogonality","kind":"node","parent":611,"result":true,"route":"../roadmap/mathlib/undergraduate/linear-algebra/duality/index.html#orthogonality","scope":"mathlib/undergraduate/linear-algebra/duality","status":"mathlib","summary":"The `dualAnnihilator` of a submodule `W` is the set of linear maps `\u03c6` such that `\u03c6 w = 0` for all `w \u2208 W`.","title":"Orthogonality"},{"data_url":"scopes/mathlib/undergraduate/linear-algebra/finite-dimensional-vector-spaces.json","id":"mathlib/undergraduate/linear-algebra/finite-dimensional-vector-spaces","kind":"scope","parent":598,"result":false,"route":"../roadmap/mathlib/undergraduate/linear-algebra/finite-dimensional-vector-spaces/index.html","scope":"mathlib/undergraduate/linear-algebra/finite-dimensional-vector-spaces","status":"mathlib","title":"Finite-dimensional vector spaces"},{"id":"mathlib/undergraduate/linear-algebra/finite-dimensional-vector-spaces/rank-of-a-linear-map","kind":"node","parent":614,"result":true,"route":"../roadmap/mathlib/undergraduate/linear-algebra/finite-dimensional-vector-spaces/index.html#rank-of-a-linear-map","scope":"mathlib/undergraduate/linear-algebra/finite-dimensional-vector-spaces","status":"mathlib","summary":"`rank f` is the rank of a `LinearMap` `f`, defined as the dimension of `f.range`.","title":"Rank of a linear map"},{"id":"mathlib/undergraduate/linear-algebra/finite-dimensional-vector-spaces/rank-of-a-set-of-vectors","kind":"node","parent":614,"result":true,"route":"../roadmap/mathlib/undergraduate/linear-algebra/finite-dimensional-vector-spaces/index.html#rank-of-a-set-of-vectors","scope":"mathlib/undergraduate/linear-algebra/finite-dimensional-vector-spaces","status":"mathlib","summary":"The rank of a set of vectors as a natural number.","title":"Rank of a set of vectors"},{"data_url":"scopes/mathlib/undergraduate/linear-algebra/multilinearity.json","id":"mathlib/undergraduate/linear-algebra/multilinearity","kind":"scope","parent":598,"result":false,"route":"../roadmap/mathlib/undergraduate/linear-algebra/multilinearity/index.html","scope":"mathlib/undergraduate/linear-algebra/multilinearity","status":"mathlib","title":"Multilinearity"},{"id":"mathlib/undergraduate/linear-algebra/multilinearity/determinant-of-vectors","kind":"node","parent":617,"result":true,"route":"../roadmap/mathlib/undergraduate/linear-algebra/multilinearity/index.html#determinant-of-vectors","scope":"mathlib/undergraduate/linear-algebra/multilinearity","status":"mathlib","summary":"The determinant of a family of vectors with respect to some basis, as an alternating multilinear map.","title":"Determinant of vectors"},{"id":"mathlib/undergraduate/linear-algebra/multilinearity/determinant-of-endomorphisms","kind":"node","parent":617,"result":true,"route":"../roadmap/mathlib/undergraduate/linear-algebra/multilinearity/index.html#determinant-of-endomorphisms","scope":"mathlib/undergraduate/linear-algebra/multilinearity","status":"mathlib","summary":"The determinant of an endomorphism independent of basis.","title":"Determinant of endomorphisms"},{"id":"mathlib/undergraduate/linear-algebra/multilinearity/special-linear-group","kind":"node","parent":617,"result":true,"route":"../roadmap/mathlib/undergraduate/linear-algebra/multilinearity/index.html#special-linear-group","scope":"mathlib/undergraduate/linear-algebra/multilinearity","status":"mathlib","summary":"`SpecialLinearGroup n R` is the group of `n` by `n` `R`-matrices with determinant equal to 1.","title":"Special linear group"},{"id":"mathlib/undergraduate/linear-algebra/multilinearity/orientation-of-a-r-vector-space","kind":"node","parent":617,"reading_title":{"format":"inline-parts-v1","parts":[["text","Orientation of a "],["math","\\(\\R\\)"],["text","-vector space"]],"source":"Orientation of a $\\R$-vector space"},"result":true,"route":"../roadmap/mathlib/undergraduate/linear-algebra/multilinearity/index.html#orientation-of-a-r-vector-space","scope":"mathlib/undergraduate/linear-algebra/multilinearity","status":"mathlib","summary":"An orientation of a module, intended to be used when `\u03b9` is a `Fintype` with the same cardinality as a basis.","title":"Orientation of a $\\R$-vector space"},{"data_url":"scopes/mathlib/undergraduate/linear-algebra/matrices.json","id":"mathlib/undergraduate/linear-algebra/matrices","kind":"scope","parent":598,"result":false,"route":"../roadmap/mathlib/undergraduate/linear-algebra/matrices/index.html","scope":"mathlib/undergraduate/linear-algebra/matrices","status":"mathlib","title":"Matrices"},{"id":"mathlib/undergraduate/linear-algebra/matrices/change-of-basis","kind":"node","parent":622,"result":true,"route":"../roadmap/mathlib/undergraduate/linear-algebra/matrices/index.html#change-of-basis","scope":"mathlib/undergraduate/linear-algebra/matrices","status":"mathlib","summary":"Mathlib states this as `basis_toMatrix_mul_linearMap_toMatrix_mul_basis_toMatrix`.","title":"Change of basis"},{"id":"mathlib/undergraduate/linear-algebra/matrices/rank-of-a-matrix","kind":"node","parent":622,"result":true,"route":"../roadmap/mathlib/undergraduate/linear-algebra/matrices/index.html#rank-of-a-matrix","scope":"mathlib/undergraduate/linear-algebra/matrices","status":"mathlib","summary":"The rank of a matrix is the rank of its image.","title":"Rank of a matrix"},{"id":"mathlib/undergraduate/linear-algebra/matrices/gaussian-elimination","kind":"node","parent":622,"result":true,"route":"../roadmap/mathlib/undergraduate/linear-algebra/matrices/index.html#gaussian-elimination","scope":"mathlib/undergraduate/linear-algebra/matrices","status":"mathlib","summary":"Any matrix can be reduced to diagonal form by elementary operations.","title":"Gaussian elimination"},{"data_url":"scopes/mathlib/undergraduate/linear-algebra/endomorphism-polynomials.json","id":"mathlib/undergraduate/linear-algebra/endomorphism-polynomials","kind":"scope","parent":598,"result":false,"route":"../roadmap/mathlib/undergraduate/linear-algebra/endomorphism-polynomials/index.html","scope":"mathlib/undergraduate/linear-algebra/endomorphism-polynomials","status":"mathlib","title":"Endomorphism polynomials"},{"id":"mathlib/undergraduate/linear-algebra/endomorphism-polynomials/annihilating-polynomials","kind":"node","parent":626,"result":true,"route":"../roadmap/mathlib/undergraduate/linear-algebra/endomorphism-polynomials/index.html#annihilating-polynomials","scope":"mathlib/undergraduate/linear-algebra/endomorphism-polynomials","status":"mathlib","summary":"`annIdeal R a` is the *annihilating ideal* of all `p : R[X]` such that `p(a) = 0`.","title":"Annihilating polynomials"},{"data_url":"scopes/mathlib/undergraduate/linear-algebra/structure-theory-of-endomorphisms.json","id":"mathlib/undergraduate/linear-algebra/structure-theory-of-endomorphisms","kind":"scope","parent":598,"result":false,"route":"../roadmap/mathlib/undergraduate/linear-algebra/structure-theory-of-endomorphisms/index.html","scope":"mathlib/undergraduate/linear-algebra/structure-theory-of-endomorphisms","status":"mathlib","title":"Structure theory of endomorphisms"},{"id":"mathlib/undergraduate/linear-algebra/structure-theory-of-endomorphisms/invariant-subspaces-of-an-endomorphism","kind":"node","parent":628,"result":true,"route":"../roadmap/mathlib/undergraduate/linear-algebra/structure-theory-of-endomorphisms/index.html#invariant-subspaces-of-an-endomorphism","scope":"mathlib/undergraduate/linear-algebra/structure-theory-of-endomorphisms","status":"mathlib","summary":"Given an endomorphism, `f` of some module, this is the sublattice of all `f`-invariant submodules.","title":"Invariant subspaces of an endomorphism"},{"id":"mathlib/undergraduate/linear-algebra/structure-theory-of-endomorphisms/generalized-eigenspaces","kind":"node","parent":628,"result":true,"route":"../roadmap/mathlib/undergraduate/linear-algebra/structure-theory-of-endomorphisms/index.html#generalized-eigenspaces","scope":"mathlib/undergraduate/linear-algebra/structure-theory-of-endomorphisms","status":"mathlib","summary":"The submodule `genEigenspace f \u03bc k` for a linear map `f`, a scalar `\u03bc`, and a number `k : \u2115\u221e` is the kernel of `(f - \u03bc \u2022 id) ^ k` if `k` is a natural number, or the union of all these kernels if `k = \u221e`. (`k = \u221e` corresponds to Def 8.19 of axler2024.) A generalized eigenspace for some exponent `k` is contained in the generalized eigenspace for exponents larger than `k`.","title":"Generalized eigenspaces"},{"id":"mathlib/undergraduate/linear-algebra/structure-theory-of-endomorphisms/jordan-chevalley-dunford-decomposition","kind":"node","parent":628,"result":true,"route":"../roadmap/mathlib/undergraduate/linear-algebra/structure-theory-of-endomorphisms/index.html#jordan-chevalley-dunford-decomposition","scope":"mathlib/undergraduate/linear-algebra/structure-theory-of-endomorphisms","status":"mathlib","summary":"Jordan-Chevalley-Dunford decomposition: an endomorphism of a finite-dimensional vector space over a perfect field may be written as a sum of nilpotent and semisimple endomorphisms. Moreover these nilpotent and semisimple components are polynomial expressions in the original endomorphism.","title":"Jordan-Chevalley-Dunford decomposition"},{"data_url":"scopes/mathlib/undergraduate/linear-algebra/linear-representations.json","id":"mathlib/undergraduate/linear-algebra/linear-representations","kind":"scope","parent":598,"result":false,"route":"../roadmap/mathlib/undergraduate/linear-algebra/linear-representations/index.html","scope":"mathlib/undergraduate/linear-algebra/linear-representations","status":"mathlib","title":"Linear representations"},{"id":"mathlib/undergraduate/linear-algebra/linear-representations/irreducible-representation","kind":"node","parent":632,"result":true,"route":"../roadmap/mathlib/undergraduate/linear-algebra/linear-representations/index.html#irreducible-representation","scope":"mathlib/undergraduate/linear-algebra/linear-representations","status":"mathlib","summary":"A representation `\u03c1` is irreducible if it is non-trivial and has no proper non-trivial subrepresentations.","title":"Irreducible representation"},{"id":"mathlib/undergraduate/linear-algebra/linear-representations/schurs-lemma","kind":"node","parent":632,"result":true,"route":"../roadmap/mathlib/undergraduate/linear-algebra/linear-representations/index.html#schurs-lemma","scope":"mathlib/undergraduate/linear-algebra/linear-representations","status":"mathlib","summary":"Schur's Lemma: the dimension of the `Hom`-space between two irreducible representation is `0` if they are not isomorphic, and `1` if they are.","title":"Schur's lemma (Linear representations)"},{"data_url":"scopes/mathlib/undergraduate/linear-algebra/exponential.json","id":"mathlib/undergraduate/linear-algebra/exponential","kind":"scope","parent":598,"result":false,"route":"../roadmap/mathlib/undergraduate/linear-algebra/exponential/index.html","scope":"mathlib/undergraduate/linear-algebra/exponential","status":"mathlib","title":"Exponential"},{"id":"mathlib/undergraduate/linear-algebra/exponential/matrix-exponential","kind":"node","parent":635,"result":true,"route":"../roadmap/mathlib/undergraduate/linear-algebra/exponential/index.html#matrix-exponential","scope":"mathlib/undergraduate/linear-algebra/exponential","status":"mathlib","summary":"`NormedSpace.exp : \ud835\udd38 \u2192 \ud835\udd38` is the exponential map. It is defined as the sum of the `FormalMultilinearSeries` `expSeries \u211a \ud835\udd38`.","title":"Matrix exponential"},{"data_url":"scopes/mathlib/undergraduate/group-theory.json","id":"mathlib/undergraduate/group-theory","kind":"scope","parent":597,"result":false,"route":"../roadmap/mathlib/undergraduate/group-theory/index.html","scope":"mathlib/undergraduate/group-theory","status":"mathlib","title":"Group Theory"},{"data_url":"scopes/mathlib/undergraduate/group-theory/basic-definitions.json","id":"mathlib/undergraduate/group-theory/basic-definitions","kind":"scope","parent":637,"result":false,"route":"../roadmap/mathlib/undergraduate/group-theory/basic-definitions/index.html","scope":"mathlib/undergraduate/group-theory/basic-definitions","status":"mathlib","title":"Basic definitions"},{"id":"mathlib/undergraduate/group-theory/basic-definitions/direct-product-of-groups","kind":"node","parent":638,"result":true,"route":"../roadmap/mathlib/undergraduate/group-theory/basic-definitions/index.html#direct-product-of-groups","scope":"mathlib/undergraduate/group-theory/basic-definitions","status":"mathlib","summary":"The direct product of two groups $G$ and $H$ is the group $G \\times H$ whose elements are ordered pairs $(g, h)$ with $g \\in G$ and $h \\in H$, multiplied one coordinate at a time. It is a basic way to build new groups from old ones: large groups are assembled from small ones as products, and many groups are understood by recognizing them as products of simpler factors, as in the classification of finite abelian\u2026","title":"Direct product of groups"},{"id":"mathlib/undergraduate/group-theory/basic-definitions/order-of-an-element","kind":"node","parent":638,"result":true,"route":"../roadmap/mathlib/undergraduate/group-theory/basic-definitions/index.html#order-of-an-element","scope":"mathlib/undergraduate/group-theory/basic-definitions","status":"mathlib","summary":"The order of an element $g$ of a group is the smallest positive integer $n$ such that $g^n = e$, where $e$ is the identity; if no positive power of $g$ equals $e$, the element has infinite order. The order records how long the sequence of powers $g, g^2, g^3, \\dots$ runs before it returns to the identity, and it equals the number of elements of the subgroup generated by $g$. It is a basic numerical invariant of a\u2026","title":"Order of an element"},{"id":"mathlib/undergraduate/group-theory/basic-definitions/stabilizer-of-a-point","kind":"node","parent":638,"result":true,"route":"../roadmap/mathlib/undergraduate/group-theory/basic-definitions/index.html#stabilizer-of-a-point","scope":"mathlib/undergraduate/group-theory/basic-definitions","status":"mathlib","summary":"The stabilizer of a point under a group action is the set of group elements that leave the point where it is. It is always a subgroup of the acting group, and it measures how much symmetry the point retains: the larger the stabilizer, the smaller the orbit. This trade-off, the orbit\u2013stabilizer theorem, turns questions about group actions into questions about subgroups and their indices, and it is the basic counting\u2026","title":"Stabilizer of a point"},{"id":"mathlib/undergraduate/group-theory/basic-definitions/orbit","kind":"node","parent":638,"result":true,"route":"../roadmap/mathlib/undergraduate/group-theory/basic-definitions/index.html#orbit","scope":"mathlib/undergraduate/group-theory/basic-definitions","status":"mathlib","summary":"The orbit of a point under a group action is the set of all points the group can move it to. When a group acts on a set, the orbits cut the set into disjoint pieces, and the size of each piece is tied to the structure of the group by the orbit\u2013stabilizer theorem. Counting arguments built on orbits underlie much of finite group theory, including the class equation, Burnside's lemma and the Sylow theorems.","title":"Orbit"},{"id":"mathlib/undergraduate/group-theory/basic-definitions/quotient-space","kind":"node","parent":638,"result":true,"route":"../roadmap/mathlib/undergraduate/group-theory/basic-definitions/index.html#quotient-space","scope":"mathlib/undergraduate/group-theory/basic-definitions","status":"mathlib","summary":"The quotient space $G/H$ of a group $G$ by a subgroup $H$ is the set of left cosets $aH$ of $H$ in $G$. It is not a group in general, but $G$ acts on it by left multiplication, and this action is the model for every transitive group action: by the orbit-stabilizer theorem, each orbit of an action of $G$ is a copy of the quotient of $G$ by the stabilizer of any of its points. Counting arguments in finite group\u2026","title":"Quotient space (Basic definitions)"},{"id":"mathlib/undergraduate/group-theory/basic-definitions/conjugacy-classes","kind":"node","parent":638,"result":true,"route":"../roadmap/mathlib/undergraduate/group-theory/basic-definitions/index.html#conjugacy-classes","scope":"mathlib/undergraduate/group-theory/basic-definitions","status":"mathlib","summary":"The conjugacy classes of a group are the pieces into which the group splits when elements that differ only by a change of viewpoint, $a$ and $gag^{-1}$, are counted as the same. They are the orbits of the group acting on itself by conjugation. They carry much of the structure of a finite group: the class equation built from their sizes shows that nontrivial finite groups of prime-power order have a nontrivial\u2026","title":"Conjugacy classes"},{"data_url":"scopes/mathlib/undergraduate/group-theory/abelian-group.json","id":"mathlib/undergraduate/group-theory/abelian-group","kind":"scope","parent":637,"result":false,"route":"../roadmap/mathlib/undergraduate/group-theory/abelian-group/index.html","scope":"mathlib/undergraduate/group-theory/abelian-group","status":"mathlib","title":"Abelian group"},{"id":"mathlib/undergraduate/group-theory/abelian-group/complex-roots-of-unity","kind":"node","parent":645,"result":true,"route":"../roadmap/mathlib/undergraduate/group-theory/abelian-group/index.html#complex-roots-of-unity","scope":"mathlib/undergraduate/group-theory/abelian-group","status":"mathlib","summary":"The complex roots of unity are the complex numbers $z$ that satisfy $z^n = 1$ for some positive integer $n$. For each $n \\ge 1$ the equation $z^n = 1$ has exactly $n$ solutions, the numbers $e^{2\\pi i k/n}$ with $0 \\le k < n$, and they form a cyclic group of order $n$ under multiplication. They are the standard model of a finite cyclic group, they describe the regular polygons inscribed in the unit circle, and they\u2026","title":"Complex roots of unity"},{"id":"mathlib/undergraduate/group-theory/abelian-group/primitive-complex-roots-of-unity","kind":"node","parent":645,"result":true,"route":"../roadmap/mathlib/undergraduate/group-theory/abelian-group/index.html#primitive-complex-roots-of-unity","scope":"mathlib/undergraduate/group-theory/abelian-group","status":"mathlib","summary":"For a positive integer $n$, a primitive $n$-th root of unity is a complex number $\\zeta$ with $\\zeta^n = 1$ whose powers $\\zeta, \\zeta^2, \\dots, \\zeta^{n-1}$ all differ from $1$. The primitive $n$-th roots are the generators of the cyclic group of complex $n$-th roots of unity, and they have an explicit description: they are the numbers $e^{2\\pi i k/n}$ with $k$ coprime to $n$. They are the roots of the cyclotomic\u2026","title":"Primitive complex roots of unity"},{"data_url":"scopes/mathlib/undergraduate/group-theory/permutation-group.json","id":"mathlib/undergraduate/group-theory/permutation-group","kind":"scope","parent":637,"result":false,"route":"../roadmap/mathlib/undergraduate/group-theory/permutation-group/index.html","scope":"mathlib/undergraduate/group-theory/permutation-group","status":"mathlib","title":"Permutation group"},{"id":"mathlib/undergraduate/group-theory/permutation-group/decomposition-into-transpositions","kind":"node","parent":648,"result":true,"route":"../roadmap/mathlib/undergraduate/group-theory/permutation-group/index.html#decomposition-into-transpositions","scope":"mathlib/undergraduate/group-theory/permutation-group","status":"mathlib","summary":"The decomposition into transpositions expresses every permutation of a finite set as a product of transpositions, permutations that exchange two elements and fix all the others. It shows that the transpositions generate the symmetric group, which makes them the basic building blocks for computing with permutations and for proving facts about all of them at once. The decomposition is far from unique, but the parity\u2026","title":"Decomposition into transpositions"},{"id":"mathlib/undergraduate/group-theory/permutation-group/decomposition-into-cycles-with-disjoint-support","kind":"node","parent":648,"result":true,"route":"../roadmap/mathlib/undergraduate/group-theory/permutation-group/index.html#decomposition-into-cycles-with-disjoint-support","scope":"mathlib/undergraduate/group-theory/permutation-group","status":"mathlib","summary":"The decomposition into cycles with disjoint support says that every permutation of a finite set is a product of cycles that move pairwise disjoint sets of points, and that these cycles are determined by the permutation up to the order in which they are written. It is the basis of cycle notation, the standard way of writing elements of a permutation group, and it reduces questions about an arbitrary permutation (its\u2026","title":"Decomposition into cycles with disjoint support"},{"id":"mathlib/undergraduate/group-theory/permutation-group/signature","kind":"node","parent":648,"result":true,"route":"../roadmap/mathlib/undergraduate/group-theory/permutation-group/index.html#signature","scope":"mathlib/undergraduate/group-theory/permutation-group","status":"mathlib","summary":"The signature (also called the sign) of a permutation $\\sigma$ of a finite set is the number $\\operatorname{sgn}(\\sigma) \\in \\{1, -1\\}$ that records whether $\\sigma$ is a product of an even or an odd number of transpositions. Permutations of signature $1$ are called even and those of signature $-1$ odd. The signature is a group homomorphism from the symmetric group to the two-element group $\\{1, -1\\}$. It supplies\u2026","title":"Signature"},{"data_url":"scopes/mathlib/undergraduate/group-theory/classical-automorphism-groups.json","id":"mathlib/undergraduate/group-theory/classical-automorphism-groups","kind":"scope","parent":637,"result":false,"route":"../roadmap/mathlib/undergraduate/group-theory/classical-automorphism-groups/index.html","scope":"mathlib/undergraduate/group-theory/classical-automorphism-groups","status":"mathlib","title":"Classical automorphism groups"},{"id":"mathlib/undergraduate/group-theory/classical-automorphism-groups/general-linear-group","kind":"node","parent":652,"result":true,"route":"../roadmap/mathlib/undergraduate/group-theory/classical-automorphism-groups/index.html#general-linear-group","scope":"mathlib/undergraduate/group-theory/classical-automorphism-groups","status":"mathlib","summary":"The general linear group $\\mathrm{GL}_n(R)$ is the group of invertible $n \\times n$ matrices with entries in a ring $R$, under matrix multiplication. Over a field $K$ it is the group of all invertible linear transformations of $K^n$ written in coordinates, so once a basis is chosen, every group acting linearly on an $n$-dimensional vector space over $K$ maps homomorphically into $\\mathrm{GL}_n(K)$. It is the basic\u2026","title":"General linear group (Classical automorphism groups)"},{"id":"mathlib/undergraduate/group-theory/classical-automorphism-groups/orthogonal-group","kind":"node","parent":652,"result":true,"route":"../roadmap/mathlib/undergraduate/group-theory/classical-automorphism-groups/index.html#orthogonal-group","scope":"mathlib/undergraduate/group-theory/classical-automorphism-groups","status":"mathlib","summary":"The orthogonal group $\\mathrm{O}(n)$ is the group of $n \\times n$ matrices whose transpose is their inverse. Over the real numbers these are exactly the matrices of the linear maps of $\\mathbb{R}^n$ that preserve lengths and angles: the rotations, the reflections and their composites. It is one of the classical groups, it is the symmetry group of Euclidean geometry with a fixed origin, and it serves as a basic\u2026","title":"Orthogonal group"},{"id":"mathlib/undergraduate/group-theory/classical-automorphism-groups/special-orthogonal-group","kind":"node","parent":652,"result":true,"route":"../roadmap/mathlib/undergraduate/group-theory/classical-automorphism-groups/index.html#special-orthogonal-group","scope":"mathlib/undergraduate/group-theory/classical-automorphism-groups","status":"mathlib","summary":"The special orthogonal group $\\mathrm{SO}_n(R)$ is the group of $n \\times n$ orthogonal matrices over a commutative ring $R$ whose determinant equals $1$. Over the real numbers it is the group of rotations of $n$-dimensional Euclidean space: the linear maps that preserve lengths, angles and orientation. It is one of the classical matrix groups, next to the general linear, special linear and unitary groups, and its\u2026","title":"Special orthogonal group"},{"id":"mathlib/undergraduate/group-theory/classical-automorphism-groups/unitary-group","kind":"node","parent":652,"result":true,"route":"../roadmap/mathlib/undergraduate/group-theory/classical-automorphism-groups/index.html#unitary-group","scope":"mathlib/undergraduate/group-theory/classical-automorphism-groups","status":"mathlib","summary":"The unitary group $U(n)$ is the group of $n \\times n$ complex matrices $U$ satisfying $U^*U = I$, where $U^*$ is the conjugate transpose of $U$. These are exactly the matrices of the linear maps of $\\mathbb{C}^n$ that preserve the standard Hermitian inner product, so $U(n)$ plays for complex space the role that the orthogonal group plays for real Euclidean space. It is a compact Lie group, and it is basic to linear\u2026","title":"Unitary group"},{"id":"mathlib/undergraduate/group-theory/classical-automorphism-groups/special-unitary-group","kind":"node","parent":652,"result":true,"route":"../roadmap/mathlib/undergraduate/group-theory/classical-automorphism-groups/index.html#special-unitary-group","scope":"mathlib/undergraduate/group-theory/classical-automorphism-groups","status":"mathlib","summary":"The special unitary group $\\mathrm{SU}(n)$ is the group of complex $n \\times n$ unitary matrices whose determinant equals $1$. It is the subgroup of the unitary group $\\mathrm{U}(n)$ cut out by the single condition $\\det A = 1$, the kernel of the determinant, and it is one of the basic examples of a compact Lie group. The group $\\mathrm{SU}(2)$ is a double cover of the rotation group $\\mathrm{SO}(3)$ of\u2026","title":"Special unitary group"},{"data_url":"scopes/mathlib/undergraduate/group-theory/representation-theory-of-finite-groups.json","id":"mathlib/undergraduate/group-theory/representation-theory-of-finite-groups","kind":"scope","parent":637,"result":false,"route":"../roadmap/mathlib/undergraduate/group-theory/representation-theory-of-finite-groups/index.html","scope":"mathlib/undergraduate/group-theory/representation-theory-of-finite-groups","status":"mathlib","title":"Representation theory of finite groups"},{"id":"mathlib/undergraduate/group-theory/representation-theory-of-finite-groups/maschke-theorem","kind":"node","parent":658,"result":true,"route":"../roadmap/mathlib/undergraduate/group-theory/representation-theory-of-finite-groups/index.html#maschke-theorem","scope":"mathlib/undergraduate/group-theory/representation-theory-of-finite-groups","status":"mathlib","summary":"Maschke's theorem says that a representation of a finite group $G$ over a field $k$ whose characteristic does not divide the order of $G$ splits along every invariant subspace: each subrepresentation has an invariant complement. It is the starting point of the representation theory of finite groups. It reduces the study of finite-dimensional representations to that of irreducible representations, and so makes the\u2026","title":"Maschke theorem"},{"data_url":"scopes/mathlib/undergraduate/ring-theory.json","id":"mathlib/undergraduate/ring-theory","kind":"scope","parent":597,"result":false,"route":"../roadmap/mathlib/undergraduate/ring-theory/index.html","scope":"mathlib/undergraduate/ring-theory","status":"mathlib","title":"Ring Theory"},{"data_url":"scopes/mathlib/undergraduate/ring-theory/fundamentals.json","id":"mathlib/undergraduate/ring-theory/fundamentals","kind":"scope","parent":660,"result":false,"route":"../roadmap/mathlib/undergraduate/ring-theory/fundamentals/index.html","scope":"mathlib/undergraduate/ring-theory/fundamentals","status":"mathlib","title":"Fundamentals"},{"id":"mathlib/undergraduate/ring-theory/fundamentals/ring-structure-z","kind":"node","parent":661,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ring structure "],["math","\\(\\Z\\)"]],"source":"Ring structure $\\Z$"},"result":true,"route":"../roadmap/mathlib/undergraduate/ring-theory/fundamentals/index.html#ring-structure-z","scope":"mathlib/undergraduate/ring-theory/fundamentals","status":"mathlib","summary":"Mathlib states this as `Int.instCommRing`.","title":"Ring structure $\\Z$"},{"id":"mathlib/undergraduate/ring-theory/fundamentals/product-of-rings","kind":"node","parent":661,"result":true,"route":"../roadmap/mathlib/undergraduate/ring-theory/fundamentals/index.html#product-of-rings","scope":"mathlib/undergraduate/ring-theory/fundamentals","status":"mathlib","summary":"Mathlib states this as `Pi.ring`.","title":"Product of rings"},{"id":"mathlib/undergraduate/ring-theory/ideals-and-quotients","kind":"node","parent":660,"result":false,"route":"../roadmap/mathlib/undergraduate/ring-theory/ideals-and-quotients/index.html","scope":"mathlib/undergraduate/ring-theory","status":"can_state","title":"Ideals and Quotients"},{"id":"mathlib/undergraduate/ring-theory/algebra","kind":"node","parent":660,"result":false,"route":"../roadmap/mathlib/undergraduate/ring-theory/algebra/index.html","scope":"mathlib/undergraduate/ring-theory","status":"can_state","title":"Algebra"},{"data_url":"scopes/mathlib/undergraduate/ring-theory/divisibility-in-integral-domains.json","id":"mathlib/undergraduate/ring-theory/divisibility-in-integral-domains","kind":"scope","parent":660,"result":false,"route":"../roadmap/mathlib/undergraduate/ring-theory/divisibility-in-integral-domains/index.html","scope":"mathlib/undergraduate/ring-theory/divisibility-in-integral-domains","status":"mathlib","title":"Divisibility in integral domains"},{"id":"mathlib/undergraduate/ring-theory/divisibility-in-integral-domains/invertible-elements","kind":"node","parent":666,"result":true,"route":"../roadmap/mathlib/undergraduate/ring-theory/divisibility-in-integral-domains/index.html#invertible-elements","scope":"mathlib/undergraduate/ring-theory/divisibility-in-integral-domains","status":"mathlib","summary":"`Invertible a` gives a two-sided multiplicative inverse of `a`.","title":"Invertible elements"},{"id":"mathlib/undergraduate/ring-theory/divisibility-in-integral-domains/z-is-a-euclidean-ring","kind":"node","parent":666,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\Z\\)"],["text"," is a Euclidean ring"]],"source":"$\\Z$ is a Euclidean ring"},"result":true,"route":"../roadmap/mathlib/undergraduate/ring-theory/divisibility-in-integral-domains/index.html#z-is-a-euclidean-ring","scope":"mathlib/undergraduate/ring-theory/divisibility-in-integral-domains","status":"mathlib","summary":"Mathlib states this as `Int.euclideanDomain`.","title":"$\\Z$ is a Euclidean ring"},{"id":"mathlib/undergraduate/ring-theory/divisibility-in-integral-domains/congruence-in-z","kind":"node","parent":666,"reading_title":{"format":"inline-parts-v1","parts":[["text","Congruence in "],["math","\\(\\Z\\)"]],"source":"Congruence in $\\Z$"},"result":true,"route":"../roadmap/mathlib/undergraduate/ring-theory/divisibility-in-integral-domains/index.html#congruence-in-z","scope":"mathlib/undergraduate/ring-theory/divisibility-in-integral-domains","status":"mathlib","summary":"`a \u2261 b [ZMOD n]` when `a % n = b % n`.","title":"Congruence in $\\Z$"},{"id":"mathlib/undergraduate/ring-theory/divisibility-in-integral-domains/prime-numbers","kind":"node","parent":666,"result":true,"route":"../roadmap/mathlib/undergraduate/ring-theory/divisibility-in-integral-domains/index.html#prime-numbers","scope":"mathlib/undergraduate/ring-theory/divisibility-in-integral-domains","status":"mathlib","summary":"An element `p` of a commutative monoid with zero (e.g., a ring) is called *prime*, if it's not zero, not a unit, and `p \u2223 a * b \u2192 p \u2223 a \u2228 p \u2223 b` for all `a`, `b`.","title":"Prime numbers"},{"id":"mathlib/undergraduate/ring-theory/divisibility-in-integral-domains/bezouts-identity","kind":"node","parent":666,"result":true,"route":"../roadmap/mathlib/undergraduate/ring-theory/divisibility-in-integral-domains/index.html#bezouts-identity","scope":"mathlib/undergraduate/ring-theory/divisibility-in-integral-domains","status":"mathlib","summary":"B\u00e9zout's lemma","title":"B\u00e9zout's identity"},{"id":"mathlib/undergraduate/ring-theory/divisibility-in-integral-domains/z-n-z-and-its-invertible-elements","kind":"node","parent":666,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\Z/n\\Z\\)"],["text"," and its invertible elements"]],"source":"$\\Z/n\\Z$ and its invertible elements"},"result":true,"route":"../roadmap/mathlib/undergraduate/ring-theory/divisibility-in-integral-domains/index.html#z-n-z-and-its-invertible-elements","scope":"mathlib/undergraduate/ring-theory/divisibility-in-integral-domains","status":"mathlib","summary":"`unitOfCoprime` makes an element of `(ZMod n)\u02e3` given a natural number `x` and a proof that `x` is coprime to `n`","title":"$\\Z/n\\Z$ and its invertible elements"},{"data_url":"scopes/mathlib/undergraduate/ring-theory/polynomial-rings.json","id":"mathlib/undergraduate/ring-theory/polynomial-rings","kind":"scope","parent":660,"result":false,"route":"../roadmap/mathlib/undergraduate/ring-theory/polynomial-rings/index.html","scope":"mathlib/undergraduate/ring-theory/polynomial-rings","status":"mathlib","title":"Polynomial rings"},{"id":"mathlib/undergraduate/ring-theory/polynomial-rings/cyclotomic-polynomials-in-q-x","kind":"node","parent":673,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cyclotomic polynomials in "],["math","\\(\\Q[X]\\)"]],"source":"Cyclotomic polynomials in $\\Q[X]$"},"result":true,"route":"../roadmap/mathlib/undergraduate/ring-theory/polynomial-rings/index.html#cyclotomic-polynomials-in-q-x","scope":"mathlib/undergraduate/ring-theory/polynomial-rings","status":"mathlib","summary":"The `n`-th cyclotomic polynomial with coefficients in `R`.","title":"Cyclotomic polynomials in $\\Q[X]$"},{"id":"mathlib/undergraduate/ring-theory/polynomial-rings/relationship-between-the-coefficients-and-the-ro","kind":"node","parent":673,"result":true,"route":"../roadmap/mathlib/undergraduate/ring-theory/polynomial-rings/index.html#relationship-between-the-coefficients-and-the-ro","scope":"mathlib/undergraduate/ring-theory/polynomial-rings","status":"mathlib","summary":"Vieta's formula for the coefficients and the roots of a polynomial over an integral domain with as many roots as its degree.","title":"Relationship between the coefficients and the roots of a split polynomial"},{"id":"mathlib/undergraduate/ring-theory/polynomial-rings/newtons-identities","kind":"node","parent":673,"result":true,"route":"../roadmap/mathlib/undergraduate/ring-theory/polynomial-rings/index.html#newtons-identities","scope":"mathlib/undergraduate/ring-theory/polynomial-rings","status":"mathlib","summary":"Newton's identities give a recurrence relation for the kth elementary symmetric polynomial in terms of lower degree elementary symmetric polynomials and power sums.","title":"Newton's identities"},{"id":"mathlib/undergraduate/ring-theory/polynomial-rings/polynomial-derivative","kind":"node","parent":673,"result":true,"route":"../roadmap/mathlib/undergraduate/ring-theory/polynomial-rings/index.html#polynomial-derivative","scope":"mathlib/undergraduate/ring-theory/polynomial-rings","status":"mathlib","summary":"`derivative p` is the formal derivative of the polynomial `p`","title":"Polynomial derivative"},{"id":"mathlib/undergraduate/ring-theory/polynomial-rings/decomposition-into-sums-of-homogeneous-polynomia","kind":"node","parent":673,"result":true,"route":"../roadmap/mathlib/undergraduate/ring-theory/polynomial-rings/index.html#decomposition-into-sums-of-homogeneous-polynomia","scope":"mathlib/undergraduate/ring-theory/polynomial-rings","status":"mathlib","summary":"Mathlib states this as `MvPolynomial.sum_homogeneousComponent`.","title":"Decomposition into sums of homogeneous polynomials"},{"id":"mathlib/undergraduate/ring-theory/polynomial-rings/symmetric-polynomials","kind":"node","parent":673,"result":true,"route":"../roadmap/mathlib/undergraduate/ring-theory/polynomial-rings/index.html#symmetric-polynomials","scope":"mathlib/undergraduate/ring-theory/polynomial-rings","status":"mathlib","summary":"A `MvPolynomial \u03c6` is symmetric if it is invariant under permutations of its variables by the `rename` operation","title":"Symmetric polynomials"},{"data_url":"scopes/mathlib/undergraduate/ring-theory/field-theory.json","id":"mathlib/undergraduate/ring-theory/field-theory","kind":"scope","parent":660,"result":false,"route":"../roadmap/mathlib/undergraduate/ring-theory/field-theory/index.html","scope":"mathlib/undergraduate/ring-theory/field-theory","status":"mathlib","title":"Field Theory"},{"id":"mathlib/undergraduate/ring-theory/field-theory/field-q-of-rational-numbers","kind":"node","parent":680,"reading_title":{"format":"inline-parts-v1","parts":[["text","Field "],["math","\\(\\Q\\)"],["text"," of rational numbers"]],"source":"Field $\\Q$ of rational numbers"},"result":true,"route":"../roadmap/mathlib/undergraduate/ring-theory/field-theory/index.html#field-q-of-rational-numbers","scope":"mathlib/undergraduate/ring-theory/field-theory","status":"mathlib","summary":"Mathlib states this as `Rat.instField`.","title":"Field $\\Q$ of rational numbers"},{"id":"mathlib/undergraduate/ring-theory/field-theory/field-r-of-real-numbers","kind":"node","parent":680,"reading_title":{"format":"inline-parts-v1","parts":[["text","Field "],["math","\\(\\R\\)"],["text"," of real numbers"]],"source":"Field $\\R$ of real numbers"},"result":true,"route":"../roadmap/mathlib/undergraduate/ring-theory/field-theory/index.html#field-r-of-real-numbers","scope":"mathlib/undergraduate/ring-theory/field-theory","status":"mathlib","summary":"Mathlib states this as `Real.instField`.","title":"Field $\\R$ of real numbers"},{"id":"mathlib/undergraduate/ring-theory/field-theory/field-c-of-complex-numbers","kind":"node","parent":680,"reading_title":{"format":"inline-parts-v1","parts":[["text","Field "],["math","\\(\\C\\)"],["text"," of complex numbers"]],"source":"Field $\\C$ of complex numbers"},"result":true,"route":"../roadmap/mathlib/undergraduate/ring-theory/field-theory/index.html#field-c-of-complex-numbers","scope":"mathlib/undergraduate/ring-theory/field-theory","status":"mathlib","summary":"Mathlib states this as `Complex.instField`.","title":"Field $\\C$ of complex numbers"},{"id":"mathlib/undergraduate/ring-theory/field-theory/algebraic-elements","kind":"node","parent":680,"result":true,"route":"../roadmap/mathlib/undergraduate/ring-theory/field-theory/index.html#algebraic-elements","scope":"mathlib/undergraduate/ring-theory/field-theory","status":"mathlib","summary":"An element of an R-algebra is algebraic over R if it is a root of a nonzero polynomial with coefficients in R.","title":"Algebraic elements"},{"id":"mathlib/undergraduate/ring-theory/field-theory/transcendental-elements","kind":"node","parent":680,"result":true,"route":"../roadmap/mathlib/undergraduate/ring-theory/field-theory/index.html#transcendental-elements","scope":"mathlib/undergraduate/ring-theory/field-theory","status":"mathlib","summary":"An element of an R-algebra is transcendental over R if it is not algebraic over R.","title":"Transcendental elements"},{"id":"mathlib/undergraduate/ring-theory/field-theory/finite-fields","kind":"node","parent":680,"result":true,"route":"../roadmap/mathlib/undergraduate/ring-theory/field-theory/index.html#finite-fields","scope":"mathlib/undergraduate/ring-theory/field-theory","status":"mathlib","summary":"This file contains basic results about finite fields. Throughout most of this file, `K` denotes a finite field and `q` is notation for the cardinality of `K`.","title":"Finite fields"},{"id":"mathlib/undergraduate/ring-theory/field-theory/rational-fraction-fields-with-one-indeterminate","kind":"node","parent":680,"result":true,"route":"../roadmap/mathlib/undergraduate/ring-theory/field-theory/index.html#rational-fraction-fields-with-one-indeterminate","scope":"mathlib/undergraduate/ring-theory/field-theory","status":"mathlib","summary":"`RatFunc K` is `K(X)`, the field of rational functions over `K`.","title":"Rational fraction fields with one indeterminate over a field"},{"data_url":"scopes/mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space.json","id":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space","kind":"scope","parent":597,"result":false,"route":"../roadmap/mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/index.html","scope":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space","status":"mathlib","title":"Bilinear and Quadratic Forms Over a Vector Space"},{"data_url":"scopes/mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/bilinear-forms.json","id":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/bilinear-forms","kind":"scope","parent":688,"result":false,"route":"../roadmap/mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/bilinear-forms/index.html","scope":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/bilinear-forms","status":"mathlib","title":"Bilinear forms"},{"id":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/bilinear-forms/nondegenerate-forms","kind":"node","parent":689,"result":true,"route":"../roadmap/mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/bilinear-forms/index.html#nondegenerate-forms","scope":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/bilinear-forms","status":"mathlib","summary":"A nondegenerate bilinear form is a bilinear form that is both left-separating and right-separating, i.e. if `B x y = 0` for all `y` then `x = 0`, and if `B x y = 0` for all `x` then `y = 0`.","title":"Nondegenerate forms"},{"id":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/bilinear-forms/change-of-coordinates","kind":"node","parent":689,"result":true,"route":"../roadmap/mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/bilinear-forms/index.html#change-of-coordinates","scope":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/bilinear-forms","status":"mathlib","summary":"Mathlib states this as `LinearMap.BilinForm.toMatrix_comp`.","title":"Change of coordinates"},{"id":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/quadratic-forms","kind":"node","parent":688,"result":false,"route":"../roadmap/mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/quadratic-forms/index.html","scope":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space","status":"can_state","title":"Quadratic forms"},{"data_url":"scopes/mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/orthogonality.json","id":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/orthogonality","kind":"scope","parent":688,"result":false,"route":"../roadmap/mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/orthogonality/index.html","scope":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/orthogonality","status":"mathlib","title":"Orthogonality"},{"id":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/orthogonality/orthogonal-elements","kind":"node","parent":693,"result":true,"route":"../roadmap/mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/orthogonality/index.html#orthogonal-elements","scope":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/orthogonality","status":"mathlib","summary":"A set of vectors `v` is orthogonal with respect to some bilinear form `B` if and only if for all `i \u2260 j`, `B (v i) (v j) = 0`.","title":"Orthogonal elements"},{"id":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/orthogonality/adjoint-endomorphism","kind":"node","parent":693,"result":true,"route":"../roadmap/mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/orthogonality/index.html#adjoint-endomorphism","scope":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/orthogonality","status":"mathlib","summary":"Given the nondegenerate bilinear form `B` and the linear map `\u03c6`, `leftAdjointOfNondegenerate` provides the left adjoint of `\u03c6` with respect to `B`. The lemma proving this property is `BilinForm.isAdjointPairLeftAdjointOfNondegenerate`.","title":"Adjoint endomorphism"},{"id":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/orthogonality/sylvesters-law-of-inertia-existence","kind":"node","parent":693,"result":true,"route":"../roadmap/mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/orthogonality/index.html#sylvesters-law-of-inertia-existence","scope":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/orthogonality","status":"mathlib","summary":"Sylvester's law of inertia: A real quadratic form is equivalent to a weighted sum of squares with the weights being \u00b11 or 0.","title":"Sylvester's law of inertia (existence)"},{"id":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/orthogonality/sylvesters-law-of-inertia-uniqueness","kind":"node","parent":693,"result":true,"route":"../roadmap/mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/orthogonality/index.html#sylvesters-law-of-inertia-uniqueness","scope":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/orthogonality","status":"mathlib","summary":"Uniqueness part of Sylvester's law of inertia (positive part): for any weighted sum of squares equivalent to `Q`, the number of strictly positive weights is equal to `sigPos Q`.","title":"Sylvester's law of inertia (uniqueness)"},{"id":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/orthogonality/complex-classification","kind":"node","parent":693,"result":true,"route":"../roadmap/mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/orthogonality/index.html#complex-classification","scope":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/orthogonality","status":"mathlib","summary":"A nondegenerate quadratic form on an algebraically closed field of characteristic not equal to 2 is equivalent to the sum of squares, i.e. `weightedSumSquares` with weight `fun (i : \u03b9) => 1`.","title":"Complex classification"},{"id":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/orthogonality/gram-schmidt-orthogonalisation","kind":"node","parent":693,"result":true,"route":"../roadmap/mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/orthogonality/index.html#gram-schmidt-orthogonalisation","scope":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/orthogonality","status":"mathlib","summary":"Gram-Schmidt Orthogonalisation: `gramSchmidt` produces an orthogonal system of vectors.","title":"Gram-Schmidt orthogonalisation"},{"data_url":"scopes/mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/euclidean-and-hermitian-spaces.json","id":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/euclidean-and-hermitian-spaces","kind":"scope","parent":688,"result":false,"route":"../roadmap/mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/euclidean-and-hermitian-spaces/index.html","scope":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/euclidean-and-hermitian-spaces","status":"mathlib","title":"Euclidean and Hermitian spaces"},{"id":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/euclidean-and-hermitian-spaces/norm","kind":"node","parent":700,"result":true,"route":"../roadmap/mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/euclidean-and-hermitian-spaces/index.html#norm","scope":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/euclidean-and-hermitian-spaces","status":"mathlib","summary":"(Semi)norm constructed from a `PreInnerProductSpace.Core` structure, defined to be the square root of the scalar product.","title":"Norm"},{"id":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/euclidean-and-hermitian-spaces/orthonormal-bases","kind":"node","parent":700,"result":true,"route":"../roadmap/mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/euclidean-and-hermitian-spaces/index.html#orthonormal-bases","scope":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/euclidean-and-hermitian-spaces","status":"mathlib","summary":"An orthonormal set in a finite-dimensional `InnerProductSpace` is maximal, if and only if it is a basis.","title":"Orthonormal bases"},{"data_url":"scopes/mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/endomorphisms.json","id":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/endomorphisms","kind":"scope","parent":688,"result":false,"route":"../roadmap/mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/endomorphisms/index.html","scope":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/endomorphisms","status":"mathlib","title":"Endomorphisms"},{"id":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/endomorphisms/normal-endomorphism","kind":"node","parent":703,"result":true,"route":"../roadmap/mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/endomorphisms/index.html#normal-endomorphism","scope":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/endomorphisms","status":"mathlib","summary":"An element of a star monoid is normal if it commutes with its adjoint.","title":"Normal endomorphism"},{"id":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/endomorphisms/decomposition-of-an-orthogonal-transformation-as","kind":"node","parent":703,"result":true,"route":"../roadmap/mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/endomorphisms/index.html#decomposition-of-an-orthogonal-transformation-as","scope":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/endomorphisms","status":"mathlib","summary":"The orthogonal group of `F` is generated by reflections.","title":"Decomposition of an orthogonal transformation as a product of reflections"},{"data_url":"scopes/mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/low-dimensions.json","id":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/low-dimensions","kind":"scope","parent":688,"result":false,"route":"../roadmap/mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/low-dimensions/index.html","scope":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/low-dimensions","status":"mathlib","title":"Low dimensions"},{"id":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/low-dimensions/cross-product","kind":"node","parent":706,"result":true,"route":"../roadmap/mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/low-dimensions/index.html#cross-product","scope":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/low-dimensions","status":"mathlib","summary":"The cross product of two vectors in $R^3$ for $R$ a commutative ring.","title":"Cross product"},{"id":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/low-dimensions/triple-product","kind":"node","parent":706,"result":true,"route":"../roadmap/mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/low-dimensions/index.html#triple-product","scope":"mathlib/undergraduate/bilinear-and-quadratic-forms-over-a-vector-space/low-dimensions","status":"mathlib","summary":"The triple product of `u`, `v`, and `w` is equal to the determinant of the matrix with those vectors as its rows.","title":"Triple product"},{"data_url":"scopes/mathlib/undergraduate/affine-and-euclidean-geometry.json","id":"mathlib/undergraduate/affine-and-euclidean-geometry","kind":"scope","parent":597,"result":false,"route":"../roadmap/mathlib/undergraduate/affine-and-euclidean-geometry/index.html","scope":"mathlib/undergraduate/affine-and-euclidean-geometry","status":"mathlib","title":"Affine and Euclidean Geometry"},{"data_url":"scopes/mathlib/undergraduate/affine-and-euclidean-geometry/general-definitions.json","id":"mathlib/undergraduate/affine-and-euclidean-geometry/general-definitions","kind":"scope","parent":709,"result":false,"route":"../roadmap/mathlib/undergraduate/affine-and-euclidean-geometry/general-definitions/index.html","scope":"mathlib/undergraduate/affine-and-euclidean-geometry/general-definitions","status":"mathlib","title":"General definitions"},{"id":"mathlib/undergraduate/affine-and-euclidean-geometry/general-definitions/affine-groups","kind":"node","parent":710,"result":true,"route":"../roadmap/mathlib/undergraduate/affine-and-euclidean-geometry/general-definitions/index.html#affine-groups","scope":"mathlib/undergraduate/affine-and-euclidean-geometry/general-definitions","status":"mathlib","summary":"Mathlib states this as `AffineEquiv.group`.","title":"Affine groups"},{"data_url":"scopes/mathlib/undergraduate/affine-and-euclidean-geometry/convexity.json","id":"mathlib/undergraduate/affine-and-euclidean-geometry/convexity","kind":"scope","parent":709,"result":false,"route":"../roadmap/mathlib/undergraduate/affine-and-euclidean-geometry/convexity/index.html","scope":"mathlib/undergraduate/affine-and-euclidean-geometry/convexity","status":"mathlib","title":"Convexity"},{"id":"mathlib/undergraduate/affine-and-euclidean-geometry/convexity/convex-subsets","kind":"node","parent":712,"result":true,"route":"../roadmap/mathlib/undergraduate/affine-and-euclidean-geometry/convexity/index.html#convex-subsets","scope":"mathlib/undergraduate/affine-and-euclidean-geometry/convexity","status":"mathlib","summary":"Convexity of sets.","title":"Convex subsets"},{"id":"mathlib/undergraduate/affine-and-euclidean-geometry/convexity/convex-hull-of-a-subset-of-an-affine-real-space","kind":"node","parent":712,"result":true,"route":"../roadmap/mathlib/undergraduate/affine-and-euclidean-geometry/convexity/index.html#convex-hull-of-a-subset-of-an-affine-real-space","scope":"mathlib/undergraduate/affine-and-euclidean-geometry/convexity","status":"mathlib","summary":"The convex hull of a set `s` is the minimal convex set that includes `s`.","title":"Convex hull of a subset of an affine real space"},{"id":"mathlib/undergraduate/affine-and-euclidean-geometry/convexity/extreme-point","kind":"node","parent":712,"result":true,"route":"../roadmap/mathlib/undergraduate/affine-and-euclidean-geometry/convexity/index.html#extreme-point","scope":"mathlib/undergraduate/affine-and-euclidean-geometry/convexity","status":"mathlib","summary":"A point `x` is an extreme point of a set `A` if `x` belongs to no open segment with ends in `A`, except for the obvious `openSegment x x`.","title":"Extreme point"},{"data_url":"scopes/mathlib/undergraduate/affine-and-euclidean-geometry/euclidean-affine-spaces.json","id":"mathlib/undergraduate/affine-and-euclidean-geometry/euclidean-affine-spaces","kind":"scope","parent":709,"result":false,"route":"../roadmap/mathlib/undergraduate/affine-and-euclidean-geometry/euclidean-affine-spaces/index.html","scope":"mathlib/undergraduate/affine-and-euclidean-geometry/euclidean-affine-spaces","status":"mathlib","title":"Euclidean affine spaces"},{"id":"mathlib/undergraduate/affine-and-euclidean-geometry/euclidean-affine-spaces/isometries-of-a-euclidean-affine-space","kind":"node","parent":716,"result":true,"route":"../roadmap/mathlib/undergraduate/affine-and-euclidean-geometry/euclidean-affine-spaces/index.html#isometries-of-a-euclidean-affine-space","scope":"mathlib/undergraduate/affine-and-euclidean-geometry/euclidean-affine-spaces","status":"mathlib","summary":"An affine isometric equivalence between two normed vector spaces, denoted `f : P \u2243\u1d43\u2071[\ud835\udd5c] P\u2082`.","title":"Isometries of a Euclidean affine space"},{"id":"mathlib/undergraduate/affine-and-euclidean-geometry/euclidean-affine-spaces/group-of-isometries-of-a-euclidean-affine-space","kind":"node","parent":716,"result":true,"route":"../roadmap/mathlib/undergraduate/affine-and-euclidean-geometry/euclidean-affine-spaces/index.html#group-of-isometries-of-a-euclidean-affine-space","scope":"mathlib/undergraduate/affine-and-euclidean-geometry/euclidean-affine-spaces","status":"mathlib","summary":"The group of affine isometries of a `NormedAddTorsor`, `P`.","title":"Group of isometries of a Euclidean affine space"},{"id":"mathlib/undergraduate/affine-and-euclidean-geometry/euclidean-affine-spaces/inscribed-angle-theorem","kind":"node","parent":716,"result":true,"route":"../roadmap/mathlib/undergraduate/affine-and-euclidean-geometry/euclidean-affine-spaces/index.html#inscribed-angle-theorem","scope":"mathlib/undergraduate/affine-and-euclidean-geometry/euclidean-affine-spaces","status":"mathlib","summary":"Oriented vector angle version of \"angles in same segment are equal\" and \"opposite angles of a cyclic quadrilateral add to \u03c0\", for oriented angles mod \u03c0 (for which those are the same result), represented here as equality of twice the angles.","title":"Inscribed angle theorem"},{"id":"mathlib/undergraduate/affine-and-euclidean-geometry/euclidean-affine-spaces/cocyclicity","kind":"node","parent":716,"result":true,"route":"../roadmap/mathlib/undergraduate/affine-and-euclidean-geometry/euclidean-affine-spaces/index.html#cocyclicity","scope":"mathlib/undergraduate/affine-and-euclidean-geometry/euclidean-affine-spaces","status":"mathlib","summary":"A set of points is concyclic if it is cospherical and coplanar. (Most results are stated directly in terms of `Cospherical` instead of using `Concyclic`.)","title":"Cocyclicity"},{"data_url":"scopes/mathlib/undergraduate/single-variable-real-analysis.json","id":"mathlib/undergraduate/single-variable-real-analysis","kind":"scope","parent":597,"result":false,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/index.html","scope":"mathlib/undergraduate/single-variable-real-analysis","status":"mathlib","title":"Single Variable Real Analysis"},{"data_url":"scopes/mathlib/undergraduate/single-variable-real-analysis/real-numbers.json","id":"mathlib/undergraduate/single-variable-real-analysis/real-numbers","kind":"scope","parent":721,"result":false,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/real-numbers/index.html","scope":"mathlib/undergraduate/single-variable-real-analysis/real-numbers","status":"mathlib","title":"Real numbers"},{"id":"mathlib/undergraduate/single-variable-real-analysis/real-numbers/order","kind":"node","parent":722,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/real-numbers/index.html#order","scope":"mathlib/undergraduate/single-variable-real-analysis/real-numbers","status":"mathlib","summary":"Mathlib states this as `Real.linearOrder`.","title":"Order"},{"data_url":"scopes/mathlib/undergraduate/single-variable-real-analysis/sequences-of-real-numbers.json","id":"mathlib/undergraduate/single-variable-real-analysis/sequences-of-real-numbers","kind":"scope","parent":721,"result":false,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/sequences-of-real-numbers/index.html","scope":"mathlib/undergraduate/single-variable-real-analysis/sequences-of-real-numbers","status":"mathlib","title":"Sequences of real numbers"},{"id":"mathlib/undergraduate/single-variable-real-analysis/sequences-of-real-numbers/limit-point","kind":"node","parent":724,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/sequences-of-real-numbers/index.html#limit-point","scope":"mathlib/undergraduate/single-variable-real-analysis/sequences-of-real-numbers","status":"mathlib","summary":"A point `x` is a cluster point of a sequence `u` along a filter `F` if it is a cluster point of `map u F`.","title":"Limit point"},{"data_url":"scopes/mathlib/undergraduate/single-variable-real-analysis/topology-of-r.json","id":"mathlib/undergraduate/single-variable-real-analysis/topology-of-r","kind":"scope","parent":721,"result":false,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/topology-of-r/index.html","scope":"mathlib/undergraduate/single-variable-real-analysis/topology-of-r","status":"mathlib","title":"Topology of R"},{"id":"mathlib/undergraduate/single-variable-real-analysis/topology-of-r/metric-structure","kind":"node","parent":726,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/topology-of-r/index.html#metric-structure","scope":"mathlib/undergraduate/single-variable-real-analysis/topology-of-r","status":"mathlib","summary":"Instantiate the reals as a metric space.","title":"Metric structure"},{"id":"mathlib/undergraduate/single-variable-real-analysis/topology-of-r/completeness-of-r","kind":"node","parent":726,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/topology-of-r/index.html#completeness-of-r","scope":"mathlib/undergraduate/single-variable-real-analysis/topology-of-r","status":"mathlib","summary":"Mathlib states this as `Real.instCompleteSpace`.","title":"Completeness of R"},{"id":"mathlib/undergraduate/single-variable-real-analysis/topology-of-r/bolzano-weierstrass-theorem","kind":"node","parent":726,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/topology-of-r/index.html#bolzano-weierstrass-theorem","scope":"mathlib/undergraduate/single-variable-real-analysis/topology-of-r","status":"mathlib","summary":"A version of Bolzano-Weierstrass: in a proper metric space (e.g. $\u211d^n$), every bounded sequence has a converging subsequence.","title":"Bolzano-Weierstrass theorem (Topology of R)"},{"id":"mathlib/undergraduate/single-variable-real-analysis/topology-of-r/connected-subsets-of-r","kind":"node","parent":726,"reading_title":{"format":"inline-parts-v1","parts":[["text","Connected subsets of "],["math","\\(\\R\\)"]],"source":"Connected subsets of $\\R$"},"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/topology-of-r/index.html#connected-subsets-of-r","scope":"mathlib/undergraduate/single-variable-real-analysis/topology-of-r","status":"mathlib","summary":"In a dense conditionally complete linear order, the set of preconnected sets is exactly the set of the intervals `Icc`, `Ico`, `Ioc`, `Ioo`, `Ici`, `Ioi`, `Iic`, `Iio`, `(-\u221e, +\u221e)`, or `\u2205`. Though one can represent `\u2205` as `(sInf s, sInf s)`, we include it into the list of possible cases to improve readability.","title":"Connected subsets of $\\R$"},{"id":"mathlib/undergraduate/single-variable-real-analysis/topology-of-r/additive-subgroups-of-r","kind":"node","parent":726,"reading_title":{"format":"inline-parts-v1","parts":[["text","Additive subgroups of "],["math","\\(\\R\\)"]],"source":"Additive subgroups of $\\R$"},"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/topology-of-r/index.html#additive-subgroups-of-r","scope":"mathlib/undergraduate/single-variable-real-analysis/topology-of-r","status":"mathlib","summary":"An additive subgroup of an archimedean linear ordered additive commutative group `G` with order topology either is dense in `G` or is a cyclic subgroup.","title":"Additive subgroups of $\\R$"},{"data_url":"scopes/mathlib/undergraduate/single-variable-real-analysis/numerical-series.json","id":"mathlib/undergraduate/single-variable-real-analysis/numerical-series","kind":"scope","parent":721,"result":false,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/numerical-series/index.html","scope":"mathlib/undergraduate/single-variable-real-analysis/numerical-series","status":"mathlib","title":"Numerical series"},{"id":"mathlib/undergraduate/single-variable-real-analysis/numerical-series/geometric-series","kind":"node","parent":732,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/numerical-series/index.html#geometric-series","scope":"mathlib/undergraduate/single-variable-real-analysis/numerical-series","status":"mathlib","summary":"Mathlib states this as `tsum_geometric_of_norm_lt_one`.","title":"Geometric series"},{"id":"mathlib/undergraduate/single-variable-real-analysis/numerical-series/convergence-of-p-series-for-p-1","kind":"node","parent":732,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convergence of "],["math","\\(p\\)"],["text","-series for "],["math","\\(p>1\\)"]],"source":"Convergence of $p$-series for $p>1$"},"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/numerical-series/index.html#convergence-of-p-series-for-p-1","scope":"mathlib/undergraduate/single-variable-real-analysis/numerical-series","status":"mathlib","summary":"Test for convergence of the `p`-series: the real-valued series `\u2211' n : \u2115, 1 / n ^ p` converges if and only if `1 < p`.","title":"Convergence of $p$-series for $p>1$"},{"id":"mathlib/undergraduate/single-variable-real-analysis/numerical-series/comparison-of-a-series-and-an-integral","kind":"node","parent":732,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/numerical-series/index.html#comparison-of-a-series-and-an-integral","scope":"mathlib/undergraduate/single-variable-real-analysis/numerical-series","status":"mathlib","summary":"Integral test: bounds the sum of a nonnegative antitone function by an integral.","title":"Comparison of a series and an integral"},{"id":"mathlib/undergraduate/single-variable-real-analysis/numerical-series/alternating-series","kind":"node","parent":732,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/numerical-series/index.html#alternating-series","scope":"mathlib/undergraduate/single-variable-real-analysis/numerical-series","status":"mathlib","summary":"The alternating series test for antitone sequences.","title":"Alternating series"},{"data_url":"scopes/mathlib/undergraduate/single-variable-real-analysis/real-valued-functions-defined-on-a-subset-of-r.json","id":"mathlib/undergraduate/single-variable-real-analysis/real-valued-functions-defined-on-a-subset-of-r","kind":"scope","parent":721,"reading_title":{"format":"inline-parts-v1","parts":[["text","Real-valued functions defined on a subset of "],["math","\\(\\R\\)"]],"source":"Real-valued functions defined on a subset of $\\R$"},"result":false,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/real-valued-functions-defined-on-a-subset-of-r/index.html","scope":"mathlib/undergraduate/single-variable-real-analysis/real-valued-functions-defined-on-a-subset-of-r","status":"mathlib","title":"Real-valued functions defined on a subset of $\\R$"},{"id":"mathlib/undergraduate/single-variable-real-analysis/real-valued-functions-defined-on-a-subset-of-r/image-of-a-segment","kind":"node","parent":737,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/real-valued-functions-defined-on-a-subset-of-r/index.html#image-of-a-segment","scope":"mathlib/undergraduate/single-variable-real-analysis/real-valued-functions-defined-on-a-subset-of-r","status":"mathlib","summary":"Mathlib states this as `ContinuousOn.image_Icc`.","title":"Image of a segment"},{"id":"mathlib/undergraduate/single-variable-real-analysis/real-valued-functions-defined-on-a-subset-of-r/continuity-of-monotone-functions","kind":"node","parent":737,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/real-valued-functions-defined-on-a-subset-of-r/index.html#continuity-of-monotone-functions","scope":"mathlib/undergraduate/single-variable-real-analysis/real-valued-functions-defined-on-a-subset-of-r","status":"mathlib","summary":"Mathlib states this as `OrderIso.continuous`.","title":"Continuity of monotone functions"},{"id":"mathlib/undergraduate/single-variable-real-analysis/real-valued-functions-defined-on-a-subset-of-r/continuity-of-inverse-functions","kind":"node","parent":737,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/real-valued-functions-defined-on-a-subset-of-r/index.html#continuity-of-inverse-functions","scope":"mathlib/undergraduate/single-variable-real-analysis/real-valued-functions-defined-on-a-subset-of-r","status":"mathlib","summary":"An order isomorphism between two linear order `OrderTopology` spaces is a homeomorphism.","title":"Continuity of inverse functions"},{"data_url":"scopes/mathlib/undergraduate/single-variable-real-analysis/differentiability.json","id":"mathlib/undergraduate/single-variable-real-analysis/differentiability","kind":"scope","parent":721,"result":false,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/differentiability/index.html","scope":"mathlib/undergraduate/single-variable-real-analysis/differentiability","status":"mathlib","title":"Differentiability"},{"id":"mathlib/undergraduate/single-variable-real-analysis/differentiability/derivative-at-a-point","kind":"node","parent":741,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/differentiability/index.html#derivative-at-a-point","scope":"mathlib/undergraduate/single-variable-real-analysis/differentiability","status":"mathlib","summary":"`f` has the derivative `f'` at the point `x`.","title":"Derivative at a point"},{"id":"mathlib/undergraduate/single-variable-real-analysis/differentiability/derivative-of-a-composition-of-functions","kind":"node","parent":741,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/differentiability/index.html#derivative-of-a-composition-of-functions","scope":"mathlib/undergraduate/single-variable-real-analysis/differentiability","status":"mathlib","summary":"Mathlib states this as `deriv_comp`.","title":"Derivative of a composition of functions (Single Variable Real Analysis)"},{"id":"mathlib/undergraduate/single-variable-real-analysis/differentiability/derivative-of-the-inverse-of-a-function","kind":"node","parent":741,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/differentiability/index.html#derivative-of-the-inverse-of-a-function","scope":"mathlib/undergraduate/single-variable-real-analysis/differentiability","status":"mathlib","summary":"If `f (g y) = y` for `y` in some neighborhood of `a`, `g` is continuous at `a`, and `f` has an invertible derivative `f'` at `g a` in the strict sense, then `g` has the derivative `f'\u207b\u00b9` at `a` in the strict sense.","title":"Derivative of the inverse of a function (Single Variable Real Analysis)"},{"id":"mathlib/undergraduate/single-variable-real-analysis/differentiability/higher-order-derivatives-of-functions","kind":"node","parent":741,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/differentiability/index.html#higher-order-derivatives-of-functions","scope":"mathlib/undergraduate/single-variable-real-analysis/differentiability","status":"mathlib","summary":"The `n`-th iterated derivative of a function from `\ud835\udd5c` to `F`, as a function from `\ud835\udd5c` to `F`.","title":"Higher-order derivatives of functions"},{"id":"mathlib/undergraduate/single-variable-real-analysis/differentiability/leibniz-formula","kind":"node","parent":741,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/differentiability/index.html#leibniz-formula","scope":"mathlib/undergraduate/single-variable-real-analysis/differentiability","status":"mathlib","summary":"Mathlib states this as `deriv_mul`.","title":"Leibniz formula (Single Variable Real Analysis)"},{"data_url":"scopes/mathlib/undergraduate/single-variable-real-analysis/taylor-like-theorems.json","id":"mathlib/undergraduate/single-variable-real-analysis/taylor-like-theorems","kind":"scope","parent":721,"result":false,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/taylor-like-theorems/index.html","scope":"mathlib/undergraduate/single-variable-real-analysis/taylor-like-theorems","status":"mathlib","title":"Taylor-like theorems"},{"id":"mathlib/undergraduate/single-variable-real-analysis/taylor-like-theorems/taylors-theorem-with-integral-form-for-remainder","kind":"node","parent":747,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/taylor-like-theorems/index.html#taylors-theorem-with-integral-form-for-remainder","scope":"mathlib/undergraduate/single-variable-real-analysis/taylor-like-theorems","status":"mathlib","summary":"Taylor's theorem with the Integral form of the remainder.","title":"Taylor's theorem with integral form for remainder (Taylor-like theorems)"},{"id":"mathlib/undergraduate/single-variable-real-analysis/taylor-like-theorems/taylors-theorem-with-lagrange-form-for-remainder","kind":"node","parent":747,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/taylor-like-theorems/index.html#taylors-theorem-with-lagrange-form-for-remainder","scope":"mathlib/undergraduate/single-variable-real-analysis/taylor-like-theorems","status":"mathlib","summary":"Taylor's theorem with the Lagrange form of the remainder.","title":"Taylor's theorem with Lagrange form for remainder"},{"data_url":"scopes/mathlib/undergraduate/single-variable-real-analysis/elementary-functions-trigonometric-rational-exp.json","id":"mathlib/undergraduate/single-variable-real-analysis/elementary-functions-trigonometric-rational-exp","kind":"scope","parent":721,"reading_title":{"format":"inline-parts-v1","parts":[["text","Elementary functions (trigonometric, rational, "],["math","\\(\\exp\\)"],["text",", "],["math","\\(\\log\\)"],["text",", etc)"]],"source":"Elementary functions (trigonometric, rational, $\\exp$, $\\log$, etc)"},"result":false,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/elementary-functions-trigonometric-rational-exp/index.html","scope":"mathlib/undergraduate/single-variable-real-analysis/elementary-functions-trigonometric-rational-exp","status":"mathlib","title":"Elementary functions (trigonometric, rational, $\\exp$, $\\log$, etc)"},{"id":"mathlib/undergraduate/single-variable-real-analysis/elementary-functions-trigonometric-rational-exp/polynomial-functions","kind":"node","parent":750,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/elementary-functions-trigonometric-rational-exp/index.html#polynomial-functions","scope":"mathlib/undergraduate/single-variable-real-analysis/elementary-functions-trigonometric-rational-exp","status":"mathlib","summary":"`eval x p` is the evaluation of the polynomial `p` at `x`","title":"Polynomial functions"},{"id":"mathlib/undergraduate/single-variable-real-analysis/elementary-functions-trigonometric-rational-exp/rational-functions","kind":"node","parent":750,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/elementary-functions-trigonometric-rational-exp/index.html#rational-functions","scope":"mathlib/undergraduate/single-variable-real-analysis/elementary-functions-trigonometric-rational-exp","status":"mathlib","summary":"Evaluate a rational function `p` given a ring hom `f` from the scalar field to the target and a value `x` for the variable in the target.","title":"Rational functions"},{"id":"mathlib/undergraduate/single-variable-real-analysis/elementary-functions-trigonometric-rational-exp/power-functions","kind":"node","parent":750,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/elementary-functions-trigonometric-rational-exp/index.html#power-functions","scope":"mathlib/undergraduate/single-variable-real-analysis/elementary-functions-trigonometric-rational-exp","status":"mathlib","summary":"The real power function `x ^ y`, defined as the real part of the complex power function. For `x > 0`, it is equal to `exp (y log x)`. For `x = 0`, one sets `0 ^ 0=1` and `0 ^ y=0` for `y \u2260 0`. For `x < 0`, the definition is somewhat arbitrary as it depends on the choice of a complex determination of the logarithm. With our conventions, it is equal to `exp (y log x) cos (\u03c0 y)`.","title":"Power functions"},{"data_url":"scopes/mathlib/undergraduate/single-variable-real-analysis/integration.json","id":"mathlib/undergraduate/single-variable-real-analysis/integration","kind":"scope","parent":721,"result":false,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/integration/index.html","scope":"mathlib/undergraduate/single-variable-real-analysis/integration","status":"mathlib","title":"Integration"},{"id":"mathlib/undergraduate/single-variable-real-analysis/integration/riemann-sums","kind":"node","parent":754,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/integration/index.html#riemann-sums","scope":"mathlib/undergraduate/single-variable-real-analysis/integration","status":"mathlib","summary":"The integral sum of `f : \u211d\u207f \u2192 E` over a tagged prepartition `\u03c0` w.r.t. box-additive volume `vol` with codomain `E \u2192L[\u211d] F` is the sum of `vol J (f (\u03c0.tag J))` over all boxes of `\u03c0`.","title":"Riemann sums"},{"id":"mathlib/undergraduate/single-variable-real-analysis/integration/antiderivative-of-a-continuous-function","kind":"node","parent":754,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/integration/index.html#antiderivative-of-a-continuous-function","scope":"mathlib/undergraduate/single-variable-real-analysis/integration","status":"mathlib","summary":"Fundamental theorem of calculus-1, derivative in the right endpoint.","title":"Antiderivative of a continuous function"},{"id":"mathlib/undergraduate/single-variable-real-analysis/integration/change-of-variable","kind":"node","parent":754,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/integration/index.html#change-of-variable","scope":"mathlib/undergraduate/single-variable-real-analysis/integration","status":"mathlib","summary":"Change of variables, most common version. If `f` has continuous derivative `f'` on `[a, b]`, and `g` is continuous, then we can substitute `u = f x` to get `\u222b x in a..b, f' x \u2022 (g \u2218 f) x = \u222b u in f a..f b, g u`.","title":"Change of variable"},{"id":"mathlib/undergraduate/single-variable-real-analysis/integration/integration-by-parts","kind":"node","parent":754,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/integration/index.html#integration-by-parts","scope":"mathlib/undergraduate/single-variable-real-analysis/integration","status":"mathlib","summary":"Integration by parts. Special case of `intervalIntegrable.integral_mul_deriv_eq_deriv_mul_of_hasDeriv_right` where the functions have a derivative also at the endpoints. For improper integrals, see `MeasureTheory.integral_mul_deriv_eq_deriv_mul`, `MeasureTheory.integral_Ioi_mul_deriv_eq_deriv_mul`, and `MeasureTheory.integral_Iic_mul_deriv_eq_deriv_mul`.","title":"Integration by parts"},{"data_url":"scopes/mathlib/undergraduate/single-variable-real-analysis/sequences-and-series-of-functions.json","id":"mathlib/undergraduate/single-variable-real-analysis/sequences-and-series-of-functions","kind":"scope","parent":721,"result":false,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/sequences-and-series-of-functions/index.html","scope":"mathlib/undergraduate/single-variable-real-analysis/sequences-and-series-of-functions","status":"mathlib","title":"Sequences and series of functions"},{"id":"mathlib/undergraduate/single-variable-real-analysis/sequences-and-series-of-functions/pointwise-convergence","kind":"node","parent":759,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/sequences-and-series-of-functions/index.html#pointwise-convergence","scope":"mathlib/undergraduate/single-variable-real-analysis/sequences-and-series-of-functions","status":"mathlib","summary":"Mathlib states this as `tendsto_pi_nhds`.","title":"Pointwise convergence"},{"id":"mathlib/undergraduate/single-variable-real-analysis/sequences-and-series-of-functions/continuity-of-the-limit-of-a-sequence-of-functio","kind":"node","parent":759,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/sequences-and-series-of-functions/index.html#continuity-of-the-limit-of-a-sequence-of-functio","scope":"mathlib/undergraduate/single-variable-real-analysis/sequences-and-series-of-functions","status":"mathlib","summary":"A function which can be uniformly approximated by continuous functions is continuous.","title":"Continuity of the limit of a sequence of functions"},{"id":"mathlib/undergraduate/single-variable-real-analysis/sequences-and-series-of-functions/continuity-of-the-sum-of-a-series-of-functions","kind":"node","parent":759,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/sequences-and-series-of-functions/index.html#continuity-of-the-sum-of-a-series-of-functions","scope":"mathlib/undergraduate/single-variable-real-analysis/sequences-and-series-of-functions","status":"mathlib","summary":"An infinite sum of functions with summable sup norm is continuous if each individual function is.","title":"Continuity of the sum of a series of functions"},{"id":"mathlib/undergraduate/single-variable-real-analysis/sequences-and-series-of-functions/differentiability-of-the-limit-of-a-sequence-of","kind":"node","parent":759,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/sequences-and-series-of-functions/index.html#differentiability-of-the-limit-of-a-sequence-of","scope":"mathlib/undergraduate/single-variable-real-analysis/sequences-and-series-of-functions","status":"mathlib","summary":"`(d/dx) lim_{n \u2192 \u221e} f n x = lim_{n \u2192 \u221e} f' n x` when the `f' n` converge _uniformly_ to their limit.","title":"Differentiability of the limit of a sequence of functions"},{"id":"mathlib/undergraduate/single-variable-real-analysis/sequences-and-series-of-functions/differentiability-of-the-sum-of-a-series-of-func","kind":"node","parent":759,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/sequences-and-series-of-functions/index.html#differentiability-of-the-sum-of-a-series-of-func","scope":"mathlib/undergraduate/single-variable-real-analysis/sequences-and-series-of-functions","status":"mathlib","summary":"Consider a series of functions `\u2211' n, f n x`. If all functions in the series are differentiable with a summable bound on the derivatives, then the series is differentiable. Note that our assumptions do not ensure the pointwise convergence, but if there is no pointwise convergence then the series is zero everywhere so the result still holds.","title":"Differentiability of the sum of a series of functions"},{"id":"mathlib/undergraduate/single-variable-real-analysis/sequences-and-series-of-functions/weierstrass-polynomial-approximation-theorem","kind":"node","parent":759,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/sequences-and-series-of-functions/index.html#weierstrass-polynomial-approximation-theorem","scope":"mathlib/undergraduate/single-variable-real-analysis/sequences-and-series-of-functions","status":"mathlib","summary":"The Weierstrass Approximation Theorem: polynomials functions on `[a, b] \u2286 \u211d` are dense in `C([a,b],\u211d)`","title":"Weierstrass polynomial approximation theorem"},{"id":"mathlib/undergraduate/single-variable-real-analysis/sequences-and-series-of-functions/weierstrass-trigonometric-approximation-theorem","kind":"node","parent":759,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/sequences-and-series-of-functions/index.html#weierstrass-trigonometric-approximation-theorem","scope":"mathlib/undergraduate/single-variable-real-analysis/sequences-and-series-of-functions","status":"mathlib","summary":"The linear span of the monomials `fourier n` is dense in `C(AddCircle T, \u2102)`.","title":"Weierstrass trigonometric approximation theorem"},{"data_url":"scopes/mathlib/undergraduate/single-variable-real-analysis/convexity.json","id":"mathlib/undergraduate/single-variable-real-analysis/convexity","kind":"scope","parent":721,"result":false,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/convexity/index.html","scope":"mathlib/undergraduate/single-variable-real-analysis/convexity","status":"mathlib","title":"Convexity"},{"data_url":"scopes/mathlib/undergraduate/single-variable-real-analysis/convexity/continuity-and-differentiability-of-convex-funct.json","id":"mathlib/undergraduate/single-variable-real-analysis/convexity/continuity-and-differentiability-of-convex-funct","kind":"scope","parent":767,"result":false,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/convexity/continuity-and-differentiability-of-convex-funct/index.html","scope":"mathlib/undergraduate/single-variable-real-analysis/convexity/continuity-and-differentiability-of-convex-funct","status":"mathlib","title":"Continuity and differentiability of convex functions"},{"id":"mathlib/undergraduate/single-variable-real-analysis/convexity/continuity-and-differentiability-of-convex-funct/continuity","kind":"node","parent":768,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-real-analysis/convexity/continuity-and-differentiability-of-convex-funct/index.html#continuity","scope":"mathlib/undergraduate/single-variable-real-analysis/convexity/continuity-and-differentiability-of-convex-funct","status":"mathlib","summary":"Mathlib states this as `ConvexOn.continuousOn`.","title":"Continuity (continuity and differentiability of convex functions)"},{"data_url":"scopes/mathlib/undergraduate/single-variable-complex-analysis.json","id":"mathlib/undergraduate/single-variable-complex-analysis","kind":"scope","parent":597,"result":false,"route":"../roadmap/mathlib/undergraduate/single-variable-complex-analysis/index.html","scope":"mathlib/undergraduate/single-variable-complex-analysis","status":"mathlib","title":"Single Variable Complex Analysis"},{"data_url":"scopes/mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series.json","id":"mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series","kind":"scope","parent":770,"result":false,"route":"../roadmap/mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/index.html","scope":"mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series","status":"mathlib","title":"Complex-valued series"},{"id":"mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/radius-of-convergence","kind":"node","parent":771,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/index.html#radius-of-convergence","scope":"mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series","status":"mathlib","summary":"The radius of a formal multilinear series is the largest `r` such that the sum `\u03a3 \u2016p\u2099\u2016 \u2016y\u2016\u207f` converges for all `\u2016y\u2016 < r`. This implies that `\u03a3 p\u2099 y\u207f` converges for all `\u2016y\u2016 < r`, but these definitions are *not* equivalent in general.","title":"Radius of convergence"},{"id":"mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/continuity","kind":"node","parent":771,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/index.html#continuity","scope":"mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series","status":"mathlib","summary":"If a function admits a power series expansion on a ball, then it is continuous there.","title":"Continuity (Complex-valued series)"},{"id":"mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/differentiability-with-respect-to-the-complex-va","kind":"node","parent":771,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/index.html#differentiability-with-respect-to-the-complex-va","scope":"mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series","status":"mathlib","summary":"Mathlib states this as `HasFPowerSeriesOnBall.differentiableOn`.","title":"Differentiability with respect to the complex variable"},{"id":"mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/complex-exponential","kind":"node","parent":771,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/index.html#complex-exponential","scope":"mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series","status":"mathlib","summary":"The complex exponential function, defined via its Taylor series","title":"Complex exponential"},{"data_url":"scopes/mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/extension-of-trigonometric-functions-to-the-comp.json","id":"mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/extension-of-trigonometric-functions-to-the-comp","kind":"scope","parent":771,"result":false,"route":"../roadmap/mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/extension-of-trigonometric-functions-to-the-comp/index.html","scope":"mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/extension-of-trigonometric-functions-to-the-comp","status":"mathlib","title":"Extension of trigonometric functions to the complex plane"},{"id":"mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/extension-of-trigonometric-functions-to-the-comp/cos","kind":"node","parent":776,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/extension-of-trigonometric-functions-to-the-comp/index.html#cos","scope":"mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/extension-of-trigonometric-functions-to-the-comp","status":"mathlib","summary":"The complex cosine function, defined via `exp`","title":"Cos (extension of trigonometric functions to the complex plane)"},{"id":"mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/extension-of-trigonometric-functions-to-the-comp/sin","kind":"node","parent":776,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/extension-of-trigonometric-functions-to-the-comp/index.html#sin","scope":"mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/extension-of-trigonometric-functions-to-the-comp","status":"mathlib","summary":"The complex sine function, defined via `exp`","title":"Sin (extension of trigonometric functions to the complex plane)"},{"data_url":"scopes/mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/power-series-expansion-of-elementary-functions.json","id":"mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/power-series-expansion-of-elementary-functions","kind":"scope","parent":771,"result":false,"route":"../roadmap/mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/power-series-expansion-of-elementary-functions/index.html","scope":"mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/power-series-expansion-of-elementary-functions","status":"mathlib","title":"Power series expansion of elementary functions"},{"id":"mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/power-series-expansion-of-elementary-functions/cos","kind":"node","parent":779,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/power-series-expansion-of-elementary-functions/index.html#cos","scope":"mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/power-series-expansion-of-elementary-functions","status":"mathlib","summary":"The power series expansion of `Complex.cos`.","title":"Cos (power series expansion of elementary functions)"},{"id":"mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/power-series-expansion-of-elementary-functions/sin","kind":"node","parent":779,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/power-series-expansion-of-elementary-functions/index.html#sin","scope":"mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/power-series-expansion-of-elementary-functions","status":"mathlib","summary":"The power series expansion of `Complex.sin`.","title":"Sin (power series expansion of elementary functions)"},{"id":"mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/power-series-expansion-of-elementary-functions/log","kind":"node","parent":779,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/power-series-expansion-of-elementary-functions/index.html#log","scope":"mathlib/undergraduate/single-variable-complex-analysis/complex-valued-series/power-series-expansion-of-elementary-functions","status":"mathlib","summary":"The Taylor series of the complex logarithm at `1` converges to the logarithm in the open unit disk.","title":"Log"},{"data_url":"scopes/mathlib/undergraduate/single-variable-complex-analysis/functions-on-one-complex-variable.json","id":"mathlib/undergraduate/single-variable-complex-analysis/functions-on-one-complex-variable","kind":"scope","parent":770,"result":false,"route":"../roadmap/mathlib/undergraduate/single-variable-complex-analysis/functions-on-one-complex-variable/index.html","scope":"mathlib/undergraduate/single-variable-complex-analysis/functions-on-one-complex-variable","status":"mathlib","title":"Functions on one complex variable"},{"id":"mathlib/undergraduate/single-variable-complex-analysis/functions-on-one-complex-variable/holomorphic-functions","kind":"node","parent":783,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-complex-analysis/functions-on-one-complex-variable/index.html#holomorphic-functions","scope":"mathlib/undergraduate/single-variable-complex-analysis/functions-on-one-complex-variable","status":"mathlib","summary":"`DifferentiableOn \ud835\udd5c f s` means that `f` is differentiable within `s` at any point of `s`.","title":"Holomorphic functions"},{"id":"mathlib/undergraduate/single-variable-complex-analysis/functions-on-one-complex-variable/cauchy-formulas","kind":"node","parent":783,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-complex-analysis/functions-on-one-complex-variable/index.html#cauchy-formulas","scope":"mathlib/undergraduate/single-variable-complex-analysis/functions-on-one-complex-variable","status":"mathlib","summary":"Cauchy integral formula: if `f : \u2102 \u2192 E` is continuous on a closed disc of radius `R` and is complex differentiable at all but countably many points of its interior, then for any `w` in this interior we have $\\frac{1}{2\u03c0i}\\oint_{|z-c|=R}(z-w)^{-1}f(z)\\,dz=f(w)$.","title":"Cauchy formulas"},{"id":"mathlib/undergraduate/single-variable-complex-analysis/functions-on-one-complex-variable/holomorphic-stability-under-uniform-convergence","kind":"node","parent":783,"result":true,"route":"../roadmap/mathlib/undergraduate/single-variable-complex-analysis/functions-on-one-complex-variable/index.html#holomorphic-stability-under-uniform-convergence","scope":"mathlib/undergraduate/single-variable-complex-analysis/functions-on-one-complex-variable","status":"mathlib","summary":"A locally uniform limit of holomorphic functions on an open domain of the complex plane is holomorphic (the derivatives converge locally uniformly to that of the limit, which is proved as `TendstoLocallyUniformlyOn.deriv`).","title":"Holomorphic stability under uniform convergence"},{"data_url":"scopes/mathlib/undergraduate/topology.json","id":"mathlib/undergraduate/topology","kind":"scope","parent":597,"result":false,"route":"../roadmap/mathlib/undergraduate/topology/index.html","scope":"mathlib/undergraduate/topology","status":"mathlib","title":"Topology"},{"data_url":"scopes/mathlib/undergraduate/topology/topology-and-metric-spaces.json","id":"mathlib/undergraduate/topology/topology-and-metric-spaces","kind":"scope","parent":787,"result":false,"route":"../roadmap/mathlib/undergraduate/topology/topology-and-metric-spaces/index.html","scope":"mathlib/undergraduate/topology/topology-and-metric-spaces","status":"mathlib","title":"Topology and Metric Spaces"},{"id":"mathlib/undergraduate/topology/topology-and-metric-spaces/topology-of-a-metric-space","kind":"node","parent":788,"result":true,"route":"../roadmap/mathlib/undergraduate/topology/topology-and-metric-spaces/index.html#topology-of-a-metric-space","scope":"mathlib/undergraduate/topology/topology-and-metric-spaces","status":"mathlib","summary":"Mathlib states this as `Metric.isOpen_iff`.","title":"Topology of a metric space"},{"id":"mathlib/undergraduate/topology/topology-and-metric-spaces/finite-product-of-metric-spaces","kind":"node","parent":788,"result":true,"route":"../roadmap/mathlib/undergraduate/topology/topology-and-metric-spaces/index.html#finite-product-of-metric-spaces","scope":"mathlib/undergraduate/topology/topology-and-metric-spaces","status":"mathlib","summary":"A finite product of metric spaces is a metric space, with the sup distance.","title":"Finite product of metric spaces"},{"id":"mathlib/undergraduate/topology/topology-and-metric-spaces/limits-of-sequences","kind":"node","parent":788,"result":true,"route":"../roadmap/mathlib/undergraduate/topology/topology-and-metric-spaces/index.html#limits-of-sequences","scope":"mathlib/undergraduate/topology/topology-and-metric-spaces","status":"mathlib","summary":"Mathlib states this as `Metric.tendsto_atTop`.","title":"Limits of sequences"},{"id":"mathlib/undergraduate/topology/topology-and-metric-spaces/homeomorphisms","kind":"node","parent":788,"result":true,"route":"../roadmap/mathlib/undergraduate/topology/topology-and-metric-spaces/index.html#homeomorphisms","scope":"mathlib/undergraduate/topology/topology-and-metric-spaces","status":"mathlib","summary":"Homeomorphism between `X` and `Y`, also called topological isomorphism","title":"Homeomorphisms"},{"id":"mathlib/undergraduate/topology/topology-and-metric-spaces/connected-components","kind":"node","parent":788,"result":true,"route":"../roadmap/mathlib/undergraduate/topology/topology-and-metric-spaces/index.html#connected-components","scope":"mathlib/undergraduate/topology/topology-and-metric-spaces","status":"mathlib","summary":"The connected component of a point is the maximal connected set that contains this point.","title":"Connected components"},{"id":"mathlib/undergraduate/topology/topology-and-metric-spaces/path-connectedness","kind":"node","parent":788,"result":true,"route":"../roadmap/mathlib/undergraduate/topology/topology-and-metric-spaces/index.html#path-connectedness","scope":"mathlib/undergraduate/topology/topology-and-metric-spaces","status":"mathlib","summary":"A set `F` is path connected if it contains a point that can be joined to all other in `F`.","title":"Path connectedness"},{"id":"mathlib/undergraduate/topology/topology-and-metric-spaces/complete-metric-spaces","kind":"node","parent":788,"result":true,"route":"../roadmap/mathlib/undergraduate/topology/topology-and-metric-spaces/index.html#complete-metric-spaces","scope":"mathlib/undergraduate/topology/topology-and-metric-spaces","status":"mathlib","summary":"A pseudo-metric space is complete iff every Cauchy sequence converges.","title":"Complete metric spaces"},{"data_url":"scopes/mathlib/undergraduate/topology/normed-vector-spaces-on-r-and-c.json","id":"mathlib/undergraduate/topology/normed-vector-spaces-on-r-and-c","kind":"scope","parent":787,"reading_title":{"format":"inline-parts-v1","parts":[["text","Normed vector spaces on "],["math","\\(\\R\\)"],["text"," and "],["math","\\(\\C\\)"]],"source":"Normed vector spaces on $\\R$ and $\\C$"},"result":false,"route":"../roadmap/mathlib/undergraduate/topology/normed-vector-spaces-on-r-and-c/index.html","scope":"mathlib/undergraduate/topology/normed-vector-spaces-on-r-and-c","status":"mathlib","title":"Normed vector spaces on $\\R$ and $\\C$"},{"id":"mathlib/undergraduate/topology/normed-vector-spaces-on-r-and-c/norms-lvert-cdot-rvert-p-on-r-n-and-c-n","kind":"node","parent":796,"reading_title":{"format":"inline-parts-v1","parts":[["text","Norms "],["math","\\(\\lVert\\cdot\\rVert_p\\)"],["text"," on "],["math","\\(\\R^n\\)"],["text"," and "],["math","\\(\\C^n\\)"]],"source":"Norms $\\lVert\\cdot\\rVert_p$ on $\\R^n$ and $\\C^n$"},"result":true,"route":"../roadmap/mathlib/undergraduate/topology/normed-vector-spaces-on-r-and-c/index.html#norms-lvert-cdot-rvert-p-on-r-n-and-c-n","scope":"mathlib/undergraduate/topology/normed-vector-spaces-on-r-and-c","status":"mathlib","summary":"The product of finitely many normed spaces is a normed space, with the `L^p` norm.","title":"Norms $\\lVert\\cdot\\rVert_p$ on $\\R^n$ and $\\C^n$"},{"id":"mathlib/undergraduate/topology/normed-vector-spaces-on-r-and-c/uniform-convergence-norm-sup-norm","kind":"node","parent":796,"result":true,"route":"../roadmap/mathlib/undergraduate/topology/normed-vector-spaces-on-r-and-c/index.html#uniform-convergence-norm-sup-norm","scope":"mathlib/undergraduate/topology/normed-vector-spaces-on-r-and-c","status":"mathlib","summary":"Expressing uniform convergence on a set using `edist`.","title":"Uniform convergence norm (sup-norm)"},{"id":"mathlib/undergraduate/topology/normed-vector-spaces-on-r-and-c/normed-space-of-bounded-continuous-functions","kind":"node","parent":796,"result":true,"route":"../roadmap/mathlib/undergraduate/topology/normed-vector-spaces-on-r-and-c/index.html#normed-space-of-bounded-continuous-functions","scope":"mathlib/undergraduate/topology/normed-vector-spaces-on-r-and-c","status":"mathlib","summary":"Mathlib states this as `BoundedContinuousFunction.instNormedSpace`.","title":"Normed space of bounded continuous functions"},{"id":"mathlib/undergraduate/topology/normed-vector-spaces-on-r-and-c/riesz-lemma-unit-ball-characterization-of-finite","kind":"node","parent":796,"result":true,"route":"../roadmap/mathlib/undergraduate/topology/normed-vector-spaces-on-r-and-c/index.html#riesz-lemma-unit-ball-characterization-of-finite","scope":"mathlib/undergraduate/topology/normed-vector-spaces-on-r-and-c","status":"mathlib","summary":"Riesz's theorem: if a closed ball of positive radius is compact in a vector space, then the space is finite-dimensional.","title":"Riesz' lemma (unit-ball characterization of finite dimension)"},{"data_url":"scopes/mathlib/undergraduate/topology/hilbert-spaces.json","id":"mathlib/undergraduate/topology/hilbert-spaces","kind":"scope","parent":787,"result":false,"route":"../roadmap/mathlib/undergraduate/topology/hilbert-spaces/index.html","scope":"mathlib/undergraduate/topology/hilbert-spaces","status":"mathlib","title":"Hilbert spaces"},{"id":"mathlib/undergraduate/topology/hilbert-spaces/hilbert-projection-theorem","kind":"node","parent":801,"result":true,"route":"../roadmap/mathlib/undergraduate/topology/hilbert-spaces/index.html#hilbert-projection-theorem","scope":"mathlib/undergraduate/topology/hilbert-spaces","status":"mathlib","summary":"Existence of minimizers, aka the Hilbert projection theorem.","title":"Hilbert projection theorem"},{"id":"mathlib/undergraduate/topology/hilbert-spaces/inner-product-space-l-2","kind":"node","parent":801,"reading_title":{"format":"inline-parts-v1","parts":[["text","Inner product space "],["math","\\(l^2\\)"]],"source":"Inner product space $l^2$"},"result":true,"route":"../roadmap/mathlib/undergraduate/topology/hilbert-spaces/index.html#inner-product-space-l-2","scope":"mathlib/undergraduate/topology/hilbert-spaces","status":"mathlib","summary":"Mathlib states this as `lp.instInnerProductSpace`.","title":"Inner product space $l^2$"},{"id":"mathlib/undergraduate/topology/hilbert-spaces/completeness-of-l-2","kind":"node","parent":801,"reading_title":{"format":"inline-parts-v1","parts":[["text","Completeness of "],["math","\\(l^2\\)"]],"source":"Completeness of $l^2$"},"result":true,"route":"../roadmap/mathlib/undergraduate/topology/hilbert-spaces/index.html#completeness-of-l-2","scope":"mathlib/undergraduate/topology/hilbert-spaces","status":"mathlib","summary":"Mathlib states this as `lp.completeSpace`.","title":"Completeness of $l^2$"},{"id":"mathlib/undergraduate/topology/hilbert-spaces/inner-product-space-l-2-2","kind":"node","parent":801,"reading_title":{"format":"inline-parts-v1","parts":[["text","Inner product space "],["math","\\(L^2\\)"]],"source":"Inner product space $L^2$"},"result":true,"route":"../roadmap/mathlib/undergraduate/topology/hilbert-spaces/index.html#inner-product-space-l-2-2","scope":"mathlib/undergraduate/topology/hilbert-spaces","status":"mathlib","summary":"Mathlib states this as `MeasureTheory.L2.innerProductSpace`.","title":"Inner product space $L^2$"},{"id":"mathlib/undergraduate/topology/hilbert-spaces/completeness-of-l-2-2","kind":"node","parent":801,"reading_title":{"format":"inline-parts-v1","parts":[["text","Completeness of "],["math","\\(L^2\\)"]],"source":"Completeness of $L^2$"},"result":true,"route":"../roadmap/mathlib/undergraduate/topology/hilbert-spaces/index.html#completeness-of-l-2-2","scope":"mathlib/undergraduate/topology/hilbert-spaces","status":"mathlib","summary":"`Lp` is complete for `1 \u2264 p`.","title":"Completeness of $L^2$"},{"id":"mathlib/undergraduate/topology/hilbert-spaces/hilbert-bases","kind":"node","parent":801,"result":true,"route":"../roadmap/mathlib/undergraduate/topology/hilbert-spaces/index.html#hilbert-bases","scope":"mathlib/undergraduate/topology/hilbert-spaces","status":"mathlib","summary":"A Hilbert basis on `\u03b9` for an inner product space `E` is an identification of `E` with the `lp` space `\u2113\u00b2(\u03b9, \ud835\udd5c)`.","title":"Hilbert bases"},{"id":"mathlib/undergraduate/topology/hilbert-spaces/example-the-hilbert-basis-of-trigonometric-polyn","kind":"node","parent":801,"result":true,"route":"../roadmap/mathlib/undergraduate/topology/hilbert-spaces/index.html#example-the-hilbert-basis-of-trigonometric-polyn","scope":"mathlib/undergraduate/topology/hilbert-spaces","status":"mathlib","summary":"We define `fourierBasis` to be a `\u2124`-indexed Hilbert basis for `Lp \u2102 2 haarAddCircle`, which by definition is an isometric isomorphism from `Lp \u2102 2 haarAddCircle` to `\u2113\u00b2(\u2124, \u2102)`.","title":"Example, the Hilbert basis of trigonometric polynomials"},{"data_url":"scopes/mathlib/undergraduate/multivariable-calculus.json","id":"mathlib/undergraduate/multivariable-calculus","kind":"scope","parent":597,"result":false,"route":"../roadmap/mathlib/undergraduate/multivariable-calculus/index.html","scope":"mathlib/undergraduate/multivariable-calculus","status":"mathlib","title":"Multivariable calculus"},{"data_url":"scopes/mathlib/undergraduate/multivariable-calculus/differential-calculus.json","id":"mathlib/undergraduate/multivariable-calculus/differential-calculus","kind":"scope","parent":809,"result":false,"route":"../roadmap/mathlib/undergraduate/multivariable-calculus/differential-calculus/index.html","scope":"mathlib/undergraduate/multivariable-calculus/differential-calculus","status":"mathlib","title":"Differential calculus"},{"id":"mathlib/undergraduate/multivariable-calculus/differential-calculus/differentials-linear-tangent-functions","kind":"node","parent":810,"result":true,"route":"../roadmap/mathlib/undergraduate/multivariable-calculus/differential-calculus/index.html#differentials-linear-tangent-functions","scope":"mathlib/undergraduate/multivariable-calculus/differential-calculus","status":"mathlib","summary":"If `f` has a derivative at `x`, then `fderiv \ud835\udd5c f x` is such a derivative. Otherwise, it is set to `0`.","title":"Differentials (linear tangent functions)"},{"id":"mathlib/undergraduate/multivariable-calculus/differential-calculus/directional-derivative","kind":"node","parent":810,"result":true,"route":"../roadmap/mathlib/undergraduate/multivariable-calculus/differential-calculus/index.html#directional-derivative","scope":"mathlib/undergraduate/multivariable-calculus/differential-calculus","status":"mathlib","summary":"Line derivative of `f` at the point `x` in the direction `v`, if it exists. Zero otherwise.","title":"Directional derivative"},{"id":"mathlib/undergraduate/multivariable-calculus/differential-calculus/gradient-vector","kind":"node","parent":810,"result":true,"route":"../roadmap/mathlib/undergraduate/multivariable-calculus/differential-calculus/index.html#gradient-vector","scope":"mathlib/undergraduate/multivariable-calculus/differential-calculus","status":"mathlib","summary":"Gradient of `f` at the point `x`, if it exists. Zero otherwise. Denoted as `\u2207` within the Gradient namespace.","title":"Gradient vector"},{"id":"mathlib/undergraduate/multivariable-calculus/differential-calculus/chain-rule","kind":"node","parent":810,"result":true,"route":"../roadmap/mathlib/undergraduate/multivariable-calculus/differential-calculus/index.html#chain-rule","scope":"mathlib/undergraduate/multivariable-calculus/differential-calculus","status":"mathlib","summary":"Mathlib states this as `fderiv_comp`.","title":"Chain rule"},{"id":"mathlib/undergraduate/multivariable-calculus/differential-calculus/differentiable-functions","kind":"node","parent":810,"result":true,"route":"../roadmap/mathlib/undergraduate/multivariable-calculus/differential-calculus/index.html#differentiable-functions","scope":"mathlib/undergraduate/multivariable-calculus/differential-calculus","status":"mathlib","summary":"`Differentiable \ud835\udd5c f` means that `f` is differentiable at any point.","title":"Differentiable functions"},{"id":"mathlib/undergraduate/multivariable-calculus/differential-calculus/partial-derivatives-commute","kind":"node","parent":810,"result":true,"route":"../roadmap/mathlib/undergraduate/multivariable-calculus/differential-calculus/index.html#partial-derivatives-commute","scope":"mathlib/undergraduate/multivariable-calculus/differential-calculus","status":"mathlib","summary":"If a function is differentiable, and has two derivatives at `x`, then the second derivative is symmetric.","title":"Partial derivatives commute"},{"id":"mathlib/undergraduate/multivariable-calculus/differential-calculus/taylors-theorem-with-integral-form-for-remainder","kind":"node","parent":810,"result":true,"route":"../roadmap/mathlib/undergraduate/multivariable-calculus/differential-calculus/index.html#taylors-theorem-with-integral-form-for-remainder","scope":"mathlib/undergraduate/multivariable-calculus/differential-calculus","status":"mathlib","summary":"*Taylor's theorem with remainder in integral form*. If `f` is `n + 1` times continuously differentiable, then `f (x + y)` is given by `\u2211 k in 0..n, D^k f(x; y,..,y) / k! + 1/n! \u222b t in 0..1, (1 - t) ^ n \u2022 D^{n+1}f (x + t \u2022 y; y,..,y)`, where `D^k f` denotes the iterated derivative of `f`.","title":"Taylor's theorem with integral form for remainder (Differential calculus)"},{"id":"mathlib/undergraduate/multivariable-calculus/differential-calculus/diffeomorphisms","kind":"node","parent":810,"result":true,"route":"../roadmap/mathlib/undergraduate/multivariable-calculus/differential-calculus/index.html#diffeomorphisms","scope":"mathlib/undergraduate/multivariable-calculus/differential-calculus","status":"mathlib","summary":"A `G`-diffeomorphism between two charted spaces is a homeomorphism which, when read in the charts, belongs to `G`. We avoid the word diffeomorph as it is too related to the smooth category, and use structomorph instead.","title":"Diffeomorphisms"},{"id":"mathlib/undergraduate/multivariable-calculus/differential-calculus/inverse-function-theorem","kind":"node","parent":810,"result":true,"route":"../roadmap/mathlib/undergraduate/multivariable-calculus/differential-calculus/index.html#inverse-function-theorem","scope":"mathlib/undergraduate/multivariable-calculus/differential-calculus","status":"mathlib","summary":"Mathlib states this as `HasStrictDerivAt.to_localInverse`.","title":"Inverse function theorem (Differential calculus)"},{"data_url":"scopes/mathlib/undergraduate/multivariable-calculus/differential-equations.json","id":"mathlib/undergraduate/multivariable-calculus/differential-equations","kind":"scope","parent":809,"result":false,"route":"../roadmap/mathlib/undergraduate/multivariable-calculus/differential-equations/index.html","scope":"mathlib/undergraduate/multivariable-calculus/differential-equations","status":"mathlib","title":"Differential equations"},{"id":"mathlib/undergraduate/multivariable-calculus/differential-equations/cauchy-lipschitz-theorem","kind":"node","parent":820,"result":true,"route":"../roadmap/mathlib/undergraduate/multivariable-calculus/differential-equations/index.html#cauchy-lipschitz-theorem","scope":"mathlib/undergraduate/multivariable-calculus/differential-equations","status":"mathlib","summary":"Picard-Lindel\u00f6f (Cauchy-Lipschitz) theorem, differential form. This version shows the existence of a local solution whose initial point `x` may be different from the centre `x\u2080` of the closed ball within which the properties of the vector field hold.","title":"Cauchy-Lipschitz Theorem"},{"id":"mathlib/undergraduate/multivariable-calculus/differential-equations/gronwall-lemma","kind":"node","parent":820,"result":true,"route":"../roadmap/mathlib/undergraduate/multivariable-calculus/differential-equations/index.html#gronwall-lemma","scope":"mathlib/undergraduate/multivariable-calculus/differential-equations","status":"mathlib","summary":"A Gr\u00f6nwall-like inequality: if `f : \u211d \u2192 E` is continuous on `[a, b]`, has right derivative `f' x` at every point `x \u2208 [a, b)`, and satisfies the inequalities `\u2016f a\u2016 \u2264 \u03b4`, `\u2200 x \u2208 [a, b), \u2016f' x\u2016 \u2264 K * \u2016f x\u2016 + \u03b5`, then `\u2016f x\u2016` is bounded by `gronwallBound \u03b4 K \u03b5 (x - a)` on `[a, b]`.","title":"Gr\u00f6nwall lemma"},{"data_url":"scopes/mathlib/undergraduate/measures-and-integral-calculus.json","id":"mathlib/undergraduate/measures-and-integral-calculus","kind":"scope","parent":597,"result":false,"route":"../roadmap/mathlib/undergraduate/measures-and-integral-calculus/index.html","scope":"mathlib/undergraduate/measures-and-integral-calculus","status":"mathlib","title":"Measures and integral calculus"},{"data_url":"scopes/mathlib/undergraduate/measures-and-integral-calculus/measure-theory.json","id":"mathlib/undergraduate/measures-and-integral-calculus/measure-theory","kind":"scope","parent":823,"result":false,"route":"../roadmap/mathlib/undergraduate/measures-and-integral-calculus/measure-theory/index.html","scope":"mathlib/undergraduate/measures-and-integral-calculus/measure-theory","status":"mathlib","title":"Measure theory"},{"id":"mathlib/undergraduate/measures-and-integral-calculus/measure-theory/product-of-sigma-algebras","kind":"node","parent":824,"result":true,"route":"../roadmap/mathlib/undergraduate/measures-and-integral-calculus/measure-theory/index.html#product-of-sigma-algebras","scope":"mathlib/undergraduate/measures-and-integral-calculus/measure-theory","status":"mathlib","summary":"Mathlib states this as `MeasurableSpace.pi`.","title":"Product of sigma-algebras"},{"id":"mathlib/undergraduate/measures-and-integral-calculus/measure-theory/counting-measure","kind":"node","parent":824,"result":true,"route":"../roadmap/mathlib/undergraduate/measures-and-integral-calculus/measure-theory/index.html#counting-measure","scope":"mathlib/undergraduate/measures-and-integral-calculus/measure-theory","status":"mathlib","summary":"Counting measure on any measurable space.","title":"Counting measure"},{"id":"mathlib/undergraduate/measures-and-integral-calculus/measure-theory/lebesgue-measure","kind":"node","parent":824,"result":true,"route":"../roadmap/mathlib/undergraduate/measures-and-integral-calculus/measure-theory/index.html#lebesgue-measure","scope":"mathlib/undergraduate/measures-and-integral-calculus/measure-theory","status":"mathlib","summary":"A measure space is a measurable space equipped with a measure, referred to as `volume`.","title":"Lebesgue measure (Measure theory)"},{"id":"mathlib/undergraduate/measures-and-integral-calculus/measure-theory/approximation-by-step-functions","kind":"node","parent":824,"result":true,"route":"../roadmap/mathlib/undergraduate/measures-and-integral-calculus/measure-theory/index.html#approximation-by-step-functions","scope":"mathlib/undergraduate/measures-and-integral-calculus/measure-theory","status":"mathlib","summary":"Mathlib states this as `MeasureTheory.SimpleFunc.tendsto_approxOn`.","title":"Approximation by step functions"},{"data_url":"scopes/mathlib/undergraduate/measures-and-integral-calculus/integration.json","id":"mathlib/undergraduate/measures-and-integral-calculus/integration","kind":"scope","parent":823,"result":false,"route":"../roadmap/mathlib/undergraduate/measures-and-integral-calculus/integration/index.html","scope":"mathlib/undergraduate/measures-and-integral-calculus/integration","status":"mathlib","title":"Integration"},{"id":"mathlib/undergraduate/measures-and-integral-calculus/integration/continuity-of-integrals-with-respect-to-paramete","kind":"node","parent":829,"result":true,"route":"../roadmap/mathlib/undergraduate/measures-and-integral-calculus/integration/index.html#continuity-of-integrals-with-respect-to-paramete","scope":"mathlib/undergraduate/measures-and-integral-calculus/integration","status":"mathlib","summary":"Continuity of interval integral with respect to a parameter. Given `F : X \u2192 \u211d \u2192 E`, assume each `F x` is ae-measurable on `[a, b]`, and assume it is bounded by a function integrable on `[a, b]` independent of `x`. If `(fun x \u21a6 F x t)` is continuous for almost every `t` in `[a, b]` then the same holds for `(fun x \u21a6 \u222b t in a..b, F x t \u2202\u03bc) s x\u2080`.","title":"Continuity of integrals with respect to parameters"},{"id":"mathlib/undergraduate/measures-and-integral-calculus/integration/differentiability-of-integrals-with-respect-to-p","kind":"node","parent":829,"result":true,"route":"../roadmap/mathlib/undergraduate/measures-and-integral-calculus/integration/index.html#differentiability-of-integrals-with-respect-to-p","scope":"mathlib/undergraduate/measures-and-integral-calculus/integration","status":"mathlib","summary":"The (local) interval version of the *Lebesgue Differentiation Theorem*: if `f : \u211d \u2192 E` is interval integrable on `a..b`, then for almost every `x \u2208 uIcc a b`, for any `c \u2208 uIcc a b`, the derivative of `\u222b (t : \u211d) in c..x, f t` at `x` is equal to `f x`.","title":"Differentiability of integrals with respect to parameters"},{"id":"mathlib/undergraduate/measures-and-integral-calculus/integration/holders-inequality","kind":"node","parent":829,"result":true,"route":"../roadmap/mathlib/undergraduate/measures-and-integral-calculus/integration/index.html#holders-inequality","scope":"mathlib/undergraduate/measures-and-integral-calculus/integration","status":"mathlib","summary":"H\u00f6lder's inequality for functions `\u03b1 \u2192 \u211d\u22650`. The integral of the product of two functions is bounded by the product of their `\u2112p` and `\u2112q` seminorms when `p` and `q` are conjugate exponents.","title":"H\u00f6lder's inequality"},{"id":"mathlib/undergraduate/measures-and-integral-calculus/integration/change-of-variables-to-polar-co-ordinates","kind":"node","parent":829,"result":true,"route":"../roadmap/mathlib/undergraduate/measures-and-integral-calculus/integration/index.html#change-of-variables-to-polar-co-ordinates","scope":"mathlib/undergraduate/measures-and-integral-calculus/integration","status":"mathlib","summary":"Mathlib states this as `integral_comp_polarCoord_symm`.","title":"Change of variables to polar co-ordinates"},{"data_url":"scopes/mathlib/undergraduate/measures-and-integral-calculus/fourier-analysis.json","id":"mathlib/undergraduate/measures-and-integral-calculus/fourier-analysis","kind":"scope","parent":823,"result":false,"route":"../roadmap/mathlib/undergraduate/measures-and-integral-calculus/fourier-analysis/index.html","scope":"mathlib/undergraduate/measures-and-integral-calculus/fourier-analysis","status":"mathlib","title":"Fourier analysis"},{"id":"mathlib/undergraduate/measures-and-integral-calculus/fourier-analysis/riemann-lebesgue-lemma","kind":"node","parent":834,"result":true,"route":"../roadmap/mathlib/undergraduate/measures-and-integral-calculus/fourier-analysis/index.html#riemann-lebesgue-lemma","scope":"mathlib/undergraduate/measures-and-integral-calculus/fourier-analysis","status":"mathlib","summary":"Riemann-Lebesgue lemma for functions on a finite-dimensional real vector space, formulated via dual space.","title":"Riemann-Lebesgue lemma (Measures and integral calculus)"},{"id":"mathlib/undergraduate/measures-and-integral-calculus/fourier-analysis/fourier-transform-on-l-2-r-d","kind":"node","parent":834,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fourier transform on "],["math","\\(\\mathrm{L}^2(\\R^d)\\)"]],"source":"Fourier transform on $\\mathrm{L}^2(\\R^d)$"},"result":true,"route":"../roadmap/mathlib/undergraduate/measures-and-integral-calculus/fourier-analysis/index.html#fourier-transform-on-l-2-r-d","scope":"mathlib/undergraduate/measures-and-integral-calculus/fourier-analysis","status":"mathlib","summary":"The Fourier transform on `L2` as a linear isometry equivalence.","title":"Fourier transform on $\\mathrm{L}^2(\\R^d)$"},{"id":"mathlib/undergraduate/measures-and-integral-calculus/fourier-analysis/plancherels-theorem","kind":"node","parent":834,"result":true,"route":"../roadmap/mathlib/undergraduate/measures-and-integral-calculus/fourier-analysis/index.html#plancherels-theorem","scope":"mathlib/undergraduate/measures-and-integral-calculus/fourier-analysis","status":"mathlib","summary":"Plancherel's theorem for Schwartz functions.","title":"Plancherel\u2019s theorem"},{"data_url":"scopes/mathlib/undergraduate/probability-theory.json","id":"mathlib/undergraduate/probability-theory","kind":"scope","parent":597,"result":false,"route":"../roadmap/mathlib/undergraduate/probability-theory/index.html","scope":"mathlib/undergraduate/probability-theory","status":"mathlib","title":"Probability Theory"},{"data_url":"scopes/mathlib/undergraduate/probability-theory/definitions-of-a-probability-space.json","id":"mathlib/undergraduate/probability-theory/definitions-of-a-probability-space","kind":"scope","parent":838,"result":false,"route":"../roadmap/mathlib/undergraduate/probability-theory/definitions-of-a-probability-space/index.html","scope":"mathlib/undergraduate/probability-theory/definitions-of-a-probability-space","status":"mathlib","title":"Definitions of a probability space"},{"id":"mathlib/undergraduate/probability-theory/definitions-of-a-probability-space/events","kind":"node","parent":839,"result":true,"route":"../roadmap/mathlib/undergraduate/probability-theory/definitions-of-a-probability-space/index.html#events","scope":"mathlib/undergraduate/probability-theory/definitions-of-a-probability-space","status":"mathlib","summary":"`MeasurableSet s` means that `s` is measurable (in the ambient measure space on `\u03b1`)","title":"Events"},{"id":"mathlib/undergraduate/probability-theory/definitions-of-a-probability-space/borel-cantelli-lemma-easy-direction","kind":"node","parent":839,"result":true,"route":"../roadmap/mathlib/undergraduate/probability-theory/definitions-of-a-probability-space/index.html#borel-cantelli-lemma-easy-direction","scope":"mathlib/undergraduate/probability-theory/definitions-of-a-probability-space","status":"mathlib","summary":"One direction of the Borel-Cantelli lemma (sometimes called the \"*first* Borel-Cantelli lemma\"): if `(s i)` is a sequence of sets such that `\u2211' i, \u03bc (s i)` is finite, then the limit superior of the `s i` along the `atTop` filter is a null set.","title":"Borel-Cantelli lemma (easy direction)"},{"id":"mathlib/undergraduate/probability-theory/definitions-of-a-probability-space/borel-cantelli-lemma-difficult-direction","kind":"node","parent":839,"result":true,"route":"../roadmap/mathlib/undergraduate/probability-theory/definitions-of-a-probability-space/index.html#borel-cantelli-lemma-difficult-direction","scope":"mathlib/undergraduate/probability-theory/definitions-of-a-probability-space","status":"mathlib","summary":"The second Borel-Cantelli lemma: Given a sequence of independent sets `(s\u2099)` such that `\u2211 n, \u03bc s\u2099 = \u221e`, `limsup s\u2099` has measure 1.","title":"Borel-Cantelli lemma (difficult direction)"},{"id":"mathlib/undergraduate/probability-theory/random-variables-and-their-laws","kind":"node","parent":838,"result":false,"route":"../roadmap/mathlib/undergraduate/probability-theory/random-variables-and-their-laws/index.html","scope":"mathlib/undergraduate/probability-theory","status":"can_state","title":"Random variables and their laws"},{"id":"mathlib/undergraduate/probability-theory/convergence-of-a-sequence-of-random-variables","kind":"node","parent":838,"result":false,"route":"../roadmap/mathlib/undergraduate/probability-theory/convergence-of-a-sequence-of-random-variables/index.html","scope":"mathlib/undergraduate/probability-theory","status":"can_state","title":"Convergence of a sequence of random variables"},{"data_url":"scopes/mathlib/undergraduate/distribution-calculus.json","id":"mathlib/undergraduate/distribution-calculus","kind":"scope","parent":597,"result":false,"route":"../roadmap/mathlib/undergraduate/distribution-calculus/index.html","scope":"mathlib/undergraduate/distribution-calculus","status":"mathlib","title":"Distribution calculus"},{"id":"mathlib/undergraduate/distribution-calculus/spaces-d-r-d","kind":"node","parent":845,"reading_title":{"format":"inline-parts-v1","parts":[["text","Spaces "],["math","\\(\\mathcal{D}(\\R^d)\\)"]],"source":"Spaces $\\mathcal{D}(\\R^d)$"},"result":false,"route":"../roadmap/mathlib/undergraduate/distribution-calculus/spaces-d-r-d/index.html","scope":"mathlib/undergraduate/distribution-calculus","status":"can_state","title":"Spaces $\\mathcal{D}(\\R^d)$"},{"id":"mathlib/undergraduate/distribution-calculus/distributions-on-r-d","kind":"node","parent":845,"reading_title":{"format":"inline-parts-v1","parts":[["text","Distributions on "],["math","\\(\\R^d\\)"]],"source":"Distributions on $\\R^d$"},"result":false,"route":"../roadmap/mathlib/undergraduate/distribution-calculus/distributions-on-r-d/index.html","scope":"mathlib/undergraduate/distribution-calculus","status":"can_state","title":"Distributions on $\\R^d$"},{"data_url":"scopes/mathlib/undergraduate/distribution-calculus/spaces-s-r-d.json","id":"mathlib/undergraduate/distribution-calculus/spaces-s-r-d","kind":"scope","parent":845,"reading_title":{"format":"inline-parts-v1","parts":[["text","Spaces "],["math","\\(\\mathcal{S}(\\R^d)\\)"]],"source":"Spaces $\\mathcal{S}(\\R^d)$"},"result":false,"route":"../roadmap/mathlib/undergraduate/distribution-calculus/spaces-s-r-d/index.html","scope":"mathlib/undergraduate/distribution-calculus/spaces-s-r-d","status":"mathlib","title":"Spaces $\\mathcal{S}(\\R^d)$"},{"id":"mathlib/undergraduate/distribution-calculus/spaces-s-r-d/stability-by-derivation","kind":"node","parent":848,"result":true,"route":"../roadmap/mathlib/undergraduate/distribution-calculus/spaces-s-r-d/index.html#stability-by-derivation","scope":"mathlib/undergraduate/distribution-calculus/spaces-s-r-d","status":"mathlib","summary":"The Fr\u00e9chet derivative on Schwartz space as a continuous `\ud835\udd5c`-linear map.","title":"Stability by derivation"},{"id":"mathlib/undergraduate/distribution-calculus/spaces-s-r-d/stability-by-multiplication-by-a-slowly-growing","kind":"node","parent":848,"result":true,"route":"../roadmap/mathlib/undergraduate/distribution-calculus/spaces-s-r-d/index.html#stability-by-multiplication-by-a-slowly-growing","scope":"mathlib/undergraduate/distribution-calculus/spaces-s-r-d","status":"mathlib","summary":"The map `f \u21a6 (x \u21a6 B (f x) (g x))` as a continuous `\ud835\udd5c`-linear map on Schwartz space, where `B` is a continuous `\ud835\udd5c`-linear map and `g` is a function of temperate growth.","title":"Stability by multiplication by a slowly growing smooth function"},{"id":"mathlib/undergraduate/distribution-calculus/spaces-s-r-d/fourier-transforms-on-s-r-d","kind":"node","parent":848,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fourier transforms on "],["math","\\(\\mathcal{S}(\\R^d)\\)"]],"source":"Fourier transforms on $\\mathcal{S}(\\R^d)$"},"result":true,"route":"../roadmap/mathlib/undergraduate/distribution-calculus/spaces-s-r-d/index.html#fourier-transforms-on-s-r-d","scope":"mathlib/undergraduate/distribution-calculus/spaces-s-r-d","status":"mathlib","summary":"The Fourier transform on a real inner product space, as a continuous linear map on the Schwartz space.","title":"Fourier transforms on $\\mathcal{S}(\\R^d)$"},{"id":"mathlib/undergraduate/distribution-calculus/spaces-s-r-d/convolution-of-two-functions-of-s-r-d","kind":"node","parent":848,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convolution of two functions of "],["math","\\(\\mathcal{S}(\\R^d)\\)"]],"source":"Convolution of two functions of $\\mathcal{S}(\\R^d)$"},"result":true,"route":"../roadmap/mathlib/undergraduate/distribution-calculus/spaces-s-r-d/index.html#convolution-of-two-functions-of-s-r-d","scope":"mathlib/undergraduate/distribution-calculus/spaces-s-r-d","status":"mathlib","summary":"The bilinear convolution of Schwartz functions.","title":"Convolution of two functions of $\\mathcal{S}(\\R^d)$"},{"data_url":"scopes/mathlib/undergraduate/distribution-calculus/tempered-distributions.json","id":"mathlib/undergraduate/distribution-calculus/tempered-distributions","kind":"scope","parent":845,"result":false,"route":"../roadmap/mathlib/undergraduate/distribution-calculus/tempered-distributions/index.html","scope":"mathlib/undergraduate/distribution-calculus/tempered-distributions","status":"mathlib","title":"Tempered distributions"},{"id":"mathlib/undergraduate/distribution-calculus/tempered-distributions/derivation-of-tempered-distributions","kind":"node","parent":853,"result":true,"route":"../roadmap/mathlib/undergraduate/distribution-calculus/tempered-distributions/index.html#derivation-of-tempered-distributions","scope":"mathlib/undergraduate/distribution-calculus/tempered-distributions","status":"mathlib","summary":"The partial derivative (or directional derivative) in the direction `m : E` as a continuous linear map on tempered distributions.","title":"Derivation of tempered distributions"},{"id":"mathlib/undergraduate/distribution-calculus/tempered-distributions/multiplication-by-a-function-c-infty-of-slow-gro","kind":"node","parent":853,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiplication by a function "],["math","\\(C^\\infty\\)"],["text"," of slow growth"]],"source":"Multiplication by a function $C^\\infty$ of slow growth"},"result":true,"route":"../roadmap/mathlib/undergraduate/distribution-calculus/tempered-distributions/index.html#multiplication-by-a-function-c-infty-of-slow-gro","scope":"mathlib/undergraduate/distribution-calculus/tempered-distributions","status":"mathlib","summary":"Multiplication with a temperate growth function as a continuous linear map on `\ud835\udce2'(E, F)`.","title":"Multiplication by a function $C^\\infty$ of slow growth"},{"id":"mathlib/undergraduate/distribution-calculus/tempered-distributions/l-p-functions","kind":"node","parent":853,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(L^p\\)"],["text"," functions"]],"source":"$L^p$ functions"},"result":true,"route":"../roadmap/mathlib/undergraduate/distribution-calculus/tempered-distributions/index.html#l-p-functions","scope":"mathlib/undergraduate/distribution-calculus/tempered-distributions","status":"mathlib","summary":"The natural embedding of L^p into tempered distributions.","title":"$L^p$ functions"},{"id":"mathlib/undergraduate/distribution-calculus/tempered-distributions/fourier-transforms","kind":"node","parent":853,"result":true,"route":"../roadmap/mathlib/undergraduate/distribution-calculus/tempered-distributions/index.html#fourier-transforms","scope":"mathlib/undergraduate/distribution-calculus/tempered-distributions","status":"mathlib","summary":"The Fourier transform as a continuous linear map.","title":"Fourier transforms"},{"id":"mathlib/undergraduate/distribution-calculus/tempered-distributions/inverse-fourier-transform","kind":"node","parent":853,"result":true,"route":"../roadmap/mathlib/undergraduate/distribution-calculus/tempered-distributions/index.html#inverse-fourier-transform","scope":"mathlib/undergraduate/distribution-calculus/tempered-distributions","status":"mathlib","summary":"The inverse Fourier transform as a continuous linear map.","title":"Inverse Fourier transform"},{"data_url":"scopes/mathlib/undergraduate/distribution-calculus/applications.json","id":"mathlib/undergraduate/distribution-calculus/applications","kind":"scope","parent":845,"result":false,"route":"../roadmap/mathlib/undergraduate/distribution-calculus/applications/index.html","scope":"mathlib/undergraduate/distribution-calculus/applications","status":"mathlib","title":"Applications"},{"id":"mathlib/undergraduate/distribution-calculus/applications/poissons-formula","kind":"node","parent":859,"result":true,"route":"../roadmap/mathlib/undergraduate/distribution-calculus/applications/index.html#poissons-formula","scope":"mathlib/undergraduate/distribution-calculus/applications","status":"mathlib","summary":"Poisson's summation formula for Schwartz functions.","title":"Poisson\u2019s formula"},{"data_url":"scopes/mathlib/undergraduate/numerical-analysis.json","id":"mathlib/undergraduate/numerical-analysis","kind":"scope","parent":597,"result":false,"route":"../roadmap/mathlib/undergraduate/numerical-analysis/index.html","scope":"mathlib/undergraduate/numerical-analysis","status":"mathlib","title":"Numerical Analysis"},{"data_url":"scopes/mathlib/undergraduate/numerical-analysis/approximation-of-numerical-functions.json","id":"mathlib/undergraduate/numerical-analysis/approximation-of-numerical-functions","kind":"scope","parent":861,"result":false,"route":"../roadmap/mathlib/undergraduate/numerical-analysis/approximation-of-numerical-functions/index.html","scope":"mathlib/undergraduate/numerical-analysis/approximation-of-numerical-functions","status":"mathlib","title":"Approximation of numerical functions"},{"id":"mathlib/undergraduate/numerical-analysis/approximation-of-numerical-functions/lagrange-interpolation","kind":"node","parent":862,"result":true,"route":"../roadmap/mathlib/undergraduate/numerical-analysis/approximation-of-numerical-functions/index.html#lagrange-interpolation","scope":"mathlib/undergraduate/numerical-analysis/approximation-of-numerical-functions","status":"mathlib","summary":"Lagrange interpolation: given a finset `s : Finset \u03b9`, a nodal map `v : \u03b9 \u2192 F` injective on `s` and a value function `r : \u03b9 \u2192 F`, `interpolate s v r` is the unique polynomial of degree `< #s` that takes value `r i` on `v i` for all `i` in `s`.","title":"Lagrange interpolation"},{"data_url":"scopes/mathlib/hundred-theorems.json","id":"mathlib/hundred-theorems","kind":"scope","parent":1,"result":false,"route":"../roadmap/mathlib/hundred-theorems/index.html","scope":"mathlib/hundred-theorems","status":"not_ready","summary":"Freek Wiedijk's list of 100 famous theorems, with Mathlib's formalizations (docs/100.yaml). Mathlib does not have 21 of its 100 items; they have no article here.","title":"Freek Wiedijk's 100 theorems"},{"data_url":"scopes/mathlib/hundred-theorems/part-01.json","id":"mathlib/hundred-theorems/part-01","kind":"scope","navigation":"transparent","parent":864,"result":false,"route":"../roadmap/mathlib/hundred-theorems/part-01/index.html","scope":"mathlib/hundred-theorems/part-01","status":"mathlib","title":"Theorems 1 to 27"},{"id":"mathlib/hundred-theorems/part-01/the-irrationality-of-the-square-root-of-2","kind":"node","parent":865,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-01/index.html#the-irrationality-of-the-square-root-of-2","scope":"mathlib/hundred-theorems/part-01","status":"mathlib","summary":"Irrationality of the Square Root of 2","title":"The Irrationality of the Square Root of 2"},{"id":"mathlib/hundred-theorems/part-01/the-denumerability-of-the-rational-numbers","kind":"node","parent":865,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-01/index.html#the-denumerability-of-the-rational-numbers","scope":"mathlib/hundred-theorems/part-01","status":"mathlib","summary":"Denumerability of the Rational Numbers","title":"The Denumerability of the Rational Numbers"},{"id":"mathlib/hundred-theorems/part-01/pythagorean-theorem","kind":"node","parent":865,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-01/index.html#pythagorean-theorem","scope":"mathlib/hundred-theorems/part-01","status":"mathlib","summary":"Pythagorean theorem, if-and-only-if angle-at-point form.","title":"Pythagorean Theorem"},{"id":"mathlib/hundred-theorems/part-01/the-area-of-a-circle","kind":"node","parent":865,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-01/index.html#the-area-of-a-circle","scope":"mathlib/hundred-theorems/part-01","status":"mathlib","summary":"Area of a Circle: The area of a disc with radius `r` is `\u03c0 * r ^ 2`.","title":"The Area of a Circle"},{"id":"mathlib/hundred-theorems/part-01/eulers-generalization-of-fermats-little-theorem","kind":"node","parent":865,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-01/index.html#eulers-generalization-of-fermats-little-theorem","scope":"mathlib/hundred-theorems/part-01","status":"mathlib","summary":"The Fermat-Euler totient theorem. `ZMod.pow_totient` is an alternative statement of the same theorem.","title":"Euler\u2019s Generalization of Fermat\u2019s Little Theorem"},{"id":"mathlib/hundred-theorems/part-01/the-infinitude-of-primes","kind":"node","parent":865,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-01/index.html#the-infinitude-of-primes","scope":"mathlib/hundred-theorems/part-01","status":"mathlib","summary":"Euclid's theorem on the infinitude of primes. Here given in the form: for every `n`, there exists a prime number `p \u2265 n`.","title":"The Infinitude of Primes"},{"id":"mathlib/hundred-theorems/part-01/eulers-summation-of-1-1-2-2-1-3-2","kind":"node","parent":865,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-01/index.html#eulers-summation-of-1-1-2-2-1-3-2","scope":"mathlib/hundred-theorems/part-01","status":"mathlib","summary":"Mathlib states this as `hasSum_zeta_two`.","title":"Euler\u2019s Summation of 1 + (1/2)^2 + (1/3)^2 + \u2026."},{"id":"mathlib/hundred-theorems/part-01/fundamental-theorem-of-integral-calculus","kind":"node","parent":865,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-01/index.html#fundamental-theorem-of-integral-calculus","scope":"mathlib/hundred-theorems/part-01","status":"mathlib","summary":"Mathlib records this with 2 declarations.","title":"Fundamental Theorem of Integral Calculus"},{"id":"mathlib/hundred-theorems/part-01/insolvability-of-general-higher-degree-equations","kind":"node","parent":865,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-01/index.html#insolvability-of-general-higher-degree-equations","scope":"mathlib/hundred-theorems/part-01","status":"mathlib","summary":"Abel-Ruffini Theorem","title":"Insolvability of General Higher Degree Equations (Abel-Ruffini Theorem)"},{"id":"mathlib/hundred-theorems/part-01/de-moivres-formula","kind":"node","parent":865,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-01/index.html#de-moivres-formula","scope":"mathlib/hundred-theorems/part-01","status":"mathlib","summary":"De Moivre's formula","title":"De Moivre\u2019s Formula"},{"id":"mathlib/hundred-theorems/part-01/liouvilles-theorem-and-the-construction-of-trans","kind":"node","parent":865,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-01/index.html#liouvilles-theorem-and-the-construction-of-trans","scope":"mathlib/hundred-theorems/part-01","status":"mathlib","summary":"Liouville's Theorem","title":"Liouville\u2019s Theorem and the Construction of Transcendental Numbers"},{"id":"mathlib/hundred-theorems/part-01/the-non-denumerability-of-the-continuum","kind":"node","parent":865,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-01/index.html#the-non-denumerability-of-the-continuum","scope":"mathlib/hundred-theorems/part-01","status":"mathlib","summary":"Mathlib states this as `Cardinal.not_countable_real`.","title":"The Non-Denumerability of the Continuum"},{"id":"mathlib/hundred-theorems/part-01/formula-for-pythagorean-triples","kind":"node","parent":865,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-01/index.html#formula-for-pythagorean-triples","scope":"mathlib/hundred-theorems/part-01","status":"mathlib","summary":"Formula for Pythagorean Triples","title":"Formula for Pythagorean Triples"},{"id":"mathlib/hundred-theorems/part-01/schroeder-bernstein-theorem","kind":"node","parent":865,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-01/index.html#schroeder-bernstein-theorem","scope":"mathlib/hundred-theorems/part-01","status":"mathlib","summary":"The Schr\u00f6der-Bernstein Theorem: Given injections `\u03b1 \u2192 \u03b2` and `\u03b2 \u2192 \u03b1`, we can get a bijection `\u03b1 \u2192 \u03b2`.","title":"Schroeder-Bernstein Theorem"},{"id":"mathlib/hundred-theorems/part-01/leibnizs-series-for-pi","kind":"node","parent":865,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-01/index.html#leibnizs-series-for-pi","scope":"mathlib/hundred-theorems/part-01","status":"mathlib","summary":"Leibniz's series for `\u03c0`. The alternating sum of odd number reciprocals is `\u03c0 / 4`, proved by using Abel's limit theorem to extend the Maclaurin series of `arctan` to 1.","title":"Leibniz\u2019s Series for Pi"},{"id":"mathlib/hundred-theorems/part-01/sum-of-the-angles-of-a-triangle","kind":"node","parent":865,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-01/index.html#sum-of-the-angles-of-a-triangle","scope":"mathlib/hundred-theorems/part-01","status":"mathlib","summary":"The sum of the angles of a triangle (possibly degenerate, where two given vertices are distinct), angle-at-point.","title":"Sum of the Angles of a Triangle"},{"data_url":"scopes/mathlib/hundred-theorems/part-02.json","id":"mathlib/hundred-theorems/part-02","kind":"scope","navigation":"transparent","parent":864,"result":false,"route":"../roadmap/mathlib/hundred-theorems/part-02/index.html","scope":"mathlib/hundred-theorems/part-02","status":"not_ready","title":"Theorems 30 to 57"},{"id":"mathlib/hundred-theorems/part-02/the-ballot-problem","kind":"node","parent":882,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-02/index.html#the-ballot-problem","scope":"mathlib/hundred-theorems/part-02","status":"mathlib","summary":"The ballot problem.","title":"The Ballot Problem"},{"id":"mathlib/hundred-theorems/part-02/fermats-last-theorem","kind":"node","parent":882,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-02/index.html#fermats-last-theorem","scope":"mathlib/hundred-theorems/part-02","status":"not_ready","summary":"Mathlib states this as `FermatLastTheorem` but does not prove it.","title":"Fermat\u2019s Last Theorem"},{"id":"mathlib/hundred-theorems/part-02/divergence-of-the-harmonic-series","kind":"node","parent":882,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-02/index.html#divergence-of-the-harmonic-series","scope":"mathlib/hundred-theorems/part-02","status":"mathlib","summary":"Divergence of the Harmonic Series","title":"Divergence of the Harmonic Series"},{"id":"mathlib/hundred-theorems/part-02/the-solution-of-a-cubic","kind":"node","parent":882,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-02/index.html#the-solution-of-a-cubic","scope":"mathlib/hundred-theorems/part-02","status":"mathlib","summary":"The Solution of Cubic. The roots of a cubic polynomial whose discriminant is nonzero.","title":"The Solution of a Cubic"},{"id":"mathlib/hundred-theorems/part-02/arithmetic-mean-geometric-mean","kind":"node","parent":882,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-02/index.html#arithmetic-mean-geometric-mean","scope":"mathlib/hundred-theorems/part-02","status":"mathlib","summary":"AM-GM inequality: The geometric mean is less than or equal to the arithmetic mean, weighted version for real-valued nonnegative functions.","title":"Arithmetic Mean/Geometric Mean"},{"id":"mathlib/hundred-theorems/part-02/solutions-to-pells-equation","kind":"node","parent":882,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-02/index.html#solutions-to-pells-equation","scope":"mathlib/hundred-theorems/part-02","status":"mathlib","summary":"Mathlib records this with 2 declarations.","title":"Solutions to Pell\u2019s Equation (Freek Wiedijk's 100 theorems)"},{"id":"mathlib/hundred-theorems/part-02/minkowskis-fundamental-theorem","kind":"node","parent":882,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-02/index.html#minkowskis-fundamental-theorem","scope":"mathlib/hundred-theorems/part-02","status":"mathlib","summary":"The Minkowski Convex Body Theorem. If `s` is a convex symmetric domain of `E` whose volume is large enough compared to the covolume of a lattice `L` of `E`, then it contains a non-zero lattice point of `L`.","title":"Minkowski\u2019s Fundamental Theorem"},{"id":"mathlib/hundred-theorems/part-02/sum-of-the-reciprocals-of-the-triangular-numbers","kind":"node","parent":882,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-02/index.html#sum-of-the-reciprocals-of-the-triangular-numbers","scope":"mathlib/hundred-theorems/part-02","status":"mathlib","summary":"Sum of the Reciprocals of the Triangular Numbers","title":"Sum of the Reciprocals of the Triangular Numbers"},{"id":"mathlib/hundred-theorems/part-02/the-binomial-theorem","kind":"node","parent":882,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-02/index.html#the-binomial-theorem","scope":"mathlib/hundred-theorems/part-02","status":"mathlib","summary":"The binomial theorem","title":"The Binomial Theorem"},{"id":"mathlib/hundred-theorems/part-02/the-partition-theorem","kind":"node","parent":882,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-02/index.html#the-partition-theorem","scope":"mathlib/hundred-theorems/part-02","status":"mathlib","summary":"Mathlib states this as `Nat.Partition.card_odds_eq_card_distincts`.","title":"The Partition Theorem"},{"id":"mathlib/hundred-theorems/part-02/the-solution-of-the-general-quartic-equation","kind":"node","parent":882,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-02/index.html#the-solution-of-the-general-quartic-equation","scope":"mathlib/hundred-theorems/part-02","status":"mathlib","summary":"The Solution of Quartic. The roots of a quartic polynomial when `q` is nonzero. See Zwillinger. Here, `u` needs to satisfy the cubic resolvent. An explicit expression of `u` is possible using the cubic formula, but would be too long.","title":"The Solution of the General Quartic Equation"},{"id":"mathlib/hundred-theorems/part-02/dirichlets-theorem","kind":"node","parent":882,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-02/index.html#dirichlets-theorem","scope":"mathlib/hundred-theorems/part-02","status":"mathlib","summary":"Dirichlet's Theorem on primes in arithmetic progression: if `q` is a positive integer and `a : ZMod q` is a unit, then there are infinitely many prime numbers `p` such that `(p : ZMod q) = a`.","title":"Dirichlet\u2019s Theorem"},{"id":"mathlib/hundred-theorems/part-02/wilsons-lemma","kind":"node","parent":882,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-02/index.html#wilsons-lemma","scope":"mathlib/hundred-theorems/part-02","status":"mathlib","summary":"Wilson's Lemma: the product of `1`, ..., `p-1` is `-1` modulo `p`.","title":"Wilson\u2019s Lemma"},{"id":"mathlib/hundred-theorems/part-02/the-number-of-subsets-of-a-set","kind":"node","parent":882,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-02/index.html#the-number-of-subsets-of-a-set","scope":"mathlib/hundred-theorems/part-02","status":"mathlib","summary":"Number of Subsets of a Set","title":"The Number of Subsets of a Set"},{"id":"mathlib/hundred-theorems/part-02/konigsberg-bridges-problem","kind":"node","parent":882,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-02/index.html#konigsberg-bridges-problem","scope":"mathlib/hundred-theorems/part-02","status":"mathlib","summary":"The K\u00f6nigsberg graph is not Eulerian.","title":"K\u00f6nigsberg Bridges Problem"},{"id":"mathlib/hundred-theorems/part-02/product-of-segments-of-chords","kind":"node","parent":882,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-02/index.html#product-of-segments-of-chords","scope":"mathlib/hundred-theorems/part-02","status":"mathlib","summary":"Intersecting Chords Theorem.","title":"Product of Segments of Chords"},{"id":"mathlib/hundred-theorems/part-02/herons-formula","kind":"node","parent":882,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-02/index.html#herons-formula","scope":"mathlib/hundred-theorems/part-02","status":"mathlib","summary":"Heron's formula: The area of a triangle with side lengths `a`, `b`, and `c` is `\u221a(s * (s - a) * (s - b) * (s - c))` where `s = (a + b + c) / 2` is the semiperimeter. We show this by equating this formula to `a * b * sin \u03b3`, where `\u03b3` is the angle opposite the side `c`.","title":"Heron\u2019s Formula"},{"data_url":"scopes/mathlib/hundred-theorems/part-03.json","id":"mathlib/hundred-theorems/part-03","kind":"scope","navigation":"transparent","parent":864,"result":false,"route":"../roadmap/mathlib/hundred-theorems/part-03/index.html","scope":"mathlib/hundred-theorems/part-03","status":"mathlib","title":"Theorems 58 to 78"},{"id":"mathlib/hundred-theorems/part-03/formula-for-the-number-of-combinations","kind":"node","parent":900,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-03/index.html#formula-for-the-number-of-combinations","scope":"mathlib/hundred-theorems/part-03","status":"mathlib","summary":"Mathlib records this with 2 declarations.","title":"Formula for the Number of Combinations"},{"id":"mathlib/hundred-theorems/part-03/bezouts-theorem","kind":"node","parent":900,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-03/index.html#bezouts-theorem","scope":"mathlib/hundred-theorems/part-03","status":"mathlib","summary":"B\u00e9zout's lemma: given `x y : \u2115`, `gcd x y = x * a + y * b`, where `a = gcd_a x y` and `b = gcd_b x y` are computed by the extended Euclidean algorithm.","title":"Bezout\u2019s Theorem"},{"id":"mathlib/hundred-theorems/part-03/theorem-of-ceva","kind":"node","parent":900,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-03/index.html#theorem-of-ceva","scope":"mathlib/hundred-theorems/part-03","status":"mathlib","summary":"Ceva's theorem for a triangle, expressed using division of distances.","title":"Theorem of Ceva"},{"id":"mathlib/hundred-theorems/part-03/cantors-theorem","kind":"node","parent":900,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-03/index.html#cantors-theorem","scope":"mathlib/hundred-theorems/part-03","status":"mathlib","summary":"Cantor's diagonal argument implies that there are no surjective functions from `\u03b1` to `Set \u03b1`.","title":"Cantor\u2019s Theorem"},{"id":"mathlib/hundred-theorems/part-03/lhopitals-rule","kind":"node","parent":900,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-03/index.html#lhopitals-rule","scope":"mathlib/hundred-theorems/part-03","status":"mathlib","summary":"L'H\u00f4pital's rule for approaching a real, `deriv` version","title":"L\u2019Hopital\u2019s Rule"},{"id":"mathlib/hundred-theorems/part-03/isosceles-triangle-theorem","kind":"node","parent":900,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-03/index.html#isosceles-triangle-theorem","scope":"mathlib/hundred-theorems/part-03","status":"mathlib","summary":"Isosceles Triangle Theorem: Pons asinorum, angle-at-point form.","title":"Isosceles Triangle Theorem"},{"id":"mathlib/hundred-theorems/part-03/sum-of-a-geometric-series","kind":"node","parent":900,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-03/index.html#sum-of-a-geometric-series","scope":"mathlib/hundred-theorems/part-03","status":"mathlib","summary":"Mathlib records this with 2 declarations.","title":"Sum of a Geometric Series"},{"id":"mathlib/hundred-theorems/part-03/sum-of-an-arithmetic-series","kind":"node","parent":900,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-03/index.html#sum-of-an-arithmetic-series","scope":"mathlib/hundred-theorems/part-03","status":"mathlib","summary":"Gauss' summation formula","title":"Sum of an arithmetic series"},{"id":"mathlib/hundred-theorems/part-03/greatest-common-divisor-algorithm","kind":"node","parent":900,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-03/index.html#greatest-common-divisor-algorithm","scope":"mathlib/hundred-theorems/part-03","status":"mathlib","summary":"Mathlib records this with 3 declarations.","title":"Greatest Common Divisor Algorithm"},{"id":"mathlib/hundred-theorems/part-03/the-perfect-number-theorem","kind":"node","parent":900,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-03/index.html#the-perfect-number-theorem","scope":"mathlib/hundred-theorems/part-03","status":"mathlib","summary":"Perfect Number Theorem: Euler's theorem that even perfect numbers can be factored as a power of two times a Mersenne prime.","title":"The Perfect Number Theorem"},{"id":"mathlib/hundred-theorems/part-03/ascending-or-descending-sequences-erdos-szekeres","kind":"node","parent":900,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-03/index.html#ascending-or-descending-sequences-erdos-szekeres","scope":"mathlib/hundred-theorems/part-03","status":"mathlib","summary":"Erd\u0151s\u2013Szekeres Theorem: Given a sequence of more than `r * s` distinct values, there is an increasing sequence of length longer than `r` or a decreasing sequence of length longer than `s`.","title":"Ascending or Descending Sequences (Erd\u0151s\u2013Szekeres Theorem)"},{"id":"mathlib/hundred-theorems/part-03/the-mean-value-theorem","kind":"node","parent":900,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-03/index.html#the-mean-value-theorem","scope":"mathlib/hundred-theorems/part-03","status":"mathlib","summary":"Lagrange's Mean Value Theorem, `deriv` version.","title":"The Mean Value Theorem (Freek Wiedijk's 100 theorems)"},{"id":"mathlib/hundred-theorems/part-03/fourier-series","kind":"node","parent":900,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-03/index.html#fourier-series","scope":"mathlib/hundred-theorems/part-03","status":"mathlib","summary":"Mathlib records this with 2 declarations.","title":"Fourier Series"},{"id":"mathlib/hundred-theorems/part-03/sum-of-kth-powers","kind":"node","parent":900,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-03/index.html#sum-of-kth-powers","scope":"mathlib/hundred-theorems/part-03","status":"mathlib","summary":"Mathlib records this with 2 declarations.","title":"Sum of kth powers"},{"data_url":"scopes/mathlib/hundred-theorems/part-04.json","id":"mathlib/hundred-theorems/part-04","kind":"scope","navigation":"transparent","parent":864,"result":false,"route":"../roadmap/mathlib/hundred-theorems/part-04/index.html","scope":"mathlib/hundred-theorems/part-04","status":"mathlib","title":"Theorems 79 to 100"},{"id":"mathlib/hundred-theorems/part-04/the-fundamental-theorem-of-arithmetic","kind":"node","parent":915,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-04/index.html#the-fundamental-theorem-of-arithmetic","scope":"mathlib/hundred-theorems/part-04","status":"mathlib","summary":"Mathlib records this with 5 declarations.","title":"The Fundamental Theorem of Arithmetic"},{"id":"mathlib/hundred-theorems/part-04/divergence-of-the-prime-reciprocal-series","kind":"node","parent":915,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-04/index.html#divergence-of-the-prime-reciprocal-series","scope":"mathlib/hundred-theorems/part-04","status":"mathlib","summary":"Mathlib records this with 2 declarations.","title":"Divergence of the Prime Reciprocal Series"},{"id":"mathlib/hundred-theorems/part-04/dissection-of-cubes-j-e-littlewoods-elegant-proo","kind":"node","parent":915,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-04/index.html#dissection-of-cubes-j-e-littlewoods-elegant-proo","scope":"mathlib/hundred-theorems/part-04","status":"mathlib","summary":"Dissection of Cubes: A cube cannot be cubed.","title":"Dissection of Cubes (J.E. Littlewood\u2019s \u2018elegant\u2019 proof)"},{"id":"mathlib/hundred-theorems/part-04/the-friendship-theorem","kind":"node","parent":915,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-04/index.html#the-friendship-theorem","scope":"mathlib/hundred-theorems/part-04","status":"mathlib","summary":"Friendship theorem: We wish to show that a friendship graph has a politician (a vertex adjacent to all others). We proceed by contradiction, and assume the graph has no politician. We have already proven that a friendship graph with no politician is `d`-regular for some `d`, and now we do casework on `d`. If the degree is at most 2, we observe by casework that it has a politician anyway. If the degree is at least\u2026","title":"The Friendship Theorem"},{"id":"mathlib/hundred-theorems/part-04/divisibility-by-3-rule","kind":"node","parent":915,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-04/index.html#divisibility-by-3-rule","scope":"mathlib/hundred-theorems/part-04","status":"mathlib","summary":"Divisibility by 3 Rule","title":"Divisibility by 3 Rule"},{"id":"mathlib/hundred-theorems/part-04/derangements-formula","kind":"node","parent":915,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-04/index.html#derangements-formula","scope":"mathlib/hundred-theorems/part-04","status":"mathlib","summary":"Mathlib records this with 2 declarations.","title":"Derangements Formula"},{"id":"mathlib/hundred-theorems/part-04/the-factor-and-remainder-theorems","kind":"node","parent":915,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-04/index.html#the-factor-and-remainder-theorems","scope":"mathlib/hundred-theorems/part-04","status":"mathlib","summary":"Mathlib records this with 2 declarations.","title":"The Factor and Remainder Theorems"},{"id":"mathlib/hundred-theorems/part-04/stirlings-formula","kind":"node","parent":915,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-04/index.html#stirlings-formula","scope":"mathlib/hundred-theorems/part-04","status":"mathlib","summary":"Stirling's Formula","title":"Stirling\u2019s Formula"},{"id":"mathlib/hundred-theorems/part-04/the-triangle-inequality","kind":"node","parent":915,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-04/index.html#the-triangle-inequality","scope":"mathlib/hundred-theorems/part-04","status":"mathlib","summary":"Triangle inequality for the norm.","title":"The Triangle Inequality"},{"id":"mathlib/hundred-theorems/part-04/the-birthday-problem","kind":"node","parent":915,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-04/index.html#the-birthday-problem","scope":"mathlib/hundred-theorems/part-04","status":"mathlib","summary":"Mathlib records this with 2 declarations.","title":"The Birthday Problem"},{"id":"mathlib/hundred-theorems/part-04/the-law-of-cosines","kind":"node","parent":915,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-04/index.html#the-law-of-cosines","scope":"mathlib/hundred-theorems/part-04","status":"mathlib","summary":"Law of cosines (cosine rule), angle-at-point form.","title":"The Law of Cosines"},{"id":"mathlib/hundred-theorems/part-04/ptolemys-theorem","kind":"node","parent":915,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-04/index.html#ptolemys-theorem","scope":"mathlib/hundred-theorems/part-04","status":"mathlib","summary":"Ptolemy\u2019s Theorem.","title":"Ptolemy\u2019s Theorem"},{"id":"mathlib/hundred-theorems/part-04/principle-of-inclusion-exclusion","kind":"node","parent":915,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-04/index.html#principle-of-inclusion-exclusion","scope":"mathlib/hundred-theorems/part-04","status":"mathlib","summary":"Mathlib records this with 4 declarations.","title":"Principle of Inclusion/Exclusion"},{"id":"mathlib/hundred-theorems/part-04/cramers-rule","kind":"node","parent":915,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-04/index.html#cramers-rule","scope":"mathlib/hundred-theorems/part-04","status":"mathlib","summary":"A stronger form of Cramer's rule that allows us to solve some instances of `A * x = b` even if the determinant is not a unit. A sufficient (but still not necessary) condition is that `A.det` divides `b`.","title":"Cramer\u2019s Rule"},{"id":"mathlib/hundred-theorems/part-04/bertrands-postulate","kind":"node","parent":915,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-04/index.html#bertrands-postulate","scope":"mathlib/hundred-theorems/part-04","status":"mathlib","summary":"Alias of `Nat.exists_prime_lt_and_le_two_mul`.","title":"Bertrand\u2019s Postulate"},{"id":"mathlib/hundred-theorems/part-04/buffon-needle-problem","kind":"node","parent":915,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-04/index.html#buffon-needle-problem","scope":"mathlib/hundred-theorems/part-04","status":"mathlib","summary":"Mathlib records this with 2 declarations.","title":"Buffon Needle Problem"},{"id":"mathlib/hundred-theorems/part-04/descartes-rule-of-signs","kind":"node","parent":915,"result":true,"route":"../roadmap/mathlib/hundred-theorems/part-04/index.html#descartes-rule-of-signs","scope":"mathlib/hundred-theorems/part-04","status":"mathlib","summary":"Descartes' Rule of Signs: the number of positive roots is at most the number of sign variations.","title":"Descartes Rule of Signs"},{"data_url":"scopes/mathlib/thousand-plus.json","id":"mathlib/thousand-plus","kind":"scope","parent":1,"result":false,"route":"../roadmap/mathlib/thousand-plus/index.html","scope":"mathlib/thousand-plus","status":"mathlib","summary":"The theorems of Wikipedia's list of theorems, as collected by the 1000+ theorems project, grouped by Mathematics Subject Classification (docs/1000.yaml). Mathlib does not have 981 of its 1,199 items; they have no article here.","title":"1000+ theorems"},{"data_url":"scopes/mathlib/thousand-plus/general-topics-history-and-foundations.json","id":"mathlib/thousand-plus/general-topics-history-and-foundations","kind":"scope","parent":933,"result":false,"route":"../roadmap/mathlib/thousand-plus/general-topics-history-and-foundations/index.html","scope":"mathlib/thousand-plus/general-topics-history-and-foundations","status":"mathlib","title":"General topics, history and foundations"},{"id":"mathlib/thousand-plus/general-topics-history-and-foundations/ax-grothendieck-theorem","kind":"node","parent":934,"result":true,"route":"../roadmap/mathlib/thousand-plus/general-topics-history-and-foundations/index.html#ax-grothendieck-theorem","scope":"mathlib/thousand-plus/general-topics-history-and-foundations","status":"mathlib","summary":"The Ax-Grothendieck theorem","title":"Ax\u2013Grothendieck theorem"},{"id":"mathlib/thousand-plus/general-topics-history-and-foundations/boolean-prime-ideal-theorem","kind":"node","parent":934,"result":true,"route":"../roadmap/mathlib/thousand-plus/general-topics-history-and-foundations/index.html#boolean-prime-ideal-theorem","scope":"mathlib/thousand-plus/general-topics-history-and-foundations","status":"mathlib","summary":"Mathlib states this as `DistribLattice.prime_ideal_of_disjoint_filter_ideal`.","title":"Boolean prime ideal theorem"},{"id":"mathlib/thousand-plus/general-topics-history-and-foundations/kleenes-recursion-theorem","kind":"node","parent":934,"result":true,"route":"../roadmap/mathlib/thousand-plus/general-topics-history-and-foundations/index.html#kleenes-recursion-theorem","scope":"mathlib/thousand-plus/general-topics-history-and-foundations","status":"mathlib","summary":"Kleene's second recursion theorem","title":"Kleene's recursion theorem"},{"id":"mathlib/thousand-plus/general-topics-history-and-foundations/konigs-theorem-set-theory","kind":"node","parent":934,"result":true,"route":"../roadmap/mathlib/thousand-plus/general-topics-history-and-foundations/index.html#konigs-theorem-set-theory","scope":"mathlib/thousand-plus/general-topics-history-and-foundations","status":"mathlib","summary":"K\u00f6nig's theorem","title":"K\u00f6nig's theorem (set theory)"},{"id":"mathlib/thousand-plus/general-topics-history-and-foundations/szpilrajn-extension-theorem","kind":"node","parent":934,"result":true,"route":"../roadmap/mathlib/thousand-plus/general-topics-history-and-foundations/index.html#szpilrajn-extension-theorem","scope":"mathlib/thousand-plus/general-topics-history-and-foundations","status":"mathlib","summary":"Szpilrajn extension theorem: any partial order can be extended to a linear order.","title":"Szpilrajn extension theorem"},{"id":"mathlib/thousand-plus/general-topics-history-and-foundations/well-ordering-theorem","kind":"node","parent":934,"result":true,"route":"../roadmap/mathlib/thousand-plus/general-topics-history-and-foundations/index.html#well-ordering-theorem","scope":"mathlib/thousand-plus/general-topics-history-and-foundations","status":"mathlib","summary":"The well-ordering theorem (or Zermelo's theorem): every type can be well-ordered.","title":"Well-ordering theorem"},{"id":"mathlib/thousand-plus/general-topics-history-and-foundations/os-theorem","kind":"node","parent":934,"result":true,"route":"../roadmap/mathlib/thousand-plus/general-topics-history-and-foundations/index.html#os-theorem","scope":"mathlib/thousand-plus/general-topics-history-and-foundations","status":"mathlib","summary":"\u0141o\u015b's Theorem: A sentence is true in an ultraproduct if and only if the set of structures it is true in is in the ultrafilter.","title":"\u0141o\u015b' theorem"},{"data_url":"scopes/mathlib/thousand-plus/combinatorics-and-ordered-structures.json","id":"mathlib/thousand-plus/combinatorics-and-ordered-structures","kind":"scope","parent":933,"result":false,"route":"../roadmap/mathlib/thousand-plus/combinatorics-and-ordered-structures/index.html","scope":"mathlib/thousand-plus/combinatorics-and-ordered-structures","status":"mathlib","title":"Combinatorics and ordered structures"},{"data_url":"scopes/mathlib/thousand-plus/combinatorics-and-ordered-structures/05-combinatorics.json","id":"mathlib/thousand-plus/combinatorics-and-ordered-structures/05-combinatorics","kind":"scope","parent":942,"result":false,"route":"../roadmap/mathlib/thousand-plus/combinatorics-and-ordered-structures/05-combinatorics/index.html","scope":"mathlib/thousand-plus/combinatorics-and-ordered-structures/05-combinatorics","status":"mathlib","title":"05 Combinatorics"},{"id":"mathlib/thousand-plus/combinatorics-and-ordered-structures/05-combinatorics/erdos-ko-rado-theorem","kind":"node","parent":943,"result":true,"route":"../roadmap/mathlib/thousand-plus/combinatorics-and-ordered-structures/05-combinatorics/index.html#erdos-ko-rado-theorem","scope":"mathlib/thousand-plus/combinatorics-and-ordered-structures/05-combinatorics","status":"mathlib","summary":"The Erd\u0151s\u2013Ko\u2013Rado theorem.","title":"Erd\u0151s\u2013Ko\u2013Rado theorem"},{"id":"mathlib/thousand-plus/combinatorics-and-ordered-structures/05-combinatorics/hindmans-theorem","kind":"node","parent":943,"result":true,"route":"../roadmap/mathlib/thousand-plus/combinatorics-and-ordered-structures/05-combinatorics/index.html#hindmans-theorem","scope":"mathlib/thousand-plus/combinatorics-and-ordered-structures/05-combinatorics","status":"mathlib","summary":"The strong form of Hindman's theorem: in any finite cover of an FP-set, one the parts contains an FP-set.","title":"Hindman's theorem (1000+ theorems)"},{"id":"mathlib/thousand-plus/combinatorics-and-ordered-structures/05-combinatorics/multinomial-theorem","kind":"node","parent":943,"result":true,"route":"../roadmap/mathlib/thousand-plus/combinatorics-and-ordered-structures/05-combinatorics/index.html#multinomial-theorem","scope":"mathlib/thousand-plus/combinatorics-and-ordered-structures/05-combinatorics","status":"mathlib","summary":"The multinomial theorem.","title":"Multinomial theorem"},{"id":"mathlib/thousand-plus/combinatorics-and-ordered-structures/05-combinatorics/tutte-theorem","kind":"node","parent":943,"result":true,"route":"../roadmap/mathlib/thousand-plus/combinatorics-and-ordered-structures/05-combinatorics/index.html#tutte-theorem","scope":"mathlib/thousand-plus/combinatorics-and-ordered-structures/05-combinatorics","status":"mathlib","summary":"Tutte's theorem","title":"Tutte theorem"},{"data_url":"scopes/mathlib/thousand-plus/combinatorics-and-ordered-structures/06-order-lattices-ordered-algebraic-structures.json","id":"mathlib/thousand-plus/combinatorics-and-ordered-structures/06-order-lattices-ordered-algebraic-structures","kind":"scope","parent":942,"result":false,"route":"../roadmap/mathlib/thousand-plus/combinatorics-and-ordered-structures/06-order-lattices-ordered-algebraic-structures/index.html","scope":"mathlib/thousand-plus/combinatorics-and-ordered-structures/06-order-lattices-ordered-algebraic-structures","status":"mathlib","title":"06 Order, lattices, ordered algebraic structures"},{"id":"mathlib/thousand-plus/combinatorics-and-ordered-structures/06-order-lattices-ordered-algebraic-structures/birkhoffs-representation-theorem","kind":"node","parent":948,"result":true,"route":"../roadmap/mathlib/thousand-plus/combinatorics-and-ordered-structures/06-order-lattices-ordered-algebraic-structures/index.html#birkhoffs-representation-theorem","scope":"mathlib/thousand-plus/combinatorics-and-ordered-structures/06-order-lattices-ordered-algebraic-structures","status":"mathlib","summary":"Birkhoff's Representation Theorem. Any finite distributive lattice can be embedded in a powerset lattice.","title":"Birkhoff's representation theorem"},{"id":"mathlib/thousand-plus/combinatorics-and-ordered-structures/06-order-lattices-ordered-algebraic-structures/bourbaki-witt-theorem","kind":"node","parent":948,"result":true,"route":"../roadmap/mathlib/thousand-plus/combinatorics-and-ordered-structures/06-order-lattices-ordered-algebraic-structures/index.html#bourbaki-witt-theorem","scope":"mathlib/thousand-plus/combinatorics-and-ordered-structures/06-order-lattices-ordered-algebraic-structures","status":"mathlib","summary":"The Bourbaki-Witt Theorem: If `\u03b1` is a chain complete partial order and `f : \u03b1 \u2192 \u03b1` is inflationary, then `f` has a fixed point","title":"Bourbaki\u2013Witt theorem"},{"id":"mathlib/thousand-plus/combinatorics-and-ordered-structures/06-order-lattices-ordered-algebraic-structures/cantors-isomorphism-theorem","kind":"node","parent":948,"result":true,"route":"../roadmap/mathlib/thousand-plus/combinatorics-and-ordered-structures/06-order-lattices-ordered-algebraic-structures/index.html#cantors-isomorphism-theorem","scope":"mathlib/thousand-plus/combinatorics-and-ordered-structures/06-order-lattices-ordered-algebraic-structures","status":"mathlib","summary":"Any two countable dense, nonempty linear orders without endpoints are order isomorphic. This is also known as Cantor's isomorphism theorem.","title":"Cantor's isomorphism theorem"},{"id":"mathlib/thousand-plus/combinatorics-and-ordered-structures/06-order-lattices-ordered-algebraic-structures/hahn-embedding-theorem","kind":"node","parent":948,"result":true,"route":"../roadmap/mathlib/thousand-plus/combinatorics-and-ordered-structures/06-order-lattices-ordered-algebraic-structures/index.html#hahn-embedding-theorem","scope":"mathlib/thousand-plus/combinatorics-and-ordered-structures/06-order-lattices-ordered-algebraic-structures","status":"mathlib","summary":"Hahn embedding theorem","title":"Hahn embedding theorem"},{"id":"mathlib/thousand-plus/combinatorics-and-ordered-structures/06-order-lattices-ordered-algebraic-structures/kleene-fixed-point-theorem","kind":"node","parent":948,"result":true,"route":"../roadmap/mathlib/thousand-plus/combinatorics-and-ordered-structures/06-order-lattices-ordered-algebraic-structures/index.html#kleene-fixed-point-theorem","scope":"mathlib/thousand-plus/combinatorics-and-ordered-structures/06-order-lattices-ordered-algebraic-structures","status":"mathlib","summary":"Kleene's fixed point Theorem: The least fixed point in a complete lattice is the supremum of iterating a function on bottom arbitrary often.","title":"Kleene fixed-point theorem"},{"id":"mathlib/thousand-plus/combinatorics-and-ordered-structures/06-order-lattices-ordered-algebraic-structures/knaster-tarski-theorem","kind":"node","parent":948,"result":true,"route":"../roadmap/mathlib/thousand-plus/combinatorics-and-ordered-structures/06-order-lattices-ordered-algebraic-structures/index.html#knaster-tarski-theorem","scope":"mathlib/thousand-plus/combinatorics-and-ordered-structures/06-order-lattices-ordered-algebraic-structures","status":"mathlib","summary":"Knaster-Tarski Theorem: The fixed points of `f` form a complete lattice.","title":"Knaster\u2013Tarski theorem"},{"data_url":"scopes/mathlib/thousand-plus/number-theory.json","id":"mathlib/thousand-plus/number-theory","kind":"scope","parent":933,"result":false,"route":"../roadmap/mathlib/thousand-plus/number-theory/index.html","scope":"mathlib/thousand-plus/number-theory","status":"mathlib","title":"Number theory"},{"data_url":"scopes/mathlib/thousand-plus/number-theory/part-01.json","id":"mathlib/thousand-plus/number-theory/part-01","kind":"scope","navigation":"transparent","parent":955,"result":false,"route":"../roadmap/mathlib/thousand-plus/number-theory/part-01/index.html","scope":"mathlib/thousand-plus/number-theory/part-01","status":"mathlib","title":"Beatty's theorem to Mahler's theorem"},{"id":"mathlib/thousand-plus/number-theory/part-01/beattys-theorem","kind":"node","parent":956,"result":true,"route":"../roadmap/mathlib/thousand-plus/number-theory/part-01/index.html#beattys-theorem","scope":"mathlib/thousand-plus/number-theory/part-01","status":"mathlib","summary":"Mathlib records this with 2 declarations.","title":"Beatty's theorem"},{"id":"mathlib/thousand-plus/number-theory/part-01/dirichlets-approximation-theorem","kind":"node","parent":956,"result":true,"route":"../roadmap/mathlib/thousand-plus/number-theory/part-01/index.html#dirichlets-approximation-theorem","scope":"mathlib/thousand-plus/number-theory/part-01","status":"mathlib","summary":"Mathlib records this with 4 declarations.","title":"Dirichlet's approximation theorem"},{"id":"mathlib/thousand-plus/number-theory/part-01/dirichlets-unit-theorem","kind":"node","parent":956,"result":true,"route":"../roadmap/mathlib/thousand-plus/number-theory/part-01/index.html#dirichlets-unit-theorem","scope":"mathlib/thousand-plus/number-theory/part-01","status":"mathlib","summary":"Mathlib records this with 2 declarations.","title":"Dirichlet's unit theorem"},{"id":"mathlib/thousand-plus/number-theory/part-01/fermats-little-theorem","kind":"node","parent":956,"result":true,"route":"../roadmap/mathlib/thousand-plus/number-theory/part-01/index.html#fermats-little-theorem","scope":"mathlib/thousand-plus/number-theory/part-01","status":"mathlib","summary":"Fermat's Little Theorem: for all nonzero `a : ZMod p`, we have `a ^ (p - 1) = 1`.","title":"Fermat's little theorem"},{"id":"mathlib/thousand-plus/number-theory/part-01/glaishers-theorem","kind":"node","parent":956,"result":true,"route":"../roadmap/mathlib/thousand-plus/number-theory/part-01/index.html#glaishers-theorem","scope":"mathlib/thousand-plus/number-theory/part-01","status":"mathlib","summary":"Mathlib states this as `Nat.Partition.card_restricted_eq_card_countRestricted`.","title":"Glaisher's theorem"},{"id":"mathlib/thousand-plus/number-theory/part-01/lucass-theorem","kind":"node","parent":956,"result":true,"route":"../roadmap/mathlib/thousand-plus/number-theory/part-01/index.html#lucass-theorem","scope":"mathlib/thousand-plus/number-theory/part-01","status":"mathlib","summary":"Alias of `Choose.choose_modEq_prod_range_choose`.","title":"Lucas's theorem"},{"id":"mathlib/thousand-plus/number-theory/part-01/mahlers-theorem","kind":"node","parent":956,"result":true,"route":"../roadmap/mathlib/thousand-plus/number-theory/part-01/index.html#mahlers-theorem","scope":"mathlib/thousand-plus/number-theory/part-01","status":"mathlib","summary":"Mahler's theorem: for any continuous function `f` from `\u2124_[p]` to a `p`-adic Banach space, the Mahler series with coefficients `n \u21a6 \u0394_[1]^[n] f 0` converges to the original function `f`.","title":"Mahler's theorem"},{"data_url":"scopes/mathlib/thousand-plus/number-theory/part-02.json","id":"mathlib/thousand-plus/number-theory/part-02","kind":"scope","navigation":"transparent","parent":955,"result":false,"route":"../roadmap/mathlib/thousand-plus/number-theory/part-02/index.html","scope":"mathlib/thousand-plus/number-theory/part-02","status":"mathlib","title":"Minkowski's theorem to Zeckendorf's theorem"},{"id":"mathlib/thousand-plus/number-theory/part-02/nivens-theorem","kind":"node","parent":964,"result":true,"route":"../roadmap/mathlib/thousand-plus/number-theory/part-02/index.html#nivens-theorem","scope":"mathlib/thousand-plus/number-theory/part-02","status":"mathlib","summary":"Niven's theorem: The only rational values of `cos` that occur at rational multiples of \u03c0 are `{-1, -1/2, 0, 1/2, 1}`.","title":"Niven's theorem"},{"id":"mathlib/thousand-plus/number-theory/part-02/ostrowskis-theorem","kind":"node","parent":964,"result":true,"route":"../roadmap/mathlib/thousand-plus/number-theory/part-02/index.html#ostrowskis-theorem","scope":"mathlib/thousand-plus/number-theory/part-02","status":"mathlib","summary":"Mathlib records this with 2 declarations.","title":"Ostrowski's theorem"},{"id":"mathlib/thousand-plus/number-theory/part-02/pentagonal-number-theorem","kind":"node","parent":964,"result":true,"route":"../roadmap/mathlib/thousand-plus/number-theory/part-02/index.html#pentagonal-number-theorem","scope":"mathlib/thousand-plus/number-theory/part-02","status":"mathlib","summary":"Mathlib records this with 2 declarations.","title":"Pentagonal number theorem"},{"id":"mathlib/thousand-plus/number-theory/part-02/von-staudt-clausen-theorem","kind":"node","parent":964,"result":true,"route":"../roadmap/mathlib/thousand-plus/number-theory/part-02/index.html#von-staudt-clausen-theorem","scope":"mathlib/thousand-plus/number-theory/part-02","status":"mathlib","summary":"von Staudt-Clausen theorem: For any natural number $k$, the sum","title":"Von Staudt\u2013Clausen theorem"},{"id":"mathlib/thousand-plus/number-theory/part-02/zeckendorfs-theorem","kind":"node","parent":964,"result":true,"route":"../roadmap/mathlib/thousand-plus/number-theory/part-02/index.html#zeckendorfs-theorem","scope":"mathlib/thousand-plus/number-theory/part-02","status":"mathlib","summary":"Zeckendorf's Theorem as an equivalence between natural numbers and Zeckendorf representations. Every natural number can be written uniquely as a sum of non-consecutive Fibonacci numbers (if we forget about the first two terms `F\u2080 = 0`, `F\u2081 = 1`).","title":"Zeckendorf's theorem"},{"data_url":"scopes/mathlib/thousand-plus/algebra.json","id":"mathlib/thousand-plus/algebra","kind":"scope","parent":933,"result":false,"route":"../roadmap/mathlib/thousand-plus/algebra/index.html","scope":"mathlib/thousand-plus/algebra","status":"mathlib","title":"Algebra"},{"data_url":"scopes/mathlib/thousand-plus/algebra/12-field-theory-and-polynomials.json","id":"mathlib/thousand-plus/algebra/12-field-theory-and-polynomials","kind":"scope","parent":970,"result":false,"route":"../roadmap/mathlib/thousand-plus/algebra/12-field-theory-and-polynomials/index.html","scope":"mathlib/thousand-plus/algebra/12-field-theory-and-polynomials","status":"mathlib","title":"12 Field theory and polynomials"},{"id":"mathlib/thousand-plus/algebra/12-field-theory-and-polynomials/chevalley-warning-theorem","kind":"node","parent":971,"result":true,"route":"../roadmap/mathlib/thousand-plus/algebra/12-field-theory-and-polynomials/index.html#chevalley-warning-theorem","scope":"mathlib/thousand-plus/algebra/12-field-theory-and-polynomials","status":"mathlib","summary":"The Chevalley\u2013Warning theorem, finitary version. Let `(f i)` be a finite family of multivariate polynomials in finitely many variables (`X s`, `s : \u03c3`) over a finite field of characteristic `p`. Assume that the sum of the total degrees of the `f i` is less than the cardinality of `\u03c3`. Then the number of common solutions of the `f i` is divisible by `p`.","title":"Chevalley\u2013Warning theorem (1000+ theorems)"},{"id":"mathlib/thousand-plus/algebra/12-field-theory-and-polynomials/integral-root-theorem","kind":"node","parent":971,"result":true,"route":"../roadmap/mathlib/thousand-plus/algebra/12-field-theory-and-polynomials/index.html#integral-root-theorem","scope":"mathlib/thousand-plus/algebra/12-field-theory-and-polynomials","status":"mathlib","summary":"Integral root theorem: if `r : f.codomain` is a root of a monic polynomial over the ufd `A`, then `r` is an integer","title":"Integral root theorem"},{"id":"mathlib/thousand-plus/algebra/12-field-theory-and-polynomials/mason-stothers-theorem","kind":"node","parent":971,"result":true,"route":"../roadmap/mathlib/thousand-plus/algebra/12-field-theory-and-polynomials/index.html#mason-stothers-theorem","scope":"mathlib/thousand-plus/algebra/12-field-theory-and-polynomials","status":"mathlib","summary":"Polynomial ABC theorem.","title":"Mason\u2013Stothers theorem"},{"id":"mathlib/thousand-plus/algebra/12-field-theory-and-polynomials/primitive-element-theorem","kind":"node","parent":971,"result":true,"route":"../roadmap/mathlib/thousand-plus/algebra/12-field-theory-and-polynomials/index.html#primitive-element-theorem","scope":"mathlib/thousand-plus/algebra/12-field-theory-and-polynomials","status":"mathlib","summary":"Primitive element theorem: a finite separable field extension `E` of `F` has a primitive element, i.e. there is an `\u03b1 \u2208 E` such that `F\u27ee\u03b1\u27ef = (\u22a4 : Subalgebra F E)`.","title":"Primitive element theorem"},{"id":"mathlib/thousand-plus/algebra/12-field-theory-and-polynomials/rational-root-theorem","kind":"node","parent":971,"result":true,"route":"../roadmap/mathlib/thousand-plus/algebra/12-field-theory-and-polynomials/index.html#rational-root-theorem","scope":"mathlib/thousand-plus/algebra/12-field-theory-and-polynomials","status":"mathlib","summary":"Mathlib records this with 2 declarations.","title":"Rational root theorem"},{"id":"mathlib/thousand-plus/algebra/12-field-theory-and-polynomials/schwartz-zippel-theorem","kind":"node","parent":971,"result":true,"route":"../roadmap/mathlib/thousand-plus/algebra/12-field-theory-and-polynomials/index.html#schwartz-zippel-theorem","scope":"mathlib/thousand-plus/algebra/12-field-theory-and-polynomials","status":"mathlib","summary":"The Schwartz-Zippel lemma","title":"Schwartz\u2013Zippel theorem"},{"data_url":"scopes/mathlib/thousand-plus/algebra/13-commutative-algebra.json","id":"mathlib/thousand-plus/algebra/13-commutative-algebra","kind":"scope","parent":970,"result":false,"route":"../roadmap/mathlib/thousand-plus/algebra/13-commutative-algebra/index.html","scope":"mathlib/thousand-plus/algebra/13-commutative-algebra","status":"mathlib","title":"13 Commutative algebra"},{"id":"mathlib/thousand-plus/algebra/13-commutative-algebra/going-up-and-going-down-theorems","kind":"node","parent":978,"result":true,"route":"../roadmap/mathlib/thousand-plus/algebra/13-commutative-algebra/index.html#going-up-and-going-down-theorems","scope":"mathlib/thousand-plus/algebra/13-commutative-algebra","status":"mathlib","summary":"Mathlib records this with 4 declarations.","title":"Going-up and going-down theorems"},{"id":"mathlib/thousand-plus/algebra/13-commutative-algebra/jacobson-density-theorem","kind":"node","parent":978,"result":true,"route":"../roadmap/mathlib/thousand-plus/algebra/13-commutative-algebra/index.html#jacobson-density-theorem","scope":"mathlib/thousand-plus/algebra/13-commutative-algebra","status":"mathlib","summary":"Mathlib states this as `jacobson_density`.","title":"Jacobson density theorem"},{"id":"mathlib/thousand-plus/algebra/13-commutative-algebra/krulls-principal-ideal-theorem","kind":"node","parent":978,"result":true,"route":"../roadmap/mathlib/thousand-plus/algebra/13-commutative-algebra/index.html#krulls-principal-ideal-theorem","scope":"mathlib/thousand-plus/algebra/13-commutative-algebra","status":"mathlib","summary":"Krull's principal ideal theorem (also known as Krullscher Hauptidealsatz) : In a commutative Noetherian ring `R`, any prime ideal that is minimal over a principal ideal has height at most 1.","title":"Krull's principal ideal theorem"},{"id":"mathlib/thousand-plus/algebra/13-commutative-algebra/lasker-noether-theorem","kind":"node","parent":978,"result":true,"route":"../roadmap/mathlib/thousand-plus/algebra/13-commutative-algebra/index.html#lasker-noether-theorem","scope":"mathlib/thousand-plus/algebra/13-commutative-algebra","status":"mathlib","summary":"Mathlib states this as `Submodule.isPrimary_decomposition_pairwise_ne_radical`.","title":"Lasker\u2013Noether theorem"},{"id":"mathlib/thousand-plus/algebra/13-commutative-algebra/vietas-formulas","kind":"node","parent":978,"result":true,"route":"../roadmap/mathlib/thousand-plus/algebra/13-commutative-algebra/index.html#vietas-formulas","scope":"mathlib/thousand-plus/algebra/13-commutative-algebra","status":"mathlib","summary":"A sum version of Vieta's formula for `Multiset`: the product of the linear terms `X + \u03bb` where `\u03bb` runs through a multiset `s` is equal to a linear combination of the symmetric functions `esymm s` of the `\u03bb`'s .","title":"Vieta's formulas"},{"id":"mathlib/thousand-plus/algebra/13-commutative-algebra/wedderburns-little-theorem","kind":"node","parent":978,"result":true,"route":"../roadmap/mathlib/thousand-plus/algebra/13-commutative-algebra/index.html#wedderburns-little-theorem","scope":"mathlib/thousand-plus/algebra/13-commutative-algebra","status":"mathlib","summary":"A finite division ring is a field. See `Finite.isDomain_to_isField` and `Fintype.divisionRingOfIsDomain` for more general statements, but these create data, and therefore may cause diamonds if used improperly.","title":"Wedderburn's little theorem"},{"id":"mathlib/thousand-plus/algebra/14-algebraic-geometry","kind":"node","parent":970,"result":false,"route":"../roadmap/mathlib/thousand-plus/algebra/14-algebraic-geometry/index.html","scope":"mathlib/thousand-plus/algebra","status":"can_state","title":"14 Algebraic geometry"},{"data_url":"scopes/mathlib/thousand-plus/algebra/15-linear-and-multilinear-algebra-matrix-theory.json","id":"mathlib/thousand-plus/algebra/15-linear-and-multilinear-algebra-matrix-theory","kind":"scope","parent":970,"result":false,"route":"../roadmap/mathlib/thousand-plus/algebra/15-linear-and-multilinear-algebra-matrix-theory/index.html","scope":"mathlib/thousand-plus/algebra/15-linear-and-multilinear-algebra-matrix-theory","status":"mathlib","title":"15 Linear and multilinear algebra; matrix theory"},{"id":"mathlib/thousand-plus/algebra/15-linear-and-multilinear-algebra-matrix-theory/binomial-inverse-theorem","kind":"node","parent":986,"result":true,"route":"../roadmap/mathlib/thousand-plus/algebra/15-linear-and-multilinear-algebra-matrix-theory/index.html#binomial-inverse-theorem","scope":"mathlib/thousand-plus/algebra/15-linear-and-multilinear-algebra-matrix-theory","status":"mathlib","summary":"The Binomial Inverse Theorem (`\u215f` version).","title":"Binomial inverse theorem"},{"id":"mathlib/thousand-plus/algebra/15-linear-and-multilinear-algebra-matrix-theory/birkhoff-von-neumann-theorem","kind":"node","parent":986,"result":true,"route":"../roadmap/mathlib/thousand-plus/algebra/15-linear-and-multilinear-algebra-matrix-theory/index.html#birkhoff-von-neumann-theorem","scope":"mathlib/thousand-plus/algebra/15-linear-and-multilinear-algebra-matrix-theory","status":"mathlib","summary":"Birkhoff's theorem The set of doubly stochastic matrices is the convex hull of the permutation matrices. Note `exists_eq_sum_perm_of_mem_doublyStochastic` gives a convex weighting of each permutation matrix directly. To show `doublyStochastic n` is convex, use `convex_doublyStochastic`.","title":"Birkhoff\u2013Von Neumann theorem"},{"id":"mathlib/thousand-plus/algebra/15-linear-and-multilinear-algebra-matrix-theory/dimension-theorem-for-vector-spaces","kind":"node","parent":986,"result":true,"route":"../roadmap/mathlib/thousand-plus/algebra/15-linear-and-multilinear-algebra-matrix-theory/index.html#dimension-theorem-for-vector-spaces","scope":"mathlib/thousand-plus/algebra/15-linear-and-multilinear-algebra-matrix-theory","status":"mathlib","summary":"The dimension theorem: if `v` and `v'` are two bases, their index types have the same cardinalities.","title":"Dimension theorem for vector spaces"},{"id":"mathlib/thousand-plus/algebra/15-linear-and-multilinear-algebra-matrix-theory/gershgorin-circle-theorem","kind":"node","parent":986,"result":true,"route":"../roadmap/mathlib/thousand-plus/algebra/15-linear-and-multilinear-algebra-matrix-theory/index.html#gershgorin-circle-theorem","scope":"mathlib/thousand-plus/algebra/15-linear-and-multilinear-algebra-matrix-theory","status":"mathlib","summary":"Gershgorin's circle theorem: for any eigenvalue `\u03bc` of a square matrix `A`, there exists an index `k` such that `\u03bc` lies in the closed ball of center the diagonal term `A k k` and of radius the sum of the norms `\u2211 j \u2260 k, \u2016A k j\u2016`.","title":"Gershgorin circle theorem"},{"id":"mathlib/thousand-plus/algebra/15-linear-and-multilinear-algebra-matrix-theory/rank-nullity-theorem","kind":"node","parent":986,"result":true,"route":"../roadmap/mathlib/thousand-plus/algebra/15-linear-and-multilinear-algebra-matrix-theory/index.html#rank-nullity-theorem","scope":"mathlib/thousand-plus/algebra/15-linear-and-multilinear-algebra-matrix-theory","status":"mathlib","summary":"The rank-nullity theorem","title":"Rank\u2013nullity theorem"},{"id":"mathlib/thousand-plus/algebra/15-linear-and-multilinear-algebra-matrix-theory/sylvesters-determinant-theorem","kind":"node","parent":986,"result":true,"route":"../roadmap/mathlib/thousand-plus/algebra/15-linear-and-multilinear-algebra-matrix-theory/index.html#sylvesters-determinant-theorem","scope":"mathlib/thousand-plus/algebra/15-linear-and-multilinear-algebra-matrix-theory","status":"mathlib","summary":"The Weinstein\u2013Aronszajn identity. Note the `1` on the LHS is of shape m\u00d7m, while the `1` on the RHS is of shape n\u00d7n.","title":"Sylvester's determinant theorem"},{"data_url":"scopes/mathlib/thousand-plus/algebra/16-associative-rings-and-algebras.json","id":"mathlib/thousand-plus/algebra/16-associative-rings-and-algebras","kind":"scope","parent":970,"result":false,"route":"../roadmap/mathlib/thousand-plus/algebra/16-associative-rings-and-algebras/index.html","scope":"mathlib/thousand-plus/algebra/16-associative-rings-and-algebras","status":"mathlib","title":"16 Associative rings and algebras"},{"id":"mathlib/thousand-plus/algebra/16-associative-rings-and-algebras/focal-subgroup-theorem","kind":"node","parent":993,"result":true,"route":"../roadmap/mathlib/thousand-plus/algebra/16-associative-rings-and-algebras/index.html#focal-subgroup-theorem","scope":"mathlib/thousand-plus/algebra/16-associative-rings-and-algebras","status":"mathlib","summary":"The Focal Subgroup Theorem","title":"Focal subgroup theorem"},{"id":"mathlib/thousand-plus/algebra/16-associative-rings-and-algebras/wedderburn-artin-theorem","kind":"node","parent":993,"result":true,"route":"../roadmap/mathlib/thousand-plus/algebra/16-associative-rings-and-algebras/index.html#wedderburn-artin-theorem","scope":"mathlib/thousand-plus/algebra/16-associative-rings-and-algebras","status":"mathlib","summary":"Mathlib records this with 2 declarations.","title":"Wedderburn\u2013Artin theorem"},{"data_url":"scopes/mathlib/thousand-plus/algebra/17-nonassociative-rings-and-algebras.json","id":"mathlib/thousand-plus/algebra/17-nonassociative-rings-and-algebras","kind":"scope","parent":970,"result":false,"route":"../roadmap/mathlib/thousand-plus/algebra/17-nonassociative-rings-and-algebras/index.html","scope":"mathlib/thousand-plus/algebra/17-nonassociative-rings-and-algebras","status":"mathlib","title":"17 Nonassociative rings and algebras"},{"id":"mathlib/thousand-plus/algebra/17-nonassociative-rings-and-algebras/lies-theorem","kind":"node","parent":996,"result":true,"route":"../roadmap/mathlib/thousand-plus/algebra/17-nonassociative-rings-and-algebras/index.html#lies-theorem","scope":"mathlib/thousand-plus/algebra/17-nonassociative-rings-and-algebras","status":"mathlib","summary":"Lie's theorem: Lie modules of solvable Lie algebras over fields of characteristic 0 have a common eigenvector for the action of all elements of the Lie algebra.","title":"Lie's theorem"},{"data_url":"scopes/mathlib/thousand-plus/algebra/18-category-theory-homological-algebra.json","id":"mathlib/thousand-plus/algebra/18-category-theory-homological-algebra","kind":"scope","parent":970,"result":false,"route":"../roadmap/mathlib/thousand-plus/algebra/18-category-theory-homological-algebra/index.html","scope":"mathlib/thousand-plus/algebra/18-category-theory-homological-algebra","status":"mathlib","title":"18 Category theory; homological algebra"},{"id":"mathlib/thousand-plus/algebra/18-category-theory-homological-algebra/becks-monadicity-theorem","kind":"node","parent":998,"result":true,"route":"../roadmap/mathlib/thousand-plus/algebra/18-category-theory-homological-algebra/index.html#becks-monadicity-theorem","scope":"mathlib/thousand-plus/algebra/18-category-theory-homological-algebra","status":"mathlib","summary":"Beck's monadicity theorem: if `G` has a left adjoint and creates coequalizers of `G`-split pairs, then it is monadic. This is the converse of `createsGSplitCoequalizersOfMonadic`.","title":"Beck's monadicity theorem"},{"id":"mathlib/thousand-plus/algebra/18-category-theory-homological-algebra/mitchells-embedding-theorem","kind":"node","parent":998,"result":true,"route":"../roadmap/mathlib/thousand-plus/algebra/18-category-theory-homological-algebra/index.html#mitchells-embedding-theorem","scope":"mathlib/thousand-plus/algebra/18-category-theory-homological-algebra","status":"mathlib","summary":"The Freyd-Mitchell embedding theorem. See also `FreydMitchell.functor` for a functor which has the relevant instances.","title":"Mitchell's embedding theorem"},{"data_url":"scopes/mathlib/thousand-plus/algebra/20-group-theory-and-generalizations.json","id":"mathlib/thousand-plus/algebra/20-group-theory-and-generalizations","kind":"scope","parent":970,"result":false,"route":"../roadmap/mathlib/thousand-plus/algebra/20-group-theory-and-generalizations/index.html","scope":"mathlib/thousand-plus/algebra/20-group-theory-and-generalizations","status":"mathlib","title":"20 Group theory and generalizations"},{"id":"mathlib/thousand-plus/algebra/20-group-theory-and-generalizations/cauchys-theorem-group-theory","kind":"node","parent":1001,"result":true,"route":"../roadmap/mathlib/thousand-plus/algebra/20-group-theory-and-generalizations/index.html#cauchys-theorem-group-theory","scope":"mathlib/thousand-plus/algebra/20-group-theory-and-generalizations","status":"mathlib","summary":"For every prime `p` dividing the order of a finite group `G` there exists an element of order `p` in `G`. This is known as Cauchy's theorem.","title":"Cauchy's theorem (group theory)"},{"id":"mathlib/thousand-plus/algebra/20-group-theory-and-generalizations/jordan-holder-theorem","kind":"node","parent":1001,"result":true,"route":"../roadmap/mathlib/thousand-plus/algebra/20-group-theory-and-generalizations/index.html#jordan-holder-theorem","scope":"mathlib/thousand-plus/algebra/20-group-theory-and-generalizations","status":"mathlib","summary":"The Jordan-H\u00f6lder theorem, stated for any `JordanHolderLattice`. If two composition series start and finish at the same place, they are equivalent.","title":"Jordan\u2013H\u00f6lder theorem"},{"id":"mathlib/thousand-plus/algebra/20-group-theory-and-generalizations/nielsen-schreier-theorem","kind":"node","parent":1001,"result":true,"route":"../roadmap/mathlib/thousand-plus/algebra/20-group-theory-and-generalizations/index.html#nielsen-schreier-theorem","scope":"mathlib/thousand-plus/algebra/20-group-theory-and-generalizations","status":"mathlib","summary":"The Nielsen-Schreier theorem: a subgroup of a free group is free.","title":"Nielsen\u2013Schreier theorem"},{"id":"mathlib/thousand-plus/algebra/20-group-theory-and-generalizations/schurs-lemma","kind":"node","parent":1001,"result":true,"route":"../roadmap/mathlib/thousand-plus/algebra/20-group-theory-and-generalizations/index.html#schurs-lemma","scope":"mathlib/thousand-plus/algebra/20-group-theory-and-generalizations","status":"mathlib","summary":"Schur's lemma for endomorphisms in `\ud835\udd5c`-linear categories.","title":"Schur's lemma (1000+ theorems)"},{"id":"mathlib/thousand-plus/algebra/20-group-theory-and-generalizations/schur-zassenhaus-theorem","kind":"node","parent":1001,"result":true,"route":"../roadmap/mathlib/thousand-plus/algebra/20-group-theory-and-generalizations/index.html#schur-zassenhaus-theorem","scope":"mathlib/thousand-plus/algebra/20-group-theory-and-generalizations","status":"mathlib","summary":"Schur-Zassenhaus for normal subgroups: If `H : Subgroup G` is normal, and has order coprime to its index, then there exists a subgroup `K` which is a (left) complement of `H`.","title":"Schur\u2013Zassenhaus theorem"},{"data_url":"scopes/mathlib/thousand-plus/analysis.json","id":"mathlib/thousand-plus/analysis","kind":"scope","parent":933,"result":false,"route":"../roadmap/mathlib/thousand-plus/analysis/index.html","scope":"mathlib/thousand-plus/analysis","status":"mathlib","title":"Analysis"},{"data_url":"scopes/mathlib/thousand-plus/analysis/26-real-functions.json","id":"mathlib/thousand-plus/analysis/26-real-functions","kind":"scope","parent":1007,"result":false,"route":"../roadmap/mathlib/thousand-plus/analysis/26-real-functions/index.html","scope":"mathlib/thousand-plus/analysis/26-real-functions","status":"mathlib","title":"26 Real functions"},{"data_url":"scopes/mathlib/thousand-plus/analysis/26-real-functions/part-01.json","id":"mathlib/thousand-plus/analysis/26-real-functions/part-01","kind":"scope","navigation":"transparent","parent":1008,"result":false,"route":"../roadmap/mathlib/thousand-plus/analysis/26-real-functions/part-01/index.html","scope":"mathlib/thousand-plus/analysis/26-real-functions/part-01","status":"mathlib","title":"Abel's theorem to Rademacher's theorem"},{"id":"mathlib/thousand-plus/analysis/26-real-functions/part-01/abels-theorem","kind":"node","parent":1009,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/26-real-functions/part-01/index.html#abels-theorem","scope":"mathlib/thousand-plus/analysis/26-real-functions/part-01","status":"mathlib","summary":"Mathlib records this with 2 declarations.","title":"Abel's theorem"},{"id":"mathlib/thousand-plus/analysis/26-real-functions/part-01/besicovitch-covering-theorem","kind":"node","parent":1009,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/26-real-functions/part-01/index.html#besicovitch-covering-theorem","scope":"mathlib/thousand-plus/analysis/26-real-functions/part-01","status":"mathlib","summary":"The measurable Besicovitch covering theorem.","title":"Besicovitch covering theorem"},{"id":"mathlib/thousand-plus/analysis/26-real-functions/part-01/bolzano-weierstrass-theorem","kind":"node","parent":1009,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/26-real-functions/part-01/index.html#bolzano-weierstrass-theorem","scope":"mathlib/thousand-plus/analysis/26-real-functions/part-01","status":"mathlib","summary":"Mathlib records this with 2 declarations.","title":"Bolzano\u2013Weierstrass theorem (1000+ theorems)"},{"id":"mathlib/thousand-plus/analysis/26-real-functions/part-01/cantors-intersection-theorem","kind":"node","parent":1009,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/26-real-functions/part-01/index.html#cantors-intersection-theorem","scope":"mathlib/thousand-plus/analysis/26-real-functions/part-01","status":"mathlib","summary":"Cantor's intersection theorem for `sInter`: the intersection of a directed family of nonempty compact closed sets is nonempty.","title":"Cantor's intersection theorem"},{"id":"mathlib/thousand-plus/analysis/26-real-functions/part-01/darbouxs-theorem-analysis","kind":"node","parent":1009,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/26-real-functions/part-01/index.html#darbouxs-theorem-analysis","scope":"mathlib/thousand-plus/analysis/26-real-functions/part-01","status":"mathlib","summary":"Mathlib records this with 10 declarations.","title":"Darboux's theorem (analysis)"},{"id":"mathlib/thousand-plus/analysis/26-real-functions/part-01/dinis-theorem","kind":"node","parent":1009,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/26-real-functions/part-01/index.html#dinis-theorem","scope":"mathlib/thousand-plus/analysis/26-real-functions/part-01","status":"mathlib","summary":"Dini's theorem: if `F n` is a monotone increasing collection of continuous functions converging pointwise to a continuous function `f`, then `F n` converges locally uniformly to `f`.","title":"Dini's theorem"},{"id":"mathlib/thousand-plus/analysis/26-real-functions/part-01/fermats-theorem-stationary-points","kind":"node","parent":1009,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/26-real-functions/part-01/index.html#fermats-theorem-stationary-points","scope":"mathlib/thousand-plus/analysis/26-real-functions/part-01","status":"mathlib","summary":"Mathlib records this with 2 declarations.","title":"Fermat's theorem (stationary points)"},{"id":"mathlib/thousand-plus/analysis/26-real-functions/part-01/lebesgue-differentiation-theorem","kind":"node","parent":1009,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/26-real-functions/part-01/index.html#lebesgue-differentiation-theorem","scope":"mathlib/thousand-plus/analysis/26-real-functions/part-01","status":"mathlib","summary":"Mathlib states this as `VitaliFamily.ae_tendsto_lintegral_div`.","title":"Lebesgue differentiation theorem"},{"id":"mathlib/thousand-plus/analysis/26-real-functions/part-01/monotone-convergence-theorem","kind":"node","parent":1009,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/26-real-functions/part-01/index.html#monotone-convergence-theorem","scope":"mathlib/thousand-plus/analysis/26-real-functions/part-01","status":"mathlib","summary":"Mathlib records this with 2 declarations.","title":"Monotone convergence theorem (1000+ theorems)"},{"id":"mathlib/thousand-plus/analysis/26-real-functions/part-01/rademachers-theorem","kind":"node","parent":1009,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/26-real-functions/part-01/index.html#rademachers-theorem","scope":"mathlib/thousand-plus/analysis/26-real-functions/part-01","status":"mathlib","summary":"*Rademacher's theorem*: a function between finite-dimensional real vector spaces which is Lipschitz on a set is differentiable almost everywhere in this set.","title":"Rademacher's theorem"},{"data_url":"scopes/mathlib/thousand-plus/analysis/26-real-functions/part-02.json","id":"mathlib/thousand-plus/analysis/26-real-functions/part-02","kind":"scope","navigation":"transparent","parent":1008,"result":false,"route":"../roadmap/mathlib/thousand-plus/analysis/26-real-functions/part-02/index.html","scope":"mathlib/thousand-plus/analysis/26-real-functions/part-02","status":"mathlib","title":"Rolle's theorem to Taylor's theorem"},{"id":"mathlib/thousand-plus/analysis/26-real-functions/part-02/squeeze-theorem","kind":"node","parent":1020,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/26-real-functions/part-02/index.html#squeeze-theorem","scope":"mathlib/thousand-plus/analysis/26-real-functions/part-02","status":"mathlib","summary":"Squeeze theorem (also known as sandwich theorem). This version assumes that inequalities hold eventually for the filter.","title":"Squeeze theorem"},{"data_url":"scopes/mathlib/thousand-plus/analysis/28-measure-and-integration.json","id":"mathlib/thousand-plus/analysis/28-measure-and-integration","kind":"scope","parent":1007,"result":false,"route":"../roadmap/mathlib/thousand-plus/analysis/28-measure-and-integration/index.html","scope":"mathlib/thousand-plus/analysis/28-measure-and-integration","status":"mathlib","title":"28 Measure and integration"},{"id":"mathlib/thousand-plus/analysis/28-measure-and-integration/bounded-convergence-theorem","kind":"node","parent":1022,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/28-measure-and-integration/index.html#bounded-convergence-theorem","scope":"mathlib/thousand-plus/analysis/28-measure-and-integration","status":"mathlib","summary":"Mathlib records this with 2 declarations.","title":"Bounded convergence theorem"},{"id":"mathlib/thousand-plus/analysis/28-measure-and-integration/caratheodorys-extension-theorem","kind":"node","parent":1022,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/28-measure-and-integration/index.html#caratheodorys-extension-theorem","scope":"mathlib/thousand-plus/analysis/28-measure-and-integration","status":"mathlib","summary":"Mathlib records this with 2 declarations.","title":"Carath\u00e9odory's extension theorem"},{"id":"mathlib/thousand-plus/analysis/28-measure-and-integration/caratheodorys-theorem-measure-theory","kind":"node","parent":1022,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/28-measure-and-integration/index.html#caratheodorys-theorem-measure-theory","scope":"mathlib/thousand-plus/analysis/28-measure-and-integration","status":"mathlib","summary":"Given an outer measure `\u03bc`, the Carath\u00e9odory-measurable space is defined such that `s` is measurable if `\u2200 t, \u03bc t = \u03bc (t \u2229 s) + \u03bc (t \\ s)`.","title":"Carath\u00e9odory's theorem (measure theory)"},{"id":"mathlib/thousand-plus/analysis/28-measure-and-integration/egorovs-theorem","kind":"node","parent":1022,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/28-measure-and-integration/index.html#egorovs-theorem","scope":"mathlib/thousand-plus/analysis/28-measure-and-integration","status":"mathlib","summary":"Egorov's theorem: If `f : \u03b9 \u2192 \u03b1 \u2192 \u03b2` is a sequence of strongly measurable functions that converges to `g : \u03b1 \u2192 \u03b2` almost everywhere on a measurable set `s` of finite measure, then for all `\u03b5 > 0`, there exists a subset `t \u2286 s` such that `\u03bc t \u2264 \u03b5` and `f` converges to `g` uniformly on `s \\ t`. We require the index type `\u03b9` to be countable, and usually `\u03b9 = \u2115`.","title":"Egorov's theorem"},{"id":"mathlib/thousand-plus/analysis/28-measure-and-integration/ferniques-theorem","kind":"node","parent":1022,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/28-measure-and-integration/index.html#ferniques-theorem","scope":"mathlib/thousand-plus/analysis/28-measure-and-integration","status":"mathlib","summary":"Fernique's theorem: for a Gaussian measure, there exists `C > 0` such that the function `x \u21a6 exp (C * \u2016x\u2016 ^ 2)` is integrable.","title":"Fernique's theorem"},{"id":"mathlib/thousand-plus/analysis/28-measure-and-integration/hahn-decomposition-theorem","kind":"node","parent":1022,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/28-measure-and-integration/index.html#hahn-decomposition-theorem","scope":"mathlib/thousand-plus/analysis/28-measure-and-integration","status":"mathlib","summary":"The Hahn decomposition theorem: Given a signed measure `s`, there exist complement measurable sets `i` and `j` such that `i` is positive, `j` is negative.","title":"Hahn decomposition theorem"},{"id":"mathlib/thousand-plus/analysis/28-measure-and-integration/prokhorovs-theorem","kind":"node","parent":1022,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/28-measure-and-integration/index.html#prokhorovs-theorem","scope":"mathlib/thousand-plus/analysis/28-measure-and-integration","status":"mathlib","summary":"Mathlib records this with 4 declarations.","title":"Prokhorov's theorem"},{"id":"mathlib/thousand-plus/analysis/28-measure-and-integration/radon-nikodym-theorem","kind":"node","parent":1022,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/28-measure-and-integration/index.html#radon-nikodym-theorem","scope":"mathlib/thousand-plus/analysis/28-measure-and-integration","status":"mathlib","summary":"The Radon-Nikodym theorem: Given two measures `\u03bc` and `\u03bd`, if `HaveLebesgueDecomposition \u03bc \u03bd`, then `\u03bc` is absolutely continuous to `\u03bd` if and only if `\u03bd.withDensity (rnDeriv \u03bc \u03bd) = \u03bc`.","title":"Radon\u2013Nikodym theorem"},{"id":"mathlib/thousand-plus/analysis/28-measure-and-integration/schroeder-bernstein-theorem-for-measurable-space","kind":"node","parent":1022,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/28-measure-and-integration/index.html#schroeder-bernstein-theorem-for-measurable-space","scope":"mathlib/thousand-plus/analysis/28-measure-and-integration","status":"mathlib","summary":"The measurable Schr\u00f6der-Bernstein Theorem: given measurable embeddings `\u03b1 \u2192 \u03b2` and `\u03b2 \u2192 \u03b1`, we can find a measurable equivalence `\u03b1 \u2243\u1d50 \u03b2`.","title":"Schroeder\u2013Bernstein theorem for measurable spaces"},{"id":"mathlib/thousand-plus/analysis/28-measure-and-integration/steinhaus-theorem","kind":"node","parent":1022,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/28-measure-and-integration/index.html#steinhaus-theorem","scope":"mathlib/thousand-plus/analysis/28-measure-and-integration","status":"mathlib","summary":"Steinhaus Theorem.","title":"Steinhaus theorem"},{"id":"mathlib/thousand-plus/analysis/28-measure-and-integration/vitali-convergence-theorem","kind":"node","parent":1022,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/28-measure-and-integration/index.html#vitali-convergence-theorem","scope":"mathlib/thousand-plus/analysis/28-measure-and-integration","status":"mathlib","summary":"Vitali's convergence theorem (non-finite measure version).","title":"Vitali convergence theorem"},{"id":"mathlib/thousand-plus/analysis/28-measure-and-integration/vitali-covering-theorem","kind":"node","parent":1022,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/28-measure-and-integration/index.html#vitali-covering-theorem","scope":"mathlib/thousand-plus/analysis/28-measure-and-integration","status":"mathlib","summary":"The measurable Vitali covering theorem.","title":"Vitali covering theorem"},{"data_url":"scopes/mathlib/thousand-plus/analysis/30-functions-of-a-complex-variable.json","id":"mathlib/thousand-plus/analysis/30-functions-of-a-complex-variable","kind":"scope","parent":1007,"result":false,"route":"../roadmap/mathlib/thousand-plus/analysis/30-functions-of-a-complex-variable/index.html","scope":"mathlib/thousand-plus/analysis/30-functions-of-a-complex-variable","status":"mathlib","title":"30 Functions of a complex variable"},{"id":"mathlib/thousand-plus/analysis/30-functions-of-a-complex-variable/borel-caratheodory-theorem","kind":"node","parent":1035,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/30-functions-of-a-complex-variable/index.html#borel-caratheodory-theorem","scope":"mathlib/thousand-plus/analysis/30-functions-of-a-complex-variable","status":"mathlib","summary":"Borel-Carath\u00e9odory theorem.","title":"Borel\u2013Carath\u00e9odory theorem"},{"id":"mathlib/thousand-plus/analysis/30-functions-of-a-complex-variable/cauchy-hadamard-theorem","kind":"node","parent":1035,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/30-functions-of-a-complex-variable/index.html#cauchy-hadamard-theorem","scope":"mathlib/thousand-plus/analysis/30-functions-of-a-complex-variable","status":"mathlib","summary":"The Cauchy-Hadamard theorem for formal multilinear series: The inverse of the radius is equal to $\\limsup_{n\\to\\infty} \\sqrt[n]{\u2016p n\u2016}$.","title":"Cauchy\u2013Hadamard theorem"},{"id":"mathlib/thousand-plus/analysis/30-functions-of-a-complex-variable/gauss-lucas-theorem","kind":"node","parent":1035,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/30-functions-of-a-complex-variable/index.html#gauss-lucas-theorem","scope":"mathlib/thousand-plus/analysis/30-functions-of-a-complex-variable","status":"mathlib","summary":"Mathlib records this with 2 declarations.","title":"Gauss\u2013Lucas theorem"},{"id":"mathlib/thousand-plus/analysis/30-functions-of-a-complex-variable/hadamard-three-lines-theorem","kind":"node","parent":1035,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/30-functions-of-a-complex-variable/index.html#hadamard-three-lines-theorem","scope":"mathlib/thousand-plus/analysis/30-functions-of-a-complex-variable","status":"mathlib","summary":"Hadamard three-line theorem: If `f` is a bounded function, continuous on the closed strip `re \u207b\u00b9' [l, u]` and differentiable on open strip `re \u207b\u00b9' (l, u)`, then for `M(x) := sup ((norm \u2218 f) '' re \u207b\u00b9' {x})` we have that for all `z` in the closed strip `re \u207b\u00b9' [a,b]` the inequality `\u2016f(z)\u2016 \u2264 M(0) ^ (1 - ((z.re - l) / (u - l))) * M(1) ^ ((z.re - l) / (u - l))` holds.","title":"Hadamard three-lines theorem"},{"id":"mathlib/thousand-plus/analysis/30-functions-of-a-complex-variable/mellin-inversion-theorem","kind":"node","parent":1035,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/30-functions-of-a-complex-variable/index.html#mellin-inversion-theorem","scope":"mathlib/thousand-plus/analysis/30-functions-of-a-complex-variable","status":"mathlib","summary":"The inverse Mellin transform of the Mellin transform applied to `x > 0` is x.","title":"Mellin inversion theorem"},{"id":"mathlib/thousand-plus/analysis/30-functions-of-a-complex-variable/moreras-theorem","kind":"node","parent":1035,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/30-functions-of-a-complex-variable/index.html#moreras-theorem","scope":"mathlib/thousand-plus/analysis/30-functions-of-a-complex-variable","status":"mathlib","summary":"Mathlib states this as `Complex.isConservativeOn_and_continuousOn_iff_isDifferentiableOn`.","title":"Morera's theorem"},{"id":"mathlib/thousand-plus/analysis/30-functions-of-a-complex-variable/open-mapping-theorem-complex-analysis","kind":"node","parent":1035,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/30-functions-of-a-complex-variable/index.html#open-mapping-theorem-complex-analysis","scope":"mathlib/thousand-plus/analysis/30-functions-of-a-complex-variable","status":"mathlib","summary":"The *open mapping theorem* for holomorphic functions, global version: if a function `g : E \u2192 \u2102` is analytic on a connected set `U`, then either it is constant on `U`, or it is open on `U` (in the sense that it maps any open set contained in `U` to an open set in `\u2102`).","title":"Open mapping theorem (complex analysis)"},{"data_url":"scopes/mathlib/thousand-plus/analysis/32-several-complex-variables-and-analytic-spaces.json","id":"mathlib/thousand-plus/analysis/32-several-complex-variables-and-analytic-spaces","kind":"scope","parent":1007,"result":false,"route":"../roadmap/mathlib/thousand-plus/analysis/32-several-complex-variables-and-analytic-spaces/index.html","scope":"mathlib/thousand-plus/analysis/32-several-complex-variables-and-analytic-spaces","status":"mathlib","title":"32 Several complex variables and analytic spaces"},{"id":"mathlib/thousand-plus/analysis/32-several-complex-variables-and-analytic-spaces/weierstrass-preparation-theorem","kind":"node","parent":1043,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/32-several-complex-variables-and-analytic-spaces/index.html#weierstrass-preparation-theorem","scope":"mathlib/thousand-plus/analysis/32-several-complex-variables-and-analytic-spaces","status":"mathlib","summary":"Weierstrass preparation theorem ([washington_cyclotomic], Theorem 7.3): let `g` be a power series over a complete local ring, such that its image in the residue field is not zero. Then there exists a distinguished polynomial `f` and a power series `h` which is a unit, such that `g = f * h`.","title":"Weierstrass preparation theorem"},{"id":"mathlib/thousand-plus/analysis/34-ordinary-differential-equations","kind":"node","parent":1007,"result":false,"route":"../roadmap/mathlib/thousand-plus/analysis/34-ordinary-differential-equations/index.html","scope":"mathlib/thousand-plus/analysis","status":"can_state","title":"34 Ordinary differential equations"},{"data_url":"scopes/mathlib/thousand-plus/analysis/35-partial-differential-equations.json","id":"mathlib/thousand-plus/analysis/35-partial-differential-equations","kind":"scope","parent":1007,"result":false,"route":"../roadmap/mathlib/thousand-plus/analysis/35-partial-differential-equations/index.html","scope":"mathlib/thousand-plus/analysis/35-partial-differential-equations","status":"mathlib","title":"35 Partial differential equations"},{"id":"mathlib/thousand-plus/analysis/35-partial-differential-equations/eulers-theorem-on-homogeneous-functions","kind":"node","parent":1046,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/35-partial-differential-equations/index.html#eulers-theorem-on-homogeneous-functions","scope":"mathlib/thousand-plus/analysis/35-partial-differential-equations","status":"mathlib","summary":"Mathlib records this with 3 declarations.","title":"Euler's theorem on homogeneous functions"},{"data_url":"scopes/mathlib/thousand-plus/analysis/37-dynamical-systems-and-ergodic-theory.json","id":"mathlib/thousand-plus/analysis/37-dynamical-systems-and-ergodic-theory","kind":"scope","parent":1007,"result":false,"route":"../roadmap/mathlib/thousand-plus/analysis/37-dynamical-systems-and-ergodic-theory/index.html","scope":"mathlib/thousand-plus/analysis/37-dynamical-systems-and-ergodic-theory","status":"mathlib","title":"37 Dynamical systems and ergodic theory"},{"id":"mathlib/thousand-plus/analysis/37-dynamical-systems-and-ergodic-theory/poincare-recurrence-theorem","kind":"node","parent":1048,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/37-dynamical-systems-and-ergodic-theory/index.html#poincare-recurrence-theorem","scope":"mathlib/thousand-plus/analysis/37-dynamical-systems-and-ergodic-theory","status":"mathlib","summary":"Poincar\u00e9 recurrence theorem: given a conservative map `f` and a measurable set `s`, almost every point `x \u2208 s` returns back to `s` infinitely many times.","title":"Poincar\u00e9 recurrence theorem"},{"id":"mathlib/thousand-plus/analysis/42-harmonic-analysis-on-euclidean-spaces","kind":"node","parent":1007,"result":false,"route":"../roadmap/mathlib/thousand-plus/analysis/42-harmonic-analysis-on-euclidean-spaces/index.html","scope":"mathlib/thousand-plus/analysis","status":"can_state","title":"42 Harmonic analysis on Euclidean spaces"},{"data_url":"scopes/mathlib/thousand-plus/analysis/43-abstract-harmonic-analysis.json","id":"mathlib/thousand-plus/analysis/43-abstract-harmonic-analysis","kind":"scope","parent":1007,"result":false,"route":"../roadmap/mathlib/thousand-plus/analysis/43-abstract-harmonic-analysis/index.html","scope":"mathlib/thousand-plus/analysis/43-abstract-harmonic-analysis","status":"mathlib","title":"43 Abstract harmonic analysis"},{"id":"mathlib/thousand-plus/analysis/43-abstract-harmonic-analysis/fourier-theorem","kind":"node","parent":1051,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/43-abstract-harmonic-analysis/index.html#fourier-theorem","scope":"mathlib/thousand-plus/analysis/43-abstract-harmonic-analysis","status":"mathlib","summary":"Mathlib records this with 2 declarations.","title":"Fourier theorem"},{"data_url":"scopes/mathlib/thousand-plus/analysis/46-functional-analysis.json","id":"mathlib/thousand-plus/analysis/46-functional-analysis","kind":"scope","parent":1007,"result":false,"route":"../roadmap/mathlib/thousand-plus/analysis/46-functional-analysis/index.html","scope":"mathlib/thousand-plus/analysis/46-functional-analysis","status":"mathlib","title":"46 Functional analysis"},{"id":"mathlib/thousand-plus/analysis/46-functional-analysis/banach-alaoglu-theorem","kind":"node","parent":1053,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/46-functional-analysis/index.html#banach-alaoglu-theorem","scope":"mathlib/thousand-plus/analysis/46-functional-analysis","status":"mathlib","summary":"The Banach-Alaoglu theorem: the polar set of a neighborhood `s` of the origin in a normed space `E` is a compact subset of `WeakDual \ud835\udd5c E`.","title":"Banach\u2013Alaoglu theorem"},{"id":"mathlib/thousand-plus/analysis/46-functional-analysis/closed-graph-theorem","kind":"node","parent":1053,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/46-functional-analysis/index.html#closed-graph-theorem","scope":"mathlib/thousand-plus/analysis/46-functional-analysis","status":"mathlib","summary":"The closed graph theorem : a linear map between two Banach spaces whose graph is closed is continuous.","title":"Closed graph theorem"},{"id":"mathlib/thousand-plus/analysis/46-functional-analysis/gelfand-mazur-theorem","kind":"node","parent":1053,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/46-functional-analysis/index.html#gelfand-mazur-theorem","scope":"mathlib/thousand-plus/analysis/46-functional-analysis","status":"mathlib","summary":"Mathlib records this with 3 declarations.","title":"Gelfand\u2013Mazur theorem"},{"id":"mathlib/thousand-plus/analysis/46-functional-analysis/hellinger-toeplitz-theorem","kind":"node","parent":1053,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/46-functional-analysis/index.html#hellinger-toeplitz-theorem","scope":"mathlib/thousand-plus/analysis/46-functional-analysis","status":"mathlib","summary":"The Hellinger--Toeplitz theorem: if a symmetric operator is defined on a complete space, then it is automatically continuous.","title":"Hellinger\u2013Toeplitz theorem"},{"id":"mathlib/thousand-plus/analysis/46-functional-analysis/m-riesz-extension-theorem","kind":"node","parent":1053,"result":true,"route":"../roadmap/mathlib/thousand-plus/analysis/46-functional-analysis/index.html#m-riesz-extension-theorem","scope":"mathlib/thousand-plus/analysis/46-functional-analysis","status":"mathlib","summary":"M. Riesz extension theorem: given a convex cone `s` in a vector space `E`, a submodule `p`, and a linear `f : p \u2192 \u211d`, assume that `f` is nonnegative on `p \u2229 s` and `p + s = E`. Then there exists a globally defined linear function `g : E \u2192 \u211d` that agrees with `f` on `p`, and is nonnegative on `s`.","title":"M. Riesz extension theorem"},{"data_url":"scopes/mathlib/thousand-plus/geometry-and-topology.json","id":"mathlib/thousand-plus/geometry-and-topology","kind":"scope","parent":933,"result":false,"route":"../roadmap/mathlib/thousand-plus/geometry-and-topology/index.html","scope":"mathlib/thousand-plus/geometry-and-topology","status":"mathlib","title":"Geometry and topology"},{"data_url":"scopes/mathlib/thousand-plus/geometry-and-topology/51-geometry.json","id":"mathlib/thousand-plus/geometry-and-topology/51-geometry","kind":"scope","parent":1059,"result":false,"route":"../roadmap/mathlib/thousand-plus/geometry-and-topology/51-geometry/index.html","scope":"mathlib/thousand-plus/geometry-and-topology/51-geometry","status":"mathlib","title":"51 Geometry"},{"id":"mathlib/thousand-plus/geometry-and-topology/51-geometry/apolloniuss-theorem","kind":"node","parent":1060,"result":true,"route":"../roadmap/mathlib/thousand-plus/geometry-and-topology/51-geometry/index.html#apolloniuss-theorem","scope":"mathlib/thousand-plus/geometry-and-topology/51-geometry","status":"mathlib","summary":"Apollonius's Theorem.","title":"Apollonius's theorem"},{"id":"mathlib/thousand-plus/geometry-and-topology/51-geometry/british-flag-theorem","kind":"node","parent":1060,"result":true,"route":"../roadmap/mathlib/thousand-plus/geometry-and-topology/51-geometry/index.html#british-flag-theorem","scope":"mathlib/thousand-plus/geometry-and-topology/51-geometry","status":"mathlib","summary":"British flag theorem: for a rectangle `a b c d` and any point `p` (not necessarily in the plane of the rectangle), the sum of the squares of the distances from `p` to the opposite corners `a` and `c` equals the sum of the squares of the distances from `p` to `b` and `d`.","title":"British flag theorem"},{"id":"mathlib/thousand-plus/geometry-and-topology/51-geometry/commandinos-theorem","kind":"node","parent":1060,"result":true,"route":"../roadmap/mathlib/thousand-plus/geometry-and-topology/51-geometry/index.html#commandinos-theorem","scope":"mathlib/thousand-plus/geometry-and-topology/51-geometry","status":"mathlib","summary":"*Commandino's theorem* : For n-simplex, the vector from a vertex to the `centroid` equals `n` times the vector from the `centroid` to the corresponding `faceOppositeCentroid`.","title":"Commandino's theorem"},{"id":"mathlib/thousand-plus/geometry-and-topology/51-geometry/cpctc","kind":"node","parent":1060,"result":true,"route":"../roadmap/mathlib/thousand-plus/geometry-and-topology/51-geometry/index.html#cpctc","scope":"mathlib/thousand-plus/geometry-and-topology/51-geometry","status":"mathlib","summary":"Mathlib records this with 2 declarations.","title":"CPCTC"},{"id":"mathlib/thousand-plus/geometry-and-topology/51-geometry/exterior-angle-theorem","kind":"node","parent":1060,"result":true,"route":"../roadmap/mathlib/thousand-plus/geometry-and-topology/51-geometry/index.html#exterior-angle-theorem","scope":"mathlib/thousand-plus/geometry-and-topology/51-geometry","status":"mathlib","summary":"The Exterior angle theorem. an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.","title":"Exterior angle theorem"},{"id":"mathlib/thousand-plus/geometry-and-topology/51-geometry/geometric-mean-theorem","kind":"node","parent":1060,"result":true,"route":"../roadmap/mathlib/thousand-plus/geometry-and-topology/51-geometry/index.html#geometric-mean-theorem","scope":"mathlib/thousand-plus/geometry-and-topology/51-geometry","status":"mathlib","summary":"Geometric mean theorem (or right triangle altitude theorem): in a right-angled triangle `p\u2081 p\u2082 p\u2083` with the right angle at `p\u2082`, the square of the altitude `dist p\u2082 p\u2084` to the hypotenuse is the product of the two segments `dist p\u2081 p\u2084` and `dist p\u2083 p\u2084` into which its foot `p\u2084` divides the hypotenuse.","title":"Geometric mean theorem"},{"id":"mathlib/thousand-plus/geometry-and-topology/51-geometry/intersecting-secants-theorem","kind":"node","parent":1060,"result":true,"route":"../roadmap/mathlib/thousand-plus/geometry-and-topology/51-geometry/index.html#intersecting-secants-theorem","scope":"mathlib/thousand-plus/geometry-and-topology/51-geometry","status":"mathlib","summary":"Intersecting Secants Theorem.","title":"Intersecting secants theorem"},{"id":"mathlib/thousand-plus/geometry-and-topology/51-geometry/stewarts-theorem","kind":"node","parent":1060,"result":true,"route":"../roadmap/mathlib/thousand-plus/geometry-and-topology/51-geometry/index.html#stewarts-theorem","scope":"mathlib/thousand-plus/geometry-and-topology/51-geometry","status":"mathlib","summary":"Stewart's Theorem.","title":"Stewart's theorem"},{"id":"mathlib/thousand-plus/geometry-and-topology/51-geometry/tangent-secant-theorem","kind":"node","parent":1060,"result":true,"route":"../roadmap/mathlib/thousand-plus/geometry-and-topology/51-geometry/index.html#tangent-secant-theorem","scope":"mathlib/thousand-plus/geometry-and-topology/51-geometry","status":"mathlib","summary":"Tangent-Secant Theorem. The square of the tangent length equals the product of secant segment lengths.","title":"Tangent-secant theorem"},{"id":"mathlib/thousand-plus/geometry-and-topology/51-geometry/thaless-theorem","kind":"node","parent":1060,"result":true,"route":"../roadmap/mathlib/thousand-plus/geometry-and-topology/51-geometry/index.html#thaless-theorem","scope":"mathlib/thousand-plus/geometry-and-topology/51-geometry","status":"mathlib","summary":"Alias of `EuclideanGeometry.Sphere.angle_eq_pi_div_two_iff_mem_sphere_of_isDiameter`.","title":"Thales's theorem"},{"data_url":"scopes/mathlib/thousand-plus/geometry-and-topology/52-convex-and-discrete-geometry.json","id":"mathlib/thousand-plus/geometry-and-topology/52-convex-and-discrete-geometry","kind":"scope","parent":1059,"result":false,"route":"../roadmap/mathlib/thousand-plus/geometry-and-topology/52-convex-and-discrete-geometry/index.html","scope":"mathlib/thousand-plus/geometry-and-topology/52-convex-and-discrete-geometry","status":"mathlib","title":"52 Convex and discrete geometry"},{"id":"mathlib/thousand-plus/geometry-and-topology/52-convex-and-discrete-geometry/hellys-theorem","kind":"node","parent":1071,"result":true,"route":"../roadmap/mathlib/thousand-plus/geometry-and-topology/52-convex-and-discrete-geometry/index.html#hellys-theorem","scope":"mathlib/thousand-plus/geometry-and-topology/52-convex-and-discrete-geometry","status":"mathlib","summary":"Helly's theorem for finite families of convex sets in its classical form.","title":"Helly's theorem"},{"id":"mathlib/thousand-plus/geometry-and-topology/52-convex-and-discrete-geometry/krein-milman-theorem","kind":"node","parent":1071,"result":true,"route":"../roadmap/mathlib/thousand-plus/geometry-and-topology/52-convex-and-discrete-geometry/index.html#krein-milman-theorem","scope":"mathlib/thousand-plus/geometry-and-topology/52-convex-and-discrete-geometry","status":"mathlib","summary":"Krein-Milman theorem: In an LCTVS, any compact convex set is the closure of the convex hull of its extreme points.","title":"Krein\u2013Milman theorem"},{"id":"mathlib/thousand-plus/geometry-and-topology/52-convex-and-discrete-geometry/radons-theorem","kind":"node","parent":1071,"result":true,"route":"../roadmap/mathlib/thousand-plus/geometry-and-topology/52-convex-and-discrete-geometry/index.html#radons-theorem","scope":"mathlib/thousand-plus/geometry-and-topology/52-convex-and-discrete-geometry","status":"mathlib","summary":"Radon's theorem on convex sets.","title":"Radon's theorem"},{"data_url":"scopes/mathlib/thousand-plus/geometry-and-topology/54-general-topology.json","id":"mathlib/thousand-plus/geometry-and-topology/54-general-topology","kind":"scope","parent":1059,"result":false,"route":"../roadmap/mathlib/thousand-plus/geometry-and-topology/54-general-topology/index.html","scope":"mathlib/thousand-plus/geometry-and-topology/54-general-topology","status":"mathlib","title":"54 General topology"},{"id":"mathlib/thousand-plus/geometry-and-topology/54-general-topology/lebesgues-decomposition-theorem","kind":"node","parent":1075,"result":true,"route":"../roadmap/mathlib/thousand-plus/geometry-and-topology/54-general-topology/index.html#lebesgues-decomposition-theorem","scope":"mathlib/thousand-plus/geometry-and-topology/54-general-topology","status":"mathlib","summary":"The Lebesgue decomposition theorem: Any s-finite measure `\u03bc` has Lebesgue decomposition with respect to any \u03c3-finite measure `\u03bd`. That is to say, there exist a measure `\u03be` and a measurable function `f`, such that `\u03be` is mutually singular with respect to `\u03bd` and `\u03bc = \u03be + \u03bd.withDensity f`","title":"Lebesgue's decomposition theorem"},{"id":"mathlib/thousand-plus/geometry-and-topology/54-general-topology/lebesgues-density-theorem","kind":"node","parent":1075,"result":true,"route":"../roadmap/mathlib/thousand-plus/geometry-and-topology/54-general-topology/index.html#lebesgues-density-theorem","scope":"mathlib/thousand-plus/geometry-and-topology/54-general-topology","status":"mathlib","summary":"Given an arbitrary set `s`, then `\u03bc (s \u2229 closedBall x r) / \u03bc (closedBall x r)` converges to `1` when `r` tends to `0`, for almost every `x` in `s`. This shows that almost every point of `s` is a Lebesgue density point for `s`. A stronger version holds for measurable sets, see `ae_tendsto_measure_inter_div_of_measurableSet`.","title":"Lebesgue's density theorem"},{"id":"mathlib/thousand-plus/geometry-and-topology/54-general-topology/metrization-theorems","kind":"node","parent":1075,"result":true,"route":"../roadmap/mathlib/thousand-plus/geometry-and-topology/54-general-topology/index.html#metrization-theorems","scope":"mathlib/thousand-plus/geometry-and-topology/54-general-topology","status":"mathlib","summary":"*Urysohn's metrization theorem* (Tychonoff's version): a T\u2083 topological space with second countable topology `X` is metrizable, i.e., there exists a metric space structure that generates the same topology.","title":"Metrization theorems"},{"id":"mathlib/thousand-plus/geometry-and-topology/54-general-topology/tietze-extension-theorem","kind":"node","parent":1075,"result":true,"route":"../roadmap/mathlib/thousand-plus/geometry-and-topology/54-general-topology/index.html#tietze-extension-theorem","scope":"mathlib/thousand-plus/geometry-and-topology/54-general-topology","status":"mathlib","summary":"Tietze extension theorem for real-valued bounded continuous maps, a version with a closed embedding and unbundled composition. If `e : C(X, Y)` is a closed embedding of a topological space into a normal topological space and `f : X \u2192\u1d47 \u211d` is a bounded continuous function, then there exists a bounded continuous function `g : Y \u2192\u1d47 \u211d` of the same norm such that `g \u2218 e = f`.","title":"Tietze extension theorem"},{"id":"mathlib/thousand-plus/geometry-and-topology/54-general-topology/tychonoffs-theorem","kind":"node","parent":1075,"result":true,"route":"../roadmap/mathlib/thousand-plus/geometry-and-topology/54-general-topology/index.html#tychonoffs-theorem","scope":"mathlib/thousand-plus/geometry-and-topology/54-general-topology","status":"mathlib","summary":"Tychonoff's theorem: product of compact sets is compact.","title":"Tychonoff's theorem"},{"data_url":"scopes/mathlib/thousand-plus/geometry-and-topology/57-manifolds-and-cell-complexes.json","id":"mathlib/thousand-plus/geometry-and-topology/57-manifolds-and-cell-complexes","kind":"scope","parent":1059,"result":false,"route":"../roadmap/mathlib/thousand-plus/geometry-and-topology/57-manifolds-and-cell-complexes/index.html","scope":"mathlib/thousand-plus/geometry-and-topology/57-manifolds-and-cell-complexes","status":"mathlib","title":"57 Manifolds and cell complexes"},{"id":"mathlib/thousand-plus/geometry-and-topology/57-manifolds-and-cell-complexes/whitney-embedding-theorem","kind":"node","parent":1081,"result":true,"route":"../roadmap/mathlib/thousand-plus/geometry-and-topology/57-manifolds-and-cell-complexes/index.html#whitney-embedding-theorem","scope":"mathlib/thousand-plus/geometry-and-topology/57-manifolds-and-cell-complexes","status":"mathlib","summary":"Baby version of the Whitney weak embedding theorem: if `M` admits a finite covering by supports of bump functions, then for some `n` it can be embedded into the `n`-dimensional Euclidean space.","title":"Whitney embedding theorem"},{"data_url":"scopes/mathlib/thousand-plus/probability-and-statistics.json","id":"mathlib/thousand-plus/probability-and-statistics","kind":"scope","parent":933,"result":false,"route":"../roadmap/mathlib/thousand-plus/probability-and-statistics/index.html","scope":"mathlib/thousand-plus/probability-and-statistics","status":"mathlib","title":"Probability and statistics"},{"id":"mathlib/thousand-plus/probability-and-statistics/bayes-theorem","kind":"node","parent":1083,"result":true,"route":"../roadmap/mathlib/thousand-plus/probability-and-statistics/index.html#bayes-theorem","scope":"mathlib/thousand-plus/probability-and-statistics","status":"mathlib","summary":"Bayes' Theorem","title":"Bayes' theorem"},{"id":"mathlib/thousand-plus/probability-and-statistics/continuous-mapping-theorem","kind":"node","parent":1083,"result":true,"route":"../roadmap/mathlib/thousand-plus/probability-and-statistics/index.html#continuous-mapping-theorem","scope":"mathlib/thousand-plus/probability-and-statistics","status":"mathlib","summary":"Continuous mapping theorem: if `X n` tends to `Z` in distribution and `g` is continuous, then `g \u2218 X n` tends to `g \u2218 Z` in distribution.","title":"Continuous mapping theorem"},{"id":"mathlib/thousand-plus/probability-and-statistics/ionescu-tulcea-theorem","kind":"node","parent":1083,"result":true,"route":"../roadmap/mathlib/thousand-plus/probability-and-statistics/index.html#ionescu-tulcea-theorem","scope":"mathlib/thousand-plus/probability-and-statistics","status":"mathlib","summary":"*Ionescu-Tulcea Theorem* : Given a family of kernels `\u03ba n` taking variables in `Iic n` with value in `X (n + 1)`, the kernel `traj \u03ba a` takes a variable `x` depending on the variables `i \u2264 a` and associates to it a kernel on trajectories depending on all variables, where the entries with index `\u2264 a` are those of `x`, and then one follows iteratively the kernels `\u03ba a`, then `\u03ba (a + 1)`, and so on.","title":"Ionescu-Tulcea theorem"},{"id":"mathlib/thousand-plus/probability-and-statistics/levy-continuity-theorem","kind":"node","parent":1083,"result":true,"route":"../roadmap/mathlib/thousand-plus/probability-and-statistics/index.html#levy-continuity-theorem","scope":"mathlib/thousand-plus/probability-and-statistics","status":"mathlib","summary":"The L\u00e9vy convergence theorem: the weak convergence of probability measures is equivalent to the pointwise convergence of their characteristic functions.","title":"L\u00e9vy continuity theorem"},{"id":"mathlib/thousand-plus/probability-and-statistics/poisson-limit-theorem","kind":"node","parent":1083,"result":true,"route":"../roadmap/mathlib/thousand-plus/probability-and-statistics/index.html#poisson-limit-theorem","scope":"mathlib/thousand-plus/probability-and-statistics","status":"mathlib","summary":"Poisson limit Theorem: If `n * p n \u2192 r` as `n \u2192 \u221e`. Then `(n.choose k) * (p n)^k * (1 - p n)^(n - k) \u2192 exp (-r) * r^k / k!`.","title":"Poisson limit theorem"},{"id":"mathlib/thousand-plus/probability-and-statistics/slutskys-theorem","kind":"node","parent":1083,"result":true,"route":"../roadmap/mathlib/thousand-plus/probability-and-statistics/index.html#slutskys-theorem","scope":"mathlib/thousand-plus/probability-and-statistics","status":"mathlib","summary":"Slutsky's theorem for a continuous function: if `X n` converges in distribution to `Z`, `Y n` converges in probability to a constant `c`, and `g` is a continuous function, then `g (X n, Y n)` converges in distribution to `g (Z, c)`.","title":"Slutsky's theorem"},{"data_url":"scopes/mathlib/thousand-plus/applied-mathematics.json","id":"mathlib/thousand-plus/applied-mathematics","kind":"scope","parent":933,"result":false,"route":"../roadmap/mathlib/thousand-plus/applied-mathematics/index.html","scope":"mathlib/thousand-plus/applied-mathematics","status":"mathlib","title":"Applied mathematics"},{"id":"mathlib/thousand-plus/applied-mathematics/akra-bazzi-theorem","kind":"node","parent":1090,"result":true,"route":"../roadmap/mathlib/thousand-plus/applied-mathematics/index.html#akra-bazzi-theorem","scope":"mathlib/thousand-plus/applied-mathematics","status":"mathlib","summary":"The Akra-Bazzi theorem: `T \u2208 O(n^p (1 + \u2211_u^n g(u) / u^{p+1}))`","title":"Akra\u2013Bazzi theorem"},{"id":"mathlib/thousand-plus/applied-mathematics/myhill-nerode-theorem","kind":"node","parent":1090,"result":true,"route":"../roadmap/mathlib/thousand-plus/applied-mathematics/index.html#myhill-nerode-theorem","scope":"mathlib/thousand-plus/applied-mathematics","status":"mathlib","summary":"Myhill\u2013Nerode theorem. A language is regular if and only if the set of left quotients is finite.","title":"Myhill\u2013Nerode theorem"},{"id":"mathlib/thousand-plus/applied-mathematics/smn-theorem","kind":"node","parent":1090,"result":true,"route":"../roadmap/mathlib/thousand-plus/applied-mathematics/index.html#smn-theorem","scope":"mathlib/thousand-plus/applied-mathematics","status":"mathlib","summary":"The $S_n^m$ theorem: There is a computable function, namely `Nat.Partrec.Code.curry`, that takes a program and a \u2115 `n`, and returns a new program using `n` as the first argument.","title":"Smn theorem"},{"id":"mathlib/thousand-plus/unclassified","kind":"node","parent":933,"result":false,"route":"../roadmap/mathlib/thousand-plus/unclassified/index.html","scope":"mathlib/thousand-plus","status":"can_state","title":"Unclassified"},{"data_url":"chapters/tauceti.json","id":"tauceti","kind":"scope","parent":0,"result":false,"route":"../roadmap/tauceti/index.html","scope":"tauceti","status":"fully_proved","summary":"Tau Ceti, read at commit ace60277 (2026-11-07), with its roadmaps at eaf17b10: 2,809 articles on 3,906 of its declarations, in parts by the library's top-level areas. 1,377 are the declarations a roadmap's status names as a result or a notable definition. A model writes the statuses and names at most five results in each, so the other 2,529 are results Tau Ceti's own docstrings name: a theorem whose docstring opens\u2026","title":"Tau Ceti"},{"data_url":"scopes/tauceti/algebra.json","id":"tauceti/algebra","kind":"scope","parent":1095,"result":false,"route":"../roadmap/tauceti/algebra/index.html","scope":"tauceti/algebra","status":"fully_proved","summary":"Results in Tau Ceti's `TauCeti.Algebra` modules, and the roadmaps whose articles are mostly there: by roadmap, then by area.","title":"Algebra"},{"data_url":"scopes/tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed.json","id":"tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed","kind":"scope","parent":1096,"result":false,"route":"../roadmap/tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed/index.html","scope":"tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Affine Lie algebras, smooth modules and completed centers"},{"id":"tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed/invariant-forms-classify-the-continuous-second-c","kind":"node","lean":"TauCeti.CurrentAlgebra.invariantFormsResidueClassEquiv","parent":1097,"result":true,"route":"../roadmap/tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed/index.html#invariant-forms-classify-the-continuous-second-c","scope":"tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed","status":"fully_proved","summary":"For a finite-dimensional Lie algebra with nondegenerate Killing form over a field of characteristic zero, for instance a semisimple Lie algebra, `\u03a6 \u21a6 [c_\u03a6]` is a linear equivalence from the invariant symmetric forms on `L` onto the continuous second cohomology of formal currents with trivial coefficients.","title":"Invariant forms classify the continuous second cohomology of formal loops"},{"id":"tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed/the-continuous-second-cohomology-of-the-formal-l","kind":"node","lean":"TauCeti.CurrentAlgebra.finrank_continuousSecondCohomology_eq_one","parent":1097,"result":true,"route":"../roadmap/tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed/index.html#the-continuous-second-cohomology-of-the-formal-l","scope":"tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed","status":"fully_proved","summary":"Here `L` is a finite-dimensional Lie algebra over an algebraically closed field of characteristic zero whose adjoint module is irreducible and whose Killing form is nondegenerate, for instance a simple Lie algebra; the residue class of the Killing form spans.","title":"The continuous second cohomology of the formal loop algebra of a simple Lie algebra is one-dimensional"},{"id":"tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed/the-second-cohomology-of-the-loop-algebra-of-a-s","kind":"node","lean":"TauCeti.LieAlgebra.LoopAlgebra.twoCocycle_eq_span_killingForm_sup_range_d\u2081\u2082","parent":1097,"result":true,"route":"../roadmap/tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed/index.html#the-second-cohomology-of-the-loop-algebra-of-a-s","scope":"tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed","status":"fully_proved","summary":"Here `L` is a finite-dimensional Lie algebra over an algebraically closed field of characteristic zero whose adjoint module is irreducible and whose Killing form is nondegenerate, for instance a simple Lie algebra. Together with `smul_twoCochainOfBilinear_mem_range_d\u2081\u2082_iff`, this says that the second cohomology of `loopAlgebra K \u2124 L` with trivial coefficients is one-dimensional.","title":"The second cohomology of the loop algebra of a simple Lie algebra is spanned by the residue class of the Killing form"},{"id":"tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed/whiteheads-second-lemma-as-an-equality-of-submod","kind":"node","lean":"LieModule.Cohomology.twoCocycle_eq_range_d\u2081\u2082, LieModule.Cohomology.exists_d\u2081\u2082_eq","parent":1097,"result":true,"route":"../roadmap/tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed/index.html#whiteheads-second-lemma-as-an-equality-of-submod","scope":"tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed","status":"fully_proved","summary":"the 2-cocycles of a Lie algebra with nondegenerate Killing form in characteristic zero, with values in a finite-dimensional module, are exactly the coboundaries.","title":"Whitehead's second lemma, as an equality of submodules"},{"id":"tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed/whiteheads-first-lemma","kind":"node","lean":"LieDerivation.IsKilling.exists_eq_inner, LieModule.Cohomology.exists_eq_lie_of_d\u2081\u2082_eq_zero","parent":1097,"result":true,"route":"../roadmap/tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed/index.html#whiteheads-first-lemma","scope":"tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed","status":"fully_proved","summary":"Over a field of characteristic zero, every Lie derivation from a Lie algebra with nondegenerate Killing form (necessarily finite-dimensional) into a finite-dimensional module is inner: it is `x \u21a6 \u2045x, m\u2046` for some `m`.","title":"Whitehead's first lemma"},{"id":"tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed/tauceti-currentalgebra-derivedseries-oftwococycl","kind":"node","lean":"TauCeti.CurrentAlgebra.derivedSeries_ofTwoCocycle_eq_top_iff","parent":1097,"result":true,"route":"../roadmap/tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed/index.html#tauceti-currentalgebra-derivedseries-oftwococycl","scope":"tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed","status":"fully_proved","summary":"The residue central extension of a perfect Lie algebra over a field is perfect if and only if its form is nonzero. No finite-dimensionality or characteristic assumption is needed.","title":"TauCeti.CurrentAlgebra.derivedSeries_ofTwoCocycle_eq_top_iff"},{"id":"tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed/tauceti-liealgebra-exists-eq-smul-killingform","kind":"node","lean":"TauCeti.LieAlgebra.exists_eq_smul_killingForm","parent":1097,"result":true,"route":"../roadmap/tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed/index.html#tauceti-liealgebra-exists-eq-smul-killingform","scope":"tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed","status":"fully_proved","summary":"Every invariant bilinear form on a Lie algebra is a multiple of the Killing form, for a finite-dimensional Lie algebra over an algebraically closed field whose adjoint module is irreducible and whose Killing form is nondegenerate, such as a simple Lie algebra in characteristic zero.","title":"TauCeti.LieAlgebra.exists_eq_smul_killingForm"},{"id":"tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed/tauceti-currentalgebra-instcompletespace","kind":"node","lean":"TauCeti.CurrentAlgebra.instCompleteSpace","parent":1097,"result":true,"route":"../roadmap/tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed/index.html#tauceti-currentalgebra-instcompletespace","scope":"tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed","status":"fully_proved","summary":"The order topology is complete over a finite free coefficient module.","title":"TauCeti.CurrentAlgebra.instCompleteSpace"},{"id":"tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed/tauceti-liemodule-cohomology-continuoussecondcoh","kind":"node","lean":"TauCeti.LieModule.Cohomology.continuousSecondCohomology","parent":1097,"result":true,"route":"../roadmap/tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed/index.html#tauceti-liemodule-cohomology-continuoussecondcoh","scope":"tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed","status":"fully_proved","summary":"Continuous second Lie algebra cohomology with discrete coefficients: continuous two-cocycles modulo the continuous coboundaries of continuous one-cochains.","title":"TauCeti.LieModule.Cohomology.continuousSecondCohomology"},{"id":"tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed/tauceti-coinduction","kind":"node","lean":"TauCeti.coinduction","parent":1097,"result":true,"route":"../roadmap/tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed/index.html#tauceti-coinduction","scope":"tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed","status":"fully_proved","summary":"The *coinduction* of `R`-modules to smooth `A`-modules along `f : R \u2192+* A`: the smooth part of the coextension of scalars `Hom_R(A, V)`, which is the module of continuous `R`-linear maps `A \u2192 V` when the `J i` form a basis of neighborhoods of `0` (`TauCeti.mem_smoothPart_coextendScalars_iff_continuous`).","title":"TauCeti.coinduction"},{"id":"tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed/tauceti-enoughinjectives-smoothmodules","kind":"node","lean":"TauCeti.enoughInjectives_smoothModules","parent":1097,"result":true,"route":"../roadmap/tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed/index.html#tauceti-enoughinjectives-smoothmodules","scope":"tauceti/algebra/affine-lie-algebras-smooth-modules-and-completed","status":"fully_proved","summary":"For a directed family `J` of left ideals of `A : Type u`, the smooth `A`-modules in universe `max u w` have enough injectives: every smooth module embeds, through the unit of the coinduction adjunction along `\u2124 \u2192+* A`, into the coinduction of an injective abelian group.","title":"TauCeti.enoughInjectives_smoothModules"},{"data_url":"scopes/tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module.json","id":"tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module","kind":"scope","parent":1096,"result":false,"route":"../roadmap/tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module/index.html","scope":"tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"DG and A-infinity algebras, categories, and modules"},{"id":"tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module/tauceti-ainfinityalgebra-minimalmodel","kind":"node","lean":"TauCeti.AInfinityAlgebra.minimalModel","parent":1109,"result":true,"route":"../roadmap/tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module/index.html#tauceti-ainfinityalgebra-minimalmodel","scope":"tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module","status":"fully_proved","summary":"The minimal `A\u221e` model on cohomology obtained by homogeneous transfer.","title":"TauCeti.AInfinityAlgebra.minimalModel"},{"id":"tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module/kadeishvilis-intrinsic-formality-criterion","kind":"node","lean":"TauCeti.AInfinityAlgebra.isIntrinsicallyFormal_of_forall_hochschildVanishesAt, TauCeti.AInfinityAlgebra.IsMinimal.exists_isIso_of_forall_hochschildVanishesAt","parent":1109,"result":true,"route":"../roadmap/tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module/index.html#kadeishvilis-intrinsic-formality-criterion","scope":"tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module","status":"fully_proved","summary":"Over a field, a graded algebra `\u210b`, presented as a minimal `A\u221e` algebra without operations above arity two, is intrinsically formal when `HH^{n, 2 - n}(H, H) = 0` for every `n \u2265 3`: every Hochschild cocycle of `\u210b` of suspended degree one concentrated on the words of length `n` is the Hochschild differential of a cochain of suspended degree zero concentrated on the words of length `n - 1`.","title":"Kadeishvili's intrinsic formality criterion"},{"id":"tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module/the-triple-massey-product-is-a-coset-of-the-mini","kind":"node","lean":"TauCeti.AInfinityAlgebra.mem_tripleMasseyProduct_iff_sub_minimalModel_m_three_mem","parent":1109,"result":true,"route":"../roadmap/tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module/index.html#the-triple-massey-product-is-a-coset-of-the-mini","scope":"tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module","status":"fully_proved","summary":"if `x y = 0` and `y z = 0`, then `w` lies in `\u27e8x, y, z\u27e9` exactly when `w - m\u2083^H (x, y, z)` lies in the indeterminacy `\u03c4x \u00b7 H + H \u00b7 z`.","title":"The triple Massey product is a coset of the minimal-model ternary operation"},{"id":"tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module/the-transferred-ternary-operation-on-a-a-b","kind":"node","lean":"TauCeti.HeisenbergDGA.minimalModel_m_three","parent":1109,"result":true,"route":"../roadmap/tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module/index.html#the-transferred-ternary-operation-on-a-a-b","scope":"tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module","status":"fully_proved","summary":"`m\u2083^H ([a], [a], [b]) = [a c]`. Of the Kontsevich--Soibelman/Merkulov tree sum only the term `-p m\u2082 (i (\u03c4 [a]), h m\u2082 (i [a], i [b]))` survives, with `h (a b) = c` and the Koszul twist `\u03c4 [a] = -[a]`.","title":"The transferred ternary operation on `([a], [a], [b])`"},{"id":"tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module/tauceti-ainfinityalgebra-transfer","kind":"node","lean":"TauCeti.AInfinityAlgebra.transfer","parent":1109,"result":true,"route":"../roadmap/tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module/index.html#tauceti-ainfinityalgebra-transfer","scope":"tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module","status":"fully_proved","summary":"The transferred `A\u221e` structure on the retract `H` of a special contraction of `(A, m\u2081)`: its Taylor map is the letter component of the transferred bar differential.","title":"TauCeti.AInfinityAlgebra.transfer"},{"id":"tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module/tauceti-ainfinityalgebra-transferhomotopy","kind":"node","lean":"TauCeti.AInfinityAlgebra.transferHomotopy","parent":1109,"result":true,"route":"../roadmap/tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module/index.html#tauceti-ainfinityalgebra-transferhomotopy","scope":"tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module","status":"fully_proved","summary":"The homotopy from the identity of an A-infinity algebra to its transferred inclusion composed with its transferred projection. On bar constructions this is `K - K X K`, where `K` is the tensor-trick homotopy and `X` the perturbation operator.","title":"TauCeti.AInfinityAlgebra.transferHomotopy"},{"id":"tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module/tauceti-ainfinitycategory-transfer","kind":"node","lean":"TauCeti.AInfinityCategory.transfer","parent":1109,"result":true,"route":"../roadmap/tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module/index.html#tauceti-ainfinitycategory-transfer","scope":"tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module","status":"fully_proved","summary":"The transferred `A\u221e` category: the `A\u221e` structure transferred from the total algebra of morphisms of `\ud835\udc9e` along a special contraction onto the total module of morphisms of `D` which is compatible with the hom modules along a bijection of objects `e : D \u2243 C`. Its differential is the differential `d_D` of the retract (`homInclusion_homDifferential_transfer`), its composition is `g \u2218 f = p m\u2082(i g, i f)`\u2026","title":"TauCeti.AInfinityCategory.transfer"},{"id":"tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module/the-yoneda-lemma-for-right-a-modules","kind":"node","lean":"TauCeti.AInfinityRightModule.yonedaHomotopyEquiv","parent":1109,"result":true,"route":"../roadmap/tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module/index.html#the-yoneda-lemma-for-right-a-modules","scope":"tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module","status":"fully_proved","summary":"Over an `A\u221e` algebra with a strict unit `e`, for a module on which `e` is a right unit for the binary operation, evaluation at `e` is a homotopy equivalence from the morphism complex out of the free module of rank one to the underlying complex of the module, with homotopy inverse the Yoneda cochains.","title":"The Yoneda lemma for right `A\u221e` modules"},{"id":"tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module/the-yoneda-lemma-for-right-dg-modules","kind":"node","lean":"TauCeti.dgYonedaEquiv","parent":1109,"result":true,"route":"../roadmap/tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module/index.html#the-yoneda-lemma-for-right-dg-modules","scope":"tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module","status":"fully_proved","summary":"DG natural transformations `Hom(-, X) \u27f6 F` are the closed degree-zero elements of `F(X)`. A transformation goes to the image of the identity of `X` under its component at `X` (`TauCeti.dgYonedaEquiv_apply_coe`), and a closed element `x` goes to the transformation sending `g : Y \u27f6 X` to `F(g)(x)` (`TauCeti.dgApp_dgYonedaEquiv_symm_v`).","title":"The Yoneda lemma for right DG modules"},{"id":"tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module/tauceti-ainfinityfunctor","kind":"node","lean":"TauCeti.AInfinityFunctor","parent":1109,"result":true,"route":"../roadmap/tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module/index.html#tauceti-ainfinityfunctor","scope":"tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module","status":"fully_proved","summary":"An `A\u221e` functor between nonunital `A\u221e` categories `\u212c` on `C` and `\ud835\udc9e` on `D`: a map of objects `obj : C \u2192 D` together with an `A\u221e` morphism from the total algebra of morphisms of `\u212c` to that of the induced category `\ud835\udc9e.induced obj`, whose components send a composable string of `C` to a morphism between the images of its endpoints and every other string to zero. Its component of arity `n` on a composable string\u2026","title":"TauCeti.AInfinityFunctor"},{"id":"tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module/the-hochschild-differential-squares-to-zero","kind":"node","lean":"TauCeti.AInfinityAlgebra.hochschildDifferential_hochschildDifferential","parent":1109,"result":true,"route":"../roadmap/tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module/index.html#the-hochschild-differential-squares-to-zero","scope":"tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module","status":"fully_proved","summary":"`[m, [m, F]] = 0` for every cochain `F` of degree `q`.","title":"The Hochschild differential squares to zero"},{"id":"tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module/tauceti-dgtwistedobject","kind":"node","lean":"TauCeti.DGTwistedObject","parent":1109,"result":true,"route":"../roadmap/tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module/index.html#tauceti-dgtwistedobject","scope":"tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module","status":"fully_proved","summary":"A twisted object of a differential graded category: an object `obj` with a degree-one endomorphism `twist` satisfying the Maurer\u2013Cartan equation `d \u03b4 = dgComp \u03b4 \u03b4`, which in Keller's order is `d \u03b4 + \u03b4 \u2218 \u03b4 = 0`. The twisted objects form a differential graded category whose Hom complexes carry the twisted differential; see `TauCeti.DGTwistedObject.homComplexData`.","title":"TauCeti.DGTwistedObject"},{"id":"tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module/tauceti-dgtwistedobject-cone","kind":"node","lean":"TauCeti.DGTwistedObject.cone","parent":1109,"result":true,"route":"../roadmap/tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module/index.html#tauceti-dgtwistedobject-cone","scope":"tauceti/algebra/dg-and-a-infinity-algebras-categories-and-module","status":"fully_proved","summary":"The cone of a closed degree-zero morphism `u : P \u27f6 Q` of twisted objects: the twisted object `(Q \u2295 P\u27e61\u27e7, \u03b4)` whose twisting matrix has the twists of `Q` and `P\u27e61\u27e7` on the diagonal and the morphism `P\u27e61\u27e7 \u27f6 P \u27f6 Q` underlying `u` in the block from `P\u27e61\u27e7` to `Q`.","title":"TauCeti.DGTwistedObject.cone"},{"data_url":"scopes/tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms.json","id":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms","kind":"scope","parent":1096,"result":false,"route":"../roadmap/tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/index.html","scope":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Grothendieck groups, Cartan maps, and Euler forms"},{"id":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/the-resolution-theorem","kind":"node","lean":"TauCeti.ExactStructure.IsResolving.resolutionEquiv","parent":1123,"result":true,"route":"../roadmap/tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/index.html#the-resolution-theorem","scope":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms","status":"fully_proved","summary":"For a resolving property `P` of an exact category, the inclusion of the full subcategory on `P` induces an isomorphism of exact Grothendieck groups. Its inverse sends the class of an object to the alternating class of any finite `P`-resolution of it, by `TauCeti.ExactStructure.IsResolving.resolutionEquiv_symm_of`.","title":"The resolution theorem"},{"id":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/cartan-matrix-unimodularity","kind":"node","lean":"TauCeti.cartanMatrix_det_eq_one_or_neg_one","parent":1123,"result":true,"route":"../roadmap/tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/index.html#cartan-matrix-unimodularity","scope":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms","status":"fully_proved","summary":"If every finitely generated module admits a finite resolution by finitely generated projectives, the Cartan determinant is `1` or `-1`. No positivity of its sign is asserted.","title":"Cartan-matrix unimodularity"},{"id":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/the-graded-cartan-matrix-of-the-dual-numbers-is","kind":"node","lean":"TauCeti.gradedCartanMatrix_dualNumber","parent":1123,"result":true,"route":"../roadmap/tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/index.html#the-graded-cartan-matrix-of-the-dual-numbers-is","scope":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms","status":"fully_proved","summary":"the regular module has the graded composition factors `S` and `S{2}`. At `q = 1` this is the ungraded Cartan matrix `2` of `TauCeti.cartanMatrix_dualNumber`.","title":"The graded Cartan matrix of the dual numbers is `1 + q\u00b2`, with `\u03b5` in degree two, in the bases `[A]` and `[S]`"},{"id":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/tauceti-gradedcartanequiv","kind":"node","lean":"TauCeti.gradedCartanEquiv","parent":1123,"result":true,"route":"../roadmap/tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/index.html#tauceti-gradedcartanequiv","scope":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms","status":"fully_proved","summary":"If every finite graded module admits a finite graded-projective resolution, the graded Cartan map is an isomorphism of `\u2124[q,q\u207b\u00b9]`-modules.","title":"TauCeti.gradedCartanEquiv"},{"id":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/the-q-euler-matrix-of-a-path-algebra-is-the-inve","kind":"node","lean":"TauCeti.PathAlgebra.gradedExtEulerForm_gradedVertexSimpleClassBasis_eq_inv","parent":1123,"result":true,"route":"../roadmap/tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/index.html#the-q-euler-matrix-of-a-path-algebra-is-the-inve","scope":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms","status":"fully_proved","summary":"`\u03c7_q(S\u1d62, S\u2c7c)` is the `(i, j)` entry of `((C.map invert)\u1d40)\u207b\u00b9`, where `C` is the graded Cartan matrix in the bases of graded vertex projectives and graded vertex simples.","title":"The q-Euler matrix of a path algebra is the inverse of its conjugate-transposed graded Cartan matrix"},{"id":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/the-graded-quiver-q-euler-form-at-q-1-is-the-ung","kind":"node","lean":"TauCeti.PathAlgebra.laurentSpecialize_gradedExtEulerForm_eq_extEulerPairing","parent":1123,"result":true,"route":"../roadmap/tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/index.html#the-graded-quiver-q-euler-form-at-q-1-is-the-ung","scope":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms","status":"fully_proved","summary":"On `G\u2080^gr(mod kQ)` specialized at `q = 1`, the q-Euler form is the Ext-Euler pairing on `G\u2080(mod kQ)` of the classes with their grading forgotten.","title":"The graded quiver q-Euler form at `q = 1` is the ungraded Ext-Euler pairing"},{"id":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/bigraded-ext-detects-linear-resolutions-over-a-s","kind":"node","lean":"TauCeti.forall_exists_isLinear_iff_bigradedExtPoincareMatrix_isDiagonal","parent":1123,"result":true,"route":"../roadmap/tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/index.html#bigraded-ext-detects-linear-resolutions-over-a-s","scope":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms","status":"fully_proved","summary":"Suppose that `A` is finite-dimensional and nonnegatively graded, with semisimple degree-zero part. For an exhaustive family of finite simple modules concentrated in degree zero, every simple admits a linear graded projective resolution if and only if every entry of the full bigraded Ext Poincare matrix has diagonal support.","title":"Bigraded Ext detects linear resolutions over a semisimple degree-zero part"},{"id":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/the-cartan-poincare-identity-over-a-semisimple-d","kind":"node","lean":"TauCeti.cartan_poincare_of_bigradedExtPoincareMatrix_isDiagonal_of_semisimple","parent":1123,"result":true,"route":"../roadmap/tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/index.html#the-cartan-poincare-identity-over-a-semisimple-d","scope":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms","status":"fully_proved","summary":"If `A` is finite-dimensional and nonnegatively graded, its degree-zero part is semisimple, the chosen simples are concentrated in degree zero, and their full bigraded Ext Poincare matrix is diagonal, then `C^T * P(-q) = diagonal (fun i \u21a6 gdim(e\u1d62 \u2022 S\u1d62))`.","title":"The Cartan--Poincare identity over a semisimple degree-zero part"},{"id":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/the-graded-cartan-matrix-in-a-projective-basis-b","kind":"node","lean":"TauCeti.gradedCartanMatrix","parent":1123,"result":true,"route":"../roadmap/tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/index.html#the-graded-cartan-matrix-in-a-projective-basis-b","scope":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms","status":"fully_proved","summary":"the matrix of the Laurent-linear graded Cartan map. Rows are indexed by `bM` and columns by `bP`.","title":"The graded Cartan matrix in a projective basis `bP` and a module basis `bM`"},{"id":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/the-graded-projectives-a-ei-form-a-basis-of-k0-g","kind":"node","lean":"TauCeti.gradedIdempotentProjectiveBasis","parent":1123,"result":true,"route":"../roadmap/tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/index.html#the-graded-projectives-a-ei-form-a-basis-of-k0-g","scope":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms","status":"fully_proved","summary":"The graded projectives `A e\u1d62` form a basis of `K\u2080^gr(proj A)` over `\u2124[q,q\u207b\u00b9]`, for a nonnegatively graded finite-dimensional algebra whose degree-zero piece is spanned by a complete family of orthogonal nonzero idempotents `e\u1d62`. Its basis vector at `i` is the class `[A e\u1d62]`. This is the projective counterpart of the head basis `TauCeti.gradedIdempotentHeadBasis`, and the case of\u2026","title":"The graded projectives `A e\u1d62` form a basis of `K\u2080^gr(proj A)`"},{"id":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/tauceti-gradedprojectiveresolution-islinear","kind":"node","lean":"TauCeti.GradedProjectiveResolution.IsLinear","parent":1123,"result":true,"route":"../roadmap/tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/index.html#tauceti-gradedprojectiveresolution-islinear","scope":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms","status":"fully_proved","summary":"A graded projective resolution is linear if its term in homological degree `n` is generated by its homogeneous piece of internal degree `n`.","title":"TauCeti.GradedProjectiveResolution.IsLinear"},{"id":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/equivalence-invariance-of-the-resolution-theorem","kind":"node","lean":"TauCeti.ExactStructure.IsResolving.mapEquiv_comp_resolutionEquiv","parent":1123,"result":true,"route":"../roadmap/tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/index.html#equivalence-invariance-of-the-resolution-theorem","scope":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms","status":"fully_proved","summary":"An exact equivalence identifying two resolving properties intertwines both the ambient exact `K\u2080` equivalence and the equivalence of the resolving subcategories with their resolution-theorem comparisons.","title":"Equivalence invariance of the resolution theorem"},{"id":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/the-graded-first-arrow-euler-formula","kind":"node","lean":"TauCeti.PathAlgebra.gradedExtEuler_gradedVertexSimple","parent":1123,"result":true,"route":"../roadmap/tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/index.html#the-graded-first-arrow-euler-formula","scope":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms","status":"fully_proved","summary":"Pairing a vertex simple against a finite-dimensional graded module subtracts the shifted outgoing-arrow coordinates from its own vertex coordinate.","title":"The graded first-arrow Euler formula"},{"id":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/the-graded-ringel-formula-on-grothendieck-classe","kind":"node","lean":"TauCeti.PathAlgebra.gradedExtEulerForm_eq_sum, TauCeti.PathAlgebra.gradedExtEuler_eq_sum_smulGradedDimension","parent":1123,"result":true,"route":"../roadmap/tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/index.html#the-graded-ringel-formula-on-grothendieck-classe","scope":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms","status":"fully_proved","summary":"The vertex coordinates are the idempotent graded dimensions, and the source coordinates are conjugated by `q \u21a6 q\u207b\u00b9`.","title":"The graded Ringel formula on Grothendieck classes"},{"id":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/the-weighted-graded-cartan-coefficient-formula","kind":"node","lean":"TauCeti.finrank_smul_simple_piece_zero_mul_coeff_gradedCartanMatrix","parent":1123,"result":true,"route":"../roadmap/tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms/index.html#the-weighted-graded-cartan-coefficient-formula","scope":"tauceti/algebra/grothendieck-groups-cartan-maps-and-euler-forms","status":"fully_proved","summary":"A coefficient in a column `[A f]`, multiplied by `dim_k(e\u1d62 \u2022 (S\u1d62)\u2080)`, is the corresponding graded corner dimension.","title":"The weighted graded Cartan coefficient formula"},{"data_url":"scopes/tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu.json","id":"tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu","kind":"scope","parent":1096,"result":false,"route":"../roadmap/tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu/index.html","scope":"tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Nilpotent orbits, Springer resolution and subregular geometry"},{"id":"tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu/the-jacobson-morozov-theorem","kind":"node","lean":"TauCeti.exists_isSl2Triple_of_isNilpotent_ad","parent":1139,"result":true,"route":"../roadmap/tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu/index.html#the-jacobson-morozov-theorem","scope":"tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu","status":"fully_proved","summary":"In a finite-dimensional Lie algebra with nondegenerate Killing form over a field of characteristic zero, every nonzero `ad`-nilpotent element `e` is the nilpositive element of an `sl\u2082` triple `(h, e, f)`.","title":"The Jacobson\u2013Morozov theorem"},{"id":"tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu/the-jacobson-morozov-theorem-for-embeddings","kind":"node","lean":"TauCeti.exists_injective_lieHom_slFinTwo_iff","parent":1139,"result":true,"route":"../roadmap/tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu/index.html#the-jacobson-morozov-theorem-for-embeddings","scope":"tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu","status":"fully_proved","summary":"In a finite-dimensional Lie algebra with nondegenerate Killing form over a field of characteristic zero, `e` is the image of the raising element `E\u2080\u2081` under an injective Lie algebra homomorphism `sl (Fin 2) K \u2192\u2097\u2045K\u2046 L` if and only if `e \u2260 0` and `ad e` is nilpotent.","title":"The Jacobson\u2013Morozov theorem for embeddings"},{"id":"tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu/kostants-conjugacy-theorem","kind":"node","lean":"IsSl2Triple.exists_lieEquiv_apply_eq, TauCeti.exists_lieEquiv_comp_eq_of_apply_eq","parent":1139,"result":true,"route":"../roadmap/tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu/index.html#kostants-conjugacy-theorem","scope":"tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu","status":"fully_proved","summary":"Two `sl\u2082` triples `(h, e, f)` and `(h', e, f')` with the same nilpositive element in a finite-dimensional Lie algebra over a field of characteristic zero are conjugate by an automorphism of `L` that fixes `e` and lies in the subgroup `TauCeti.expAdSubgroup K e` generated by the automorphisms `exp (ad y)` with `y \u2208 ker (ad e) \u2293 [e, L]`.","title":"Kostant's conjugacy theorem"},{"id":"tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu/the-transverse-decomposition-l-l-e-ker-ad-f","kind":"node","lean":"IsSl2Triple.isCompl_range_ad_e_ker_ad_f","parent":1139,"result":true,"route":"../roadmap/tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu/index.html#the-transverse-decomposition-l-l-e-ker-ad-f","scope":"tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu","status":"fully_proved","summary":"For an `sl\u2082` triple `(h, e, f)` in a finite-dimensional Lie algebra over a field of characteristic zero, the range of `ad e` and the kernel of `ad f` are complementary.","title":"The transverse decomposition `L = [L, e] \u2295 ker (ad f)`"},{"id":"tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu/the-simple-root-values-of-a-dominant-neutral-ele","kind":"node","lean":"IsSl2Triple.root_apply_eq_zero_or_eq_one_or_eq_two","parent":1139,"result":true,"route":"../roadmap/tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu/index.html#the-simple-root-values-of-a-dominant-neutral-ele","scope":"tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu","status":"fully_proved","summary":"If the neutral element `h` of an `sl\u2082` triple lies in the Cartan subalgebra `H` and every simple root takes a natural value at `h`, then each simple root takes the value `0`, `1` or `2` at `h`. These values label the weighted Dynkin diagram of the triple.","title":"The simple root values of a dominant neutral element lie in `{0, 1, 2}`"},{"id":"tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu/the-dimension-of-a-centralizer-and-of-g-x","kind":"node","lean":"TauCeti.ringKrullDim_quotient_centralizerHopfIdeal_add_finrank_range_ad","parent":1139,"result":true,"route":"../roadmap/tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu/index.html#the-dimension-of-a-centralizer-and-of-g-x","scope":"tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu","status":"fully_proved","summary":"Over a field of characteristic zero, the dimension of a finite-type affine group `G` is the dimension of the centralizer `C_G(x)` of an element `x` of its Lie algebra `\ud835\udd24` plus the rank of `ad x`, the dimension of `[\ud835\udd24, x]`.","title":"The dimension of a centralizer and of `[\ud835\udd24, x]`"},{"id":"tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu/tauceti-centralizerhopfideal","kind":"node","lean":"TauCeti.centralizerHopfIdeal","parent":1139,"result":true,"route":"../roadmap/tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu/index.html#tauceti-centralizerhopfideal","scope":"tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu","status":"fully_proved","summary":"The Hopf ideal of the centralizer of an element `x` of the Lie algebra, the stabilizer of `x` under the adjoint representation. Contravariantly, it cuts out the closed subgroup of `Spec H` fixing `x` under `Ad`.","title":"TauCeti.centralizerHopfIdeal"},{"id":"tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu/liealgebra-iscentralizerregular","kind":"node","lean":"LieAlgebra.IsCentralizerRegular","parent":1139,"result":true,"route":"../roadmap/tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu/index.html#liealgebra-iscentralizerregular","scope":"tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu","status":"fully_proved","summary":"An element `x` of a Lie algebra is regular when its centralizer `ker (ad x)` has minimal dimension among the centralizers of all elements. The dimension is the cardinal `Module.rank`, so that an infinite-dimensional centralizer is never smaller than a finite-dimensional one. Over a field of characteristic zero this is the dimension of the group centralizer `C_G(x)` when the Lie algebra is that of an affine\u2026","title":"LieAlgebra.IsCentralizerRegular"},{"id":"tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu/the-slodowy-slice-e-ker-ad-f-as-an-affine-subspa","kind":"node","lean":"TauCeti.slodowySlice","parent":1139,"result":true,"route":"../roadmap/tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu/index.html#the-slodowy-slice-e-ker-ad-f-as-an-affine-subspa","scope":"tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu","status":"fully_proved","summary":"When `(h, e, f)` is an `sl\u2082` triple in a finite-dimensional Lie algebra over a field of characteristic zero, its direction `ker (ad f)` is complementary to `[L, e]` (`IsSl2Triple.isCompl_range_ad_e_ker_ad_f`). For `f = 0`, in particular in the degenerate case `e = f = 0`, it is all of `L` (`TauCeti.slodowySlice_zero_right`).","title":"The Slodowy slice `e + ker (ad f)`, as an affine subspace of `L`"},{"id":"tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu/the-contracting-action-of-kx-on-l-attached-to-an","kind":"node","lean":"IsSl2Triple.contractingAction","parent":1139,"result":true,"route":"../roadmap/tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu/index.html#the-contracting-action-of-kx-on-l-attached-to-an","scope":"tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu","status":"fully_proved","summary":"`c` acts by `c ^ 2 \u2022 gradingAut c\u207b\u00b9`, that is, by `c ^ (2 - j)` on the `j`-eigenspace of `ad h` (`IsSl2Triple.contractingAction_apply_of_mem`). For an algebraic group with Lie algebra `L` and a cocharacter `\u03bb` with differential `h` this is `c \u21a6 c ^ 2 \u2022 Ad (\u03bb c\u207b\u00b9)`. It fixes `e` and preserves the Slodowy slice `e + ker (ad f)`, on which it extends to `c = 0` (`IsSl2Triple.exists_contractingAction_eq`).","title":"The contracting action of `K\u02e3` on `L` attached to an `sl\u2082` triple `(h, e, f)`"},{"id":"tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu/morozovs-lemma","kind":"node","lean":"TauCeti.existsUnique_lie_eq_and_lie_eq_neg_two_nsmul, TauCeti.existsUnique_isSl2Triple_of_isNilpotent_ad","parent":1139,"result":true,"route":"../roadmap/tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu/index.html#morozovs-lemma","scope":"tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu","status":"fully_proved","summary":"Let `ad e` be nilpotent, `\u2045h, e\u2046 = 2 \u2022 e` and `h \u2208 [e, L]`, over a field of characteristic zero. Then there is a unique `f` with `\u2045e, f\u2046 = h` and `\u2045h, f\u2046 = -(2 \u2022 f)`. The case `h = 0` is allowed.","title":"Morozov's lemma"},{"id":"tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu/the-rank-one-kostant-straightening-formula","kind":"node","lean":"TauCeti.Sl2.dividedPower_e_mul_dividedPower_f","parent":1139,"result":true,"route":"../roadmap/tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu/index.html#the-rank-one-kostant-straightening-formula","scope":"tauceti/algebra/nilpotent-orbits-springer-resolution-and-subregu","status":"fully_proved","summary":"If `H`, `E`, and `F` satisfy the `sl\u2082` commutator relations in an associative `\u211a`-algebra, then the product of a raising divided power and a lowering divided power is an integral sum in PBW order `F`, `H`, `E`.","title":"The rank-one Kostant straightening formula"},{"data_url":"scopes/tauceti/algebra/reductive-algebraic-groups.json","id":"tauceti/algebra/reductive-algebraic-groups","kind":"scope","parent":1096,"result":false,"route":"../roadmap/tauceti/algebra/reductive-algebraic-groups/index.html","scope":"tauceti/algebra/reductive-algebraic-groups","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Reductive algebraic groups"},{"id":"tauceti/algebra/reductive-algebraic-groups/borel-conjugacy","kind":"node","lean":"TauCeti.HopfIdeal.IsBorelOverAlgClosed.exists_conjugate_eq","parent":1152,"result":true,"route":"../roadmap/tauceti/algebra/reductive-algebraic-groups/index.html#borel-conjugacy","scope":"tauceti/algebra/reductive-algebraic-groups","status":"fully_proved","summary":"Any two Borel subgroups of a finite-type affine group over an algebraically closed field are conjugate by a rational point. The ambient group need not be smooth, reduced, or connected.","title":"Borel conjugacy"},{"id":"tauceti/algebra/reductive-algebraic-groups/existence-of-closed-orbits","kind":"node","lean":"TauCeti.SchemeAction.exists_isClosed_orbit","parent":1152,"result":true,"route":"../roadmap/tauceti/algebra/reductive-algebraic-groups/index.html#existence-of-closed-orbits","scope":"tauceti/algebra/reductive-algebraic-groups","status":"fully_proved","summary":"A nonempty scheme of finite type over an algebraically closed field, acted on by a group scheme of finite type, contains the closed orbit of a rational point.","title":"Existence of closed orbits"},{"id":"tauceti/algebra/reductive-algebraic-groups/gln-g-is-affine-for-a-faithful-reductive-represe","kind":"node","lean":"TauCeti.Comodule.isIso_fppfHomogeneousQuotientToCoinvariantsPoints_ker_coordinateBialgHom","parent":1152,"result":true,"route":"../roadmap/tauceti/algebra/reductive-algebraic-groups/index.html#gln-g-is-affine-for-a-faithful-reductive-represe","scope":"tauceti/algebra/reductive-algebraic-groups","status":"fully_proved","summary":"Let `G` be a reductive group over a field of characteristic zero and `\u03c1 : G \u2192 GL\u2099` a faithful representation with coordinate morphism `\u03c1* : O(GL\u2099) \u27f6 O(G)`. Restriction to the functions on `GL\u2099` invariant under right translation by `G` identifies the fppf homogeneous quotient `GL\u2099/G` with the sheaf of points of `Spec O(GL\u2099)^G`.","title":"`GL\u2099/G` is affine for a faithful reductive representation"},{"id":"tauceti/algebra/reductive-algebraic-groups/tauceti-algebraicgeometry-smooth-of-grpobj-of-ch","kind":"node","lean":"TauCeti.AlgebraicGeometry.smooth_of_grpObj_of_charZero","parent":1152,"result":true,"route":"../roadmap/tauceti/algebra/reductive-algebraic-groups/index.html#tauceti-algebraicgeometry-smooth-of-grpobj-of-ch","scope":"tauceti/algebra/reductive-algebraic-groups","status":"fully_proved","summary":"An affine group scheme of finite type over a field of characteristic zero is smooth.","title":"TauCeti.AlgebraicGeometry.smooth_of_grpObj_of_charZero"},{"id":"tauceti/algebra/reductive-algebraic-groups/tauceti-hopfideal-isparabolic","kind":"node","lean":"TauCeti.HopfIdeal.IsParabolic","parent":1152,"result":true,"route":"../roadmap/tauceti/algebra/reductive-algebraic-groups/index.html#tauceti-hopfideal-isparabolic","scope":"tauceti/algebra/reductive-algebraic-groups","status":"fully_proved","summary":"A Hopf ideal defines a parabolic subgroup when its base change to an algebraic closure defines a smooth closed subgroup of the geometric fibre containing a Borel subgroup.","title":"TauCeti.HopfIdeal.IsParabolic"},{"id":"tauceti/algebra/reductive-algebraic-groups/the-fppf-quotient-p-u-is-the-dynamic-levi-z","kind":"node","lean":"BialgHom.parabolicLeviFppfQuotientIso","parent":1152,"result":true,"route":"../roadmap/tauceti/algebra/reductive-algebraic-groups/index.html#the-fppf-quotient-p-u-is-the-dynamic-levi-z","scope":"tauceti/algebra/reductive-algebraic-groups","status":"fully_proved","summary":"The subgroup in the quotient is given by its actual defining ideal inside the parabolic coordinate ring.","title":"The fppf quotient `P(\u03bb)/U(\u03bb)` is the dynamic Levi `Z(\u03bb)`"},{"id":"tauceti/algebra/reductive-algebraic-groups/bialghom-algebraetale-coinvariants-centralizer-n","kind":"node","lean":"BialgHom.algebraEtale_coinvariants_centralizer_normalizer","parent":1152,"result":true,"route":"../roadmap/tauceti/algebra/reductive-algebraic-groups/index.html#bialghom-algebraetale-coinvariants-centralizer-n","scope":"tauceti/algebra/reductive-algebraic-groups","status":"fully_proved","summary":"The represented quotient of the scheme normalizer by the scheme centralizer is \u00e9tale, retaining the full subgroup schemes even when `D(X)` is nonreduced.","title":"BialgHom.algebraEtale_coinvariants_centralizer_normalizer"},{"id":"tauceti/algebra/reductive-algebraic-groups/the-centralizer-of-a-diagonalizable-subgroup-of","kind":"node","lean":"BialgHom.smoothCommHopfAlgProperty_quotient_centralizerHopfIdeal","parent":1152,"result":true,"route":"../roadmap/tauceti/algebra/reductive-algebraic-groups/index.html#the-centralizer-of-a-diagonalizable-subgroup-of","scope":"tauceti/algebra/reductive-algebraic-groups","status":"fully_proved","summary":"Formalized in Tau Ceti as `BialgHom.smoothCommHopfAlgProperty_quotient_centralizerHopfIdeal`.","title":"The centralizer of a diagonalizable subgroup of a smooth affine group of finite type is smooth, over any base ring"},{"id":"tauceti/algebra/reductive-algebraic-groups/g0-k-g-f","kind":"node","lean":"TauCeti.FiniteField.rationalPointsMulEquivFixedSubgroupGeometricFrobenius","parent":1152,"result":true,"route":"../roadmap/tauceti/algebra/reductive-algebraic-groups/index.html#g0-k-g-f","scope":"tauceti/algebra/reductive-algebraic-groups","status":"fully_proved","summary":"The `K`-rational points of `H` are isomorphic to the points of the base change `k \u2297[K] H` over a domain `k` fixed by the geometric Frobenius. The isomorphism sends a rational point `g` to `s \u2297 h \u21a6 s * g h`.","title":"`G\u2080(K) \u2243 G^F`"},{"data_url":"scopes/tauceti/algebra/representation-theory.json","id":"tauceti/algebra/representation-theory","kind":"scope","parent":1096,"result":false,"route":"../roadmap/tauceti/algebra/representation-theory/index.html","scope":"tauceti/algebra/representation-theory","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Representation theory"},{"id":"tauceti/algebra/representation-theory/the-linear-harish-chandra-isomorphism-z-u-l-a-k","kind":"node","lean":"TauCeti.harishChandraLinearIso","parent":1162,"result":true,"route":"../roadmap/tauceti/algebra/representation-theory/index.html#the-linear-harish-chandra-isomorphism-z-u-l-a-k","scope":"tauceti/algebra/representation-theory","status":"fully_proved","summary":"the linear Harish-Chandra projection `TauCeti.hcProjectionLinear b`, with its values in the Weyl invariants.","title":"The linear Harish-Chandra isomorphism `Z(U(L)) \u2243\u2090[K] S(H)^W`"},{"id":"tauceti/algebra/representation-theory/harish-chandras-theorem-on-central-characters","kind":"node","lean":"TauCeti.vermaCentralCharacter_eq_iff","parent":1162,"result":true,"route":"../roadmap/tauceti/algebra/representation-theory/index.html#harish-chandras-theorem-on-central-characters","scope":"tauceti/algebra/representation-theory","status":"fully_proved","summary":"two weights `lam` and `mu` have the same central character, `\u03c7_lam = \u03c7_mu`, exactly when `mu` lies in the dot orbit of `lam`, that is `mu = w \u00b7 lam = w (lam + \u03c1) - \u03c1` for an element `w` of the Weyl group.","title":"Harish-Chandra's theorem on central characters"},{"id":"tauceti/algebra/representation-theory/chevalleys-restriction-theorem","kind":"node","lean":"LieSubalgebra.lieChevalleyRestrictionIso","parent":1162,"result":true,"route":"../roadmap/tauceti/algebra/representation-theory/index.html#chevalleys-restriction-theorem","scope":"tauceti/algebra/representation-theory","status":"fully_proved","summary":"Restriction of polynomial functions from `L` to `H`, `TauCeti.lieCartanRestriction H`, is an isomorphism from the invariants `S(L)^L` onto the Weyl invariants `S(H)^W`.","title":"Chevalley's restriction theorem"},{"id":"tauceti/algebra/representation-theory/modulecat-exists-isalmostsplit-x3-eq-iff","kind":"node","lean":"ModuleCat.exists_isAlmostSplit_X\u2083_eq_iff","parent":1162,"result":true,"route":"../roadmap/tauceti/algebra/representation-theory/index.html#modulecat-exists-isalmostsplit-x3-eq-iff","scope":"tauceti/algebra/representation-theory","status":"fully_proved","summary":"Over a finite-dimensional algebra, a finitely generated module ends an almost-split sequence of `A`-modules exactly when it is indecomposable and non-projective.","title":"ModuleCat.exists_isAlmostSplit_X\u2083_eq_iff"},{"id":"tauceti/algebra/representation-theory/the-peter-weyl-theorem-unconditionally","kind":"node","lean":"TauCeti.stdPeterWeylBasis","parent":1162,"result":true,"route":"../roadmap/tauceti/algebra/representation-theory/index.html#the-peter-weyl-theorem-unconditionally","scope":"tauceti/algebra/representation-theory","status":"fully_proved","summary":"The normalized matrix coefficients of the models chosen in the unitary equivalence classes are a Hilbert basis of `L\u00b2(G)`, indexed by `\u03a3 i, Fin (IrrepClass.model i).dim \u00d7 Fin (IrrepClass.model i).dim`. No skeleton is assumed: `isIrrepSkeleton_model` supplies one.","title":"The Peter-Weyl theorem, unconditionally"},{"id":"tauceti/algebra/representation-theory/gabriels-theorem-tits-form-version","kind":"node","lean":"TauCeti.isFiniteRepType_iff_titsForm_posDef, TauCeti.isFiniteRepType_iff_card_hom_add_card_hom_le_one_and_exists_dynkinType_iso, TauCeti.isRepresentationFinite_pathAlgebra_iff_titsForm_posDef","parent":1162,"result":true,"route":"../roadmap/tauceti/algebra/representation-theory/index.html#gabriels-theorem-tits-form-version","scope":"tauceti/algebra/representation-theory","status":"fully_proved","summary":"A finite connected quiver has finite representation type exactly when its Tits form is positive definite.","title":"Gabriel's theorem, Tits form version"},{"id":"tauceti/algebra/representation-theory/tauceti-universalenvelopingalgebra-reesalgebra","kind":"node","lean":"TauCeti.UniversalEnvelopingAlgebra.reesAlgebra","parent":1162,"result":true,"route":"../roadmap/tauceti/algebra/representation-theory/index.html#tauceti-universalenvelopingalgebra-reesalgebra","scope":"tauceti/algebra/representation-theory","status":"fully_proved","summary":"The PBW Rees algebra: the coefficient of `t\u207f` belongs to PBW filtration degree `n`.","title":"TauCeti.UniversalEnvelopingAlgebra.reesAlgebra"},{"id":"tauceti/algebra/representation-theory/tauceti-universalenvelopingalgebra-reeszerofiber","kind":"node","lean":"TauCeti.UniversalEnvelopingAlgebra.reesZeroFiberEquiv","parent":1162,"result":true,"route":"../roadmap/tauceti/algebra/representation-theory/index.html#tauceti-universalenvelopingalgebra-reeszerofiber","scope":"tauceti/algebra/representation-theory","status":"fully_proved","summary":"The PBW zero fibre is canonically the symmetric algebra, without a choice of basis.","title":"TauCeti.UniversalEnvelopingAlgebra.reesZeroFiberEquiv"},{"id":"tauceti/algebra/representation-theory/tauceti-universalenvelopingalgebra-reesonefibere","kind":"node","lean":"TauCeti.UniversalEnvelopingAlgebra.reesOneFiberEquiv","parent":1162,"result":true,"route":"../roadmap/tauceti/algebra/representation-theory/index.html#tauceti-universalenvelopingalgebra-reesonefibere","scope":"tauceti/algebra/representation-theory","status":"fully_proved","summary":"Tensor base change of the PBW Rees algebra at one is the enveloping algebra. The polynomial parameter acts on the left tensor factor by evaluation at one.","title":"TauCeti.UniversalEnvelopingAlgebra.reesOneFiberEquiv"},{"id":"tauceti/algebra/representation-theory/tauceti-gelfandtsetlingenerator","kind":"node","lean":"TauCeti.gelfandTsetlinGenerator","parent":1162,"result":true,"route":"../roadmap/tauceti/algebra/representation-theory/index.html#tauceti-gelfandtsetlingenerator","scope":"tauceti/algebra/representation-theory","status":"fully_proved","summary":"The Gelfand-Tsetlin generator of degree `r` at level `k \u2264 n`: the image in `U(gl (Fin n))` of the Gelfand invariant `C\u1d63` of the upper-left corner `gl (Fin k)`.","title":"TauCeti.gelfandTsetlinGenerator"},{"id":"tauceti/algebra/representation-theory/the-auslander-reiten-theorem-existence","kind":"node","lean":"ModuleCat.exists_isAlmostSplit_arTranslate, ModuleCat.exists_isAlmostSplit_of_not_injective","parent":1162,"result":true,"route":"../roadmap/tauceti/algebra/representation-theory/index.html#the-auslander-reiten-theorem-existence","scope":"tauceti/algebra/representation-theory","status":"fully_proved","summary":"Over a finite-dimensional algebra, a finitely generated indecomposable non-projective module `M` ends an almost-split sequence `0 \u2192 \u03c4 M \u2192 E \u2192 M \u2192 0` of `A`-modules whose left-hand end is the translate `\u03c4 M` and whose middle term is finitely generated.","title":"The Auslander\u2013Reiten theorem, existence"},{"id":"tauceti/algebra/representation-theory/chevalleys-restriction-theorem-image-form","kind":"node","lean":"LieSubalgebra.map_lieInvariants_lieCartanRestriction, LieSubalgebra.lieCartanRestriction_injOn_lieInvariants","parent":1162,"result":true,"route":"../roadmap/tauceti/algebra/representation-theory/index.html#chevalleys-restriction-theorem-image-form","scope":"tauceti/algebra/representation-theory","status":"fully_proved","summary":"Restriction to `H` maps the invariants `S(L)^L` onto the Weyl invariants `S(H)^W`.","title":"Chevalley's restriction theorem, image form"},{"id":"tauceti/algebra/representation-theory/harish-chandra-surjectivity-with-a-degree-bound","kind":"node","lean":"TauCeti.exists_mem_pbwFiltration_hcProjectionLinear_eq, TauCeti.exists_mem_pbwFiltration_hcProjection_eq","parent":1162,"result":true,"route":"../roadmap/tauceti/algebra/representation-theory/index.html#harish-chandra-surjectivity-with-a-degree-bound","scope":"tauceti/algebra/representation-theory","status":"fully_proved","summary":"A Weyl invariant `q \u2208 S(H)^W` of degree at most `n` is the linear Harish-Chandra projection of a central element of `U(L)` of PBW degree at most `n`.","title":"Harish-Chandra surjectivity with a degree bound"},{"id":"tauceti/algebra/representation-theory/the-peter-weyl-theorem","kind":"node","lean":"TauCeti.peterWeylBasis, TauCeti.dense_representativeSubmodule","parent":1162,"result":true,"route":"../roadmap/tauceti/algebra/representation-theory/index.html#the-peter-weyl-theorem","scope":"tauceti/algebra/representation-theory","status":"fully_proved","summary":"For a skeleton of the unitary dual of a compact group, the normalized matrix coefficients `\u221a(dim V_i) \u00b7 (models i)_{ab}` are a Hilbert basis of `L\u00b2(G)`, indexed by `\u03a3 i, Fin (models i).dim \u00d7 Fin (models i).dim`.","title":"The Peter-Weyl theorem"},{"data_url":"scopes/tauceti/algebra/stable-periodic-and-curved-homological-algebra.json","id":"tauceti/algebra/stable-periodic-and-curved-homological-algebra","kind":"scope","parent":1096,"result":false,"route":"../roadmap/tauceti/algebra/stable-periodic-and-curved-homological-algebra/index.html","scope":"tauceti/algebra/stable-periodic-and-curved-homological-algebra","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Stable, periodic, and curved homological algebra"},{"id":"tauceti/algebra/stable-periodic-and-curved-homological-algebra/happels-theorem","kind":"node","lean":"TauCeti.ExactStructure.IsFrobenius.stableIsTriangulated, TauCeti.ExactStructure.IsFrobenius.stablePretriangulated","parent":1177,"result":true,"route":"../roadmap/tauceti/algebra/stable-periodic-and-curved-homological-algebra/index.html#happels-theorem","scope":"tauceti/algebra/stable-periodic-and-curved-homological-algebra","status":"fully_proved","summary":"The stable category of a Frobenius exact category, with the shift generated by stable suspension and the triangles generated by conflations, is triangulated.","title":"Happel's theorem"},{"id":"tauceti/algebra/stable-periodic-and-curved-homological-algebra/the-triangulation-of-the-periodic-homotopy-categ","kind":"node","lean":"TauCeti.PeriodicComplex.mem_distTriang_iff","parent":1177,"result":true,"route":"../roadmap/tauceti/algebra/stable-periodic-and-curved-homological-algebra/index.html#the-triangulation-of-the-periodic-homotopy-categ","scope":"tauceti/algebra/stable-periodic-and-curved-homological-algebra","status":"fully_proved","summary":"A triangle is distinguished exactly when it is isomorphic to the image of a mapping-cone triangle `X \u27f6 Y \u27f6 cone(f) \u27f6 X\u27e61\u27e7` of periodic complexes, whose last map is minus the cone projection.","title":"The triangulation of the periodic homotopy category is the mapping-cone triangulation"},{"id":"tauceti/algebra/stable-periodic-and-curved-homological-algebra/the-homotopy-category-of-matrix-factorizations-i","kind":"node","lean":"TauCeti.MatrixFactorization.HomotopyCategory.instIsTriangulated","parent":1177,"result":true,"route":"../roadmap/tauceti/algebra/stable-periodic-and-curved-homological-algebra/index.html#the-homotopy-category-of-matrix-factorizations-i","scope":"tauceti/algebra/stable-periodic-and-curved-homological-algebra","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.MatrixFactorization.HomotopyCategory.instIsTriangulated`.","title":"The homotopy category of matrix factorizations is triangulated, with the shift by `1` given by the parity shift"},{"id":"tauceti/algebra/stable-periodic-and-curved-homological-algebra/the-homotopy-category-of-curved-right-modules-is","kind":"node","lean":"TauCeti.CurvedDGRightModuleCat.homotopyIsTriangulated","parent":1177,"result":true,"route":"../roadmap/tauceti/algebra/stable-periodic-and-curved-homological-algebra/index.html#the-homotopy-category-of-curved-right-modules-is","scope":"tauceti/algebra/stable-periodic-and-curved-homological-algebra","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.CurvedDGRightModuleCat.homotopyIsTriangulated`.","title":"The homotopy category of curved right modules is triangulated, with the shift by one given by the cohomological shift"},{"id":"tauceti/algebra/stable-periodic-and-curved-homological-algebra/the-indecomposable-finitely-generated-right-modu","kind":"node","lean":"TauCeti.isIndecomposableModule_iff_exists_linearEquiv_quotient_span_op_root_X_pow_pow","parent":1177,"result":true,"route":"../roadmap/tauceti/algebra/stable-periodic-and-curved-homological-algebra/index.html#the-indecomposable-finitely-generated-right-modu","scope":"tauceti/algebra/stable-periodic-and-curved-homological-algebra","status":"fully_proved","summary":"such a module is indecomposable exactly when it is isomorphic to `A \u29f8 (x ^ i)` for some `0 < i \u2264 n`.","title":"The indecomposable finitely generated right modules over `A = K[X]/(X\u207f)` are exactly the `M_i`, up to isomorphism"},{"id":"tauceti/algebra/stable-periodic-and-curved-homological-algebra/the-periodic-free-resolution-of-m-i-over-r-x-x-n","kind":"node","lean":"TauCeti.cyclicModuleRootPowProjectiveResolution","parent":1177,"result":true,"route":"../roadmap/tauceti/algebra/stable-periodic-and-curved-homological-algebra/index.html#the-periodic-free-resolution-of-m-i-over-r-x-x-n","scope":"tauceti/algebra/stable-periodic-and-curved-homological-algebra","status":"fully_proved","summary":"With `x` the class of `X` and `M_i = A \u29f8 (x ^ i)`, the complex `\u22ef \u27f6 A --x^(n-i)--> A --x^i--> A --x^(n-i)--> A --x^i--> A \u27f6 M_i \u27f6 0` is a projective resolution of `M_i`.","title":"The periodic free resolution of `M_i` over `R[X]/(X ^ n)`"},{"id":"tauceti/algebra/stable-periodic-and-curved-homological-algebra/tauceti-orbitcategory-lift","kind":"node","lean":"TauCeti.OrbitCategory.lift","parent":1177,"result":true,"route":"../roadmap/tauceti/algebra/stable-periodic-and-curved-homological-algebra/index.html#tauceti-orbitcategory-lift","scope":"tauceti/algebra/stable-periodic-and-curved-homological-algebra","status":"fully_proved","summary":"Keller's descent of an additive functor `G` with a periodicity isomorphism `\u03c6 : [n] \u22d9 G \u2245 G` to the orbit category `D/[n]`. It is `G` on objects, and acts on morphisms by `liftMap G \u03c6`.","title":"TauCeti.OrbitCategory.lift"},{"id":"tauceti/algebra/stable-periodic-and-curved-homological-algebra/compression-is-a-triangle-functor-on-derived-cat","kind":"node","lean":"TauCeti.PeriodicComplex.instIsTriangulatedBoundedDerivedCategoryDerivedCompressionFunctor","parent":1177,"result":true,"route":"../roadmap/tauceti/algebra/stable-periodic-and-curved-homological-algebra/index.html#compression-is-a-triangle-functor-on-derived-cat","scope":"tauceti/algebra/stable-periodic-and-curved-homological-algebra","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.PeriodicComplex.instIsTriangulatedBoundedDerivedCategoryDerivedCompressionFunctor`.","title":"Compression is a triangle functor on derived categories"},{"id":"tauceti/algebra/stable-periodic-and-curved-homological-algebra/tauceti-curveddgrightmodulecat-absolutelyacyclic","kind":"node","lean":"TauCeti.CurvedDGRightModuleCat.absolutelyAcyclic","parent":1177,"result":true,"route":"../roadmap/tauceti/algebra/stable-periodic-and-curved-homological-algebra/index.html#tauceti-curveddgrightmodulecat-absolutelyacyclic","scope":"tauceti/algebra/stable-periodic-and-curved-homological-algebra","status":"fully_proved","summary":"The absolutely acyclic curved right modules: the smallest thick triangulated subcategory of the curved homotopy category containing all totalizations of short exact sequences.","title":"TauCeti.CurvedDGRightModuleCat.absolutelyAcyclic"},{"data_url":"scopes/tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive.json","id":"tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive","kind":"scope","parent":1096,"result":false,"route":"../roadmap/tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive/index.html","scope":"tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Type-A Hecke algebras, braid traces, and positive quotients"},{"id":"tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive/split-semisimplicity-separates-the-jucys-murphy","kind":"node","lean":"YoungDiagram.injOn_zpow_cellContent_of_prod_geom_sum_ne_zero","parent":1187,"result":true,"route":"../roadmap/tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive/index.html#split-semisimplicity-separates-the-jucys-murphy","scope":"tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive","status":"fully_proved","summary":"Over a commutative ring without zero divisors, if the unit `Q` is not `1` and the product `\u220f_{d=1}^{n} [d]_Q` of the `Q`-integers `[d]_Q = 1 + Q + \u22ef + Q ^ (d - 1)` is nonzero, then on every Young diagram of size less than `n` the powers `Q ^ content c` of distinct addable cells are distinct.","title":"Split semisimplicity separates the Jucys\u2013Murphy eigenvalues"},{"id":"tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive/the-separation-criterion-is-sharp","kind":"node","lean":"YoungDiagram.injOn_zpow_cellContent_iff","parent":1187,"result":true,"route":"../roadmap/tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive/index.html#the-separation-criterion-is-sharp","scope":"tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive","status":"fully_proved","summary":"The powers `Q ^ content c` separate the addable cells of every Young diagram of size less than `n` if and only if `Q ^ d \u2260 1` for `2 \u2264 d \u2264 n`.","title":"The separation criterion is sharp"},{"id":"tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive/tauceti-signedcheckedr-braid","kind":"node","lean":"TauCeti.signedCheckedR_braid","parent":1187,"result":true,"route":"../roadmap/tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive/index.html#tauceti-signedcheckedr-braid","scope":"tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive","status":"fully_proved","summary":"The braid relation for signed checked R-matrices on two pairs of strands sharing the strand `k`, the Yang-Baxter equation for the signed checked R-matrix.","title":"TauCeti.signedCheckedR_braid"},{"id":"tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive/comparison-with-the-jimbo-r-matrix","kind":"node","lean":"TauCeti.signedCheckedR_eq_jimboGenerator","parent":1187,"result":true,"route":"../roadmap/tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive/index.html#comparison-with-the-jimbo-r-matrix","scope":"tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive","status":"fully_proved","summary":"If every letter is positive, the signed checked R-matrix is the Jimbo R-matrix at the parameter `-s`, rescaled by `-s`, on the alphabet with the reversed order: it is the conjugate of `(-s) \u2022 T_{-s}` on `(Fin n \u2192 \u03b9\u1d52\u1d48) \u2192\u2080 R` by the relabelling of words along `\u03b9 \u2243 \u03b9\u1d52\u1d48`.","title":"Comparison with the Jimbo R-matrix"},{"id":"tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive/the-power-sum-basis","kind":"node","lean":"TauCeti.psumPartBasis","parent":1187,"result":true,"route":"../roadmap/tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive/index.html#the-power-sum-basis","scope":"tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive","status":"fully_proved","summary":"over a `\u211a`-algebra and in an alphabet of at least `n` letters, the power-sum products `p_\u03bc` of the partitions `\u03bc` of `n` form a basis of the symmetric polynomials of degree `n`.","title":"The power-sum basis"},{"id":"tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive/tauceti-infinitetypeaheckealgebra","kind":"node","lean":"TauCeti.InfiniteTypeAHeckeAlgebra","parent":1187,"result":true,"route":"../roadmap/tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive/index.html#tauceti-infinitetypeaheckealgebra","scope":"tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive","status":"fully_proved","summary":"The infinite type-A Hecke algebra over `R` with parameter `q`: the free `R`-algebra on `\u2115` modulo `TauCeti.InfiniteTypeAHeckeAlgebra.Rel`. It is the union of its finite stages `TauCeti.InfiniteTypeAHeckeAlgebra.stage R q n`; see the module docstring for what is proved about them.","title":"TauCeti.InfiniteTypeAHeckeAlgebra"},{"id":"tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive/tauceti-infinitetypeaheckealgebra-stage","kind":"node","lean":"TauCeti.InfiniteTypeAHeckeAlgebra.stage","parent":1187,"result":true,"route":"../roadmap/tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive/index.html#tauceti-infinitetypeaheckealgebra-stage","scope":"tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive","status":"fully_proved","summary":"The stage of the first `n` strands: the subalgebra generated by the generators `T 0, \u2026, T (n - 2)` that cross two of the strands `0, \u2026, n - 1`. It is the image of the type-A Hecke algebra on `n` strands.","title":"TauCeti.InfiniteTypeAHeckeAlgebra.stage"},{"id":"tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive/tauceti-infinitetypeaheckealgebra-shift","kind":"node","lean":"TauCeti.InfiniteTypeAHeckeAlgebra.shift","parent":1187,"result":true,"route":"../roadmap/tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive/index.html#tauceti-infinitetypeaheckealgebra-shift","scope":"tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive","status":"fully_proved","summary":"The strand shift by `k`: the algebra endomorphism `T i \u21a6 T (i + k)` that moves every braid `k` strands to the right.","title":"TauCeti.InfiniteTypeAHeckeAlgebra.shift"},{"id":"tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive/tauceti-infinitetypeaheckealgebra-blockincl","kind":"node","lean":"TauCeti.InfiniteTypeAHeckeAlgebra.blockIncl","parent":1187,"result":true,"route":"../roadmap/tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive/index.html#tauceti-infinitetypeaheckealgebra-blockincl","scope":"tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive","status":"fully_proved","summary":"The disjoint block map: the algebra map from the tensor product of the stages `a` and `b` to the stage `a + b` that places the first factor on the first `a` strands and the second factor, shifted by `a`, on the next `b` strands. Its injectivity, which would make it an embedding, needs the basis theorem and is not proved here.","title":"TauCeti.InfiniteTypeAHeckeAlgebra.blockIncl"},{"id":"tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive/tauceti-signedcheckedrep","kind":"node","lean":"TauCeti.signedCheckedRep","parent":1187,"result":true,"route":"../roadmap/tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive/index.html#tauceti-signedcheckedrep","scope":"tauceti/algebra/type-a-hecke-algebras-braid-traces-and-positive","status":"fully_proved","summary":"The signed checked representation of the braid group on the word space `(Fin n \u2192 \u03b9) \u2192\u2080 R`: the elementary braid `\u03c3 i` acts by the signed checked R-matrix on the strands `i` and `i + 1`.","title":"TauCeti.signedCheckedRep"},{"data_url":"scopes/tauceti/algebra/algebraic-group.json","id":"tauceti/algebra/algebraic-group","kind":"scope","parent":1096,"result":false,"route":"../roadmap/tauceti/algebra/algebraic-group/index.html","scope":"tauceti/algebra/algebraic-group","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Algebra.AlgebraicGroup` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Algebraic Group"},{"id":"tauceti/algebra/algebraic-group/borel-fixed-point-theorem-orbit-form","kind":"node","lean":"TauCeti.Comodule.exists_projectiveOrbitMap_eq_identity_of_isClosed_of_isSolvable, TauCeti.Comodule.exists_fixed_mem_of_isClosed_of_isSolvable","parent":1198,"result":true,"route":"../roadmap/tauceti/algebra/algebraic-group/index.html#borel-fixed-point-theorem-orbit-form","scope":"tauceti/algebra/algebraic-group","status":"fully_proved","summary":"Every nonempty closed invariant subset of projective space contains a rational line whose full scheme-point orbit is a singleton.","title":"Borel fixed-point theorem, orbit form"},{"id":"tauceti/algebra/algebraic-group/cartiers-theorem","kind":"node","lean":"TauCeti.smoothCommHopfAlgProperty_of_charZero","parent":1198,"result":true,"route":"../roadmap/tauceti/algebra/algebraic-group/index.html#cartiers-theorem","scope":"tauceti/algebra/algebraic-group","status":"fully_proved","summary":"a finite-type commutative Hopf algebra over a field of characteristic zero is smooth. Equivalently, every affine group scheme of finite type over such a field is smooth.","title":"Cartier's theorem"},{"id":"tauceti/algebra/algebraic-group/chevalleys-coset-criterion","kind":"node","lean":"TauCeti.HopfIdeal.exists_finite_subcomodule_exteriorPower_coset_matrixCoefficient","parent":1198,"result":true,"route":"../roadmap/tauceti/algebra/algebraic-group/index.html#chevalleys-coset-criterion","scope":"tauceti/algebra/algebraic-group","status":"fully_proved","summary":"A closed subgroup with finitely generated defining ideal admits a finite exterior-power representation and a unimodular vector whose matrix coefficients detect its left cosets over every commutative value algebra, including nonreduced algebras.","title":"Chevalley's coset criterion"},{"id":"tauceti/algebra/algebraic-group/chevalleys-theorem-with-the-subgroup-character","kind":"node","lean":"TauCeti.HopfIdeal.exists_finite_subcomodule_exteriorPower_line_stabilizer","parent":1198,"result":true,"route":"../roadmap/tauceti/algebra/algebraic-group/index.html#chevalleys-theorem-with-the-subgroup-character","scope":"tauceti/algebra/algebraic-group","status":"fully_proved","summary":"A closed subgroup with finitely generated defining ideal is the stabilizer, over every commutative value algebra, of a line in a finite-dimensional exterior-power representation. The line lies in the weight space of a unique character of the subgroup, including when the subgroup is nonreduced.","title":"Chevalley's theorem with the subgroup character"},{"id":"tauceti/algebra/algebraic-group/delignes-theorem-tauceti-alghom-convpow-eq-one-o","kind":"node","lean":"TauCeti.AlgHom.convPow_eq_one_of_rankAtStalk, TauCeti.AlgHom.convPow_finrank_eq_one","parent":1198,"result":true,"route":"../roadmap/tauceti/algebra/algebraic-group/index.html#delignes-theorem-tauceti-alghom-convpow-eq-one-o","scope":"tauceti/algebra/algebraic-group","status":"fully_proved","summary":"Every point of a commutative and cocommutative Hopf algebra which is finite flat of constant stalk rank `n` over `R` is killed by `n`.","title":"Deligne's theorem (`TauCeti.AlgHom.convPow_eq_one_of_rankAtStalk`)"},{"id":"tauceti/algebra/algebraic-group/the-dynamic-levi-decomposition-on-points","kind":"node","lean":"TauCeti.Cocharacter.exists_mem_unipotent_mem_levi","parent":1198,"result":true,"route":"../roadmap/tauceti/algebra/algebraic-group/index.html#the-dynamic-levi-decomposition-on-points","scope":"tauceti/algebra/algebraic-group","status":"fully_proved","summary":"every point of the dynamic parabolic is a point of its unipotent part times a point of its Levi part.","title":"The dynamic Levi decomposition, on points"},{"id":"tauceti/algebra/algebraic-group/the-fppf-first-isomorphism-theorem-for-affine-gr","kind":"node","lean":"TauCeti.CommHopfAlgCat.kernelFppfQuotientIso","parent":1198,"result":true,"route":"../roadmap/tauceti/algebra/algebraic-group/index.html#the-fppf-first-isomorphism-theorem-for-affine-gr","scope":"tauceti/algebra/algebraic-group","status":"fully_proved","summary":"If `f : H \u27f6 K` is faithfully flat and of finite presentation, then the fppf quotient of `Spec K` by the kernel of the represented homomorphism `Spec K \u27f6 Spec H` is represented by `Spec H`.","title":"The fppf first isomorphism theorem for affine groups"},{"id":"tauceti/algebra/algebraic-group/kolchins-theorem-for-ga","kind":"node","lean":"TauCeti.AdditiveGroup.exists_ne_zero_coact_eq_tmul_one","parent":1198,"result":true,"route":"../roadmap/tauceti/algebra/algebraic-group/index.html#kolchins-theorem-for-ga","scope":"tauceti/algebra/algebraic-group","status":"fully_proved","summary":"every nonzero vector of a comodule over the coordinate algebra of the additive group produces a nonzero fixed vector, over an arbitrary base and with no finiteness hypothesis. The witness is the top nonvanishing divided-power component `N\u1d62 v`: by the binomial composition rule each further component of it is a component of `v` of strictly larger index, which vanishes by maximality.","title":"Kolchin's theorem for `\ud835\udd3e\u2090`"},{"id":"tauceti/algebra/algebraic-group/the-lie-kolchin-theorem","kind":"node","lean":"TauCeti.Comodule.exists_basis_coefficientMatrix_isUpperTriangular_of_isSolvable, TauCeti.GeneralLinear.exists_map_inv_mul_mul_map_mem_upperTriangularGroup, TauCeti.Symplectic.IsotropicFlag.exists_map_inv_mul_mul_map_mem_matrixSubgroup, TauCeti.Comodule.exists_basis_coefficientMatrix_isUpperTriangular_of_geometricallySolvable, TauCeti.Comodule.exists_basis_coefficientMatrix_isUpperTriangular_of_forall_isUnipotentPoint_derived, TauCeti.Comodule.hasNonzeroWeightVector_of_geometricallySolvable, TauCeti.Comodule.hasNonzeroWeightVector_of_isSolvable","parent":1198,"result":true,"route":"../roadmap/tauceti/algebra/algebraic-group/index.html#the-lie-kolchin-theorem","scope":"tauceti/algebra/algebraic-group","status":"fully_proved","summary":"If the rational points of a reduced connected affine group of finite type over an algebraically closed field form a solvable group, every finite-dimensional representation admits a basis in which its coefficient matrix is upper triangular, with characters on the diagonal.","title":"The Lie--Kolchin theorem"},{"id":"tauceti/algebra/algebraic-group/matsushimas-criterion-affine-direction-in-charac","kind":"node","lean":"TauCeti.HopfIdeal.isIso_fppfHomogeneousQuotientToCoinvariantsPoints_of_reductive, TauCeti.HopfIdeal.isIso_fppfHomogeneousQuotientToCoinvariantsPoints_ker_of_reductive","parent":1198,"result":true,"route":"../roadmap/tauceti/algebra/algebraic-group/index.html#matsushimas-criterion-affine-direction-in-charac","scope":"tauceti/algebra/algebraic-group","status":"fully_proved","summary":"For a reductive closed subgroup `N` of an affine group `G` of finite type over a field of characteristic zero, restriction to the invariant functions identifies the fppf homogeneous quotient `G/N` with the sheaf of points of `Spec O(G)^N`.","title":"Matsushima's criterion, affine direction, in characteristic zero"},{"id":"tauceti/algebra/algebraic-group/pointwise-first-isomorphism-theorem-for-affine-g","kind":"node","lean":"TauCeti.CommHopfAlgCat.kernelPointwiseQuotientIso","parent":1198,"result":true,"route":"../roadmap/tauceti/algebra/algebraic-group/index.html#pointwise-first-isomorphism-theorem-for-affine-g","scope":"tauceti/algebra/algebraic-group","status":"fully_proved","summary":"If the morphism represented by `f` is surjective on `A`-points, the target group of points is isomorphic to the pointwise quotient of source points by the scheme-theoretic kernel.","title":"Pointwise first isomorphism theorem for affine groups"},{"id":"tauceti/algebra/algebraic-group/weyls-theorem-for-algebraic-groups","kind":"node","lean":"TauCeti.Comodule.isCompletelyReducible_of_isKilling","parent":1198,"result":true,"route":"../roadmap/tauceti/algebra/algebraic-group/index.html#weyls-theorem-for-algebraic-groups","scope":"tauceti/algebra/algebraic-group","status":"fully_proved","summary":"Over an algebraically closed field of characteristic zero, every finite-dimensional representation of a connected group whose Lie algebra has nondegenerate Killing form is completely reducible.","title":"Weyl's theorem for algebraic groups"},{"data_url":"scopes/tauceti/algebra/category.json","id":"tauceti/algebra/category","kind":"scope","parent":1096,"result":false,"route":"../roadmap/tauceti/algebra/category/index.html","scope":"tauceti/algebra/category","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Algebra.Category` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Category"},{"id":"tauceti/algebra/category/fittings-lemma-categorically","kind":"node","lean":"TauCeti.indecomposable_iff_isLocalRing_end, FGModuleCat.isLocalRing_end_of_indecomposable","parent":1211,"result":true,"route":"../roadmap/tauceti/algebra/category/index.html#fittings-lemma-categorically","scope":"tauceti/algebra/category","status":"fully_proved","summary":"an object of `ModuleCat A` whose underlying module has finite length is indecomposable exactly when its ring of endomorphisms in the category is local.","title":"Fitting's lemma, categorically"},{"id":"tauceti/algebra/category/the-general-field-cartan-poincare-identity","kind":"node","lean":"TauCeti.gradedCartanMatrix_transpose_mul_diagonalExtPoincareMatrix_map_eq_diagonal","parent":1211,"result":true,"route":"../roadmap/tauceti/algebra/category/index.html#the-general-field-cartan-poincare-identity","scope":"tauceti/algebra/category","status":"fully_proved","summary":"Let `C` be the graded Cartan matrix in a projective basis of classes `[A e\u1d62]` and the module basis `[S\u2c7c]`, with `gdim(e\u1d62 \u2022 S\u2c7c) = 0` for `i \u2260 j`. Suppose every pair of finite graded modules is graded Euler-admissible, that every `S\u1d62` is concentrated in internal degree zero and has a linear graded projective resolution, and that `b` bounds the cohomological support of Ext between any two `S\u1d62`. Then\u2026","title":"The general-field Cartan--Poincar\u00e9 identity"},{"id":"tauceti/algebra/category/the-krull-schmidt-theorem-with-no-ambient-object","kind":"node","lean":"TauCeti.exists_equiv_iso_of_biproduct_iso, TauCeti.exists_equiv_iso_of_iso_biproduct","parent":1211,"result":true,"route":"../roadmap/tauceti/algebra/category/index.html#the-krull-schmidt-theorem-with-no-ambient-object","scope":"tauceti/algebra/category","status":"fully_proved","summary":"an isomorphism between two finite biproducts of indecomposable objects of finite length matches their summands by a bijection of the index sets.","title":"The Krull-Schmidt theorem with no ambient object"},{"id":"tauceti/algebra/category/the-module-category-frobenius-theorem","kind":"node","lean":"FGModuleCat.abelian_isFrobenius","parent":1211,"result":true,"route":"../roadmap/tauceti/algebra/category/index.html#the-module-category-frobenius-theorem","scope":"tauceti/algebra/category","status":"fully_proved","summary":"For a finite-dimensional algebra `A` over a field which is self-injective on both sides, the canonical exact structure on the category of finitely generated (equivalently, finite-dimensional) `A`-modules is Frobenius.","title":"The module-category Frobenius theorem"},{"id":"tauceti/algebra/category/the-resolution-theorem-for-modules","kind":"node","lean":"TauCeti.cartanEquiv","parent":1211,"result":true,"route":"../roadmap/tauceti/algebra/category/index.html#the-resolution-theorem-for-modules","scope":"tauceti/algebra/category","status":"fully_proved","summary":"If every finitely generated `R`-module admits a finite resolution by finitely generated projective modules, then the Cartan map `K\u2080(proj R) \u27f6 G\u2080(mod R)` is an isomorphism; its inverse sends the class of a module to the alternating class of any such resolution.","title":"The resolution theorem for modules"},{"id":"tauceti/algebra/category/the-resolution-theorem-for-modules-admitting-fin","kind":"node","lean":"TauCeti.moduleResolutionEquiv","parent":1211,"result":true,"route":"../roadmap/tauceti/algebra/category/index.html#the-resolution-theorem-for-modules-admitting-fin","scope":"tauceti/algebra/category","status":"fully_proved","summary":"their exact `K\u2080` is the exact `K\u2080` of the finitely generated projective modules. This is `TauCeti.ExactStructure.resolutionEquiv` in `ModuleCat R`; its inverse sends the class of a module to the alternating class of any such resolution.","title":"The resolution theorem for modules admitting finite resolutions by finitely generated projectives"},{"data_url":"scopes/tauceti/algebra/cayley-dickson.json","id":"tauceti/algebra/cayley-dickson","kind":"scope","parent":1096,"result":false,"route":"../roadmap/tauceti/algebra/cayley-dickson/index.html","scope":"tauceti/algebra/cayley-dickson","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Algebra.CayleyDickson` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Cayley Dickson"},{"id":"tauceti/algebra/cayley-dickson/artins-theorem-for-the-double-of-the-quaternions","kind":"node","lean":"TauCeti.CayleyDickson.mul_assoc_of_mem_adjoin","parent":1218,"result":true,"route":"../roadmap/tauceti/algebra/cayley-dickson/index.html#artins-theorem-for-the-double-of-the-quaternions","scope":"tauceti/algebra/cayley-dickson","status":"fully_proved","summary":"the subalgebra generated by two elements `x` and `y` is associative.","title":"Artin's theorem for the double of the quaternions"},{"id":"tauceti/algebra/cayley-dickson/the-left-moufang-identity-tauceti-cayleydickson","kind":"node","lean":"TauCeti.CayleyDickson.moufang_left","parent":1218,"result":true,"route":"../roadmap/tauceti/algebra/cayley-dickson/index.html#the-left-moufang-identity-tauceti-cayleydickson","scope":"tauceti/algebra/cayley-dickson","status":"fully_proved","summary":"`(z * x * z) * y = z * (x * (z * y))`.","title":"The left Moufang identity (`TauCeti.CayleyDickson.moufang_left`)"},{"id":"tauceti/algebra/cayley-dickson/the-middle-moufang-identity-tauceti-cayleydickso","kind":"node","lean":"TauCeti.CayleyDickson.moufang_middle","parent":1218,"result":true,"route":"../roadmap/tauceti/algebra/cayley-dickson/index.html#the-middle-moufang-identity-tauceti-cayleydickso","scope":"tauceti/algebra/cayley-dickson","status":"fully_proved","summary":"`(z * x) * (y * z) = (z * (x * y)) * z`.","title":"The middle Moufang identity (`TauCeti.CayleyDickson.moufang_middle`)"},{"id":"tauceti/algebra/cayley-dickson/the-right-moufang-identity-tauceti-cayleydickson","kind":"node","lean":"TauCeti.CayleyDickson.moufang_right","parent":1218,"result":true,"route":"../roadmap/tauceti/algebra/cayley-dickson/index.html#the-right-moufang-identity-tauceti-cayleydickson","scope":"tauceti/algebra/cayley-dickson","status":"fully_proved","summary":"`x * (z * y * z) = ((x * z) * y) * z`.","title":"The right Moufang identity (`TauCeti.CayleyDickson.moufang_right`)"},{"data_url":"scopes/tauceti/algebra/central-simple.json","id":"tauceti/algebra/central-simple","kind":"scope","parent":1096,"result":false,"route":"../roadmap/tauceti/algebra/central-simple/index.html","scope":"tauceti/algebra/central-simple","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Algebra.CentralSimple` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Central Simple"},{"id":"tauceti/algebra/central-simple/artin-wedderburn-over-a-separably-closed-field","kind":"node","lean":"IsSimpleRing.exists_algEquiv_matrix_of_isSepClosed","parent":1223,"result":true,"route":"../roadmap/tauceti/algebra/central-simple/index.html#artin-wedderburn-over-a-separably-closed-field","scope":"tauceti/algebra/central-simple","status":"fully_proved","summary":"A finite-dimensional central simple `K`-algebra is a full matrix algebra over `K`.","title":"Artin--Wedderburn over a separably closed field"},{"id":"tauceti/algebra/central-simple/the-centralizer-theorem","kind":"node","lean":"TauCeti.finrank_mul_finrank_centralizer","parent":1223,"result":true,"route":"../roadmap/tauceti/algebra/central-simple/index.html#the-centralizer-theorem","scope":"tauceti/algebra/central-simple","status":"fully_proved","summary":"For a central simple `K`-subalgebra `B` of a finite-dimensional simple `K`-algebra `A`, the dimensions of `B` and of its centralizer are complementary:","title":"The centralizer theorem"},{"id":"tauceti/algebra/central-simple/the-dimension-count-behind-the-centralizer-theor","kind":"node","lean":"TauCeti.finrank_mul_finrank_centralizer_of_isSimpleRing_tensorProduct_mulOpposite","parent":1223,"result":true,"route":"../roadmap/tauceti/algebra/central-simple/index.html#the-dimension-count-behind-the-centralizer-theor","scope":"tauceti/algebra/central-simple","status":"fully_proved","summary":"that `A` is finite-dimensional and nonzero over the field `K` \u2014 so that the factor of `finrank K A` cancelled at the end is nonzero \u2014 and that the ring `R = B \u2297[K] A\u1d50\u1d52\u1d56` acting on `A` is simple. Then","title":"The dimension count behind the centralizer theorem, asking only for what it uses"},{"id":"tauceti/algebra/central-simple/frobenius-theorem","kind":"node","lean":"TauCeti.nonempty_algEquiv_real_or_quaternion","parent":1223,"result":true,"route":"../roadmap/tauceti/algebra/central-simple/index.html#frobenius-theorem","scope":"tauceti/algebra/central-simple","status":"fully_proved","summary":"A finite-dimensional division algebra over `\u211d` with centre `\u211d` is `\u211d`-isomorphic either to `\u211d` or to the Hamilton quaternions `\u210d[\u211d]`.","title":"Frobenius' theorem"},{"id":"tauceti/algebra/central-simple/the-skolem-noether-theorem","kind":"node","lean":"TauCeti.skolemNoether","parent":1223,"result":true,"route":"../roadmap/tauceti/algebra/central-simple/index.html#the-skolem-noether-theorem","scope":"tauceti/algebra/central-simple","status":"fully_proved","summary":"Two `K`-algebra homomorphisms `f g : B \u2192\u2090[K] A` from a finite-dimensional central simple `K`-algebra `B` to a finite-dimensional simple `K`-algebra `A` are conjugate: there is a unit `u` of `A` with `g x = u * f x * u\u207b\u00b9` for all `x : B`.","title":"The Skolem-Noether theorem"},{"id":"tauceti/algebra/central-simple/wedderburn-artin-for-central-simple-algebras","kind":"node","lean":"IsSimpleRing.exists_algEquiv_matrix_centralDivisionRing","parent":1223,"result":true,"route":"../roadmap/tauceti/algebra/central-simple/index.html#wedderburn-artin-for-central-simple-algebras","scope":"tauceti/algebra/central-simple","status":"fully_proved","summary":"A finite-dimensional central simple `K`-algebra `A` is isomorphic to a matrix algebra over a finite-dimensional central division `K`-algebra `D`, and then `finrank K A = n ^ 2 * finrank K D`.","title":"Wedderburn-Artin for central simple algebras"},{"data_url":"scopes/tauceti/algebra/homology.json","id":"tauceti/algebra/homology","kind":"scope","parent":1096,"result":false,"route":"../roadmap/tauceti/algebra/homology/index.html","scope":"tauceti/algebra/homology","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Algebra.Homology` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Homology"},{"id":"tauceti/algebra/homology/chens-theorem","kind":"node","lean":"CategoryTheory.GorensteinProjective.fullyFaithfulToSingularityCategory, CategoryTheory.GorensteinProjective.instFaithfulProjectiveStableCategoryExactStructureSingularityCategoryToSingularityCategory, CategoryTheory.GorensteinProjective.instFullProjectiveStableCategoryExactStructureSingularityCategoryToSingularityCategory","parent":1230,"result":true,"route":"../roadmap/tauceti/algebra/homology/index.html#chens-theorem","scope":"tauceti/algebra/homology","status":"fully_proved","summary":"the functor from the stable category of Gorenstein-projective objects of an abelian category to its singularity category is fully faithful.","title":"Chen's theorem"},{"id":"tauceti/algebra/homology/the-comparison-theorem-categorytheory-abelian-sp","kind":"node","lean":"CategoryTheory.Abelian.SpectralObject.isIso_hom_app_abutment_of_isIso_page","parent":1230,"result":true,"route":"../roadmap/tauceti/algebra/homology/index.html#the-comparison-theorem-categorytheory-abelian-sp","scope":"tauceti/algebra/homology","status":"fully_proved","summary":"Let `\u03c6 : X \u27f6 Y` be a morphism of spectral objects indexed by `EInt` and `a \u2264 b` such that `H^m(\u22a5 \u2264 i)` vanishes for `i \u2264 a` and `m \u2208 {n, n + 1}`, and `H^m(i \u2264 \u22a4)` vanishes for `i \u2265 b` and `m \u2208 {n - 1, n}`, for `X` and `Y`. If the induced morphism of spectral sequences is an isomorphism on every object of some page `r \u2265 2`, then `\u03c6` is an isomorphism on the abutments `H^n(\u22a5 \u2264 \u22a4)`.","title":"The comparison theorem (`CategoryTheory.Abelian.SpectralObject.isIso_hom_app_abutment_of_isIso_page`)"},{"id":"tauceti/algebra/homology/the-comparison-theorem-for-exhaustive-filtration","kind":"node","lean":"CochainComplex.isIso_homologyMap_of_isIso_spectralSequence_page_of_exhaustive","parent":1230,"result":true,"route":"../roadmap/tauceti/algebra/homology/index.html#the-comparison-theorem-for-exhaustive-filtration","scope":"tauceti/algebra/homology","status":"fully_proved","summary":"Let `\u03b1` be a morphism between the functors of filtrations `F` of `K` and `G` of `L`, and let `n` be a degree. Suppose that the stages `F_i` and `G_i` at all indices `i \u2264 a` have vanishing cohomology in degrees `n` and `n + 1` (for instance `F_a = G_a = \u22a5`), that in degrees `n - 1` and `n` every element of `K` and of `L` lies in some stage of the filtration, and that the graded pieces of both filtrations at all\u2026","title":"The comparison theorem for exhaustive filtrations"},{"id":"tauceti/algebra/homology/the-comparison-theorem-for-filtered-complexes","kind":"node","lean":"CochainComplex.isIso_homologyMap_of_isIso_spectralSequence_page","parent":1230,"result":true,"route":"../roadmap/tauceti/algebra/homology/index.html#the-comparison-theorem-for-filtered-complexes","scope":"tauceti/algebra/homology","status":"fully_proved","summary":"Let `\u03b1` be a morphism between the functors of finite increasing filtrations `F` of `K` and `G` of `L` (`F_a = G_a = \u22a5` and `F_b = G_b = \u22a4`). If the induced morphism of spectral sequences is an isomorphism on every object of some page `r \u2265 2`, then `\u03b1.app \u22a4` induces isomorphisms in cohomology.","title":"The comparison theorem for filtered complexes"},{"id":"tauceti/algebra/homology/the-comparison-theorem-on-e","kind":"node","lean":"CategoryTheory.Abelian.SpectralObject.isIso_hom_app_abutment","parent":1230,"result":true,"route":"../roadmap/tauceti/algebra/homology/index.html#the-comparison-theorem-on-e","scope":"tauceti/algebra/homology","status":"fully_proved","summary":"Let `\u03c6 : X \u27f6 Y` be a morphism of spectral objects indexed by `EInt` and `a \u2264 b` such that `H^n(\u22a5 \u2264 a)` and `H^n(b \u2264 \u22a4)` vanish for `X` and `Y` (the filtrations of the abutments in degree `n` start at `a` and end at `b`). If `\u03c6` is an isomorphism on `EInfty q n` for `a \u2264 q < b`, then it is an isomorphism on the abutment `H^n(\u22a5 \u2264 \u22a4)`.","title":"The comparison theorem on `E\u221e`"},{"id":"tauceti/algebra/homology/the-differential-graded-yoneda-lemma","kind":"node","lean":"TauCeti.dgYonedaIso, TauCeti.dgYonedaCochainEquiv, TauCeti.dgYonedaHomEquiv","parent":1230,"result":true,"route":"../roadmap/tauceti/algebra/homology/index.html#the-differential-graded-yoneda-lemma","scope":"tauceti/algebra/homology","status":"fully_proved","summary":"the Hom complex out of the free rank-one right module is the underlying cochain complex of `M`.","title":"The differential graded Yoneda lemma"},{"id":"tauceti/algebra/homology/euler-poincare-for-cubes","kind":"node","lean":"CategoryTheory.Functor.homologyEulerChar_cubeComplex, ChainComplex.eulerChar_eq_homologyEulerChar, CochainComplex.eulerChar_eq_homologyEulerChar, HomologicalComplex.eulerChar_forgetFG_eq_homologyEulerChar, ChainComplex.sum_range_finrank_X_eq_sum_range_finrank_homology","parent":1230,"result":true,"route":"../roadmap/tauceti/algebra/homology/index.html#euler-poincare-for-cubes","scope":"tauceti/algebra/homology","status":"fully_proved","summary":"Over a Noetherian ring with the rank-nullity property, such as `\u2124` or a field, the Euler characteristic of the cohomology of the cube complex of a cube of finite free modules is the sum of the ranks of the vertices `F(S)`, each with the sign `(-1) ^ #S`. Over `\u2124` the rank of a cohomology group is the rank of its free part, so torsion does not contribute.","title":"Euler\u2013Poincar\u00e9 for cubes"},{"id":"tauceti/algebra/homology/graded-nakayama-lemma-for-free-complexes-over-a","kind":"node","lean":"LinearMap.ker_le_range_of_mapRange_constantCoeff","parent":1230,"result":true,"route":"../roadmap/tauceti/algebra/homology/index.html#graded-nakayama-lemma-for-free-complexes-over-a","scope":"tauceti/algebra/homology","status":"fully_proved","summary":"Let `d` be a square-zero endomorphism of the free module `\u03b9 \u2192\u2080 R[V_v : v \u2208 \u03c3]` which is homogeneous of degree `r` when the generator `i` has degree `g i` and the variable `V_v` has negative degree `w v`, with the degrees `g i` bounded above. If the reduction `d\u2080` of `d` modulo the variables, the endomorphism of `\u03b9 \u2192\u2080 R` with `d\u2080 \u2218 \u03c1 = \u03c1 \u2218 d` for `\u03c1` the coordinatewise constant coefficient, is exact, then so is `d`.","title":"Graded Nakayama lemma for free complexes over a polynomial ring"},{"id":"tauceti/algebra/homology/the-kunneth-theorem-for-split-complexes","kind":"node","lean":"HomologicalComplex.isIso_homologyKunneth_of_isSplitMono_of_isSplitEpi, HomologicalComplex.isIso_homologyKunneth_of_d_eq_zero_of_flat, HomologicalComplex.isIso_homologyKunneth_of_d_eq_zero_of_flat_right","parent":1230,"result":true,"route":"../roadmap/tauceti/algebra/homology/index.html#the-kunneth-theorem-for-split-complexes","scope":"tauceti/algebra/homology","status":"fully_proved","summary":"for complexes `K` and `L` of modules over a commutative ring `R` whose cycles split off their terms and whose homology splits off their cycles, the K\u00fcnneth map `\u2a01_{p + q = n} H\u209a(K) \u2297 H_q(L) \u27f6 H\u2099(K \u2297 L)` is an isomorphism. This applies to complexes of semisimple modules, for instance of vector spaces over a field (`HomologicalComplex.isSplitMono_iCycles_of_isSemisimpleModule`), and to complexes whose cycles split\u2026","title":"The K\u00fcnneth theorem for split complexes"},{"id":"tauceti/algebra/homology/the-morse-polynomial-identity-chaincomplex-sum-r","kind":"node","lean":"ChainComplex.sum_range_monomial_finrank_X_eq, TauCeti.exists_sum_range_monomial_eq_add_mul","parent":1230,"result":true,"route":"../roadmap/tauceti/algebra/homology/index.html#the-morse-polynomial-identity-chaincomplex-sum-r","scope":"tauceti/algebra/homology","status":"fully_proved","summary":"If the terms of `K` through degree `n` are finite-dimensional and the differential `K\u2099\u208a\u2081 \u27f6 K\u2099` vanishes, then in `\u2115[X]` `\u2211\u1d62\u208c\u2080\u207f c\u1d62 X\u2071 = \u2211\u1d62\u208c\u2080\u207f b\u1d62 X\u2071 + (1 + X) Q`, where `c\u1d62` is the dimension of `K\u1d62`, `b\u1d62` that of `H\u1d62(K)`, and `Q = \u2211\u1d62\u208c\u2080\u207f\u207b\u00b9 q\u1d62 X\u2071` with `q\u1d62` the rank of `K\u1d62\u208a\u2081 \u27f6 K\u1d62`: the coefficient of `X\u2071` in `Q` is that rank for `i < n` and zero otherwise.","title":"The Morse polynomial identity (`ChainComplex.sum_range_monomial_finrank_X_eq`)"},{"id":"tauceti/algebra/homology/the-strong-morse-inequalities-chaincomplex-sum-r","kind":"node","lean":"ChainComplex.sum_range_neg_one_pow_mul_finrank_homology_le","parent":1230,"result":true,"route":"../roadmap/tauceti/algebra/homology/index.html#the-strong-morse-inequalities-chaincomplex-sum-r","scope":"tauceti/algebra/homology","status":"fully_proved","summary":"If the terms of `K` through degree `n` are finite-dimensional, then `b\u2099 - b\u2099\u208b\u2081 + \u22ef \u00b1 b\u2080 \u2264 c\u2099 - c\u2099\u208b\u2081 + \u22ef \u00b1 c\u2080`, where `b\u1d62` is the dimension of `H\u1d62(K)` and `c\u1d62` that of `K\u1d62`. The difference of the two sides is the rank of `K\u2099\u208a\u2081 \u27f6 K\u2099`.","title":"The strong Morse inequalities (`ChainComplex.sum_range_neg_one_pow_mul_finrank_homology_le`)"},{"id":"tauceti/algebra/homology/the-truncated-morse-polynomial-identity","kind":"node","lean":"ChainComplex.sum_range_monomial_finrank_X_add_monomial_eq","parent":1230,"result":true,"route":"../roadmap/tauceti/algebra/homology/index.html#the-truncated-morse-polynomial-identity","scope":"tauceti/algebra/homology","status":"fully_proved","summary":"If the terms of `K` through degree `n` are finite-dimensional, then in `\u2115[X]` `\u2211\u1d62\u208c\u2080\u207f c\u1d62 X\u2071 + q\u2099 X\u207f\u207a\u00b9 = \u2211\u1d62\u208c\u2080\u207f b\u1d62 X\u2071 + (1 + X) \u2211\u1d62\u208c\u2080\u207f q\u1d62 X\u2071`, where `c\u1d62` is the dimension of `K\u1d62`, `b\u1d62` that of `H\u1d62(K)` and `q\u1d62` the rank of `K\u1d62\u208a\u2081 \u27f6 K\u1d62`.","title":"The truncated Morse polynomial identity"},{"id":"tauceti/algebra/homology/the-universal-coefficient-theorem-above-a-projec","kind":"node","lean":"TauCeti.ChainComplex.kronecker_bijective_succ_of_projective_homology","parent":1230,"result":true,"route":"../roadmap/tauceti/algebra/homology/index.html#the-universal-coefficient-theorem-above-a-projec","scope":"tauceti/algebra/homology","status":"fully_proved","summary":"The universal coefficient theorem above a projective homology object, when every object has projective dimension at most one, as for modules over a principal ideal domain: for a chain complex `X` indexed by `\u2115` with projective terms whose homology `H\u2099(X)` is projective, the Kronecker map `H\u207f\u207a\u00b9(Hom(X, Y)) \u2192\u2097[k] (H\u2099\u208a\u2081(X) \u27f6 Y)` is bijective, for every object `Y`. Nothing is assumed about the homology in the other\u2026","title":"The universal coefficient theorem above a projective homology object"},{"id":"tauceti/algebra/homology/the-universal-coefficient-theorem-below-the-firs","kind":"node","lean":"TauCeti.ChainComplex.kronecker_bijective_of_projective_homology_lt","parent":1230,"result":true,"route":"../roadmap/tauceti/algebra/homology/index.html#the-universal-coefficient-theorem-below-the-firs","scope":"tauceti/algebra/homology","status":"fully_proved","summary":"for a chain complex `X` indexed by `\u2115` with projective terms whose homology objects are projective below degree `n`, the Kronecker map `H\u207f(Hom(X, Y)) \u2192\u2097[k] (H\u2099(X) \u27f6 Y)` is bijective, for every object `Y`. In particular this holds when `X` is exact below degree `n`.","title":"The universal coefficient theorem below the first non-projective homology"},{"id":"tauceti/algebra/homology/the-universal-coefficient-theorem-with-projectiv","kind":"node","lean":"TauCeti.ChainComplex.kronecker_bijective_of_projective_homology, TauCeti.ChainComplex.kronecker_bijective, TauCeti.ChainComplex.kronecker_bijective_of_projective_homology_of_add_one_eq","parent":1230,"result":true,"route":"../roadmap/tauceti/algebra/homology/index.html#the-universal-coefficient-theorem-with-projectiv","scope":"tauceti/algebra/homology","status":"fully_proved","summary":"for a chain complex `X` indexed by `\u2115` whose terms and homology objects are all projective, the Kronecker map `H\u207f(Hom(X, Y)) \u2192\u2097[k] (H\u2099(X) \u27f6 Y)` is bijective in every degree, for every object `Y`.","title":"The universal coefficient theorem with projective homology"},{"id":"tauceti/algebra/homology/the-weak-morse-inequality","kind":"node","lean":"HomologicalComplex.finrank_homology_le_finrank_X","parent":1230,"result":true,"route":"../roadmap/tauceti/algebra/homology/index.html#the-weak-morse-inequality","scope":"tauceti/algebra/homology","status":"fully_proved","summary":"In a homological complex of vector spaces, a homology group has dimension at most that of the finite-dimensional term it is computed from.","title":"The weak Morse inequality"},{"data_url":"scopes/tauceti/algebra/lie.json","id":"tauceti/algebra/lie","kind":"scope","parent":1096,"result":false,"route":"../roadmap/tauceti/algebra/lie/index.html","scope":"tauceti/algebra/lie","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Algebra.Lie` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Lie"},{"data_url":"scopes/tauceti/algebra/lie/part-01.json","id":"tauceti/algebra/lie/part-01","kind":"scope","navigation":"transparent","parent":1247,"result":false,"route":"../roadmap/tauceti/algebra/lie/part-01/index.html","scope":"tauceti/algebra/lie/part-01","status":"fully_proved","title":"Abstract Jordan decomposition\u2026 to The Racah\u2013Speiser formula, in\u2026"},{"id":"tauceti/algebra/lie/part-01/abstract-jordan-decomposition-existence","kind":"node","lean":"TauCeti.exists_isNilpotent_ad_isSemisimple_ad, TauCeti.exists_isNilpotent_ad_isSemisimple_ad_of_iSup_maxGenEigenspace_eq_top, TauCeti.isNilpotent_ad_isSemisimple_ad_unique","parent":1248,"result":true,"route":"../roadmap/tauceti/algebra/lie/part-01/index.html#abstract-jordan-decomposition-existence","scope":"tauceti/algebra/lie/part-01","status":"fully_proved","summary":"In a finite-dimensional Lie algebra with nondegenerate Killing form over an algebraically closed field, every element `x` is a sum `n + s` of commuting elements with `ad n` nilpotent and `ad s` semisimple. Both `ad n` and `ad s` are polynomials in `ad x`.","title":"Abstract Jordan decomposition, existence"},{"id":"tauceti/algebra/lie/part-01/the-brauer-klimyk-formula-in-group-algebra-form","kind":"node","lean":"TauCeti.isotypicMultiplicity_tensor_eq_coeff_formalCharacter_mul_weylNumerator, TauCeti.isotypicMultiplicity_tensor_eq_sum_weylSign_mul_formalCharacter_coeff","parent":1248,"result":true,"route":"../roadmap/tauceti/algebra/lie/part-01/index.html#the-brauer-klimyk-formula-in-group-algebra-form","scope":"tauceti/algebra/lie/part-01","status":"fully_proved","summary":"for dominant integral weights `lam` and `nu` and a finite-dimensional `L`-module `M`, the multiplicity of `L(nu)` in `L(lam) \u2297 M` is the coefficient at `nu` of `ch M \u00b7 N(lam)`.","title":"The Brauer\u2013Klimyk formula, in group-algebra form"},{"id":"tauceti/algebra/lie/part-01/cartans-formula-x-d-l-x-d-x","kind":"node","lean":"TauCeti.LieModule.Cohomology.curryLeft_d_succ","parent":1248,"result":true,"route":"../roadmap/tauceti/algebra/lie/part-01/index.html#cartans-formula-x-d-l-x-d-x","scope":"tauceti/algebra/lie/part-01","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.LieModule.Cohomology.curryLeft_d_succ`.","title":"Cartan's formula `\u03b9_x d \u03c9 = L_x \u03c9 - d \u03b9_x \u03c9`"},{"id":"tauceti/algebra/lie/part-01/chevalleys-rank-one-identity","kind":"node","lean":"TauCeti.UniversalEnvelopingAlgebra.kostantRootSubgroupParam_mul_mul_eq_kostantCoordinateCocharacter_mul_kostantWeylGL","parent":1248,"result":true,"route":"../roadmap/tauceti/algebra/lie/part-01/index.html#chevalleys-rank-one-identity","scope":"tauceti/algebra/lie/part-01","status":"fully_proved","summary":"Chevalley's rank-one identity `x\u1d62(u) x\u2c7c(-u\u207b\u00b9) x\u1d62(u) = h_c(u) n`, over every value ring and at every unit `u`. Here `(e\u1d62, e\u2c7c)` is a root pair whose `sl\u2082` triple has Cartan index `c`, `h_c(u)` is the coordinate cocharacter at `c` and `n = x\u1d62(1) x\u2c7c(-1) x\u1d62(1)` is the Weyl representative, and `\u03b1` is the weight of `e\u1d62`, so that `e\u2c7c` has weight `-\u03b1`.","title":"Chevalley's rank-one identity"},{"id":"tauceti/algebra/lie/part-01/the-chevalley-eilenberg-formula-for-the-differen","kind":"node","lean":"TauCeti.LieModule.Cohomology.d_succ_apply","parent":1248,"result":true,"route":"../roadmap/tauceti/algebra/lie/part-01/index.html#the-chevalley-eilenberg-formula-for-the-differen","scope":"tauceti/algebra/lie/part-01","status":"fully_proved","summary":"`(d \u03c9)(x\u2080, \u2026, x\u2099\u208a\u2081) = \u2211 i, (-1)^i \u2045x\u1d62, \u03c9(\u2026, x\u0302\u1d62, \u2026)\u2046 + \u2211_{i < j} (-1)^(i + j) \u03c9(\u2045x\u1d62, x\u2c7c\u2046, \u2026)`, where the pairs `i < j` are written as `(i.castSucc, j.succ)` with `i \u2264 j`.","title":"The Chevalley\u2013Eilenberg formula for the differential of a cochain of positive degree"},{"id":"tauceti/algebra/lie/part-01/the-first-isomorphism-theorem-for-a-surjective-h","kind":"node","lean":"LieHom.quotKerEquivOfSurjective","parent":1248,"result":true,"route":"../roadmap/tauceti/algebra/lie/part-01/index.html#the-first-isomorphism-theorem-for-a-surjective-h","scope":"tauceti/algebra/lie/part-01","status":"fully_proved","summary":"the quotient by its kernel is isomorphic to the target.","title":"The first isomorphism theorem for a surjective homomorphism of Lie algebras"},{"id":"tauceti/algebra/lie/part-01/freudenthals-multiplicity-formula","kind":"node","lean":"TauCeti.freudenthal_multiplicity_formula","parent":1248,"result":true,"route":"../roadmap/tauceti/algebra/lie/part-01/index.html#freudenthals-multiplicity-formula","scope":"tauceti/algebra/lie/part-01","status":"fully_proved","summary":"Let `M` be a finite-dimensional module generated by a highest weight vector of weight `\u03bb`. Then for every linear form `\u03bc` on the Cartan subalgebra,","title":"Freudenthal's multiplicity formula"},{"id":"tauceti/algebra/lie/part-01/the-frobenius-commutator-identity","kind":"node","lean":"TauCeti.LieAlgebra.ad_pow_expChar_pow","parent":1248,"result":true,"route":"../roadmap/tauceti/algebra/lie/part-01/index.html#the-frobenius-commutator-identity","scope":"tauceti/algebra/lie/part-01","status":"fully_proved","summary":"In exponential characteristic `p`, taking the `p ^ n`-th power in the algebra `A` and iterating the commutator `p ^ n` times in `Module.End R A` give the same endomorphism.","title":"The Frobenius commutator identity"},{"id":"tauceti/algebra/lie/part-01/the-jacobi-identity-for-a-positive-definite-b-wi","kind":"node","lean":"Matrix.FullShellLieAlgebra.lie_jacobi, Matrix.ShellLieAlgebra.lie_jacobi","parent":1248,"result":true,"route":"../roadmap/tauceti/algebra/lie/part-01/index.html#the-jacobi-identity-for-a-positive-definite-b-wi","scope":"tauceti/algebra/lie/part-01","status":"fully_proved","summary":"Formalized in Tau Ceti as `Matrix.FullShellLieAlgebra.lie_jacobi`.","title":"The Jacobi identity, for a positive definite `B` with even diagonal"},{"id":"tauceti/algebra/lie/part-01/jacobsons-lemma-for-the-adjoint-representation","kind":"node","lean":"TauCeti.isNilpotent_ad_lie_of_lie_lie_eq_zero","parent":1248,"result":true,"route":"../roadmap/tauceti/algebra/lie/part-01/index.html#jacobsons-lemma-for-the-adjoint-representation","scope":"tauceti/algebra/lie/part-01","status":"fully_proved","summary":"In a finite-dimensional Lie algebra over a field of characteristic zero, if `\u2045e, \u2045e, a\u2046\u2046 = 0` then `ad \u2045e, a\u2046` is nilpotent. In particular every element of `ker (ad e) \u2293 [e, L]` is `ad`-nilpotent.","title":"Jacobson's lemma for the adjoint representation"},{"id":"tauceti/algebra/lie/part-01/jordan-chevalley-decomposition-of-a-derivation","kind":"node","lean":"LieDerivation.exists_isNilpotent_isSemisimple, LieDerivation.exists_isNilpotent_isSemisimple_of_iSup_maxGenEigenspace_eq_top","parent":1248,"result":true,"route":"../roadmap/tauceti/algebra/lie/part-01/index.html#jordan-chevalley-decomposition-of-a-derivation","scope":"tauceti/algebra/lie/part-01","status":"fully_proved","summary":"Over an algebraically closed field, the nilpotent and semisimple parts of a derivation of a finite-dimensional Lie algebra are derivations. They are polynomials in the original derivation.","title":"Jordan\u2013Chevalley decomposition of a derivation"},{"id":"tauceti/algebra/lie/part-01/kostants-multiplicity-formula","kind":"node","lean":"TauCeti.finrank_weightSpace_eq_kostantMultiplicity, TauCeti.formalCharacter_coeff_eq_kostantMultiplicity","parent":1248,"result":true,"route":"../roadmap/tauceti/algebra/lie/part-01/index.html#kostants-multiplicity-formula","scope":"tauceti/algebra/lie/part-01","status":"fully_proved","summary":"The multiplicity of the weight `mu` in a finite-dimensional module generated by a highest weight vector of weight `lam` is the alternating sum `\u2211_{w \u2208 W} sgn(w) P(w(lam + \u03c1) - (mu + \u03c1))` of values of the Kostant partition function.","title":"Kostant's multiplicity formula"},{"id":"tauceti/algebra/lie/part-01/the-lattice-form-of-weyls-theorem","kind":"node","lean":"TauCeti.complementedLattice_lieSubmodule_of_isKilling","parent":1248,"result":true,"route":"../roadmap/tauceti/algebra/lie/part-01/index.html#the-lattice-form-of-weyls-theorem","scope":"tauceti/algebra/lie/part-01","status":"fully_proved","summary":"the Lie submodules of a finite-dimensional module over a Lie algebra with nondegenerate Killing form form a complemented lattice.","title":"The lattice form of Weyl's theorem"},{"id":"tauceti/algebra/lie/part-01/the-leibniz-form-of-the-jacobi-identity-for-a-po","kind":"node","lean":"Matrix.FullShellLieAlgebra.leibniz_lie, Matrix.ShellLieAlgebra.leibniz_lie","parent":1248,"result":true,"route":"../roadmap/tauceti/algebra/lie/part-01/index.html#the-leibniz-form-of-the-jacobi-identity-for-a-po","scope":"tauceti/algebra/lie/part-01","status":"fully_proved","summary":"Formalized in Tau Ceti as `Matrix.FullShellLieAlgebra.leibniz_lie`.","title":"The Leibniz form of the Jacobi identity, for a positive definite `B` with even diagonal"},{"id":"tauceti/algebra/lie/part-01/the-levi-decomposition","kind":"node","lean":"TauCeti.exists_leviDecomposition","parent":1248,"result":true,"route":"../roadmap/tauceti/algebra/lie/part-01/index.html#the-levi-decomposition","scope":"tauceti/algebra/lie/part-01","status":"fully_proved","summary":"Over a field of characteristic zero, a finite-dimensional Lie algebra is the semidirect sum of its solvable radical and a semisimple Lie subalgebra acting on the radical by the adjoint action.","title":"The Levi decomposition"},{"id":"tauceti/algebra/lie/part-01/levis-theorem","kind":"node","lean":"TauCeti.exists_leviComplement","parent":1248,"result":true,"route":"../roadmap/tauceti/algebra/lie/part-01/index.html#levis-theorem","scope":"tauceti/algebra/lie/part-01","status":"fully_proved","summary":"Over a field of characteristic zero, the solvable radical of a finite-dimensional Lie algebra has a complementary Lie subalgebra, a Levi complement.","title":"Levi's theorem"},{"id":"tauceti/algebra/lie/part-01/the-linear-independence-half-of-the-poincare-bir","kind":"node","lean":"TauCeti.UniversalEnvelopingAlgebra.pbwAssociatedGradedMap_injective","parent":1248,"result":true,"route":"../roadmap/tauceti/algebra/lie/part-01/index.html#the-linear-independence-half-of-the-poincare-bir","scope":"tauceti/algebra/lie/part-01","status":"fully_proved","summary":"The canonical map `Sym(L) \u2192 gr U(L)` is injective when `L` is a free module.","title":"The linear-independence half of the Poincar\u00e9--Birkhoff--Witt theorem"},{"id":"tauceti/algebra/lie/part-01/the-poincare-birkhoff-witt-theorem","kind":"node","lean":"TauCeti.UniversalEnvelopingAlgebra.pbwAssociatedGradedEquiv","parent":1248,"result":true,"route":"../roadmap/tauceti/algebra/lie/part-01/index.html#the-poincare-birkhoff-witt-theorem","scope":"tauceti/algebra/lie/part-01","status":"fully_proved","summary":"For a Lie algebra that is free as a module, the symmetric algebra is isomorphic to the associated graded of the PBW filtration of the enveloping algebra, by the algebra map sending `x \u2208 L` to the degree-one class of `\u03b9(x)`.","title":"The Poincar\u00e9--Birkhoff--Witt theorem"},{"id":"tauceti/algebra/lie/part-01/the-positive-simple-root-chevalley-collection-fo","kind":"node","lean":"TauCeti.G2ShortRoot.PrimeField.commutatorElement_positive_simple_eq_root_product, TauCeti.G2ShortRoot.IntegralToralClosure.commutatorElement_positive_simple_eq_root_product","parent":1248,"result":true,"route":"../roadmap/tauceti/algebra/lie/part-01/index.html#the-positive-simple-root-chevalley-collection-fo","scope":"tauceti/algebra/lie/part-01","status":"fully_proved","summary":"The four factors have roots \u03b1\u2081+\u03b1\u2082, 2\u03b1\u2081+\u03b1\u2082, 3\u03b1\u2081+\u03b1\u2082, and 3\u03b1\u2081+2\u03b1\u2082, with their parameters normalized by the displayed conjugating words. The formula includes nonreduced coefficient algebras.","title":"The positive-simple-root Chevalley collection formula over \ud835\udd3d\u2083"},{"id":"tauceti/algebra/lie/part-01/the-racah-speiser-formula-in-group-algebra-form","kind":"node","lean":"TauCeti.tensorMultiplicity_eq_coeff_formalCharacter_mul_weylNumerator, TauCeti.tensorMultiplicity_eq_sum_weylSign_mul_formalCharacter_coeff","parent":1248,"result":true,"route":"../roadmap/tauceti/algebra/lie/part-01/index.html#the-racah-speiser-formula-in-group-algebra-form","scope":"tauceti/algebra/lie/part-01","status":"fully_proved","summary":"for dominant integral weights `lam`, `mu`, and `nu`, the tensor multiplicity `c^nu_{lam mu}` of `L(nu)` in `L(lam) \u2297 L(mu)` is the coefficient at `nu` of `ch L(mu) \u00b7 N(lam)`.","title":"The Racah\u2013Speiser formula, in group-algebra form"},{"data_url":"scopes/tauceti/algebra/lie/part-02.json","id":"tauceti/algebra/lie/part-02","kind":"scope","navigation":"transparent","parent":1247,"result":false,"route":"../roadmap/tauceti/algebra/lie/part-02/index.html","scope":"tauceti/algebra/lie/part-02","status":"fully_proved","title":"Schur's lemma for Lie modules to Weyl's theorem as a\u2026"},{"id":"tauceti/algebra/lie/part-02/schurs-lemma-for-lie-modules","kind":"node","lean":"TauCeti.exists_forall_apply_eq_smul, TauCeti.LieModule.finrank_lieModuleHom, TauCeti.LieModule.bijective_of_ne_zero","parent":1269,"result":true,"route":"../roadmap/tauceti/algebra/lie/part-02/index.html#schurs-lemma-for-lie-modules","scope":"tauceti/algebra/lie/part-02","status":"fully_proved","summary":"Over an algebraically closed field, a morphism of a finite-dimensional irreducible `L`-module to itself is multiplication by a scalar: an eigenvalue exists, and the corresponding eigenspace is a nonzero Lie submodule, hence everything.","title":"Schur's lemma for Lie modules"},{"id":"tauceti/algebra/lie/part-02/the-structure-theorem-for-reductive-lie-algebras","kind":"node","lean":"TauCeti.hasCentralRadical_iff_isCompl_center_derivedSeries","parent":1269,"result":true,"route":"../roadmap/tauceti/algebra/lie/part-02/index.html#the-structure-theorem-for-reductive-lie-algebras","scope":"tauceti/algebra/lie/part-02","status":"fully_proved","summary":"Over a field of characteristic zero, a finite-dimensional Lie algebra is reductive (its solvable radical is its centre) if and only if it is the direct sum of its centre and its derived ideal, with the derived ideal semisimple.","title":"The structure theorem for reductive Lie algebras"},{"id":"tauceti/algebra/lie/part-02/the-weyl-character-formula","kind":"node","lean":"TauCeti.formalCharacter_mul_weylDenominator_eq_weylNumerator","parent":1269,"result":true,"route":"../roadmap/tauceti/algebra/lie/part-02/index.html#the-weyl-character-formula","scope":"tauceti/algebra/lie/part-02","status":"fully_proved","summary":"For a finite-dimensional module `M` generated by a highest weight vector of weight `lam`, the formal character of `M` times the Weyl denominator is the Weyl numerator of `lam`: `ch M \u00b7 \u220f_{\u03b1>0} (1 - e^{-\u03b1}) = \u2211_{w \u2208 W} sgn(w) e^{w(lam + \u03c1) - \u03c1}`.","title":"The Weyl character formula"},{"id":"tauceti/algebra/lie/part-02/the-weyl-dimension-formula-tauceti-finrank-eq-pr","kind":"node","lean":"TauCeti.finrank_eq_prod_coweightPairing_div, TauCeti.finrank_mul_prod_coweightPairing_weylVector_eq","parent":1269,"result":true,"route":"../roadmap/tauceti/algebra/lie/part-02/index.html#the-weyl-dimension-formula-tauceti-finrank-eq-pr","scope":"tauceti/algebra/lie/part-02","status":"fully_proved","summary":"For a finite-dimensional module `M` generated by a highest weight vector of weight `lam`,","title":"The Weyl dimension formula (`TauCeti.finrank_eq_prod_coweightPairing_div`)"},{"id":"tauceti/algebra/lie/part-02/weyls-complete-reducibility-theorem","kind":"node","lean":"TauCeti.exists_isCompl_of_isKilling","parent":1269,"result":true,"route":"../roadmap/tauceti/algebra/lie/part-02/index.html#weyls-complete-reducibility-theorem","scope":"tauceti/algebra/lie/part-02","status":"fully_proved","summary":"Over a field of characteristic zero, every Lie submodule of a finite-dimensional module over a Lie algebra with nondegenerate Killing form has a complement, so the module is a direct sum of irreducibles.","title":"Weyl's complete reducibility theorem"},{"id":"tauceti/algebra/lie/part-02/weyls-theorem-as-a-decomposition","kind":"node","lean":"TauCeti.exists_isInternal_isIrreducible_of_isKilling, LieSubmodule.exists_isCompl_of_le_ker","parent":1269,"result":true,"route":"../roadmap/tauceti/algebra/lie/part-02/index.html#weyls-theorem-as-a-decomposition","scope":"tauceti/algebra/lie/part-02","status":"fully_proved","summary":"A finite-dimensional module over a Lie algebra with nondegenerate Killing form, over a field of characteristic zero, is the internal direct sum of a finite family of irreducible Lie submodules.","title":"Weyl's theorem as a decomposition"},{"data_url":"scopes/tauceti/algebra/module.json","id":"tauceti/algebra/module","kind":"scope","parent":1096,"result":false,"route":"../roadmap/tauceti/algebra/module/index.html","scope":"tauceti/algebra/module","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Algebra.Module` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Module"},{"id":"tauceti/algebra/module/the-free-part-of-the-structure-theorem","kind":"node","lean":"TauCeti.InternalGrading.exists_linearIndependent_isCompl_torsion","parent":1276,"result":true,"route":"../roadmap/tauceti/algebra/module/index.html#the-free-part-of-the-structure-theorem","scope":"tauceti/algebra/module","status":"fully_proved","summary":"In a finitely generated graded `k[X]`-module over a field on which `X` lowers degree by `d \u2260 0`, the torsion submodule has a complement spanned by a `k[X]`-linearly independent family of homogeneous elements: `M` is its torsion submodule plus a graded free module, one `X`-tower for each element of the family.","title":"The free part of the structure theorem"},{"id":"tauceti/algebra/module/the-ikeda-nakayama-annihilator-criterion-for-a-l","kind":"node","lean":"TauCeti.exists_eq_sum_mul_of_forall_mul_eq_zero, TauCeti.exists_eq_sum_mul_of_forall_mul_eq_zero'","parent":1276,"result":true,"route":"../roadmap/tauceti/algebra/module/index.html#the-ikeda-nakayama-annihilator-criterion-for-a-l","scope":"tauceti/algebra/module","status":"fully_proved","summary":"if every `r` with `r * a i = 0` for all `i` also satisfies `r * x = 0`, then `x` lies in the right ideal generated by the `a i`.","title":"The Ikeda--Nakayama annihilator criterion, for a left self-injective ring"},{"id":"tauceti/algebra/module/nakayamas-lemma-for-a-nilpotent-ideal","kind":"node","lean":"TauCeti.isSuperfluous_smul_top_of_isNilpotent","parent":1276,"result":true,"route":"../roadmap/tauceti/algebra/module/index.html#nakayamas-lemma-for-a-nilpotent-ideal","scope":"tauceti/algebra/module","status":"fully_proved","summary":"If `I` is a nilpotent ideal then `I \u2022 M` is superfluous in `M`.","title":"Nakayama's lemma for a nilpotent ideal"},{"id":"tauceti/algebra/module/schanuels-lemma-in-automorphism-form","kind":"node","lean":"TauCeti.exists_linearEquiv_comp_fst_eq_comp_snd_comp","parent":1276,"result":true,"route":"../roadmap/tauceti/algebra/module/index.html#schanuels-lemma-in-automorphism-form","scope":"tauceti/algebra/module","status":"fully_proved","summary":"If `a : A \u2192 E` and `b : B \u2192 E` are maps from projective modules with the same range, then `a \u2218 fst` and `b \u2218 snd`, as maps `A \u00d7 B \u2192 E`, differ by an automorphism of `A \u00d7 B`.","title":"Schanuel's lemma in automorphism form"},{"id":"tauceti/algebra/module/serres-lemma-on-the-tate-module","kind":"node","lean":"TauCeti.TateModule.eq_one_of_isOfFinOrder_of_forall_proj_eq","parent":1276,"result":true,"route":"../roadmap/tauceti/algebra/module/index.html#serres-lemma-on-the-tate-module","scope":"tauceti/algebra/module","status":"fully_proved","summary":"An endomorphism of finite order of `T_p A` that induces the identity on `A[p ^ k]`, with `p ^ k \u2265 3`, is the identity.","title":"Serre's lemma on the Tate module"},{"id":"tauceti/algebra/module/structure-theorem-for-finitely-generated-modules","kind":"node","lean":"TauCeti.Module.equiv_pi_prod_pi_quotient_span_pow","parent":1276,"result":true,"route":"../roadmap/tauceti/algebra/module/index.html#structure-theorem-for-finitely-generated-modules","scope":"tauceti/algebra/module","status":"fully_proved","summary":"A finitely generated module over a discrete valuation ring `R` with uniformizer `\u03d6` is isomorphic to `R ^ n \u00d7 \u220f i : Fin m, R \u29f8 (\u03d6 ^ e i)` with every exponent `e i` positive.","title":"Structure theorem for finitely generated modules over a discrete valuation ring"},{"data_url":"scopes/tauceti/algebra/octonion.json","id":"tauceti/algebra/octonion","kind":"scope","parent":1096,"result":false,"route":"../roadmap/tauceti/algebra/octonion/index.html","scope":"tauceti/algebra/octonion","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Algebra.Octonion` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Octonion"},{"id":"tauceti/algebra/octonion/the-left-moufang-identity-tauceti-octonion-moufa","kind":"node","lean":"TauCeti.Octonion.moufang_left","parent":1283,"result":true,"route":"../roadmap/tauceti/algebra/octonion/index.html#the-left-moufang-identity-tauceti-octonion-moufa","scope":"tauceti/algebra/octonion","status":"fully_proved","summary":"`(z * x * z) * y = z * (x * (z * y))`.","title":"The left Moufang identity (`TauCeti.Octonion.moufang_left`)"},{"id":"tauceti/algebra/octonion/the-middle-moufang-identity-tauceti-octonion-mou","kind":"node","lean":"TauCeti.Octonion.moufang_middle","parent":1283,"result":true,"route":"../roadmap/tauceti/algebra/octonion/index.html#the-middle-moufang-identity-tauceti-octonion-mou","scope":"tauceti/algebra/octonion","status":"fully_proved","summary":"`(z * x) * (y * z) = (z * (x * y)) * z`.","title":"The middle Moufang identity (`TauCeti.Octonion.moufang_middle`)"},{"id":"tauceti/algebra/octonion/the-right-moufang-identity-tauceti-octonion-mouf","kind":"node","lean":"TauCeti.Octonion.moufang_right","parent":1283,"result":true,"route":"../roadmap/tauceti/algebra/octonion/index.html#the-right-moufang-identity-tauceti-octonion-mouf","scope":"tauceti/algebra/octonion","status":"fully_proved","summary":"`x * (z * y * z) = ((x * z) * y) * z`.","title":"The right Moufang identity (`TauCeti.Octonion.moufang_right`)"},{"data_url":"scopes/tauceti/algebra/polynomial.json","id":"tauceti/algebra/polynomial","kind":"scope","parent":1096,"result":false,"route":"../roadmap/tauceti/algebra/polynomial/index.html","scope":"tauceti/algebra/polynomial","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Algebra.Polynomial` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Polynomial"},{"id":"tauceti/algebra/polynomial/sturm-tarski-on-a-b","kind":"node","lean":"TauCeti.Sturm.sum_sign_Icc, TauCeti.Sturm.sum_sign, TauCeti.Sturm.sum_sign_Ioc, TauCeti.Sturm.sum_sign_Ico, Polynomial.tarskiQuery_eq_signVariationsAtBot_sub_signVariationsAtTop, TauCeti.Sturm.sum_sign_Ioo","parent":1287,"result":true,"route":"../roadmap/tauceti/algebra/polynomial/index.html#sturm-tarski-on-a-b","scope":"tauceti/algebra/polynomial","status":"fully_proved","summary":"The contributions of both closed endpoints are added explicitly to the open-interval formula. For `a = b` this is the contribution of a point.","title":"Sturm\u2013Tarski on `[a, b]`"},{"id":"tauceti/algebra/polynomial/taylors-estimate","kind":"node","lean":"Polynomial.mul_dvd_eval_sub_eval_sub_mul","parent":1287,"result":true,"route":"../roadmap/tauceti/algebra/polynomial/index.html#taylors-estimate","scope":"tauceti/algebra/polynomial","status":"fully_proved","summary":"If `d \u2223 x - t`, `d \u2223 e` and `e \u2223 x' - x`, then the derivative at `t` linearises the polynomial `f` from `x` to `x'` up to an error divisible by `d * e`.","title":"Taylor's estimate"},{"id":"tauceti/algebra/polynomial/thoms-lemma","kind":"node","lean":"Polynomial.preimage_derivativeSign_subsingleton_or_eq_Ioo, Polynomial.preimage_thomEncoding_subsingleton_or_eq_Ioo","parent":1287,"result":true,"route":"../roadmap/tauceti/algebra/polynomial/index.html#thoms-lemma","scope":"tauceti/algebra/polynomial","status":"fully_proved","summary":"For any signs `\u03c3 k` prescribed for the derivatives of every order of `p`, including order zero, the set of points where `p` realizes them is empty, a single point, or an open interval with endpoints `a \u2208 WithBot R` and `b \u2208 WithTop R`.","title":"Thom's lemma"},{"data_url":"scopes/tauceti/algebra/other-results.json","id":"tauceti/algebra/other-results","kind":"scope","parent":1096,"result":false,"route":"../roadmap/tauceti/algebra/other-results/index.html","scope":"tauceti/algebra/other-results","status":"fully_proved","summary":"Results named in the rest of Tau Ceti's `TauCeti.Algebra` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Other results"},{"id":"tauceti/algebra/other-results/abels-inequality-comparison-form","kind":"node","lean":"TauCeti.sum_range_mul_le_sum_range_mul","parent":1291,"result":true,"route":"../roadmap/tauceti/algebra/other-results/index.html#abels-inequality-comparison-form","scope":"tauceti/algebra/other-results","status":"fully_proved","summary":"If the partial sums of `f` are dominated by those of `g` up to `N`, and `w` is nonnegative and antitone on the first `N` indices, then `\u2211_{i < N} w i * f i \u2264 \u2211_{i < N} w i * g i`. Only the last weight is required to be nonnegative explicitly; the other signs follow from the successive comparisons when `N > 0`.","title":"Abel's inequality, comparison form"},{"id":"tauceti/algebra/other-results/the-basis-theorem","kind":"node","lean":"TauCeti.TypeAHeckeAlgebra.basisTw","parent":1291,"result":true,"route":"../roadmap/tauceti/algebra/other-results/index.html#the-basis-theorem","scope":"tauceti/algebra/other-results","status":"fully_proved","summary":"the elements `T_w`, `w : W`, form a basis of the type-A Hecke algebra over every commutative ring and for every unit parameter.","title":"The basis theorem"},{"id":"tauceti/algebra/other-results/the-conway-sloane-exponent-identity","kind":"node","lean":"Finset.two_mul_sum_range_choose_add_sum_sum_tsub","parent":1291,"result":true,"route":"../roadmap/tauceti/algebra/other-results/index.html#the-conway-sloane-exponent-identity","scope":"tauceti/algebra/other-results","status":"fully_proved","summary":"Twice `\u2211_{j < max k} binom(N\u2c7c + 1, 2)`, where `N\u2c7c = #{i \u2208 s | j < k\u1d62}`, plus `\u2211_{i, i' \u2208 s} (k_{i'} - k\u1d62)` (truncated subtraction) is `(#s + 1) \u2211_{i \u2208 s} k\u1d62`. The first sum is the exponent of `p` in the local density of the diagonal Jordan form `diag(u\u1d62 p\u1d4f\u2071)` at an odd prime, the second is the exponent of `\u221ap` in the cross terms of its Conway\u2013Sloane `p`-mass.","title":"The Conway\u2013Sloane exponent identity"},{"id":"tauceti/algebra/other-results/dedekinds-modular-law","kind":"node","lean":"AddSubgroup.inf_sup_assoc_of_le, Subgroup.inf_sup_assoc_of_le","parent":1291,"result":true,"route":"../roadmap/tauceti/algebra/other-results/index.html#dedekinds-modular-law","scope":"tauceti/algebra/other-results","status":"fully_proved","summary":"if the join `H \u2294 N` is the sum set `H + N` (for example when `H` normalizes `N`) and `N \u2264 K`, then `(K \u2293 H) \u2294 N = K \u2293 (H \u2294 N)`.","title":"Dedekind's modular law"},{"id":"tauceti/algebra/other-results/hermites-identity-in-truncated-ceiling-division","kind":"node","lean":"Nat.sum_range_sub_ceilDiv","parent":1291,"result":true,"route":"../roadmap/tauceti/algebra/other-results/index.html#hermites-identity-in-truncated-ceiling-division","scope":"tauceti/algebra/other-results","status":"fully_proved","summary":"for `0 < n`, the ceilings of `(r - i) / n` over a complete block `0 \u2264 i < n` of shifts add up to `r`. The subtraction is truncated, so the terms with `r \u2264 i` vanish. Compare the floor form `Int.sum_range_add_ediv`.","title":"Hermite's identity in truncated ceiling-division form"},{"id":"tauceti/algebra/other-results/hilberts-finiteness-theorem","kind":"node","lean":"TauCeti.ComoduleAlgebra.fg_coinvariants_symmetricAlgebra","parent":1291,"result":true,"route":"../roadmap/tauceti/algebra/other-results/index.html#hilberts-finiteness-theorem","scope":"tauceti/algebra/other-results","status":"fully_proved","summary":"For a finite-dimensional comodule `V` over a linearly reductive commutative bialgebra, the invariants of the symmetric algebra `Sym V` form a finitely generated algebra.","title":"Hilbert's finiteness theorem"},{"id":"tauceti/algebra/other-results/multinomial-theorem-for-square-zero-elements","kind":"node","lean":"Finset.sum_pow_of_commute_of_mul_self_eq_zero","parent":1291,"result":true,"route":"../roadmap/tauceti/algebra/other-results/index.html#multinomial-theorem-for-square-zero-elements","scope":"tauceti/algebra/other-results","status":"fully_proved","summary":"If the `x i` commute pairwise and square to zero on `s`, then `(\u2211 i \u2208 s, x i) ^ n` is `n !` times the sum, over the `n`-element subsets `t` of `s`, of the products of the `x i` with `i \u2208 t`.","title":"Multinomial theorem for square-zero elements"},{"id":"tauceti/algebra/other-results/nakayamas-lemma-for-r-g","kind":"node","lean":"TauCeti.MonoidAlgebra.bijective_of_forall_exists_mapRingHom_residue_eq","parent":1291,"result":true,"route":"../roadmap/tauceti/algebra/other-results/index.html#nakayamas-lemma-for-r-g","scope":"tauceti/algebra/other-results","status":"fully_proved","summary":"An `R[G]`-linear endomorphism of `R[G]^\u03b9` that is onto modulo the maximal ideal is bijective.","title":"Nakayama's lemma for `R[G]^\u03b9`"},{"id":"tauceti/algebra/other-results/newtons-formula-for-the-determinant","kind":"node","lean":"TauCeti.AlbertAlgebra.det_eq_invOf_two_mul_invOf_three_mul, TauCeti.AlbertAlgebra.six_mul_det","parent":1291,"result":true,"route":"../roadmap/tauceti/algebra/other-results/index.html#newtons-formula-for-the-determinant","scope":"tauceti/algebra/other-results","status":"fully_proved","summary":"when `2` and `3` are invertible, `det A = (T(A)\u00b3 - 3 T(A) T(A \u2218 A) + 2 T(A \u2218 (A \u2218 A))) / 6`.","title":"Newton's formula for the determinant"},{"id":"tauceti/algebra/other-results/ocneanus-theorem","kind":"node","lean":"TauCeti.InfiniteTypeAHeckeAlgebra.IsMarkovTrace.eq_ocneanuTrace, TauCeti.InfiniteTypeAHeckeAlgebra.existsUnique_isMarkovTrace","parent":1291,"result":true,"route":"../roadmap/tauceti/algebra/other-results/index.html#ocneanus-theorem","scope":"tauceti/algebra/other-results","status":"fully_proved","summary":"Every Markov trace with parameter `z` is the Ocneanu trace.","title":"Ocneanu's theorem"},{"id":"tauceti/algebra/other-results/siegels-identity","kind":"node","lean":"TauCeti.siegel_identity","parent":1291,"result":true,"route":"../roadmap/tauceti/algebra/other-results/index.html#siegels-identity","scope":"tauceti/algebra/other-results","status":"fully_proved","summary":"For `\u03b8\u2081 \u2260 \u03b8\u2080` and `x - \u03b8\u2082 y \u2260 0` in a field, `(\u03b8\u2081 - \u03b8\u2082) / (\u03b8\u2081 - \u03b8\u2080) \u00b7 (x - \u03b8\u2080 y) / (x - \u03b8\u2082 y) + (\u03b8\u2082 - \u03b8\u2080) / (\u03b8\u2081 - \u03b8\u2080) \u00b7 (x - \u03b8\u2081 y) / (x - \u03b8\u2082 y)` equals `1`. Applied to the linear factors of a binary form, it turns a Thue equation into unit equations.","title":"Siegel's identity"},{"data_url":"scopes/tauceti/algebraic-geometry.json","id":"tauceti/algebraic-geometry","kind":"scope","parent":1095,"result":false,"route":"../roadmap/tauceti/algebraic-geometry/index.html","scope":"tauceti/algebraic-geometry","status":"fully_proved","summary":"Results in Tau Ceti's `TauCeti.AlgebraicGeometry` modules, and the roadmaps whose articles are mostly there: by roadmap, then by area.","title":"Algebraic Geometry"},{"data_url":"scopes/tauceti/algebraic-geometry/algebraic-vector-bundles.json","id":"tauceti/algebraic-geometry/algebraic-vector-bundles","kind":"scope","parent":1303,"result":false,"route":"../roadmap/tauceti/algebraic-geometry/algebraic-vector-bundles/index.html","scope":"tauceti/algebraic-geometry/algebraic-vector-bundles","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Algebraic vector bundles"},{"id":"tauceti/algebraic-geometry/algebraic-vector-bundles/tauceti-algebraicgeometry-relativespecequiv","kind":"node","lean":"TauCeti.AlgebraicGeometry.relativeSpecEquiv","parent":1304,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/algebraic-vector-bundles/index.html#tauceti-algebraicgeometry-relativespecequiv","scope":"tauceti/algebraic-geometry/algebraic-vector-bundles","status":"fully_proved","summary":"Relative Spec and the algebra of regular functions give an anti-equivalence between quasi-coherent commutative algebras and affine schemes over an arbitrary base scheme. Its unit and counit are the canonical section and affine-chart comparisons.","title":"TauCeti.AlgebraicGeometry.relativeSpecEquiv"},{"id":"tauceti/algebraic-geometry/algebraic-vector-bundles/relative-spec-commutes-with-base-change-naturall","kind":"node","lean":"TauCeti.AlgebraicGeometry.relativeSpecBaseChangeIso","parent":1304,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/algebraic-vector-bundles/index.html#relative-spec-commutes-with-base-change-naturall","scope":"tauceti/algebraic-geometry/algebraic-vector-bundles","status":"fully_proved","summary":"For `f : Y \u27f6 X`, taking the relative spectrum of the pullback of a quasicoherent algebra is naturally isomorphic to base changing its relative spectrum along `f`.","title":"Relative Spec commutes with base change, naturally in the algebra"},{"id":"tauceti/algebraic-geometry/algebraic-vector-bundles/relative-spec-commutes-with-base-change-compatib","kind":"node","lean":"TauCeti.AlgebraicGeometry.relativeSpecPullbackIso_comp","parent":1304,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/algebraic-vector-bundles/index.html#relative-spec-commutes-with-base-change-compatib","scope":"tauceti/algebraic-geometry/algebraic-vector-bundles","status":"fully_proved","summary":"For `g : Z \u27f6 Y` and `f : Y \u27f6 X`, the isomorphism `Spec_Z ((g \u226b f)^* A) \u2245 Z \u00d7_X Spec_X A` is the composite of the identification `Spec_Z ((g \u226b f)^* A) \u2245 Spec_Z (g^* f^* A)`, the isomorphisms for `g` and for `f`, and the canonical isomorphism `Z \u00d7_Y (Y \u00d7_X Spec_X A) \u2245 Z \u00d7_X Spec_X A`.","title":"Relative Spec commutes with base change, compatibly with composition"},{"id":"tauceti/algebraic-geometry/algebraic-vector-bundles/spec-x-sym-f-represents-linear-functionals","kind":"node","lean":"TauCeti.AlgebraicGeometry.relativeSpecSymmetricAlgebraPullbackHomEquiv","parent":1304,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/algebraic-vector-bundles/index.html#spec-x-sym-f-represents-linear-functionals","scope":"tauceti/algebraic-geometry/algebraic-vector-bundles","status":"fully_proved","summary":"For a quasicoherent sheaf `F` on `X` and a scheme `q : T \u27f6 X` over `X`, morphisms `T \u27f6 Spec_X (Sym F)` over `X` correspond to morphisms of `\ud835\udcaa_T`-modules `q^* F \u27f6 \ud835\udcaa_T`. It is `relativeSpecSymmetricAlgebraHomEquiv` followed by the adjunction between pullback and pushforward of modules.","title":"`Spec_X (Sym F)` represents linear functionals"},{"id":"tauceti/algebraic-geometry/algebraic-vector-bundles/tauceti-algebraicgeometry-quasicoherentsheaf-non","kind":"node","lean":"TauCeti.AlgebraicGeometry.QuasicoherentSheaf.nonempty_hasLeftDual_iff_isFiniteLocallyFree","parent":1304,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/algebraic-vector-bundles/index.html#tauceti-algebraicgeometry-quasicoherentsheaf-non","scope":"tauceti/algebraic-geometry/algebraic-vector-bundles","status":"fully_proved","summary":"A quasicoherent sheaf on any scheme has a left dual in `QuasicoherentSheaf X` if and only if it is finite locally free.","title":"TauCeti.AlgebraicGeometry.QuasicoherentSheaf.nonempty_hasLeftDual_iff_isFiniteLocallyFree"},{"id":"tauceti/algebraic-geometry/algebraic-vector-bundles/tauceti-algebraicgeometry-finitelocallyfreesheaf","kind":"node","lean":"TauCeti.AlgebraicGeometry.FiniteLocallyFreeSheaf.evalNatIso","parent":1304,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/algebraic-vector-bundles/index.html#tauceti-algebraicgeometry-finitelocallyfreesheaf","scope":"tauceti/algebraic-geometry/algebraic-vector-bundles","status":"fully_proved","summary":"The identity functor on finite locally free sheaves is naturally isomorphic to internal-Hom dualization applied twice.","title":"TauCeti.AlgebraicGeometry.FiniteLocallyFreeSheaf.evalNatIso"},{"id":"tauceti/algebraic-geometry/algebraic-vector-bundles/symmetric-powers-commute-with-pullback-of-quasic","kind":"node","lean":"TauCeti.AlgebraicGeometry.isIso_pullbackSymmetricPowerComparison_app","parent":1304,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/algebraic-vector-bundles/index.html#symmetric-powers-commute-with-pullback-of-quasic","scope":"tauceti/algebraic-geometry/algebraic-vector-bundles","status":"fully_proved","summary":"for every morphism of schemes `f : X \u27f6 Y` and every quasicoherent `\ud835\udcaa_Y`-module `M`, the canonical comparison `f^* (Sym\u207f M) \u27f6 Sym\u207f (f^* M)` is an isomorphism.","title":"Symmetric powers commute with pullback of quasicoherent sheaves"},{"id":"tauceti/algebraic-geometry/algebraic-vector-bundles/tauceti-algebraicgeometry-finitelocallyfreesheaf-2","kind":"node","lean":"TauCeti.AlgebraicGeometry.FiniteLocallyFreeSheaf.pullbackDeterminantIso","parent":1304,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/algebraic-vector-bundles/index.html#tauceti-algebraicgeometry-finitelocallyfreesheaf-2","scope":"tauceti/algebraic-geometry/algebraic-vector-bundles","status":"fully_proved","summary":"Pulling back the determinant of a finite locally free sheaf of constant rank `r` is naturally isomorphic to taking the determinant of its pullback: `f^* (det E) \u2245 det (f^* E)`. The forward map is the canonical exterior-power comparison `f^* (\u22c0\u02b3 E) \u27f6 \u22c0\u02b3 (f^* E)`, which is invertible because `E` is quasicoherent.","title":"TauCeti.AlgebraicGeometry.FiniteLocallyFreeSheaf.pullbackDeterminantIso"},{"id":"tauceti/algebraic-geometry/algebraic-vector-bundles/tauceti-algebraicgeometry-gradedsymmetricalgebra","kind":"node","lean":"TauCeti.AlgebraicGeometry.gradedSymmetricAlgebraAdjunction","parent":1304,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/algebraic-vector-bundles/index.html#tauceti-algebraicgeometry-gradedsymmetricalgebra","scope":"tauceti/algebraic-geometry/algebraic-vector-bundles","status":"fully_proved","summary":"The graded symmetric algebra functor is left adjoint to the degree-one functor: morphisms of graded quasicoherent algebras `Sym F \u27f6 A` correspond to morphisms of quasicoherent sheaves `F \u27f6 A\u2081`, by restriction to degree one along `F \u2245 Sym\u00b9 F` and by `gradedSymmetricAlgebraLift`.","title":"TauCeti.AlgebraicGeometry.gradedSymmetricAlgebraAdjunction"},{"id":"tauceti/algebraic-geometry/algebraic-vector-bundles/tauceti-algebraicgeometry-gradedsymmetricalgebra-2","kind":"node","lean":"TauCeti.AlgebraicGeometry.gradedSymmetricAlgebraEquiv","parent":1304,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/algebraic-vector-bundles/index.html#tauceti-algebraicgeometry-gradedsymmetricalgebra-2","scope":"tauceti/algebraic-geometry/algebraic-vector-bundles","status":"fully_proved","summary":"The graded symmetric algebra functor is an equivalence between quasicoherent sheaves and graded quasicoherent algebras freely generated in degree one. Its inverse takes the degree-one component, and its unit is the identification `F \u2245 Sym\u00b9 F`.","title":"TauCeti.AlgebraicGeometry.gradedSymmetricAlgebraEquiv"},{"id":"tauceti/algebraic-geometry/algebraic-vector-bundles/tauceti-algebraicgeometry-gradedrelativespecequi","kind":"node","lean":"TauCeti.AlgebraicGeometry.gradedRelativeSpecEquiv","parent":1304,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/algebraic-vector-bundles/index.html#tauceti-algebraicgeometry-gradedrelativespecequi","scope":"tauceti/algebraic-geometry/algebraic-vector-bundles","status":"fully_proved","summary":"Graded relative Spec is an anti-equivalence between graded quasicoherent algebras and graded affine schemes over `X`. The functor sends `A` to the relative spectrum of `\u2a01\u2099 A\u2099`, graded through the relative-Spec unit `p_* \ud835\udcaa \u2245 \u2a01\u2099 A\u2099`, and the inverse reads off the grading. The unit is the identity, and the counit is, on underlying schemes, relative Spec of the grading isomorphism followed by the relative-Spec counit.","title":"TauCeti.AlgebraicGeometry.gradedRelativeSpecEquiv"},{"id":"tauceti/algebraic-geometry/algebraic-vector-bundles/tauceti-algebraicgeometry-finitelocallyfreesheaf-3","kind":"node","lean":"TauCeti.AlgebraicGeometry.FiniteLocallyFreeSheaf","parent":1304,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/algebraic-vector-bundles/index.html#tauceti-algebraicgeometry-finitelocallyfreesheaf-3","scope":"tauceti/algebraic-geometry/algebraic-vector-bundles","status":"fully_proved","summary":"The full category of finite locally free sheaves on a scheme, that is, of locally free and finitely presented `\ud835\udcaa_X`-modules. Its morphisms are morphisms of `\ud835\udcaa_X`-modules.","title":"TauCeti.AlgebraicGeometry.FiniteLocallyFreeSheaf"},{"data_url":"scopes/tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves.json","id":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves","kind":"scope","parent":1303,"result":false,"route":"../roadmap/tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/index.html","scope":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Analytic families and moduli of complex curves"},{"id":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/wieners-lemma-for-h-s-a-s-1-2","kind":"node","lean":"TauCeti.CircleSobolev.isUnit_iff_forall_isUnit, TauCeti.CircleSobolev.isUnit_iff_forall_ne_zero","parent":1317,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/index.html#wieners-lemma-for-h-s-a-s-1-2","scope":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves","status":"fully_proved","summary":"An element of `H^s(A)` is a unit if and only if its continuous representative takes invertible values at every point of the circle.","title":"Wiener's lemma for `H^s(A)`, `s > 1/2`"},{"id":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/the-product-estimate-in-h-s-a-s-1-2","kind":"node","lean":"TauCeti.CircleSobolev.norm_mul_le","parent":1317,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/index.html#the-product-estimate-in-h-s-a-s-1-2","scope":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.CircleSobolev.norm_mul_le`.","title":"The product estimate in `H^s(A)`, `s > 1/2`"},{"id":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/the-holomorphic-nemytskii-map-is-holomorphic","kind":"node","lean":"TauCeti.CircleSobolev.analyticOnNhd_nemytskii","parent":1317,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/index.html#the-holomorphic-nemytskii-map-is-holomorphic","scope":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves","status":"fully_proved","summary":"If `F` is holomorphic on an open set `U`, then `u \u21a6 nemytskii F u` is analytic on the open subset of `H^s`, `s > 1/2`, of elements with values in `U` (`isOpen_setOf_forall_toContinuousMap_mem`).","title":"The holomorphic Nemytskii map is holomorphic"},{"id":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/stalks-at-the-nodes-of-a-glued-nodal-curve","kind":"node","lean":"TauCeti.NodalGluing.quotientStalkToNode_bijective","parent":1317,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/index.html#stalks-at-the-nodes-of-a-glued-nodal-curve","scope":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves","status":"fully_proved","summary":"At the node of a flag pair, the restrictions to the two branches identify the stalk of the glued curve with the branch-pair node ring.","title":"Stalks at the nodes of a glued nodal curve"},{"id":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/stalks-at-smooth-points-of-a-glued-nodal-curve","kind":"node","lean":"TauCeti.NodalGluing.quotientStalkBranch_bijective","parent":1317,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/index.html#stalks-at-smooth-points-of-a-glued-nodal-curve","scope":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves","status":"fully_proved","summary":"Away from the flag points, restriction to the normalization identifies the stalks of the glued curve with the holomorphic stalks of the normalization.","title":"Stalks at smooth points of a glued nodal curve"},{"id":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/smoothing-preserves-the-arithmetic-genus","kind":"node","lean":"TauCeti.DualGraph.arithmeticGenus_smoothing","parent":1317,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/index.html#smoothing-preserves-the-arithmetic-genus","scope":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves","status":"fully_proved","summary":"The first Betti number lost by smoothing the nodes in `S` reappears in the genera of the smoothed components.","title":"Smoothing preserves the arithmetic genus"},{"id":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/genus-of-a-partial-normalization","kind":"node","lean":"TauCeti.DualGraph.sum_arithmeticGenus_cutComponent_add_card","parent":1317,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/index.html#genus-of-a-partial-normalization","scope":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves","status":"fully_proved","summary":"The sum of the arithmetic genera of its connected pieces, plus the number of cut nodes plus one, equals the original genus plus the number of pieces. This includes separating, nonseparating and simultaneous cuts, without subtraction.","title":"Genus of a partial normalization"},{"id":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/the-glued-space","kind":"node","lean":"TauCeti.NodalGluing.GluedCurve","parent":1317,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/index.html#the-glued-space","scope":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves","status":"fully_proved","summary":"The quotient of the disjoint union of the normalization components by the generated relation: the underlying topological space of the nodal curve. The topology is the quotient topology, i.e. the finest topology making the quotient map continuous.","title":"The glued space"},{"id":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/tauceti-holomorphicstructuresheaf-quotientsheaf","kind":"node","lean":"TauCeti.holomorphicStructureSheaf.quotientSheaf","parent":1317,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/index.html#tauceti-holomorphicstructuresheaf-quotientsheaf","scope":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves","status":"fully_proved","summary":"The sheaf of holomorphic functions on the normalization which have matching values on each fibre of the quotient map. For paired node flags this is the normalization construction of the structure sheaf of a nodal curve.","title":"TauCeti.holomorphicStructureSheaf.quotientSheaf"},{"id":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/an-isomorphism-of-nodal-curves-presented-by-flag","kind":"node","lean":"TauCeti.NodalGluing.Iso","parent":1317,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/index.html#an-isomorphism-of-nodal-curves-presented-by-flag","scope":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves","status":"fully_proved","summary":"A `C^n` diffeomorphism of the disjoint unions of the normalization components which carries the two points of a flag pair of `d` exactly to the two points of a flag pair of `d'`. Over `\u2102` with `n = \u221e` this is a biholomorphism of the normalizations matching the nodes.","title":"An isomorphism of nodal curves presented by flag-pair data"},{"id":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/the-automorphism-group-of-a-nodal-curve-presente","kind":"node","lean":"TauCeti.NodalGluing.Aut","parent":1317,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/index.html#the-automorphism-group-of-a-nodal-curve-presente","scope":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves","status":"fully_proved","summary":"its isomorphisms with itself, with composition as multiplication.","title":"The automorphism group of a nodal curve presented by flag-pair data"},{"id":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/the-groupoid-of-curves-with-unordered-auxiliary","kind":"node","lean":"TauCeti.NodalGluing.AuxiliaryMarkedCurve.unorderedAuxiliaryIsomorphismGroupoid","parent":1317,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/index.html#the-groupoid-of-curves-with-unordered-auxiliary","scope":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves","status":"fully_proved","summary":"Its arrows are the pairs `(\u03c0, f)` of `AuxiliaryIso`, composed by `(\u03c1, h) \u2218 (\u03c0, f) = (\u03c1 \u2218 \u03c0, h \u2218 f)`. Its objects are the same as those of the groupoid of isomorphisms preserving every label, of which it is the label-permutation extension, a finite extension when `R` is finite.","title":"The groupoid of curves with unordered auxiliary marks"},{"id":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/the-groupoid-of-curves-with-partitioned-auxiliar","kind":"node","lean":"TauCeti.NodalGluing.AuxiliaryMarkedCurve.partitionedAuxiliaryIsomorphismGroupoid","parent":1317,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/index.html#the-groupoid-of-curves-with-partitioned-auxiliar","scope":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves","status":"fully_proved","summary":"For a colouring `\u03ba : R \u2192 Cl` of the auxiliary labels, this is the subgroupoid of the groupoid of curves with unordered auxiliary marks consisting of the colour-preserving arrows `(\u03c0, f)`, those whose permutation `\u03c0` lies in `\u220f_c S(\u03ba \u207b\u00b9' {c})`. It is wide (`isWide_partitionedAuxiliaryIsomorphismGroupoid`): it has the same objects, but in general fewer arrows between them.","title":"The groupoid of curves with partitioned auxiliary marks"},{"id":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/tauceti-circlesobolev-mulfourierseriesequiv","kind":"node","lean":"TauCeti.CircleSobolev.mulFourierSeriesEquiv","parent":1317,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/index.html#tauceti-circlesobolev-mulfourierseriesequiv","scope":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves","status":"fully_proved","summary":"For a smooth function `f` on the circle that vanishes nowhere, multiplication by `f` as a continuous linear automorphism of `H^s`, for every real `s`; its inverse is multiplication by `f\u207b\u00b9`.","title":"TauCeti.CircleSobolev.mulFourierSeriesEquiv"},{"id":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/ehrlings-inequality-on-the-circle","kind":"node","lean":"TauCeti.CircleSobolev.exists_norm_inclusion_le","parent":1317,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/index.html#ehrlings-inequality-on-the-circle","scope":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves","status":"fully_proved","summary":"for `r \u2264 t < s`, the `H^t` norm is bounded by an arbitrarily small multiple of the `H^s` norm plus a multiple of the weaker `H^r` norm.","title":"Ehrling's inequality on the circle"},{"id":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/the-nonlinear-hardy-graph-theorem-at-a-plumbing","kind":"node","lean":"TauCeti.CircleSobolev.exists_analyticAt_hardyGraph","parent":1317,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/index.html#the-nonlinear-hardy-graph-theorem-at-a-plumbing","scope":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves","status":"fully_proved","summary":"For `s > 1/2` there are a neighbourhood `W` of `0` in `H^s \u00d7 H^s` and a map `\u03c8`, analytic at the origin with `\u03c8 0 = 0`, such that for all data `y = (a, p, q)` near `0`, with `p` and `q` of nonnegative frequencies, `\u03c8 y` is the unique `(F, G) \u2208 W` with","title":"The nonlinear Hardy graph theorem at a plumbing node"},{"id":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/peetres-inequality-tauceti-circlesobolev-weight","kind":"node","lean":"TauCeti.CircleSobolev.weight_add_le","parent":1317,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/index.html#peetres-inequality-tauceti-circlesobolev-weight","scope":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves","status":"fully_proved","summary":"`(1 + |m + n|) ^ s \u2264 (1 + |m|) ^ |s| (1 + |n|) ^ s` for every real `s`. Shifting frequencies by `m` therefore changes the weights of every order `s` by at most the factor `(1 + |m|) ^ |s|`.","title":"Peetre's inequality (`TauCeti.CircleSobolev.weight_add_le`)"},{"id":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/rellichs-theorem-on-the-circle","kind":"node","lean":"TauCeti.CircleSobolev.isCompactOperator_inclusion","parent":1317,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves/index.html#rellichs-theorem-on-the-circle","scope":"tauceti/algebraic-geometry/analytic-families-and-moduli-of-complex-curves","status":"fully_proved","summary":"the inclusion `H^s \u2192 H^t` is a compact operator for `t < s` when `E` is finite-dimensional.","title":"Rellich's theorem on the circle"},{"data_url":"scopes/tauceti/algebraic-geometry/elliptic-curves.json","id":"tauceti/algebraic-geometry/elliptic-curves","kind":"scope","parent":1303,"result":false,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curves/index.html","scope":"tauceti/algebraic-geometry/elliptic-curves","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Elliptic curves"},{"id":"tauceti/algebraic-geometry/elliptic-curves/the-hasse-bound-silverman-v-1-1","kind":"node","lean":"WeierstrassCurve.hasse_bound, WeierstrassCurve.frobeniusTrace_sq_le_four_mul_card","parent":1336,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curves/index.html#the-hasse-bound-silverman-v-1-1","scope":"tauceti/algebraic-geometry/elliptic-curves","status":"fully_proved","summary":"an elliptic curve `E` over a field with `q` elements satisfies `(#E(\ud835\udd3d_q) - (q + 1))\u00b2 \u2264 4q`, that is `|#E(\ud835\udd3d_q) - (q + 1)| \u2264 2\u221aq`, where the count includes the point at infinity. `frobeniusTrace_sq_le_four_mul_card` is the same bound for the trace `a_q`.","title":"The Hasse bound (Silverman V.1.1)"},{"id":"tauceti/algebraic-geometry/elliptic-curves/isogenous-curves-over-a-finite-field-have-the-sa","kind":"node","lean":"TauCeti.Isogeny.zetaFunction_eq","parent":1336,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curves/index.html#isogenous-curves-over-a-finite-field-have-the-sa","scope":"tauceti/algebraic-geometry/elliptic-curves","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.Isogeny.zetaFunction_eq`.","title":"Isogenous curves over a finite field have the same zeta function `Z(W/F, T) = exp (\u2211_{n \u2265 1} #W(\ud835\udd3d_{q\u207f}) T\u207f / n)`"},{"id":"tauceti/algebraic-geometry/elliptic-curves/the-mordell-weil-theorem","kind":"node","lean":"WeierstrassCurve.Affine.fg_point_of_numberField, WeierstrassCurve.Affine.fg_point, WeierstrassCurve.Affine.fg_point_of_variableChange, WeierstrassCurve.Affine.moduleFinite_point_of_numberField","parent":1336,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curves/index.html#the-mordell-weil-theorem","scope":"tauceti/algebraic-geometry/elliptic-curves","status":"fully_proved","summary":"the group `E(K)` of `K`-rational points of an elliptic curve `E` over a number field `K` is finitely generated.","title":"The Mordell\u2013Weil theorem"},{"id":"tauceti/algebraic-geometry/elliptic-curves/the-mordell-weil-group-of-y2-x3-x-1-over-q-is-z","kind":"node","lean":"TauCeti.WorkedExamples.nonempty_addEquiv_int_point_curveX3SubXAddOne","parent":1336,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curves/index.html#the-mordell-weil-group-of-y2-x3-x-1-over-q-is-z","scope":"tauceti/algebraic-geometry/elliptic-curves","status":"fully_proved","summary":"It is finitely generated, torsion-free and of rank one.","title":"The Mordell\u2013Weil group of `y\u00b2 = x\u00b3 - x + 1` over `\u211a` is `\u2124`"},{"id":"tauceti/algebraic-geometry/elliptic-curves/the-m-selmer-group-of-an-elliptic-curve-over-a-n","kind":"node","lean":"WeierstrassCurve.finite_selmerGroup","parent":1336,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curves/index.html#the-m-selmer-group-of-an-elliptic-curve-over-a-n","scope":"tauceti/algebraic-geometry/elliptic-curves","status":"fully_proved","summary":"The local conditions are the images of the Kummer maps over the completions at all finite places. The conclusion holds for every nonzero natural number `m`.","title":"The `m`-Selmer group of an elliptic curve over a number field is finite (Silverman X.4.2)"},{"id":"tauceti/algebraic-geometry/elliptic-curves/the-criterion-of-neron-ogg-and-shafarevich-for-p","kind":"node","lean":"WeierstrassCurve.hasPotentialGoodReduction_iff_finite_image_tateModuleGaloisRepresentation","parent":1336,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curves/index.html#the-criterion-of-neron-ogg-and-shafarevich-for-p","scope":"tauceti/algebraic-geometry/elliptic-curves","status":"fully_proved","summary":"Let `W` be an elliptic curve over a nonarchimedean local field `K`, and let `\u2113` be a prime different from the residue characteristic of `K`. Then `W` has potential good reduction exactly when the inertia subgroup of `Gal(K^{alg}/K)` acts on the `\u2113`-adic Tate module of `W` through a finite quotient.","title":"The criterion of N\u00e9ron, Ogg and Shafarevich for potential good reduction (Silverman AEC VII.7.3)"},{"id":"tauceti/algebraic-geometry/elliptic-curves/the-criterion-of-neron-ogg-and-shafarevich-with","kind":"node","lean":"WeierstrassCurve.HasPotentialGoodReduction.hasGoodReduction_iff_forall_mem_inertiaSubgroup","parent":1336,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curves/index.html#the-criterion-of-neron-ogg-and-shafarevich-with","scope":"tauceti/algebraic-geometry/elliptic-curves","status":"fully_proved","summary":"Let `W` be an elliptic curve over a nonarchimedean local field `K`, given by a minimal equation, with potential good reduction, and let `N \u2265 3` be an integer that is nonzero in the residue field of `K`. Then `W` has good reduction exactly when the inertia subgroup of `Gal(K^{alg}/K)` fixes every point of `E(K^{alg})[N]`.","title":"The criterion of N\u00e9ron, Ogg and Shafarevich with finite coefficients (Serre\u2013Tate)"},{"id":"tauceti/algebraic-geometry/elliptic-curves/pointed-twists-are-classified-by-nonabelian-galo","kind":"node","lean":"WeierstrassCurve.pointedTwistEquivNonabelianH1","parent":1336,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curves/index.html#pointed-twists-are-classified-by-nonabelian-galo","scope":"tauceti/algebraic-geometry/elliptic-curves","status":"fully_proved","summary":"For a Galois extension `L/K`, the pointed twists of `W` split by `L` are in bijection with `H\u00b9(Gal(L/K), Aut(W_L))`. The equivalence sends a trivialisation `C` to the class of the cocycle `\u03c3 \u21a6 C (\u03c3 C)\u207b\u00b9`.","title":"Pointed twists are classified by nonabelian Galois cohomology"},{"id":"tauceti/algebraic-geometry/elliptic-curves/the-descent-sequence-is-exact-at-h1-g-k-e1","kind":"node","lean":"TauCeti.Isogeny.exact_kummerClassMap_kerCoeffInclH1","parent":1336,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curves/index.html#the-descent-sequence-is-exact-at-h1-g-k-e1","scope":"tauceti/algebraic-geometry/elliptic-curves","status":"fully_proved","summary":"the kernel of `H\u00b9(G_K, E\u2081[\u03c6]) \u2192 H\u00b9(G_K, E\u2081)[\u03c6]` is the image of `E\u2082(K) \u29f8 \u03c6(E\u2081(K))` under the Kummer map. Together with `TauCeti.Isogeny.kummerClassMap_injective` and `TauCeti.Isogeny.kerCoeffInclH1_surjective` this is the short exact sequence `0 \u2192 E\u2082(K) \u29f8 \u03c6(E\u2081(K)) \u2192 H\u00b9(G_K, E\u2081[\u03c6]) \u2192 H\u00b9(G_K, E\u2081)[\u03c6] \u2192 0`.","title":"The `\u03c6`-descent sequence is exact at `H\u00b9(G_K, E\u2081[\u03c6])`"},{"id":"tauceti/algebraic-geometry/elliptic-curves/tates-local-index-is-the-point-group-index-silve","kind":"node","lean":"WeierstrassCurve.Affine.index_nonsingularReduction_eq_tateLocalIndex","parent":1336,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curves/index.html#tates-local-index-is-the-point-group-index-silve","scope":"tauceti/algebraic-geometry/elliptic-curves","status":"fully_proved","summary":"Suppose that the valuation ring of `v` is Henselian with finite residue field, for instance the ring of integers of a nonarchimedean local field, and that Tate's algorithm, run on the integral model of `W` with a generator `\u03d6` of the maximal ideal, stops with the symbol `s`. Then the local index the algorithm reports is `[W(F) : E\u2080(F)]`, the index of the subgroup of points with nonsingular reduction.","title":"Tate's local index is the point-group index (Silverman, *Advanced Topics*, IV.9, Table 4.1)"},{"id":"tauceti/algebraic-geometry/elliptic-curves/dirichlets-unit-theorem-weierstrasscurve-affine","kind":"node","lean":"WeierstrassCurve.Affine.fg_units_ringOfIntegersFactor","parent":1336,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curves/index.html#dirichlets-unit-theorem-weierstrasscurve-affine","scope":"tauceti/algebraic-geometry/elliptic-curves","status":"fully_proved","summary":"Dirichlet's unit theorem for the field factors of the \u00e9tale algebra of a curve over a number field: the ring of integers of each factor `F[X] \u29f8 (p)` has finitely generated unit group.","title":"Dirichlet's unit theorem (`WeierstrassCurve.Affine.fg_units_ringOfIntegersFactor`)"},{"id":"tauceti/algebraic-geometry/elliptic-curves/serres-lemma-for-the-l-adic-representation","kind":"node","lean":"WeierstrassCurve.tateModuleGaloisRepresentation_eq_one_of_isOfFinOrder","parent":1336,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curves/index.html#serres-lemma-for-the-l-adic-representation","scope":"tauceti/algebraic-geometry/elliptic-curves","status":"fully_proved","summary":"An automorphism `\u03c3` fixing every point of `E[\u2113 ^ k]`, with `\u2113 ^ k \u2265 3`, and acting on `T_\u2113 E` with finite order, acts trivially on `T_\u2113 E`.","title":"Serre's lemma for the `\u2113`-adic representation"},{"data_url":"scopes/tauceti/algebraic-geometry/foundations-of-adic-spaces.json","id":"tauceti/algebraic-geometry/foundations-of-adic-spaces","kind":"scope","parent":1303,"result":false,"route":"../roadmap/tauceti/algebraic-geometry/foundations-of-adic-spaces/index.html","scope":"tauceti/algebraic-geometry/foundations-of-adic-spaces","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Foundations of adic spaces"},{"id":"tauceti/algebraic-geometry/foundations-of-adic-spaces/the-interval-ring-b-i-is-strongly-noetherian","kind":"node","lean":"TauCeti.WittVector.isStronglyNoetherian_intervalRing","parent":1349,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/foundations-of-adic-spaces/index.html#the-interval-ring-b-i-is-strongly-noetherian","scope":"tauceti/algebraic-geometry/foundations-of-adic-spaces","status":"fully_proved","summary":"The interval ring `B^I` is strongly noetherian (Kedlaya, Theorem 4.10 and Remark 4.11), for all radii `\u03c1\u2081 \u2264 \u03c1\u2082`: each completed Huber algebra `B^I\u27e8X\u2081, \u2026, X\u2099\u27e9` is the Gauss-normed ring `B^I{X\u2081, \u2026, X\u2099}` of unit-radius restricted series, which is noetherian by `TauCeti.WittVector.isNoetherianRing_isRestricted_subring_intervalRing`.","title":"The interval ring `B^I` is strongly noetherian"},{"id":"tauceti/algebraic-geometry/foundations-of-adic-spaces/the-interval-ring-b-i-is-stably-sheafy-for-all-r","kind":"node","lean":"TauCeti.WittVector.isStablySheafyRing_intervalRing","parent":1349,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/foundations-of-adic-spaces/index.html#the-interval-ring-b-i-is-stably-sheafy-for-all-r","scope":"tauceti/algebraic-geometry/foundations-of-adic-spaces","status":"fully_proved","summary":"every complete Hausdorff Huber ring topologically of finite type over `B^I` is sheafy. The ring `B^I` is a complete Hausdorff Tate ring that is strongly noetherian by Kedlaya's Theorem 4.10 (`TauCeti.WittVector.isStronglyNoetherian_intervalRing`), so this is Wedhorn's Corollary 8.35 (`TauCeti.Huber.isStablySheafyRing_of_isStronglyNoetherian`). In particular `B^I` is sheafy\u2026","title":"The interval ring `B^I` is stably sheafy for all radii `0 < \u03c1\u2081 \u2264 \u03c1\u2082 < 1`"},{"id":"tauceti/algebraic-geometry/foundations-of-adic-spaces/o-y-is-a-sheaf","kind":"node","lean":"TauCeti.FarguesFontaine.isSheaf_spaYPresheaf","parent":1349,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/foundations-of-adic-spaces/index.html#o-y-is-a-sheaf","scope":"tauceti/algebraic-geometry/foundations-of-adic-spaces","status":"fully_proved","summary":"let `R` be the ring of integers of a valuation `v` on a field `K`, perfect of characteristic `p`, and `\u03d6 \u2208 R` with `0 < v(\u03d6) < 1`; for `R = \ud835\udcaa_F` this is the Fargues\u2013Fontaine space `\ud835\udcb4 \u2286 Spa(A_inf, A_inf)`. Then the structure presheaf `\ud835\udcaa_\ud835\udcb4` is a sheaf, so `\ud835\udcb4` and `\ud835\udcb3 = \ud835\udcb4 / \u03c6^\u2124` are adic spaces (`isAdic_spaYPreAdicSpace_iff`, `isAdic_spaXPreAdicSpace`). The window `W_n = {p ^ (-n) \u2264 \u03ba \u2264 p ^ n}` is `Spa(B^I, B^{I,+})`\u2026","title":"`\ud835\udcaa_\ud835\udcb4` is a sheaf"},{"id":"tauceti/algebraic-geometry/foundations-of-adic-spaces/the-image-in-x-of-a-wandering-window-is-an-open","kind":"node","lean":"TauCeti.FarguesFontaine.quotientMap_image_window_mem_affinoidOpens","parent":1349,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/foundations-of-adic-spaces/index.html#the-image-in-x-of-a-wandering-window-is-an-open","scope":"tauceti/algebraic-geometry/foundations-of-adic-spaces","status":"fully_proved","summary":"Let `O` be the ring of integers of `v`, perfect of characteristic `p`, with pseudouniformiser `\u03d6`, and suppose `\ud835\udcaa_\ud835\udcb4` is a sheaf. If `0 < q \u2264 q'` and the window `W = {v \u2208 \ud835\udcb4 : q \u2264 \u03ba(v) \u2264 q'}` meets none of its nontrivial integer Frobenius translates, then `\ud835\udcb3` restricted to `q(W)` is affinoid: it is isomorphic to `Spa(B^I, B^{I,+})` for the interval `I = [v(\u03d6) ^ (1 / q), v(\u03d6) ^ (1 / q')]`, through the window chart\u2026","title":"The image in `\ud835\udcb3` of a wandering window is an open affinoid subspace"},{"id":"tauceti/algebraic-geometry/foundations-of-adic-spaces/the-adic-fargues-fontaine-curves-x-y-z-for-two-p","kind":"node","lean":"TauCeti.FarguesFontaine.spaXAdicSpaceCongr","parent":1349,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/foundations-of-adic-spaces/index.html#the-adic-fargues-fontaine-curves-x-y-z-for-two-p","scope":"tauceti/algebraic-geometry/foundations-of-adic-spaces","status":"fully_proved","summary":"If `\u03d6` and `\u03d6'` define the same subset `\ud835\udcb4` of `Spa(\ud835\udd4e R, \ud835\udd4e R)`, as they do when they generate ideals with the same radical (`spaY_eq_of_radical_eq`), and `\ud835\udcaa_\ud835\udcb4` is a sheaf, then the isomorphism `spaXPreAdicSpaceCongr` of pre-adic spaces is an isomorphism in the full subcategory `(Adic)`. It is the isomorphism induced by `spaYAdicSpaceCongr` on the quotients by Frobenius\u2026","title":"The adic Fargues\u2013Fontaine curves `\ud835\udcb3 = \ud835\udcb4 / \u03c6^\u2124` for two pseudouniformisers are isomorphic"},{"id":"tauceti/algebraic-geometry/foundations-of-adic-spaces/every-strongly-noetherian-tate-ring-is-sheafy","kind":"node","lean":"TauCeti.Huber.isSheafyRing_of_isStronglyNoetherian","parent":1349,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/foundations-of-adic-spaces/index.html#every-strongly-noetherian-tate-ring-is-sheafy","scope":"tauceti/algebraic-geometry/foundations-of-adic-spaces","status":"fully_proved","summary":"Every strongly noetherian Tate ring is sheafy in the sense of Wedhorn's Definition 8.26 (`TauCeti.Huber.IsSheafyRing`). This is Wedhorn's Theorem 8.28(b); `A` need not be complete or Hausdorff. The identity of `\u00c2` is topologically of finite type, so this is the case `B = \u00c2` of `TauCeti.Huber.isStablySheafyRing_of_isStronglyNoetherian`.","title":"Every strongly noetherian Tate ring is sheafy"},{"id":"tauceti/algebraic-geometry/foundations-of-adic-spaces/affinoid-algebras-are-the-algebras-topologically","kind":"node","lean":"TauCeti.Huber.isTopologicallyFiniteType_algebraMap_iff_isAffinoidAlgebra","parent":1349,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/foundations-of-adic-spaces/index.html#affinoid-algebras-are-the-algebras-topologically","scope":"tauceti/algebraic-geometry/foundations-of-adic-spaces","status":"fully_proved","summary":"A complete ultrametric normed `K`-algebra `B` is topologically of finite type over `K` in Huber's sense (Wedhorn, Definition 5.57) exactly when it is a quotient of a Tate algebra `K\u27e8X\u2081, \u2026, X\u2099\u27e9` as a `K`-algebra.","title":"Affinoid algebras are the algebras topologically of finite type"},{"id":"tauceti/algebraic-geometry/foundations-of-adic-spaces/the-adic-spectrum-of-a-classical-affinoid-algebr","kind":"node","lean":"TauCeti.ValuationSpectrum.isAdic_presentationLimitPreAdicSpace_of_isAffinoidAlgebra","parent":1349,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/foundations-of-adic-spaces/index.html#the-adic-spectrum-of-a-classical-affinoid-algebr","scope":"tauceti/algebraic-geometry/foundations-of-adic-spaces","status":"fully_proved","summary":"Let `K` be a complete nonarchimedean field with a nontrivial absolute value and `B` a quotient of a Tate algebra `K\u27e8X\u2081, \u2026, X\u2099\u27e9`, carrying a complete ultrametric algebra norm, for instance the quotient Gauss norm. Then for any pair of definition `P` of `B` and any plus ring `B\u207a` of power-bounded elements containing its ring of definition, such as `B\u00b0`, `Spa(B, B\u207a)` is an affinoid adic space.","title":"The adic spectrum of a classical affinoid algebra is an affinoid adic space"},{"id":"tauceti/algebraic-geometry/foundations-of-adic-spaces/every-point-of-the-closed-unit-disc-is-of-one-of","kind":"node","lean":"TauCeti.ValuationSpectrum.exists_eq_classicalPoint_or_discPoint_or_discPointAbove_or_discPointBelow_or_nestedDiscPoint","parent":1349,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/foundations-of-adic-spaces/index.html#every-point-of-the-closed-unit-disc-is-of-one-of","scope":"tauceti/algebraic-geometry/foundations-of-adic-spaces","status":"fully_proved","summary":"Every point of the closed unit disc is of one of Wedhorn's five types, over an algebraically closed complete nonarchimedean field (Example 7.57): a classical point `x_a` (type (1)), a point `\u03b7_{a,r}` (types (2) and (3)), the point attached to a decreasing sequence of discs `D(a\u2099, |c\u2099|)` with no common point in `K` (type (4)), or a rank-two point `\u03b7_{a,r\u207a}` or `\u03b7_{a,r\u207b}` with `r` the norm of an element of `K` (type\u2026","title":"Every point of the closed unit disc is of one of Wedhorn's five types"},{"id":"tauceti/algebraic-geometry/foundations-of-adic-spaces/adic-spaces-wedhorn-definition-8-22","kind":"node","lean":"TauCeti.PreAdicSpace.isAdic","parent":1349,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/foundations-of-adic-spaces/index.html#adic-spaces-wedhorn-definition-8-22","scope":"tauceti/algebraic-geometry/foundations-of-adic-spaces","status":"fully_proved","summary":"the objects of `\ud835\udcb1`, the sheafy objects of `\ud835\udcb1^pre`, that admit an open cover by affinoid adic spaces. Since the restriction of a sheafy object to an open is sheafy (`isSheafy_restrict`), an open cover by affinoid adic spaces is the same as an open cover by open affinoid subspaces: an adic space is a sheafy locally affinoid object of `\ud835\udcb1^pre`. Their full subcategory of `\ud835\udcb1^pre` is Wedhorn's category `(Adic)`,\u2026","title":"Adic spaces (Wedhorn, Definition 8.22)"},{"id":"tauceti/algebraic-geometry/foundations-of-adic-spaces/closed-immersions-of-adic-spaces-are-monomorphis","kind":"node","lean":"TauCeti.AdicSpace.mono_of_isClosedImmersion","parent":1349,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/foundations-of-adic-spaces/index.html#closed-immersions-of-adic-spaces-are-monomorphis","scope":"tauceti/algebraic-geometry/foundations-of-adic-spaces","status":"fully_proved","summary":"a morphism into an adic space factors through a closed immersion in at most one way.","title":"Closed immersions of adic spaces are monomorphisms"},{"id":"tauceti/algebraic-geometry/foundations-of-adic-spaces/wedhorn-corollary-6-4-2","kind":"node","lean":"TauCeti.Huber.IsHuberRing.subring","parent":1349,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/foundations-of-adic-spaces/index.html#wedhorn-corollary-6-4-2","scope":"tauceti/algebraic-geometry/foundations-of-adic-spaces","status":"fully_proved","summary":"An open subring of a Huber ring, with the subspace topology, is a Huber ring.","title":"Wedhorn Corollary 6.4(2)"},{"data_url":"scopes/tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers.json","id":"tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers","kind":"scope","parent":1303,"result":false,"route":"../roadmap/tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers/index.html","scope":"tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"The Jacobian challenge (Christian Merten's AG version)"},{"id":"tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers/the-riemann-roch-theorem","kind":"node","lean":"TauCeti.AlgebraicGeometry.SchemeWeilDivisor.eulerCharBelow_sheaf_eq_relativeDegree_add_one_sub_genus, TauCeti.AlgebraicGeometry.SchemeWeilDivisor.finrank_cohomology_zero_sheaf_eq_relativeDegree_of_genus_eq_one, TauCeti.AlgebraicGeometry.InvertibleSheaf.eulerCharBelow_eq_relativeDegree_add_one_sub_genus, TauCeti.AlgebraicGeometry.SchemeWeilDivisor.finrank_cohomology_zero_sheaf_sub_finrank_cohomology_one_sheaf","parent":1362,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers/index.html#the-riemann-roch-theorem","scope":"tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers","status":"fully_proved","summary":"On a proper integral curve over a field `k` whose codimension-one local rings are discrete valuation rings, with a `k`-rational point and with `H\u00b9(X, \ud835\udcaa_X)` finite-dimensional, every Weil divisor `D` satisfies","title":"The Riemann\u2013Roch theorem"},{"id":"tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers/tauceti-algebraicgeometry-nonempty-cohomologyone","kind":"node","lean":"TauCeti.AlgebraicGeometry.nonempty_cohomologyOneDualEquivCohomologyZero_differentials_tensor_dual","parent":1362,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers/index.html#tauceti-algebraicgeometry-nonempty-cohomologyone","scope":"tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers","status":"fully_proved","summary":"Over an exact constant field, the dual of `H\u00b9(X, L)` is linearly equivalent to `H\u2070(X, \u03a9_{X/k} \u2297 L\u207b\u00b9)`, by the identification of the differential line bundle with the canonical bundle. This asserts only the existence of a noncanonical linear equivalence, not a natural duality pairing.","title":"TauCeti.AlgebraicGeometry.nonempty_cohomologyOneDualEquivCohomologyZero_differentials_tensor_dual"},{"id":"tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers/the-cartier-picard-dictionary","kind":"node","lean":"TauCeti.AlgebraicGeometry.Scheme.CartierDivisor.classGroupAddEquivLineBundleClass","parent":1362,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers/index.html#the-cartier-picard-dictionary","scope":"tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers","status":"fully_proved","summary":"On any integral scheme, Cartier divisors modulo principal divisors form the Picard group of line-bundle classes under tensor product.","title":"The Cartier\u2013Picard dictionary"},{"id":"tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers/the-points-of-an-elliptic-curve-form-pic0-of-its","kind":"node","lean":"WeierstrassCurve.pointAddEquivPicZero","parent":1362,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers/index.html#the-points-of-an-elliptic-curve-form-pic0-of-its","scope":"tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers","status":"fully_proved","summary":"For an elliptic curve `W` over a field `K`, the group `W.toAffine.Point` of Mathlib, with its chord-and-tangent group law, is isomorphic to the group `Pic\u2070 (projModel W)` of degree-zero line bundles on the projective model. The point at infinity goes to the trivial bundle, and an affine point `P` to the class of `\ud835\udcaa(x_P - x_O)` (`coe_pointAddEquivPicZero_some`).","title":"The points of an elliptic curve form `Pic\u2070` of its projective model"},{"id":"tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers/the-abel-jacobi-map-of-a-curve-of-positive-genus","kind":"node","lean":"TauCeti.AlgebraicGeometry.abelJacobiMap_app_injective_of_genus_pos","parent":1362,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers/index.html#the-abel-jacobi-map-of-a-curve-of-positive-genus","scope":"tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers","status":"fully_proved","summary":"Let `X` be a smooth proper geometrically connected curve of genus `g \u2265 1` over a field `k`, with a rational point `x\u2080`. For every field extension `K / k`, two `K`-points of `X` with the same class `\ud835\udcaa(\u0393\u2093 - x\u2080)` in `Pic(X_K) / Pic(Spec K)` are equal. Taking `K` algebraically closed, the Abel\u2013Jacobi map is injective on geometric points.","title":"The Abel\u2013Jacobi map of a curve of positive genus separates points over every field extension"},{"id":"tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers/the-degree-zero-picard-functor-pic0-x-s","kind":"node","lean":"TauCeti.AlgebraicGeometry.relativePicardDegreeZeroFunctor","parent":1362,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers/index.html#the-degree-zero-picard-functor-pic0-x-s","scope":"tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers","status":"fully_proved","summary":"The degree-zero Picard functor `Pic\u2070_{X/S}` of a smooth proper geometrically connected family of curves `f : X \u27f6 S`, as a presheaf of commutative groups on schemes over `S`: it sends `T` to the subgroup `relativePicardDegreeZeroSubgroup f T` of `Pic(X_T) / Pic(T)`, and a morphism over `S` to the restriction of the pullback map of the relative Picard presheaf.","title":"The degree-zero Picard functor `Pic\u2070_{X/S}`"},{"id":"tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers/tauceti-algebraicgeometry-relativepicardfppfshea","kind":"node","lean":"TauCeti.AlgebraicGeometry.relativePicardFppfSheaf","parent":1362,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers/index.html#tauceti-algebraicgeometry-relativepicardfppfshea","scope":"tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers","status":"fully_proved","summary":"The relative Picard fppf sheaf of `f : X \u27f6 S`, obtained by sheafifying the presheaf `T \u21a6 Pic(X_T) / Pic(T)` on the fppf site of schemes over `S`.","title":"TauCeti.AlgebraicGeometry.relativePicardFppfSheaf"},{"id":"tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers/the-abel-jacobi-map-into-the-degree-zero-picard","kind":"node","lean":"TauCeti.AlgebraicGeometry.abelJacobiMapToPicZeroFppf","parent":1362,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers/index.html#the-abel-jacobi-map-into-the-degree-zero-picard","scope":"tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers","status":"fully_proved","summary":"It sends an `S`-point `x : T \u27f6 X` to the fppf-local class of `\ud835\udcaa(\u0393\u2093 - x\u2080,T)`.","title":"The Abel\u2013Jacobi map into the degree-zero Picard fppf sheaf"},{"id":"tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers/tauceti-algebraicgeometry-abelianvariety-isalban","kind":"node","lean":"TauCeti.AlgebraicGeometry.AbelianVariety.IsAlbanese","parent":1362,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers/index.html#tauceti-algebraicgeometry-abelianvariety-isalban","scope":"tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers","status":"fully_proved","summary":"A pointed morphism `a : X \u27f6 J.toOver` from a scheme `X` over `K` with a `K`-rational point `x\u2080` to an abelian variety `J` has the Albanese property if every morphism `f : X \u27f6 A.toOver` to an abelian variety sending `x\u2080` to the identity factors as `f = a \u226b \u03c6` through a unique homomorphism `\u03c6 : J \u27f6 A` of abelian varieties.","title":"TauCeti.AlgebraicGeometry.AbelianVariety.IsAlbanese"},{"id":"tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers/riemanns-inequality","kind":"node","lean":"TauCeti.AlgebraicGeometry.SchemeWeilDivisor.relativeDegree_add_one_sub_genus_le_finrank_cohomology_zero_sheaf","parent":1362,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers/index.html#riemanns-inequality","scope":"tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers","status":"fully_proved","summary":"On a proper integral curve over a field `k` whose codimension-one local rings are discrete valuation rings, with all global functions constant and with `H\u00b9(X, \ud835\udcaa_X)` finite-dimensional, `dim_k H\u2070(X, \ud835\udcaa_X(D)) \u2265 deg D + 1 - g`.","title":"Riemann's inequality"},{"id":"tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers/the-riemann-roch-theorem-with-the-sheaf-of-diffe","kind":"node","lean":"TauCeti.AlgebraicGeometry.finrank_cohomology_zero_sub_finrank_cohomology_zero_differentials_tensor_dual","parent":1362,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers/index.html#the-riemann-roch-theorem-with-the-sheaf-of-diffe","scope":"tauceti/algebraic-geometry/the-jacobian-challenge-christian-mertens-ag-vers","status":"fully_proved","summary":"Over an exact constant field, every line bundle `L` satisfies","title":"The Riemann\u2013Roch theorem with the sheaf of differentials"},{"data_url":"scopes/tauceti/algebraic-geometry/modular-curves-following-katz-mazur.json","id":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur","kind":"scope","parent":1303,"result":false,"route":"../roadmap/tauceti/algebraic-geometry/modular-curves-following-katz-mazur/index.html","scope":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Modular curves, following Katz\u2013Mazur"},{"id":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur/y1-n-is-a-fine-moduli-scheme-for-n-4","kind":"node","lean":"TauCeti.AlgebraicGeometry.ModuliProblem.gammaOneNaiveRepresentableBy","parent":1374,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/modular-curves-following-katz-mazur/index.html#y1-n-is-a-fine-moduli-scheme-for-n-4","scope":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur","status":"fully_proved","summary":"For `N \u2265 4`, the naive moduli problem `[\u0393\u2081(N)]` over `Ell/B` is represented by the universal elliptic curve over `Y\u2081(N)` (`EllObj.yOne B N`): the naive `[\u0393\u2081(N)]`-structures on an object `X` of `Ell/B` correspond to the arrows `X \u27f6 EllObj.yOne B N`, by pulling back the point `(0, 0)` (`naiveLevelEquiv_gammaOneNaiveRepresentableBy_homEquiv`).","title":"`Y\u2081(N)` is a fine moduli scheme for `N \u2265 4`"},{"id":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur/y1-n-is-smooth-over-b","kind":"node","lean":"TauCeti.AlgebraicGeometry.EllObj.smooth_yOne_structureMap","parent":1374,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/modular-curves-following-katz-mazur/index.html#y1-n-is-smooth-over-b","scope":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur","status":"fully_proved","summary":"For `N \u2265 4` invertible on `B`, the structure morphism of the fine moduli scheme `Y\u2081(N)` to `B` is smooth; for example `Y\u2081(N)` is smooth over `\u2124[1/N]`.","title":"`Y\u2081(N)` is smooth over `B`"},{"id":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur/tauceti-algebraicgeometry-moduliproblem-gammaone","kind":"node","lean":"TauCeti.AlgebraicGeometry.ModuliProblem.gammaOneRepresentableBy","parent":1374,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/modular-curves-following-katz-mazur/index.html#tauceti-algebraicgeometry-moduliproblem-gammaone","scope":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur","status":"fully_proved","summary":"For `N \u2265 4` invertible on `B`, the existing fine curve `Y\u2081(N)` also represents the integral Drinfeld `[\u0393\u2081(N)]` problem restricted to `B`.","title":"TauCeti.AlgebraicGeometry.ModuliProblem.gammaOneRepresentableBy"},{"id":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur/drinfeld-and-naive-n-structures-coincide-where-n","kind":"node","lean":"TauCeti.AlgebraicGeometry.ModuliProblem.gammaIsoGammaNaive","parent":1374,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/modular-curves-following-katz-mazur/index.html#drinfeld-and-naive-n-structures-coincide-where-n","scope":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur","status":"fully_proved","summary":"over a base `B` on which `N` is invertible, `gammaToGammaNaive` and `gammaNaiveToGamma` are mutually inverse isomorphisms between `[\u0393(N)]` and `[\u0393(N)]_naive`.","title":"Drinfeld and naive `[\u0393(N)]`-structures coincide where `N` is invertible (Katz\u2013Mazur, \u00a73.7)"},{"id":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur/the-moduli-problem-n-is-etale-over-ell-b-when-n","kind":"node","lean":"TauCeti.AlgebraicGeometry.ModuliProblem.hasProperty_etale_gammaRelRepData","parent":1374,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/modular-curves-following-katz-mazur/index.html#the-moduli-problem-n-is-etale-over-ell-b-when-n","scope":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur","status":"fully_proved","summary":"where `N` is invertible on `B`, `[\u0393(N)]` is isomorphic to `[\u0393(N)]_naive`, whose representing schemes are \u00e9tale over their bases. Together with `ModuliProblem.hasProperty_isFinite_gammaRelRepData`, this makes `[\u0393(N)]` finite \u00e9tale over `Ell/B`.","title":"The moduli problem `[\u0393(N)]` is \u00e9tale over `Ell/B` when `N` is invertible on `B` (Katz\u2013Mazur, 3.7.1)"},{"id":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur/n-is-rigid-for-n-3-where-n-is-invertible-katz-ma","kind":"node","lean":"TauCeti.AlgebraicGeometry.ModuliProblem.isRigid_gamma","parent":1374,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/modular-curves-following-katz-mazur/index.html#n-is-rigid-for-n-3-where-n-is-invertible-katz-ma","scope":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur","status":"fully_proved","summary":"Over a base `B` on which `N` is invertible, `[\u0393(N)]` is isomorphic to the rigid naive problem `[\u0393(N)]_naive` (`ModuliProblem.isRigid_gammaNaive`).","title":"`[\u0393(N)]` is rigid for `N \u2265 3` where `N` is invertible (Katz\u2013Mazur, 2.7.2)"},{"id":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur/the-integral-moduli-problem-n-is-relatively-repr","kind":"node","lean":"TauCeti.AlgebraicGeometry.ModuliProblem.gammaRelRepData","parent":1374,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/modular-curves-following-katz-mazur/index.html#the-integral-moduli-problem-n-is-relatively-repr","scope":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur","status":"fully_proved","summary":"For an elliptic curve `E/S`, it is represented by the closed Drinfeld-basis locus in `E[N] \u00d7_S E[N]`.","title":"The integral moduli problem `[\u0393(N)]` is relatively representable (Katz\u2013Mazur, 3.6.0)"},{"id":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur/the-moduli-problem-1-n-is-relatively-representab","kind":"node","lean":"TauCeti.AlgebraicGeometry.ModuliProblem.gammaOneRelRepData","parent":1374,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/modular-curves-following-katz-mazur/index.html#the-moduli-problem-1-n-is-relatively-representab","scope":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur","status":"fully_proved","summary":"For an object `X = (E/S)` of `Ell/B`, the functor `T \u21a6 \u0393\u2081(N)` on schemes over `S` is represented by the exact-order scheme `E.exactOrderScheme N`, the closed subscheme of `E` of points of exact order `N`, through `gammaOneBaseChangeEquiv`. No hypothesis is placed on `B`, and `N` need not be invertible on it.","title":"The moduli problem `[\u0393\u2081(N)]` is relatively representable (Katz\u2013Mazur, 3.6.0)"},{"id":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur/multiplication-by-a-nonzero-integer-n-is-an-isog","kind":"node","lean":"TauCeti.AlgebraicGeometry.EllipticCurveGeom.isogenyOfDegree_intCast","parent":1374,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/modular-curves-following-katz-mazur/index.html#multiplication-by-a-nonzero-integer-n-is-an-isog","scope":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.AlgebraicGeometry.EllipticCurveGeom.isogenyOfDegree_intCast`.","title":"Multiplication by a nonzero integer `n` is an isogeny of degree `n\u00b2`"},{"id":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur/the-n-torsion-of-an-elliptic-curve-is-etale-wher","kind":"node","lean":"TauCeti.AlgebraicGeometry.EllipticCurveGeom.etale_torsion_hom_of_forall_intCast_ne_zero","parent":1374,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/modular-curves-following-katz-mazur/index.html#the-n-torsion-of-an-elliptic-curve-is-etale-wher","scope":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur","status":"fully_proved","summary":"For an elliptic curve `E` over a scheme `S` and an integer `n` invertible in every residue field of `S`, the `n`-torsion `E[n]` is \u00e9tale over `S`. It is also finite over `S` (`EllipticCurveGeom.isFinite_torsion_hom`), so `E[n]` is finite \u00e9tale over `S`.","title":"The `n`-torsion of an elliptic curve is \u00e9tale where `n` is invertible (Katz\u2013Mazur 2.3.1)"},{"id":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur/galois-theory-for-finite-etale-schemes-over-a-fi","kind":"node","lean":"TauCeti.FiniteEtaleScheme.galoisEquivalence","parent":1374,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/modular-curves-following-katz-mazur/index.html#galois-theory-for-finite-etale-schemes-over-a-fi","scope":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur","status":"fully_proved","summary":"For a separable closure `\u03a9` of `K`, finite \u00e9tale `K`-schemes are equivalent to finite sets with a continuous action of `Gal(\u03a9/K)`. A finite \u00e9tale scheme `Spec A` goes to the set of `\u03a9`-valued points of `A` (`specCompGaloisEquivalenceFunctorIso`).","title":"Galois theory for finite \u00e9tale schemes over a field"},{"id":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur/finite-etale-commutative-group-schemes-over-a-fi","kind":"node","lean":"TauCeti.FiniteEtaleScheme.galoisEquivalenceCommGrp","parent":1374,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/modular-curves-following-katz-mazur/index.html#finite-etale-commutative-group-schemes-over-a-fi","scope":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur","status":"fully_proved","summary":"Commutative group objects in finite \u00e9tale `K`-schemes are equivalent to commutative group objects in finite continuous `Gal(\u03a9/K)`-sets, that is, to finite abelian groups with a continuous action of `Gal(\u03a9/K)` by group automorphisms.","title":"Finite \u00e9tale commutative group schemes over a field are finite Galois modules"},{"id":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur/gl2-z-nz-acts-freely-on-m-p-n-for-every-rigid-p","kind":"node","lean":"TauCeti.AlgebraicGeometry.ModuliProblem.IsRigid.eq_one_of_comp_simultaneousAction_gammaNaiveAction_base_eq","parent":1374,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/modular-curves-following-katz-mazur/index.html#gl2-z-nz-acts-freely-on-m-p-n-for-every-rigid-p","scope":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur","status":"fully_proved","summary":"Let `V` represent the simultaneous problem `(\ud835\udcab, [\u0393(N)])` with the naive full level-`N` problem, on which `GL\u2082(\u2124/N\u2124)` acts by change of basis. If a point of `V.base` with values in a nonempty scheme is fixed by a matrix `g`, then `g = 1`. For `N = 3` this is the freeness of the action of `GL\u2082(\ud835\udd3d\u2083)` on the moduli scheme of `\ud835\udcab` rigidified by naive level-three structures.","title":"`GL\u2082(\u2124/N\u2124)` acts freely on `\ud835\udd44(\ud835\udcab, [\u0393(N)])` for every rigid `\ud835\udcab`"},{"id":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur/categorytheory-commmon-invariantsubalgebrapreshe","kind":"node","lean":"CategoryTheory.CommMon.invariantSubalgebraPresheaf","parent":1374,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/modular-curves-following-katz-mazur/index.html#categorytheory-commmon-invariantsubalgebrapreshe","scope":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur","status":"fully_proved","summary":"The presheaf of subalgebras of invariant sections of `A`.","title":"CategoryTheory.CommMon.invariantSubalgebraPresheaf"},{"id":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur/tauceti-descentdatum-descended","kind":"node","lean":"TauCeti.DescentDatum.descended","parent":1374,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/modular-curves-following-katz-mazur/index.html#tauceti-descentdatum-descended","scope":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur","status":"fully_proved","summary":"The scheme descended from a descent datum `D` on an affine morphism `X \u27f6 S'`: the relative spectrum over `S` of the functions on `X` whose pullbacks along the action and along the second projection `S' \u00d7_S X \u27f6 X` agree.","title":"TauCeti.DescentDatum.descended"},{"id":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur/tauceti-stricthenselization-algebra","kind":"node","lean":"TauCeti.StrictHenselization.algebra","parent":1374,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/modular-curves-following-katz-mazur/index.html#tauceti-stricthenselization-algebra","scope":"tauceti/algebraic-geometry/modular-curves-following-katz-mazur","status":"fully_proved","summary":"The colimit of pointed \u00e9tale algebras. For a local base ring and a separable closure of its residue field as target, this is the strict henselization.","title":"TauCeti.StrictHenselization.algebra"},{"data_url":"scopes/tauceti/algebraic-geometry/stable-reduction-of-curves-and-stable-maps.json","id":"tauceti/algebraic-geometry/stable-reduction-of-curves-and-stable-maps","kind":"scope","parent":1303,"result":false,"route":"../roadmap/tauceti/algebraic-geometry/stable-reduction-of-curves-and-stable-maps/index.html","scope":"tauceti/algebraic-geometry/stable-reduction-of-curves-and-stable-maps","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Stable reduction of curves and stable maps"},{"id":"tauceti/algebraic-geometry/stable-reduction-of-curves-and-stable-maps/classification-of-minimal-numerical-types-of-gen","kind":"node","lean":"TauCeti.NumericalType.IsMinimal.isGenusOneMinimalShape","parent":1391,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/stable-reduction-of-curves-and-stable-maps/index.html#classification-of-minimal-numerical-types-of-gen","scope":"tauceti/algebraic-geometry/stable-reduction-of-curves-and-stable-maps","status":"fully_proved","summary":"Classification of minimal numerical types of genus one (Stacks, Lemma 55.6.2). Every minimal numerical type of arithmetic genus one belongs to one of the shape families of `TauCeti.NumericalType.IsGenusOneMinimalShape`.","title":"Classification of minimal numerical types of genus one"},{"id":"tauceti/algebraic-geometry/stable-reduction-of-curves-and-stable-maps/prime-torsion-of-a-numerical-type","kind":"node","lean":"TauCeti.NumericalType.finrank_torsion_le_arithmeticGenus","parent":1391,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/stable-reduction-of-curves-and-stable-maps/index.html#prime-torsion-of-a-numerical-type","scope":"tauceti/algebraic-geometry/stable-reduction-of-curves-and-stable-maps","status":"fully_proved","summary":"In a numerical type of genus `g \u2265 2`, every prime `\u2113 > 768g - 768` satisfies `dim_{\ud835\udd3d_\u2113} Pic(T)[\u2113] \u2264 g`. In particular this holds for every prime `\u2113 > 768g`, which is Stacks, Proposition 55.7.4.","title":"Prime torsion of a numerical type"},{"id":"tauceti/algebraic-geometry/stable-reduction-of-curves-and-stable-maps/a-curve-of-genus-zero-with-a-divisor-of-degree-o","kind":"node","lean":"TauCeti.HasNodalReduction.of_genus_eq_zero_of_divisor_degree_eq_one","parent":1391,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/stable-reduction-of-curves-and-stable-maps/index.html#a-curve-of-genus-zero-with-a-divisor-of-degree-o","scope":"tauceti/algebraic-geometry/stable-reduction-of-curves-and-stable-maps","status":"fully_proved","summary":"Let `C` be a regular proper curve over the fraction field `K` of a discrete valuation ring `R`, with `K` integrally closed in `K(C)`. If `dim_K H\u00b9(C, \ud835\udcaa_C) = 0` and some divisor on `C` has degree one, then `C` has a proper at-worst-nodal model over `R`; indeed `C \u2245 \u2119\u00b9_K`, which has the smooth proper model `\u2119\u00b9_R`.","title":"A curve of genus zero with a divisor of degree one has nodal reduction"},{"id":"tauceti/algebraic-geometry/stable-reduction-of-curves-and-stable-maps/algebraicgeometry-scheme-exists-split-singularlo","kind":"node","lean":"AlgebraicGeometry.Scheme.exists_split_singularLocus","parent":1391,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/stable-reduction-of-curves-and-stable-maps/index.html#algebraicgeometry-scheme-exists-split-singularlo","scope":"tauceti/algebraic-geometry/stable-reduction-of-curves-and-stable-maps","status":"fully_proved","summary":"All nodes of a proper nodal curve split over one finite separable field extension. For any chosen base change of the curve, its singular subscheme is the spectrum of a product of copies of the extended field, as a scheme over that field. This includes the smooth case, in which the node set is empty.","title":"AlgebraicGeometry.Scheme.exists_split_singularLocus"},{"id":"tauceti/algebraic-geometry/stable-reduction-of-curves-and-stable-maps/the-universal-property-of-a-contraction","kind":"node","lean":"AlgebraicGeometry.Scheme.Hom.existsUnique_comp_eq_of_universallyClosed","parent":1391,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/stable-reduction-of-curves-and-stable-maps/index.html#the-universal-property-of-a-contraction","scope":"tauceti/algebraic-geometry/stable-reduction-of-curves-and-stable-maps","status":"fully_proved","summary":"The universal property of a contraction, for a universally closed surjective morphism, such as a proper surjective one, for which `\ud835\udcaa_Y \u2192 c_* \ud835\udcaa_X` is an isomorphism.","title":"The universal property of a contraction"},{"id":"tauceti/algebraic-geometry/stable-reduction-of-curves-and-stable-maps/tauceti-algebraicgeometry-atworstnodalofrelative","kind":"node","lean":"TauCeti.AlgebraicGeometry.AtWorstNodalOfRelativeDimensionOne","parent":1391,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/stable-reduction-of-curves-and-stable-maps/index.html#tauceti-algebraicgeometry-atworstnodalofrelative","scope":"tauceti/algebraic-geometry/stable-reduction-of-curves-and-stable-maps","status":"fully_proved","summary":"Being at worst nodal of relative dimension one is the affine-local extension of the scheme-theoretic criterion over spectra.","title":"TauCeti.AlgebraicGeometry.AtWorstNodalOfRelativeDimensionOne"},{"id":"tauceti/algebraic-geometry/stable-reduction-of-curves-and-stable-maps/the-relative-proj-is-proper","kind":"node","lean":"CategoryTheory.CommMon.isProper_relativeProjToBase","parent":1391,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/stable-reduction-of-curves-and-stable-maps/index.html#the-relative-proj-is-proper","scope":"tauceti/algebraic-geometry/stable-reduction-of-curves-and-stable-maps","status":"fully_proved","summary":"If the graded algebra `A` is of finite type and its degree-zero part `A\u2080` is finite over `X`, then `Proj_X(A) \u27f6 X` is proper.","title":"The relative `Proj` is proper"},{"id":"tauceti/algebraic-geometry/stable-reduction-of-curves-and-stable-maps/stabilization-terminates","kind":"node","lean":"TauCeti.DualGraph.exists_reflTransGen_stabilizationStep_isStable","parent":1391,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/stable-reduction-of-curves-and-stable-maps/index.html#stabilization-terminates","scope":"tauceti/algebraic-geometry/stable-reduction-of-curves-and-stable-maps","status":"fully_proved","summary":"In the stable range `2g - 2 + n > 0`, iterating stabilization steps from any graph reaches a stable graph, of the same arithmetic genus by `TauCeti.DualGraph.arithmeticGenus_eq_of_reflTransGen`. Termination holds because every step removes a vertex.","title":"Stabilization terminates"},{"data_url":"scopes/tauceti/algebraic-geometry/etale-cohomology-foundations.json","id":"tauceti/algebraic-geometry/etale-cohomology-foundations","kind":"scope","parent":1303,"result":false,"route":"../roadmap/tauceti/algebraic-geometry/etale-cohomology-foundations/index.html","scope":"tauceti/algebraic-geometry/etale-cohomology-foundations","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"\u00c9tale cohomology foundations"},{"id":"tauceti/algebraic-geometry/etale-cohomology-foundations/tauceti-compactification-commonrefinement","kind":"node","lean":"TauCeti.Compactification.commonRefinement","parent":1400,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/etale-cohomology-foundations/index.html#tauceti-compactification-commonrefinement","scope":"tauceti/algebraic-geometry/etale-cohomology-foundations","status":"fully_proved","summary":"The common refinement of two compactifications: the scheme-theoretic closure of the source in the fibre product of their ambient schemes over the base.","title":"TauCeti.Compactification.commonRefinement"},{"id":"tauceti/algebraic-geometry/etale-cohomology-foundations/tauceti-densecompactification-islimitcommonrefin","kind":"node","lean":"TauCeti.DenseCompactification.isLimitCommonRefinementFan","parent":1400,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/etale-cohomology-foundations/index.html#tauceti-densecompactification-islimitcommonrefin","scope":"tauceti/algebraic-geometry/etale-cohomology-foundations","status":"fully_proved","summary":"The common refinement is the binary product in the category of dense compactifications.","title":"TauCeti.DenseCompactification.isLimitCommonRefinementFan"},{"id":"tauceti/algebraic-geometry/etale-cohomology-foundations/tauceti-compactification-exists-forall-nonempty","kind":"node","lean":"TauCeti.Compactification.exists_forall_nonempty_hom","parent":1400,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/etale-cohomology-foundations/index.html#tauceti-compactification-exists-forall-nonempty","scope":"tauceti/algebraic-geometry/etale-cohomology-foundations","status":"fully_proved","summary":"Every nonempty finite family of compactifications has a dense common refinement.","title":"TauCeti.Compactification.exists_forall_nonempty_hom"},{"id":"tauceti/algebraic-geometry/etale-cohomology-foundations/tauceti-compactification-hom-unique","kind":"node","lean":"TauCeti.Compactification.hom_unique","parent":1400,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/etale-cohomology-foundations/index.html#tauceti-compactification-hom-unique","scope":"tauceti/algebraic-geometry/etale-cohomology-foundations","status":"fully_proved","summary":"Maps from a compactification with schematically dense interior to any fixed compactification are unique. No density assumption on the target is needed.","title":"TauCeti.Compactification.hom_unique"},{"id":"tauceti/algebraic-geometry/etale-cohomology-foundations/tauceti-finiteetalescheme-exists-iscolimit-binar","kind":"node","lean":"TauCeti.FiniteEtaleScheme.exists_isColimit_binaryCofanMk_of_mono","parent":1400,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/etale-cohomology-foundations/index.html#tauceti-finiteetalescheme-exists-iscolimit-binar","scope":"tauceti/algebraic-geometry/etale-cohomology-foundations","status":"fully_proved","summary":"Every monomorphism `i : X \u27f6 Y` of finite \u00e9tale schemes over `S` is the inclusion of a direct summand: there is a finite \u00e9tale `Z` over `S` with a morphism `u : Z \u27f6 Y` exhibiting `Y` as the coproduct of `X` and `Z`. Here `Z` is the open complement of the image of `i`.","title":"TauCeti.FiniteEtaleScheme.exists_isColimit_binaryCofanMk_of_mono"},{"id":"tauceti/algebraic-geometry/etale-cohomology-foundations/tauceti-finiteetalescheme-epi-iff-surjective","kind":"node","lean":"TauCeti.FiniteEtaleScheme.epi_iff_surjective","parent":1400,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/etale-cohomology-foundations/index.html#tauceti-finiteetalescheme-epi-iff-surjective","scope":"tauceti/algebraic-geometry/etale-cohomology-foundations","status":"fully_proved","summary":"A morphism of finite \u00e9tale schemes is an epimorphism if and only if its underlying morphism of schemes is surjective.","title":"TauCeti.FiniteEtaleScheme.epi_iff_surjective"},{"id":"tauceti/algebraic-geometry/etale-cohomology-foundations/the-invariants-of-a-finite-etale-algebra-under-a","kind":"node","lean":"TauCeti.FixedPoints.etale_subalgebra","parent":1400,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/etale-cohomology-foundations/index.html#the-invariants-of-a-finite-etale-algebra-under-a","scope":"tauceti/algebraic-geometry/etale-cohomology-foundations","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.FixedPoints.etale_subalgebra`.","title":"The invariants of a finite \u00e9tale algebra under a finite monoid of algebra endomorphisms form an \u00e9tale algebra"},{"id":"tauceti/algebraic-geometry/etale-cohomology-foundations/tauceti-compactification","kind":"node","lean":"TauCeti.Compactification","parent":1400,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/etale-cohomology-foundations/index.html#tauceti-compactification","scope":"tauceti/algebraic-geometry/etale-cohomology-foundations","status":"fully_proved","summary":"Compactifications of `f`, with maps fixing `X` and lying over `S`.","title":"TauCeti.Compactification"},{"id":"tauceti/algebraic-geometry/etale-cohomology-foundations/tauceti-densecompactification","kind":"node","lean":"TauCeti.DenseCompactification","parent":1400,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/etale-cohomology-foundations/index.html#tauceti-densecompactification","scope":"tauceti/algebraic-geometry/etale-cohomology-foundations","status":"fully_proved","summary":"The full subcategory of compactifications with scheme-theoretically dense interior.","title":"TauCeti.DenseCompactification"},{"id":"tauceti/algebraic-geometry/etale-cohomology-foundations/tauceti-finiteetalescheme-fiber","kind":"node","lean":"TauCeti.FiniteEtaleScheme.fiber","parent":1400,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/etale-cohomology-foundations/index.html#tauceti-finiteetalescheme-fiber","scope":"tauceti/algebraic-geometry/etale-cohomology-foundations","status":"fully_proved","summary":"The geometric fibre functor at a geometric point `x : Spec \u03a9 \u27f6 S`, for `\u03a9` separably closed: a finite \u00e9tale `S`-scheme `X` goes to the finite set classifying `X \u00d7_S Spec \u03a9`.","title":"TauCeti.FiniteEtaleScheme.fiber"},{"data_url":"scopes/tauceti/algebraic-geometry/adic-space.json","id":"tauceti/algebraic-geometry/adic-space","kind":"scope","parent":1303,"result":false,"route":"../roadmap/tauceti/algebraic-geometry/adic-space/index.html","scope":"tauceti/algebraic-geometry/adic-space","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.AlgebraicGeometry.AdicSpace` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Adic Space"},{"id":"tauceti/algebraic-geometry/adic-space/buzzard-verberkmoes-for-the-presheaf-of-sets","kind":"node","lean":"TauCeti.ValuationSpectrum.isSheaf_underlying_presentationLimitPresheaf_of_isStablyUniform, TauCeti.ValuationSpectrum.isSheaf_presentationLimitPresheaf_of_isStablyUniform","parent":1411,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/adic-space/index.html#buzzard-verberkmoes-for-the-presheaf-of-sets","scope":"tauceti/algebraic-geometry/adic-space","status":"fully_proved","summary":"Let `A` be a stably uniform Tate ring, `P` a pair of definition whose ring of definition lies in `A\u207a`, and `A\u207a` a subring of power-bounded elements. The presheaf of sets underlying the presentation-limit structure presheaf of `Spa(A, A\u207a)` is a sheaf on all opens. The ring need not be complete or Hausdorff.","title":"Buzzard\u2013Verberkmoes for the presheaf of sets"},{"id":"tauceti/algebraic-geometry/adic-space/the-remainder-theorem-in-kt","kind":"node","lean":"TauCeti.ValuationSpectrum.weightedX_sub_weightedC_dvd_sub_weightedC_evalAtHom","parent":1411,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/adic-space/index.html#the-remainder-theorem-in-kt","scope":"tauceti/algebraic-geometry/adic-space","status":"fully_proved","summary":"For `a` in the closed unit disc (power-bounded, that is `\u2016a\u2016 \u2264 1`), `T - a` divides `f - f(a)` for every restricted power series `f` in one variable over a complete nonarchimedean field. This is the analogue of Mathlib's `Polynomial.X_sub_C_dvd_sub_C_eval`.","title":"The remainder theorem in `K\u27e8T\u27e9`"},{"id":"tauceti/algebraic-geometry/adic-space/tates-acyclicity-theorem-for-the-standard-lauren","kind":"node","lean":"TauCeti.ValuationSpectrum.quasiIso_cechAugmentation_closedDisc_spaBasicOpen","parent":1411,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/adic-space/index.html#tates-acyclicity-theorem-for-the-standard-lauren","scope":"tauceti/algebraic-geometry/adic-space","status":"fully_proved","summary":"Let `K` be a complete nonarchimedean field and `a, c \u2208 K` with `c \u2260 0`. The closed unit disc `Spa(K\u27e8T\u27e9, K\u27e8T\u27e9\u00b0)` is covered by the closed subdisc `R({T - a, c} / c) = {|T - a| \u2264 |c|}` and by `R({c} / (T - a)) = {|T - a| \u2265 |c|}`, the Laurent cover generated by `c\u207b\u00b9 (T - a)`. The augmented \u010cech complex of the structure presheaf, as a presheaf of abelian groups, for this cover is exact in every degree.","title":"Tate's acyclicity theorem for the standard Laurent cover of the closed unit disc"},{"data_url":"scopes/tauceti/algebraic-geometry/elliptic-curve.json","id":"tauceti/algebraic-geometry/elliptic-curve","kind":"scope","parent":1303,"result":false,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curve/index.html","scope":"tauceti/algebraic-geometry/elliptic-curve","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.AlgebraicGeometry.EllipticCurve` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Elliptic Curve: other results"},{"data_url":"scopes/tauceti/algebraic-geometry/elliptic-curve/part-01.json","id":"tauceti/algebraic-geometry/elliptic-curve/part-01","kind":"scope","navigation":"transparent","parent":1415,"result":false,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curve/part-01/index.html","scope":"tauceti/algebraic-geometry/elliptic-curve/part-01","status":"fully_proved","title":"The criterion behind the weak\u2026 to The weak Mordell\u2013Weil theorem"},{"id":"tauceti/algebraic-geometry/elliptic-curve/part-01/the-criterion-behind-the-weak-mordell-weil-theor","kind":"node","lean":"WeierstrassCurve.Affine.finiteIndex_range_nsmulAddMonoidHom_two_iff","parent":1416,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curve/part-01/index.html#the-criterion-behind-the-weak-mordell-weil-theor","scope":"tauceti/algebraic-geometry/elliptic-curve/part-01","status":"fully_proved","summary":"`2E(K)` has finite index in `E(K)` exactly when the image of the descent map is finite.","title":"The criterion behind the weak Mordell\u2013Weil theorem"},{"id":"tauceti/algebraic-geometry/elliptic-curve/part-01/delignes-theorem-weierstrasscurve-isintegralmodu","kind":"node","lean":"WeierstrassCurve.IsIntegralModularForm.mem_adjoin","parent":1416,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curve/part-01/index.html#delignes-theorem-weierstrasscurve-isintegralmodu","scope":"tauceti/algebraic-geometry/elliptic-curve/part-01","status":"fully_proved","summary":"Every integral modular form lies in the subring `\u2124[c\u2084, c\u2086, \u0394]`.","title":"Deligne's theorem (`WeierstrassCurve.IsIntegralModularForm.mem_adjoin`)"},{"id":"tauceti/algebraic-geometry/elliptic-curve/part-01/the-factorisation-theorem","kind":"node","lean":"TauCeti.Isogeny.existsUnique_comp_eq_iff_fieldRange_le","parent":1416,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curve/part-01/index.html#the-factorisation-theorem","scope":"tauceti/algebraic-geometry/elliptic-curve/part-01","status":"fully_proved","summary":"The factorisation theorem (Silverman, *The Arithmetic of Elliptic Curves*, III.4.11 in its separable form): an isogeny `\u03c8 : W\u2081 \u2192 W\u2083` factors through an isogeny `\u03c6 : W\u2081 \u2192 W\u2082`, by a unique isogeny `\u03bb : W\u2082 \u2192 W\u2083`, exactly when the function field `\u03c8` pulls back sits inside the one `\u03c6` pulls back.","title":"The factorisation theorem"},{"id":"tauceti/algebraic-geometry/elliptic-curve/part-01/the-factorisation-theorem-for-isogenies-silverma","kind":"node","lean":"TauCeti.Isogeny.existsUnique_comp_iterateRelativeFrobeniusIsogeny_eq","parent":1416,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curve/part-01/index.html#the-factorisation-theorem-for-isogenies-silverma","scope":"tauceti/algebraic-geometry/elliptic-curve/part-01","status":"fully_proved","summary":"when `p ^ r` is the inseparable degree of `\u03c6 : W\u2081 \u2192 W\u2082`, it factors through the `r`-fold relative Frobenius `F^r : W\u2081 \u2192 W\u2081\u207d\u1d56\u02b3\u207e` as `\u03c6 = \u03c7 \u2218 F^r` for a unique isogeny `\u03c7 : W\u2081\u207d\u1d56\u02b3\u207e \u2192 W\u2082`. The factor is separable (`isSeparable_of_comp_iterateRelativeFrobeniusIsogeny_eq`), of degree the separable degree of `\u03c6` (`degree_eq_separableDegree_of_comp_iterateRelativeFrobeniusIsogeny_eq`).","title":"The factorisation theorem for isogenies (Silverman II.2.12)"},{"id":"tauceti/algebraic-geometry/elliptic-curve/part-01/the-galois-cocycle-carried-by-that-isomorphism","kind":"node","lean":"WeierstrassCurve.map_quadraticTwistOfTraceNormVariableChange","parent":1416,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curve/part-01/index.html#the-galois-cocycle-carried-by-that-isomorphism","scope":"tauceti/algebraic-geometry/elliptic-curve/part-01","status":"fully_proved","summary":"Conjugating `quadraticTwistOfTraceNormVariableChange` by the nontrivial `\u03c3 \u2208 Gal(L/K)` changes it by `negVariableChange`, the change of variables `[-1]`: this is the cocycle identity for `H\u00b9(Gal(L/K), Aut E)`.","title":"The Galois cocycle carried by that isomorphism"},{"id":"tauceti/algebraic-geometry/elliptic-curve/part-01/the-hurwitz-genus-formula-for-y2-x3-x-over-k-x-a","kind":"node","lean":"TauCeti.CurveX3SubX.hurwitz_genus_formula","parent":1416,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curve/part-01/index.html#the-hurwitz-genus-formula-for-y2-x3-x-over-k-x-a","scope":"tauceti/algebraic-geometry/elliptic-curve/part-01","status":"fully_proved","summary":"with `e - 1` summed over the four ramified places,","title":"The Hurwitz genus formula for `y\u00b2 = x\u00b3 - x` over `K(x)`, as a numerical identity"},{"id":"tauceti/algebraic-geometry/elliptic-curve/part-01/the-jacobian-criterion-as-two-equalities-the-for","kind":"node","lean":"WeierstrassCurve.Affine.isSingular_iff","parent":1416,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curve/part-01/index.html#the-jacobian-criterion-as-two-equalities-the-for","scope":"tauceti/algebraic-geometry/elliptic-curve/part-01","status":"fully_proved","summary":"Formalized in Tau Ceti as `WeierstrassCurve.Affine.isSingular_iff`.","title":"The Jacobian criterion as two equalities, the form `Nonsingular` negates"},{"id":"tauceti/algebraic-geometry/elliptic-curve/part-01/the-kernel-form-of-the-factorisation-theorem-sil","kind":"node","lean":"TauCeti.Isogeny.existsUnique_comp_eq_iff_ker_le","parent":1416,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curve/part-01/index.html#the-kernel-form-of-the-factorisation-theorem-sil","scope":"tauceti/algebraic-geometry/elliptic-curve/part-01","status":"fully_proved","summary":"When the kernel of `\u03c6 : W\u2081 \u2192 W\u2082` has `deg \u03c6` points, an isogeny `\u03c8 : W\u2081 \u2192 W\u2083` factors through `\u03c6`, by a unique isogeny, exactly when `ker \u03c6 \u2264 ker \u03c8`.","title":"The kernel form of the factorisation theorem (Silverman III.4.11)"},{"id":"tauceti/algebraic-geometry/elliptic-curve/part-01/krauss-global-criterion","kind":"node","lean":"TauCeti.krausGlobalCondition_iff_exists_integralModel","parent":1416,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curve/part-01/index.html#krauss-global-criterion","scope":"tauceti/algebraic-geometry/elliptic-curve/part-01","status":"fully_proved","summary":"Let `O` be a Dedekind domain with fraction field `K`, in which `2` and `3` are invertible, and assume `O` has a height-one prime, as the ring of integers of a number field does. Then the pair `(c\u2084, c\u2086)` is the pair of `c`-invariants of a nonsingular Weierstrass equation with coefficients in `O` exactly when Kraus's local condition holds at every height-one prime.","title":"Kraus's global criterion"},{"id":"tauceti/algebraic-geometry/elliptic-curve/part-01/krauss-local-criterion","kind":"node","lean":"TauCeti.krausLocalCondition_iff_exists_integralModel","parent":1416,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curve/part-01/index.html#krauss-local-criterion","scope":"tauceti/algebraic-geometry/elliptic-curve/part-01","status":"fully_proved","summary":"For a local ring `R` with an algebra map to a field `K` in which `2` and `3` are invertible, the pair `(c\u2084, c\u2086)` is the pair of `c`-invariants of a nonsingular Weierstrass equation with coefficients in `R` exactly when Kraus's local condition holds.","title":"Kraus's local criterion"},{"id":"tauceti/algebraic-geometry/elliptic-curve/part-01/the-modified-szpiro-conjecture-implies-szpiros-c","kind":"node","lean":"TauCeti.AlgorithmicModifiedSzpiroConjecture.algorithmicSzpiroConjecture","parent":1416,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curve/part-01/index.html#the-modified-szpiro-conjecture-implies-szpiros-c","scope":"tauceti/algebraic-geometry/elliptic-curve/part-01","status":"fully_proved","summary":"the relation `1728 \u0394 = c\u2084\u00b3 - c\u2086\u00b2` on the reduced minimal model gives `|\u0394_min| \u2264 max (|c\u2084|\u00b3, c\u2086\u00b2)`.","title":"The modified Szpiro conjecture implies Szpiro's conjecture, with the same constants"},{"id":"tauceti/algebraic-geometry/elliptic-curve/part-01/the-mordell-weil-rank-bound-from-the-2-selmer-gr","kind":"node","lean":"WeierstrassCurve.Affine.pow_rank_le_card_selmerGroup\u2082, WeierstrassCurve.Affine.pow_rank_add_one_le_prod","parent":1416,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curve/part-01/index.html#the-mordell-weil-rank-bound-from-the-2-selmer-gr","scope":"tauceti/algebraic-geometry/elliptic-curve/part-01","status":"fully_proved","summary":"for an elliptic curve over a number field `F`, `2 ^ rank W(F) * #W(F)[2] \u2264 #Sel\u2082(W/F)`. The rank is that of the finitely generated group `W(F)` (`fg_point_of_numberField`), and the right-hand side is finite by `finite_selmerGroup\u2082`.","title":"The Mordell\u2013Weil rank bound from the 2-Selmer group"},{"id":"tauceti/algebraic-geometry/elliptic-curve/part-01/the-nagell-lutz-theorem","kind":"node","lean":"WeierstrassCurve.lutz_nagell, WeierstrassCurve.isInteger_or_order_two_of_torsion_rat, WeierstrassCurve.y_eq_zero_or_sq_dvd_\u0394_of_torsion, WeierstrassCurve.isInteger_of_torsion","parent":1416,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curve/part-01/index.html#the-nagell-lutz-theorem","scope":"tauceti/algebraic-geometry/elliptic-curve/part-01","status":"fully_proved","summary":"Let `A B : \u2124` and let `(x, y)` be a nonzero rational point of finite order on `y\u00b2 = x\u00b3 + Ax + B`. Then `x` and `y` are integers, and either `y = 0` or `y\u00b2 \u2223 \u0394`.","title":"The Nagell\u2013Lutz theorem"},{"id":"tauceti/algebraic-geometry/elliptic-curve/part-01/perfect-local-tate-duality-for-e-m","kind":"node","lean":"WeierstrassCurve.localTateWeilPairing_bijective, TauCeti.Isogeny.localTateKerWeilPairing_bijective","parent":1416,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curve/part-01/index.html#perfect-local-tate-duality-for-e-m","scope":"tauceti/algebraic-geometry/elliptic-curve/part-01","status":"fully_proved","summary":"Pairing with a class gives a bijection from `H\u00b9(K, E[m])` to its additive characters with values in `\u2124/m`.","title":"Perfect local Tate duality for `E[m]`"},{"id":"tauceti/algebraic-geometry/elliptic-curve/part-01/suyamas-exact-ellipticity-criterion","kind":"node","lean":"TauCeti.Montgomery.isElliptic_model_suyama_iff","parent":1416,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curve/part-01/index.html#suyamas-exact-ellipticity-criterion","scope":"tauceti/algebraic-geometry/elliptic-curve/part-01","status":"fully_proved","summary":"Under the denominator restrictions, precisely these five factors detect singular models. No odd characteristic is excluded.","title":"Suyama's exact ellipticity criterion"},{"id":"tauceti/algebraic-geometry/elliptic-curve/part-01/szpiros-conjecture-for-frey-curves","kind":"node","lean":"TauCeti.AlgorithmicSzpiroConjecture.exists_sq_le, TauCeti.AlgorithmicSzpiroConjecture.exists_sq_le_mul_radical_rpow","parent":1416,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curve/part-01/index.html#szpiros-conjecture-for-frey-curves","scope":"tauceti/algebraic-geometry/elliptic-curve/part-01","status":"fully_proved","summary":"Under `TauCeti.AlgorithmicSzpiroConjecture`, for every `\u03b5 > 0` there is a constant `C` such that `(A B (A + B))\u00b2 \u2264 C rad (A B (A + B)) ^ (6 + \u03b5)` for all coprime integers `A` and `B` with `A B (A + B) \u2260 0`. This applies Szpiro's conjecture to the curve of `TauCeti.exists_natAbs_sq_le_absNorm_minimalDiscriminantIdeal`.","title":"Szpiro's conjecture for Frey curves"},{"id":"tauceti/algebraic-geometry/elliptic-curve/part-01/szpiros-conjecture-implies-asymptotic-fermat","kind":"node","lean":"TauCeti.AlgorithmicSzpiroConjecture.asymptoticFermat","parent":1416,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curve/part-01/index.html#szpiros-conjecture-implies-asymptotic-fermat","scope":"tauceti/algebraic-geometry/elliptic-curve/part-01","status":"fully_proved","summary":"If `TauCeti.AlgorithmicSzpiroConjecture` holds, there is an `N` such that `a\u207f + b\u207f = c\u207f` has no solution in positive integers for any exponent `n \u2265 N`. The bound `N` depends on the constant of Szpiro's conjecture for `\u03b5 = 1`, so it is ineffective.","title":"Szpiro's conjecture implies asymptotic Fermat"},{"id":"tauceti/algebraic-geometry/elliptic-curve/part-01/szpiros-conjecture-implies-the-abc-conjecture-wi","kind":"node","lean":"TauCeti.AlgorithmicSzpiroConjecture.abcConjectureWithExponent","parent":1416,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curve/part-01/index.html#szpiros-conjecture-implies-the-abc-conjecture-wi","scope":"tauceti/algebraic-geometry/elliptic-curve/part-01","status":"fully_proved","summary":"under `TauCeti.AlgorithmicSzpiroConjecture`, for every `\u03b5 > 0` there is a constant `C` with `max (|a|, |b|, |c|) \u2264 C rad (a b c) ^ (3 / 2 + \u03b5)` for all coprime `a + b = c`. For coprime positive integers `a + b = c`, Szpiro's bound for Frey curves gives `(a b c)\u00b2 \u226a_\u03b5 rad (a b c) ^ (6 + 4 \u03b5)` (`TauCeti.AlgorithmicSzpiroConjecture.exists_sq_le`), and `c\u00b2 \u2264 2 a b c` because `a + b \u2264 2 a b`.","title":"Szpiro's conjecture implies the abc conjecture with exponent `3 / 2`"},{"id":"tauceti/algebraic-geometry/elliptic-curve/part-01/vietas-formula-at-a-pair-of-parameters-cleared-o","kind":"node","lean":"WeierstrassCurve.formalThirdRootEval_relation","parent":1416,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curve/part-01/index.html#vietas-formula-at-a-pair-of-parameters-cleared-o","scope":"tauceti/algebraic-geometry/elliptic-curve/part-01","status":"fully_proved","summary":"the third root satisfies `(1 + a\u2082\u03bb + a\u2084\u03bb\u00b2 + a\u2086\u03bb\u00b3)(t\u2083 + t\u2081 + t\u2082) = -(a\u2081\u03bb + a\u2082\u03bd + a\u2083\u03bb\u00b2 + 2a\u2084\u03bb\u03bd + 3a\u2086\u03bb\u00b2\u03bd)`.","title":"Vieta's formula at a pair of parameters, cleared of the inverse of the cubic's leading coefficient"},{"id":"tauceti/algebraic-geometry/elliptic-curve/part-01/the-weak-mordell-weil-theorem","kind":"node","lean":"WeierstrassCurve.Affine.finiteIndex_range_nsmulAddMonoidHom_two","parent":1416,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curve/part-01/index.html#the-weak-mordell-weil-theorem","scope":"tauceti/algebraic-geometry/elliptic-curve/part-01","status":"fully_proved","summary":"`E(K)/2E(K)` is finite, for an elliptic curve in the normal form `y\u00b2 = x\u00b3 + a\u2082x\u00b2 + a\u2084x + a\u2086` over the fraction field `K` of a Dedekind domain `R`, provided that for each irreducible factor `p` of the cubic the ring of integers of `K[X] \u29f8 (p)` has finite class group and finitely generated unit group.","title":"The weak Mordell\u2013Weil theorem"},{"data_url":"scopes/tauceti/algebraic-geometry/elliptic-curve/part-02.json","id":"tauceti/algebraic-geometry/elliptic-curve/part-02","kind":"scope","navigation":"transparent","parent":1415,"result":false,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curve/part-02/index.html","scope":"tauceti/algebraic-geometry/elliptic-curve/part-02","status":"fully_proved","title":"The Wronskian identity of an\u2026"},{"id":"tauceti/algebraic-geometry/elliptic-curve/part-02/the-wronskian-identity-of-an-isogeny","kind":"node","lean":"TauCeti.Isogeny.aeval_wronskian_xRatFunc_mul_invariantDifferentialDenom, TauCeti.Isogeny.wronskian_xRatFunc_sq_mul_\u03a8\u2082Sq","parent":1437,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/elliptic-curve/part-02/index.html#the-wronskian-identity-of-an-isogeny","scope":"tauceti/algebraic-geometry/elliptic-curve/part-02","status":"fully_proved","summary":"if `\u03c6^*\u03c9\u2082 = c \u03c9\u2081`, then","title":"The Wronskian identity of an isogeny"},{"data_url":"scopes/tauceti/algebraic-geometry/other-results.json","id":"tauceti/algebraic-geometry/other-results","kind":"scope","parent":1303,"result":false,"route":"../roadmap/tauceti/algebraic-geometry/other-results/index.html","scope":"tauceti/algebraic-geometry/other-results","status":"fully_proved","summary":"Results named in the rest of Tau Ceti's `TauCeti.AlgebraicGeometry` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Other results"},{"id":"tauceti/algebraic-geometry/other-results/algebraic-hartogs-principle","kind":"node","lean":"TauCeti.AlgebraicGeometry.StructureSheaf.toOpen\u2097_bijective_of_isWeaklyRegular","parent":1439,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/other-results/index.html#algebraic-hartogs-principle","scope":"tauceti/algebraic-geometry/other-results","status":"fully_proved","summary":"Let `a, b` be a weakly `M`-regular sequence. If `D(a)` and `D(b)` lie in the open subset `U` of `Spec R`, then `M \u2192 M~(U)` is bijective: every section of `M~` over `U` comes from a unique element of `M`.","title":"Algebraic Hartogs principle"},{"id":"tauceti/algebraic-geometry/other-results/delignes-theorem-tauceti-algebraicgeometry-finit","kind":"node","lean":"TauCeti.AlgebraicGeometry.FiniteFlatCommGroupScheme.pow_eq_one_of_finrank","parent":1439,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/other-results/index.html#delignes-theorem-tauceti-algebraicgeometry-finit","scope":"tauceti/algebraic-geometry/other-results","status":"fully_proved","summary":"Let `G` be a finite locally free commutative group scheme over `S` whose rank is `n` at every point of `S`. Then every point `x : X \u27f6 G` over `S` is killed by `n`: its `n`-th power in the group of `X`-valued points of `G` is the unit.","title":"Deligne's theorem (`TauCeti.AlgebraicGeometry.FiniteFlatCommGroupScheme.pow_eq_one_of_finrank`)"},{"id":"tauceti/algebraic-geometry/other-results/krulls-principal-ideal-theorem-on-a-scheme","kind":"node","lean":"TauCeti.AlgebraicGeometry.Scheme.coheight_eq_one_of_maximal_mem_zeroLocus","parent":1439,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/other-results/index.html#krulls-principal-ideal-theorem-on-a-scheme","scope":"tauceti/algebraic-geometry/other-results","status":"fully_proved","summary":"A maximal point of the zero locus of a nonzero global function on an integral locally Noetherian scheme, that is, the generic point of an irreducible component of that zero locus, has coheight one.","title":"Krull's principal ideal theorem on a scheme"},{"id":"tauceti/algebraic-geometry/other-results/the-riemann-roch-theorem-for-line-bundles-in-ser","kind":"node","lean":"TauCeti.AlgebraicGeometry.InvertibleSheaf.finrank_cohomology_zero_sub_finrank_cohomology_zero_tensor_dual","parent":1439,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/other-results/index.html#the-riemann-roch-theorem-for-line-bundles-in-ser","scope":"tauceti/algebraic-geometry/other-results","status":"fully_proved","summary":"For every line bundle `L` and every representative `\u03c9` of the canonical class,","title":"The Riemann\u2013Roch theorem for line bundles, in Serre-duality form"},{"id":"tauceti/algebraic-geometry/other-results/the-riemann-roch-theorem-on-x-f-in-dimension-for","kind":"node","lean":"TauCeti.AlgebraicGeometry.twoChartModel.finrank_cohomology_zero_sheaf_sub_finrank_cohomology_one_sheaf, TauCeti.AlgebraicGeometry.twoChartModel.eulerCharBelow_sheaf_eq_degree_add_one_sub_genus","parent":1439,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/other-results/index.html#the-riemann-roch-theorem-on-x-f-in-dimension-for","scope":"tauceti/algebraic-geometry/other-results","status":"fully_proved","summary":"when `k` is integrally closed in `F`, `dim_k H\u2070(X_F, \ud835\udcaa(D)) - dim_k H\u00b9(X_F, \ud835\udcaa(D)) = deg D + 1 - g(F)`.","title":"The Riemann\u2013Roch theorem on `X_F` in dimension form"},{"id":"tauceti/algebraic-geometry/other-results/serre-duality-in-degree-zero","kind":"node","lean":"TauCeti.AlgebraicGeometry.InvertibleSheaf.nonempty_cohomologyZeroDualEquivCohomologyOne_tensor_dual, TauCeti.AlgebraicGeometry.InvertibleSheaf.nonempty_cohomologyOneDualEquivCohomologyZero_tensor_dual","parent":1439,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/other-results/index.html#serre-duality-in-degree-zero","scope":"tauceti/algebraic-geometry/other-results","status":"fully_proved","summary":"The dual of `H\u2070(X, L)` is linearly isomorphic to `H\u00b9(X, \u03c9 \u2297 L\u207b\u00b9)` for any representative `\u03c9` of the canonical class.","title":"Serre duality in degree zero"},{"id":"tauceti/algebraic-geometry/other-results/serre-duality-on-x-f-in-dimensions","kind":"node","lean":"TauCeti.AlgebraicGeometry.twoChartModel.finrank_cohomology_one_sheaf_eq_finrank_cohomology_zero_sheaf_sub","parent":1439,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/other-results/index.html#serre-duality-on-x-f-in-dimensions","scope":"tauceti/algebraic-geometry/other-results","status":"fully_proved","summary":"for every divisor `K` on `X_F` whose image in `F` represents `canonicalClass hF hk`, `dim_k H\u00b9(X_F, \ud835\udcaa(D)) = dim_k H\u2070(X_F, \ud835\udcaa(K - D))`.","title":"Serre duality on `X_F`, in dimensions"},{"id":"tauceti/algebraic-geometry/other-results/serre-duality-in-dimensions","kind":"node","lean":"TauCeti.AlgebraicGeometry.SchemeWeilDivisor.finrank_cohomology_one_sheaf_eq_finrank_cohomology_zero_sheaf_sub","parent":1439,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/other-results/index.html#serre-duality-in-dimensions","scope":"tauceti/algebraic-geometry/other-results","status":"fully_proved","summary":"Assume `k` is integrally closed in `k(X)`. For every canonical divisor `K` on the curve `X` (a divisor whose function-field divisor represents the canonical class) and every Weil divisor `D`, `dim_k H\u00b9(X, \ud835\udcaa_X(D)) = dim_k H\u2070(X, \ud835\udcaa_X(K - D))`.","title":"Serre duality, in dimensions"},{"id":"tauceti/algebraic-geometry/other-results/serres-vanishing-theorem-on-affine-opens","kind":"node","lean":"AlgebraicGeometry.Scheme.Modules.subsingleton_cohomologyOn_succ_of_isAffineOpen, AlgebraicGeometry.Scheme.Modules.subsingleton_cohomology_succ_of_isAffine","parent":1439,"result":true,"route":"../roadmap/tauceti/algebraic-geometry/other-results/index.html#serres-vanishing-theorem-on-affine-opens","scope":"tauceti/algebraic-geometry/other-results","status":"fully_proved","summary":"Serre's vanishing theorem on affine opens (Hartshorne, *Algebraic Geometry*, Theorem III.3.5, without the Noetherian hypothesis): a sheaf of modules that is quasi-coherent over an affine open subset `U` of a scheme (for example, a quasi-coherent sheaf on the scheme) has no cohomology in positive degrees on `U`.","title":"Serre's vanishing theorem on affine opens"},{"data_url":"scopes/tauceti/algebraic-topology.json","id":"tauceti/algebraic-topology","kind":"scope","parent":1095,"result":false,"route":"../roadmap/tauceti/algebraic-topology/index.html","scope":"tauceti/algebraic-topology","status":"fully_proved","summary":"Results in Tau Ceti's `TauCeti.AlgebraicTopology` modules, and the roadmaps whose articles are mostly there: by roadmap, then by area.","title":"Algebraic Topology"},{"data_url":"scopes/tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds.json","id":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds","kind":"scope","parent":1449,"result":false,"route":"../roadmap/tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds/index.html","scope":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Algebraic topology of spaces and manifolds"},{"id":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds/the-groupoid-van-kampen-theorem","kind":"node","lean":"TauCeti.FundamentalGroupoid.isColimitCechCocone","parent":1450,"result":true,"route":"../roadmap/tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds/index.html#the-groupoid-van-kampen-theorem","scope":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds","status":"fully_proved","summary":"For an open cover of `X`, the fundamental groupoid of `X` is the colimit of the fundamental groupoids of the nonempty finite intersections of members of the cover.","title":"The groupoid van Kampen theorem"},{"id":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds/excision-of-a-set-with-closure-in-the-interior-o","kind":"node","lean":"TopPair.isIso_singularHomologyMap_excisionMap_compl","parent":1450,"result":true,"route":"../roadmap/tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds/index.html#excision-of-a-set-with-closure-in-the-interior-o","scope":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds","status":"fully_proved","summary":"If `closure Z \u2286 interior B`, the inclusion of pairs `(X \u2216 Z, B \u2216 Z) \u27f6 (X, B)` induces isomorphisms on relative singular homology.","title":"Excision of a set with closure in the interior of the subspace"},{"id":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds/the-hurewicz-theorem","kind":"node","lean":"HomotopyGroup.bijective_hurewicz, TauCeti.HurewiczOne.hurewiczAddEquiv, TauCeti.HurewiczOne.hurewiczHomologyAddEquiv","parent":1450,"result":true,"route":"../roadmap/tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds/index.html#the-hurewicz-theorem","scope":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds","status":"fully_proved","summary":"If `\u03c0\u2096(X, x)` is trivial for every `k < n + 2`, then the Hurewicz map `\u03c0\u2099\u208a\u2082(X, x) \u2192 H\u2099\u208a\u2082(X; \u2124)`, `[p] \u21a6 p_*[I\u207f\u207a\u00b2]`, is bijective.","title":"The Hurewicz theorem"},{"id":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds/the-relative-hurewicz-isomorphism-for-highly-con","kind":"node","lean":"TauCeti.RelHomotopyGroup.bijective_hurewicz_of_subsingleton_homotopyGroup","parent":1450,"result":true,"route":"../roadmap/tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds/index.html#the-relative-hurewicz-isomorphism-for-highly-con","scope":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds","status":"fully_proved","summary":"Let `B` be a subspace of `Y` and `b \u2208 B`. If `\u03c0\u2096(B, b)` and `\u03c0\u2096(Y, b)` are trivial for every `k < n + 2`, then the relative Hurewicz map `\u03c0\u2099\u208a\u2082(Y, B, b) \u2192 H\u2099\u208a\u2082(Y, B; \u2124)`, `[p] \u21a6 p_*[I\u207f\u207a\u00b2]`, is bijective. In degree two, that is for `n = 0`, the hypotheses say that `B` and `Y` are simply connected.","title":"The relative Hurewicz isomorphism for highly connected spaces"},{"id":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds/whiteheads-theorem-hatcher-theorem-4-5","kind":"node","lean":"ContinuousMap.IsWeakHomotopyEquivalence.homotopyEquiv","parent":1450,"result":true,"route":"../roadmap/tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds/index.html#whiteheads-theorem-hatcher-theorem-4-5","scope":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds","status":"fully_proved","summary":"A weak homotopy equivalence `f` between spaces of CW type is a homotopy equivalence. This is the homotopy equivalence whose forward map is `f` itself (`ContinuousMap.IsWeakHomotopyEquivalence.homotopyEquiv_toFun`); its homotopy inverse is a lift of the identity through `f` given by `ContinuousMap.IsWeakHomotopyEquivalence.exists_comp_homotopic`.","title":"Whitehead's theorem (Hatcher, Theorem 4.5)"},{"id":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds/the-kunneth-short-exact-sequence-over-a-heredita","kind":"node","lean":"TopCat.singularHomologyKunnethUnitSplitting","parent":1450,"result":true,"route":"../roadmap/tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds/index.html#the-kunneth-short-exact-sequence-over-a-heredita","scope":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds","status":"fully_proved","summary":"The K\u00fcnneth short exact sequence over a hereditary commutative ring splits, so that `H\u2098\u208a\u2081(X \u00d7 Y; k)` is the direct sum of `\u2a01_{p + q = m + 1} H\u209a(X; k) \u2297 H_q(Y; k)` and `\u2a01_{p + q = m} Tor\u2081(H\u209a(X; k), H_q(Y; k))` (`CategoryTheory.ShortComplex.Splitting.isoBinaryBiproduct`). The splitting depends on chosen splittings of the cycles of the singular chain complexes, and need not be natural in `X` and `Y`.","title":"The K\u00fcnneth short exact sequence over a hereditary commutative ring splits"},{"id":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds/the-splitting-of-the-universal-coefficient-seque","kind":"node","lean":"TopCat.singularUniversalCoefficientEquiv","parent":1450,"result":true,"route":"../roadmap/tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds/index.html#the-splitting-of-the-universal-coefficient-seque","scope":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds","status":"fully_proved","summary":"a `k`-linear equivalence `H\u207f\u207a\u00b9(X; R, M) \u2243\u2097[k] Ext\u00b9(H\u2099(X; R), M) \u00d7 Hom(H\u2099\u208a\u2081(X; R), M)` whose second component is the Kronecker map (`TopCat.singularUniversalCoefficientEquiv_apply_snd`) and whose inverse restricts to `TopCat.singularExtToCohomology` on the first factor (`TopCat.singularUniversalCoefficientEquiv_symm_inl`). It is built from the section `TopCat.singularKroneckerSection` and is not natural in `X`.","title":"The splitting of the universal coefficient sequence of singular cohomology over a hereditary commutative ring"},{"id":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds/a-classical-relative-cw-complex-in-a-hausdorff-s","kind":"node","lean":"TauCeti.toRelativeCWComplex","parent":1450,"result":true,"route":"../roadmap/tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds/index.html#a-classical-relative-cw-complex-in-a-hausdorff-s","scope":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds","status":"fully_proved","summary":"The inclusion of the base `D` into `C` is the transfinite composition of the inclusions of the skeleta `skeletonLT C n`, and the step from `skeletonLT C n` to `skeletonLT C (n + 1)` attaches the `n`-cells of `C` along their attaching maps. The body is exposed so that the `n`-th space is the skeleton `skeletonObj C n` and the cells attached at step `n` are `cell C n` by definition.","title":"A classical relative CW complex in a Hausdorff space is a categorical relative CW complex"},{"id":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds/the-top-homology-of-a-compact-manifold-is-the-mo","kind":"node","lean":"TauCeti.isLimitLocalHomologySystemCone","parent":1450,"result":true,"route":"../roadmap/tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds/index.html#the-top-homology-of-a-compact-manifold-is-the-mo","scope":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds","status":"fully_proved","summary":"For a compact Hausdorff space modelled on a real normed space of finite dimension `n`, the restrictions `H\u2099(M, M \u2216 M; N) \u27f6 H\u2099(M, M \u2216 {x}; N)` exhibit `H\u2099(M, M \u2216 M; N)` as the limit of the local homology system: every transport-invariant family of local classes is the family of restrictions of exactly one class.","title":"The top homology of a compact manifold is the module of parallel families of local classes"},{"id":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds/exactness-of-the-milnor-sequence-at-hj-lim-c","kind":"node","lean":"TauCeti.ComplexTower.exact_limOneToHomology_homologyToLim","parent":1450,"result":true,"route":"../roadmap/tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds/index.html#exactness-of-the-milnor-sequence-at-hj-lim-c","scope":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds","status":"fully_proved","summary":"For a tower whose shift is surjective in every degree, the image of `lim\u00b9 H\u1d62(C\u2099) \u2192 H\u2c7c(lim C)` is the kernel of `H\u2c7c(lim C) \u2192 lim H\u2c7c(C\u2099)`.","title":"Exactness of the Milnor sequence at `H\u2c7c(lim C)`"},{"id":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds/inductive-hurewicz-vanishing-criterion-singular","kind":"node","lean":"TauCeti.subsingleton_homotopyGroup_of_isZero_homology_singularChains, TauCeti.subsingleton_homotopyGroup_of_isZero_homology_normalizedCubicalChainComplex","parent":1450,"result":true,"route":"../roadmap/tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds/index.html#inductive-hurewicz-vanishing-criterion-singular","scope":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds","status":"fully_proved","summary":"If `X` is path connected with trivial `\u03c0_ 1 X x` and its integral singular homology vanishes in every degree `n + 2`, then all homotopy groups `\u03c0_ k X x` are trivial.","title":"Inductive Hurewicz vanishing criterion (singular homology)"},{"id":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds/the-relative-universal-coefficient-theorem-above","kind":"node","lean":"TopPair.singularKronecker_bijective_succ_of_projective_homology","parent":1450,"result":true,"route":"../roadmap/tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds/index.html#the-relative-universal-coefficient-theorem-above","scope":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds","status":"fully_proved","summary":"for a projective coefficient module `R`, if `H\u2099(X, A; R)` is projective, the relative Kronecker map `H\u207f\u207a\u00b9(X, A; R, M) \u2192\u2097[k] (H\u2099\u208a\u2081(X, A; R) \u27f6 M)` is bijective, for every module `M`. Nothing is assumed about the relative homology in the other degrees.","title":"The relative universal coefficient theorem above a projective homology module, over a hereditary commutative ring"},{"id":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds/the-relative-universal-coefficient-theorem-below","kind":"node","lean":"TopPair.singularKronecker_bijective_of_projective_homology_lt","parent":1450,"result":true,"route":"../roadmap/tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds/index.html#the-relative-universal-coefficient-theorem-below","scope":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds","status":"fully_proved","summary":"for a projective coefficient module `R`, if the relative singular homology modules `H\u2c7c(X, A; R)` of a pair are projective for `j < n`, for instance zero, the relative Kronecker map `H\u207f(X, A; R, M) \u2192\u2097[k] (H\u2099(X, A; R) \u27f6 M)` is bijective, for every module `M`.","title":"The relative universal coefficient theorem below the first non-projective homology, over any commutative ring `k`"},{"id":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds/the-universal-coefficient-theorem-above-a-projec","kind":"node","lean":"TopCat.singularKronecker_bijective_succ_of_projective_homology","parent":1450,"result":true,"route":"../roadmap/tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds/index.html#the-universal-coefficient-theorem-above-a-projec","scope":"tauceti/algebraic-topology/algebraic-topology-of-spaces-and-manifolds","status":"fully_proved","summary":"for a projective coefficient module `R`, if `H\u2099(X; R)` is projective, the Kronecker map `H\u207f\u207a\u00b9(X; R, M) \u2192\u2097[k] (H\u2099\u208a\u2081(X; R) \u27f6 M)` is bijective, for every module `M`. The `Ext\u00b9(H\u2099(X; R), M)` term of the universal coefficient sequence vanishes.","title":"The universal coefficient theorem above a projective homology module, over a hereditary commutative ring"},{"data_url":"scopes/tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove.json","id":"tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove","kind":"scope","parent":1449,"result":false,"route":"../roadmap/tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove/index.html","scope":"tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Belyi maps, dessins d'enfants, and three-point covers"},{"id":"tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove/analytic-belyi-pairs-are-classified-by-finite-co","kind":"node","lean":"TauCeti.AnalyticBelyiPair.isoClassEquivConnectedCoverClass","parent":1465,"result":true,"route":"../roadmap/tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove/index.html#analytic-belyi-pairs-are-classified-by-finite-co","scope":"tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove","status":"fully_proved","summary":"Restricting the Belyi map over `\u2102 \u2216 {0, 1}` and filling the punctures are inverse bijections between isomorphism classes of analytic Belyi pairs of degree `n` and isomorphism classes of connected degree-`n` covers of `\u2102 \u2216 {0, 1}`.","title":"Analytic Belyi pairs are classified by finite covers of the thrice-punctured sphere"},{"id":"tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove/connected-covers-of-c-0-1-of-degree-n-up-to-isom","kind":"node","lean":"TauCeti.ConnectedCoverClass.isoClassEquiv","parent":1465,"result":true,"route":"../roadmap/tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove/index.html#connected-covers-of-c-0-1-of-degree-n-up-to-isom","scope":"tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.ConnectedCoverClass.isoClassEquiv`.","title":"Connected covers of `\u2102 \u2216 {0, 1}` of degree `n` up to isomorphism are classified by the isomorphism classes of connected triples of degree `n`"},{"id":"tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove/polynomial-belyification-of-algebraic-numbers","kind":"node","lean":"TauCeti.exists_polynomial_aeval_mem_criticalValues_subset","parent":1465,"result":true,"route":"../roadmap/tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove/index.html#polynomial-belyification-of-algebraic-numbers","scope":"tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove","status":"fully_proved","summary":"Every finite set `S` of elements algebraic over `\u211a` in a field `K`, such as a finite subset of `AlgebraicClosure \u211a`, is carried into `{0, 1}` by a nonconstant polynomial `q` with rational coefficients whose critical values, in every field of characteristic zero, lie in `{0, 1}`.","title":"Polynomial Belyification of algebraic numbers"},{"id":"tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove/belyis-degree-reduction","kind":"node","lean":"TauCeti.exists_polynomial_aeval_mem_range","parent":1465,"result":true,"route":"../roadmap/tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove/index.html#belyis-degree-reduction","scope":"tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove","status":"fully_proved","summary":"Every finite set `S` of elements algebraic over a perfect field `F`, such as a finite subset of `AlgebraicClosure \u211a`, is carried into `F` by a polynomial `q \u2208 F[X]` with `q\u2032 \u2260 0`, in particular nonconstant, whose critical values in every field extension of `F` also lie in `F`.","title":"Belyi's degree reduction"},{"id":"tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove/the-superelliptic-curve-is-the-analytic-belyi-pa","kind":"node","lean":"TauCeti.AnalyticBelyiPair.nonempty_iso_superelliptic_iff","parent":1465,"result":true,"route":"../roadmap/tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove/index.html#the-superelliptic-curve-is-the-analytic-belyi-pa","scope":"tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove","status":"fully_proved","summary":"an analytic Belyi pair is isomorphic to the superelliptic curve exactly when its monodromy triple is isomorphic to `TauCeti.PermutationTriple.torusTriple`, for any numbering of its fibre over `1/2`.","title":"The superelliptic curve is the analytic Belyi pair of the torus triple"},{"id":"tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove/tauceti-riemannsurface-meromorphicfunction","kind":"node","lean":"TauCeti.RiemannSurface.MeromorphicFunction","parent":1465,"result":true,"route":"../roadmap/tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove/index.html#tauceti-riemannsurface-meromorphicfunction","scope":"tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove","status":"fully_proved","summary":"A meromorphic function on a Riemann surface `X`: a holomorphic map `X \u2192 OnePoint \u2102` to the Riemann sphere which is not the constant map `\u221e`. On a connected `X` these form a field (`TauCeti.RiemannSurface.MeromorphicFunction.instField`).","title":"TauCeti.RiemannSurface.MeromorphicFunction"},{"id":"tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove/tauceti-isalgebraicbelyipair","kind":"node","lean":"TauCeti.IsAlgebraicBelyiPair","parent":1465,"result":true,"route":"../roadmap/tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove/index.html#tauceti-isalgebraicbelyipair","scope":"tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove","status":"fully_proved","summary":"An algebraic Belyi pair over a field `k` of characteristic zero: a function field `F / k` with exact constant field `k`, finite over the rational function field `k(t)` through the given `Algebra (RatFunc k) F` structure, in which every place lying over a place of `k(t)` other than `t = 0`, `t = 1` and `t = \u221e` has ramification index `1`.","title":"TauCeti.IsAlgebraicBelyiPair"},{"id":"tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove/the-galois-conjugate-of-an-algebraic-belyi-pair","kind":"node","lean":"TauCeti.IsAlgebraicBelyiPair.of_mapRingEquiv","parent":1465,"result":true,"route":"../roadmap/tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove/index.html#the-galois-conjugate-of-an-algebraic-belyi-pair","scope":"tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove","status":"fully_proved","summary":"if `F` is an algebraic Belyi pair over `k`, and `\u03c4 : F \u2243+* F'` carries `k(t)` onto `k'(t)` by the coefficient map of `\u03c3 : k \u2243+* k'`, then `F'` is an algebraic Belyi pair over `k'`.","title":"The Galois conjugate of an algebraic Belyi pair is an algebraic Belyi pair"},{"id":"tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove/tauceti-permutationtriple-lmfdb-passportrecord-i","kind":"node","lean":"TauCeti.PermutationTriple.LMFDB.PassportRecord.IsValidAt","parent":1465,"result":true,"route":"../roadmap/tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove/index.html#tauceti-permutationtriple-lmfdb-passportrecord-i","scope":"tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove","status":"fully_proved","summary":"The record `r` is a valid row for the passport of the connected triple `t`: the triple `t` is one of the stored triples, the certificate data of `r` are correct on the sheets of `t`, and every other certified field is the corresponding invariant of `t`.","title":"TauCeti.PermutationTriple.LMFDB.PassportRecord.IsValidAt"},{"id":"tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove/tauceti-permutationtriple-lmfdb-belyirecord-isva","kind":"node","lean":"TauCeti.PermutationTriple.LMFDB.BelyiRecord.IsValidFor","parent":1465,"result":true,"route":"../roadmap/tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove/index.html#tauceti-permutationtriple-lmfdb-belyirecord-isva","scope":"tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove","status":"fully_proved","summary":"A row is valid for the passport `P` when its stored degree is the degree, its stored orbit size is its number of stored embedding entries, it stores at least one triple, every stored triple denotes a connected triple with passport `P`, its stored cycle notation writes the stored triples position by position, the stored partitions are those of `P`, and the stored genus is the genus of every stored triple.","title":"TauCeti.PermutationTriple.LMFDB.BelyiRecord.IsValidFor"},{"id":"tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove/belyis-theorem-existence-direction","kind":"node","lean":"TauCeti.IsFunctionField.exists_isAlgebraicBelyiPair, TauCeti.IsFunctionField.exists_polynomial_isAlgebraicBelyiPair","parent":1465,"result":true,"route":"../roadmap/tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove/index.html#belyis-theorem-existence-direction","scope":"tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove","status":"fully_proved","summary":"over a field `k` algebraic over `\u211a`, such as an algebraic closure of `\u211a` or a number field, every function field `F / k` with exact constant field `k` admits a Belyi map: there is `\u03b2 \u2208 F` transcendental over `k` such that `F` over `k(t)`, with `t` acting as `\u03b2`, is an algebraic Belyi pair.","title":"Belyi's theorem, existence direction"},{"id":"tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove/the-riemann-hurwitz-formula-for-a-filled-cover","kind":"node","lean":"TauCeti.FilledCover.two_mul_genus_sub_two","parent":1465,"result":true,"route":"../roadmap/tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove/index.html#the-riemann-hurwitz-formula-for-a-filled-cover","scope":"tauceti/algebraic-topology/belyi-maps-dessins-denfants-and-three-point-cove","status":"fully_proved","summary":"For a connected cover of degree `n`, whose monodromy triple has genus `g`, the total ramification `\u2211 (e - 1)` over the added points satisfies `2 g - 2 = -2 n + \u2211 (e - 1)`. This is the shape of the Riemann\u2013Hurwitz formula for a degree-`n` branched cover of the Riemann sphere, which has genus `0`.","title":"The Riemann\u2013Hurwitz formula for a filled cover"},{"data_url":"scopes/tauceti/algebraic-topology/classifying-spaces-of-discrete-groups.json","id":"tauceti/algebraic-topology/classifying-spaces-of-discrete-groups","kind":"scope","parent":1449,"result":false,"route":"../roadmap/tauceti/algebraic-topology/classifying-spaces-of-discrete-groups/index.html","scope":"tauceti/algebraic-topology/classifying-spaces-of-discrete-groups","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Classifying spaces of discrete groups"},{"id":"tauceti/algebraic-topology/classifying-spaces-of-discrete-groups/uniqueness-of-aspherical-spaces-of-cw-type","kind":"node","lean":"TauCeti.IsAspherical.exists_homotopyEquiv","parent":1478,"result":true,"route":"../roadmap/tauceti/algebraic-topology/classifying-spaces-of-discrete-groups/index.html#uniqueness-of-aspherical-spaces-of-cw-type","scope":"tauceti/algebraic-topology/classifying-spaces-of-discrete-groups","status":"fully_proved","summary":"Two aspherical spaces of CW type are homotopy equivalent by a homotopy equivalence realizing any prescribed isomorphism `\u03c6` of their fundamental groups: there is a homotopy equivalence `e` and a path `\u03b3` from `e x` to `y` such that the map induced by `e` on fundamental groups, followed by basepoint change along `\u03b3`, is `\u03c6`.","title":"Uniqueness of aspherical spaces of CW type"},{"id":"tauceti/algebraic-topology/classifying-spaces-of-discrete-groups/the-cellular-and-the-simplicial-classifying-spac","kind":"node","lean":"TauCeti.cellularHomotopyEquivClassifyingSpace","parent":1478,"result":true,"route":"../roadmap/tauceti/algebraic-topology/classifying-spaces-of-discrete-groups/index.html#the-cellular-and-the-simplicial-classifying-spac","scope":"tauceti/algebraic-topology/classifying-spaces-of-discrete-groups","status":"fully_proved","summary":"For a presentation `e : PresentedGroup R \u2243* G` of `G`, this is a homotopy equivalence from the cellular classifying space `cellularClassifyingSpace R` to Milnor's classifying space `classifyingSpace G`. Its forward map is pointed, induces `e` under the identifications of both fundamental groups (`TauCeti.classifyingSpaceFundamentalGroupMulEquiv_mapOfEq_cellularHomotopyEquivClassifyingSpace`), and is unique up to\u2026","title":"The cellular and the simplicial classifying spaces agree"},{"id":"tauceti/algebraic-topology/classifying-spaces-of-discrete-groups/every-group-has-a-k-g-1","kind":"node","lean":"TauCeti.exists_isEilenbergMacLaneSpaceOne","parent":1478,"result":true,"route":"../roadmap/tauceti/algebraic-topology/classifying-spaces-of-discrete-groups/index.html#every-group-has-a-k-g-1","scope":"tauceti/algebraic-topology/classifying-spaces-of-discrete-groups","status":"fully_proved","summary":"Every group has a `K(G, 1)`, in its own universe, satisfying the standing hypotheses of covering-space theory: local path-connectedness and semilocal simple connectivity. No hypothesis on the group is needed.","title":"Every group has a `K(G, 1)`"},{"id":"tauceti/algebraic-topology/classifying-spaces-of-discrete-groups/the-singular-homology-of-the-classifying-space-i","kind":"node","lean":"TauCeti.classifyingSpaceSingularHomologyIsoGroupHomology","parent":1478,"result":true,"route":"../roadmap/tauceti/algebraic-topology/classifying-spaces-of-discrete-groups/index.html#the-singular-homology-of-the-classifying-space-i","scope":"tauceti/algebraic-topology/classifying-spaces-of-discrete-groups","status":"fully_proved","summary":"for every group `G`, the singular homology `H\u2099(BG; k)` of Milnor's classifying space with coefficients in a commutative ring `k` is Mathlib's group homology `H\u2099(G, k)` of the trivial representation. It comes from the free resolution of `k` by the singular chains of `universalGSpace G`.","title":"The singular homology of the classifying space is group homology"},{"id":"tauceti/algebraic-topology/classifying-spaces-of-discrete-groups/the-singular-cohomology-of-the-classifying-space","kind":"node","lean":"TauCeti.classifyingSpaceSingularCohomologyIsoGroupCohomology","parent":1478,"result":true,"route":"../roadmap/tauceti/algebraic-topology/classifying-spaces-of-discrete-groups/index.html#the-singular-cohomology-of-the-classifying-space","scope":"tauceti/algebraic-topology/classifying-spaces-of-discrete-groups","status":"fully_proved","summary":"for every group `G`, the singular cohomology `H\u207f(BG; M)` of Milnor's classifying space with coefficients in a module `M` over a commutative ring `k` is Mathlib's group cohomology `H\u207f(G, M)` of the trivial representation `M`. It comes from the free resolution of `k` by the singular chains of `universalGSpace G`.","title":"The singular cohomology of the classifying space is group cohomology"},{"id":"tauceti/algebraic-topology/classifying-spaces-of-discrete-groups/milnors-join-resolution-is-homotopy-equivalent-t","kind":"node","lean":"TauCeti.universalGSpaceStandardHomotopyEquiv","parent":1478,"result":true,"route":"../roadmap/tauceti/algebraic-topology/classifying-spaces-of-discrete-groups/index.html#milnors-join-resolution-is-homotopy-equivalent-t","scope":"tauceti/algebraic-topology/classifying-spaces-of-discrete-groups","status":"fully_proved","summary":"Milnor's join resolution is homotopy equivalent to the standard resolution, by a homotopy equivalence whose forward map is the explicit join-to-standard chain map. Its homotopy inverse is the lift of the identity of the trivial representation from the standard resolution to the join resolution.","title":"Milnor's join resolution is homotopy equivalent to the standard resolution"},{"id":"tauceti/algebraic-topology/classifying-spaces-of-discrete-groups/the-join-to-standard-comparison-is-natural-in-th","kind":"node","lean":"MonoidHom.universalGSpaceChainComplexToStandardComplex_naturality","parent":1478,"result":true,"route":"../roadmap/tauceti/algebraic-topology/classifying-spaces-of-discrete-groups/index.html#the-join-to-standard-comparison-is-natural-in-th","scope":"tauceti/algebraic-topology/classifying-spaces-of-discrete-groups","status":"fully_proved","summary":"Relabelling simplices before forgetting their indices agrees with first forgetting indices and then relabelling the tuples. This is an equality of chain maps in `Rep k G`, with target actions restricted along `\u03c6`.","title":"The join-to-standard comparison is natural in the group"},{"id":"tauceti/algebraic-topology/classifying-spaces-of-discrete-groups/the-cellular-classifying-space-is-a-cw-complex","kind":"node","lean":"TauCeti.instCWComplexCellularClassifyingSpace","parent":1478,"result":true,"route":"../roadmap/tauceti/algebraic-topology/classifying-spaces-of-discrete-groups/index.html#the-cellular-classifying-space-is-a-cw-complex","scope":"tauceti/algebraic-topology/classifying-spaces-of-discrete-groups","status":"fully_proved","summary":"Its cells are those of the presentation complex, one `0`-cell, one `1`-cell for each generator and one `2`-cell for each relator, together with the cells of dimension at least `3` attached to kill the higher homotopy groups (`TauCeti.killHomotopyGroupsCWComplexCellEquiv`).","title":"The cellular classifying space is a CW complex"},{"id":"tauceti/algebraic-topology/classifying-spaces-of-discrete-groups/a-cw-structure-lifts-along-a-covering-map","kind":"node","lean":"TauCeti.cwComplexPreimage","parent":1478,"result":true,"route":"../roadmap/tauceti/algebraic-topology/classifying-spaces-of-discrete-groups/index.html#a-cw-structure-lifts-along-a-covering-map","scope":"tauceti/algebraic-topology/classifying-spaces-of-discrete-groups","status":"fully_proved","summary":"If `C` is a CW complex in the Hausdorff space `B` and `p : E \u2192 B` is a covering map, then `p \u207b\u00b9' C` is a CW complex in `E`. Its `n`-cells are the pairs of an `n`-cell of `C` and a point of `E` over the centre of that cell (`TauCeti.cellPreimageEquiv`); the characteristic map of such a pair is the lift of the characteristic map of the cell through `p` sending the centre to the chosen point.","title":"A CW structure lifts along a covering map"},{"id":"tauceti/algebraic-topology/classifying-spaces-of-discrete-groups/the-orbit-space-of-a-free-cellular-action-is-a-c","kind":"node","lean":"IsQuotientCoveringMap.cwComplexQuotient","parent":1478,"result":true,"route":"../roadmap/tauceti/algebraic-topology/classifying-spaces-of-discrete-groups/index.html#the-orbit-space-of-a-free-cellular-action-is-a-c","scope":"tauceti/algebraic-topology/classifying-spaces-of-discrete-groups","status":"fully_proved","summary":"Let `p : E \u2192 B` be the quotient covering map of an action of `G` on the Hausdorff CW complex `E` that carries every open `n`-cell onto an open `n`-cell, and carries an open cell onto itself only by `1`. Then `B` is a CW complex whose `n`-cells are the `G`-orbits of `n`-cells of `E` (`IsQuotientCoveringMap.cellQuotientEquiv`). The characteristic map of an orbit is `p \u2218 \u03c6`, for `\u03c6` the characteristic map of the\u2026","title":"The orbit space of a free cellular action is a CW complex"},{"id":"tauceti/algebraic-topology/classifying-spaces-of-discrete-groups/the-cellular-chains-of-a-contractible-free-cover","kind":"node","lean":"TauCeti.cellularResolution","parent":1478,"result":true,"route":"../roadmap/tauceti/algebraic-topology/classifying-spaces-of-discrete-groups/index.html#the-cellular-chains-of-a-contractible-free-cover","scope":"tauceti/algebraic-topology/classifying-spaces-of-discrete-groups","status":"fully_proved","summary":"if the lift `p \u207b\u00b9' C` of the CW complex `C` is contractible, its cellular chain complex, on which `G` acts through translation of the lifted cells, is a complex of free representations whose augmentation is a quasi-isomorphism.","title":"The cellular chains of a contractible free cover are a projective resolution of the trivial representation of `G` over `R`"},{"id":"tauceti/algebraic-topology/classifying-spaces-of-discrete-groups/the-singular-chains-of-a-contractible-free-g-spa","kind":"node","lean":"TauCeti.singularChainsResolution","parent":1478,"result":true,"route":"../roadmap/tauceti/algebraic-topology/classifying-spaces-of-discrete-groups/index.html#the-singular-chains-of-a-contractible-free-g-spa","scope":"tauceti/algebraic-topology/classifying-spaces-of-discrete-groups","status":"fully_proved","summary":"`G` acts freely on singular simplices, so the terms are free over the group ring, and the augmented singular chain complex of a contractible space is exact.","title":"The singular chains of a contractible free `G`-space are a projective resolution of the trivial representation of `G` over `k`"},{"data_url":"scopes/tauceti/algebraic-topology/enhanced-derived-categories-of-sheaves.json","id":"tauceti/algebraic-topology/enhanced-derived-categories-of-sheaves","kind":"scope","parent":1449,"result":false,"route":"../roadmap/tauceti/algebraic-topology/enhanced-derived-categories-of-sheaves/index.html","scope":"tauceti/algebraic-topology/enhanced-derived-categories-of-sheaves","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Enhanced derived categories of sheaves"},{"id":"tauceti/algebraic-topology/enhanced-derived-categories-of-sheaves/the-lax-monoidal-structure-of-the-dold-kan-funct","kind":"node","lean":"CategoryTheory.Idempotents.DoldKan.laxMonoidal\u0393","parent":1491,"result":true,"route":"../roadmap/tauceti/algebraic-topology/enhanced-derived-categories-of-sheaves/index.html#the-lax-monoidal-structure-of-the-dold-kan-funct","scope":"tauceti/algebraic-topology/enhanced-derived-categories-of-sheaves","status":"fully_proved","summary":"it is the mate, along the Dold\u2013Kan equivalence, of the Alexander\u2013Whitney oplax monoidal structure of the normalized chain functor `N` (`CategoryTheory.Idempotents.DoldKan.\u0393_\u03b5` and `CategoryTheory.Idempotents.DoldKan.\u0393_\u03bc`).","title":"The lax monoidal structure of the Dold\u2013Kan functor `\u0393` from chain complexes to simplicial objects"},{"id":"tauceti/algebraic-topology/enhanced-derived-categories-of-sheaves/coassociativity-of-the-alexander-whitney-map","kind":"node","lean":"CategoryTheory.SimplicialObject.alexanderWhitney_coassoc","parent":1491,"result":true,"route":"../roadmap/tauceti/algebraic-topology/enhanced-derived-categories-of-sheaves/index.html#coassociativity-of-the-alexander-whitney-map","scope":"tauceti/algebraic-topology/enhanced-derived-categories-of-sheaves","status":"fully_proved","summary":"the two ways of splitting a simplex into a front, a middle and a back face agree, up to the associators of simplicial objects and of chain complexes.","title":"Coassociativity of the Alexander\u2013Whitney map"},{"id":"tauceti/algebraic-topology/enhanced-derived-categories-of-sheaves/frobenius-root-ideals-are-radicals","kind":"node","lean":"TauCeti.frobeniusRootIdeal_eq_radical","parent":1491,"result":true,"route":"../roadmap/tauceti/algebraic-topology/enhanced-derived-categories-of-sheaves/index.html#frobenius-root-ideals-are-radicals","scope":"tauceti/algebraic-topology/enhanced-derived-categories-of-sheaves","status":"fully_proved","summary":"In a perfect ring, the Frobenius-root ideal of `f` is the radical of the principal ideal `(f)`.","title":"Frobenius-root ideals are radicals"},{"id":"tauceti/algebraic-topology/enhanced-derived-categories-of-sheaves/frobenius-root-ideals-are-flat-also-when-r-has-z","kind":"node","lean":"TauCeti.flat_frobeniusRootIdeal","parent":1491,"result":true,"route":"../roadmap/tauceti/algebraic-topology/enhanced-derived-categories-of-sheaves/index.html#frobenius-root-ideals-are-flat-also-when-r-has-z","scope":"tauceti/algebraic-topology/enhanced-derived-categories-of-sheaves","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.flat_frobeniusRootIdeal`.","title":"Frobenius-root ideals are flat, also when `R` has zero divisors"},{"id":"tauceti/algebraic-topology/enhanced-derived-categories-of-sheaves/the-multiplication-tower-is-pro-zero-exactly-for","kind":"node","lean":"TauCeti.ModuleTower.isProZero_multiplicationTower_iff","parent":1491,"result":true,"route":"../roadmap/tauceti/algebraic-topology/enhanced-derived-categories-of-sheaves/index.html#the-multiplication-tower-is-pro-zero-exactly-for","scope":"tauceti/algebraic-topology/enhanced-derived-categories-of-sheaves","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.ModuleTower.isProZero_multiplicationTower_iff`.","title":"The multiplication tower is pro-zero exactly for nilpotent elements"},{"id":"tauceti/algebraic-topology/enhanced-derived-categories-of-sheaves/categorytheory-functor-isprozero","kind":"node","lean":"CategoryTheory.Functor.IsProZero","parent":1491,"result":true,"route":"../roadmap/tauceti/algebraic-topology/enhanced-derived-categories-of-sheaves/index.html#categorytheory-functor-isprozero","scope":"tauceti/algebraic-topology/enhanced-derived-categories-of-sheaves","status":"fully_proved","summary":"A diagram `F : J \u2964 C` is *pro-zero* if every object `F j` is killed by some map of the diagram: for every `j` there is `f : i \u27f6 j` with `F.map f = 0`. For towers `\u2115\u1d52\u1d56 \u2964 C` this says that every term is killed by some later composite transition (`isProZero_iff_forall_exists_le`).","title":"CategoryTheory.Functor.IsProZero"},{"id":"tauceti/algebraic-topology/enhanced-derived-categories-of-sheaves/categorytheory-functor-proisomorphisms","kind":"node","lean":"CategoryTheory.Functor.proIsomorphisms","parent":1491,"result":true,"route":"../roadmap/tauceti/algebraic-topology/enhanced-derived-categories-of-sheaves/index.html#categorytheory-functor-proisomorphisms","scope":"tauceti/algebraic-topology/enhanced-derived-categories-of-sheaves","status":"fully_proved","summary":"The *pro-isomorphisms* of diagrams `J \u2964 C`: the morphisms whose kernel and cokernel are pro-zero (`proIsomorphisms_iff`). This is Mathlib's class `ObjectProperty.isoModSerre` of isomorphisms modulo the Serre class of pro-zero diagrams, so pro-isomorphisms are stable under composition and retracts and satisfy two-out-of-three.","title":"CategoryTheory.Functor.proIsomorphisms"},{"id":"tauceti/algebraic-topology/enhanced-derived-categories-of-sheaves/sset-fullenhancedsubcategory","kind":"node","lean":"SSet.fullEnhancedSubcategory","parent":1491,"result":true,"route":"../roadmap/tauceti/algebraic-topology/enhanced-derived-categories-of-sheaves/index.html#sset-fullenhancedsubcategory","scope":"tauceti/algebraic-topology/enhanced-derived-categories-of-sheaves","status":"fully_proved","summary":"The full enhanced subcategory of a simplicial set `X` on a predicate `P` on its vertices: the subcomplex of the simplices all of whose vertices satisfy `P`.","title":"SSet.fullEnhancedSubcategory"},{"data_url":"scopes/tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations.json","id":"tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations","kind":"scope","parent":1449,"result":false,"route":"../roadmap/tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations/index.html","scope":"tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Generalized cohomology, spectra and orientations"},{"id":"tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations/the-homotopy-fibre-of-the-fibre-inclusion-is-the","kind":"node","lean":"TauCeti.FiberInclusionPathFiber.loopHomotopyEquiv","parent":1500,"result":true,"route":"../roadmap/tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations/index.html#the-homotopy-fibre-of-the-fibre-inclusion-is-the","scope":"tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations","status":"fully_proved","summary":"For a Hurewicz fibration `p`, the space of paths in `E` from the fibre through `e\u2080` to `e\u2080` is homotopy equivalent to the loop space `\u03a9 B (p e\u2080)`, by composing the paths with `p`.","title":"The homotopy fibre of the fibre inclusion is the loop space of the base"},{"id":"tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations/tauceti-ishurewiczfibration-iff-nonempty-lifting","kind":"node","lean":"TauCeti.isHurewiczFibration_iff_nonempty_liftingFunction","parent":1500,"result":true,"route":"../roadmap/tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations/index.html#tauceti-ishurewiczfibration-iff-nonempty-lifting","scope":"tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations","status":"fully_proved","summary":"A map is a Hurewicz fibration for test spaces in `Type (max u v)` if and only if it has a lifting function.","title":"TauCeti.isHurewiczFibration_iff_nonempty_liftingFunction"},{"id":"tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations/tauceti-mappingpathspace-ishurewiczfibration-end","kind":"node","lean":"TauCeti.MappingPathSpace.isHurewiczFibration_endpoint","parent":1500,"result":true,"route":"../roadmap/tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations/index.html#tauceti-mappingpathspace-ishurewiczfibration-end","scope":"tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations","status":"fully_proved","summary":"The endpoint map of a mapping-path replacement is a Hurewicz fibration, for test spaces in every universe.","title":"TauCeti.MappingPathSpace.isHurewiczFibration_endpoint"},{"id":"tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations/tauceti-mappingpathfiber-projfiberhomotopyequivl","kind":"node","lean":"TauCeti.MappingPathFiber.projFiberHomotopyEquivLoopSpace","parent":1500,"result":true,"route":"../roadmap/tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations/index.html#tauceti-mappingpathfiber-projfiberhomotopyequivl","scope":"tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations","status":"fully_proved","summary":"A reference path `\u03b1` from `f b` to `k` makes the fibre of the source projection over `b` homotopy equivalent to the loop space `\u03a9 K k`. A path `\u03b3` from `f b` to `k` is sent to the loop `\u03b1.symm.trans \u03b3`, and the homotopy inverse sends a loop `\u2113` to the path `\u03b1.trans \u2113`.","title":"TauCeti.MappingPathFiber.projFiberHomotopyEquivLoopSpace"},{"id":"tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations/categorytheory-monoidalcategory-dayfunctor-iscom","kind":"node","lean":"CategoryTheory.MonoidalCategory.DayFunctor.isCommMonObj","parent":1500,"result":true,"route":"../roadmap/tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations/index.html#categorytheory-monoidalcategory-dayfunctor-iscom","scope":"tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations","status":"fully_proved","summary":"The Day convolution monoid of a lax braided functor out of a symmetric monoidal category is commutative.","title":"CategoryTheory.MonoidalCategory.DayFunctor.isCommMonObj"},{"id":"tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations/tauceti-pointedsset-suspensionloopadjunction","kind":"node","lean":"TauCeti.PointedSSet.suspensionLoopAdjunction","parent":1500,"result":true,"route":"../roadmap/tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations/index.html#tauceti-pointedsset-suspensionloopadjunction","scope":"tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations","status":"fully_proved","summary":"Smash with the simplicial circle is left adjoint to the pointed mapping simplicial set out of the circle.","title":"TauCeti.PointedSSet.suspensionLoopAdjunction"},{"id":"tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations/tauceti-symmetricspectrum","kind":"node","lean":"TauCeti.SymmetricSpectrum","parent":1500,"result":true,"route":"../roadmap/tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations/index.html#tauceti-symmetricspectrum","scope":"tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations","status":"fully_proved","summary":"Symmetric spectra of pointed simplicial sets: the modules over the commutative monoid `SymmetricSequence.sphere` of symmetric sequences. A symmetric spectrum is a symmetric sequence `X` with an action `S \u2297 X \u27f6 X` for the Day convolution product, satisfying the unit and associativity laws.","title":"TauCeti.SymmetricSpectrum"},{"id":"tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations/tauceti-symmetricspectrum-islimitoflevelwise","kind":"node","lean":"TauCeti.SymmetricSpectrum.isLimitOfLevelwise","parent":1500,"result":true,"route":"../roadmap/tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations/index.html#tauceti-symmetricspectrum-islimitoflevelwise","scope":"tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations","status":"fully_proved","summary":"A cone of symmetric spectra is limiting if its cone of pointed simplicial sets is limiting at every level. In particular, the limiting maps respect the sphere action.","title":"TauCeti.SymmetricSpectrum.isLimitOfLevelwise"},{"id":"tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations/tauceti-symmetricsequence","kind":"node","lean":"TauCeti.SymmetricSequence","parent":1500,"result":true,"route":"../roadmap/tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations/index.html#tauceti-symmetricsequence","scope":"tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations","status":"fully_proved","summary":"Symmetric sequences of pointed simplicial sets: functors from the finite-bijection groupoid to pointed simplicial sets, with the Day convolution product along the block sum.","title":"TauCeti.SymmetricSequence"},{"id":"tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations/tauceti-pointedsset-instbraidedfinitebijectionsm","kind":"node","lean":"TauCeti.PointedSSet.instBraidedFiniteBijectionSmashPowFunctor","parent":1500,"result":true,"route":"../roadmap/tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations/index.html#tauceti-pointedsset-instbraidedfinitebijectionsm","scope":"tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations","status":"fully_proved","summary":"The block swap of `FiniteBijection` acts on `X^\u2227(k + l)` as the swap of the two smash factors `X^\u2227k \u2227 X^\u2227l`, so the symmetric sequence of smash powers is a braided monoidal functor.","title":"TauCeti.PointedSSet.instBraidedFiniteBijectionSmashPowFunctor"},{"id":"tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations/tauceti-pointedsset-monoidalclosed","kind":"node","lean":"TauCeti.PointedSSet.monoidalClosed","parent":1500,"result":true,"route":"../roadmap/tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations/index.html#tauceti-pointedsset-monoidalclosed","scope":"tauceti/algebraic-topology/generalized-cohomology-spectra-and-orientations","status":"fully_proved","summary":"The smash product of pointed simplicial sets is closed: `X \u2297 -` is left adjoint to the pointed mapping simplicial set functor `Map_*(X, -)`.","title":"TauCeti.PointedSSet.monoidalClosed"},{"data_url":"scopes/tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie.json","id":"tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie","kind":"scope","parent":1449,"result":false,"route":"../roadmap/tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie/index.html","scope":"tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Morse and Floer homology with DG local coefficients"},{"id":"tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie/cubical-singular-homology-is-naturally-isomorphi","kind":"node","lean":"TauCeti.singularCubicalHomologyNatIso","parent":1512,"result":true,"route":"../roadmap/tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie/index.html#cubical-singular-homology-is-naturally-isomorphi","scope":"tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.singularCubicalHomologyNatIso`.","title":"Cubical singular homology is naturally isomorphic to singular homology, as functors on topological spaces"},{"id":"tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie/cubical-and-simplicial-chains-with-local-coeffic","kind":"node","lean":"TauCeti.LocalCoefficientSystem.twistedCubicalSingularHomotopyEquiv","parent":1512,"result":true,"route":"../roadmap/tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie/index.html#cubical-and-simplicial-chains-with-local-coeffic","scope":"tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.LocalCoefficientSystem.twistedCubicalSingularHomotopyEquiv`.","title":"Cubical and simplicial chains with local coefficients are chain homotopy equivalent"},{"id":"tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie/cw-approximation-hatcher-proposition-4-13","kind":"node","lean":"TauCeti.isWeakHomotopyEquivalence_cwApproximationMap","parent":1512,"result":true,"route":"../roadmap/tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie/index.html#cw-approximation-hatcher-proposition-4-13","scope":"tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie","status":"fully_proved","summary":"For a path-connected space `X`, the map `TauCeti.cwApproximationMap X x` from the CW complex `TauCeti.cwApproximation X x` to `X` is a weak homotopy equivalence.","title":"CW approximation (Hatcher, Proposition 4.13)"},{"id":"tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie/a-weak-homotopy-equivalence-induces-isomorphisms","kind":"node","lean":"ContinuousMap.IsWeakHomotopyEquivalence.isIso_twistedHomologyMap","parent":1512,"result":true,"route":"../roadmap/tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie/index.html#a-weak-homotopy-equivalence-induces-isomorphisms","scope":"tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie","status":"fully_proved","summary":"for every local coefficient system `L` on `Y`, the map `H_n(X; f\u204eL) \u27f6 H_n(Y; L)` is an isomorphism.","title":"A weak homotopy equivalence induces isomorphisms on singular homology with local coefficients, the local-coefficient form of Hatcher, Proposition 4.21"},{"id":"tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie/barraud-damian-humiliere-oancea-lemma-3-41-moore","kind":"node","lean":"TauCeti.MooreLoopSpace.isAspherical_iff_double_singularHomology","parent":1512,"result":true,"route":"../roadmap/tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie/index.html#barraud-damian-humiliere-oancea-lemma-3-41-moore","scope":"tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie","status":"fully_proved","summary":"A path-connected space `X` is aspherical if and only if its double Moore loop space is path connected with vanishing positive-degree integral singular homology.","title":"Barraud--Damian--Humili\u00e8re--Oancea, Lemma 3.41 (Moore singular homology)"},{"id":"tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie/homologous-twisting-cocycles-give-homotopy-equiv","kind":"node","lean":"TauCeti.Homologous.homotopyEquiv","parent":1512,"result":true,"route":"../roadmap/tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie/index.html#homologous-twisting-cocycles-give-homotopy-equiv","scope":"tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie","status":"fully_proved","summary":"for every differential graded right module `(\u2133, dM)`, the twisted complexes of `mP` and `mQ` are homotopy equivalent.","title":"Homologous twisting cocycles give homotopy equivalent twisted complexes"},{"id":"tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie/the-second-page-of-the-index-spectral-sequence-i","kind":"node","lean":"TauCeti.TwistingCocycle.indexSpectralSequenceSecondPageXIso","parent":1512,"result":true,"route":"../roadmap/tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie/index.html#the-second-page-of-the-index-spectral-sequence-i","scope":"tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie","status":"fully_proved","summary":"the object at `(a, b)` of the page `3` of the index spectral sequence (the classical `E\u2082` page, in Mathlib's numbering of pages) is the homology in degree `b + 1` of the classical complex in coefficient degree `a + 2 * b + 1`. It is the homology of the first page at `(a, b)`, identified with the homology of the classical complex by `indexSecondPageShortComplexIso` (see `indexSpectralSequenceSecondPageXIso_eq`).","title":"The second page of the index spectral sequence is the cohomology of the classical complex (BDHO, Definition 3.38 and Ch. 3 \u00a74.2)"},{"id":"tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie/the-unit-codisc-bundle-is-a-liouville-domain","kind":"node","lean":"TauCeti.CotangentBundle.isLiouvilleDomain_codiscBundle","parent":1512,"result":true,"route":"../roadmap/tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie/index.html#the-unit-codisc-bundle-is-a-liouville-domain","scope":"tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie","status":"fully_proved","summary":"For a compact smooth Riemannian manifold `M` without boundary, the unit codisc bundle `D*M` is a Liouville domain in `T*M` for the canonical symplectic form `\u03c9 = -d\u03bb`, with primitive `-\u03bb`: its Liouville field, the radial field `p \u2202_p`, points strictly outwards along the unit cosphere bundle, the level `\u2016p\u2016 = 1` of the norm of covectors.","title":"The unit codisc bundle is a Liouville domain"},{"id":"tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie/tauceti-loopchainalgebra","kind":"node","lean":"TauCeti.loopChainAlgebra","parent":1512,"result":true,"route":"../roadmap/tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie/index.html#tauceti-loopchainalgebra","scope":"tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie","status":"fully_proved","summary":"The chain algebra of the Moore loop space: `C_*(\u03a9_b B; R)`, the normalized cubical chains of the Moore loops at `b`, with the Pontryagin product.","title":"TauCeti.loopChainAlgebra"},{"id":"tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie/tauceti-fibrationlocalsystem","kind":"node","lean":"TauCeti.fibrationLocalSystem","parent":1512,"result":true,"route":"../roadmap/tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie/index.html#tauceti-fibrationlocalsystem","scope":"tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie","status":"fully_proved","summary":"The DG local system of a map `p : E \u2192 B` at `b`: the normalized cubical chains of the fibre `F'` of the Moore-path replacement, a right DG module over `C_*(\u03a9_b B; R)` through the strict action of the Moore loops.","title":"TauCeti.fibrationLocalSystem"},{"id":"tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie/tauceti-isliouvilledomain","kind":"node","lean":"TauCeti.IsLiouvilleDomain","parent":1512,"result":true,"route":"../roadmap/tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie/index.html#tauceti-isliouvilledomain","scope":"tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie","status":"fully_proved","summary":"A subset `W` of a manifold `M` is a Liouville domain for the smooth two-form `\u03c9`, with primitive `\u03b1`, when `\u03c9` is nondegenerate, `d\u03b1 = \u03c9`, `W` is compact and lies in the interior of `M`, and the Liouville field of `\u03b1` points strictly outwards along the boundary of `W`: some smooth vector field `Z` with `\u03b9_Z \u03c9 = \u03b1` and some continuous function `f` and value `c` satisfy `W = {f \u2264 c}`, with `f` smooth and `df(Z) > 0`\u2026","title":"TauCeti.IsLiouvilleDomain"},{"id":"tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie/the-boundary-of-a-liouville-domain-is-of-restric","kind":"node","lean":"TauCeti.IsLiouvilleDomain.exists_isRestrictedContactType","parent":1512,"result":true,"route":"../roadmap/tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie/index.html#the-boundary-of-a-liouville-domain-is-of-restric","scope":"tauceti/algebraic-topology/morse-and-floer-homology-with-dg-local-coefficie","status":"fully_proved","summary":"On a finite-dimensional manifold without boundary, a Liouville domain `W` is the sublevel set `{f \u2264 c}` of a function `f`, `C\u00b9` along the level `f\u207b\u00b9(c)`, of which `c` is a regular value; the frontier of `W` is exactly the level; and the level, with its atlas `TauCeti.regularLevelChartedSpace`, is a hypersurface of restricted contact type for the primitive `\u03b1`.","title":"The boundary of a Liouville domain is of restricted contact type"},{"data_url":"scopes/tauceti/algebraic-topology/universal-covers.json","id":"tauceti/algebraic-topology/universal-covers","kind":"scope","parent":1449,"result":false,"route":"../roadmap/tauceti/algebraic-topology/universal-covers/index.html","scope":"tauceti/algebraic-topology/universal-covers","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Universal covers"},{"id":"tauceti/algebraic-topology/universal-covers/every-pointed-connected-cover-of-x-x0-is-realise","kind":"node","lean":"TauCeti.UniversalCover.existsUnique_subgroup_homeomorph_subgroupQuotient","parent":1525,"result":true,"route":"../roadmap/tauceti/algebraic-topology/universal-covers/index.html#every-pointed-connected-cover-of-x-x0-is-realise","scope":"tauceti/algebraic-topology/universal-covers","status":"fully_proved","summary":"The subgroups of `\u03c0\u2081(X, x\u2080)` therefore parametrise the pointed connected covers of `(X, x\u2080)` bijectively, up to isomorphism over `X` respecting the chosen lifts.","title":"Every pointed connected cover of `(X, x\u2080)` is realised by exactly one subgroup of `\u03c0\u2081(X, x\u2080)`"},{"id":"tauceti/algebraic-topology/universal-covers/the-classification-of-covering-spaces-by-fundame","kind":"node","lean":"TauCeti.CoveringSpace.monodromyEquivalence","parent":1525,"result":true,"route":"../roadmap/tauceti/algebraic-topology/universal-covers/index.html#the-classification-of-covering-spaces-by-fundame","scope":"tauceti/algebraic-topology/universal-covers","status":"fully_proved","summary":"Over a path-connected, locally path-connected, semilocally simply connected base, monodromy is an equivalence from covering spaces to functors from the fundamental groupoid to types.","title":"The classification of covering spaces by fundamental-groupoid actions"},{"id":"tauceti/algebraic-topology/universal-covers/the-fundamental-group-of-x-at-x0-is-the-automorp","kind":"node","lean":"TauCeti.CoveringSpace.autFiberFunctorMulEquiv","parent":1525,"result":true,"route":"../roadmap/tauceti/algebraic-topology/universal-covers/index.html#the-fundamental-group-of-x-at-x0-is-the-automorp","scope":"tauceti/algebraic-topology/universal-covers","status":"fully_proved","summary":"A loop class is sent to the natural automorphism acting on every fibre by monodromy; that this is a bijection onto all natural automorphisms is Tannaka duality for `\u03c0\u2081(X, x\u2080)`-sets, transported along the classification of covering spaces by `\u03c0\u2081(X, x\u2080)`-sets.","title":"The fundamental group of `X` at `x\u2080` is the automorphism group of the fibre functor over `x\u2080`"},{"id":"tauceti/algebraic-topology/universal-covers/the-profinite-completion-of-1-x-x0-is-the-automo","kind":"node","lean":"TauCeti.FiniteCoveringSpace.profiniteCompletionAutFiberFunctorMulEquiv","parent":1525,"result":true,"route":"../roadmap/tauceti/algebraic-topology/universal-covers/index.html#the-profinite-completion-of-1-x-x0-is-the-automo","scope":"tauceti/algebraic-topology/universal-covers","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.FiniteCoveringSpace.profiniteCompletionAutFiberFunctorMulEquiv`.","title":"The profinite completion of `\u03c0\u2081(X, x\u2080)` is the automorphism group of the finite-cover fibre functor"},{"id":"tauceti/algebraic-topology/universal-covers/a-path-connected-locally-path-connected-space-is","kind":"node","lean":"TauCeti.semilocallySimplyConnectedSpace_iff_exists_isCoveringMap_and_simplyConnectedSpace","parent":1525,"result":true,"route":"../roadmap/tauceti/algebraic-topology/universal-covers/index.html#a-path-connected-locally-path-connected-space-is","scope":"tauceti/algebraic-topology/universal-covers","status":"fully_proved","summary":"The forward direction is the universal-cover construction, the reverse direction is `TauCeti.SemilocallySimplyConnectedSpace.of_isCoveringMap`; surjectivity of the covering map is automatic here, by `IsCoveringMap.comp_subtypeVal_pathComponent_surjective`, because the base is path-connected and a simply connected total space is nonempty.","title":"A path-connected, locally path-connected space is semilocally simply connected if and only if it admits a simply connected covering space"},{"id":"tauceti/algebraic-topology/universal-covers/the-fundamental-group-of-real-projective-space-r","kind":"node","lean":"TauCeti.RealProjectiveSpace.fundamentalGroupMulEquivAt","parent":1525,"result":true,"route":"../roadmap/tauceti/algebraic-topology/universal-covers/index.html#the-fundamental-group-of-real-projective-space-r","scope":"tauceti/algebraic-topology/universal-covers","status":"fully_proved","summary":"`FundamentalGroup (RealProjectiveSpace n) x \u2243* \u2124\u02e3`.","title":"The fundamental group of real projective space `RP\u207f` (for `2 \u2264 n`) is isomorphic to `\u2124\u02e3` for any basepoint `x`"},{"id":"tauceti/algebraic-topology/universal-covers/tauceti-simplyconnectedspace-sphere","kind":"node","lean":"TauCeti.simplyConnectedSpace_sphere","parent":1525,"result":true,"route":"../roadmap/tauceti/algebraic-topology/universal-covers/index.html#tauceti-simplyconnectedspace-sphere","scope":"tauceti/algebraic-topology/universal-covers","status":"fully_proved","summary":"Every sphere of nonnegative radius in a real normed space of rank greater than two is simply connected.","title":"TauCeti.simplyConnectedSpace_sphere"},{"id":"tauceti/algebraic-topology/universal-covers/tauceti-coveringspace","kind":"node","lean":"TauCeti.CoveringSpace","parent":1525,"result":true,"route":"../roadmap/tauceti/algebraic-topology/universal-covers/index.html#tauceti-coveringspace","scope":"tauceti/algebraic-topology/universal-covers","status":"fully_proved","summary":"The category of covering spaces over `X`. Its objects are covering maps to `X`, and its morphisms are continuous maps commuting with the projections to `X`.","title":"TauCeti.CoveringSpace"},{"id":"tauceti/algebraic-topology/universal-covers/tauceti-coveringspace-mk","kind":"node","lean":"TauCeti.CoveringSpace.mk","parent":1525,"result":true,"route":"../roadmap/tauceti/algebraic-topology/universal-covers/index.html#tauceti-coveringspace-mk","scope":"tauceti/algebraic-topology/universal-covers","status":"fully_proved","summary":"Construct a covering space over `X` from a covering map `p`. The definition is `@[expose]`d so that its total space is `E` and its projection is `p` by `rfl` in downstream modules.","title":"TauCeti.CoveringSpace.mk"},{"id":"tauceti/algebraic-topology/universal-covers/the-fundamental-group-at-x-acts-on-every-homotop","kind":"node","lean":"TauCeti.homotopyGroupMulAction","parent":1525,"result":true,"route":"../roadmap/tauceti/algebraic-topology/universal-covers/index.html#the-fundamental-group-at-x-acts-on-every-homotop","scope":"tauceti/algebraic-topology/universal-covers","status":"fully_proved","summary":"The fundamental group at `x` acts on every homotopy group based at `x`, an element of `\u03c0\u2081(X, x)` acting by transport around the loop it represents.","title":"The fundamental group at `x` acts on every homotopy group based at `x`"},{"id":"tauceti/algebraic-topology/universal-covers/addcircle-fundamentalgroupmulequiv-eq-windingnum","kind":"node","lean":"AddCircle.fundamentalGroupMulEquiv_eq_windingNumberIso","parent":1525,"result":true,"route":"../roadmap/tauceti/algebraic-topology/universal-covers/index.html#addcircle-fundamentalgroupmulequiv-eq-windingnum","scope":"tauceti/algebraic-topology/universal-covers","status":"fully_proved","summary":"For positive periods, the chosen-lift equivalence agrees with Mathlib's winding-number isomorphism. In particular it is independent of the chosen lift and has the same orientation.","title":"AddCircle.fundamentalGroupMulEquiv_eq_windingNumberIso"},{"id":"tauceti/algebraic-topology/universal-covers/the-isomorphism-1-s1-z-is-the-degree","kind":"node","lean":"Circle.fundamentalGroupMulEquiv_fromPath","parent":1525,"result":true,"route":"../roadmap/tauceti/algebraic-topology/universal-covers/index.html#the-isomorphism-1-s1-z-is-the-degree","scope":"tauceti/algebraic-topology/universal-covers","status":"fully_proved","summary":"The class of a loop is sent to its degree `Circle.degree`, the number of full turns of any continuous angle lift.","title":"The isomorphism `\u03c0\u2081(S\u00b9) \u2243* \u2124` is the degree"},{"data_url":"scopes/tauceti/algebraic-topology/cellular.json","id":"tauceti/algebraic-topology/cellular","kind":"scope","parent":1449,"result":false,"route":"../roadmap/tauceti/algebraic-topology/cellular/index.html","scope":"tauceti/algebraic-topology/cellular","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.AlgebraicTopology.Cellular` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Cellular"},{"id":"tauceti/algebraic-topology/cellular/euler-poincare-for-cellular-chains","kind":"node","lean":"TauCeti.sum_range_finrank_cellularHomology, TauCeti.eulerChar_eq_finsum_finrank_singularHomology, TauCeti.homologyEulerChar_cellularChainComplex, TauCeti.finsum_finrank_singularHomology_of_finite_cwComplex, TauCeti.finsum_finrank_singularHomology_complexBasePair, TauCeti.eulerChar_eq_finsum_finrank_singularCohomology","parent":1538,"result":true,"route":"../roadmap/tauceti/algebraic-topology/cellular/index.html#euler-poincare-for-cellular-chains","scope":"tauceti/algebraic-topology/cellular","status":"fully_proved","summary":"For a relative CW complex with finitely many cells through degree `n` and a finite-dimensional coefficient module `M`, if there are no `(n + 1)`-cells, then the alternating sum of the dimensions of the cellular homology groups in degrees at most `n` is `dim M` times the alternating count of the cells of dimension at most `n`.","title":"Euler--Poincar\u00e9 for cellular chains"},{"id":"tauceti/algebraic-topology/cellular/the-morse-polynomial-identity-for-a-cw-pair","kind":"node","lean":"TauCeti.sum_range_monomial_natCard_cell_mul_finrank_eq, TauCeti.sum_range_monomial_natCard_cell_mul_finrank_eq_of_cwComplex","parent":1538,"result":true,"route":"../roadmap/tauceti/algebraic-topology/cellular/index.html#the-morse-polynomial-identity-for-a-cw-pair","scope":"tauceti/algebraic-topology/cellular","status":"fully_proved","summary":"If there are finitely many relative cells through degree `n` and none in degree `n + 1`, then in `\u2115[X]` `\u2211\u1d62\u208c\u2080\u207f c\u1d62 dim M \u00b7 X\u2071 = \u2211\u1d62\u208c\u2080\u207f b\u1d62 X\u2071 + (1 + X) Q`, where `c\u1d62` is the number of relative `i`-cells, `b\u1d62` the dimension of the relative singular homology, and the coefficient of `X\u2071` in `Q` is the rank of the cellular differential in degree `i + 1` for `i < n`, and zero otherwise.","title":"The Morse polynomial identity for a CW pair"},{"id":"tauceti/algebraic-topology/cellular/the-strong-morse-inequalities-for-a-cw-pair","kind":"node","lean":"TauCeti.sum_range_neg_one_pow_mul_finrank_singularHomology_complexBasePair_le, TauCeti.sum_range_neg_one_pow_mul_finrank_singularHomology_le_of_cwComplex","parent":1538,"result":true,"route":"../roadmap/tauceti/algebraic-topology/cellular/index.html#the-strong-morse-inequalities-for-a-cw-pair","scope":"tauceti/algebraic-topology/cellular","status":"fully_proved","summary":"If there are finitely many relative cells through degree `n`, then `b\u2099 - b\u2099\u208b\u2081 + \u22ef \u00b1 b\u2080 \u2264 (c\u2099 - c\u2099\u208b\u2081 + \u22ef \u00b1 c\u2080) \u00b7 dim M`, where `b\u1d62` is the dimension of the relative singular homology and `c\u1d62` the number of relative `i`-cells.","title":"The strong Morse inequalities for a CW pair"},{"id":"tauceti/algebraic-topology/cellular/the-weak-morse-inequality-for-a-cw-pair","kind":"node","lean":"TauCeti.finrank_singularHomology_complexBasePair_le, TauCeti.finrank_singularHomology_le_of_cwComplex","parent":1538,"result":true,"route":"../roadmap/tauceti/algebraic-topology/cellular/index.html#the-weak-morse-inequality-for-a-cw-pair","scope":"tauceti/algebraic-topology/cellular","status":"fully_proved","summary":"In a degree with finitely many relative cells, the relative singular homology has dimension at most the number of cells times `dim M`.","title":"The weak Morse inequality for a CW pair"},{"data_url":"scopes/tauceti/algebraic-topology/cohomology.json","id":"tauceti/algebraic-topology/cohomology","kind":"scope","parent":1449,"result":false,"route":"../roadmap/tauceti/algebraic-topology/cohomology/index.html","scope":"tauceti/algebraic-topology/cohomology","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.AlgebraicTopology.Cohomology` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Cohomology"},{"id":"tauceti/algebraic-topology/cohomology/the-kunneth-theorem-for-cohomology","kind":"node","lean":"TopCat.singularCohomologyKunneth_bijective","parent":1543,"result":true,"route":"../roadmap/tauceti/algebraic-topology/cohomology/index.html#the-kunneth-theorem-for-cohomology","scope":"tauceti/algebraic-topology/cohomology","status":"fully_proved","summary":"over a commutative semisimple ring `k`, for instance a field, if the homology `H\u209a(X; k)` is finitely generated in degrees `p \u2264 n`, the cohomology K\u00fcnneth map `\u2a01_{p + q = n} H\u1d56(X; k) \u2297 H^q(Y; k) \u27f6 H\u207f(X \u00d7 Y; k)` is bijective.","title":"The K\u00fcnneth theorem for cohomology"},{"id":"tauceti/algebraic-topology/cohomology/the-relative-small-chain-theorem-for-cochains","kind":"node","lean":"TopPair.quasiIso_cochainComplexMap_restrict\u03b9_smallSingularSubcomplex","parent":1543,"result":true,"route":"../roadmap/tauceti/algebraic-topology/cohomology/index.html#the-relative-small-chain-theorem-for-cochains","scope":"tauceti/algebraic-topology/cohomology","status":"fully_proved","summary":"Restricting relative singular cochains of a topological pair to the simplices subordinate to an open cover of the ambient space is a quasi-isomorphism.","title":"The relative small-chain theorem for cochains"},{"id":"tauceti/algebraic-topology/cohomology/the-relative-universal-coefficient-theorem-over","kind":"node","lean":"TopPair.singularCohomologyDualEquiv, TopPair.singularKronecker_bijective_of_projective_homology","parent":1543,"result":true,"route":"../roadmap/tauceti/algebraic-topology/cohomology/index.html#the-relative-universal-coefficient-theorem-over","scope":"tauceti/algebraic-topology/cohomology","status":"fully_proved","summary":"the relative singular cohomology of a pair with coefficients in a field `k` is the dual of its relative singular homology, `H\u207f(X, A; k) \u2243\u2097[k] Module.Dual k (H\u2099(X, A; k))`. It sends a class to the linear form underlying its relative Kronecker pairing (`TopPair.singularCohomologyDualEquiv_apply`).","title":"The relative universal coefficient theorem over a field"},{"id":"tauceti/algebraic-topology/cohomology/the-universal-coefficient-theorem-over-a-field","kind":"node","lean":"TopCat.singularCohomologyDualEquiv, TopCat.singularKroneckerEquiv, TopCat.singularKronecker_bijective_zero, TopCat.singularKronecker_bijective_of_projective_homology","parent":1543,"result":true,"route":"../roadmap/tauceti/algebraic-topology/cohomology/index.html#the-universal-coefficient-theorem-over-a-field","scope":"tauceti/algebraic-topology/cohomology","status":"fully_proved","summary":"singular cohomology with coefficients in a field `k` is the dual of singular homology, `H\u207f(X; k) \u2243\u2097[k] Module.Dual k (H\u2099(X; k))`. It sends a class to the linear form underlying its Kronecker pairing (`TopCat.singularCohomologyDualEquiv_apply`).","title":"The universal coefficient theorem over a field"},{"data_url":"scopes/tauceti/algebraic-topology/fundamental-group.json","id":"tauceti/algebraic-topology/fundamental-group","kind":"scope","parent":1449,"result":false,"route":"../roadmap/tauceti/algebraic-topology/fundamental-group/index.html","scope":"tauceti/algebraic-topology/fundamental-group","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.AlgebraicTopology.FundamentalGroup` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Fundamental Group"},{"id":"tauceti/algebraic-topology/fundamental-group/the-based-seifert-van-kampen-theorem","kind":"node","lean":"TauCeti.isPushout_fundamentalGroup, TauCeti.vanKampenLift_bijective","parent":1548,"result":true,"route":"../roadmap/tauceti/algebraic-topology/fundamental-group/index.html#the-based-seifert-van-kampen-theorem","scope":"tauceti/algebraic-topology/fundamental-group","status":"fully_proved","summary":"If the interiors of `A` and `B` cover `X`, the sets `A`, `B` and `A \u2229 B` are path connected, and all three contain the basepoint `x`, then the square of fundamental groups induced by the inclusions of `A \u2229 B` into `A` and `B` and of `A` and `B` into `X` is a pushout of groups. That is, `\u03c0\u2081(X, x)` is the free product of `\u03c0\u2081(A, x)` and `\u03c0\u2081(B, x)` amalgamated over `\u03c0\u2081(A \u2229 B, x)`.","title":"The based Seifert--van Kampen theorem"},{"id":"tauceti/algebraic-topology/fundamental-group/the-based-van-kampen-theorem-for-a-neighborhood","kind":"node","lean":"TauCeti.existsUnique_vanKampenDesc","parent":1548,"result":true,"route":"../roadmap/tauceti/algebraic-topology/fundamental-group/index.html#the-based-van-kampen-theorem-for-a-neighborhood","scope":"tauceti/algebraic-topology/fundamental-group","status":"fully_proved","summary":"Pairwise compatible homomorphisms out of the fundamental groups of the cover members extend uniquely to the ambient fundamental group. Repeated indices in the triple-intersection hypothesis imply path connectedness of the double intersections and individual members. No common pairwise intersection is required, and the target can be any monoid.","title":"The based van Kampen theorem for a neighborhood cover, in universal-property form"},{"id":"tauceti/algebraic-topology/fundamental-group/the-generation-half-of-van-kampens-theorem-for-a","kind":"node","lean":"TauCeti.vanKampenWideLift_surjective","parent":1548,"result":true,"route":"../roadmap/tauceti/algebraic-topology/fundamental-group/index.html#the-generation-half-of-van-kampens-theorem-for-a","scope":"tauceti/algebraic-topology/fundamental-group","status":"fully_proved","summary":"Every loop class is the image of an element of the indexed free product when all pairwise intersections of the cover members are path connected.","title":"The generation half of van Kampen's theorem for a family"},{"id":"tauceti/algebraic-topology/fundamental-group/the-relations-half-of-van-kampens-theorem-for-a","kind":"node","lean":"TauCeti.ker_vanKampenWideLift","parent":1548,"result":true,"route":"../roadmap/tauceti/algebraic-topology/fundamental-group/index.html#the-relations-half-of-van-kampens-theorem-for-a","scope":"tauceti/algebraic-topology/fundamental-group","status":"fully_proved","summary":"The kernel of the canonical map from the indexed free product is exactly the normal closure of the common-intersection relators.","title":"The relations half of van Kampen's theorem for a family"},{"id":"tauceti/algebraic-topology/fundamental-group/the-seifert-van-kampen-theorem-for-a-family-with","kind":"node","lean":"TauCeti.isColimitFundamentalGroupWideCocone, TauCeti.vanKampenWideEquiv","parent":1548,"result":true,"route":"../roadmap/tauceti/algebraic-topology/fundamental-group/index.html#the-seifert-van-kampen-theorem-for-a-family-with","scope":"tauceti/algebraic-topology/fundamental-group","status":"fully_proved","summary":"If the interiors of the path-connected sets `U i` cover `X`, all of them contain the path-connected set `C \u220b x`, and two distinct members meet inside `C`, then `\u03c0\u2081(X, x)` is the wide pushout in the category of groups of the groups `\u03c0\u2081(U i, x)` over `\u03c0\u2081(C, x)`, along the maps induced by the inclusions.","title":"The Seifert--van Kampen theorem for a family with a common pairwise intersection"},{"id":"tauceti/algebraic-topology/fundamental-group/two-set-van-kampen-criterion-for-simple-connecte","kind":"node","lean":"TauCeti.simplyConnectedSpace_of_interior_union","parent":1548,"result":true,"route":"../roadmap/tauceti/algebraic-topology/fundamental-group/index.html#two-set-van-kampen-criterion-for-simple-connecte","scope":"tauceti/algebraic-topology/fundamental-group","status":"fully_proved","summary":"A space covered by the interiors of two simply connected sets with path-connected intersection is simply connected.","title":"Two-set van Kampen criterion for simple connectedness"},{"id":"tauceti/algebraic-topology/fundamental-group/uniqueness-in-the-seifert-van-kampen-theorem-for","kind":"node","lean":"TauCeti.vanKampenWide_hom_ext","parent":1548,"result":true,"route":"../roadmap/tauceti/algebraic-topology/fundamental-group/index.html#uniqueness-in-the-seifert-van-kampen-theorem-for","scope":"tauceti/algebraic-topology/fundamental-group","status":"fully_proved","summary":"If the interiors of the sets `U i \u220b x` cover `X` and their pairwise intersections are path connected, then two homomorphisms out of `\u03c0\u2081(X, x)` which agree on the image of every `\u03c0\u2081(U i, x)` are equal.","title":"Uniqueness in the Seifert--van Kampen theorem for a family"},{"data_url":"scopes/tauceti/algebraic-topology/singular.json","id":"tauceti/algebraic-topology/singular","kind":"scope","parent":1449,"result":false,"route":"../roadmap/tauceti/algebraic-topology/singular/index.html","scope":"tauceti/algebraic-topology/singular","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.AlgebraicTopology.Singular` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Singular"},{"id":"tauceti/algebraic-topology/singular/brouwers-fixed-point-theorem","kind":"node","lean":"TauCeti.brouwerFixedPoint_topCatDisk, TauCeti.brouwerFixedPoint_closedBall, TauCeti.brouwerFixedPoint_compactConvex, TauCeti.brouwerFixedPoint_compactConvex_continuousOn, TauCeti.brouwerFixedPoint_compactConvex_finiteSpan","parent":1556,"result":true,"route":"../roadmap/tauceti/algebraic-topology/singular/index.html#brouwers-fixed-point-theorem","scope":"tauceti/algebraic-topology/singular","status":"fully_proved","summary":"every continuous self-map of the disk `D\u207f` has a fixed point.","title":"Brouwer's fixed-point theorem"},{"id":"tauceti/algebraic-topology/singular/the-cubical-eilenberg-zilber-theorem","kind":"node","lean":"TauCeti.quasiIso_cubicalCrossProduct","parent":1556,"result":true,"route":"../roadmap/tauceti/algebraic-topology/singular/index.html#the-cubical-eilenberg-zilber-theorem","scope":"tauceti/algebraic-topology/singular","status":"fully_proved","summary":"the cross product of normalized cubical chains `N(X; k) \u2297 N(Y; k) \u27f6 N(X \u00d7 Y; k)` is a quasi-isomorphism.","title":"The cubical Eilenberg\u2013Zilber theorem"},{"id":"tauceti/algebraic-topology/singular/the-eilenberg-zilber-theorem-for-singular-chains","kind":"node","lean":"TopCat.eilenbergZilberHomotopyEquiv","parent":1556,"result":true,"route":"../roadmap/tauceti/algebraic-topology/singular/index.html#the-eilenberg-zilber-theorem-for-singular-chains","scope":"tauceti/algebraic-topology/singular","status":"fully_proved","summary":"the Alexander\u2013Whitney map `C(X \u00d7 Y; R \u2297 S) \u27f6 C(X; R) \u2297 C(Y; S)` is a chain homotopy equivalence, with homotopy inverse the shuffle map.","title":"The Eilenberg\u2013Zilber theorem for singular chains"},{"id":"tauceti/algebraic-topology/singular/the-kunneth-theorem-for-cubical-homology","kind":"node","lean":"TauCeti.isIso_cubicalHomologyKunneth, TauCeti.isIso_cubicalHomologyKunneth_of_isSemisimpleRing, TauCeti.isIso_cubicalHomologyKunneth_of_projective, TopCat.isIso_singularHomologyKunneth, TopCat.isIso_singularHomologyKunneth_of_projective, TopCat.isIso_singularHomologyKunneth_of_projective_homology","parent":1556,"result":true,"route":"../roadmap/tauceti/algebraic-topology/singular/index.html#the-kunneth-theorem-for-cubical-homology","scope":"tauceti/algebraic-topology/singular","status":"fully_proved","summary":"if the cycles of the normalized cubical chain complexes split off their terms and their homology splits off the cycles, the K\u00fcnneth map `\u2a01_{p + q = n} H\u209a(X; k) \u2297 H_q(Y; k) \u27f6 H\u2099(X \u00d7 Y; k)` is an isomorphism. This is the K\u00fcnneth theorem for the chain complexes `N(X; k)` and `N(Y; k)`, transported along the cubical Eilenberg\u2013Zilber quasi-isomorphism.","title":"The K\u00fcnneth theorem for cubical homology"},{"id":"tauceti/algebraic-topology/singular/the-mayer-vietoris-isomorphism-of-a-sphere","kind":"node","lean":"TauCeti.isIso_reducedMayerVietoris\u03b4_sphere","parent":1556,"result":true,"route":"../roadmap/tauceti/algebraic-topology/singular/index.html#the-mayer-vietoris-isomorphism-of-a-sphere","scope":"tauceti/algebraic-topology/singular","status":"fully_proved","summary":"The reduced Mayer\u2013Vietoris connecting morphism `H\u2096\u208a\u2081(S) \u27f6 H_red\u2096(S \u2216 {p, -p})` of the cover of the unit sphere `S` by the complements of `p` and `-p` is an isomorphism in every degree, since both complements are contractible.","title":"The Mayer\u2013Vietoris isomorphism of a sphere"},{"id":"tauceti/algebraic-topology/singular/the-no-retraction-theorem","kind":"node","lean":"TauCeti.noRetraction_diskBoundary","parent":1556,"result":true,"route":"../roadmap/tauceti/algebraic-topology/singular/index.html#the-no-retraction-theorem","scope":"tauceti/algebraic-topology/singular","status":"fully_proved","summary":"the boundary sphere `\u2202D\u207f` of the disk is not a retract of the disk `D\u207f`. That is, there are no maps `i : \u2202D\u207f \u27f6 D\u207f` and `r : D\u207f \u27f6 \u2202D\u207f` with `i \u226b r = \ud835\udfd9`; in particular the boundary inclusion `TopCat.diskBoundaryInclusion n` has no retraction. This includes `n = 0`, where the boundary is empty.","title":"The no-retraction theorem"},{"id":"tauceti/algebraic-topology/singular/the-relative-small-chain-theorem","kind":"node","lean":"TopPair.isIso_homologyMap_restrict\u03b9_smallSingularSubcomplex","parent":1556,"result":true,"route":"../roadmap/tauceti/algebraic-topology/singular/index.html#the-relative-small-chain-theorem","scope":"tauceti/algebraic-topology/singular","status":"fully_proved","summary":"Restricting the singular pair of a topological pair to the simplices subordinate to an open cover of the ambient space does not change relative homology.","title":"The relative small-chain theorem"},{"data_url":"scopes/tauceti/algebraic-topology/other-results.json","id":"tauceti/algebraic-topology/other-results","kind":"scope","parent":1449,"result":false,"route":"../roadmap/tauceti/algebraic-topology/other-results/index.html","scope":"tauceti/algebraic-topology/other-results","status":"fully_proved","summary":"Results named in the rest of Tau Ceti's `TauCeti.AlgebraicTopology` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Other results"},{"id":"tauceti/algebraic-topology/other-results/the-eilenberg-zilber-theorem","kind":"node","lean":"SSet.eilenbergZilberHomotopyEquiv","parent":1564,"result":true,"route":"../roadmap/tauceti/algebraic-topology/other-results/index.html#the-eilenberg-zilber-theorem","scope":"tauceti/algebraic-topology/other-results","status":"fully_proved","summary":"the Alexander\u2013Whitney map `C(K \u00d7 L; R \u2297 S) \u27f6 C(K; R) \u2297 C(L; S)` is a chain homotopy equivalence, with homotopy inverse the shuffle map.","title":"The Eilenberg\u2013Zilber theorem"},{"id":"tauceti/algebraic-topology/other-results/joyals-theorem-on-special-outer-horns-htt-propos","kind":"node","lean":"SSet.Quasicategory.hornFilling_zero","parent":1564,"result":true,"route":"../roadmap/tauceti/algebraic-topology/other-results/index.html#joyals-theorem-on-special-outer-horns-htt-propos","scope":"tauceti/algebraic-topology/other-results","status":"fully_proved","summary":"in a quasicategory, a horn `\u039b[n + 2, 0]` whose initial edge, from `0` to `1`, is an equivalence can be filled.","title":"Joyal's theorem on special outer horns (HTT Proposition 1.2.4.3)"},{"id":"tauceti/algebraic-topology/other-results/moores-theorem","kind":"node","lean":"SSet.kanComplex_of_group, SSet.kanComplex_of_addGroup","parent":1564,"result":true,"route":"../roadmap/tauceti/algebraic-topology/other-results/index.html#moores-theorem","scope":"tauceti/algebraic-topology/other-results","status":"fully_proved","summary":"a simplicial set whose sets of simplices are groups, with simplicial operators group homomorphisms, is a Kan complex.","title":"Moore's theorem"},{"id":"tauceti/algebraic-topology/other-results/the-two-set-van-kampen-theorem-for-fundamental-g","kind":"node","lean":"TauCeti.isPushout_fundamentalGroupoid, TauCeti.isPushout_fundamentalGroupoidOn","parent":1564,"result":true,"route":"../roadmap/tauceti/algebraic-topology/other-results/index.html#the-two-set-van-kampen-theorem-for-fundamental-g","scope":"tauceti/algebraic-topology/other-results","status":"fully_proved","summary":"If the interiors of `A` and `B` cover `X`, the square of fundamental groupoids induced by the inclusions of `A \u2229 B` into `A` and `B` and of `A` and `B` into `X` is a pushout in the category of groupoids.","title":"The two-set van Kampen theorem for fundamental groupoids"},{"data_url":"scopes/tauceti/analysis.json","id":"tauceti/analysis","kind":"scope","parent":1095,"result":false,"route":"../roadmap/tauceti/analysis/index.html","scope":"tauceti/analysis","status":"fully_proved","summary":"Results in Tau Ceti's `TauCeti.Analysis` modules, and the roadmaps whose articles are mostly there: by roadmap, then by area.","title":"Analysis"},{"data_url":"scopes/tauceti/analysis/part-01.json","id":"tauceti/analysis/part-01","kind":"scope","navigation":"transparent","parent":1569,"result":false,"route":"../roadmap/tauceti/analysis/part-01/index.html","scope":"tauceti/analysis/part-01","status":"fully_proved","title":"Analytic Heegaard Floer homology to Bounded Variation"},{"data_url":"scopes/tauceti/analysis/part-01/analytic-heegaard-floer-homology.json","id":"tauceti/analysis/part-01/analytic-heegaard-floer-homology","kind":"scope","parent":1570,"result":false,"route":"../roadmap/tauceti/analysis/part-01/analytic-heegaard-floer-homology/index.html","scope":"tauceti/analysis/part-01/analytic-heegaard-floer-homology","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Analytic Heegaard Floer homology"},{"id":"tauceti/analysis/part-01/analytic-heegaard-floer-homology/the-sard-smale-theorem","kind":"node","lean":"TauCeti.isMeagre_image_criticalPoints_of_isFredholm, TauCeti.dense_compl_image_criticalPoints_of_isFredholm, TauCeti.exists_mem_nhds_isClosed_isNowhereDense_image_criticalPoints","parent":1571,"result":true,"route":"../roadmap/tauceti/analysis/part-01/analytic-heegaard-floer-homology/index.html#the-sard-smale-theorem","scope":"tauceti/analysis/part-01/analytic-heegaard-floer-homology","status":"fully_proved","summary":"The critical values of a sufficiently smooth Fredholm map on a Lindel\u00f6f subset of a Banach space form a meagre set. The subset need not be open; smoothness is assumed at each of its points in the ambient space. Every subset of a separable Banach space is Lindel\u00f6f.","title":"The Sard--Smale theorem"},{"id":"tauceti/analysis/part-01/analytic-heegaard-floer-homology/the-morse-sard-theorem","kind":"node","lean":"ContDiff.addHaar_image_criticalPoints_eq_zero, TauCeti.addHaar_image_criticalPoints_eq_zero, TauCeti.interior_image_criticalPoints_eq_empty, TauCeti.addHaar_image_criticalPoints_eq_zero_of_finrank_le_add_two, TauCeti.interior_image_criticalPoints_eq_empty_of_finrank_le_add_two","parent":1571,"result":true,"route":"../roadmap/tauceti/analysis/part-01/analytic-heegaard-floer-homology/index.html#the-morse-sard-theorem","scope":"tauceti/analysis/part-01/analytic-heegaard-floer-homology","status":"fully_proved","summary":"The critical values of a sufficiently smooth map between finite-dimensional real normed spaces, that is the values it takes at the points where its Fr\u00e9chet derivative is not surjective, form a set of additive Haar measure zero. No relation between the two dimensions is required.","title":"The Morse--Sard theorem"},{"id":"tauceti/analysis/part-01/analytic-heegaard-floer-homology/the-index-of-the-trajectory-operator-is-a-differ","kind":"node","lean":"TauCeti.WeakDerivStep.index_trajectoryOperator_eq_finrank_pointSpectralSubspace_sub_finrank","parent":1571,"result":true,"route":"../roadmap/tauceti/analysis/part-01/analytic-heegaard-floer-homology/index.html#the-index-of-the-trajectory-operator-is-a-differ","scope":"tauceti/analysis/part-01/analytic-heegaard-floer-homology","status":"fully_proved","summary":"Let `A` be a continuous family of operators on a finite-dimensional real inner product space with invertible symmetric limits `A\u00b1` at `\u00b1\u221e`. Then the index of the trajectory operator `u \u21a6 u' + A u` from `W^{1,2}(\u211d, E)` to `L\u00b2(\u211d, E)` is the number of negative eigenvalues of `A\u207b` minus the number of negative eigenvalues of `A\u207a`, counted with multiplicity: the dimensions of the sums of their eigenspaces at negative\u2026","title":"The index of the trajectory operator is a difference of Morse indices"},{"id":"tauceti/analysis/part-01/analytic-heegaard-floer-homology/tauceti-isnondegeneratecriticalpoint-exists-stab","kind":"node","lean":"TauCeti.IsNondegenerateCriticalPoint.exists_stableSet_isManifold","parent":1571,"result":true,"route":"../roadmap/tauceti/analysis/part-01/analytic-heegaard-floer-homology/index.html#tauceti-isnondegeneratecriticalpoint-exists-stab","scope":"tauceti/analysis/part-01/analytic-heegaard-floer-homology","status":"fully_proved","summary":"The global stable set of a Morse critical point of a globally `C^(n+1)` function has a `C^n` manifold structure modelled on the stable Hessian subspace, with its subspace topology and a `C^n` inclusion into the ambient space. The dimension of the model plus the Morse index equals the ambient dimension.","title":"TauCeti.IsNondegenerateCriticalPoint.exists_stableSet_isManifold"},{"id":"tauceti/analysis/part-01/analytic-heegaard-floer-homology/darbouxs-theorem","kind":"node","lean":"TauCeti.exists_diffeomorph_eventually_compContinuousLinearMap_fderiv_eq, TauCeti.SmoothTwoForm.IsSymplectic.exists_darbouxChart, TauCeti.SmoothTwoForm.IsSymplectic.exists_darbouxChart_of_isInteriorPoint, TauCeti.SmoothTwoForm.IsSymplectic.exists_darbouxChart_of_symplecticForm, TauCeti.SmoothTwoForm.IsSymplectic.exists_darbouxChart_of_symplecticForm_of_isInteriorPoint","parent":1571,"result":true,"route":"../roadmap/tauceti/analysis/part-01/analytic-heegaard-floer-homology/index.html#darbouxs-theorem","scope":"tauceti/analysis/part-01/analytic-heegaard-floer-homology","status":"fully_proved","summary":"A closed `C^r` two-form `\u03c9` (`r \u2265 1`) on an open subset of a finite-dimensional real normed space, nondegenerate at `x\u2080`, is constant in suitable coordinates near `x\u2080`: there is a `C^r` diffeomorphism `\u03c8` of the space fixing `x\u2080` such that the pullback `\u03c8^* \u03c9` equals the constant form `\u03c9(x\u2080)` on a neighbourhood of `x\u2080`.","title":"Darboux's theorem"},{"id":"tauceti/analysis/part-01/analytic-heegaard-floer-homology/the-symmetric-power-of-a-complex-curve-is-a-comp","kind":"node","lean":"TauCeti.isManifold_symChartedSpace","parent":1571,"result":true,"route":"../roadmap/tauceti/analysis/part-01/analytic-heegaard-floer-homology/index.html#the-symmetric-power-of-a-complex-curve-is-a-comp","scope":"tauceti/analysis/part-01/analytic-heegaard-floer-homology","status":"fully_proved","summary":"For a Hausdorff one-dimensional complex manifold `\u03b1`, the elementary-symmetric charts of `Sym \u03b1 n` have analytic transition maps, so `TauCeti.symChartedSpace` makes `Sym \u03b1 n` an analytic manifold modelled on `Fin n \u2192 \u2102`.","title":"The symmetric power of a complex curve is a complex analytic manifold"},{"id":"tauceti/analysis/part-01/analytic-heegaard-floer-homology/generic-regular-zeros","kind":"node","lean":"TauCeti.eventually_residual_exists_isManifold_sectionZero","parent":1571,"result":true,"route":"../roadmap/tauceti/analysis/part-01/analytic-heegaard-floer-homology/index.html#generic-regular-zeros","scope":"tauceti/analysis/part-01/analytic-heegaard-floer-homology","status":"fully_proved","summary":"For a residual set of parameters `l`, the zeros of `s (\u00b7, l)` form a `C^n` manifold of dimension the index `d`, with a `C^n` immersion into `M` whose tangent spaces are the kernels of the linearizations in the `M` direction.","title":"Generic regular zeros"},{"id":"tauceti/analysis/part-01/analytic-heegaard-floer-homology/generic-absence-of-negative-index-zeros","kind":"node","lean":"TauCeti.mem_residual_setOf_forall_section_ne_zero_of_index_neg","parent":1571,"result":true,"route":"../roadmap/tauceti/analysis/part-01/analytic-heegaard-floer-homology/index.html#generic-absence-of-negative-index-zeros","scope":"tauceti/analysis/part-01/analytic-heegaard-floer-homology","status":"fully_proved","summary":"If the slice linearization is Fredholm of negative index at each zero and the total section is transverse to zero, zero-free parameters form a residual set. Second countability is needed only for the universal zero set.","title":"Generic absence of negative-index zeros"},{"id":"tauceti/analysis/part-01/analytic-heegaard-floer-homology/tauceti-morseboundary-comp-morseboundary-eq-zero","kind":"node","lean":"TauCeti.morseBoundary_comp_morseBoundary_eq_zero_iff","parent":1571,"result":true,"route":"../roadmap/tauceti/analysis/part-01/analytic-heegaard-floer-homology/index.html#tauceti-morseboundary-comp-morseboundary-eq-zero","scope":"tauceti/analysis/part-01/analytic-heegaard-floer-homology","status":"fully_proved","summary":"The mod-two Morse boundary squares to zero exactly when, between any two critical points, the number of broken index-two trajectories is even.","title":"TauCeti.morseBoundary_comp_morseBoundary_eq_zero_iff"},{"id":"tauceti/analysis/part-01/analytic-heegaard-floer-homology/mosers-lemma-local-form","kind":"node","lean":"TauCeti.exists_diffeomorph_eventually_compContinuousLinearMap_fderiv_eq_of_extDeriv_eq","parent":1571,"result":true,"route":"../roadmap/tauceti/analysis/part-01/analytic-heegaard-floer-homology/index.html#mosers-lemma-local-form","scope":"tauceti/analysis/part-01/analytic-heegaard-floer-homology","status":"fully_proved","summary":"Let `\u03c9\u2080`, `\u03c9\u2081` be closed `C^r` two-forms (`r \u2265 1`) on an open subset `W` of a finite-dimensional real normed space, with `\u03c9\u2081 - \u03c9\u2080 = d\u03b1` on `W` for a `C^r` one-form `\u03b1` vanishing at `x\u2080 \u2208 W`. If `\u03c9\u2080` and `\u03c9\u2081` agree at `x\u2080` and are nondegenerate there, then some `C^r` diffeomorphism `\u03c8` of the space pulls `\u03c9\u2081` back to `\u03c9\u2080` on a neighbourhood of `x\u2080`. Moreover `\u03c8` fixes, near `x\u2080`, every point at which `\u03b1` vanishes;\u2026","title":"Moser's lemma, local form"},{"data_url":"scopes/tauceti/analysis/part-01/banach-algebras.json","id":"tauceti/analysis/part-01/banach-algebras","kind":"scope","parent":1570,"result":false,"route":"../roadmap/tauceti/analysis/part-01/banach-algebras/index.html","scope":"tauceti/analysis/part-01/banach-algebras","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Banach algebras"},{"id":"tauceti/analysis/part-01/banach-algebras/cohen-hewitt-factorization","kind":"node","lean":"ContinuousLinearMap.exists_apply_eq_of_mem_topologicalClosure_span","parent":1582,"result":true,"route":"../roadmap/tauceti/analysis/part-01/banach-algebras/index.html#cohen-hewitt-factorization","scope":"tauceti/analysis/part-01/banach-algebras","status":"fully_proved","summary":"Let `A` be a Banach algebra with a left approximate identity `l` bounded by `C`, acting boundedly on the left of a Banach space `E` through `L`. Every `x` in the essential part `closure (span (A \u00b7 E))` factors as `x = L a y` with `\u2016a\u2016 \u2264 C`, where `y` lies in the closure of `A \u00b7 x` and within any prescribed distance `\u03b5` of `x`.","title":"Cohen\u2013Hewitt factorization"},{"id":"tauceti/analysis/part-01/banach-algebras/sinclairs-stability-lemma","kind":"node","lean":"TauCeti.eventuallyConst_separatingSpace","parent":1582,"result":true,"route":"../roadmap/tauceti/analysis/part-01/banach-algebras/index.html#sinclairs-stability-lemma","scope":"tauceti/analysis/part-01/banach-algebras","status":"fully_proved","summary":"Let `E n` and `F` be Banach spaces, let `R n : E (n + 1) \u2192 E n` be continuous linear maps, and let `T n : E n \u2192\u2097 F` be linear maps, not assumed continuous, with `T (n + 1) = T n \u2218 R n`, so that `T n = T 0 \u2218 R 0 \u2218 \u22ef \u2218 R (n - 1)`. Then the decreasing sequence of separating spaces of the `T n` is eventually constant.","title":"Sinclair's stability lemma"},{"id":"tauceti/analysis/part-01/banach-algebras/linearmap-continuous-iff-separatingspace-eq-bot","kind":"node","lean":"LinearMap.continuous_iff_separatingSpace_eq_bot","parent":1582,"result":true,"route":"../roadmap/tauceti/analysis/part-01/banach-algebras/index.html#linearmap-continuous-iff-separatingspace-eq-bot","scope":"tauceti/analysis/part-01/banach-algebras","status":"fully_proved","summary":"A linear map between Banach spaces is continuous if and only if its separating space is trivial.","title":"LinearMap.continuous_iff_separatingSpace_eq_bot"},{"id":"tauceti/analysis/part-01/banach-algebras/tauceti-ring-mem-jacobson-iff-forall-gelfandradi","kind":"node","lean":"TauCeti.Ring.mem_jacobson_iff_forall_gelfandRadius_mul_eq_zero","parent":1582,"result":true,"route":"../roadmap/tauceti/analysis/part-01/banach-algebras/index.html#tauceti-ring-mem-jacobson-iff-forall-gelfandradi","scope":"tauceti/analysis/part-01/banach-algebras","status":"fully_proved","summary":"An element of a complex Banach algebra lies in its Jacobson radical exactly when every left multiple `b * a` is quasinilpotent, that is, `\u2016(b * a) ^ n\u2016 ^ (1 / n)` tends to zero.","title":"TauCeti.Ring.mem_jacobson_iff_forall_gelfandRadius_mul_eq_zero"},{"id":"tauceti/analysis/part-01/banach-algebras/the-homology-cauchy-theorem-for-contour-cycles","kind":"node","lean":"TauCeti.Contour.Cycle.homologyCauchyTheorem","parent":1582,"result":true,"route":"../roadmap/tauceti/analysis/part-01/banach-algebras/index.html#the-homology-cauchy-theorem-for-contour-cycles","scope":"tauceti/analysis/part-01/banach-algebras","status":"fully_proved","summary":"Let `C` be a finite formal integer combination of closed piecewise-`C\u00b9` curves whose trace lies in an open set `U`. If `C` is null-homologous in `U` and `f` is holomorphic on `U`, with values in any complex normed space, then the contour integral of `f` over the whole cycle vanishes.","title":"The homology Cauchy theorem for contour cycles"},{"id":"tauceti/analysis/part-01/banach-algebras/linearmap-separatingspace","kind":"node","lean":"LinearMap.separatingSpace","parent":1582,"result":true,"route":"../roadmap/tauceti/analysis/part-01/banach-algebras/index.html#linearmap-separatingspace","scope":"tauceti/analysis/part-01/banach-algebras","status":"fully_proved","summary":"The *separating space* of a linear map `T : E \u2192\u2097[R] F` between topological modules: the vertical fibre over `0` of the closure of the graph of `T`. For first countable spaces it consists of the limits of `T x\u2099` along sequences `x\u2099 \u2192 0` (`LinearMap.mem_separatingSpace_iff_exists_seq`).","title":"LinearMap.separatingSpace"},{"id":"tauceti/analysis/part-01/banach-algebras/tauceti-nonunitaljacobson","kind":"node","lean":"TauCeti.nonUnitalJacobson","parent":1582,"result":true,"route":"../roadmap/tauceti/analysis/part-01/banach-algebras/index.html#tauceti-nonunitaljacobson","scope":"tauceti/analysis/part-01/banach-algebras","status":"fully_proved","summary":"The Jacobson radical of a nonunital algebra, defined through its external unitization.","title":"TauCeti.nonUnitalJacobson"},{"id":"tauceti/analysis/part-01/banach-algebras/continuouslinearmap-unitizationaction","kind":"node","lean":"ContinuousLinearMap.unitizationAction","parent":1582,"result":true,"route":"../roadmap/tauceti/analysis/part-01/banach-algebras/index.html#continuouslinearmap-unitizationaction","scope":"tauceti/analysis/part-01/banach-algebras","status":"fully_proved","summary":"The unital action on `E` of the `\u2113\u00b9` unitization of `A` extending the left action `L`.","title":"ContinuousLinearMap.unitizationAction"},{"id":"tauceti/analysis/part-01/banach-algebras/cohens-factorization-theorem-for-banach-algebras","kind":"node","lean":"TauCeti.exists_mul_eq_of_tendsto_mul_right","parent":1582,"result":true,"route":"../roadmap/tauceti/analysis/part-01/banach-algebras/index.html#cohens-factorization-theorem-for-banach-algebras","scope":"tauceti/analysis/part-01/banach-algebras","status":"fully_proved","summary":"if `A` has a left approximate identity bounded by `C`, then every element of `A` is a product `a * y` with `\u2016a\u2016 \u2264 C`.","title":"Cohen's factorization theorem for Banach algebras"},{"data_url":"scopes/tauceti/analysis/part-01/banach-space-theory.json","id":"tauceti/analysis/part-01/banach-space-theory","kind":"scope","parent":1570,"result":false,"route":"../roadmap/tauceti/analysis/part-01/banach-space-theory/index.html","scope":"tauceti/analysis/part-01/banach-space-theory","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Banach space theory"},{"id":"tauceti/analysis/part-01/banach-space-theory/pietsch-domination-theorem","kind":"node","lean":"ContinuousLinearMap.summingENorm_le_ofReal_iff_exists_isProbabilityMeasure","parent":1592,"result":true,"route":"../roadmap/tauceti/analysis/part-01/banach-space-theory/index.html#pietsch-domination-theorem","scope":"tauceti/analysis/part-01/banach-space-theory","status":"fully_proved","summary":"For `1 \u2264 p < \u221e` and `C \u2265 0`, an operator satisfies `\u03c0\u209a(T) \u2264 C` if and only if there is a regular Borel probability measure `\u03bd` on the weak-star compact dual unit ball with `\u2016T x\u2016 \u2264 C (\u222b |f x|\u1d56 d\u03bd(f))^(1/p)` for all `x`.","title":"Pietsch domination theorem"},{"id":"tauceti/analysis/part-01/banach-space-theory/pietsch-factorization-through-the-closure-of-the","kind":"node","lean":"ContinuousLinearMap.summingENorm_le_ofReal_iff_exists_eq_comp_codRestrict_topologicalClosure","parent":1592,"result":true,"route":"../roadmap/tauceti/analysis/part-01/banach-space-theory/index.html#pietsch-factorization-through-the-closure-of-the","scope":"tauceti/analysis/part-01/banach-space-theory","status":"fully_proved","summary":"for a Banach space `F`, `1 \u2264 p < \u221e` and `C \u2265 0`, `\u03c0\u209a(T) \u2264 C` iff there are a regular Borel probability measure `\u03bd` on the weak-star compact dual unit ball and an operator `S` of norm at most `C` on the closure of `range j\u209a` in `L\u1d56(\u03bd)` such that `T = S \u2218 j\u209a`.","title":"Pietsch factorization through the closure of the canonical image"},{"id":"tauceti/analysis/part-01/banach-space-theory/closed-range-theorem","kind":"node","lean":"ContinuousLinearMap.isClosed_range_precomp_iff, ContinuousLinearMap.range_precomp_eq_orthogonalBilin_ker, ContinuousLinearMap.isClosed_range_of_isClosed_range_precomp","parent":1592,"result":true,"route":"../roadmap/tauceti/analysis/part-01/banach-space-theory/index.html#closed-range-theorem","scope":"tauceti/analysis/part-01/banach-space-theory","status":"fully_proved","summary":"A bounded operator between Banach spaces has closed range if and only if its adjoint has closed range.","title":"Closed-range theorem"},{"id":"tauceti/analysis/part-01/banach-space-theory/the-bishop-phelps-theorem","kind":"node","lean":"TauCeti.dense_setOf_exists_norm_apply_eq_norm, Convex.dense_setOf_ne_zero_isMaxOn_re, Convex.mem_closure_setOf_ne_zero_isMaxOn_re_of_bddAbove, Convex.frontier_subset_closure_setOf_isMaxOn_re","parent":1592,"result":true,"route":"../roadmap/tauceti/analysis/part-01/banach-space-theory/index.html#the-bishop-phelps-theorem","scope":"tauceti/analysis/part-01/banach-space-theory","status":"fully_proved","summary":"The continuous linear functionals on a Banach space which attain their norm on the closed unit ball are norm dense in the dual.","title":"The Bishop\u2013Phelps theorem"},{"id":"tauceti/analysis/part-01/banach-space-theory/milman-pettis-theorem","kind":"node","lean":"TauCeti.surjective_inclusionInDoubleDual_of_uniformConvexSpace","parent":1592,"result":true,"route":"../roadmap/tauceti/analysis/part-01/banach-space-theory/index.html#milman-pettis-theorem","scope":"tauceti/analysis/part-01/banach-space-theory","status":"fully_proved","summary":"A uniformly convex Banach space is reflexive: its canonical map into the continuous bidual is surjective.","title":"Milman\u2013Pettis theorem"},{"id":"tauceti/analysis/part-01/banach-space-theory/clarksons-inequality","kind":"node","lean":"MeasureTheory.Lp.norm_add_rpow_add_norm_sub_rpow_le","parent":1592,"result":true,"route":"../roadmap/tauceti/analysis/part-01/banach-space-theory/index.html#clarksons-inequality","scope":"tauceti/analysis/part-01/banach-space-theory","status":"fully_proved","summary":"For `2 \u2264 p < \u221e` and `f g : Lp E p \u03bc`, where the norm of `E` satisfies the parallelogram identity, `\u2016f + g\u2016 ^ p + \u2016f - g\u2016 ^ p \u2264 2 ^ (p - 1) * (\u2016f\u2016 ^ p + \u2016g\u2016 ^ p)`.","title":"Clarkson's inequality"},{"id":"tauceti/analysis/part-01/banach-space-theory/jamess-basis-criterion","kind":"node","lean":"SchauderBasis.surjective_inclusionInDoubleDual_iff","parent":1592,"result":true,"route":"../roadmap/tauceti/analysis/part-01/banach-space-theory/index.html#jamess-basis-criterion","scope":"tauceti/analysis/part-01/banach-space-theory","status":"fully_proved","summary":"A Banach space with a Schauder basis is reflexive if and only if that basis is shrinking and boundedly complete.","title":"James's basis criterion"},{"id":"tauceti/analysis/part-01/banach-space-theory/tauceti-absolutelysumming","kind":"node","lean":"TauCeti.AbsolutelySumming","parent":1592,"result":true,"route":"../roadmap/tauceti/analysis/part-01/banach-space-theory/index.html#tauceti-absolutelysumming","scope":"tauceti/analysis/part-01/banach-space-theory","status":"fully_proved","summary":"The operator ideal `\u03a0\u209a(E, F)` of absolutely `p`-summing operators, normed by the `p`-summing norm `\u03c0\u209a`. It is a Banach space when `F` is complete, and its inclusion `OperatorGauge.Carrier.toContinuousLinearMapL` into the bounded operators is a contraction.","title":"TauCeti.AbsolutelySumming"},{"id":"tauceti/analysis/part-01/banach-space-theory/tauceti-hasapproximationproperty","kind":"node","lean":"TauCeti.HasApproximationProperty","parent":1592,"result":true,"route":"../roadmap/tauceti/analysis/part-01/banach-space-theory/index.html#tauceti-hasapproximationproperty","scope":"tauceti/analysis/part-01/banach-space-theory","status":"fully_proved","summary":"The identity can be approximated uniformly on every norm-compact set by bounded operators with finite-dimensional range.","title":"TauCeti.HasApproximationProperty"},{"id":"tauceti/analysis/part-01/banach-space-theory/tauceti-projectivetensor","kind":"node","lean":"TauCeti.ProjectiveTensor","parent":1592,"result":true,"route":"../roadmap/tauceti/analysis/part-01/banach-space-theory/index.html#tauceti-projectivetensor","scope":"tauceti/analysis/part-01/banach-space-theory","status":"fully_proved","summary":"The algebraic tensor product `E \u2297[\ud835\udd5c] F` carrying the projective seminorm.","title":"TauCeti.ProjectiveTensor"},{"id":"tauceti/analysis/part-01/banach-space-theory/tauceti-injectivetensor","kind":"node","lean":"TauCeti.InjectiveTensor","parent":1592,"result":true,"route":"../roadmap/tauceti/analysis/part-01/banach-space-theory/index.html#tauceti-injectivetensor","scope":"tauceti/analysis/part-01/banach-space-theory","status":"fully_proved","summary":"The algebraic tensor product `E \u2297[\ud835\udd5c] F` carrying the injective seminorm.","title":"TauCeti.InjectiveTensor"},{"id":"tauceti/analysis/part-01/banach-space-theory/the-bishop-phelps-lemma","kind":"node","lean":"StrongDual.exists_pos_norm_sub_ofReal_smul_le","parent":1592,"result":true,"route":"../roadmap/tauceti/analysis/part-01/banach-space-theory/index.html#the-bishop-phelps-lemma","scope":"tauceti/analysis/part-01/banach-space-theory","status":"fully_proved","summary":"Let `0 \u2264 k < 1` and `f \u2260 0`. If the real part of `h` is positive on the open cone `{x | k * \u2016f\u2016 * \u2016x\u2016 < re (f x)}`, then some positive real multiple of `h` lies within `k * \u2016f\u2016` of `f`.","title":"The Bishop\u2013Phelps lemma"},{"id":"tauceti/analysis/part-01/banach-space-theory/bishop-phelps-support-lemma","kind":"node","lean":"Convex.exists_isMaxOn_re_of_bddAbove","parent":1592,"result":true,"route":"../roadmap/tauceti/analysis/part-01/banach-space-theory/index.html#bishop-phelps-support-lemma","scope":"tauceti/analysis/part-01/banach-space-theory","status":"fully_proved","summary":"Let `C` be a complete convex subset of a normed space, let `f` be a nonzero functional whose real part is bounded above on `C`, and let `0 < k < 1`. From every `z \u2208 C` one reaches a point `x \u2208 C` with `k * \u2016f\u2016 * \u2016x - z\u2016 \u2264 re (f (x - z))` at which `C` has a support functional `h`, nonzero and with real part positive on the open cone `{w | k * \u2016f\u2016 * \u2016w\u2016 < re (f w)}`.","title":"Bishop\u2013Phelps support lemma"},{"id":"tauceti/analysis/part-01/banach-space-theory/goldstines-theorem","kind":"node","lean":"NormedSpace.closure_image_inclusionInDoubleDualWeak_closedBall, NormedSpace.closure_image_inclusionInDoubleDualWeak_closedBall_zero_one","parent":1592,"result":true,"route":"../roadmap/tauceti/analysis/part-01/banach-space-theory/index.html#goldstines-theorem","scope":"tauceti/analysis/part-01/banach-space-theory","status":"fully_proved","summary":"The weak-star closure of the image of the closed ball `closedBall x r` under the canonical embedding into the bidual is the closed ball of radius `r` about the image of `x`.","title":"Goldstine's theorem"},{"id":"tauceti/analysis/part-01/banach-space-theory/grothendiecks-theorem","kind":"node","lean":"ContinuousLinearMap.summingENorm_one_le_ofReal_sixteen_mul_norm, ContinuousLinearMap.summingENorm_one_le_ofReal_eight_mul_norm","parent":1592,"result":true,"route":"../roadmap/tauceti/analysis/part-01/banach-space-theory/index.html#grothendiecks-theorem","scope":"tauceti/analysis/part-01/banach-space-theory","status":"fully_proved","summary":"Every bounded operator `T` from `L\u00b9(\u03bc)` into an inner product space over an `RCLike` field is absolutely `1`-summing, with `\u03c0\u2081(T) \u2264 16 \u2016T\u2016`. Over `\u211d`, `ContinuousLinearMap.summingENorm_one_le_ofReal_eight_mul_norm` gives the constant `8`.","title":"Grothendieck's theorem"},{"id":"tauceti/analysis/part-01/banach-space-theory/hellys-criterion","kind":"node","lean":"TauCeti.exists_norm_lt_and_forall_apply_eq_iff","parent":1592,"result":true,"route":"../roadmap/tauceti/analysis/part-01/banach-space-theory/index.html#hellys-criterion","scope":"tauceti/analysis/part-01/banach-space-theory","status":"fully_proved","summary":"Let `f i` be finitely many continuous linear functionals on `E`, let `c i` be scalars and let `0 \u2264 M`. For every `\u03b5 > 0` some `x : E` with `\u2016x\u2016 < M + \u03b5` satisfies `f i x = c i` for every `i` if and only if `\u2016\u2211 i, a i * c i\u2016 \u2264 M * \u2016\u2211 i, a i \u2022 f i\u2016` for all scalars `a i`.","title":"Helly's criterion"},{"id":"tauceti/analysis/part-01/banach-space-theory/hellys-lemma","kind":"node","lean":"StrongDual.exists_norm_lt_and_forall_apply_eq, StrongDual.exists_norm_lt_and_forall_mem_apply_eq, StrongDual.norm_sum_apply_comp_single_le","parent":1592,"result":true,"route":"../roadmap/tauceti/analysis/part-01/banach-space-theory/index.html#hellys-lemma","scope":"tauceti/analysis/part-01/banach-space-theory","status":"fully_proved","summary":"For an element `\u03c6` of the bidual of `E`, finitely many continuous linear functionals `f i` on `E` and `\u03b5 > 0`, some `x : E` with `\u2016x\u2016 < \u2016\u03c6\u2016 + \u03b5` satisfies `f i x = \u03c6 (f i)` for every `i`.","title":"Helly's lemma"},{"id":"tauceti/analysis/part-01/banach-space-theory/krein-smulian-theorem","kind":"node","lean":"Convex.isClosed_iff_forall_isClosed_inter_closedBall","parent":1592,"result":true,"route":"../roadmap/tauceti/analysis/part-01/banach-space-theory/index.html#krein-smulian-theorem","scope":"tauceti/analysis/part-01/banach-space-theory","status":"fully_proved","summary":"A convex subset of the dual of a Banach space is weak-star closed if and only if its intersection with every closed ball centred at zero is weak-star closed.","title":"Krein\u2013\u0160mulian theorem"},{"id":"tauceti/analysis/part-01/banach-space-theory/krein-smulian-theorem-for-subspaces","kind":"node","lean":"Submodule.isClosed_iff_isClosed_inter_closedBall_zero_one","parent":1592,"result":true,"route":"../roadmap/tauceti/analysis/part-01/banach-space-theory/index.html#krein-smulian-theorem-for-subspaces","scope":"tauceti/analysis/part-01/banach-space-theory","status":"fully_proved","summary":"A subspace of the dual of a Banach space is weak-star closed if and only if its intersection with the closed unit ball is weak-star closed.","title":"Krein\u2013\u0160mulian theorem for subspaces"},{"id":"tauceti/analysis/part-01/banach-space-theory/separation-lemma-of-the-krein-smulian-theorem","kind":"node","lean":"Convex.exists_forall_one_le_re_apply","parent":1592,"result":true,"route":"../roadmap/tauceti/analysis/part-01/banach-space-theory/index.html#separation-lemma-of-the-krein-smulian-theorem","scope":"tauceti/analysis/part-01/banach-space-theory","status":"fully_proved","summary":"Let `C` be a convex set of functionals on a Banach space whose intersection with every closed ball centred at zero is weak-star closed. If every element of `C` has norm greater than some `r > 0`, then evaluation at a single point `x` separates `C` from zero: `1 \u2264 re (f x)` for every `f \u2208 C`.","title":"Separation lemma of the Krein\u2013\u0160mulian theorem"},{"data_url":"scopes/tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th.json","id":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th","kind":"scope","parent":1570,"result":false,"route":"../roadmap/tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/index.html","scope":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Compact operators and nonarchimedean Fredholm theory"},{"id":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/serres-criterion-over-a-field","kind":"node","lean":"ContinuousLinearMap.isRowCompact_iff_isCompletelyContinuous, ContinuousLinearMap.IsCompletelyContinuous.isRowCompact_of_forall_fg_isClosed, ZeroAtInftyContinuousMap.isCompletelyContinuous_diagonalCLM_iff, TauCeti.isONable_iff_free_residueModule","parent":1613,"result":true,"route":"../roadmap/tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/index.html#serres-criterion-over-a-field","scope":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th","status":"fully_proved","summary":"over a complete ultrametric nontrivially normed field `K`, a continuous linear map between model spaces is row compact exactly when it is completely continuous. Neither a discrete valuation nor spherical completeness is needed.","title":"Serre's criterion over a field"},{"id":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/row-compactness-is-compactoid-image","kind":"node","lean":"ContinuousLinearMap.isRowCompact_iff_hasCompactoidImage","parent":1613,"result":true,"route":"../roadmap/tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/index.html#row-compactness-is-compactoid-image","scope":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th","status":"fully_proved","summary":"over an ultrametric nontrivially normed field, a map between model spaces is row compact exactly when it maps the closed unit ball to a compactoid set.","title":"Row compactness is compactoid image"},{"id":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/the-fredholm-determinant-is-entire","kind":"node","lean":"ContinuousLinearMap.IsRowCompact.isEntire_charPowerSeries","parent":1613,"result":true,"route":"../roadmap/tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/index.html#the-fredholm-determinant-is-entire","scope":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th","status":"fully_proved","summary":"For a row-compact `u`, the principal-minor series `det (1 - T u)` is a restricted power series at every radius.","title":"The Fredholm determinant is entire"},{"id":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/invertibility-and-the-fredholm-series","kind":"node","lean":"ContinuousLinearMap.IsRowCompact.isUnit_one_sub_smul_iff","parent":1613,"result":true,"route":"../roadmap/tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/index.html#invertibility-and-the-fredholm-series","scope":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th","status":"fully_proved","summary":"For a row-compact endomorphism `u` of `C\u2080(I, R)`, the operator `1 - a \u2022 u` is invertible if and only if the value `H_u(a)` of the Fredholm series `det (1 - T u)` is a unit of `R`.","title":"Invertibility and the Fredholm series"},{"id":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/the-nonzero-spectrum-of-a-row-compact-operator","kind":"node","lean":"ContinuousLinearMap.IsRowCompact.mem_spectrum_iff_evalAt_inv_eq_zero","parent":1613,"result":true,"route":"../roadmap/tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/index.html#the-nonzero-spectrum-of-a-row-compact-operator","scope":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th","status":"fully_proved","summary":"For a row-compact endomorphism `u` of `C\u2080(I, K)` over a complete ultrametric field and `\u03bc \u2260 0`, `\u03bc` lies in the spectrum of `u` if and only if `\u03bc\u207b\u00b9` is a zero of the Fredholm series `det (1 - T u)`.","title":"The nonzero spectrum of a row-compact operator"},{"id":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/continuouslinearmap-iscompletelycontinuous-isfre","kind":"node","lean":"ContinuousLinearMap.IsCompletelyContinuous.isFredholm_one_sub","parent":1613,"result":true,"route":"../roadmap/tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/index.html#continuouslinearmap-iscompletelycontinuous-isfre","scope":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th","status":"fully_proved","summary":"A completely continuous perturbation of the identity is a Fredholm operator, over any complete nontrivially normed field.","title":"ContinuousLinearMap.IsCompletelyContinuous.isFredholm_one_sub"},{"id":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/continuouslinearmap-isrowcompact-evalat-charpowe","kind":"node","lean":"ContinuousLinearMap.IsRowCompact.evalAt_charPowerSeries_eq_zero_iff_hasEigenvalue","parent":1613,"result":true,"route":"../roadmap/tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/index.html#continuouslinearmap-isrowcompact-evalat-charpowe","scope":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th","status":"fully_proved","summary":"The zeros of the Fredholm series of a row-compact operator are exactly the inverses of its nonzero eigenvalues. This conclusion concerns eigenvalues in the given coefficient field.","title":"ContinuousLinearMap.IsRowCompact.evalAt_charPowerSeries_eq_zero_iff_hasEigenvalue"},{"id":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/the-riesz-projector-at-a-good-zero","kind":"node","lean":"ContinuousLinearMap.IsRowCompact.isRieszProjection_rieszProjection","parent":1613,"result":true,"route":"../roadmap/tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/index.html#the-riesz-projector-at-a-good-zero","scope":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th","status":"fully_proved","summary":"Let `u` be row compact, and let `a` be a good zero of order `h` of its Fredholm determinant `H_u`: `\u0394\u02b2H_u(a) = 0` for `j < h` and `\u0394\u02b0H_u(a)` is a unit. Then `1 - (C b) ^ h` is a Riesz projection of `u` at `a` of order `h`.","title":"The Riesz projector at a good zero"},{"id":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/continuouslinearmap-isrowcompact","kind":"node","lean":"ContinuousLinearMap.IsRowCompact","parent":1613,"result":true,"route":"../roadmap/tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/index.html#continuouslinearmap-isrowcompact","scope":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th","status":"fully_proved","summary":"A continuous linear map between model spaces is row compact if the entries of its matrix are uniformly bounded and its row norms tend to zero along the cofinite filter of output indices. The bound is automatic over Tate scalars (`ContinuousLinearMap.isRowCompact_iff_tendsto_rowNorm`); it is part of the definition so that the real suprema defining the row norms are meaningful.","title":"ContinuousLinearMap.IsRowCompact"},{"id":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/a-row-compact-idempotent-has-projective-range","kind":"node","lean":"ContinuousLinearMap.IsRowCompact.projective_range_of_isIdempotentElem","parent":1613,"result":true,"route":"../roadmap/tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/index.html#a-row-compact-idempotent-has-projective-range","scope":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th","status":"fully_proved","summary":"Formalized in Tau Ceti as `ContinuousLinearMap.IsRowCompact.projective_range_of_isIdempotentElem`.","title":"A row-compact idempotent has projective range"},{"id":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/continuouslinearmap-charpowerseries","kind":"node","lean":"ContinuousLinearMap.charPowerSeries","parent":1613,"result":true,"route":"../roadmap/tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/index.html#continuouslinearmap-charpowerseries","scope":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th","status":"fully_proved","summary":"The Fredholm series `det (1 - T u)` of an endomorphism `u` of a module `M` with property (Pr): the principal-minor series of the extension `i \u2218 u \u2218 r` of `u` through a retract presentation `r \u2218 i = id` of `M` by a model space `C\u2080(J, R)`, fixed by choice. If `u` is row compact, it does not depend on the presentation (`ContinuousLinearMap.IsPrCompact.charPowerSeries_eq`). If `M` does not have property (Pr) the series\u2026","title":"ContinuousLinearMap.charPowerSeries"},{"id":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/independence-of-the-presentation","kind":"node","lean":"ContinuousLinearMap.IsPrCompact.charPowerSeries_eq","parent":1613,"result":true,"route":"../roadmap/tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/index.html#independence-of-the-presentation","scope":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th","status":"fully_proved","summary":"The Fredholm series of a row-compact endomorphism `u` of a module with property (Pr) is the principal-minor series of its extension `i \u2218 u \u2218 r` through any retract presentation `r \u2218 i = id` of the module by a model space.","title":"Independence of the presentation"},{"id":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/powerseries-isgoodzero","kind":"node","lean":"PowerSeries.IsGoodZero","parent":1613,"result":true,"route":"../roadmap/tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/index.html#powerseries-isgoodzero","scope":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th","status":"fully_proved","summary":"A power series `f` has a good zero of order `h` at `a` if its Hasse derivatives of order less than `h` vanish at `a` and its Hasse derivative of order `h` is a unit at `a`. For `h = 0` this says that `f(a)` is a unit. Values at `a` are the infinite sums `PowerSeries.evalAt`, so the notion is meaningful for entire series over a complete normed ring.","title":"PowerSeries.IsGoodZero"},{"id":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/the-riesz-identity-at-a-good-zero","kind":"node","lean":"ContinuousLinearMap.IsRowCompact.one_sub_smul_pow_succ_mul_inverse_smul_resolventHasseDerivAt","parent":1613,"result":true,"route":"../roadmap/tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/index.html#the-riesz-identity-at-a-good-zero","scope":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th","status":"fully_proved","summary":"Let `u` be row compact, put `C = 1 - a \u2022 u`, and suppose `\u0394\u02b2H_u(a) = 0` for `j < h` while `c = \u0394\u02b0H_u(a)` is a unit, so that `a` is a good zero of order `h` of the Fredholm determinant. Then `b = c\u207b\u00b9 \u2022 \u0394\u02b0V(a)` satisfies `C ^ (h + 1) b = C ^ h`, that is, `C ^ h (C b - 1) = 0`. This is the identity from which the Riesz projector `1 - (C b) ^ h` at `a` and its complementary projector `(C b) ^ h` are built.","title":"The Riesz identity at a good zero"},{"id":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/row-hodge-lower-bound-for-a-row-compact-operator","kind":"node","lean":"ContinuousLinearMap.IsRowCompact.rowHodgePolygon_rowNorm_le_newtonPolygon_charPowerSeries","parent":1613,"result":true,"route":"../roadmap/tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/index.html#row-hodge-lower-bound-for-a-row-compact-operator","scope":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th","status":"fully_proved","summary":"Let `v` be a real additive valuation on an ultrametric normed field with `\u2016x\u2016 = b ^ (-v x)` for `x \u2260 0`, where `b > 1`. For a row-compact endomorphism `u` of `C\u2080(I, K)`, the row Hodge polygon of its row norms lies below the Newton polygon of its principal-minor series `u.matrixCoeff.charPowerSeries`. When `K` is complete, these principal-minor sums converge and the series is the Fredholm series `det (1 - T u)`.","title":"Row Hodge lower bound for a row-compact operator"},{"id":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/strassmanns-bound-for-the-spectrum","kind":"node","lean":"ContinuousLinearMap.IsRowCompact.encard_setOf_mem_spectrum_mapCoeffs_le, ContinuousLinearMap.IsRowCompact.encard_setOf_mem_spectrum_le","parent":1613,"result":true,"route":"../roadmap/tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th/index.html#strassmanns-bound-for-the-spectrum","scope":"tauceti/analysis/part-01/compact-operators-and-nonarchimedean-fredholm-th","status":"fully_proved","summary":"Let `u` be a row-compact endomorphism of `C\u2080(I, K)`, and let its Fredholm series `H_u` be distinguished of degree `s` at the radius `\u03b5\u207b\u00b9`. Then the extension of `u` to `C\u2080(I, L)`, for a complete normed field extension `L` of `K`, has at most `s` spectral values of norm at least `\u03b5`.","title":"Strassmann's bound for the spectrum"},{"data_url":"scopes/tauceti/analysis/part-01/complex-morse-theory-affine-topology-and-weak-le.json","id":"tauceti/analysis/part-01/complex-morse-theory-affine-topology-and-weak-le","kind":"scope","parent":1570,"result":false,"route":"../roadmap/tauceti/analysis/part-01/complex-morse-theory-affine-topology-and-weak-le/index.html","scope":"tauceti/analysis/part-01/complex-morse-theory-affine-topology-and-weak-le","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Complex Morse theory, affine topology and weak Lefschetz"},{"id":"tauceti/analysis/part-01/complex-morse-theory-affine-topology-and-weak-le/the-index-bound-for-strongly-q-pseudoconvex-func","kind":"node","lean":"TauCeti.IsManifoldStronglyPseudoconvexAt.manifoldMorseIndex_le","parent":1630,"result":true,"route":"../roadmap/tauceti/analysis/part-01/complex-morse-theory-affine-topology-and-weak-le/index.html#the-index-bound-for-strongly-q-pseudoconvex-func","scope":"tauceti/analysis/part-01/complex-morse-theory-affine-topology-and-weak-le","status":"fully_proved","summary":"At a critical point `x` of a function `f` which is `C\u00b2` at `x` and strongly `q`-pseudoconvex at `x`, the Morse index of `f` is at most `dim_\u2102 E + q`.","title":"The index bound for strongly `q`-pseudoconvex functions on a complex manifold"},{"id":"tauceti/analysis/part-01/complex-morse-theory-affine-topology-and-weak-le/quadraticform-istotallyreal-of-add-comp-posdef","kind":"node","lean":"QuadraticForm.isTotallyReal_of_add_comp_posDef","parent":1630,"result":true,"route":"../roadmap/tauceti/analysis/part-01/complex-morse-theory-affine-topology-and-weak-le/index.html#quadraticform-istotallyreal-of-add-comp-posdef","scope":"tauceti/analysis/part-01/complex-morse-theory-affine-topology-and-weak-le","status":"fully_proved","summary":"A submodule on which `Q` is nonpositive is totally real for a linear endomorphism `J` if `Q(v) + Q(Jv)` is positive for every nonzero `v`.","title":"QuadraticForm.isTotallyReal_of_add_comp_posDef"},{"id":"tauceti/analysis/part-01/complex-morse-theory-affine-topology-and-weak-le/the-complex-hessian-identity-for-squared-distanc","kind":"node","lean":"TauCeti.fderiv_fderiv_norm_sq_sub_comp_add_I_smul","parent":1630,"result":true,"route":"../roadmap/tauceti/analysis/part-01/complex-morse-theory-affine-topology-and-weak-le/index.html#the-complex-hessian-identity-for-squared-distanc","scope":"tauceti/analysis/part-01/complex-morse-theory-affine-topology-and-weak-le","status":"fully_proved","summary":"For a holomorphic `C\u00b2` map, the sum of the real Hessian on `v` and `I \u2022 v` is four times the squared norm of the complex derivative. The identity holds at every point; at a critical point it is the intrinsic Hessian identity used in the squared-distance Morse-index estimate.","title":"The complex Hessian identity for squared distance"},{"id":"tauceti/analysis/part-01/complex-morse-theory-affine-topology-and-weak-le/paired-principal-curvatures","kind":"node","lean":"ContDiffAt.finrank_genEigenspace_shapeOperator_neg","parent":1630,"result":true,"route":"../roadmap/tauceti/analysis/part-01/complex-morse-theory-affine-topology-and-weak-le/index.html#paired-principal-curvatures","scope":"tauceti/analysis/part-01/complex-morse-theory-affine-topology-and-weak-le","status":"fully_proved","summary":"For a `C\u00b2` holomorphic map, multiplication by `I` identifies the generalized `\u03ba`- and `-\u03ba`-eigenspaces of each shape operator, so they have the same dimension. For `k = 1` these are the geometric multiplicities of the principal curvatures `\u03ba` and `-\u03ba`.","title":"Paired principal curvatures"},{"id":"tauceti/analysis/part-01/complex-morse-theory-affine-topology-and-weak-le/symmetric-focal-points","kind":"node","lean":"ContDiffAt.finrank_ker_fderiv_fderiv_norm_sq_sub_sub_comp","parent":1630,"result":true,"route":"../roadmap/tauceti/analysis/part-01/complex-morse-theory-affine-topology-and-weak-le/index.html#symmetric-focal-points","scope":"tauceti/analysis/part-01/complex-morse-theory-affine-topology-and-weak-le","status":"fully_proved","summary":"For a holomorphic immersion and a normal vector `\u03bd`, the points `i x + \u03bd` and `i x - \u03bd` have the same focal multiplicity, possibly zero: one is a focal point of `i` at `x` exactly when the other is, and the Hessians at `x` of squared distance from them have null spaces of the same dimension.","title":"Symmetric focal points"},{"id":"tauceti/analysis/part-01/complex-morse-theory-affine-topology-and-weak-le/tauceti-leviform","kind":"node","lean":"TauCeti.leviForm","parent":1630,"result":true,"route":"../roadmap/tauceti/analysis/part-01/complex-morse-theory-affine-topology-and-weak-le/index.html#tauceti-leviform","scope":"tauceti/analysis/part-01/complex-morse-theory-affine-topology-and-weak-le","status":"fully_proved","summary":"The Levi form of a real function `f` at `x`: the sesquilinear form `(D\u00b2f(x)(v, w) + D\u00b2f(x)(I \u2022 v, I \u2022 w) + I * (D\u00b2f(x)(I \u2022 v, w) - D\u00b2f(x)(v, I \u2022 w))) / 4`, conjugate-linear in `v` and complex-linear in `w`. Its matrix entries in holomorphic coordinates are the Wirtinger derivatives, with the arguments reversed: for `C\u00b2` `f` and coordinate vectors `e_j`, `leviForm f x (e_k) (e_j) = \u2202\u00b2f / \u2202z_j \u2202(conj z_k)`.","title":"TauCeti.leviForm"},{"id":"tauceti/analysis/part-01/complex-morse-theory-affine-topology-and-weak-le/tauceti-shapeoperator","kind":"node","lean":"TauCeti.shapeOperator","parent":1630,"result":true,"route":"../roadmap/tauceti/analysis/part-01/complex-morse-theory-affine-topology-and-weak-le/index.html#tauceti-shapeoperator","scope":"tauceti/analysis/part-01/complex-morse-theory-affine-topology-and-weak-le","status":"fully_proved","summary":"The shape operator `S_\u03bd` of `i : V \u2192 E` at `x` in the direction `\u03bd`: the endomorphism of `V` with `\u27eaDi(x) (S_\u03bd v), Di(x) w\u27eb = \u27ea\u03bd, B(v, w)\u27eb` for all `v` and `w`, where `B` is the second fundamental form. It is defined when `Di(x)` is injective, so that the induced metric `\u27eaDi(x) v, Di(x) w\u27eb` is nondegenerate, and is `0` otherwise.","title":"TauCeti.shapeOperator"},{"id":"tauceti/analysis/part-01/complex-morse-theory-affine-topology-and-weak-le/tauceti-complexprojectivezerolocus","kind":"node","lean":"TauCeti.complexProjectiveZeroLocus","parent":1630,"result":true,"route":"../roadmap/tauceti/analysis/part-01/complex-morse-theory-affine-topology-and-weak-le/index.html#tauceti-complexprojectivezerolocus","scope":"tauceti/analysis/part-01/complex-morse-theory-affine-topology-and-weak-le","status":"fully_proved","summary":"The common zero locus of a family of polynomials in `N` variables, as a set of complex lines in `\u2102^N`: a line belongs to it when every polynomial of the family vanishes at every vector of the line. The line of `L` is `L.submodule`, which is the underlying submodule of `Projectivization.equivSubmodule \u2102 (Fin N \u2192 \u2102) L`.","title":"TauCeti.complexProjectiveZeroLocus"},{"data_url":"scopes/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho.json","id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho","kind":"scope","parent":1570,"result":false,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/index.html","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Conformal mapping and the geometric theory of holomorphic functions"},{"data_url":"scopes/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01.json","id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01","kind":"scope","navigation":"transparent","parent":1639,"result":false,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/index.html","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01","status":"fully_proved","title":"The Riemann mapping theorem to Janiszewski's theorem"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/the-riemann-mapping-theorem","kind":"node","lean":"TauCeti.riemannMapping, TauCeti.exists_bijOn_ball_differentiableOn_invFunOn, TauCeti.riemannMapping_openPartialHomeomorph, TauCeti.riemannMapping_homeomorph, TauCeti.riemannMapping_of_hasHolomorphicSquareRoots, TauCeti.riemannMapping_normalized","parent":1640,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/index.html#the-riemann-mapping-theorem","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01","status":"fully_proved","summary":"A simply connected open proper subset `\u03a9` of `\u2102` admits a holomorphic bijection onto the open unit disc, with nonvanishing derivative throughout `\u03a9`.","title":"The Riemann mapping theorem"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/the-riemann-map-of-a-jordan-domain-extends-to-a","kind":"node","lean":"TauCeti.exists_homeomorph_closedBall_closure_of_isJordanCurve_frontier","parent":1640,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/index.html#the-riemann-map-of-a-jordan-domain-extends-to-a","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.exists_homeomorph_closedBall_closure_of_isJordanCurve_frontier`.","title":"The Riemann map of a Jordan domain extends to a homeomorphism of the closures"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/caratheodorys-continuity-theorem-as-a-characteri","kind":"node","lean":"TauCeti.exists_continuousOn_closedBall_eqOn_iff_locallyConnectedSpace_frontier, TauCeti.exists_continuousOn_closedBall_eqOn_of_locallyConnectedSpace_frontier, TauCeti.exists_continuousOn_bijOn_upperHalfPlaneSet_of_locallyConnectedSpace_frontier","parent":1640,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/index.html#caratheodorys-continuity-theorem-as-a-characteri","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01","status":"fully_proved","summary":"A holomorphic injection of a disc onto a bounded region extends continuously to the closed disc if and only if the boundary of the region is locally connected.","title":"Carath\u00e9odory's continuity theorem, as a characterization"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/the-schwarz-christoffel-theorem-for-a-bounded-po","kind":"node","lean":"TauCeti.exists_bijOn_const_mul_schwarzChristoffelPrimitive_add_of_isJordanCurve_frontier, TauCeti.exists_bijOn_const_mul_schwarzChristoffelPrimitive_add_of_isJordanCurve_of_halfStripExterior, TauCeti.exists_bijOn_const_mul_schwarzChristoffelPrimitive_add_of_isJordanCurve_of_halfStrip, TauCeti.exists_bijOn_const_mul_schwarzChristoffelPrimitive_add_of_polygonal_ends","parent":1640,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/index.html#the-schwarz-christoffel-theorem-for-a-bounded-po","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01","status":"fully_proved","summary":"Let `U` be a bounded, connected open set whose frontier is a Jordan curve. Suppose that `U` coincides near each frontier point other than the distinct vertices `v i` with an open half-plane, and near the vertex `v i` with the open sector of opening `(e i + 1) * \u03c0` at `v i`, where `e i \u2208 (-1, 1)`. Then there are distinct real prevertices `a i` and constants `A \u2260 0` and `B` such that `z \u21a6 A * F z + B` maps the upper\u2026","title":"The Schwarz--Christoffel theorem for a bounded polygonal Jordan domain"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/the-schwarz-christoffel-theorem-for-an-unbounded","kind":"node","lean":"TauCeti.exists_bijOn_const_mul_schwarzChristoffelPrimitive_add_of_isJordanCurve_insert_infty","parent":1640,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/index.html#the-schwarz-christoffel-theorem-for-an-unbounded","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01","status":"fully_proved","summary":"Let `U` be a connected open set whose frontier, together with the point at infinity, is a Jordan curve of the Riemann sphere. Suppose that `U` coincides near each frontier point other than the distinct vertices `v i` with an open half-plane, near the vertex `v i` with the open sector of opening `(e i + 1) * \u03c0` at `v i`, where `e i \u2208 (-1, 1)`, and far from a point `c` with the open sector\u2026","title":"The Schwarz--Christoffel theorem for an unbounded polygonal Jordan domain"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/schwarz-reflection-principle-real-axis","kind":"node","lean":"TauCeti.differentiableOn_schwarzReflection_of_symmetric, TauCeti.exists_differentiableOn_eqOn_conj_of_symmetric, TauCeti.differentiableOn_circleSchwarzReflection_of_symmetric, TauCeti.exists_differentiableOn_eqOn_lineReflection_of_symmetric, TauCeti.differentiableOn_lineSchwarzReflection_of_symmetric","parent":1640,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/index.html#schwarz-reflection-principle-real-axis","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01","status":"fully_proved","summary":"On a conjugation-symmetric open set `\u03a9`, if `f` is continuous on the closed upper part, holomorphic on the open upper part, and real-valued on the real axis of `\u03a9`, then the explicit reflection extension `schwarzReflection f` is holomorphic on all of `\u03a9` \u2014 in particular across the real axis.","title":"Schwarz reflection principle (real axis)"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/rigidity-of-the-pre-schwarzian-derivative","kind":"node","lean":"TauCeti.exists_eqOn_const_mul_add_iff_logDeriv_deriv_eqOn","parent":1640,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/index.html#rigidity-of-the-pre-schwarzian-derivative","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01","status":"fully_proved","summary":"Two holomorphic functions with nonvanishing derivatives on a domain have equal pre-Schwarzian derivatives exactly when one is obtained from the other by postcomposition with `w \u21a6 a * w + b` for a nonzero constant `a`.","title":"Rigidity of the pre-Schwarzian derivative"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/tauceti-primeend","kind":"node","lean":"TauCeti.PrimeEnd","parent":1640,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/index.html#tauceti-primeend","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01","status":"fully_proved","summary":"A prime end of `U \u2286 \u2102` is an equivalence class of null chains of `U`.","title":"TauCeti.PrimeEnd"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/the-prime-ends-of-a-jordan-domain-are-the-points","kind":"node","lean":"TauCeti.exists_bijective_primeEnd_map_unitDiscPrimeEnd_of_isJordanCurve_frontier","parent":1640,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/index.html#the-prime-ends-of-a-jordan-domain-are-the-points","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01","status":"fully_proved","summary":"Let `\u03a9` be a bounded domain whose frontier is a Jordan curve. There is a holomorphic bijection `g` of the unit disc onto `\u03a9` that is continuous and injective on the closed disc, and the map of prime ends it induces sends the radial prime ends of the disc at the points of the unit circle bijectively onto the prime ends of `\u03a9`.","title":"The prime ends of a Jordan domain are the points of the circle"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/a-crosscut-of-a-simply-connected-domain-has-exac","kind":"node","lean":"Path.IsCrosscut.natCard_connectedComponents_sdiff_range_eq_two","parent":1640,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/index.html#a-crosscut-of-a-simply-connected-domain-has-exac","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01","status":"fully_proved","summary":"If `\u03b3` is a crosscut of a simply connected open set `U \u2286 \u2102`, then `U \\ range \u03b3` has exactly two connected components.","title":"A crosscut of a simply connected domain has exactly two sides"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/the-jordan-separation-theorem","kind":"node","lean":"TauCeti.IsJordanCurve.not_isPreconnected_compl, TauCeti.IsJordanCurve.not_isPreconnected_compl_onePoint","parent":1640,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/index.html#the-jordan-separation-theorem","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01","status":"fully_proved","summary":"The complement of a Jordan curve in the plane is not preconnected.","title":"The Jordan separation theorem"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/the-frontier-of-a-complementary-component-of-a-j","kind":"node","lean":"TauCeti.IsJordanCurve.frontier_connectedComponentIn","parent":1640,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/index.html#the-frontier-of-a-complementary-component-of-a-j","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01","status":"fully_proved","summary":"for `x \u2209 C`, the component of `C\u1d9c` containing `x` has frontier `C`.","title":"The frontier of a complementary component of a Jordan curve is the whole curve"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/the-jordan-brouwer-separation-theorem","kind":"node","lean":"TauCeti.natCard_zerothHomotopy_sphere_compl_range_eq_two, TauCeti.not_isPreconnected_compl_range_sphere","parent":1640,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/index.html#the-jordan-brouwer-separation-theorem","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01","status":"fully_proved","summary":"The complement of the image of a continuous injection of the unit sphere of a finite-dimensional real normed space `E` into the unit sphere of a real normed space of dimension `dim E + 1` has exactly two path components (Hatcher, *Algebraic Topology*, Corollary 2B.2).","title":"The Jordan\u2013Brouwer separation theorem"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/the-borsuk-criterion","kind":"node","lean":"TauCeti.hasContinuousLogOn_sub_div_sub_iff","parent":1640,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/index.html#the-borsuk-criterion","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01","status":"fully_proved","summary":"For two points outside a bounded closed set `K \u2286 \u2102`, the Borsuk map `z \u21a6 (z - a) / (z - b)` has a continuous logarithm on `K` exactly when `K` fails to separate the two points; both sides already force `a` and `b` outside `K`. The easy direction is `TauCeti.hasContinuousLogOn_sub_div_sub`, the hard one `TauCeti.mem_connectedComponentIn_of_hasContinuousLogOn`.","title":"The Borsuk criterion"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/borsuks-separation-theorem","kind":"node","lean":"TauCeti.mem_connectedComponentIn_of_hasContinuousLogOn","parent":1640,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/index.html#borsuks-separation-theorem","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01","status":"fully_proved","summary":"If the Borsuk map `z \u21a6 (z - a) / (z - b)` of two points outside a bounded closed set `K \u2286 \u2102` has a continuous logarithm on `K`, then `K` does not separate them: `b` lies in the connected component of `a` in `K\u1d9c`.","title":"Borsuk's separation theorem"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/the-converse-schwarz-christoffel-theorem-from-th","kind":"node","lean":"TauCeti.eqOn_const_mul_schwarzChristoffelPrimitive_add_of_tendsto","parent":1640,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/index.html#the-converse-schwarz-christoffel-theorem-from-th","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01","status":"fully_proved","summary":"Let `f` be holomorphic with nonvanishing derivative on the upper half-plane and suppose its pre-Schwarzian derivative continues to a function `\u03c6` holomorphic off the distinct real prevertices `a i`, with singularities at worst simple poles of residues `e i`, and decaying at infinity. Then `f` is the affine image `A * F + B` of the normalized Schwarz--Christoffel primitive `F` for the data `a` and `e`, with `A` and\u2026","title":"The converse Schwarz--Christoffel theorem from the prevertex residues"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/the-equicontinuity-half-of-montels-theorem","kind":"node","lean":"TauCeti.IsLocallyBoundedOn.equicontinuousOn","parent":1640,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/index.html#the-equicontinuity-half-of-montels-theorem","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01","status":"fully_proved","summary":"A locally bounded family of holomorphic functions on an open set `U \u2286 \u2102` is equicontinuous on `U`.","title":"The equicontinuity half of Montel's theorem"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/hurwitzs-theorem","kind":"node","lean":"TauCeti.hurwitz, TauCeti.hurwitz_forall_ne, TauCeti.hurwitz_forall_ne_or_forall_eq, TauCeti.hurwitz_injOn, TauCeti.eventually_finsum_analyticOrderNatAt_ball_eq, TauCeti.hurwitz_eventually_exists_eq, TauCeti.hurwitz_eventually_exists_eq_zero","parent":1640,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/index.html#hurwitzs-theorem","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01","status":"fully_proved","summary":"On a connected open set, a locally uniform limit of functions that are holomorphic for all sufficiently large `i` and nowhere zero for infinitely many `i` is itself either nowhere zero or identically zero.","title":"Hurwitz's theorem"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/janiszewskis-second-theorem","kind":"node","lean":"TauCeti.not_isPreconnected_compl_union_of_not_isPreconnected_inter","parent":1640,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/index.html#janiszewskis-second-theorem","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01","status":"fully_proved","summary":"If two closed preconnected subsets `S`, `T` of the plane have a bounded intersection that is not preconnected, then their union separates the plane: the complement of `S \u222a T` is not preconnected.","title":"Janiszewski's second theorem"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/janiszewskis-theorem","kind":"node","lean":"TauCeti.janiszewski","parent":1640,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01/index.html#janiszewskis-theorem","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-01","status":"fully_proved","summary":"If two bounded closed subsets `S`, `T` of the plane have preconnected intersection and neither separates `a` from `b`, then their union does not separate them either.","title":"Janiszewski's theorem"},{"data_url":"scopes/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02.json","id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02","kind":"scope","navigation":"transparent","parent":1639,"result":false,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02/index.html","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02","status":"fully_proved","title":"The Jordan curve theorem to Wolff's lemma\u2026"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02/the-jordan-curve-theorem","kind":"node","lean":"TauCeti.IsJordanCurve.natCard_connectedComponents_compl_eq_two, TauCeti.IsJordanCurve.natCard_zerothHomotopy_compl_eq_two_onePoint","parent":1661,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02/index.html#the-jordan-curve-theorem","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02","status":"fully_proved","summary":"The complement of a Jordan curve in the plane has exactly two connected components: by `TauCeti.IsJordanCurve.connectedComponentIn_eq_filledHull_sdiff_or_compl` they are its inside `filledHull C \\ C` and its unbounded outside `(filledHull C)\u1d9c`.","title":"The Jordan curve theorem"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02/the-monodromy-theorem","kind":"node","lean":"TauCeti.monodromy_theorem, TauCeti.continuesInside_iff_exists_analyticOnNhd","parent":1661,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02/index.html#the-monodromy-theorem","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02","status":"fully_proved","summary":"Let `h` be a homotopy rel endpoints between two paths from `z\u2080` to `z\u2081` in `\u2102`, and suppose a germ at `z\u2080` continues along every path `h (t, \u00b7)` of the homotopy, all the continuations starting from that one germ. Then they all end at one and the same germ at `z\u2081`: the result of the continuation depends on the path only through its homotopy class.","title":"The monodromy theorem"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02/the-monodromy-theorem-for-a-free-homotopy-of-pat","kind":"node","lean":"TauCeti.monodromy_theorem_of_free_homotopy","parent":1661,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02/index.html#the-monodromy-theorem-for-a-free-homotopy-of-pat","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02","status":"fully_proved","summary":"Let `h` be a continuous map of the square, read as a family of paths `h (t, \u00b7)` whose *endpoints are allowed to move*, and suppose a family of germs continues along each of those paths. If the initial germs \u2014 the germ of `f t 0` at `h (t, 0)` \u2014 themselves continue along the initial edge `t \u21a6 h (t, 0)`, then the terminal germs continue along the terminal edge `t \u21a6 h (t, 1)`.","title":"The monodromy theorem for a free homotopy of paths"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02/the-monodromy-theorem-for-a-simply-connected-dom","kind":"node","lean":"TauCeti.ContinuesInside.exists_analyticOnNhd","parent":1661,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02/index.html#the-monodromy-theorem-for-a-simply-connected-dom","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02","status":"fully_proved","summary":"A germ that continues along every path of a simply connected open set `U` issuing from `z\u2080` is the germ at `z\u2080` of a single function analytic on all of `U`.","title":"The monodromy theorem for a simply connected domain"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02/moreras-theorem","kind":"node","lean":"TauCeti.morera","parent":1661,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02/index.html#moreras-theorem","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02","status":"fully_proved","summary":"A continuous function with values in a complex Banach space, defined on an open set of `\u2102` and conservative there (its integrals over the boundaries of rectangles in the set vanish), is complex differentiable on that set.","title":"Morera's theorem"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02/poincare-defect-identity-norm-form","kind":"node","lean":"TauCeti.norm_sq_one_sub_conj_mul_sub_norm_sq_sub","parent":1661,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02/index.html#poincare-defect-identity-norm-form","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02","status":"fully_proved","summary":"The difference of the squared norms of the Moebius denominator and numerator factors is the product of the two hyperbolic defects: `\u20161 - conj w * z\u2016 ^ 2 - \u2016z - w\u2016 ^ 2 = (1 - \u2016z\u2016 ^ 2) * (1 - \u2016w\u2016 ^ 2)`.","title":"Poincar\u00e9 defect identity (norm form)"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02/rouches-theorem-for-a-disc-classical-form","kind":"node","lean":"TauCeti.rouche, TauCeti.rouche_add, TauCeti.rouche_add_exists_eq_zero_iff, TauCeti.rouche_exists_eq_zero_iff, TauCeti.rouche_symm_exists_eq_zero_iff, TauCeti.rouche_windingNumber_comp, TauCeti.rouche_add_windingNumber_comp, TauCeti.rouche_symm_windingNumber_comp, TauCeti.rouche_add_nullHomologous, TauCeti.rouche_nullHomologous, TauCeti.rouche_symm_nullHomologous_of_analyticOnNhd, TauCeti.rouche_symm_nullHomologous, TauCeti.rouche_symm","parent":1661,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02/index.html#rouches-theorem-for-a-disc-classical-form","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02","status":"fully_proved","summary":"If `f` and `g` are holomorphic on the closed disc `closedBall c R` and `\u2016f z - g z\u2016 < \u2016f z\u2016` at every point `z` of the bounding circle, then `f` and `g` have the same number of zeros in `ball c R`, each counted with multiplicity.","title":"Rouch\u00e9's theorem for a disc, classical form"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02/the-schwarz-christoffel-formula","kind":"node","lean":"TauCeti.eqOn_const_mul_schwarzChristoffelPrimitive_add_of_polygonal_boundary, TauCeti.eqOn_const_mul_schwarzChristoffelPrimitive_add_of_polygonal_domain, TauCeti.eqOn_const_mul_schwarzChristoffelPrimitive_add_of_halfStripExterior_polygonal_domain, TauCeti.eqOn_const_mul_schwarzChristoffelPrimitive_add_of_halfStrip_polygonal_domain, TauCeti.eqOn_const_mul_schwarzChristoffelPrimitive_add_of_unbounded_polygonal_domain","parent":1661,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02/index.html#the-schwarz-christoffel-formula","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02","status":"fully_proved","summary":"Let `f` be holomorphic with nonvanishing derivative on the upper half-plane. Assume that away from the distinct real prevertices `a i` its boundary values run along affine lines with the upper half-plane on one side, and that each `a i` is carried to a vertex of one of four kinds: on the two bounding rays, where `-1 < e i < 1`; `b / (f - c)` extends across `a i` with value `0`, opens the sector of angle `\u03b2 * \u03c0`\u2026","title":"The Schwarz--Christoffel formula"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02/strict-schwarz-lemma","kind":"node","lean":"TauCeti.norm_deriv_lt_div_of_not_injOn, TauCeti.norm_deriv_lt_one_of_not_injOn","parent":1661,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02/index.html#strict-schwarz-lemma","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02","status":"fully_proved","summary":"A holomorphic map of `ball c R\u2081` into `closedBall (g c) R\u2082` that is not injective has `\u2016deriv g c\u2016 < R\u2082 / R\u2081`.","title":"Strict Schwarz lemma"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02/vitalis-convergence-theorem","kind":"node","lean":"TauCeti.vitali","parent":1661,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02/index.html#vitalis-convergence-theorem","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02","status":"fully_proved","summary":"Let `\u03a9` be an open preconnected subset of `\u2102`. A locally bounded sequence of holomorphic maps from `\u03a9` into a proper complex normed space `E` (for instance `\u2102`) which converges pointwise on a subset `A \u2286 \u03a9` having an accumulation point in `\u03a9` converges locally uniformly on `\u03a9` to a holomorphic map.","title":"Vitali's convergence theorem"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02/vitalis-theorem-with-prescribed-pointwise-values","kind":"node","lean":"TauCeti.vitali_of_tendsto","parent":1661,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02/index.html#vitalis-theorem-with-prescribed-pointwise-values","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02","status":"fully_proved","summary":"Under the hypotheses of `vitali`, if the sequence converges pointwise on `A` to a specified function `g`, its locally uniform holomorphic limit agrees with `g` throughout `A`.","title":"Vitali's theorem with prescribed pointwise values"},{"id":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02/wolffs-lemma-tauceti-exists-circleimagelength-sq","kind":"node","lean":"TauCeti.exists_circleImageLength_sq_lt, TauCeti.exists_circleImageLength_sq_lt_of_volume_image","parent":1661,"result":true,"route":"../roadmap/tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02/index.html#wolffs-lemma-tauceti-exists-circleimagelength-sq","scope":"tauceti/analysis/part-01/conformal-mapping-and-the-geometric-theory-of-ho/part-02","status":"fully_proved","summary":"If `2 \u03c0` times the Dirichlet integral of `f` over `s` is smaller than `c * log (R / r)`, then some circle of radius `\u03c1` between `r` and `R` has `\u2113 \u03c1 ^ 2 < c`.","title":"Wolff's lemma (`TauCeti.exists_circleImageLength_sq_lt`)"},{"data_url":"scopes/tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g.json","id":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g","kind":"scope","parent":1570,"result":false,"route":"../roadmap/tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/index.html","scope":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Contour integration and the Hungerb\u00fchler\u2013Wasem generalized residue theorem"},{"id":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/the-hungerbuhler-wasem-generalized-residue-theor","kind":"node","lean":"TauCeti.Contour.Cycle.hungerbuhlerWasem_residueTheorem, TauCeti.Contour.hungerbuhlerWasem_residueTheorem","parent":1674,"result":true,"route":"../roadmap/tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/index.html#the-hungerbuhler-wasem-generalized-residue-theor","scope":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g","status":"fully_proved","summary":"Let `U` be open, `S \u2286 U` finite, `f` holomorphic on `U \u2216 S` and meromorphic at each point of `S`, and let `C` be a contour cycle in `U`, null-homologous in `U`, whose curves are piecewise-`C\u00b9` immersions rooted off `S` and satisfy conditions (A\u2032) and (B). Then the Cauchy principal value of `f` along `C` exists and","title":"The Hungerb\u00fchler\u2013Wasem generalized residue theorem for a contour cycle (HW Thm 3.3)"},{"id":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/hungerbuhler-wasem-proposition-2-2-winding-decom","kind":"node","lean":"TauCeti.Contour.IsPwC1ImmersionOn.exists_crossingDecomposition","parent":1674,"result":true,"route":"../roadmap/tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/index.html#hungerbuhler-wasem-proposition-2-2-winding-decom","scope":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g","status":"fully_proved","summary":"Let `\u03b3` be a closed piecewise-`C\u00b9` immersion on `[a, b]`, with a basepoint away from `s`. There is a finite ordered family of disjoint crossing windows whose circular-cap excision is again piecewise `C\u00b9`, is closed, and avoids `s`. Its winding number accounts for the integer part of the original winding number; each crossing contributes its geometric angle divided by `2\u03c0`.","title":"Hungerb\u00fchler--Wasem Proposition 2.2 (winding decomposition)"},{"id":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/the-real-bounded-integrand-formula-allowing-cros","kind":"node","lean":"TauCeti.Contour.windingNumber_eq_real_integral_of_closed_interior_crossings","parent":1674,"result":true,"route":"../roadmap/tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/index.html#the-real-bounded-integrand-formula-allowing-cros","scope":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g","status":"fully_proved","summary":"For a closed piecewise-`C\u00b9` immersion `\u03b3` on `[[a, b]]` that avoids `s` at the basepoint `a` (`hsa`, so also at `b` by closedness \u2014 every other crossing of `s`, if any, is automatically interior to `[[a, b]]`) and is `C^{1,1}` on each side of any such crossing (`derivWithin \u03b3` Lipschitz on a one-sided closed piece ending or starting there, `h\u03b3_lip` \u2014 the two sides need not agree, so a crossing may coincide with a\u2026","title":"The real bounded-integrand formula, allowing crossings (Hungerb\u00fchler\u2013Wasem Prop 2.3)"},{"id":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/hungerbuhler-wasem-proposition-2-3-crossing-valu","kind":"node","lean":"TauCeti.Contour.tendsto_realWindingIntegrand_at_crossing_of_contDiffAt","parent":1674,"result":true,"route":"../roadmap/tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/index.html#hungerbuhler-wasem-proposition-2-3-crossing-valu","scope":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g","status":"fully_proved","summary":"For a curve that is `C\u00b2` at the crossing parameter `t\u2080` with nonvanishing velocity there, the real winding integrand about `z\u2080 = \u03b3 t\u2080` tends to `(\u1e8b \u00ff - \u1e8f \u1e8d) / (2 (\u1e8b\u00b2 + \u1e8f\u00b2))`, where `\u1e8b + i \u1e8f` and `\u1e8d + i \u00ff` are the first and second derivatives of `\u03b3` at `t\u2080`.","title":"Hungerb\u00fchler\u2013Wasem Proposition 2.3, crossing value, under a `C\u00b2` hypothesis"},{"id":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/the-homology-cauchy-theorem-roadmap-homologycauc","kind":"node","lean":"TauCeti.Contour.homologyCauchyTheorem, TauCeti.Contour.homologyCauchyTheorem_of_point_off_curve","parent":1674,"result":true,"route":"../roadmap/tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/index.html#the-homology-cauchy-theorem-roadmap-homologycauc","scope":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g","status":"fully_proved","summary":"Let `\u03b3` be a closed piecewise-`C\u00b9` curve on `[[a, b]]`, null-homologous in an open set `\u03a9` containing it, and let `f` be holomorphic on `\u03a9`. Then the contour integral of `f` along `\u03b3` vanishes: `\u222b t in a..b, deriv \u03b3 t \u2022 f (\u03b3 t) = 0`.","title":"The homology Cauchy theorem (roadmap `homologyCauchyTheorem`, Layer 3)"},{"id":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/cauchys-integral-formula-for-the-k-th-derivative","kind":"node","lean":"TauCeti.Contour.cauchyIntegralFormula_iteratedDeriv_nullHomologous, TauCeti.Contour.Cycle.cauchyIntegralFormula_nullHomologous, TauCeti.Contour.Cycle.cauchyIntegralFormula_deriv_nullHomologous, TauCeti.Contour.Cycle.cauchyIntegralFormula_deriv_smul_nullHomologous, TauCeti.Contour.Cycle.cauchyIntegralFormula_smul_nullHomologous, TauCeti.Contour.Cycle.cauchyIntegralFormula_iteratedDeriv_nullHomologous, TauCeti.Contour.Cycle.cauchyIntegralFormula_iteratedDeriv_smul_nullHomologous, TauCeti.Contour.cauchyIntegralFormula_deriv_nullHomologous, TauCeti.Contour.dixonH2_eq_windingNumber_mul_f_of_dixonFunction_eq_zero, TauCeti.Contour.cauchyIntegralFormula_ball, TauCeti.Contour.cauchyIntegralFormula_iteratedDeriv_ball, TauCeti.Contour.cauchyIntegralFormula_starConvex, TauCeti.Contour.cauchyIntegralFormula_iteratedDeriv_starConvex, TauCeti.Contour.cauchyIntegralFormula_nullHomologous","parent":1674,"result":true,"route":"../roadmap/tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/index.html#cauchys-integral-formula-for-the-k-th-derivative","scope":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g","status":"fully_proved","summary":"Let `f` be holomorphic on an open `U`, let `\u03b3` be a closed piecewise-`C\u00b9` curve in `U` that is null-homologous there, and let `z \u2208 U` lie off the curve. Then for every `k`,","title":"Cauchy's integral formula for the `k`-th derivative, homology form"},{"id":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/the-classical-residue-theorem-for-a-null-homolog","kind":"node","lean":"TauCeti.Contour.Cycle.classicalResidueTheorem_nullHomologous, TauCeti.Contour.Cycle.classicalResidueTheorem_starConvex, TauCeti.Contour.classicalResidueTheorem_nullHomologous","parent":1674,"result":true,"route":"../roadmap/tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/index.html#the-classical-residue-theorem-for-a-null-homolog","scope":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g","status":"fully_proved","summary":"Let `U` be open, `S` a finite set, `f` differentiable on `U \u2216 S` and meromorphic at each point of `S` lying in `U`, and let `C` be a contour cycle in `U`, null-homologous in `U`, whose trace avoids `S`. Then","title":"The classical residue theorem for a null-homologous contour cycle"},{"id":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/the-classical-residue-theorem-on-a-circle","kind":"node","lean":"TauCeti.Contour.classicalResidueTheorem_circle, TauCeti.Contour.classicalResidueTheorem_circle_of_meromorphicOrderAt_neg, TauCeti.Contour.classicalResidueTheorem_starConvex","parent":1674,"result":true,"route":"../roadmap/tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/index.html#the-classical-residue-theorem-on-a-circle","scope":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g","status":"fully_proved","summary":"If `f` is meromorphic on the closed disc `C(c, R)` (`R > 0`) and every point of nonzero meromorphic order lies in a finite set `S` inside the open disc, then `\u222e_{C(c,R)} f = 2\u03c0i \u00b7 \u2211_{s \u2208 S} residue f s`. Since residues at non-poles vanish, `classicalResidueTheorem_circle_of_meromorphicOrderAt_neg` proves the same conclusion asking only the poles (points of negative order) to lie in `S`.","title":"The classical residue theorem on a circle"},{"id":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/tauceti-contour-cycle","kind":"node","lean":"TauCeti.Contour.Cycle","parent":1674,"result":true,"route":"../roadmap/tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/index.html#tauceti-contour-cycle","scope":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g","status":"fully_proved","summary":"A contour cycle is a finite formal `\u2124`-linear combination of closed piecewise-`C\u00b9` curves.","title":"TauCeti.Contour.Cycle"},{"id":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/generalized-winding-number-hw-def-2-1","kind":"node","lean":"TauCeti.Contour.windingNumber","parent":1674,"result":true,"route":"../roadmap/tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/index.html#generalized-winding-number-hw-def-2-1","scope":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g","status":"fully_proved","summary":"`n_{z\u2080}(\u03b3) = (2\u03c0i)\u207b\u00b9 \u00b7 PV \u222e_\u03b3 dz/(z \u2212 z\u2080)`, the principal-value normalization of the index integral for a curve `\u03b3 : \u211d \u2192 \u2102` on `[a, b]` and any point `z\u2080`. See `windingNumber_eq_integral_of_avoidance` for its reduction to the ordinary index integral; as a `limUnder`-based value it is junk when the principal value does not exist.","title":"Generalized winding number (HW Def 2.1)"},{"id":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/first-exit-time-at-radius-left-side","kind":"node","lean":"TauCeti.Contour.firstExitTimeLeft","parent":1674,"result":true,"route":"../roadmap/tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/index.html#first-exit-time-at-radius-left-side","scope":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g","status":"fully_proved","summary":"the `sSup` of the times `t \u2208 [t\u2080 - \u03b4, t\u2080]` with `\u03b5 \u2264 \u2016\u03b3 t - s\u2016` \u2014 the latest outside-the-ball time before `t\u2080`, which is the first exit when moving left from `t\u2080`. The junk value is `sSup \u2205` when the defining set is empty.","title":"First exit time at radius `\u03b5` (left side)"},{"id":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/the-exact-local-angle-contribution-of-one-exit-t","kind":"node","lean":"TauCeti.Contour.windingNumber_sub_cap_exitCapWindow_eq_crossingAngle_div_two_pi","parent":1674,"result":true,"route":"../roadmap/tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/index.html#the-exact-local-angle-contribution-of-one-exit-t","scope":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g","status":"fully_proved","summary":"The winding number of the curve over the window minus that of its cap is `crossingAngle \u03b3 t\u2080 / 2\u03c0`. Beyond the standing continuity, tangent, and radius hypotheses -- which already give equal endpoint radii and endpoint matching -- the only extra input is the principal value of `(z - s)\u207b\u00b9` along the generally asymmetric exit-time interval.","title":"The exact local angle contribution of one exit-time cap window"},{"id":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/the-residue-of-the-cauchy-kernel","kind":"node","lean":"TauCeti.Contour.residue_div_sub_pow_of_analyticAt","parent":1674,"result":true,"route":"../roadmap/tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/index.html#the-residue-of-the-cauchy-kernel","scope":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g","status":"fully_proved","summary":"If `g` is analytic at `z\u2080`, then the function `g z / (z - z\u2080) ^ (k + 1)` has residue the Taylor coefficient `iteratedDeriv k g z\u2080 / k !` at `z\u2080`: the whole Laurent principal part comes from the first `k + 1` Taylor terms of `g`, and only the one of index `k` sits at order `\u22121`. This is the residue behind Cauchy's integral formula for the `k`-th derivative; the constant case is\u2026","title":"The residue of the Cauchy kernel"},{"id":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/null-homologous-cauchy-integral-formula-at-an-of","kind":"node","lean":"TauCeti.Contour.dixonH2_eq_windingNumber_mul_f_of_nullHomologous","parent":1674,"result":true,"route":"../roadmap/tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/index.html#null-homologous-cauchy-integral-formula-at-an-of","scope":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g","status":"fully_proved","summary":"Let `\u03b3` be a closed curve in an open set `U`, continuous on `uIcc a b`, differentiable off a countable set, with interval-integrable derivative, and null-homologous in `U`. If `f` is holomorphic on `U` and `w \u2208 U` is not on the curve, then the Cauchy-type integral evaluates to `2\u03c0i \u00b7 windingNumber \u03b3 a b w \u00b7 f w`.","title":"Null-homologous Cauchy integral formula at an off-curve point"},{"id":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/the-residue-theorem-for-a-fixed-polar-part-decom","kind":"node","lean":"TauCeti.Contour.PolarPartDecomposition.intervalIntegral_deriv_smul_eq_sum_windingNumber_mul_residue","parent":1674,"result":true,"route":"../roadmap/tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g/index.html#the-residue-theorem-for-a-fixed-polar-part-decom","scope":"tauceti/analysis/part-01/contour-integration-and-the-hungerbuhler-wasem-g","status":"fully_proved","summary":"Once `f` is presented on `U` as an analytic remainder plus finite Laurent tails at the points of `S`, a closed null-homologous piecewise-`C\u00b9` curve in `U` avoiding `S` integrates to the winding-weighted residue sum. This is the sum of the two preceding facts: the remainder contributes nothing (`intervalIntegral_deriv_smul_analyticRemainder_eq_zero`) and the tails contribute the residue sum\u2026","title":"The residue theorem for a fixed polar-part decomposition"},{"data_url":"scopes/tauceti/analysis/part-01/four-dimensional-seiberg-witten-theory.json","id":"tauceti/analysis/part-01/four-dimensional-seiberg-witten-theory","kind":"scope","parent":1570,"result":false,"route":"../roadmap/tauceti/analysis/part-01/four-dimensional-seiberg-witten-theory/index.html","scope":"tauceti/analysis/part-01/four-dimensional-seiberg-witten-theory","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Four-dimensional Seiberg\u2013Witten theory"},{"id":"tauceti/analysis/part-01/four-dimensional-seiberg-witten-theory/tauceti-seibergwitten-hasstrictfderivat-quadrati","kind":"node","lean":"TauCeti.SeibergWitten.hasStrictFDerivAt_quadraticEnd","parent":1690,"result":true,"route":"../roadmap/tauceti/analysis/part-01/four-dimensional-seiberg-witten-theory/index.html#tauceti-seibergwitten-hasstrictfderivat-quadrati","scope":"tauceti/analysis/part-01/four-dimensional-seiberg-witten-theory","status":"fully_proved","summary":"The real Fr\u00e9chet derivative of `Q\u2080` at `\u03c8` is `B(\u03c8, \u00b7)`, that is `DQ\u2080|_\u03c8 (\u03c6) = \u03c6\u03c8* + \u03c8\u03c6* - Re\u27ea\u03c8, \u03c6\u27eb Id`; this holds in the strict sense.","title":"TauCeti.SeibergWitten.hasStrictFDerivAt_quadraticEnd"},{"id":"tauceti/analysis/part-01/four-dimensional-seiberg-witten-theory/tauceti-seibergwitten-trace-quadraticendpolar-mu","kind":"node","lean":"TauCeti.SeibergWitten.trace_quadraticEndPolar_mul_of_mem_traceFreeSelfAdjoint","parent":1690,"result":true,"route":"../roadmap/tauceti/analysis/part-01/four-dimensional-seiberg-witten-theory/index.html#tauceti-seibergwitten-trace-quadraticendpolar-mu","scope":"tauceti/analysis/part-01/four-dimensional-seiberg-witten-theory","status":"fully_proved","summary":"Against a trace-free self-adjoint `T`, the polarization of `Q\u2080` pairs as `tr (B(\u03c8, \u03c6) T) = 2 Re\u27eaT \u03c8, \u03c6\u27eb`: the formal adjoint of the derivative `B(\u03c8, \u00b7)` of `Q\u2080` is `T \u21a6 2 T \u03c8`.","title":"TauCeti.SeibergWitten.trace_quadraticEndPolar_mul_of_mem_traceFreeSelfAdjoint"},{"id":"tauceti/analysis/part-01/four-dimensional-seiberg-witten-theory/frame-independence-of-the-chirality-element","kind":"node","lean":"CliffordAlgebra.chirality_eq","parent":1690,"result":true,"route":"../roadmap/tauceti/analysis/part-01/four-dimensional-seiberg-witten-theory/index.html#frame-independence-of-the-chirality-element","scope":"tauceti/analysis/part-01/four-dimensional-seiberg-witten-theory","status":"fully_proved","summary":"`\u0393 = -c(e\u2081) c(e\u2082) c(e\u2083) c(e\u2084)` for every positively oriented orthonormal basis `e`.","title":"Frame independence of the chirality element"},{"id":"tauceti/analysis/part-01/four-dimensional-seiberg-witten-theory/the-self-duality-relation","kind":"node","lean":"CliffordAlgebra.bivectorExterior_hodgeStar","parent":1690,"result":true,"route":"../roadmap/tauceti/analysis/part-01/four-dimensional-seiberg-witten-theory/index.html#the-self-duality-relation","scope":"tauceti/analysis/part-01/four-dimensional-seiberg-witten-theory","status":"fully_proved","summary":"the Clifford action of the Hodge dual of a two-form is the chirality element times the action of the two-form, `c\u2082(\u22c6\u03b1) = \u0393 c\u2082(\u03b1)`.","title":"The self-duality relation"},{"id":"tauceti/analysis/part-01/four-dimensional-seiberg-witten-theory/the-anticommutator-of-two-two-forms-in-the-cliff","kind":"node","lean":"CliffordAlgebra.bivectorExterior_mul_bivectorExterior_add_swap","parent":1690,"result":true,"route":"../roadmap/tauceti/analysis/part-01/four-dimensional-seiberg-witten-theory/index.html#the-anticommutator-of-two-two-forms-in-the-cliff","scope":"tauceti/analysis/part-01/four-dimensional-seiberg-witten-theory","status":"fully_proved","summary":"`c\u2082(\u03b1) c\u2082(\u03b2) + c\u2082(\u03b2) c\u2082(\u03b1) = -2 \u27ea\u03b1, \u03b2\u27eb - 2 \u27ea\u22c6\u03b1, \u03b2\u27eb \u0393`. Since `\u03b1 \u2227 \u03b2 = \u27ea\u22c6\u03b1, \u03b2\u27eb vol`, the coefficient of `\u0393` is minus twice the volume of `\u03b1 \u2227 \u03b2`.","title":"The anticommutator of two two-forms in the Clifford algebra of an oriented four-space"},{"id":"tauceti/analysis/part-01/four-dimensional-seiberg-witten-theory/orientation-hodgestar","kind":"node","lean":"Orientation.hodgeStar","parent":1690,"result":true,"route":"../roadmap/tauceti/analysis/part-01/four-dimensional-seiberg-witten-theory/index.html#orientation-hodgestar","scope":"tauceti/analysis/part-01/four-dimensional-seiberg-witten-theory","status":"fully_proved","summary":"The Hodge star `\u22c0\u1d4f V \u2243 \u22c0\u02e1 V`, `k + l = finrank \u211d V`, of an oriented real inner product space: Mathlib's `exteriorPower.hodgeStar` for the inner product and `Orientation.volumeEquiv`. It is characterized by `\u27ea\u22c6x, y\u27eb = vol (x \u2227 y)` (`Orientation.inner_hodgeStar_apply`).","title":"Orientation.hodgeStar"},{"id":"tauceti/analysis/part-01/four-dimensional-seiberg-witten-theory/tauceti-tracefreeselfadjoint","kind":"node","lean":"TauCeti.traceFreeSelfAdjoint","parent":1690,"result":true,"route":"../roadmap/tauceti/analysis/part-01/four-dimensional-seiberg-witten-theory/index.html#tauceti-tracefreeselfadjoint","scope":"tauceti/analysis/part-01/four-dimensional-seiberg-witten-theory","status":"fully_proved","summary":"The real subspace `{T : T* = T, tr T = 0}` of trace-free self-adjoint endomorphisms of a complex inner product space.","title":"TauCeti.traceFreeSelfAdjoint"},{"data_url":"scopes/tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces.json","id":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces","kind":"scope","parent":1570,"result":false,"route":"../roadmap/tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/index.html","scope":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Fuchsian groups and orbifold Riemann surfaces"},{"id":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/tauceti-modulargroup-jbiholomorph","kind":"node","lean":"TauCeti.ModularGroup.jBiholomorph","parent":1698,"result":true,"route":"../roadmap/tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/index.html#tauceti-modulargroup-jbiholomorph","scope":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces","status":"fully_proved","summary":"The normalized modular invariant identifies the compactified level-one modular quotient biholomorphically with the Riemann sphere.","title":"TauCeti.ModularGroup.jBiholomorph"},{"id":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/poincares-presentation-theorem","kind":"node","lean":"TauCeti.UpperHalfPlane.ConvexPolygon.SidePairing.relatorQuotientTo_bijective, TauCeti.UpperHalfPlane.ConvexPolygon.SidePairing.ker_lift_generatorsIn_eq_normalClosure_relators, TauCeti.UpperHalfPlane.ConvexPolygon.SidePairing.relatorQuotientTo_injective","parent":1698,"result":true,"route":"../roadmap/tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/index.html#poincares-presentation-theorem","scope":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces","status":"fully_proved","summary":"The group of a locally finite fundamental polygon is presented by the side generators, the side relations, and the finite-vertex cycle relations.","title":"Poincar\u00e9's presentation theorem"},{"id":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/dirichlet-presentations-of-cocompact-fuchsian-gr","kind":"node","lean":"Subgroup.exists_sidePairing_relatorQuotientTo_bijective_of_compactSpace","parent":1698,"result":true,"route":"../roadmap/tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/index.html#dirichlet-presentations-of-cocompact-fuchsian-gr","scope":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces","status":"fully_proved","summary":"Let `\u0393 \u2264 PSL(2, \u211d)` be discrete with compact orbit space, and let `p \u2208 \u210d` have trivial stabilizer. The Dirichlet domain of `\u0393` centred at `p` is a compact convex polygon `P` with a side pairing by elements of `\u0393`, and `\u0393` is presented by the side-pairing maps, subject to the side relations and the vertex-cycle relations.","title":"Dirichlet presentations of cocompact Fuchsian groups"},{"id":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/the-orbifold-area-formula-for-a-fundamental-poly","kind":"node","lean":"TauCeti.UpperHalfPlane.ConvexPolygon.SidePairing.covolume_eq_two_pi_mul","parent":1698,"result":true,"route":"../roadmap/tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/index.html#the-orbifold-area-formula-for-a-fundamental-poly","scope":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces","status":"fully_proved","summary":"Let the translates of a side-paired convex `n`-gon `P` by a group `\u0393` containing the side-pairing maps be locally finite, with disjoint interiors. Then the covolume of `\u0393` is `2\u03c0 ((n - k) / 2 - 1 - V + \u2211\u2091 (1 - 1 / m\u2091))`, where `k` is the number of self-paired sides, `V` the number of finite vertex cycles, and `e` runs over the elliptic orbits of `\u0393`, with stabilizer orders `m\u2091`.","title":"The orbifold area formula for a fundamental polygon"},{"id":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/the-number-of-cusps-of-a-locally-finite-fundamen","kind":"node","lean":"TauCeti.UpperHalfPlane.ConvexPolygon.SidePairing.natCard_cuspOrbit","parent":1698,"result":true,"route":"../roadmap/tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/index.html#the-number-of-cusps-of-a-locally-finite-fundamen","scope":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces","status":"fully_proved","summary":"The number of cusp orbits of `\u0393` is the number of ideal vertex cycles of `P`.","title":"The number of cusps of a locally finite fundamental polygon"},{"id":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/a-degree-one-map-is-a-biholomorphism","kind":"node","lean":"TauCeti.RiemannSurface.biholomorphOfDegreeEqOne","parent":1698,"result":true,"route":"../roadmap/tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/index.html#a-degree-one-map-is-a-biholomorphism","scope":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces","status":"fully_proved","summary":"construct a biholomorphism from a finite holomorphic map of degree one from a compact Riemann surface to a connected one.","title":"A degree-one map is a biholomorphism"},{"id":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/the-degree-of-a-finite-index-map-of-compactified","kind":"node","lean":"Subgroup.degree_compactifiedQuotientMap","parent":1698,"result":true,"route":"../roadmap/tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/index.html#the-degree-of-a-finite-index-map-of-compactified","scope":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces","status":"fully_proved","summary":"For `\u0394 \u2264 \u0393` discrete of finite index, the map `X(\u0394) \u2192 X(\u0393)` induced by the inclusion has degree `[\u0393 : \u0394]`.","title":"The degree of a finite-index map of compactified Fuchsian quotients is the index"},{"id":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/jrgensens-inequality","kind":"node","lean":"Subgroup.one_le_abs_trace_sq_sub_four_add_abs_trace_commutatorElement_sub_two, Subgroup.one_le_four_mul_sinh_sq_mul_of_dilation_mem, Subgroup.one_le_four_sub_trace_sq_mul_cosh_sq_of_smul_eq_self, Subgroup.one_le_abs_trace_sq_sub_four_add_abs_trace_commutatorElement_sub_two_of_abs_trace_lt_two, Subgroup.one_le_abs_trace_sq_sub_four_add_abs_trace_commutatorElement_sub_two_of_two_lt_abs_trace, Subgroup.one_le_abs_trace_commutatorElement_sub_two_of_isParabolic","parent":1698,"result":true,"route":"../roadmap/tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/index.html#jrgensens-inequality","scope":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces","status":"fully_proved","summary":"Let `\u0393 \u2264 PSL(2, \u211d)` be a discrete subgroup and let `A, B \u2208 SL(2, \u211d)` lift elements of `\u0393`. Unless `B` conjugates the class of `A` to itself or to its inverse, `|tr\u00b2 A - 4| + |tr [A, B] - 2| \u2265 1`.","title":"J\u00f8rgensen's inequality"},{"id":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/the-margulis-lemma-for-fuchsian-groups","kind":"node","lean":"Subgroup.conj_eq_or_conj_eq_inv_of_dist_le","parent":1698,"result":true,"route":"../roadmap/tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/index.html#the-margulis-lemma-for-fuchsian-groups","scope":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces","status":"fully_proved","summary":"Let `\u0393 \u2264 PSL(2, \u211d)` be a discrete subgroup and let `g, h \u2208 \u0393` both move a point `z \u2208 \u210d` a distance at most `1 / 4`. If `g` fixes no point of `\u210d`, that is, if `g` is parabolic or hyperbolic, then `h g h\u207b\u00b9 = g` or `h g h\u207b\u00b9 = g\u207b\u00b9`.","title":"The Margulis lemma for Fuchsian groups"},{"id":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/subgroup-compactifiedquotient","kind":"node","lean":"Subgroup.CompactifiedQuotient","parent":1698,"result":true,"route":"../roadmap/tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/index.html#subgroup-compactifiedquotient","scope":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces","status":"fully_proved","summary":"The compactified quotient of `\u0393 \u2264 PSL(2, \u211d)`: the coarse orbit space `\u0393 \\ \u210d` with one point adjoined for each cusp orbit of `\u0393`.","title":"Subgroup.CompactifiedQuotient"},{"id":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/compactness-from-a-dirichlet-domain-with-a-bound","kind":"node","lean":"Subgroup.compactSpace_compactifiedQuotient_of_isBounded_truncation","parent":1698,"result":true,"route":"../roadmap/tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/index.html#compactness-from-a-dirichlet-domain-with-a-bound","scope":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces","status":"fully_proved","summary":"If a discrete subgroup `\u0393` has finitely many cusp orbits, and some Dirichlet domain of `\u0393` is bounded after removing finitely many cusp horodiscs of positive heights, then the quotient `\u0393 \\ \u210d` with its cusp orbits adjoined is compact.","title":"Compactness from a Dirichlet domain with a bounded cusp truncation"},{"id":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/tauceti-thickpart","kind":"node","lean":"TauCeti.thickPart","parent":1698,"result":true,"route":"../roadmap/tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/index.html#tauceti-thickpart","scope":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces","status":"fully_proved","summary":"The `\u03b5`-thick part of an action on a metric space: the points moved a distance at least `\u03b5` by every nonidentity element. For `0 < \u03b5` it excludes every point with nontrivial stabilizer.","title":"TauCeti.thickPart"},{"id":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/the-thick-part-of-the-quotient-of-a-cofinite-gro","kind":"node","lean":"Subgroup.IsCofinite.isCompact_image_thickPart","parent":1698,"result":true,"route":"../roadmap/tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/index.html#the-thick-part-of-the-quotient-of-a-cofinite-gro","scope":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces","status":"fully_proved","summary":"For a cofinite `\u0393 \u2264 PSL(2, \u211d)` and `\u03b5 > 0`, the image in `\u0393 \\ \u210d` of the points moved a hyperbolic distance at least `\u03b5` by every nonidentity element of `\u0393` is compact.","title":"The thick part of the quotient of a cofinite group is compact"},{"id":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/tauceti-islocaltiling","kind":"node","lean":"TauCeti.IsLocalTiling","parent":1698,"result":true,"route":"../roadmap/tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/index.html#tauceti-islocaltiling","scope":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces","status":"fully_proved","summary":"A local tiling by a closed path-connected tile `P`: for each `x \u2208 P`, `N x` is a set of local neighbours, the elements `h` such that `h \u2022 P` is one of the tiles around `x`. The identity is a neighbour, `x \u2208 h \u2022 P` for each neighbour `h`, the neighbours of `x` seen from the tile `h \u2022 P` are the translates `h\u207b\u00b9 \u2022 N x`, and each point of `P` lies in a tiling chart made of its own neighbours.","title":"TauCeti.IsLocalTiling"},{"id":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/the-converse-of-poincares-polygon-theorem","kind":"node","lean":"Subgroup.IsCofinite.exists_sidePairing_closure_range_map_eq","parent":1698,"result":true,"route":"../roadmap/tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/index.html#the-converse-of-poincares-polygon-theorem","scope":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces","status":"fully_proved","summary":"Every cofinite `\u0393 \u2264 PSL(2, \u211d)` arises from Poincar\u00e9's polygon theorem. For a centre `p` with trivial stabilizer, the Dirichlet domain of `\u0393` centred at `p` is the carrier of a convex polygon with finitely many vertices and a side pairing whose maps generate `\u0393`. The cycle transformation of each finite vertex has finite order `m` and the interior angles along its cycle add up to `2\u03c0 / m`, and the cycle\u2026","title":"The converse of Poincar\u00e9's polygon theorem"},{"id":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/shimizus-lemma","kind":"node","lean":"Subgroup.inv_le_abs_apply_one_zero_of_upperRightHom_mem, Subgroup.im_smul_mul_im_le_abs_mul_of_upperRightHom_mem, Subgroup.im_smul_mul_im_le_sq_of_upperRightHom_mem","parent":1698,"result":true,"route":"../roadmap/tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces/index.html#shimizus-lemma","scope":"tauceti/analysis/part-01/fuchsian-groups-and-orbifold-riemann-surfaces","status":"fully_proved","summary":"Let `\u0393 \u2264 PSL(2, \u211d)` be a discrete subgroup containing the translation `z \u21a6 z + w` with `w \u2260 0`, and let `A \u2208 SL(2, \u211d)` lift an element of `\u0393`. If the lower-left entry of `A` does not vanish, that is, if the element does not fix `\u221e`, then that entry has absolute value at least `|w|\u207b\u00b9`. Both lifts of the element give the same absolute value.","title":"Shimizu's lemma"},{"data_url":"scopes/tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f.json","id":"tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f","kind":"scope","parent":1570,"result":false,"route":"../roadmap/tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f/index.html","scope":"tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Geometric measure theory: rectifiability, BV, and finite perimeter"},{"id":"tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f/the-weighted-area-formula-with-multiplicity","kind":"node","lean":"LipschitzWith.lintegral_normDet_fderiv_mul_eq_lintegral_tsum","parent":1715,"result":true,"route":"../roadmap/tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f/index.html#the-weighted-area-formula-with-multiplicity","scope":"tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f","status":"fully_proved","summary":"Integrating the geometric Jacobian of a Lipschitz map against a nonnegative measurable weight equals the normalized Hausdorff integral of the sum of the weights over each fiber. The parameter set may be Lebesgue measurable rather than Borel, and infinite fibers use extended sums.","title":"The weighted area formula with multiplicity"},{"id":"tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f/the-coarea-formula-for-total-variation","kind":"node","lean":"TauCeti.totalVariationOn_eq_lintegral_perimeter","parent":1715,"result":true,"route":"../roadmap/tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f/index.html#the-coarea-formula-for-total-variation","scope":"tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f","status":"fully_proved","summary":"For `u` locally integrable on `\u03a9`, against an additive Haar measure, the total variation of `u` is the integral of the perimeters of its superlevel sets: `|Du|(\u03a9) = \u222b_\u211d P({u > t}; \u03a9) dt`, finite or infinite.","title":"The coarea formula for total variation"},{"id":"tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f/banach-indicatrix-formula","kind":"node","lean":"ContinuousOn.eVariationOn_eq_lintegral_encard","parent":1715,"result":true,"route":"../roadmap/tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f/index.html#banach-indicatrix-formula","scope":"tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f","status":"fully_proved","summary":"The variation of a continuous curve on `[a, b]` is the raw one-dimensional Hausdorff integral of the extended cardinality of its fibers. No Lipschitz or bounded-variation hypothesis is needed: both sides are infinite together. Pauses and repeated traversals are allowed, and infinite fibers have infinite multiplicity.","title":"Banach indicatrix formula"},{"id":"tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f/tauceti-euclideanhausdorffmeasure-sphere","kind":"node","lean":"TauCeti.euclideanHausdorffMeasure_sphere","parent":1715,"result":true,"route":"../roadmap/tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f/index.html#tauceti-euclideanhausdorffmeasure-sphere","scope":"tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f","status":"fully_proved","summary":"A sphere of radius `r > 0` in an `n`-dimensional real inner product space has `(n - 1)`-dimensional Hausdorff measure `n \u03c9_n r ^ (n - 1)`, where `\u03c9_n` is the volume of the unit ball.","title":"TauCeti.euclideanHausdorffMeasure_sphere"},{"id":"tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f/the-sobolev-inequality-for-functions-of-bounded","kind":"node","lean":"TauCeti.eLpNorm_le_totalVariationOn","parent":1715,"result":true,"route":"../roadmap/tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f/index.html#the-sobolev-inequality-for-functions-of-bounded","scope":"tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f","status":"fully_proved","summary":"Let `E` have dimension `n \u2265 2` and let `p = n / (n - 1)`. Every integrable `u : E \u2192 \u211d` satisfies `\u2016u\u2016_p \u2264 C |Du|(E)`, where `|Du|(E)` is the total variation of `u` on the whole space, finite or infinite, and `C` is the constant `MeasureTheory.eLpNormLESNormFDerivOneConst \u03bc p` of the `C\u00b9` Gagliardo\u2013Nirenberg\u2013 Sobolev inequality `MeasureTheory.eLpNorm_le_eLpNorm_fderiv_one`.","title":"The Sobolev inequality for functions of bounded variation"},{"id":"tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f/the-nonsharp-isoperimetric-inequality","kind":"node","lean":"TauCeti.measure_rpow_le_perimeter","parent":1715,"result":true,"route":"../roadmap/tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f/index.html#the-nonsharp-isoperimetric-inequality","scope":"tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f","status":"fully_proved","summary":"Let `E` have dimension `n \u2265 2` and let `p = n / (n - 1)`. Every null-measurable set `s` of finite measure satisfies `\u03bc(s) ^ (1/p) \u2264 C P(s; E)`, where `1/p = 1 - 1/n`, `P(s; E)` is the perimeter of `s` in the whole space, finite or infinite, and `C` is the constant `MeasureTheory.eLpNormLESNormFDerivOneConst \u03bc p` of the Gagliardo\u2013Nirenberg\u2013Sobolev inequality.","title":"The nonsharp isoperimetric inequality"},{"id":"tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f/tauceti-totalvariationon","kind":"node","lean":"TauCeti.totalVariationOn","parent":1715,"result":true,"route":"../roadmap/tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f/index.html#tauceti-totalvariationon","scope":"tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f","status":"fully_proved","summary":"The total variation of `u : E \u2192 \u211d` on the open set `\u03a9`, relative to the measure `\u03bc`: the extended supremum of `\u222b div \u03c6 * u \u2202\u03bc` over test vector fields `\u03c6 \u2208 \ud835\udcd3(\u03a9, E)` with `\u2016\u03c6 x\u2016 \u2264 1` everywhere. For `\u03bc` an additive Haar measure this is the variation `|Du|(\u03a9)` of the distributional derivative of `u` on `\u03a9`, defined by duality.","title":"TauCeti.totalVariationOn"},{"id":"tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f/tauceti-perimeter","kind":"node","lean":"TauCeti.perimeter","parent":1715,"result":true,"route":"../roadmap/tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f/index.html#tauceti-perimeter","scope":"tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f","status":"fully_proved","summary":"The perimeter of a set `s` in the open set `\u03a9`, relative to the measure `\u03bc`: the total variation of the indicator of `s` on `\u03a9`. For a null-measurable `s` it is the extended supremum of `\u222b_s div \u03c6 \u2202\u03bc` over test vector fields `\u03c6 \u2208 \ud835\udcd3(\u03a9, E)` with `\u2016\u03c6 x\u2016 \u2264 1` everywhere (`TauCeti.perimeter_eq_iSup`).","title":"TauCeti.perimeter"},{"id":"tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f/boundedvariationon-exists-arclength-factorizatio","kind":"node","lean":"BoundedVariationOn.exists_arclength_factorization","parent":1715,"result":true,"route":"../roadmap/tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f/index.html#boundedvariationon-exists-arclength-factorizatio","scope":"tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f","status":"fully_proved","summary":"A continuous rectifiable curve factors through a 1-Lipschitz curve on its length interval. The factor has exactly the parameter difference as variation on every subinterval, including when the original curve has pauses or zero length. Values of the factor outside its length interval are unspecified.","title":"BoundedVariationOn.exists_arclength_factorization"},{"id":"tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f/the-directional-derivatives-of-a-function-of-loc","kind":"node","lean":"TauCeti.exists_regular_lineDeriv_ofFun_apply_eq","parent":1715,"result":true,"route":"../roadmap/tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f/index.html#the-directional-derivatives-of-a-function-of-loc","scope":"tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f","status":"fully_proved","summary":"If every compact `K \u2286 \u03a9` has an open neighbourhood `U` on which `u` has finite total variation, then for every direction `v` there are regular measures `\u03bd\u2081` and `\u03bd\u2082` on `\u03a9` with `\u2202_v u (\u03c6) = \u222b \u03c6 d\u03bd\u2081 - \u222b \u03c6 d\u03bd\u2082` for every test function `\u03c6`, where `\u2202_v u` is the derivative of the distribution `Distribution.ofFun \u03a9 u \u03bc \u22a4`. If `u` is locally integrable on `\u03a9`, this is the distributional derivative of `u`,\u2026","title":"The directional derivatives of a function of locally finite variation are signed Radon measures"},{"id":"tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f/banach-indicatrix-inequality","kind":"node","lean":"ContinuousOn.eVariationOn_le_lintegral_encard","parent":1715,"result":true,"route":"../roadmap/tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f/index.html#banach-indicatrix-inequality","scope":"tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f","status":"fully_proved","summary":"The variation of a continuous curve on `[a, b]` is at most the raw one-dimensional Hausdorff integral of the extended cardinality of its fibers. No injectivity or rectifiability hypothesis is needed; repeated traversals are counted by the fiber cardinality.","title":"Banach indicatrix inequality"},{"id":"tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f/the-sharp-bv-sobolev-bound","kind":"node","lean":"TauCeti.eLpNorm_le_inv_mul_totalVariationOn","parent":1715,"result":true,"route":"../roadmap/tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f/index.html#the-sharp-bv-sobolev-bound","scope":"tauceti/analysis/part-01/geometric-measure-theory-rectifiability-bv-and-f","status":"fully_proved","summary":"In dimension `n \u2265 2`, every integrable real-valued function satisfies `\u2016u\u2016\u209a \u2264 (n \u03c9\u2099^(1/n))\u207b\u00b9 |Du|(E)` for `p = n / (n - 1)`. The total variation may be infinite.","title":"The sharp BV Sobolev bound"},{"data_url":"scopes/tauceti/analysis/part-01/integral-currents-geometric-homology-and-minimiz.json","id":"tauceti/analysis/part-01/integral-currents-geometric-homology-and-minimiz","kind":"scope","parent":1570,"result":false,"route":"../roadmap/tauceti/analysis/part-01/integral-currents-geometric-homology-and-minimiz/index.html","scope":"tauceti/analysis/part-01/integral-currents-geometric-homology-and-minimiz","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Integral currents, geometric homology and minimizing hypersurfaces"},{"id":"tauceti/analysis/part-01/integral-currents-geometric-homology-and-minimiz/stokes-theorem-for-test-forms","kind":"node","lean":"Orientation.integrationCurrent_extDerivCLM","parent":1728,"result":true,"route":"../roadmap/tauceti/analysis/part-01/integral-currents-geometric-homology-and-minimiz/index.html#stokes-theorem-for-test-forms","scope":"tauceti/analysis/part-01/integral-currents-geometric-homology-and-minimiz","status":"fully_proved","summary":"the integration current annihilates every exact test form.","title":"Stokes' theorem for test forms"},{"id":"tauceti/analysis/part-01/integral-currents-geometric-homology-and-minimiz/d-d-0-on-test-forms-as-an-identity-of-continuous","kind":"node","lean":"TestFunction.extDerivCLM_comp_extDerivCLM","parent":1728,"result":true,"route":"../roadmap/tauceti/analysis/part-01/integral-currents-geometric-homology-and-minimiz/index.html#d-d-0-on-test-forms-as-an-identity-of-continuous","scope":"tauceti/analysis/part-01/integral-currents-geometric-homology-and-minimiz","status":"fully_proved","summary":"Formalized in Tau Ceti as `TestFunction.extDerivCLM_comp_extDerivCLM`.","title":"`d \u2218 d = 0` on test forms, as an identity of continuous linear maps"},{"id":"tauceti/analysis/part-01/integral-currents-geometric-homology-and-minimiz/the-derivative-of-a-regularization","kind":"node","lean":"Distribution.fderiv_translatePairing_apply","parent":1728,"result":true,"route":"../roadmap/tauceti/analysis/part-01/integral-currents-geometric-homology-and-minimiz/index.html#the-derivative-of-a-regularization","scope":"tauceti/analysis/part-01/integral-currents-geometric-homology-and-minimiz","status":"fully_proved","summary":"At every admissible translation, the derivative of `x \u21a6 T (\u03c1 (\u00b7 - x))` in the direction `v` is the regularization of the distributional derivative `\u2202_{v} T`.","title":"The derivative of a regularization"},{"id":"tauceti/analysis/part-01/integral-currents-geometric-homology-and-minimiz/distributions-with-vanishing-derivatives-have-lo","kind":"node","lean":"Distribution.translatePairing_eq_of_lineDerivOp_eq_zero","parent":1728,"result":true,"route":"../roadmap/tauceti/analysis/part-01/integral-currents-geometric-homology-and-minimiz/index.html#distributions-with-vanishing-derivatives-have-lo","scope":"tauceti/analysis/part-01/integral-currents-geometric-homology-and-minimiz","status":"fully_proved","summary":"If every directional derivative of `T` vanishes, then `x \u21a6 T (\u03c1 (\u00b7 - x))` is constant on every preconnected open set of translations for which `\u03c1 (\u00b7 - x)` is supported in `\u03a9`.","title":"Distributions with vanishing derivatives have locally constant regularizations"},{"id":"tauceti/analysis/part-01/integral-currents-geometric-homology-and-minimiz/a-distribution-has-finite-order-on-each-compact","kind":"node","lean":"Distribution.exists_norm_apply_le","parent":1728,"result":true,"route":"../roadmap/tauceti/analysis/part-01/integral-currents-geometric-homology-and-minimiz/index.html#a-distribution-has-finite-order-on-each-compact","scope":"tauceti/analysis/part-01/integral-currents-geometric-homology-and-minimiz","status":"fully_proved","summary":"For a compact `K \u2286 \u03a9` there are an order `N` and a constant `C` such that `\u2016T \u03c6\u2016 \u2264 C * M` for every test function `\u03c6` supported in `K` whose derivatives of order at most `N` are all bounded by `M`.","title":"A distribution has finite order on each compact set"},{"id":"tauceti/analysis/part-01/integral-currents-geometric-homology-and-minimiz/orientation-integrationcurrent","kind":"node","lean":"Orientation.integrationCurrent","parent":1728,"result":true,"route":"../roadmap/tauceti/analysis/part-01/integral-currents-geometric-homology-and-minimiz/index.html#orientation-integrationcurrent","scope":"tauceti/analysis/part-01/integral-currents-geometric-homology-and-minimiz","status":"fully_proved","summary":"The oriented integration current `[[\u03a9]]` of an open set `\u03a9` of an oriented real vector space `E` of dimension `m`: the continuous linear functional `\u03c9 \u21a6 \u222b_\u03a9 \u03c9` on test `m`-forms on `\u03a9`. For any basis `b` of orientation `o` it is `\u03c9 \u21a6 \u222b x, \u03c9 x b \u2202b.addHaar` (`Orientation.integrationCurrent_apply`). If `m` is not the dimension of `E`, it is zero.","title":"Orientation.integrationCurrent"},{"id":"tauceti/analysis/part-01/integral-currents-geometric-homology-and-minimiz/testfunction-extderivclm","kind":"node","lean":"TestFunction.extDerivCLM","parent":1728,"result":true,"route":"../roadmap/tauceti/analysis/part-01/integral-currents-geometric-homology-and-minimiz/index.html#testfunction-extderivclm","scope":"tauceti/analysis/part-01/integral-currents-geometric-homology-and-minimiz","status":"fully_proved","summary":"The exterior derivative of test `k`-forms on `\u03a9` with coefficients in `F`, as a continuous linear map into the test `(k + 1)`-forms. It is the alternatization `ContinuousAlternatingMap.alternatizeUncurryFinCLM` of the Fr\u00e9chet derivative `TestFunction.fderivCLM`, so pointwise it is Mathlib's `extDeriv`.","title":"TestFunction.extDerivCLM"},{"id":"tauceti/analysis/part-01/integral-currents-geometric-homology-and-minimiz/maps-on-vector-valued-tests-are-operator-valued","kind":"node","lean":"TestFunction.dualEquivDistribution","parent":1728,"result":true,"route":"../roadmap/tauceti/analysis/part-01/integral-currents-geometric-homology-and-minimiz/index.html#maps-on-vector-valued-tests-are-operator-valued","scope":"tauceti/analysis/part-01/integral-currents-geometric-homology-and-minimiz","status":"fully_proved","summary":"For a finite-dimensional real normed space `F` and a real normed space `V`, the continuous linear maps from the `F`-valued test functions on `\u03a9` to `V` correspond to the `F \u2192L[\u211d] V`-valued distributions on `\u03a9`: a map `T` corresponds to the distribution `\u03c8 \u21a6 (v \u21a6 T (\u03c8 \u2022 v))`. Both spaces carry the topology of uniform convergence on compact sets, and the correspondence is a homeomorphism for these topologies.","title":"Maps on vector-valued tests are operator-valued distributions"},{"data_url":"scopes/tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and.json","id":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and","kind":"scope","parent":1570,"result":false,"route":"../roadmap/tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and/index.html","scope":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"One-parameter semigroups, completely monotone and positive-definite functions, and Bochner-type representations"},{"id":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and/hille-yosida-characterization","kind":"node","lean":"TauCeti.Semigroups.hilleYosida_generation_iff","parent":1737,"result":true,"route":"../roadmap/tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and/index.html#hille-yosida-characterization","scope":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and","status":"fully_proved","summary":"An unbounded operator is the generator of a strongly continuous semigroup with growth bound `(omega, M)` if and only if `1 \u2264 M`, its domain is dense, the half-line `(omega, \u221e)` lies in its resolvent set, and its resolvent powers satisfy the sharp Hille--Yosida estimates.","title":"Hille--Yosida characterization"},{"id":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and/hille-yosida-for-contraction-semigroups","kind":"node","lean":"TauCeti.Semigroups.exists_contractionSemigroup_generator_eq_iff_forall_norm_resolvent_le, TauCeti.Semigroups.exists_generator_eq_and_hasGrowthBound_one_iff_forall_norm_resolvent_le, TauCeti.Semigroups.hilleYosida_generation","parent":1737,"result":true,"route":"../roadmap/tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and/index.html#hille-yosida-for-contraction-semigroups","scope":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and","status":"fully_proved","summary":"An unbounded operator `A` on a real Banach space generates a strongly continuous contraction semigroup if and only if its domain is dense, every `lambda > 0` lies in its resolvent set, and `\u2016R(lambda, A)\u2016 \u2264 1 / lambda` for every `lambda > 0`.","title":"Hille--Yosida for contraction semigroups"},{"id":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and/stones-theorem","kind":"node","lean":"LinearPMap.isSelfAdjoint_iff_exists_isUnitary_complexGenerator_eq_I_smul, IsSelfAdjoint.existsUnique_isUnitary_complexGenerator_eq_I_smul, IsSelfAdjoint.exists_isUnitary_complexGenerator_eq_I_smul","parent":1737,"result":true,"route":"../roadmap/tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and/index.html#stones-theorem","scope":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and","status":"fully_proved","summary":"An operator `A` on a complex Hilbert space is self-adjoint if and only if `i \u2022 A` is the complex generator of a unitary C\u2080-group.","title":"Stone's theorem"},{"id":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and/hausdorff-bernstein-widder-theorem-finite-measur","kind":"node","lean":"TauCeti.hausdorff_bernstein_widder, TauCeti.hausdorff_bernstein_widder_difference, TauCeti.hausdorff_bernstein_widder_onIoi","parent":1737,"result":true,"route":"../roadmap/tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and/index.html#hausdorff-bernstein-widder-theorem-finite-measur","scope":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and","status":"fully_proved","summary":"A function is continuous on `[0, \u221e)` and completely monotone on `(0, \u221e)` if and only if it is the Laplace transform of a finite positive measure on `\u211d\u22650`.","title":"Hausdorff--Bernstein--Widder theorem, finite-measure version on `\u211d\u22650`"},{"id":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and/bochners-characterization-without-countability-a","kind":"node","lean":"TauCeti.continuous_and_isPositiveDefiniteSub_iff_existsUnique_innerRegular_pontryaginMeasureTransform_eq","parent":1737,"result":true,"route":"../roadmap/tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and/index.html#bochners-characterization-without-countability-a","scope":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and","status":"fully_proved","summary":"A function on a locally compact abelian group is continuous and positive definite if and only if it is the Fourier\u2013Stieltjes transform of a unique finite inner regular measure on the Pontryagin dual.","title":"Bochner's characterization without countability assumptions"},{"id":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and/bochners-theorem","kind":"node","lean":"TauCeti.bochner, TauCeti.bochner_probabilityMeasure, TauCeti.bochner_euclideanSpace, TauCeti.isPositiveDefiniteSub_iff_exists_pontryaginMeasureTransform_eq, TauCeti.IsPositiveDefiniteSub.exists_pontryaginMeasureTransform_eq_of_continuousAt, TauCeti.bochner_charFun_real, TauCeti.bochner_charFun, TauCeti.bochner_posSemidef","parent":1737,"result":true,"route":"../roadmap/tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and/index.html#bochners-theorem","scope":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and","status":"fully_proved","summary":"A function `F` on a finite-dimensional real inner-product space is continuous and positive definite if and only if it is the Fourier-convention transform `v \u21a6 \u222b q, fourierAtom v q \u2202\u03bc` of a unique finite Borel measure `\u03bc`. Bochner (1932); Rudin, *Fourier Analysis on Groups*, Theorem 1.4.3.","title":"Bochner's theorem"},{"id":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and/the-berg-christensen-ressel-representation-theor","kind":"node","lean":"TauCeti.bcr_semigroup_bochner","parent":1737,"result":true,"route":"../roadmap/tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and/index.html#the-berg-christensen-ressel-representation-theor","scope":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and","status":"fully_proved","summary":"A function on the involutive semigroup `\u211d\u22650 \u00d7 V`, for `V` a finite-dimensional real inner-product space, is positive definite, continuous and bounded if and only if it is the Laplace--Fourier transform of a unique finite measure on `\u211d\u22650 \u00d7 V`:","title":"The Berg--Christensen--Ressel representation theorem (Berg--Christensen--Ressel 4.1.13)"},{"id":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and/tauceti-semigroups-stronglycontinuoussemigroup-o","kind":"node","lean":"TauCeti.Semigroups.StronglyContinuousSemigroup.ofBounded_resolvent_eq_inv_smul_tsum_pow","parent":1737,"result":true,"route":"../roadmap/tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and/index.html#tauceti-semigroups-stronglycontinuoussemigroup-o","scope":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and","status":"fully_proved","summary":"For `\u03bb > \u2016A\u2016`, the Laplace-transform resolvent of `t \u21a6 exp (tA)` is the Neumann series `\u03bb\u207b\u00b9 \u2211' n, (\u03bb\u207b\u00b9 A)\u207f`.","title":"TauCeti.Semigroups.StronglyContinuousSemigroup.ofBounded_resolvent_eq_inv_smul_tsum_pow"},{"id":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and/tauceti-complexification","kind":"node","lean":"TauCeti.Complexification","parent":1737,"result":true,"route":"../roadmap/tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and/index.html#tauceti-complexification","scope":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and","status":"fully_proved","summary":"The complexification `X \u2295 i X` of a real vector space `X`: an element with real part `re` and imaginary part `im` stands for the formal sum `re + i im`.","title":"TauCeti.Complexification"},{"id":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and/the-laplace-transform-turns-convolution-into-mul","kind":"node","lean":"TauCeti.laplaceTransform_conv","parent":1737,"result":true,"route":"../roadmap/tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and/index.html#the-laplace-transform-turns-convolution-into-mul","scope":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and","status":"fully_proved","summary":"The identity needs no finiteness or integrability hypothesis: under the junk-value convention for non-integrable Bochner integrals it holds for all s-finite measures and every real parameter `t`.","title":"The Laplace transform turns convolution into multiplication"},{"id":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and/converse-half-of-bochners-theorem","kind":"node","lean":"TauCeti.continuous_and_isPositiveDefiniteSub_of_integral_fourierAtom_eq","parent":1737,"result":true,"route":"../roadmap/tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and/index.html#converse-half-of-bochners-theorem","scope":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and","status":"fully_proved","summary":"The Fourier-convention transform of a finite Borel measure is continuous and positive definite.","title":"Converse half of Bochner's theorem"},{"id":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and/existence-half-of-bochners-theorem","kind":"node","lean":"TauCeti.exists_isFiniteMeasure_integral_fourierAtom_eq","parent":1737,"result":true,"route":"../roadmap/tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and/index.html#existence-half-of-bochners-theorem","scope":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and","status":"fully_proved","summary":"A continuous positive-definite function `F` on a finite-dimensional real inner-product space is the Fourier-convention transform `v \u21a6 \u222b q, fourierAtom v q \u2202\u03bc` of a finite Borel measure `\u03bc`. Rudin, *Fourier Analysis on Groups*, Theorem 1.4.3.","title":"Existence half of Bochner's theorem"},{"id":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and/existence-half-of-the-hausdorff-bernstein-widder","kind":"node","lean":"TauCeti.exists_representsLaplace_of_isContinuousCompletelyMonotoneOnIoi, TauCeti.exists_representsLaplace_of_isDifferenceCompletelyMonotone_of_continuousWithinAt","parent":1737,"result":true,"route":"../roadmap/tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and/index.html#existence-half-of-the-hausdorff-bernstein-widder","scope":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and","status":"fully_proved","summary":"a function continuous on `[0, \u221e)` and completely monotone on `(0, \u221e)` is the Laplace transform of a finite positive measure on `\u211d\u22650`.","title":"Existence half of the Hausdorff--Bernstein--Widder theorem"},{"id":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and/the-representing-measure-of-bochners-theorem","kind":"node","lean":"TauCeti.bochnerMeasure","parent":1737,"result":true,"route":"../roadmap/tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and/index.html#the-representing-measure-of-bochners-theorem","scope":"tauceti/analysis/part-01/one-parameter-semigroups-completely-monotone-and","status":"fully_proved","summary":"the unique finite Borel measure whose Fourier-convention transform is `F`, when `F` is continuous and positive definite, and the zero measure otherwise.","title":"The representing measure of Bochner's theorem"},{"data_url":"scopes/tauceti/analysis/part-01/operator-theory.json","id":"tauceti/analysis/part-01/operator-theory","kind":"scope","parent":1570,"result":false,"route":"../roadmap/tauceti/analysis/part-01/operator-theory/index.html","scope":"tauceti/analysis/part-01/operator-theory","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Operator theory"},{"id":"tauceti/analysis/part-01/operator-theory/polar-decomposition-of-a-bounded-operator","kind":"node","lean":"ContinuousLinearMap.polarFactor_comp_modulus","parent":1752,"result":true,"route":"../roadmap/tauceti/analysis/part-01/operator-theory/index.html#polar-decomposition-of-a-bounded-operator","scope":"tauceti/analysis/part-01/operator-theory","status":"fully_proved","summary":"`U |T| = T`.","title":"Polar decomposition of a bounded operator"},{"id":"tauceti/analysis/part-01/operator-theory/the-pairwise-sylvester-bound","kind":"node","lean":"TauCeti.UnitarilyInvariantSeminorm.mul_map_le_pi_div_two_mul_map_sylvester_apply, TauCeti.UnitarilyInvariantSeminorm.mul_map_le_pi_div_two_mul_map_sylvester_apply_of_mem","parent":1752,"result":true,"route":"../roadmap/tauceti/analysis/part-01/operator-theory/index.html#the-pairwise-sylvester-bound","scope":"tauceti/analysis/part-01/operator-theory","status":"fully_proved","summary":"Let `A` and `B` be symmetric operators on finite-dimensional inner product spaces. If every eigenvalue of `A` is at distance at least `\u03b4` from every eigenvalue of `B`, then `\u03b4 N(X) \u2264 (\u03c0 / 2) N(A X - X B)` for every unitarily invariant seminorm `N`. In particular every solution of the Sylvester equation `A X - X B = C` satisfies `\u03b4 N(X) \u2264 (\u03c0 / 2) N(C)`.","title":"The pairwise Sylvester bound"},{"id":"tauceti/analysis/part-01/operator-theory/the-perturbation-davis-kahan-sin-theorem","kind":"node","lean":"TauCeti.UnitarilyInvariantSeminorm.mul_map_starProjection_comp_starProjection_le_map_sub, TauCeti.UnitarilyInvariantSeminorm.mul_map_starProjection_comp_starProjection_le_map_sub_swap","parent":1752,"result":true,"route":"../roadmap/tauceti/analysis/part-01/operator-theory/index.html#the-perturbation-davis-kahan-sin-theorem","scope":"tauceti/analysis/part-01/operator-theory","status":"fully_proved","summary":"Let `A` and `B` be symmetric operators on a finite-dimensional inner product space, let `U` be invariant under `A` and `V` invariant under `B`. If the eigenvalues of `A` on `U` lie in `[a, b]` and those of `B` on `V\u15ee` lie outside `(a - \u03b4, b + \u03b4)`, then the directed sine map `P_{V\u15ee} P_U` from `U` towards `V` satisfies `\u03b4 N(P_{V\u15ee} P_U) \u2264 N(B - A)` for every unitarily invariant seminorm `N`.","title":"The perturbation Davis\u2013Kahan `sin \u0398` theorem"},{"id":"tauceti/analysis/part-01/operator-theory/fan-dominance","kind":"node","lean":"TauCeti.UnitarilyInvariantSeminorm.map_le_of_kyFanSum_le","parent":1752,"result":true,"route":"../roadmap/tauceti/analysis/part-01/operator-theory/index.html#fan-dominance","scope":"tauceti/analysis/part-01/operator-theory","status":"fully_proved","summary":"If every Ky Fan sum of `A` is at most the corresponding Ky Fan sum of `C`, then `N A \u2264 N C` for every unitarily invariant seminorm `N`.","title":"Fan dominance"},{"id":"tauceti/analysis/part-01/operator-theory/the-kato-rellich-theorem","kind":"node","lean":"IsSelfAdjoint.add_of_isRelativelyBounded","parent":1752,"result":true,"route":"../roadmap/tauceti/analysis/part-01/operator-theory/index.html#the-kato-rellich-theorem","scope":"tauceti/analysis/part-01/operator-theory","status":"fully_proved","summary":"Let `A` be a self-adjoint partial linear map on a Hilbert space and `V` a symmetric partial linear map that is relatively bounded with respect to `A` with constants `a` and `b < 1`, that is `dom A \u2286 dom V` and `\u2016V x\u2016 \u2264 a * \u2016x\u2016 + b * \u2016A x\u2016` on `dom A`. Then `A + V`, whose domain is `dom A`, is self-adjoint.","title":"The Kato\u2013Rellich theorem"},{"id":"tauceti/analysis/part-01/operator-theory/tauceti-projvalmeasure","kind":"node","lean":"TauCeti.ProjValMeasure","parent":1752,"result":true,"route":"../roadmap/tauceti/analysis/part-01/operator-theory/index.html#tauceti-projvalmeasure","scope":"tauceti/analysis/part-01/operator-theory","status":"fully_proved","summary":"A projection-valued measure on the measurable sets of `X`, acting on a complex Hilbert space `H`.","title":"TauCeti.ProjValMeasure"},{"id":"tauceti/analysis/part-01/operator-theory/tauceti-projvalmeasure-integral","kind":"node","lean":"TauCeti.ProjValMeasure.integral","parent":1752,"result":true,"route":"../roadmap/tauceti/analysis/part-01/operator-theory/index.html#tauceti-projvalmeasure-integral","scope":"tauceti/analysis/part-01/operator-theory","status":"fully_proved","summary":"The integral of a bounded Borel symbol against a projection-valued measure `P`, `P.integral f = \u222b f dP`, as a unital star algebra homomorphism. It is characterised by `integral_ofSimpleFunc` on simple symbols and its continuity (`continuous_integral`); see `eq_integral_of_continuous`.","title":"TauCeti.ProjValMeasure.integral"},{"id":"tauceti/analysis/part-01/operator-theory/tauceti-operatoridealfamily-traceclass","kind":"node","lean":"TauCeti.OperatorIdealFamily.traceClass","parent":1752,"result":true,"route":"../roadmap/tauceti/analysis/part-01/operator-theory/index.html#tauceti-operatoridealfamily-traceclass","scope":"tauceti/analysis/part-01/operator-theory","status":"fully_proved","summary":"The trace-class operator-ideal family, whose gauge is the nuclear gauge `\u03bd(T) = \u2211' n, a\u2099(T)`: the extended sum of the approximation numbers.","title":"TauCeti.OperatorIdealFamily.traceClass"},{"id":"tauceti/analysis/part-01/operator-theory/tauceti-operatoridealfamily-hilbertschmidt","kind":"node","lean":"TauCeti.OperatorIdealFamily.hilbertSchmidt","parent":1752,"result":true,"route":"../roadmap/tauceti/analysis/part-01/operator-theory/index.html#tauceti-operatoridealfamily-hilbertschmidt","scope":"tauceti/analysis/part-01/operator-theory","status":"fully_proved","summary":"The Hilbert\u2013Schmidt operator-ideal family. Its gauge is the square root `E_b(T)^(1/2)` of the Hilbert\u2013Schmidt energy, computed in a Hilbert basis `b` of the source chosen once and for all; by `gauge_hilbertSchmidt` every Hilbert basis gives the same value.","title":"TauCeti.OperatorIdealFamily.hilbertSchmidt"},{"id":"tauceti/analysis/part-01/operator-theory/continuouslinearmap-singularvalue","kind":"node","lean":"ContinuousLinearMap.singularValue","parent":1752,"result":true,"route":"../roadmap/tauceti/analysis/part-01/operator-theory/index.html#continuouslinearmap-singularvalue","scope":"tauceti/analysis/part-01/operator-theory","status":"fully_proved","summary":"The `n`-th approximation number `a\u2099(T)` of a bounded linear map `T`: the infimum of the operator norms `\u2016T - R\u2016` over all bounded linear maps `R` of rank at most `n`. The indexing is zero-based, so `a\u2080(T) = \u2016T\u2016` (`singularValue_zero`).","title":"ContinuousLinearMap.singularValue"},{"id":"tauceti/analysis/part-01/operator-theory/the-exact-mass-of-the-haagerup-zsido-kernel","kind":"node","lean":"TauCeti.HaagerupZsido.integral_norm_reciprocalKernel","parent":1752,"result":true,"route":"../roadmap/tauceti/analysis/part-01/operator-theory/index.html#the-exact-mass-of-the-haagerup-zsido-kernel","scope":"tauceti/analysis/part-01/operator-theory","status":"fully_proved","summary":"`\u222b_\u211d \u2016k(t)\u2016 dt = \u03c0 / 2`.","title":"The exact mass of the Haagerup\u2013Zsid\u00f3 kernel"},{"id":"tauceti/analysis/part-01/operator-theory/ky-fan-two-sided-ideal-inequality","kind":"node","lean":"ContinuousLinearMap.kyFanGauge_comp_comp_le","parent":1752,"result":true,"route":"../roadmap/tauceti/analysis/part-01/operator-theory/index.html#ky-fan-two-sided-ideal-inequality","scope":"tauceti/analysis/part-01/operator-theory","status":"fully_proved","summary":"`K\u2096(A T B) \u2264 \u2016A\u2016 K\u2096(T) \u2016B\u2016`.","title":"Ky Fan two-sided ideal inequality"},{"id":"tauceti/analysis/part-01/operator-theory/the-residual-davis-kahan-sin-theorem","kind":"node","lean":"TauCeti.UnitarilyInvariantSeminorm.mul_map_starProjection_comp_le_map_sylvester_apply","parent":1752,"result":true,"route":"../roadmap/tauceti/analysis/part-01/operator-theory/index.html#the-residual-davis-kahan-sin-theorem","scope":"tauceti/analysis/part-01/operator-theory","status":"fully_proved","summary":"Let `A` and `M` be symmetric operators on finite-dimensional inner product spaces `E` and `F`, let `U` be invariant under `A`, and let `X : F \u2192 E` be linear. If the eigenvalues of `M` lie in `[a, b]` and those of `A` on `U\u15ee` lie outside `(a - \u03b4, b + \u03b4)`, then `\u03b4 N(P_{U\u15ee} X) \u2264 N(A X - X M)` for every unitarily invariant seminorm `N`.","title":"The residual Davis\u2013Kahan `sin \u0398` theorem"},{"id":"tauceti/analysis/part-01/operator-theory/the-simultaneous-ky-fan-estimate-for-the-sylvest","kind":"node","lean":"ContinuousLinearMap.mul_kyFanSum_le_pi_div_two_mul_kyFanSum_sylvester_apply","parent":1752,"result":true,"route":"../roadmap/tauceti/analysis/part-01/operator-theory/index.html#the-simultaneous-ky-fan-estimate-for-the-sylvest","scope":"tauceti/analysis/part-01/operator-theory","status":"fully_proved","summary":"Let `A` and `B` be symmetric operators on finite-dimensional inner product spaces. If every eigenvalue of `A` is at distance at least `\u03b4` from every eigenvalue of `B`, then for every `k` the Ky Fan `k`-sums satisfy `\u03b4 K\u2096(X) \u2264 (\u03c0 / 2) K\u2096(A X - X B)`. In particular every solution of the Sylvester equation `A X - X B = C` satisfies `\u03b4 K\u2096(X) \u2264 (\u03c0 / 2) K\u2096(C)` for all `k` at once.","title":"The simultaneous Ky Fan estimate for the Sylvester equation"},{"data_url":"scopes/tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e.json","id":"tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e","kind":"scope","parent":1570,"result":false,"route":"../roadmap/tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e/index.html","scope":"tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Operator-algebra dynamics, crossed products, and equilibrium"},{"id":"tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e/dense-analytic-kms-criterion","kind":"node","lean":"TauCeti.CStarAction.isKMSState_iff_of_dense","parent":1767,"result":true,"route":"../roadmap/tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e/index.html#dense-analytic-kms-criterion","scope":"tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e","status":"fully_proved","summary":"A state is \u03b2-KMS if and only if the analytic KMS identity holds for pairs in a norm-dense invariant set of entire analytic elements. In particular, the set can be a norm-dense invariant star subalgebra.","title":"Dense analytic KMS criterion"},{"id":"tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e/kms-states-are-invariant","kind":"node","lean":"TauCeti.CStarAction.IsKMSState.apply_action","parent":1767,"result":true,"route":"../roadmap/tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e/index.html#kms-states-are-invariant","scope":"tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e","status":"fully_proved","summary":"For `\u03b2 \u2260 0`, every \u03b2-KMS state `\u03c9` is invariant under the dynamics: `\u03c9 (\u03b1_t(a)) = \u03c9 a`. At `\u03b2 = 0` the KMS states are the tracial states, which need not be invariant.","title":"KMS states are invariant"},{"id":"tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e/convexity-of-k","kind":"node","lean":"TauCeti.CStarAction.convex_kmsStates","parent":1767,"result":true,"route":"../roadmap/tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e/index.html#convexity-of-k","scope":"tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e","status":"fully_proved","summary":"Convex combinations of \u03b2-KMS states are \u03b2-KMS states.","title":"Convexity of `K_\u03b2`"},{"id":"tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e/weak-compactness-of-k","kind":"node","lean":"TauCeti.CStarAction.isCompact_kmsStates","parent":1767,"result":true,"route":"../roadmap/tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e/index.html#weak-compactness-of-k","scope":"tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e","status":"fully_proved","summary":"For unital `B`, the set of \u03b2-KMS states is weak-* compact.","title":"Weak-* compactness of `K_\u03b2`"},{"id":"tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e/pointwise-limits-of-kms-states","kind":"node","lean":"TauCeti.CStarAction.IsKMSState.of_tendsto","parent":1767,"result":true,"route":"../roadmap/tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e/index.html#pointwise-limits-of-kms-states","scope":"tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e","status":"fully_proved","summary":"If \u03b2-KMS states converge pointwise to a positive functional of norm one, then the limit is a \u03b2-KMS state.","title":"Pointwise limits of KMS states"},{"id":"tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e/tauceti-cstaraction-covariantrepresentation-inte","kind":"node","lean":"TauCeti.CStarAction.CovariantRepresentation.integratedOperator_convolution","parent":1767,"result":true,"route":"../roadmap/tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e/index.html#tauceti-cstaraction-covariantrepresentation-inte","scope":"tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e","status":"fully_proved","summary":"The integrated form of a covariant pair is multiplicative for twisted convolution with respect to a left-invariant measure: `(\u03c0 \u22ca U) (f \u22c6 g) = (\u03c0 \u22ca U) f * (\u03c0 \u22ca U) g`.","title":"TauCeti.CStarAction.CovariantRepresentation.integratedOperator_convolution"},{"id":"tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e/tauceti-cstaraction-covariantrepresentation-norm","kind":"node","lean":"TauCeti.CStarAction.CovariantRepresentation.norm_integratedOperator_le","parent":1767,"result":true,"route":"../roadmap/tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e/index.html#tauceti-cstaraction-covariantrepresentation-norm","scope":"tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e","status":"fully_proved","summary":"The integrated form is bounded by the coefficient function's L\u00b9 norm.","title":"TauCeti.CStarAction.CovariantRepresentation.norm_integratedOperator_le"},{"id":"tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e/tauceti-cstaraction-convolutionalgebra","kind":"node","lean":"TauCeti.CStarAction.ConvolutionAlgebra","parent":1767,"result":true,"route":"../roadmap/tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e/index.html#tauceti-cstaraction-convolutionalgebra","scope":"tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e","status":"fully_proved","summary":"The convolution \u22c6-algebra of the C\u22c6-dynamical system `(A, G, \u03b1)` with respect to `\u03bc`: the compactly supported continuous functions `G \u2192 A`, with twisted convolution as product and the twisted involution as star. Use `ConvolutionAlgebra.toConv` and `ConvolutionAlgebra.ofConv` to pass to and from `C_c(G, A)`, whose own product and star are pointwise.","title":"TauCeti.CStarAction.ConvolutionAlgebra"},{"id":"tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e/tauceti-cstaraction-covariantrepresentation-regu","kind":"node","lean":"TauCeti.CStarAction.CovariantRepresentation.regular","parent":1767,"result":true,"route":"../roadmap/tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e/index.html#tauceti-cstaraction-covariantrepresentation-regu","scope":"tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e","status":"fully_proved","summary":"The regular covariant representation on `L\u00b2(G, H)` induced by a representation `\u03c1` of `A` on `H`: `a` acts by `\u03be \u21a6 (r \u21a6 \u03c1 (\u03b1_{r\u207b\u00b9}(a)) (\u03be r))` and `s` by left translation `\u03be \u21a6 (r \u21a6 \u03be (s\u207b\u00b9 r))`.","title":"TauCeti.CStarAction.CovariantRepresentation.regular"},{"id":"tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e/tauceti-cstaraction-entireanalytic","kind":"node","lean":"TauCeti.CStarAction.entireAnalytic","parent":1767,"result":true,"route":"../roadmap/tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e/index.html#tauceti-cstaraction-entireanalytic","scope":"tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e","status":"fully_proved","summary":"The entire analytic elements of a real action, as a nonunital star subalgebra.","title":"TauCeti.CStarAction.entireAnalytic"},{"id":"tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e/density-of-entire-analytic-elements","kind":"node","lean":"TauCeti.CStarAction.dense_entireAnalytic","parent":1767,"result":true,"route":"../roadmap/tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e/index.html#density-of-entire-analytic-elements","scope":"tauceti/analysis/part-01/operator-algebra-dynamics-crossed-products-and-e","status":"fully_proved","summary":"The entire analytic elements of a point-norm continuous real action are dense, since every element is a limit of its Gaussian smearings.","title":"Density of entire analytic elements"},{"data_url":"scopes/tauceti/analysis/part-01/partial-differential-equations.json","id":"tauceti/analysis/part-01/partial-differential-equations","kind":"scope","parent":1570,"result":false,"route":"../roadmap/tauceti/analysis/part-01/partial-differential-equations/index.html","scope":"tauceti/analysis/part-01/partial-differential-equations","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Partial differential equations"},{"id":"tauceti/analysis/part-01/partial-differential-equations/existence-and-uniqueness-for-the-dirichlet-probl","kind":"node","lean":"TauCeti.PDE.UniformlyEllipticOn.existsUnique_isWeakSolutionDirichlet_of_mass_nonneg","parent":1779,"result":true,"route":"../roadmap/tauceti/analysis/part-01/partial-differential-equations/index.html#existence-and-uniqueness-for-the-dirichlet-probl","scope":"tauceti/analysis/part-01/partial-differential-equations","status":"fully_proved","summary":"Let `\u03a9 \u2286 \u211d\u207f`, `n \u2265 1`, be a bounded open set, and let `L u = -\u2202\u1d62(a\u2071\u02b2 \u2202\u2c7cu) + b\u2071 \u2202\u1d62u + c u` have measurable, uniformly elliptic `a`, measurable `b` with `\u2016b\u2016 \u2264 \u03b2` and measurable `c` with `0 \u2264 c \u2264 \u03b3`. Then for every `f \u2208 L\u00b2(\u03a9)` there is exactly one `u \u2208 H\u00b9\u2080(\u03a9)` with `a(u, v) = \u222b_\u03a9 f v` for all `v \u2208 H\u00b9\u2080(\u03a9)`. Since `\u03b2` is arbitrary, the energy form of `L` need not be coercive.","title":"Existence and uniqueness for the Dirichlet problem with `c \u2265 0`"},{"id":"tauceti/analysis/part-01/partial-differential-equations/de-giorgis-theorem-holder-continuity-of-weak-sol","kind":"node","lean":"TauCeti.PDE.exists_holderOnWith_preciseRepresentative, TauCeti.PDE.exists_holderOnWith_preciseRepresentative_jumpCoefficient, TauCeti.PDE.exists_holderOnWith_preciseRepresentative_of_isCompact, TauCeti.PDE.exists_holderOnWith_preciseRepresentative_of_isCompact_drift, TauCeti.PDE.exists_holderOnWith_preciseRepresentative_drift, TauCeti.PDE.exists_holderOnWith_preciseRepresentative_of_memLp","parent":1779,"result":true,"route":"../roadmap/tauceti/analysis/part-01/partial-differential-equations/index.html#de-giorgis-theorem-holder-continuity-of-weak-sol","scope":"tauceti/analysis/part-01/partial-differential-equations","status":"fully_proved","summary":"Let `n \u2265 1` be the dimension. There are `\u03b1 \u2208 (0, 1]` and `C > 0`, depending only on `\u03bb`, `\u039b`, the dimension and the normalization of the additive Haar measure `mu`, such that the following holds. Let `a` be measurable and uniformly elliptic on `\u03a9` with constants `\u03bb, \u039b`, and let `u \u2208 H\u00b9(\u03a9)` be a weak solution of `-\u2202\u1d62(a\u2071\u02b2 \u2202\u2c7cu) = 0`. If the closed `R`-thickening of a set `K` lies in `\u03a9`, then the precise\u2026","title":"De Giorgi's theorem: H\u00f6lder continuity of weak solutions"},{"id":"tauceti/analysis/part-01/partial-differential-equations/mosers-harnack-inequality","kind":"node","lean":"TauCeti.PDE.exists_ae_ae_value_le_mul_value","parent":1779,"result":true,"route":"../roadmap/tauceti/analysis/part-01/partial-differential-equations/index.html#mosers-harnack-inequality","scope":"tauceti/analysis/part-01/partial-differential-equations","status":"fully_proved","summary":"Let `n \u2265 1` be the dimension. There is `C > 0`, depending only on `\u03bb`, `\u039b`, the dimension and the normalization of the additive Haar measure `mu`, with the following property. Let `a` be measurable and uniformly elliptic on `\u03a9` with constants `\u03bb, \u039b`, and let `u \u2208 H\u00b9(\u03a9)` be a weak solution of `-\u2202\u1d62(a\u2071\u02b2 \u2202\u2c7cu) = 0`, that is `a(u, v) = 0` for every `v \u2208 H\u00b9\u2080(\u03a9)`, with `u \u2265 0` almost everywhere. Then on every ball with\u2026","title":"Moser's Harnack inequality"},{"id":"tauceti/analysis/part-01/partial-differential-equations/the-mihlin-hormander-multiplier-theorem-on-lp","kind":"node","lean":"TauCeti.exists_fourierMultiplierCLM_eq_and_norm_le_of_norm_iteratedFDeriv_le, TauCeti.exists_eLpNorm_fourierMultiplierCLM_le_of_norm_iteratedFDeriv_le","parent":1779,"result":true,"route":"../roadmap/tauceti/analysis/part-01/partial-differential-equations/index.html#the-mihlin-hormander-multiplier-theorem-on-lp","scope":"tauceti/analysis/part-01/partial-differential-equations","status":"fully_proved","summary":"Let `V` be a real inner product space of finite dimension `n \u2265 1`, let `2N > n` and `1 < p < \u221e`. There is a constant `C` such that for every symbol `m` of temperate growth satisfying Mihlin's condition of order `N` with constant `A`,","title":"The Mihlin\u2013H\u00f6rmander multiplier theorem on `L\u1d56`"},{"id":"tauceti/analysis/part-01/partial-differential-equations/the-calderon-zygmund-estimate-d2u-p-c-u-p","kind":"node","lean":"TauCeti.exists_eLpNorm_iteratedFDeriv_two_le_eLpNorm_laplacian, TauCeti.exists_eLpNorm_iteratedFDeriv_two_le_eLpNorm_laplacian_real, TauCeti.exists_eLpNorm_lineDerivOp_lineDerivOp_le_eLpNorm_laplacian, TauCeti.exists_norm_iteratedGradient_le_mul_norm_of_sum_mul_inner_eq, TauCeti.PDE.exists_eLpNorm_iteratedFDeriv_two_le_eLpNorm_sum_smul_fderiv_fderiv, TauCeti.PDE.exists_eLpNorm_iteratedFDeriv_two_le_eLpNorm_sum_smul_fderiv_fderiv_real, TauCeti.exists_norm_iteratedGradient_le_mul_norm_of_traceCLM_compLpL_eq, TauCeti.exists_eLpNorm_fourierMultiplierCLM_hessianSymbolLp_le, TauCeti.PDE.exists_norm_iteratedGradient_le_of_sum_mul_inner_eq_of_dist_lt_abs_sub_le, TauCeti.PDE.exists_norm_iteratedGradient_le_mul_norm_of_sum_mul_inner_eq_of_abs_sub_le, TauCeti.PDE.exists_norm_iteratedGradient_le_of_sum_mul_inner_eq_of_uniformContinuous, TauCeti.exists_norm_iteratedGradient_le_mul_norm_of_integral_laplacian_smul_eq, TauCeti.exists_norm_iteratedGradient_le_mul_norm_of_traceCLM_comp_eq_of_norm_sub_le_of_normalHalfSpace, TauCeti.exists_norm_iteratedGradient_le_mul_norm_of_integral_laplacian_smul_eq_of_normalHalfSpace, TauCeti.exists_norm_iteratedGradient_le_mul_norm_of_traceCLM_comp_eq_of_normalHalfSpace, TauCeti.exists_norm_iteratedGradient_le_mul_norm_of_traceCLM_compLpL_eq_of_normalHalfSpace","parent":1779,"result":true,"route":"../roadmap/tauceti/analysis/part-01/partial-differential-equations/index.html#the-calderon-zygmund-estimate-d2u-p-c-u-p","scope":"tauceti/analysis/part-01/partial-differential-equations","status":"fully_proved","summary":"Let `1 < p < \u221e`, and let `V` be a finite-dimensional real inner product space with its volume measure. There is a constant `C` with","title":"The Calder\u00f3n\u2013Zygmund estimate `\u2016D\u00b2u\u2016_p \u2264 C \u2016\u0394u\u2016_p`"},{"id":"tauceti/analysis/part-01/partial-differential-equations/existence-of-weak-solutions-for-linear-parabolic","kind":"node","lean":"TauCeti.exists_hasWeakLineDerivOn_gelfandDual","parent":1779,"result":true,"route":"../roadmap/tauceti/analysis/part-01/partial-differential-equations/index.html#existence-of-weak-solutions-for-linear-parabolic","scope":"tauceti/analysis/part-01/partial-differential-equations","status":"fully_proved","summary":"Let `V \u21aa H \u21aa V*` be a Gelfand triple: `V` a separable Hilbert space, `H` a Hilbert space and `\u03b9 : V \u2192L[\u211d] H` with dense range. Let `A(t) : V \u2192L[\u211d] V*` be weakly measurable (`t \u21a6 A(t) u v` almost everywhere strongly measurable on `(a, b)` for all `u, v`), bounded by `\u039b` and satisfy G\u00e5rding's inequality `\u03b1 \u2016v\u2016\u00b2 - \u03b2 \u2016\u03b9 v\u2016\u00b2 \u2264 A(t) v v` with `\u03b1 > 0`, for almost every `t \u2208 (a, b)`. Then for every `f \u2208 L\u00b2(a, b; V*)` and\u2026","title":"Existence of weak solutions for linear parabolic equations"},{"id":"tauceti/analysis/part-01/partial-differential-equations/tauceti-wkp","kind":"node","lean":"TauCeti.Wkp","parent":1779,"result":true,"route":"../roadmap/tauceti/analysis/part-01/partial-differential-equations/index.html#tauceti-wkp","scope":"tauceti/analysis/part-01/partial-differential-equations","status":"fully_proved","summary":"The arbitrary-order, real-valued weak Sobolev space `W^{k,p}(\u03a9)`. At order zero this is `L\u1d56(\u03a9)`; order one is `W1p`; every further order adjoins the weak derivative of the highest derivative field from the preceding order.","title":"TauCeti.Wkp"},{"id":"tauceti/analysis/part-01/partial-differential-equations/the-trace-operator-of-a-bounded-lipschitz-domain","kind":"node","lean":"TauCeti.W1p.frontierTrace","parent":1779,"result":true,"route":"../roadmap/tauceti/analysis/part-01/partial-differential-equations/index.html#the-trace-operator-of-a-bounded-lipschitz-domain","scope":"tauceti/analysis/part-01/partial-differential-equations","status":"fully_proved","summary":"The trace operator of a bounded Lipschitz domain, `W^{1,p}(\u03a9) \u2192 L\u1d56(frontier \u03a9)` for `p < \u221e` and normalized `dim E`-dimensional Euclidean Hausdorff measure on the frontier. It is characterized by `TauCeti.W1p.frontierTrace_ae_eq_of_continuousOn` and `TauCeti.W1p.eq_frontierTrace`: it is the unique bounded linear operator sending every function continuous up to the boundary to its restriction to the frontier.","title":"The trace operator of a bounded Lipschitz domain"},{"id":"tauceti/analysis/part-01/partial-differential-equations/tauceti-ckholderspace","kind":"node","lean":"TauCeti.CkHolderSpace","parent":1779,"result":true,"route":"../roadmap/tauceti/analysis/part-01/partial-differential-equations/index.html#tauceti-ckholderspace","scope":"tauceti/analysis/part-01/partial-differential-equations","status":"fully_proved","summary":"The space of bounded `C^k` maps whose iterated Fr\u00e9chet derivatives through order `k` are bounded and whose `k`-th iterated derivative is globally `\u03b1`-H\u00f6lder. Its norm is","title":"TauCeti.CkHolderSpace"},{"id":"tauceti/analysis/part-01/partial-differential-equations/de-giorgis-interior-oscillation-estimate","kind":"node","lean":"TauCeti.PDE.exists_ae_value_mem_Icc_add_mul_rpow_mul_rpow_mul_sqrt_setIntegral, TauCeti.PDE.exists_ae_value_mem_Icc_add_mul_rpow_mul_rpow_mul_sqrt_setIntegral_drift, TauCeti.PDE.exists_ae_value_mem_Icc_add_mul_rpow_mul_add_of_memLp","parent":1779,"result":true,"route":"../roadmap/tauceti/analysis/part-01/partial-differential-equations/index.html#de-giorgis-interior-oscillation-estimate","scope":"tauceti/analysis/part-01/partial-differential-equations","status":"fully_proved","summary":"Let `n \u2265 1` be the dimension. There are `\u03b1 \u2208 (0, 1]` and `C > 0`, depending only on `\u03bb`, `\u039b`, the dimension and the normalization of the additive Haar measure `mu`, such that the following holds. Let `a` be measurable and uniformly elliptic on `\u03a9` with constants `\u03bb, \u039b`, and let `u \u2208 H\u00b9(\u03a9)` be a weak solution of `-\u2202\u1d62(a\u2071\u02b2 \u2202\u2c7cu) = 0`. If `B(x\u2080, R) \u2286 \u03a9` and `0 < r \u2264 R/2`, then almost everywhere on `B(x\u2080, r)` the\u2026","title":"De Giorgi's interior oscillation estimate"},{"id":"tauceti/analysis/part-01/partial-differential-equations/the-logarithmic-caccioppoli-inequality","kind":"node","lean":"TauCeti.PDE.UniformlyEllipticOn.setIntegral_sq_mul_norm_gradient_div_value_sq_le, TauCeti.PDE.UniformlyEllipticOn.setIntegral_sq_mul_norm_gradient_sq_div_sq_le, TauCeti.PDE.exists_setIntegral_ball_norm_gradient_sq_div_sq_le","parent":1779,"result":true,"route":"../roadmap/tauceti/analysis/part-01/partial-differential-equations/index.html#the-logarithmic-caccioppoli-inequality","scope":"tauceti/analysis/part-01/partial-differential-equations","status":"fully_proved","summary":"Let `a` be measurable and uniformly elliptic on `\u03a9` with constants `0 < \u03bb \u2264 \u039b`, and let `u \u2208 H\u00b9(\u03a9)` be a weak supersolution of `-\u2202\u1d62(a\u2071\u02b2 \u2202\u2c7cu) \u2265 0`, that is `a(u, v) \u2265 0` for every nonnegative `v \u2208 H\u00b9\u2080(\u03a9)`, with `u \u2265 \u03b5` almost everywhere for some `\u03b5 > 0`. Then for every smooth `\u03c8` compactly supported in `\u03a9`,","title":"The logarithmic Caccioppoli inequality"},{"id":"tauceti/analysis/part-01/partial-differential-equations/mosers-crossover-estimate-tauceti-pde-exists-set","kind":"node","lean":"TauCeti.PDE.exists_setAverage_rpow_mul_setAverage_rpow_neg_le","parent":1779,"result":true,"route":"../roadmap/tauceti/analysis/part-01/partial-differential-equations/index.html#mosers-crossover-estimate-tauceti-pde-exists-set","scope":"tauceti/analysis/part-01/partial-differential-equations","status":"fully_proved","summary":"There are constants `c > 0` and `K`, depending only on the dimension `n \u2265 1`, with the following property. Let `a` be measurable and uniformly elliptic on `\u03a9` with constants `0 < \u03bb \u2264 \u039b`, and let `u \u2208 H\u00b9(\u03a9)` be a weak supersolution of `-\u2202\u1d62(a\u2071\u02b2 \u2202\u2c7cu) \u2265 0` with `u \u2265 \u03b5` almost everywhere for some `\u03b5 > 0`. Then for every ball with `B(x\u2080, 4\u221an r) \u2286 \u03a9` and every exponent `|p| \u2264 c \u03bb/\u039b`,","title":"Moser's crossover estimate (`TauCeti.PDE.exists_setAverage_rpow_mul_setAverage_rpow_neg_le`)"},{"id":"tauceti/analysis/part-01/partial-differential-equations/mosers-reverse-holder-inequality-for-powers-of-a","kind":"node","lean":"TauCeti.PDE.exists_eLpNorm_rpow_le_mul_eLpNorm_rpow","parent":1779,"result":true,"route":"../roadmap/tauceti/analysis/part-01/partial-differential-equations/index.html#mosers-reverse-holder-inequality-for-powers-of-a","scope":"tauceti/analysis/part-01/partial-differential-equations","status":"fully_proved","summary":"There is a constant `C \u2265 0`, depending only on the dimension, with the following property. Let `a` be measurable and uniformly elliptic on `\u03a9` with constants `0 < \u03bb \u2264 \u039b`, and suppose that `W^{1,2}_0(\u03a9)` satisfies the Sobolev inequality `\u2016v\u2016_q \u2264 S \u2016\u2207v\u2016\u2082`. Let `u \u2208 H\u00b9(\u03a9)` be a weak supersolution of `-\u2202\u1d62(a\u2071\u02b2 \u2202\u2c7cu) \u2265 0`, that is `a(u, v) \u2265 0` for every nonnegative `v \u2208 H\u00b9\u2080(\u03a9)`, with `u \u2265 \u03b5` almost everywhere for some\u2026","title":"Moser's reverse H\u00f6lder inequality for powers of a positive supersolution"},{"id":"tauceti/analysis/part-01/partial-differential-equations/mosers-weak-harnack-inequality","kind":"node","lean":"TauCeti.PDE.exists_ae_setAverage_rpow_rpow_inv_le_mul_value, TauCeti.PDE.exists_pos_ae_setAverage_rpow_rpow_inv_le_mul_value","parent":1779,"result":true,"route":"../roadmap/tauceti/analysis/part-01/partial-differential-equations/index.html#mosers-weak-harnack-inequality","scope":"tauceti/analysis/part-01/partial-differential-equations","status":"fully_proved","summary":"Let `n \u2265 1` be the dimension. For every exponent `s > 0` with `(n - 2) s < n`, that is `s < n/(n - 2)` when `n \u2265 3` and `s` arbitrary when `n \u2264 2`, there is `C > 0`, depending only on `\u03bb`, `\u039b`, `s`, the dimension and the normalization of the additive Haar measure `mu`, with the following property. Let `a` be measurable and uniformly elliptic on `\u03a9` with constants `\u03bb, \u039b`, and let `u \u2208 H\u00b9(\u03a9)` be a nonnegative weak\u2026","title":"Moser's weak Harnack inequality"},{"data_url":"scopes/tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul.json","id":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul","kind":"scope","parent":1570,"result":false,"route":"../roadmap/tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/index.html","scope":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Real-analytic geometry and semianalytic triangulation"},{"id":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/complexification-of-a-real-analytic-function-on","kind":"node","lean":"AnalyticOnNhd.exists_complexification","parent":1794,"result":true,"route":"../roadmap/tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/index.html#complexification-of-a-real-analytic-function-on","scope":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul","status":"fully_proved","summary":"A function of finitely many real variables that is analytic on an open set `U` extends to a function `F` of as many complex variables that is complex analytic on an open set `W` whose real points are exactly those of `U`. The set `W` is invariant under complex conjugation, `F` agrees with `f` at the real points of `U`, and `F` commutes with complex conjugation.","title":"Complexification of a real analytic function on an open set"},{"id":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/complexification-of-a-real-analytic-map-on-an-op","kind":"node","lean":"AnalyticOnNhd.exists_complexification_pi","parent":1794,"result":true,"route":"../roadmap/tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/index.html#complexification-of-a-real-analytic-map-on-an-op","scope":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul","status":"fully_proved","summary":"A map from `\u03b9 \u2192 \u211d` to `\u03ba \u2192 \u211d`, with `\u03b9` and `\u03ba` finite, that is analytic on an open set `U` extends to a map `F : (\u03b9 \u2192 \u2102) \u2192 (\u03ba \u2192 \u2102)` that is complex analytic on an open set `W` whose real points are exactly those of `U`. The set `W` is invariant under complex conjugation, `F` agrees with `f` at the real points of `U`, and `F` commutes with complex conjugation.","title":"Complexification of a real analytic map on an open set"},{"id":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/uniqueness-of-complexifications","kind":"node","lean":"AnalyticOnNhd.exists_eqOn_of_forall_real","parent":1794,"result":true,"route":"../roadmap/tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/index.html#uniqueness-of-complexifications","scope":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul","status":"fully_proved","summary":"Two complex analytic functions of finitely many variables on an open neighborhood `W` of the real points of an open set `U`, agreeing at those real points, agree on a smaller open set `V \u2286 W` that is invariant under complex conjugation and whose real points are exactly those of `U`.","title":"Uniqueness of complexifications"},{"id":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/equivalence-of-presentations","kind":"node","lean":"TauCeti.semianalyticAt_iff_exists_iUnion","parent":1794,"result":true,"route":"../roadmap/tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/index.html#equivalence-of-presentations","scope":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul","status":"fully_proved","summary":"A set is semianalytic at `x` if and only if, on some neighbourhood `U` of `x`, it agrees with a finite union of sets `{y | f y = 0 \u2227 0 < g\u2081 y \u2227 \u22ef \u2227 0 < g\u2096 y}` with `f, g\u2081, \u2026, g\u2096` analytic on `U`.","title":"Equivalence of presentations"},{"id":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/analytic-preimages","kind":"node","lean":"TauCeti.IsSemianalytic.preimage","parent":1794,"result":true,"route":"../roadmap/tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/index.html#analytic-preimages","scope":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul","status":"fully_proved","summary":"The preimage of a semianalytic set under a real analytic map is semianalytic.","title":"Analytic preimages"},{"id":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/analytic-isomorphisms","kind":"node","lean":"Equiv.isSemianalytic_image_iff","parent":1794,"result":true,"route":"../roadmap/tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/index.html#analytic-isomorphisms","scope":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul","status":"fully_proved","summary":"For a bijection `e` that is analytic with analytic inverse, the image `e '' s` is semianalytic if and only if `s` is semianalytic.","title":"Analytic isomorphisms"},{"id":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/geometry-simplicialcomplex-space-cone-of-space-e","kind":"node","lean":"Geometry.SimplicialComplex.space_cone_of_space_eq_intrinsicFrontier","parent":1794,"result":true,"route":"../roadmap/tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/index.html#geometry-simplicialcomplex-space-cone-of-space-e","scope":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul","status":"fully_proved","summary":"Coning a triangulation of the intrinsic frontier of a compact convex set `C` to a point of the intrinsic interior of `C` triangulates `C`: every point of `C` is either the apex or lies on a segment from the apex to a point of the intrinsic frontier.","title":"Geometry.SimplicialComplex.space_cone_of_space_eq_intrinsicFrontier"},{"id":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/tauceti-semianalyticat","kind":"node","lean":"TauCeti.SemianalyticAt","parent":1794,"result":true,"route":"../roadmap/tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/index.html#tauceti-semianalyticat","scope":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul","status":"fully_proved","summary":"A set `s \u2286 E` is *semianalytic at* `x` if on some neighbourhood `U` of `x` it agrees with a finite Boolean combination of zero sets and positivity sets of real functions analytic on `U`.","title":"TauCeti.SemianalyticAt"},{"id":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/tauceti-issemianalytic","kind":"node","lean":"TauCeti.IsSemianalytic","parent":1794,"result":true,"route":"../roadmap/tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/index.html#tauceti-issemianalytic","scope":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul","status":"fully_proved","summary":"A set `s \u2286 E` is *semianalytic* if it is semianalytic at every point of `E`: near every point, it agrees with a finite Boolean combination of zero sets and positivity sets of real analytic functions defined near that point.","title":"TauCeti.IsSemianalytic"},{"id":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/locally-finite-unions","kind":"node","lean":"LocallyFinite.isSemianalytic_iUnion","parent":1794,"result":true,"route":"../roadmap/tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/index.html#locally-finite-unions","scope":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul","status":"fully_proved","summary":"The union of a locally finite family of semianalytic sets is semianalytic.","title":"Locally finite unions"},{"id":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/tauceti-issemialgebraic-issemianalytic","kind":"node","lean":"TauCeti.IsSemialgebraic.isSemianalytic","parent":1794,"result":true,"route":"../roadmap/tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/index.html#tauceti-issemialgebraic-issemianalytic","scope":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul","status":"fully_proved","summary":"A semialgebraic subset of `\u03c3 \u2192 \u211d`, for a finite type `\u03c3`, is semianalytic: polynomial functions are analytic.","title":"TauCeti.IsSemialgebraic.isSemianalytic"},{"id":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/geometry-simplicialcomplex-cone","kind":"node","lean":"Geometry.SimplicialComplex.cone","parent":1794,"result":true,"route":"../roadmap/tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/index.html#geometry-simplicialcomplex-cone","scope":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul","status":"fully_proved","summary":"The cone `pK` on a simplicial complex `K` with apex `p`. Its faces are the nonempty finite sets whose vertices other than `p` form a face of `K` or are empty: the faces of `K`, the faces of `K` with `p` adjoined, and the vertex `{p}`.","title":"Geometry.SimplicialComplex.cone"},{"id":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/geometry-simplicialcomplex-isconeapex","kind":"node","lean":"Geometry.SimplicialComplex.IsConeApex","parent":1794,"result":true,"route":"../roadmap/tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/index.html#geometry-simplicialcomplex-isconeapex","scope":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul","status":"fully_proved","summary":"A point `p` is a *cone apex* for a geometric simplicial complex `K` if `p \u2209 |K|` and every segment from `p` to a point `y \u2208 |K|` meets `|K|` only at `y`. These are the conditions under which joining `p` to the faces of `K` gives a simplicial complex, the cone `K.cone p h`.","title":"Geometry.SimplicialComplex.IsConeApex"},{"id":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/identity-theorem-on-the-real-points","kind":"node","lean":"AnalyticAt.eventually_eq_zero_of_eventually_real","parent":1794,"result":true,"route":"../roadmap/tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/index.html#identity-theorem-on-the-real-points","scope":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul","status":"fully_proved","summary":"A complex analytic function of finitely many variables that vanishes at the real points near a real point vanishes near that point.","title":"Identity theorem on the real points"},{"id":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/real-analytic-weierstrass-preparation-theorem","kind":"node","lean":"AnalyticAt.exists_weierstrassPreparation_real","parent":1794,"result":true,"route":"../roadmap/tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul/index.html#real-analytic-weierstrass-preparation-theorem","scope":"tauceti/analysis/part-01/real-analytic-geometry-and-semianalytic-triangul","status":"fully_proved","summary":"Let `f : (\u03b9 \u2192 \u211d) \u00d7 \u211d \u2192 \u211d` be real analytic at `(x\u2080, t\u2080)`, with `\u03b9` finite, and suppose the vertical slice `t \u21a6 f (x\u2080, t)` has finite order of vanishing `d` at `t\u2080`. Then there exist real analytic coefficient functions `c : Fin d \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d` vanishing at `x\u2080` and a real analytic function `u` nonvanishing at `(x\u2080, t\u2080)` such that near `(x\u2080, t\u2080)`,\u2026","title":"Real analytic Weierstrass preparation theorem"},{"data_url":"scopes/tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber.json","id":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber","kind":"scope","parent":1570,"result":false,"route":"../roadmap/tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/index.html","scope":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Weighted orthogonal L\u00b2 bases \u2014 completeness, Hilbert-basis structure, and product bases of orthogonal systems"},{"id":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/the-hermite-functions-are-a-hilbert-basis-of-l2","kind":"node","lean":"TauCeti.hermiteHilbertBasis","parent":1810,"result":true,"route":"../roadmap/tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/index.html#the-hermite-functions-are-a-hilbert-basis-of-l2","scope":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber","status":"fully_proved","summary":"Orthonormality comes from the weighted-measure machinery and completeness from moment determinacy (`TauCeti.orthogonal_span_range_bareNormalizedLp_eq_bot`), applied to the exact-degree family `H\u2099(x\u221a2)` against the Gaussian weight `e^{-x\u00b2}`.","title":"The Hermite functions are a Hilbert basis of `L\u00b2(\u211d)`"},{"id":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/parsevals-identity-for-the-hermite-basis-norm-sq","kind":"node","lean":"TauCeti.tsum_norm_sq_inner_hermiteFunctionLp, TauCeti.tsum_norm_sq_integral_normalizedChebyshevT_mul_measureT, TauCeti.tsum_norm_sq_integral_normalizedChebyshevCosine_mul, TauCeti.tsum_norm_sq_intervalIntegral_normalizedChebyshevCosine_mul, TauCeti.tsum_star_integral_normalizedChebyshevCosine_mul_mul_integral, TauCeti.tsum_star_intervalIntegral_normalizedChebyshevCosine_mul_mul_intervalIntegral, TauCeti.tsum_inner_mul_inner_hermiteFunctionLp, TauCeti.tsum_inner_mul_inner_L2piMul_hermiteFunctionLp, TauCeti.tsum_norm_sq_inner_L2piMul_hermiteFunctionLp","parent":1810,"result":true,"route":"../roadmap/tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/index.html#parsevals-identity-for-the-hermite-basis-norm-sq","scope":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber","status":"fully_proved","summary":"the squared Hermite coordinates of `f` sum to `\u2016f\u2016\u00b2`.","title":"Parseval's identity for the Hermite basis (norm-square form)"},{"id":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/the-chebyshev-t-polynomials-are-a-hilbert-basis","kind":"node","lean":"TauCeti.chebyshevTHilbertBasis","parent":1810,"result":true,"route":"../roadmap/tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/index.html#the-chebyshev-t-polynomials-are-a-hilbert-basis","scope":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber","status":"fully_proved","summary":"Orthonormality is `orthonormal_normalizedChebyshevTLp`; completeness is `orthogonal_span_normalizedChebyshevTLp_eq_bot`.","title":"The Chebyshev `T` polynomials are a Hilbert basis of `L\u00b2(measureT)`"},{"id":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/the-chebyshev-expansion","kind":"node","lean":"TauCeti.hasSum_chebyshevT_expansion","parent":1810,"result":true,"route":"../roadmap/tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/index.html#the-chebyshev-expansion","scope":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber","status":"fully_proved","summary":"Every `f \u2208 L\u00b2(measureT)` is the sum of its normalized Chebyshev-polynomial series.","title":"The Chebyshev expansion"},{"id":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/moment-determinacy-exponential-moment-form-the-r","kind":"node","lean":"TauCeti.Measure.ext_of_forall_integral_pow_eq_of_exists_integrable_exp","parent":1810,"result":true,"route":"../roadmap/tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/index.html#moment-determinacy-exponential-moment-form-the-r","scope":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber","status":"fully_proved","summary":"A finite measure on `\u211d` for which some exponential moment `\u222b e^{a|x|} d\u03bc` with `a > 0` is finite is determined by its polynomial moments. This is the form the completeness argument consumes: the hypothesis holds for Gaussian decay and automatically for compactly supported measures.","title":"Moment determinacy, exponential-moment form (the roadmap's B1 hypothesis)"},{"id":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/the-rescaled-hermite-functions-are-a-hilbert-bas","kind":"node","lean":"TauCeti.twoPiHermiteHilbertBasis","parent":1810,"result":true,"route":"../roadmap/tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/index.html#the-rescaled-hermite-functions-are-a-hilbert-bas","scope":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber","status":"fully_proved","summary":"This is the Hermite basis in the normalization Mathlib's `e^{-2\u03c0ix\u03be}` Fourier convention forces: the same bridge `TauCeti.hilbertBasisOfOrthogonalSystem`, run at the weight `e^{-2\u03c0x\u00b2}` and the polynomials `H\u2099(2\u221a\u03c0\u00b7x)`.","title":"The rescaled Hermite functions are a Hilbert basis of `L\u00b2(\u211d)`"},{"id":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/the-fourier-eigenrelation-for-the-unitary-fourie","kind":"node","lean":"TauCeti.fourier_twoPiHermiteFunctionLp","parent":1810,"result":true,"route":"../roadmap/tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/index.html#the-fourier-eigenrelation-for-the-unitary-fourie","scope":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber","status":"fully_proved","summary":"The `n`-th rescaled Hermite function is an eigenvector of `MeasureTheory.Lp.fourierTransform\u2097\u1d62` with eigenvalue `(-i)\u207f`.","title":"The Fourier eigenrelation for the unitary Fourier transform of `L\u00b2(\u211d; \u2102)`"},{"id":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/canonical-commutation-relation-ccr-for-the-hermi","kind":"node","lean":"TauCeti.hermiteAnnihilationCLM_comp_hermiteCreationCLM_sub_hermiteCreationCLM_comp_hermiteAnnihilationCLM","parent":1810,"result":true,"route":"../roadmap/tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/index.html#canonical-commutation-relation-ccr-for-the-hermi","scope":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber","status":"fully_proved","summary":"`[a, a\u2020] = a \u2218L a\u2020 - a\u2020 \u2218L a = id`.","title":"Canonical Commutation Relation (CCR) for the Hermite ladder operators"},{"id":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/the-weight-measure-isometry","kind":"node","lean":"TauCeti.weightL2Isometry","parent":1810,"result":true,"route":"../roadmap/tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/index.html#the-weight-measure-isometry","scope":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber","status":"fully_proved","summary":"For an almost-everywhere-positive weight `w`, multiplication by `\u221aw` is a linear isometric equivalence `L\u00b2(w\u00b7\u03bc; E) \u2243\u2097\u1d62[\ud835\udd5c] L\u00b2(\u03bc; E)`.","title":"The weight \u2194 measure isometry"},{"id":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/the-chebyshev-envelope-functions-are-a-hilbert-b","kind":"node","lean":"TauCeti.chebyshevTEnvelopeHilbertBasis","parent":1810,"result":true,"route":"../roadmap/tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/index.html#the-chebyshev-envelope-functions-are-a-hilbert-b","scope":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber","status":"fully_proved","summary":"It is the weight-in-the-function normalization, assembled from the same bridge as the weight-in-the-measure basis `TauCeti.chebyshevTHilbertBasis`: orthonormality from the Chebyshev orthogonality relation, completeness from moment determinacy applied to the exact-degree family `T\u2099`.","title":"The Chebyshev envelope functions are a Hilbert basis of `L\u00b2((-1, 1]; dx)`"},{"id":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/the-weighted-measure-basis","kind":"node","lean":"TauCeti.hilbertBasisOfWeightedMeasure","parent":1810,"result":true,"route":"../roadmap/tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/index.html#the-weighted-measure-basis","scope":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber","status":"fully_proved","summary":"The normalized bare functions, orthonormal by the orthogonality relation and complete by hypothesis, form a Hilbert basis of `L\u00b2(w\u00b7\u03bc)` \u2014 the textbook statement that a family of orthogonal functions is an orthonormal basis of its own weighted `L\u00b2` space.","title":"The weighted-measure basis"},{"id":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/the-product-hilbert-basis","kind":"node","lean":"HilbertBasis.prod","parent":1810,"result":true,"route":"../roadmap/tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/index.html#the-product-hilbert-basis","scope":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber","status":"fully_proved","summary":"Pointwise products of two Hilbert bases form a Hilbert basis of `L\u00b2(\u03bc \u2297 \u03bd)`, indexed by the product of the index types.","title":"The product Hilbert basis"},{"id":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/the-fintype-indexed-product-hilbert-basis","kind":"node","lean":"TauCeti.piHilbertBasis","parent":1810,"result":true,"route":"../roadmap/tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/index.html#the-fintype-indexed-product-hilbert-basis","scope":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber","status":"fully_proved","summary":"Pointwise products of coordinatewise Hilbert bases form a Hilbert basis of `L\u00b2(Measure.pi \u03bc)`, indexed by the dependent function type.","title":"The `Fintype`-indexed product Hilbert basis"},{"id":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/the-tensor-as-a-bounded-bilinear-map","kind":"node","lean":"MeasureTheory.Lp.prodMulL","parent":1810,"result":true,"route":"../roadmap/tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/index.html#the-tensor-as-a-bounded-bilinear-map","scope":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber","status":"fully_proved","summary":"`prodMulL F G = prodMul F G`, packaged so that either operand can be fixed: `prodMulL F` fixes the left factor, `prodMulL.flip G` the right. Its operator norm is at most `1`; neither measure needs to be s-finite.","title":"The tensor as a bounded bilinear map"},{"id":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/fourier-eigenfunction-theorem","kind":"node","lean":"TauCeti.fourier_twoPiHermiteFunction","parent":1810,"result":true,"route":"../roadmap/tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber/index.html#fourier-eigenfunction-theorem","scope":"tauceti/analysis/part-01/weighted-orthogonal-l2-bases-completeness-hilber","status":"fully_proved","summary":"For Mathlib's `exp (-2\u03c0ix\u03be)` convention, the rescaled Hermite function `\u03a6\u2099(x) = \u221a(\u221a(2\u03c0)) \u03c8\u2099(\u221a(2\u03c0)x)` has eigenvalue `(-i)\u207f`.","title":"Fourier eigenfunction theorem"},{"data_url":"scopes/tauceti/analysis/part-01/analytic.json","id":"tauceti/analysis/part-01/analytic","kind":"scope","parent":1570,"result":false,"route":"../roadmap/tauceti/analysis/part-01/analytic/index.html","scope":"tauceti/analysis/part-01/analytic","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Analysis.Analytic` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Analytic"},{"id":"tauceti/analysis/part-01/analytic/the-analytic-implicit-function-theorem","kind":"node","lean":"HasStrictFDerivAt.analyticAt_implicitFunctionOfProdDomain, ImplicitFunctionData.analyticAt_implicitFunction","parent":1826,"result":true,"route":"../roadmap/tauceti/analysis/part-01/analytic/index.html#the-analytic-implicit-function-theorem","scope":"tauceti/analysis/part-01/analytic","status":"fully_proved","summary":"If `f : E\u2081 \u00d7 E \u2192 F` is analytic at `u` and its partial derivative in the second variable is invertible there, the implicit function `\u03c8` with `f (x, \u03c8 x) = f u` near `u.1` is analytic at `u.1`.","title":"The analytic implicit function theorem"},{"id":"tauceti/analysis/part-01/analytic/the-analytic-implicit-root-theorem","kind":"node","lean":"AnalyticAt.exists_analyticAt_eventually_eq_zero_iff","parent":1826,"result":true,"route":"../roadmap/tauceti/analysis/part-01/analytic/index.html#the-analytic-implicit-root-theorem","scope":"tauceti/analysis/part-01/analytic","status":"fully_proved","summary":"Let `f (w, z)` be analytic at `u = (w\u2080, z\u2080)`, with `f u = 0` and `\u2202f/\u2202z \u2260 0` at `u`. Then there is a function `g`, analytic at `w\u2080` with `g w\u2080 = z\u2080`, such that near `u` the zeros of `f` are exactly the points `(w, g w)`; in particular `f (w, g w) = 0` for all `w` near `w\u2080`.","title":"The analytic implicit root theorem"},{"id":"tauceti/analysis/part-01/analytic/the-analytic-inverse-function-theorem","kind":"node","lean":"AnalyticAt.exists_openPartialHomeomorph","parent":1826,"result":true,"route":"../roadmap/tauceti/analysis/part-01/analytic/index.html#the-analytic-inverse-function-theorem","scope":"tauceti/analysis/part-01/analytic","status":"fully_proved","summary":"A map analytic at `a`, whose derivative at `a` is a continuous linear equivalence, coincides with an `OpenPartialHomeomorph` whose source is a neighbourhood of `a` inside any prescribed neighbourhood `s`, which is analytic on its source, and whose inverse is analytic on its target.","title":"The analytic inverse function theorem"},{"id":"tauceti/analysis/part-01/analytic/hadamards-lemma-along-a-graph","kind":"node","lean":"AnalyticAt.exists_eventuallyEq_sub_smul_of_graph, AnalyticAt.exists_analyticAt_eventually_sub_eq, AnalyticAt.exists_eventuallyEq_sub_smul","parent":1826,"result":true,"route":"../roadmap/tauceti/analysis/part-01/analytic/index.html#hadamards-lemma-along-a-graph","scope":"tauceti/analysis/part-01/analytic","status":"fully_proved","summary":"a function `G` analytic at `(x\u2080, g x\u2080)` that vanishes on the graph `y = g x` of a function `g` analytic at `x\u2080` is `(y - g x) \u2022 H (x, y)` near `(x\u2080, g x\u2080)`, with `H` analytic at `(x\u2080, g x\u2080)`.","title":"Hadamard's lemma along a graph"},{"data_url":"scopes/tauceti/analysis/part-01/bochner.json","id":"tauceti/analysis/part-01/bochner","kind":"scope","parent":1570,"result":false,"route":"../roadmap/tauceti/analysis/part-01/bochner/index.html","scope":"tauceti/analysis/part-01/bochner","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Analysis.Bochner` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Bochner"},{"id":"tauceti/analysis/part-01/bochner/the-bochner-khinchin-theorem","kind":"node","lean":"TauCeti.bochner_charFun_probabilityMeasure","parent":1831,"result":true,"route":"../roadmap/tauceti/analysis/part-01/bochner/index.html#the-bochner-khinchin-theorem","scope":"tauceti/analysis/part-01/bochner","status":"fully_proved","summary":"A function `F` on a finite-dimensional real inner-product space is continuous, positive definite and normalized by `F 0 = 1` if and only if it is the characteristic function of a unique Borel probability measure.","title":"The Bochner\u2013Khinchin theorem"},{"id":"tauceti/analysis/part-01/bochner/herglotzs-theorem","kind":"node","lean":"TauCeti.exists_isFiniteMeasure_integral_zpow_eq, TauCeti.isPositiveDefiniteSub_iff_exists_isFiniteMeasure_integral_zpow_eq","parent":1831,"result":true,"route":"../roadmap/tauceti/analysis/part-01/bochner/index.html#herglotzs-theorem","scope":"tauceti/analysis/part-01/bochner","status":"fully_proved","summary":"A positive-definite function `\u03c6` on `\u2124` is the moment sequence `\u03c6 n = \u222b z\u207f d\u03bc(z)` of a finite positive measure `\u03bc` on the unit circle.","title":"Herglotz's theorem"},{"id":"tauceti/analysis/part-01/bochner/the-inversion-theorem","kind":"node","lean":"MeasureTheory.Measure.eq_withDensity_charFunDensity","parent":1831,"result":true,"route":"../roadmap/tauceti/analysis/part-01/bochner/index.html#the-inversion-theorem","scope":"tauceti/analysis/part-01/bochner","status":"fully_proved","summary":"A finite measure on `\u211d` with integrable characteristic function `\u03c6` has the bounded continuous density `x \u21a6 (2\u03c0)\u207b\u00b9 * Re \u222b t, exp (-i t x) * \u03c6(t) dt` with respect to Lebesgue measure.","title":"The inversion theorem"},{"id":"tauceti/analysis/part-01/bochner/uniqueness-half-of-bochners-theorem","kind":"node","lean":"TauCeti.Measure.ext_of_forall_integral_fourierAtom_eq","parent":1831,"result":true,"route":"../roadmap/tauceti/analysis/part-01/bochner/index.html#uniqueness-half-of-bochners-theorem","scope":"tauceti/analysis/part-01/bochner","status":"fully_proved","summary":"Two finite Borel measures with the same Fourier-convention transform coincide; this is Mathlib's characteristic-function uniqueness theorem, transported through the `-2\u03c0` rescaling. Stated on any complete second-countable real inner-product space; no finite-dimensionality is needed.","title":"Uniqueness half of Bochner's theorem"},{"data_url":"scopes/tauceti/analysis/part-01/bounded-variation.json","id":"tauceti/analysis/part-01/bounded-variation","kind":"scope","parent":1570,"result":false,"route":"../roadmap/tauceti/analysis/part-01/bounded-variation/index.html","scope":"tauceti/analysis/part-01/bounded-variation","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Analysis.BoundedVariation` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Bounded Variation"},{"id":"tauceti/analysis/part-01/bounded-variation/the-gauss-green-formula-d-s-s-d-s","kind":"node","lean":"TauCeti.integral_trace_fderiv_mul_indicator_eq_integral_inner_outerNormal","parent":1836,"result":true,"route":"../roadmap/tauceti/analysis/part-01/bounded-variation/index.html#the-gauss-green-formula-d-s-s-d-s","scope":"tauceti/analysis/part-01/bounded-variation","status":"fully_proved","summary":"For a set `s` of locally finite perimeter in `\u03a9`, the outer normal `\u03bd_s` represents the distributional derivative of the indicator against the perimeter measure: `\u222b_s div \u03c6 d\u03bc = \u222b \u27ea\u03c6, \u03bd_s\u27eb d|D\u03c7_s|` for every `C\u00b9` vector field `\u03c6` with compact support in `\u03a9`, where `div \u03c6 = tr (D\u03c6)`.","title":"The Gauss\u2013Green formula, `D\u03c7_s = -\u03bd_s |D\u03c7_s|`"},{"id":"tauceti/analysis/part-01/bounded-variation/lebesgue-decomposition-of-the-bv-derivative","kind":"node","lean":"TauCeti.integral_divergenceCLM_mul_eq_neg_integral_inner_acDerivativeDensity_sub_singularPart","parent":1836,"result":true,"route":"../roadmap/tauceti/analysis/part-01/bounded-variation/index.html#lebesgue-decomposition-of-the-bv-derivative","scope":"tauceti/analysis/part-01/bounded-variation","status":"fully_proved","summary":"The distributional derivative is the sum of the density `acDerivativeDensity \u03bc \u03a9 u \u03bd` against `\u03bd` and the polar field against the singular part of `|Du|`. Both pairings are integrable because the test field has compact support.","title":"Lebesgue decomposition of the BV derivative"},{"id":"tauceti/analysis/part-01/bounded-variation/the-poincare-inequality-for-functions-of-bounded","kind":"node","lean":"TauCeti.setLIntegral_enorm_sub_setAverage_le_mul_totalVariationOn_of_eq_ball, TauCeti.setLIntegral_enorm_sub_setAverage_le_mul_totalVariationOn_of_convex, TauCeti.setLIntegral_enorm_sub_setAverage_le_mul_totalVariationOn","parent":1836,"result":true,"route":"../roadmap/tauceti/analysis/part-01/bounded-variation/index.html#the-poincare-inequality-for-functions-of-bounded","scope":"tauceti/analysis/part-01/bounded-variation","status":"fully_proved","summary":"A function locally integrable on a ball of radius `R` in a space of dimension `n` deviates from its mean over the ball in `L\u00b9` by at most `2 ^ (n + 1) R` times its total variation on the ball. The constant is proportional to `R`, as the scaling of both sides forces.","title":"The Poincar\u00e9 inequality for functions of bounded variation on a ball"},{"id":"tauceti/analysis/part-01/bounded-variation/the-poincare-wirtinger-inequality-for-functions","kind":"node","lean":"TauCeti.setLIntegral_enorm_sub_setAverage_le_mul_measure_rpow_mul_totalVariationOn_of_convex","parent":1836,"result":true,"route":"../roadmap/tauceti/analysis/part-01/bounded-variation/index.html#the-poincare-wirtinger-inequality-for-functions","scope":"tauceti/analysis/part-01/bounded-variation","status":"fully_proved","summary":"Let `\u03a9` be a bounded convex open set in a finite-dimensional real inner product space of dimension `n`, and let `S \u2286 \u03a9` have positive measure. For every `u` locally integrable on `\u03a9` and every `A \u2286 \u03a9`, the integral over `A` of the deviation of `u` from its mean over `S` is at most `\u03bc(B(0, 1)) ^ (1 - 1/n) * (diam \u03a9) ^ n / \u03bc(S) * \u03bc(A) ^ (1/n)` times the total variation `|Du|(\u03a9)`, finite or infinite.","title":"The Poincar\u00e9\u2013Wirtinger inequality for functions of bounded variation, localized to a subset"},{"data_url":"scopes/tauceti/analysis/part-02.json","id":"tauceti/analysis/part-02","kind":"scope","navigation":"transparent","parent":1569,"result":false,"route":"../roadmap/tauceti/analysis/part-02/index.html","scope":"tauceti/analysis/part-02","status":"fully_proved","title":"Calculus to Other results"},{"data_url":"scopes/tauceti/analysis/part-02/calculus.json","id":"tauceti/analysis/part-02/calculus","kind":"scope","parent":1841,"result":false,"route":"../roadmap/tauceti/analysis/part-02/calculus/index.html","scope":"tauceti/analysis/part-02/calculus","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Analysis.Calculus` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Calculus"},{"data_url":"scopes/tauceti/analysis/part-02/calculus/part-01.json","id":"tauceti/analysis/part-02/calculus/part-01","kind":"scope","navigation":"transparent","parent":1842,"result":false,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-01/index.html","scope":"tauceti/analysis/part-02/calculus/part-01","status":"fully_proved","title":"Cartan's formula to The relative Poincar\u00e9 lemma"},{"id":"tauceti/analysis/part-02/calculus/part-01/cartans-formula","kind":"node","lean":"TauCeti.lieDerivFormWithin_eq_curryLeft_extDerivWithin_add, TauCeti.lieDerivForm_eq_curryLeft_extDeriv_add","parent":1843,"result":true,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-01/index.html#cartans-formula","scope":"tauceti/analysis/part-02/calculus/part-01","status":"fully_proved","summary":"Cartan's formula within a set, `L_V \u03c9 = \u03b9_V d\u03c9 + d (\u03b9_V \u03c9)`, where the interior product `\u03b9_V` inserts `V` into the first slot.","title":"Cartan's formula"},{"id":"tauceti/analysis/part-02/calculus/part-01/duhamels-formula-at-a-small-parameter","kind":"node","lean":"TauCeti.fst_fderiv_perturbedFlow_apply_eq_neg_integral_of_norm_lt, TauCeti.fst_fderiv_perturbedFlow_apply_eq_neg_integral","parent":1843,"result":true,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-01/index.html#duhamels-formula-at-a-small-parameter","scope":"tauceti/analysis/part-02/calculus/part-01","status":"fully_proved","summary":"For a parameter `s\u2080` of norm less than `1`, the derivative of the perturbed flow of `f` at `(x, s\u2080)` in a parameter direction `e` is `-D\u03c6_T(x) \u222b\u2080\u1d40 D\u03c6_{-\u03c3}(\u03c6_\u03c3 x) (\u2211\u1d62 e\u1d62 \u2207h\u1d62(\u03c6_\u03c3 x)) d\u03c3`, where `\u03c6` is now the negative-gradient flow of the perturbation `f_{s\u2080}`.","title":"Duhamel's formula at a small parameter"},{"id":"tauceti/analysis/part-02/calculus/part-01/greens-theorem-on-a-rectangle","kind":"node","lean":"TauCeti.integral_fderiv_sub_fderiv_Icc","parent":1843,"result":true,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-01/index.html#greens-theorem-on-a-rectangle","scope":"tauceti/analysis/part-02/calculus/part-01","status":"fully_proved","summary":"For a one-form `f_s ds + f_t dt` that is `C\u00b9` over an open neighbourhood of a rectangle `[a\u2081, b\u2081] \u00d7 [a\u2082, b\u2082]`, the integral of its exterior derivative `\u2202_s f_t - \u2202_t f_s` over the rectangle is its integral over the positively oriented boundary.","title":"Green's theorem on a rectangle"},{"id":"tauceti/analysis/part-02/calculus/part-01/the-inverse-function-theorem-produces-a-c-n-inve","kind":"node","lean":"HasStrictFDerivAt.contDiffOn_toOpenPartialHomeomorph_symm","parent":1843,"result":true,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-01/index.html#the-inverse-function-theorem-produces-a-c-n-inve","scope":"tauceti/analysis/part-02/calculus/part-01","status":"fully_proved","summary":"If `f` is `C^n` on the source of the homeomorphism built by the inverse function theorem, then the inverse is `C^n` on the target.","title":"The inverse function theorem produces a `C^n` inverse"},{"id":"tauceti/analysis/part-02/calculus/part-01/the-inverse-function-theorem-with-a-c-n-inverse","kind":"node","lean":"TauCeti.ContDiffOn.exists_openPartialHomeomorph","parent":1843,"result":true,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-01/index.html#the-inverse-function-theorem-with-a-c-n-inverse","scope":"tauceti/analysis/part-02/calculus/part-01","status":"fully_proved","summary":"A map which is `C^n` on an open set `s`, with `1 \u2264 n`, and whose derivative at `a \u2208 s` is a continuous linear equivalence, coincides on a neighbourhood of `a` with an `OpenPartialHomeomorph` whose source is contained in `s` and whose inverse is `C^n` on its target.","title":"The inverse function theorem, with a `C^n` inverse on the whole target"},{"id":"tauceti/analysis/part-02/calculus/part-01/iterated-rolles-theorem","kind":"node","lean":"TauCeti.exists_card_le_card_add_of_hasDerivAt","parent":1843,"result":true,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-01/index.html#iterated-rolles-theorem","scope":"tauceti/analysis/part-02/calculus/part-01","status":"fully_proved","summary":"Let `F (n + 1)` be the derivative of `F n` for every `n < k`. Then every finite set `Z` of zeros of `F 0` has at most `k` more points than some finite set of zeros of `F k`.","title":"Iterated Rolle's theorem"},{"id":"tauceti/analysis/part-02/calculus/part-01/jacobis-adjugate-formula-valid-at-singular-as-we","kind":"node","lean":"Matrix.fderiv_det_apply_adjugate","parent":1843,"result":true,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-01/index.html#jacobis-adjugate-formula-valid-at-singular-as-we","scope":"tauceti/analysis/part-02/calculus/part-01","status":"fully_proved","summary":"Formalized in Tau Ceti as `Matrix.fderiv_det_apply_adjugate`.","title":"Jacobi's adjugate formula, valid at singular as well as invertible matrices"},{"id":"tauceti/analysis/part-02/calculus/part-01/jacobis-inverse-formula","kind":"node","lean":"Matrix.hasFDerivAt_det, Matrix.fderiv_det_apply","parent":1843,"result":true,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-01/index.html#jacobis-inverse-formula","scope":"tauceti/analysis/part-02/calculus/part-01","status":"fully_proved","summary":"At a matrix with invertible determinant, the differential is `H \u21a6 det A * trace (A\u207b\u00b9 * H)`.","title":"Jacobi's inverse formula"},{"id":"tauceti/analysis/part-02/calculus/part-01/the-lefschetz-whitehead-rank-estimate","kind":"node","lean":"HasFDerivAt.finrank_range_lt_of_infinite, HasFDerivAt.finrank_range_lt_of_infinite_of_norm_eq","parent":1843,"result":true,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-01/index.html#the-lefschetz-whitehead-rank-estimate","scope":"tauceti/analysis/part-02/calculus/part-01","status":"fully_proved","summary":"Let `g` and `h` be differentiable at `x`, with `h x` outside the range of the derivative `h'` of `h`. If the swept map `(y, t) \u21a6 g y + t \u2022 h y` has rank at most `p` at `(x, t)` for infinitely many `t`, for instance for every `t` near `0`, then `h` has rank less than `p` at `x`.","title":"The Lefschetz\u2013Whitehead rank estimate"},{"id":"tauceti/analysis/part-02/calculus/part-01/the-local-morse-lemma","kind":"node","lean":"TauCeti.IsNondegenerateCriticalPoint.exists_morse_chart_of_contDiffOn","parent":1843,"result":true,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-01/index.html#the-local-morse-lemma","scope":"tauceti/analysis/part-02/calculus/part-01","status":"fully_proved","summary":"If `g` is `C^{n+2}` on a neighbourhood `U` of a nondegenerate critical point `a`, with `1 \u2264 n \u2264 \u221e`, there is a chart `\u03c8` with source in `U`, sending `a` to `0`, `C^n` in both directions, on whose source `g` is its Hessian quadratic form read in the chart.","title":"The local Morse lemma"},{"id":"tauceti/analysis/part-02/calculus/part-01/the-morse-form-of-the-convergence-theorem","kind":"node","lean":"IsIntegralCurveOn.exists_tendsto_atTop_of_hasNondegenerateCriticalPointsOn","parent":1843,"result":true,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-01/index.html#the-morse-form-of-the-convergence-theorem","scope":"tauceti/analysis/part-02/calculus/part-01","status":"fully_proved","summary":"A negative gradient trajectory confined to a compact set on which `fderiv \u211d f` is continuous and every critical point of `f` is nondegenerate converges to a critical point of `f` in that set.","title":"The Morse form of the convergence theorem"},{"id":"tauceti/analysis/part-02/calculus/part-01/the-morse-form-of-ojasiewiczs-gradient-inequalit","kind":"node","lean":"TauCeti.IsNondegenerateCriticalPoint.exists_mul_abs_sub_le_norm_gradient_sq, TauCeti.IsNondegenerateCriticalPoint.exists_mul_abs_sub_le_norm_fderiv_sq","parent":1843,"result":true,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-01/index.html#the-morse-form-of-ojasiewiczs-gradient-inequalit","scope":"tauceti/analysis/part-02/calculus/part-01","status":"fully_proved","summary":"Near a nondegenerate critical point the absolute energy difference is bounded by a multiple of the squared norm of the gradient; equivalently the \u0141ojasiewicz inequality holds there with exponent `1 / 2`. Smoothness alone does not guarantee such an inequality, and a gradient trajectory can spiral forever without converging.","title":"The Morse form of \u0141ojasiewicz's gradient inequality"},{"id":"tauceti/analysis/part-02/calculus/part-01/the-morse-lemma","kind":"node","lean":"TauCeti.IsNondegenerateCriticalPoint.exists_normal_form, TauCeti.IsNondegenerateCriticalPoint.exists_morse_chart","parent":1843,"result":true,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-01/index.html#the-morse-lemma","scope":"tauceti/analysis/part-02/calculus/part-01","status":"fully_proved","summary":"Near a nondegenerate critical point `x` of a `C^{n+2}` function `f`, with `1 \u2264 n \u2264 \u221e`, there is a `C^n` map `\u03c6`, fixing `0` and with derivative the identity there, in terms of which `f` is exactly its Hessian quadratic form: `f (x + v) = f x + 2\u207b\u00b9 * fderiv \u211d (fderiv \u211d f) x (\u03c6 v) (\u03c6 v)` for `v` near `0`.","title":"The Morse lemma"},{"id":"tauceti/analysis/part-02/calculus/part-01/mosers-identity","kind":"node","lean":"TauCeti.compContinuousLinearMap_fderiv_eq_of_curryLeft_eq_neg","parent":1843,"result":true,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-01/index.html#mosers-identity","scope":"tauceti/analysis/part-02/calculus/part-01","status":"fully_proved","summary":"Let `\u03c9\u2080`, `\u03c9\u2081` be forms that are closed along the orbit `t \u21a6 \u03c8\u209c x`, `t \u2208 [0, 1]`, with `\u03c9\u2081 - \u03c9\u2080 = d\u03b1` there. If `\u03c8\u209c` moves along a time-dependent field `X\u209c` with `\u03b9_{X\u209c} ((1 - t) \u2022 \u03c9\u2080 + t \u2022 \u03c9\u2081) = -\u03b1` near the orbit, and the derivative of `\u03c8\u209c` at `x` solves the variational equation, then `\u03c8\u2081^* \u03c9\u2081 = \u03c8\u2080^* \u03c9\u2080` at `x`.","title":"Moser's identity"},{"id":"tauceti/analysis/part-02/calculus/part-01/nijenhuis-lie-derivative-identity","kind":"node","lean":"TauCeti.lieDerivEndomorphism_apply_sub_apply","parent":1843,"result":true,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-01/index.html#nijenhuis-lie-derivative-identity","scope":"tauceti/analysis/part-02/calculus/part-01","status":"fully_proved","summary":"if `A x` squares to `-1`, then for every differentiable vector field `X`, the derivative `DX(x)` cancels out of `(L_{A X} A)(x) w - A x ((L_X A)(x) w)`, leaving the pointwise Nijenhuis expression of `A` on `(X x, w)`.","title":"Nijenhuis\u2013Lie-derivative identity"},{"id":"tauceti/analysis/part-02/calculus/part-01/the-parametric-morse-lemma-as-a-local-chart","kind":"node","lean":"TauCeti.exists_parametric_morse_chart","parent":1843,"result":true,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-01/index.html#the-parametric-morse-lemma-as-a-local-chart","scope":"tauceti/analysis/part-02/calculus/part-01","status":"fully_proved","summary":"The chart has a product source containing `(a, 0)`, preserves the base parameter in both directions, and fixes the zero section. Its quadratic equation uses the vertical Hessian on that same zero section. The base and the fiber are Banach spaces, so the ordinary inverse function theorem applies to their product.","title":"The parametric Morse lemma, as a local chart"},{"id":"tauceti/analysis/part-02/calculus/part-01/peanos-theorem-on-mixed-partial-derivatives-diff","kind":"node","lean":"TauCeti.hasFDerivAt_timeDeriv_of_continuousWithinAt, TauCeti.hasDerivWithinAt_spatialDeriv_of_continuousWithinAt","parent":1843,"result":true,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-01/index.html#peanos-theorem-on-mixed-partial-derivatives-diff","scope":"tauceti/analysis/part-02/calculus/part-01","status":"fully_proved","summary":"Let `u : E \u2192 \u211d \u2192 F`, let `s \u2286 \u211d` be convex, and let `t \u2208 s` be a point at which derivatives within `s` are unique. Suppose that near `x` in space and near `t` within `s`, the spatial derivative of `u(\u00b7, \u03c4)` at `y` is `v(y, \u03c4)`, and that `v(y, \u00b7)` has time derivative `w(y, \u03c4)` within `s`, with `w` jointly continuous at `(x, t)`. If `g y` is the time derivative of `u(y, \u00b7)` within `s` at `t` for every `y` near `x`,\u2026","title":"Peano's theorem on mixed partial derivatives, differentiating a time derivative in space"},{"id":"tauceti/analysis/part-02/calculus/part-01/the-poincare-lemma","kind":"node","lean":"StarConvex.exists_contDiffOn_forall_extDeriv_eq, StarConvex.extDeriv_poincareHomotopy, Convex.exists_invariant_contDiffOn_forall_extDerivWithin_eq, StarConvex.exists_invariant_contDiffOn_forall_extDeriv_eq","parent":1843,"result":true,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-01/index.html#the-poincare-lemma","scope":"tauceti/analysis/part-02/calculus/part-01","status":"fully_proved","summary":"A closed `C^r` form, `r \u2265 1`, on an open star-convex set is the exterior derivative of a `C^r` form, which can be chosen to vanish at the centre.","title":"The Poincar\u00e9 lemma"},{"id":"tauceti/analysis/part-02/calculus/part-01/rademachers-theorem-for-locally-lipschitz-functi","kind":"node","lean":"LocallyLipschitzOn.ae_differentiableWithinAt_of_mem, LocallyLipschitzOn.ae_differentiableAt_of_mem","parent":1843,"result":true,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-01/index.html#rademachers-theorem-for-locally-lipschitz-functi","scope":"tauceti/analysis/part-02/calculus/part-01","status":"fully_proved","summary":"a function between finite-dimensional real normed spaces which is locally Lipschitz on a set is differentiable within that set at almost every point of it.","title":"Rademacher's theorem for locally Lipschitz functions"},{"id":"tauceti/analysis/part-02/calculus/part-01/the-relative-poincare-lemma","kind":"node","lean":"Convex.exists_contDiffOn_forall_extDerivWithin_eq, Convex.extDerivWithin_poincareHomotopy, Convex.exists_contDiffOn_relative_primitive","parent":1843,"result":true,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-01/index.html#the-relative-poincare-lemma","scope":"tauceti/analysis/part-02/calculus/part-01","status":"fully_proved","summary":"A `C^r` form, `r \u2265 1`, closed within a convex set with unique derivatives is the exterior derivative within the set of a `C^r` form, which can be chosen to vanish at any given point of the set. The set need not be open: a closed half-ball, the image of a boundary chart, is allowed.","title":"The relative Poincar\u00e9 lemma"},{"data_url":"scopes/tauceti/analysis/part-02/calculus/part-02.json","id":"tauceti/analysis/part-02/calculus/part-02","kind":"scope","navigation":"transparent","parent":1842,"result":false,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-02/index.html","scope":"tauceti/analysis/part-02/calculus/part-02","status":"fully_proved","title":"Rolle's theorem for finitely\u2026 to Uniform first-order Taylor\u2026"},{"id":"tauceti/analysis/part-02/calculus/part-02/rolles-theorem-for-finitely-many-zeros","kind":"node","lean":"TauCeti.exists_finset_deriv_eq_zero","parent":1864,"result":true,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-02/index.html#rolles-theorem-for-finitely-many-zeros","scope":"tauceti/analysis/part-02/calculus/part-02","status":"fully_proved","summary":"If a function `f : \u211d \u2192 \u211d`, continuous between the smallest and largest points of a nonempty finite set `Z`, vanishes on `Z`, then `deriv f` vanishes on a finite set `Z'` with one point fewer that separates the points of `Z`: every point of `Z'` lies strictly between two points of `Z`, and strictly between any two points of `Z` lies a point of `Z'`.","title":"Rolle's theorem for finitely many zeros"},{"id":"tauceti/analysis/part-02/calculus/part-02/sards-lemma-on-the-flat-stratum","kind":"node","lean":"TauCeti.addHaar_image_eq_zero_of_iteratedFDeriv_eq_zero, TauCeti.addHaar_image_eq_zero_of_not_surjective_fderivWithin, TauCeti.ContDiff.addHaar_image_flatStratum_eq_zero, TauCeti.addHaar_image_eq_zero_of_iteratedFDeriv_eq_zero_of_finrank_le_mul, TauCeti.dimH_image_le_of_iteratedFDeriv_eq_zero, TauCeti.hausdorffMeasure_image_eq_zero_of_iteratedFDeriv_eq_zero","parent":1864,"result":true,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-02/index.html#sards-lemma-on-the-flat-stratum","scope":"tauceti/analysis/part-02/calculus/part-02","status":"fully_proved","summary":"Let `f` be `C^{k+1}` on an open set `U` of a finite-dimensional real normed space, and let `s \u2286 U` be a set at each point of which every iterated derivative of order `1 \u2264 i \u2264 k` vanishes. If `finrank \u211d E < (k + 1) * finrank \u211d F`, then `f '' s` has additive Haar measure zero.","title":"Sard's lemma on the flat stratum"},{"id":"tauceti/analysis/part-02/calculus/part-02/sards-theorem-on-the-locus-where-the-derivative","kind":"node","lean":"TauCeti.addHaar_image_eq_zero_of_fderiv_eq_zero, TauCeti.addHaar_image_eq_zero_of_fderiv_eq_zero_of_finrank_le_add_two","parent":1864,"result":true,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-02/index.html#sards-theorem-on-the-locus-where-the-derivative","scope":"tauceti/analysis/part-02/calculus/part-02","status":"fully_proved","summary":"Let `f` be a map between finite-dimensional real normed spaces with nontrivial target, and let `s` be a set at each point of which `f` is `C^n` and the Fr\u00e9chet derivative of `f` vanishes. If `((finrank \u211d E * finrank \u211d E + 1 : \u2115) : \u2115\u221e\u03c9) \u2264 n`, then `f '' s` has additive Haar measure zero.","title":"Sard's theorem on the locus where the derivative vanishes"},{"id":"tauceti/analysis/part-02/calculus/part-02/stokes-formula-on-the-boundary-of-a-cube-for-a-c","kind":"node","lean":"TauCeti.integral2_slice_sub_eq_sides","parent":1864,"result":true,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-02/index.html#stokes-formula-on-the-boundary-of-a-cube-for-a-c","scope":"tauceti/analysis/part-02/calculus/part-02","status":"fully_proved","summary":"its integrals over the top and bottom faces `[a, b] \u00d7 {1}` and `[a, b] \u00d7 {0}` differ by the signed sum of its integrals over the four side faces, each parametrized by the direction of its edge followed by the last coordinate. All integrals are iterated integrals over oriented intervals, so the corners `a` and `b` may come in either order.","title":"Stokes' formula on the boundary of a cube for a closed two-form `\u03c8` that is C\u00b9 on `(\u211d \u00d7 \u211d) \u00d7 \u211d`"},{"id":"tauceti/analysis/part-02/calculus/part-02/stokes-theorem-on-the-standard-simplex","kind":"node","lean":"ContDiffOn.integral_extDerivWithin_fullSimplex","parent":1864,"result":true,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-02/index.html#stokes-theorem-on-the-standard-simplex","scope":"tauceti/analysis/part-02/calculus/part-02","status":"fully_proved","summary":"For a `p`-form `\u03c9` that is `C\u00b9` on the standard simplex `\u0394\u1d56\u207a\u00b9`, the integral of its exterior derivative over `\u0394\u1d56\u207a\u00b9` is the alternating sum over `i` of the integrals over `\u0394\u1d56` of its pullbacks along the face maps `simplexMapAmbient i.succAbove`, all forms being evaluated on the standard basis.","title":"Stokes' theorem on the standard simplex"},{"id":"tauceti/analysis/part-02/calculus/part-02/taylors-formula-to-second-order","kind":"node","lean":"TauCeti.map_add_eq_add_hessianAverage, TauCeti.sum_add_pow_sub_smul_iterate_dslope","parent":1864,"result":true,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-02/index.html#taylors-formula-to-second-order","scope":"tauceti/analysis/part-02/calculus/part-02","status":"fully_proved","summary":"Taylor's formula to second order, with the remainder written as the averaged Hessian evaluated twice on the increment.","title":"Taylor's formula to second order"},{"id":"tauceti/analysis/part-02/calculus/part-02/taylors-theorem-with-a-bounded-remainder","kind":"node","lean":"TauCeti.abs_sub_taylorWithinEval_le","parent":1864,"result":true,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-02/index.html#taylors-theorem-with-a-bounded-remainder","scope":"tauceti/analysis/part-02/calculus/part-02","status":"fully_proved","summary":"If `f` is `C^(n+1)` and its `(n+1)`st derivative is bounded in absolute value by `M`, then its Taylor polynomial of order `n` about `x\u2080` approximates `f x` with error at most `M * |x - x\u2080|^(n+1) / (n+1)!`.","title":"Taylor's theorem with a bounded remainder"},{"id":"tauceti/analysis/part-02/calculus/part-02/uniform-first-order-taylor-estimate-in-a-paramet","kind":"node","lean":"HasCompactSupport.exists_norm_iteratedFDeriv_prod_right_sub_sub_le, HasCompactSupport.exists_norm_iteratedFDeriv_sub_sub_add_fderiv_le","parent":1864,"result":true,"route":"../roadmap/tauceti/analysis/part-02/calculus/part-02/index.html#uniform-first-order-taylor-estimate-in-a-paramet","scope":"tauceti/analysis/part-02/calculus/part-02","status":"fully_proved","summary":"If `g : P \u00d7 E \u2192 F` is `C^{i+1}` with compact support, then the `i`-th derivative in the second variable of the first-order Taylor remainder `x \u21a6 g (p + h, x) - g (p, x) - Dg(p, x) (h, 0)` in the first variable is `o(\u2016h\u2016)` as `h \u2192 0`, uniformly on the whole space.","title":"Uniform first-order Taylor estimate in a parameter"},{"data_url":"scopes/tauceti/analysis/part-02/completely-monotone.json","id":"tauceti/analysis/part-02/completely-monotone","kind":"scope","parent":1841,"result":false,"route":"../roadmap/tauceti/analysis/part-02/completely-monotone/index.html","scope":"tauceti/analysis/part-02/completely-monotone","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Analysis.CompletelyMonotone` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Completely Monotone"},{"id":"tauceti/analysis/part-02/completely-monotone/bernsteins-theorem-existence-half","kind":"node","lean":"TauCeti.exists_representsLaplace_of_isCompletelyMonotone","parent":1873,"result":true,"route":"../roadmap/tauceti/analysis/part-02/completely-monotone/index.html#bernsteins-theorem-existence-half","scope":"tauceti/analysis/part-02/completely-monotone","status":"fully_proved","summary":"A completely monotone function on `[0, \u221e)` is the Laplace transform of a finite positive measure on `\u211d\u22650`.","title":"Bernstein's theorem, existence half"},{"id":"tauceti/analysis/part-02/completely-monotone/the-complete-bernstein-representation-theorem-un","kind":"node","lean":"TauCeti.IsCompleteBernsteinFunction.existsUnique_representsCompleteBernstein","parent":1873,"result":true,"route":"../roadmap/tauceti/analysis/part-02/completely-monotone/index.html#the-complete-bernstein-representation-theorem-un","scope":"tauceti/analysis/part-02/completely-monotone","status":"fully_proved","summary":"A complete Bernstein function has exactly one representing triple `(a, b, \u03bc)`.","title":"The complete Bernstein representation theorem, uniqueness form"},{"id":"tauceti/analysis/part-02/completely-monotone/the-stieltjes-representation-theorem-uniqueness","kind":"node","lean":"TauCeti.IsStieltjesFunction.existsUnique_representsStieltjes","parent":1873,"result":true,"route":"../roadmap/tauceti/analysis/part-02/completely-monotone/index.html#the-stieltjes-representation-theorem-uniqueness","scope":"tauceti/analysis/part-02/completely-monotone","status":"fully_proved","summary":"A Stieltjes function has exactly one representing triple `(a, b, \u03bc)`.","title":"The Stieltjes representation theorem, uniqueness form"},{"data_url":"scopes/tauceti/analysis/part-02/complex.json","id":"tauceti/analysis/part-02/complex","kind":"scope","parent":1841,"result":false,"route":"../roadmap/tauceti/analysis/part-02/complex/index.html","scope":"tauceti/analysis/part-02/complex","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Analysis.Complex` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Complex"},{"data_url":"scopes/tauceti/analysis/part-02/complex/part-01.json","id":"tauceti/analysis/part-02/complex/part-01","kind":"scope","navigation":"transparent","parent":1877,"result":false,"route":"../roadmap/tauceti/analysis/part-02/complex/part-01/index.html","scope":"tauceti/analysis/part-02/complex/part-01","status":"fully_proved","title":"The angle condition of\u2026 to The Herglotz formula on a disc"},{"id":"tauceti/analysis/part-02/complex/part-01/the-angle-condition-of-poincares-theorem","kind":"node","lean":"TauCeti.UpperHalfPlane.ConvexPolygon.orderOf_doubleTriangleSidePairing_cycleMap_mul_cycleAngleSum","parent":1878,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-01/index.html#the-angle-condition-of-poincares-theorem","scope":"tauceti/analysis/part-02/complex/part-01","status":"fully_proved","summary":"If every angle of `P` is `\u03c0 / m` for a natural number `m`, then at each finite vertex of the double the order of the cycle transformation times the cycle angle sum is `2\u03c0`.","title":"The angle condition of Poincar\u00e9's theorem"},{"id":"tauceti/analysis/part-02/complex/part-01/the-apollonius-identity-for-the-moebius-factor","kind":"node","lean":"TauCeti.sq_norm_sub_sub_mul_sq_norm_one_sub_conj_mul","parent":1878,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-01/index.html#the-apollonius-identity-for-the-moebius-factor","scope":"tauceti/analysis/part-02/complex/part-01","status":"fully_proved","summary":"For every `z : \u2102`,","title":"The Apollonius identity for the Moebius factor"},{"id":"tauceti/analysis/part-02/complex/part-01/the-bounded-image-form-of-the-small-jordan-curve","kind":"node","lean":"TauCeti.exists_isJordanCurve_superset_closure_image_ball_inter_sphere_diam_le_of_isBounded","parent":1878,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-01/index.html#the-bounded-image-form-of-the-small-jordan-curve","scope":"tauceti/analysis/part-02/complex/part-01","status":"fully_proved","summary":"A holomorphic injection of a disc onto a bounded domain has finite Dirichlet integral by the conformal area formula, so for every positive tolerance and radius bound, some closed image crosscut can be chosen on a Jordan curve in the closure of the image domain whose diameter is at most the tolerance.","title":"The bounded-image form of the small-Jordan-curve theorem"},{"id":"tauceti/analysis/part-02/complex/part-01/the-calderon-zygmund-inequality-for-the-cauchy-r","kind":"node","lean":"TauCeti.exists_eLpNorm_fderiv_le_eLpNorm_conjWirtingerDeriv, TauCeti.exists_eLpNorm_fderiv_le_eLpNorm_conjWirtingerDeriv_of_hasCompactSupport","parent":1878,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-01/index.html#the-calderon-zygmund-inequality-for-the-cauchy-r","scope":"tauceti/analysis/part-02/complex/part-01","status":"fully_proved","summary":"Let `1 < p < \u221e`. There is a constant `C` with","title":"The Calder\u00f3n\u2013Zygmund inequality for the Cauchy\u2013Riemann operator"},{"id":"tauceti/analysis/part-02/complex/part-01/caratheodorys-theorem-for-the-inside-of-a-jordan","kind":"node","lean":"TauCeti.IsJordanCurve.exists_homeomorph_closedBall_filledHull, TauCeti.IsJordanCurve.exists_homeomorph_compl_ball_compl_filledHull_sdiff, TauCeti.exists_continuousOn_bijOn_upperHalfPlaneSet_of_isJordanCurve_insert_infty","parent":1878,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-01/index.html#caratheodorys-theorem-for-the-inside-of-a-jordan","scope":"tauceti/analysis/part-02/complex/part-01","status":"fully_proved","summary":"For a Jordan curve `C \u2286 \u2102` and a point `p` of its inside `filledHull C \\ C`, there is a map `g`, holomorphic on the open unit disc with `g 0 = p`, carrying the disc bijectively onto the inside and the unit circle bijectively onto `C`, which restricts to a homeomorphism of the closed disc onto `filledHull C`.","title":"Carath\u00e9odory's theorem for the inside of a Jordan curve"},{"id":"tauceti/analysis/part-02/complex/part-01/cauchy-estimate","kind":"node","lean":"TauCeti.BoundedHolomorphicFunctions.dist_apply_le_of_ball_subset, TauCeti.norm_fderiv_le_of_forall_mem_ball_norm_le, TauCeti.norm_deriv_le_of_forall_mem_closedBall_norm_le, TauCeti.norm_deriv_le_of_forall_mem_closedBall_two_mul_norm_le","parent":1878,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-01/index.html#cauchy-estimate","scope":"tauceti/analysis/part-02/complex/part-01","status":"fully_proved","summary":"On a ball of radius `R` inside `U`, a bounded holomorphic map `f` is Lipschitz at the centre with constant `2 * \u2016f\u2016 / R`.","title":"Cauchy estimate"},{"id":"tauceti/analysis/part-02/complex/part-01/cauchy-laurent-coefficient-formula","kind":"node","lean":"TauCeti.HasLaurentSeriesOnAnnulus.circleLaurentCoeff_eq","parent":1878,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-01/index.html#cauchy-laurent-coefficient-formula","scope":"tauceti/analysis/part-02/complex/part-01","status":"fully_proved","summary":"For any convergent Laurent series on `Metric.eball c R \\ Metric.closedBall c r` and any interior circle radius `\u03c1 \u2208 (r, R)`, the `n`-th Laurent coefficient is `circleLaurentCoeff f c \u03c1 n`.","title":"Cauchy-Laurent coefficient formula"},{"id":"tauceti/analysis/part-02/complex/part-01/the-cauchy-pompeiu-formula-for-a-compactly-suppo","kind":"node","lean":"HasCompactSupport.two_pi_inv_smul_integral_sub_inv_smul_fderiv_apply_one_add_I_smul_apply_I","parent":1878,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-01/index.html#the-cauchy-pompeiu-formula-for-a-compactly-suppo","scope":"tauceti/analysis/part-02/complex/part-01","status":"fully_proved","summary":"`u` is the Cauchy transform of `\u2202\u2093 u + i \u2202\u1d67 u = 2 \\bar\u2202 u`, `u w = (2\u03c0)\u207b\u00b9 \u222b (w - z)\u207b\u00b9 \u2022 (\u2202\u2093 u z + i \u2202\u1d67 u z)`, the integral being over the whole plane.","title":"The Cauchy\u2013Pompeiu formula for a compactly supported `C\u00b9` map `u : \u2102 \u2192 F`"},{"id":"tauceti/analysis/part-02/complex/part-01/complex-analytic-weierstrass-division-theorem","kind":"node","lean":"AnalyticAt.exists_weierstrassDivision","parent":1878,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-01/index.html#complex-analytic-weierstrass-division-theorem","scope":"tauceti/analysis/part-02/complex/part-01","status":"fully_proved","summary":"Let `f, g : E \u00d7 \u2102 \u2192 \u2102` be analytic at `(x\u2080, w\u2080)` on the product of a finite-dimensional complex normed space `E` with `\u2102`, and suppose the vertical slice `w \u21a6 f (x\u2080, w)` has finite order of vanishing `d` at `w\u2080`. Then there exist `q` analytic at `(x\u2080, w\u2080)` and `r : Fin d \u2192 E \u2192 \u2102` analytic at `x\u2080` such that near `(x\u2080, w\u2080)`, `g (x, w) = f (x, w) * q (x, w) + \u2211 i : Fin d, r i x * (w - w\u2080) ^ (i : \u2115)`.","title":"Complex analytic Weierstrass division theorem"},{"id":"tauceti/analysis/part-02/complex/part-01/complex-analytic-weierstrass-preparation-theorem","kind":"node","lean":"AnalyticAt.exists_weierstrassPreparation","parent":1878,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-01/index.html#complex-analytic-weierstrass-preparation-theorem","scope":"tauceti/analysis/part-02/complex/part-01","status":"fully_proved","summary":"Let `f : E \u00d7 \u2102 \u2192 \u2102` be analytic at `(x\u2080, w\u2080)` on the product of a finite-dimensional complex normed space `E` with `\u2102`, and suppose the vertical slice `w \u21a6 f (x\u2080, w)` has finite order of vanishing `d` at `w\u2080`. Then there exist analytic coefficient functions `c : Fin d \u2192 E \u2192 \u2102` vanishing at `x\u2080` and an analytic function `u` nonvanishing at `(x\u2080, w\u2080)` such that near `(x\u2080, w\u2080)`,\u2026","title":"Complex analytic Weierstrass preparation theorem"},{"id":"tauceti/analysis/part-02/complex/part-01/the-component-mapping-theorem-from-one-boundary","kind":"node","lean":"TauCeti.bijOn_schwarzChristoffelPrimitive_connectedComponentIn_of_unique_boundary_value","parent":1878,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-01/index.html#the-component-mapping-theorem-from-one-boundary","scope":"tauceti/analysis/part-02/complex/part-01","status":"fully_proved","summary":"Under boundary avoidance, one uniquely parametrized regular boundary value makes the primitive a bijection onto the complementary component of the finite boundary trace containing its base-point image. Neither simple connectivity of that component nor global boundary injectivity is assumed.","title":"The component mapping theorem from one boundary sheet"},{"id":"tauceti/analysis/part-02/complex/part-01/the-cusp-condition-of-poincares-theorem","kind":"node","lean":"TauCeti.UpperHalfPlane.ConvexPolygon.isParabolic_doubleTriangleSidePairing_cycleMap","parent":1878,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-01/index.html#the-cusp-condition-of-poincares-theorem","scope":"tauceti/analysis/part-02/complex/part-01","status":"fully_proved","summary":"The cycle transformations at the ideal vertices of the double are parabolic.","title":"The cusp condition of Poincar\u00e9's theorem"},{"id":"tauceti/analysis/part-02/complex/part-01/the-disc-schwarz-christoffel-theorem-for-an-unbo","kind":"node","lean":"TauCeti.exists_bijOn_const_mul_schwarzChristoffelDiscPrimitive_add_of_isJordanCurve_insert_infty","parent":1878,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-01/index.html#the-disc-schwarz-christoffel-theorem-for-an-unbo","scope":"tauceti/analysis/part-02/complex/part-01","status":"fully_proved","summary":"Suppose a connected open set has Jordan frontier on the Riemann sphere, finitely many distinct corners of angles `(e i + 1) * \u03c0`, straight sides away from those corners, and a sector of opening `\u03b2 * \u03c0` at infinity. Then it is the bijective image of the unit disc under an affine image of a disc Schwarz--Christoffel primitive. The finite prevertices have exponents `e i` and tend to the prescribed vertices; the\u2026","title":"The disc Schwarz--Christoffel theorem for an unbounded polygonal Jordan domain"},{"id":"tauceti/analysis/part-02/complex/part-01/the-discreteness-part-of-poincares-polygon-theor","kind":"node","lean":"TauCeti.UpperHalfPlane.ConvexPolygon.SidePairing.isCofinite_of_isParabolic","parent":1878,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-01/index.html#the-discreteness-part-of-poincares-polygon-theor","scope":"tauceti/analysis/part-02/complex/part-01","status":"fully_proved","summary":"If a side-paired polygon satisfies the angle condition at every finite vertex cycle and has parabolic cycle transformations at its ideal vertices, then the group generated by its side-pairing maps is a cofinite Fuchsian group: it is discrete and has finite covolume.","title":"The discreteness part of Poincar\u00e9's polygon theorem"},{"id":"tauceti/analysis/part-02/complex/part-01/the-dolbeault-lemma-on-a-disc","kind":"node","lean":"TauCeti.exists_contDiffOn_conjWirtingerDeriv_apply_one_eq_of_eball, TauCeti.exists_contDiffOn_conjWirtingerDeriv_apply_one_eq_of_ball, TauCeti.exists_contDiff_conjWirtingerDeriv_apply_one_eq","parent":1878,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-01/index.html#the-dolbeault-lemma-on-a-disc","scope":"tauceti/analysis/part-02/complex/part-01","status":"fully_proved","summary":"Let `f` be `C\u207f`, `n \u2265 1`, on the open disc `eball c R` of radius `R \u2264 \u221e`. Then there is a function `g`, `C\u207f` on the disc, with `\\bar\u2202 g = f` on the disc.","title":"The Dolbeault lemma on a disc"},{"id":"tauceti/analysis/part-02/complex/part-01/the-explicit-schwarz-christoffel-formula-for-a-s","kind":"node","lean":"TauCeti.const_mul_schwarzChristoffelPrimitive_two_neg_one_add_eq_log","parent":1878,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-01/index.html#the-explicit-schwarz-christoffel-formula-for-a-s","scope":"tauceti/analysis/part-02/complex/part-01","status":"fully_proved","summary":"Two distinct prevertices of exponent `-1` give the principal logarithm of their fractional-linear ratio, after scaling by their separation and translating by the value at the base point.","title":"The explicit Schwarz--Christoffel formula for a strip"},{"id":"tauceti/analysis/part-02/complex/part-01/the-fundamental-set-part-of-poincares-polygon-th","kind":"node","lean":"TauCeti.UpperHalfPlane.ConvexPolygon.SidePairing.isFundamentalDomain_carrier_of_isParabolic","parent":1878,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-01/index.html#the-fundamental-set-part-of-poincares-polygon-th","scope":"tauceti/analysis/part-02/complex/part-01","status":"fully_proved","summary":"If a side-paired polygon satisfies the angle condition at every finite vertex cycle and has parabolic cycle transformations at its ideal vertices, then it is a measurable fundamental domain for the group generated by its side-pairing maps.","title":"The fundamental-set part of Poincar\u00e9's polygon theorem"},{"id":"tauceti/analysis/part-02/complex/part-01/gauss-bonnet-for-the-triangles-of-the-fan-from","kind":"node","lean":"TauCeti.UpperHalfPlane.ConvexPolygon.volume_carrier_inter_strip","parent":1878,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-01/index.html#gauss-bonnet-for-the-triangles-of-the-fan-from","scope":"tauceti/analysis/part-02/complex/part-01","status":"fully_proved","summary":"If `vertex 0` is `\u221e`, the part of the carrier between the verticals through `vertex i` and `vertex (i + 1)`, for an side `i` not ending at `\u221e`, has area `\u03c0` minus the angles at `vertex i` and `vertex (i + 1)` of the triangle with vertices `\u221e`, `vertex i`, `vertex (i + 1)`. That part of the carrier is this triangle by `exists_carrier_inter_strip_eq`.","title":"Gauss\u2013Bonnet for the triangles of the fan from `\u221e`"},{"id":"tauceti/analysis/part-02/complex/part-01/the-gauss-bonnet-formula","kind":"node","lean":"TauCeti.UpperHalfPlane.volume_triangle, TauCeti.UpperHalfPlane.CompactConvexPolygon.volume_carrier, TauCeti.UpperHalfPlane.ConvexPolygon.volume_carrier","parent":1878,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-01/index.html#the-gauss-bonnet-formula","scope":"tauceti/analysis/part-02/complex/part-01","status":"fully_proved","summary":"the invariant area of a hyperbolic triangle is its angular defect `\u03c0 - \u03b1 - \u03b2 - \u03b3`.","title":"The Gauss\u2013Bonnet formula"},{"id":"tauceti/analysis/part-02/complex/part-01/the-herglotz-formula-on-a-disc","kind":"node","lean":"DiffContOnCl.circleAverage_herglotzRieszKernel_smul_re_add","parent":1878,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-01/index.html#the-herglotz-formula-on-a-disc","scope":"tauceti/analysis/part-02/complex/part-01","status":"fully_proved","summary":"a function holomorphic on a disc and continuous up to its boundary is the Herglotz\u2013Riesz integral of its boundary real part, plus the imaginary constant `(f c).im * I`. Unlike the Poisson formula, which recovers `f` from all of its boundary values, this recovers `f` from the boundary values of its real part alone.","title":"The Herglotz formula on a disc"},{"data_url":"scopes/tauceti/analysis/part-02/complex/part-02.json","id":"tauceti/analysis/part-02/complex/part-02","kind":"scope","navigation":"transparent","parent":1877,"result":false,"route":"../roadmap/tauceti/analysis/part-02/complex/part-02/index.html","scope":"tauceti/analysis/part-02/complex/part-02","status":"fully_proved","title":"The Herglotz representation\u2026 to Schwarz's lemma with prescribed\u2026"},{"id":"tauceti/analysis/part-02/complex/part-02/the-herglotz-representation-theorem","kind":"node","lean":"TauCeti.exists_isFiniteMeasure_eq_herglotzTransform_add, TauCeti.differentiableOn_and_re_nonneg_iff_exists_eq_herglotzTransform_add","parent":1899,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-02/index.html#the-herglotz-representation-theorem","scope":"tauceti/analysis/part-02/complex/part-02","status":"fully_proved","summary":"A function holomorphic on the unit disc with nonnegative real part is the Herglotz transform `\u222b (z + w) / (z - w) d\u03bc(z)` of a finite positive measure `\u03bc` on the unit circle, plus the imaginary constant `(F 0).im * I`.","title":"The Herglotz representation theorem"},{"id":"tauceti/analysis/part-02/complex/part-02/identity-theorem","kind":"node","lean":"TauCeti.BoundedHolomorphicFunctions.restrict_injective","parent":1899,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-02/index.html#identity-theorem","scope":"tauceti/analysis/part-02/complex/part-02","status":"fully_proved","summary":"If `V` is finite-dimensional, `E` is complete, and every connected component of `U` meets `U'`, then restriction `H^\u221e(U, E) \u2192 H^\u221e(U', E)` is injective.","title":"Identity theorem"},{"id":"tauceti/analysis/part-02/complex/part-02/the-identity-theorem-for-riemann-surfaces","kind":"node","lean":"TauCeti.RiemannSurface.apply_eq_of_eventuallyConst","parent":1899,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-02/index.html#the-identity-theorem-for-riemann-surfaces","scope":"tauceti/analysis/part-02/complex/part-02","status":"fully_proved","summary":"A holomorphic map on a connected Riemann surface which is constant near one point is constant.","title":"The identity theorem for Riemann surfaces"},{"id":"tauceti/analysis/part-02/complex/part-02/identity-theorem-on-a-right-half-plane","kind":"node","lean":"DifferentiableOn.eqOn_of_eqOn_re_gt","parent":1899,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-02/index.html#identity-theorem-on-a-right-half-plane","scope":"tauceti/analysis/part-02/complex/part-02","status":"fully_proved","summary":"Two functions holomorphic on `Re s > a` that agree on some right half-plane `Re s > b` agree throughout `Re s > a`.","title":"Identity theorem on a right half-plane"},{"id":"tauceti/analysis/part-02/complex/part-02/the-infinitesimal-schwarz-pick-inequality","kind":"node","lean":"TauCeti.norm_deriv_div_one_sub_norm_sq_le","parent":1899,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-02/index.html#the-infinitesimal-schwarz-pick-inequality","scope":"tauceti/analysis/part-02/complex/part-02","status":"fully_proved","summary":"A holomorphic self-map `f` of the open unit disc contracts the Poincar\u00e9 metric: at every point `z` of the disc, `\u2016deriv f z\u2016 / (1 - \u2016f z\u2016 ^ 2) \u2264 1 / (1 - \u2016z\u2016 ^ 2)`.","title":"The infinitesimal Schwarz--Pick inequality"},{"id":"tauceti/analysis/part-02/complex/part-02/jensens-inequality-from-below","kind":"node","lean":"AnalyticOnNhd.le_circleAverage_log_norm","parent":1899,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-02/index.html#jensens-inequality-from-below","scope":"tauceti/analysis/part-02/complex/part-02","status":"fully_proved","summary":"If `f` is analytic on the closed disc of radius `|r|` about `c`, with `r \u2260 0`, the circle average of `log \u2016f\u2016` is at least `k * log r + log \u2016\u03c4\u2016`, where `k` is the order of `f` at `c` and `\u03c4` its trailing coefficient there.","title":"Jensen's inequality from below"},{"id":"tauceti/analysis/part-02/complex/part-02/the-joint-holomorphic-implicit-function-theorem","kind":"node","lean":"TauCeti.exists_jointImplicitFunction_of_eventually_differentiableAt","parent":1899,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-02/index.html#the-joint-holomorphic-implicit-function-theorem","scope":"tauceti/analysis/part-02/complex/part-02","status":"fully_proved","summary":"Let `G : E\u2081 \u00d7 E\u2082 \u2192 F` be complex differentiable near `u`, with invertible partial derivative `\u2202\u2082G(u) : E\u2082 \u2192L[\u2102] F`. Then there is a map `\u03c8 : E\u2081 \u00d7 F \u2192 E\u2082`, holomorphic (indeed analytic) near `(u.1, G u)`, with `\u03c8 (u.1, G u) = u.2` and `G (z, \u03c8 (z, w)) = w` near `(u.1, G u)`; moreover, for `(z, v)` near `u` and `w` near `G u`, `G (z, v) = w` holds exactly when `\u03c8 (z, w) = v`. One choice of `\u03c8` is\u2026","title":"The joint holomorphic implicit function theorem"},{"id":"tauceti/analysis/part-02/complex/part-02/laurents-theorem-on-an-open-concentric-annulus","kind":"node","lean":"DifferentiableOn.hasLaurentSeriesOnAnnulus, TauCeti.hasLaurentSeriesOnAnnulus_of_differentiable_off_countable","parent":1899,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-02/index.html#laurents-theorem-on-an-open-concentric-annulus","scope":"tauceti/analysis/part-02/complex/part-02","status":"fully_proved","summary":"Every function `f : \u2102 \u2192 E` into a complex Banach space that is complex-differentiable on `Metric.eball c R \\ Metric.closedBall c r` (`0 \u2264 r`, `r < \u03c1`, `ENNReal.ofReal \u03c1 < R`) expands in a convergent Laurent series with coefficients `circleLaurentCoeff f c \u03c1`.","title":"Laurent's theorem on an open concentric annulus"},{"id":"tauceti/analysis/part-02/complex/part-02/liouvilles-theorem-for-functions-of-negative-ord","kind":"node","lean":"TauCeti.OrderLE.isConst","parent":1899,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-02/index.html#liouvilles-theorem-for-functions-of-negative-ord","scope":"tauceti/analysis/part-02/complex/part-02","status":"fully_proved","summary":"An entire function of order at most `A < 0` is constant.","title":"Liouville's theorem for functions of negative order"},{"id":"tauceti/analysis/part-02/complex/part-02/montels-selection-theorem","kind":"node","lean":"TauCeti.montel","parent":1899,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-02/index.html#montels-selection-theorem","scope":"tauceti/analysis/part-02/complex/part-02","status":"fully_proved","summary":"A locally bounded sequence of holomorphic maps on an open set `\u03a9 \u2286 \u2102` into a proper complex normed space `E` has a subsequence converging locally uniformly on `\u03a9` to a holomorphic map.","title":"Montel's selection theorem"},{"id":"tauceti/analysis/part-02/complex/part-02/montels-theorem-as-an-equivalence","kind":"node","lean":"TauCeti.isCompact_closure_range_iff_isLocallyBoundedOn, TauCeti.BoundedHolomorphicFunctions.isCompactOperator_restrict, TauCeti.isCompact_closure_range_of_isLocallyBoundedOn","parent":1899,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-02/index.html#montels-theorem-as-an-equivalence","scope":"tauceti/analysis/part-02/complex/part-02","status":"fully_proved","summary":"For a family of holomorphic maps of an open set `\u03a9 \u2286 \u2102` into a proper complex normed space `E`, being locally bounded on `\u03a9` and being relatively compact in `C(\u21a5\u03a9, E)` are the same condition.","title":"Montel's theorem as an equivalence"},{"id":"tauceti/analysis/part-02/complex/part-02/the-morse-index-bound-n","kind":"node","lean":"TauCeti.IsStrictlyPlurisubharmonicAt.morseIndex_le, TauCeti.IsStronglyPseudoconvexAt.morseIndex_le","parent":1899,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-02/index.html#the-morse-index-bound-n","scope":"tauceti/analysis/part-02/complex/part-02","status":"fully_proved","summary":"The real Hessian of a strictly plurisubharmonic function has negative index of inertia at most the complex dimension.","title":"The Morse index bound `\u03bb \u2264 n`"},{"id":"tauceti/analysis/part-02/complex/part-02/the-open-mapping-theorem","kind":"node","lean":"TauCeti.RiemannSurface.isOpenMap_of_forall_not_eventuallyConst, TauCeti.RiemannSurface.nhds_le_map_nhds_of_not_eventuallyConst","parent":1899,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-02/index.html#the-open-mapping-theorem","scope":"tauceti/analysis/part-02/complex/part-02","status":"fully_proved","summary":"A holomorphic map between Riemann surfaces which is constant near no point is an open map.","title":"The open mapping theorem"},{"id":"tauceti/analysis/part-02/complex/part-02/phragmen-lindelof-principle-on-the-upper-half-pl","kind":"node","lean":"TauCeti.PhragmenLindelof.upper_half_plane_of_isBounded","parent":1899,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-02/index.html#phragmen-lindelof-principle-on-the-upper-half-pl","scope":"tauceti/analysis/part-02/complex/part-02","status":"fully_proved","summary":"If `f` is bounded on the open upper half-plane, holomorphic there, continuous on its closure, and bounded by `C` on the real axis, then `f` is bounded by `C` on the closed upper half-plane.","title":"Phragm\u00e9n\u2013Lindel\u00f6f principle on the upper half-plane"},{"id":"tauceti/analysis/part-02/complex/part-02/the-poisson-lower-bound-of-a-nevanlinna-represen","kind":"node","lean":"TauCeti.measureReal_Icc_le_of_eq_nevanlinnaKernel_add","parent":1899,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-02/index.html#the-poisson-lower-bound-of-a-nevanlinna-represen","scope":"tauceti/analysis/part-02/complex/part-02","status":"fully_proved","summary":"The mass a Nevanlinna measure gives to the interval of centre `u` and half-width `v` is at most `2 v` times the imaginary part of the represented function at `u + i v`.","title":"The Poisson lower bound of a Nevanlinna representation"},{"id":"tauceti/analysis/part-02/complex/part-02/the-presentation-part-of-poincares-polygon-theor","kind":"node","lean":"TauCeti.UpperHalfPlane.ConvexPolygon.SidePairing.relatorQuotientTo_bijective_of_isParabolic","parent":1899,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-02/index.html#the-presentation-part-of-poincares-polygon-theor","scope":"tauceti/analysis/part-02/complex/part-02","status":"fully_proved","summary":"If a side-paired polygon satisfies the angle condition at every finite vertex cycle and has parabolic cycle transformations at its ideal vertices, then evaluating the side generators at the side-pairing maps is an isomorphism from the group presented by the side relations and the finite vertex cycle relations onto the group generated by the side-pairing maps.","title":"The presentation part of Poincar\u00e9's polygon theorem"},{"id":"tauceti/analysis/part-02/complex/part-02/the-riemann-extension-theorem-across-a-coordinat","kind":"node","lean":"TauCeti.exists_analyticOnNhd_prod_eqOn","parent":1899,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-02/index.html#the-riemann-extension-theorem-across-a-coordinat","scope":"tauceti/analysis/part-02/complex/part-02","status":"fully_proved","summary":"Let `U \u2286 V` and `s \u2286 \u2102` be open. If `f` is analytic on `U \u00d7 (s \\ {c})` and, when `c \u2208 s`, each slice `z \u21a6 f (w, z)`, for `w \u2208 U`, is bounded on a punctured neighbourhood of `c`, then some `g` analytic on `U \u00d7 s` agrees with `f` on `U \u00d7 (s \\ {c})`. Such a `g` is unique on `U \u00d7 s` by `eqOn_prod_of_eqOn_prod_diff_singleton`.","title":"The Riemann extension theorem across a coordinate hyperplane"},{"id":"tauceti/analysis/part-02/complex/part-02/rigidity-in-the-schwarz-pick-theorem","kind":"node","lean":"TauCeti.exists_norm_eq_one_forall_eq_of_pseudoHyperbolicExpr_map_eq","parent":1899,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-02/index.html#rigidity-in-the-schwarz-pick-theorem","scope":"tauceti/analysis/part-02/complex/part-02","status":"fully_proved","summary":"If a holomorphic self-map `f` of the open unit disc preserves the pseudo-hyperbolic expression at one pair of distinct points `z \u2260 w`, then on the whole disc `f` is the Moebius conjugate of a rotation: there is a unimodular `C` with","title":"Rigidity in the Schwarz--Pick theorem"},{"id":"tauceti/analysis/part-02/complex/part-02/the-schoenflies-theorem","kind":"node","lean":"TauCeti.IsJordanCurve.exists_homeomorph_image_sphere, TauCeti.IsJordanCurve.exists_homeomorph_image_inner_eq_zero, TauCeti.IsJordanCurve.exists_homeomorph_image_sphere_of_mem","parent":1899,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-02/index.html#the-schoenflies-theorem","scope":"tauceti/analysis/part-02/complex/part-02","status":"fully_proved","summary":"For every Jordan curve `C \u2286 \u2102` there is a homeomorphism of `\u2102` carrying the unit circle onto `C`, the open unit disc onto the inside `filledHull C \\ C` of `C`, and the closed unit disc onto `filledHull C`.","title":"The Schoenflies theorem"},{"id":"tauceti/analysis/part-02/complex/part-02/schwarzs-lemma-with-prescribed-zeros","kind":"node","lean":"TauCeti.norm_le_mul_prod_div_pow, TauCeti.norm_le_mul_three_mul_div_pow","parent":1899,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-02/index.html#schwarzs-lemma-with-prescribed-zeros","scope":"tauceti/analysis/part-02/complex/part-02","status":"fully_proved","summary":"If `f` is analytic on a neighbourhood of `closedBall c T`, bounded by `M` on `sphere c T`, and vanishes to order at least `k x` at each point `x` of a finite set `s \u2286 ball c T`, then on `closedBall c T` `\u2016f z\u2016 \u2264 M * \u220f x \u2208 s, (\u2016z - x\u2016 / (T - \u2016x - c\u2016)) ^ k x`.","title":"Schwarz's lemma with prescribed zeros"},{"data_url":"scopes/tauceti/analysis/part-02/complex/part-03.json","id":"tauceti/analysis/part-02/complex/part-03","kind":"scope","navigation":"transparent","parent":1877,"result":false,"route":"../roadmap/tauceti/analysis/part-02/complex/part-03/index.html","scope":"tauceti/analysis/part-02/complex/part-03","status":"fully_proved","title":"Schwarz\u2013Pick, Lipschitz form to Weyl's lemma for \\bar\u2202"},{"id":"tauceti/analysis/part-02/complex/part-03/schwarz-pick-lipschitz-form","kind":"node","lean":"TauCeti.PoincareDisc.lipschitzWith_one, TauCeti.hyperbolicLength_comp_le, TauCeti.hyperbolicDist_map_le, TauCeti.PoincareDisc.dist_map_le","parent":1920,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-03/index.html#schwarz-pick-lipschitz-form","scope":"tauceti/analysis/part-02/complex/part-03","status":"fully_proved","summary":"A holomorphic self-map of the unit disc is a nonexpanding map of the metric space `TauCeti.PoincareDisc`.","title":"Schwarz\u2013Pick, Lipschitz form"},{"id":"tauceti/analysis/part-02/complex/part-03/shimizus-inequality-at-two-cusps","kind":"node","lean":"TauCeti.Subgroup.CuspDatum.im_scaling_smul_smul_mul_im_scaling_smul_le","parent":1920,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-03/index.html#shimizus-inequality-at-two-cusps","scope":"tauceti/analysis/part-02/complex/part-03","status":"fully_proved","summary":"Let `D` and `D'` be normalized cusp data of a discrete `\u0393 \u2264 PSL(2, \u211d)`. If `g \u2208 \u0393` does not carry the cusp of `D'` to the cusp of `D`, then the height of `g \u2022 z` above the cusp of `D` and the height of `z` above the cusp of `D'`, each measured in the scaling coordinate of its datum, have product at most `D.width * D'.width`.","title":"Shimizu's inequality at two cusps"},{"id":"tauceti/analysis/part-02/complex/part-03/the-stieltjes-perron-vanishing-theorem","kind":"node","lean":"TauCeti.measure_Icc_eq_zero_of_eq_nevanlinnaKernel_add","parent":1920,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-03/index.html#the-stieltjes-perron-vanishing-theorem","scope":"tauceti/analysis/part-02/complex/part-03","status":"fully_proved","summary":"A Nevanlinna measure gives no mass to a compact interval over which the imaginary part of the represented function is continuous up to the real axis and vanishes there. Continuity is asked for on the closed rectangle of any positive height `d` over the interval, as a function holomorphic near the interval supplies it.","title":"The Stieltjes--Perron vanishing theorem"},{"id":"tauceti/analysis/part-02/complex/part-03/uniqueness-in-the-riemann-mapping-theorem-up-to","kind":"node","lean":"TauCeti.exists_eqOn_unitDiscStandardAutomorphismFormula_comp","parent":1920,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-03/index.html#uniqueness-in-the-riemann-mapping-theorem-up-to","scope":"tauceti/analysis/part-02/complex/part-03","status":"fully_proved","summary":"If `f` and `g` are holomorphic injections from an open set `U` onto the open unit disc, then there are `u` on the unit circle and `a` in the unit disc such that","title":"Uniqueness in the Riemann mapping theorem, up to a disc automorphism"},{"id":"tauceti/analysis/part-02/complex/part-03/weyls-lemma-for-bar","kind":"node","lean":"TauCeti.HasWeakConjWirtingerDerivOn.exists_differentiableOn_ae_eq","parent":1920,"result":true,"route":"../roadmap/tauceti/analysis/part-02/complex/part-03/index.html#weyls-lemma-for-bar","scope":"tauceti/analysis/part-02/complex/part-03","status":"fully_proved","summary":"A distributional solution with finite-dimensional complex coefficients agrees almost everywhere on its domain with a holomorphic function.","title":"Weyl's lemma for \\bar\u2202"},{"data_url":"scopes/tauceti/analysis/part-02/contour.json","id":"tauceti/analysis/part-02/contour","kind":"scope","parent":1841,"result":false,"route":"../roadmap/tauceti/analysis/part-02/contour/index.html","scope":"tauceti/analysis/part-02/contour","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Analysis.Contour` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Contour"},{"id":"tauceti/analysis/part-02/contour/the-canonical-laurent-decomposition","kind":"node","lean":"TauCeti.Contour.eventuallyEq_meromorphicAnalyticPartAt_add_meromorphicPolarPartAt","parent":1926,"result":true,"route":"../roadmap/tauceti/analysis/part-02/contour/index.html#the-canonical-laurent-decomposition","scope":"tauceti/analysis/part-02/contour","status":"fully_proved","summary":"near `s`, `f` is the analytic part plus the polar part.","title":"The canonical Laurent decomposition"},{"id":"tauceti/analysis/part-02/contour/cauchys-theorem-on-a-disc","kind":"node","lean":"TauCeti.Contour.cauchyTheorem_ball, TauCeti.Contour.Cycle.cauchyTheorem_starConvex, TauCeti.Contour.cauchyTheorem_of_pathHomotopy_refl, TauCeti.Contour.cauchyTheorem_starConvex","parent":1926,"result":true,"route":"../roadmap/tauceti/analysis/part-02/contour/index.html#cauchys-theorem-on-a-disc","scope":"tauceti/analysis/part-02/contour","status":"fully_proved","summary":"Cauchy's theorem on a disc \u2014 the `Metric.ball` case of `TauCeti.Contour.cauchyTheorem_starConvex`: a function holomorphic on a disc integrates to zero along every closed piecewise-`C\u00b9` curve inside it, not only along the boundary circles.","title":"Cauchy's theorem on a disc"},{"id":"tauceti/analysis/part-02/contour/cauchy-goursat-for-a-pole-free-meromorphic-funct","kind":"node","lean":"TauCeti.Contour.circleIntegral_eq_zero_of_meromorphicOrderAt_nonneg","parent":1926,"result":true,"route":"../roadmap/tauceti/analysis/part-02/contour/index.html#cauchy-goursat-for-a-pole-free-meromorphic-funct","scope":"tauceti/analysis/part-02/contour","status":"fully_proved","summary":"If `A` is meromorphic on the closed disc `closedBall c R` (`R \u2265 0`) and has non-negative meromorphic order at every point of the disc, then `\u222e z in C(c, R), A z = 0`.","title":"Cauchy\u2013Goursat for a pole-free meromorphic function"},{"id":"tauceti/analysis/part-02/contour/fubinis-theorem-for-cycles","kind":"node","lean":"TauCeti.Contour.Cycle.integral_integral_swap","parent":1926,"result":true,"route":"../roadmap/tauceti/analysis/part-02/contour/index.html#fubinis-theorem-for-cycles","scope":"tauceti/analysis/part-02/contour","status":"fully_proved","summary":"For a function `F` continuous on the product of the traces of two cycles `C` and `D`,","title":"Fubini's theorem for cycles"},{"id":"tauceti/analysis/part-02/contour/the-generalized-residue-theorem-on-a-star-shaped","kind":"node","lean":"TauCeti.Contour.hungerbuhlerWasem_residueTheorem_of_simple_poles_starConvex, TauCeti.Contour.Cycle.hungerbuhlerWasem_residueTheorem_of_simple_poles, TauCeti.Contour.Cycle.hungerbuhlerWasem_residueTheorem_of_simple_poles_starConvex, TauCeti.Contour.hungerbuhlerWasem_residueTheorem_of_simple_poles","parent":1926,"result":true,"route":"../roadmap/tauceti/analysis/part-02/contour/index.html#the-generalized-residue-theorem-on-a-star-shaped","scope":"tauceti/analysis/part-02/contour","status":"fully_proved","summary":"Let `\u03a9` be open and star-shaped about `x`, `S \u2286 \u03a9` finite, `f` holomorphic on `\u03a9 \u2216 S` with at worst a simple pole at each point of `S`, and `\u03b3` a closed piecewise-`C\u00b9` immersion in `\u03a9` based off `S`. Then the Cauchy principal value of the contour integral is","title":"The generalized residue theorem on a star-shaped open set, simple-pole regime"},{"id":"tauceti/analysis/part-02/contour/the-half-disc-residue-theorem-with-enclosed-poss","kind":"node","lean":"TauCeti.Contour.hasCauchyPV_halfDiscBoundary_of_simple_poles","parent":1926,"result":true,"route":"../roadmap/tauceti/analysis/part-02/contour/index.html#the-half-disc-residue-theorem-with-enclosed-poss","scope":"tauceti/analysis/part-02/contour","status":"fully_proved","summary":"Let `f` be holomorphic off the finite set `insert 0 S`, with at worst a simple singularity at each of its points, and let every point of `S` lie in the open upper half-disc of radius `R`. Then the Cauchy principal value of `\u222e f` along the half-disc boundary is","title":"The half-disc residue theorem with enclosed possible singularities"},{"id":"tauceti/analysis/part-02/contour/the-half-residue-theorem-at-a-simple-pole","kind":"node","lean":"TauCeti.Contour.hasCauchyPV_half_residue_of_simple_pole","parent":1926,"result":true,"route":"../roadmap/tauceti/analysis/part-02/contour/index.html#the-half-residue-theorem-at-a-simple-pole","scope":"tauceti/analysis/part-02/contour","status":"fully_proved","summary":"the winding-`\u00bd` case with the conditions discharged automatically \u2014 an on-cycle simple pole crossed by the immersion contributes `\u03c0i \u00b7 Res_s f`. The acceptance form for the valence formula's `i` and `\u03c1`.","title":"The half-residue theorem at a simple pole"},{"id":"tauceti/analysis/part-02/contour/the-half-residue-theorem-on-the-half-disc-bounda","kind":"node","lean":"TauCeti.Contour.hasCauchyPV_halfDiscBoundary_of_simple_pole","parent":1926,"result":true,"route":"../roadmap/tauceti/analysis/part-02/contour/index.html#the-half-residue-theorem-on-the-half-disc-bounda","scope":"tauceti/analysis/part-02/contour","status":"fully_proved","summary":"If `f` is holomorphic off the origin and has at worst a simple pole there (the hypotheses also permit a removable singularity, or `f` holomorphic at `0`, in which case the residue is `0`), then along the half-disc boundary \u2014 which passes *through* the origin \u2014 the Cauchy principal value of `\u222b f` is `\u03c0 i \u00b7 residue f 0`: half of what the classical residue theorem would give for a pole enclosed by the contour, because\u2026","title":"The half-residue theorem on the half-disc boundary"},{"id":"tauceti/analysis/part-02/contour/jordan-index-theorem-orientation-sign-dichotomy","kind":"node","lean":"AbsolutelyContinuousOnInterval.jordanSign_eq_one_or_neg_one, BoundedVariationOn.jordanSign_naturalParameterization_eq_one_or_neg_one, AbsolutelyContinuousOnInterval.windingNumber_eq_jordanSign_of_mem_filledHull_sdiff, BoundedVariationOn.windingNumber_naturalParameterization_eq_jordanSign","parent":1926,"result":true,"route":"../roadmap/tauceti/analysis/part-02/contour/index.html#jordan-index-theorem-orientation-sign-dichotomy","scope":"tauceti/analysis/part-02/contour","status":"fully_proved","summary":"for a simple closed absolutely continuous curve on `[a, b]` (`a < b`), `jordanSign \u03b3 a b` is either `1` or `-1`, and equals the winding number at every point of `filledHull (\u03b3 '' uIcc a b) \\ (\u03b3 '' uIcc a b)`.","title":"Jordan index theorem (orientation sign dichotomy)"},{"id":"tauceti/analysis/part-02/contour/jordans-lemma","kind":"node","lean":"TauCeti.Contour.norm_integral_semicircle_exp_mul_le","parent":1926,"result":true,"route":"../roadmap/tauceti/analysis/part-02/contour/index.html#jordans-lemma","scope":"tauceti/analysis/part-02/contour","status":"fully_proved","summary":"If `\u2016f\u2016 \u2264 M` on the upper semicircle of radius `R`, then for `a > 0` the arc contribution of `f z \u00b7 e^{iaz}` is at most `\u03c0 M / a` in norm \u2014 a bound independent of `R`, obtained without assuming any decay of `f`.","title":"Jordan's lemma"},{"id":"tauceti/analysis/part-02/contour/the-pointwise-homology-cauchy-theorem-f-0","kind":"node","lean":"TauCeti.Contour.intervalIntegral_deriv_smul_eq_zero_of_dixonFunction_eq_zero","parent":1926,"result":true,"route":"../roadmap/tauceti/analysis/part-02/contour/index.html#the-pointwise-homology-cauchy-theorem-f-0","scope":"tauceti/analysis/part-02/contour","status":"fully_proved","summary":"Applying the Cauchy integral formula to the twist `g z = (z - w\u2080) \u00b7 f z`, which satisfies `g w\u2080 = 0`, forces `dixonH2 g \u03b3 a b w\u2080 = 0`; and since `g (\u03b3 t) / (\u03b3 t - w\u2080) = f (\u03b3 t)` for `\u03b3` off `w\u2080`, that Cauchy-type integral is the contour integral of `f`. Hence `\u222b t in a..b, deriv \u03b3 t \u2022 f (\u03b3 t) = 0`, given the pointwise vanishing `dixonFunction (fun z \u21a6 (z - w\u2080) * f z) U \u03b3 a b w\u2080 = 0`, interval-integrability of the\u2026","title":"The pointwise homology Cauchy theorem `\u222e_\u03b3 f = 0`"},{"data_url":"scopes/tauceti/analysis/part-02/convex.json","id":"tauceti/analysis/part-02/convex","kind":"scope","parent":1841,"result":false,"route":"../roadmap/tauceti/analysis/part-02/convex/index.html","scope":"tauceti/analysis/part-02/convex","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Analysis.Convex` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Convex"},{"id":"tauceti/analysis/part-02/convex/the-aleksandrov-maximum-principle","kind":"node","lean":"TauCeti.ofReal_neg_pow_mul_addHaar_ball_le_mul_mongeAmpereMeasure","parent":1938,"result":true,"route":"../roadmap/tauceti/analysis/part-02/convex/index.html#the-aleksandrov-maximum-principle","scope":"tauceti/analysis/part-02/convex","status":"fully_proved","summary":"Let `\u03a9` be a bounded open convex subset of a real inner product space `E` of finite dimension `n \u2265 1`, and let `u` be convex on `\u03a9`, continuous on its closure, and nonnegative on its frontier. Then at every `x\u2080 \u2208 \u03a9`,","title":"The Aleksandrov maximum principle"},{"id":"tauceti/analysis/part-02/convex/alexandrovs-theorem","kind":"node","lean":"TauCeti.ae_existsUnique_isPositive_isLittleO, ConvexOn.ae_existsUnique_isPositive_isLittleO, TauCeti.ae_existsUnique_isPositive_forall_mem_subdifferential_norm_sub_le","parent":1938,"result":true,"route":"../roadmap/tauceti/analysis/part-02/convex/index.html#alexandrovs-theorem","scope":"tauceti/analysis/part-02/convex","status":"fully_proved","summary":"Let `f : E \u2192 EReal` be convex (convex real epigraph, never `\u22a5`) on a finite-dimensional real inner product space with an additive Haar measure `\u03bc`, and let `F x = (f x).toReal`. At `\u03bc`-almost every point `x` of the effective domain, `F` has a second-order Taylor expansion `F x' = F x + \u27ea\u2207F x, x' - x\u27eb + \u27eaA (x' - x), x' - x\u27eb / 2 + o(\u2016x' - x\u2016 ^ 2)` with a positive operator `A`, the only positive operator giving such\u2026","title":"Alexandrov's theorem"},{"id":"tauceti/analysis/part-02/convex/the-fenchel-moreau-theorem","kind":"node","lean":"TauCeti.fenchelConjugate_flip_fenchelConjugate_eq, TauCeti.fenchelConjugate_inner\u2097_fenchelConjugate_inner\u2097, TauCeti.fenchelConjugate_flip_fenchelConjugate_eq_iff","parent":1938,"result":true,"route":"../roadmap/tauceti/analysis/part-02/convex/index.html#the-fenchel-moreau-theorem","scope":"tauceti/analysis/part-02/convex","status":"fully_proved","summary":"Let the pairing `B` represent every continuous linear functional on the locally convex space `E`. If `f : E \u2192 EReal` has convex real epigraph, is lower semicontinuous and never takes the value `\u22a5`, then `f` equals its biconjugate `f\u22c6\u22c6`, the second conjugate being taken for the transposed pairing.","title":"The Fenchel\u2013Moreau theorem"},{"id":"tauceti/analysis/part-02/convex/the-fenchel-young-inequality","kind":"node","lean":"TauCeti.le_add_fenchelConjugate, TauCeti.sub_le_fenchelConjugate","parent":1938,"result":true,"route":"../roadmap/tauceti/analysis/part-02/convex/index.html#the-fenchel-young-inequality","scope":"tauceti/analysis/part-02/convex","status":"fully_proved","summary":"The Fenchel\u2013Young inequality in additive form, `\u27eax, y\u27eb \u2264 f x + f\u22c6 y`, valid whenever neither summand is `\u22a5`; the second summand is `\u22a5` only when `f \u2261 \u22a4`.","title":"The Fenchel\u2013Young inequality"},{"id":"tauceti/analysis/part-02/convex/grothendiecks-compactness-lemma","kind":"node","lean":"TotallyBounded.exists_tendsto_zero_subset_closure_convexHull","parent":1938,"result":true,"route":"../roadmap/tauceti/analysis/part-02/convex/index.html#grothendiecks-compactness-lemma","scope":"tauceti/analysis/part-02/convex","status":"fully_proved","summary":"A totally bounded subset of a real seminormed space lies in the closed convex hull of the range of a sequence tending to `0`.","title":"Grothendieck's compactness lemma"},{"id":"tauceti/analysis/part-02/convex/jensens-theorem","kind":"node","lean":"TauCeti.convexOn_of_midpoint","parent":1938,"result":true,"route":"../roadmap/tauceti/analysis/part-02/convex/index.html#jensens-theorem","scope":"tauceti/analysis/part-02/convex","status":"fully_proved","summary":"On a convex subset of a real topological vector space, a continuous function that is midpoint convex is convex.","title":"Jensen's theorem"},{"id":"tauceti/analysis/part-02/convex/lebesgue-decomposition-of-the-monge-ampere-measu","kind":"node","lean":"TauCeti.mongeAmpereMeasure_eq_singularPart_add_withDensity, TauCeti.exists_mongeAmpereMeasure_eq_singularPart_add_withDensity, TauCeti.mongeAmpereMeasure_withDensity_eq_singularPart_add_withDensity","parent":1938,"result":true,"route":"../roadmap/tauceti/analysis/part-02/convex/index.html#lebesgue-decomposition-of-the-monge-ampere-measu","scope":"tauceti/analysis/part-02/convex","status":"fully_proved","summary":"Let `f : E \u2192 EReal` be convex (convex real epigraph, never `\u22a5`) with `D` the interior of its effective domain, and let `H x` be, for almost every `x \u2208 D`, a first-order expansion of the subdifferential of `f` at `x` (the Alexandrov Hessian, see `rnDeriv_mongeAmpereMeasure_ae_eq`). Then for every additive Haar measure `\u03bc`, the Aleksandrov Monge\u2013Amp\u00e8re measure splits as `MA_f = (MA_f)\u02e2 + det (H x) dx` on `D`, where\u2026","title":"Lebesgue decomposition of the Monge\u2013Amp\u00e8re measure"},{"id":"tauceti/analysis/part-02/convex/rademachers-theorem-for-extended-real-convex-fun","kind":"node","lean":"TauCeti.ae_eventually_ne_top_and_differentiableAt_toReal","parent":1938,"result":true,"route":"../roadmap/tauceti/analysis/part-02/convex/index.html#rademachers-theorem-for-extended-real-convex-fun","scope":"tauceti/analysis/part-02/convex","status":"fully_proved","summary":"Let `f : E \u2192 EReal` have convex real epigraph and never take the value `\u22a5`, on a finite-dimensional real normed space `E` with an additive Haar measure `\u03bc`. Then at `\u03bc`-almost every point `x` of the effective domain, `f` is finite on a neighbourhood of `x` and its real representative `x \u21a6 (f x).toReal` is differentiable at `x`.","title":"Rademacher's theorem for extended-real convex functions"},{"id":"tauceti/analysis/part-02/convex/rados-theorem","kind":"node","lean":"TauCeti.mem_convexHull_range_comp_perm_iff, TauCeti.mem_convexHull_range_comp_perm_iff_of_antitone","parent":1938,"result":true,"route":"../roadmap/tauceti/analysis/part-02/convex/index.html#rados-theorem","scope":"tauceti/analysis/part-02/convex","status":"fully_proved","summary":"For antitone `w`, a tuple `x` lies in the convex hull of the permutation orbit of `w` exactly when `x` is majorized by `w`: the prefix sums of the decreasing rearrangement of `x` are at most those of `w`, and the total sums agree.","title":"Rado's theorem"},{"id":"tauceti/analysis/part-02/convex/the-vitali-covering-lemma-for-sections","kind":"node","lean":"TauCeti.exists_countable_pairwiseDisjoint_convexSection, TauCeti.exists_countable_pairwiseDisjoint_convexSection_measure_le","parent":1938,"result":true,"route":"../roadmap/tauceti/analysis/part-02/convex/index.html#the-vitali-covering-lemma-for-sections","scope":"tauceti/analysis/part-02/convex","status":"fully_proved","summary":"There is a constant `C`, depending only on `E` and `\u03bc`, with the following property. Let `u` be convex on an open set `\u03a9`, with `\u03bb \u03bc \u2264 MA_u \u2264 \u039b \u03bc` on `\u03a9` for the Monge\u2013Amp\u00e8re measure of `u` extended by `\u22a4` off `\u03a9` and `0 < \u03bb`, and write `\u03b8 = 1 + C \u039b / \u03bb`. Suppose that every section of height `H` centred at a point of `D`, with any supporting slope, is bounded with closure inside `\u03a9`. Consider a family of sections\u2026","title":"The Vitali covering lemma for sections"},{"data_url":"scopes/tauceti/analysis/part-02/distribution.json","id":"tauceti/analysis/part-02/distribution","kind":"scope","parent":1841,"result":false,"route":"../roadmap/tauceti/analysis/part-02/distribution/index.html","scope":"tauceti/analysis/part-02/distribution","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Analysis.Distribution` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Distribution"},{"id":"tauceti/analysis/part-02/distribution/the-constancy-theorem","kind":"node","lean":"Orientation.existsUnique_eq_smul_integrationCurrent_of_comp_extDerivCLM_eq_zero","parent":1949,"result":true,"route":"../roadmap/tauceti/analysis/part-02/distribution/index.html#the-constancy-theorem","scope":"tauceti/analysis/part-02/distribution","status":"fully_proved","summary":"On a connected open subset `\u03a9` of an oriented real vector space of dimension `k + 1`, a top-degree current with zero boundary, `T \u2218 d = 0`, is a unique real multiple of the oriented integration current `[[\u03a9]]`.","title":"The constancy theorem"},{"id":"tauceti/analysis/part-02/distribution/the-constancy-theorem-in-degree-zero","kind":"node","lean":"Orientation.existsUnique_eq_smul_integrationCurrent_of_finrank_eq_zero","parent":1949,"result":true,"route":"../roadmap/tauceti/analysis/part-02/distribution/index.html#the-constancy-theorem-in-degree-zero","scope":"tauceti/analysis/part-02/distribution","status":"fully_proved","summary":"On a zero-dimensional oriented real vector space every `0`-current on a connected open set, which is then the whole point, is a unique real multiple of the integration current, the signed Dirac current at the origin. No boundary condition enters: the boundary of a `0`-current is zero.","title":"The constancy theorem in degree zero"},{"id":"tauceti/analysis/part-02/distribution/the-du-bois-reymond-lemma","kind":"node","lean":"MeasureTheory.LocallyIntegrableOn.exists_ae_eq_const_Ioo_of_integral_deriv_smul_eq_zero, ContinuousOn.exists_eqOn_const_Ioo_of_integral_deriv_smul_eq_zero","parent":1949,"result":true,"route":"../roadmap/tauceti/analysis/part-02/distribution/index.html#the-du-bois-reymond-lemma","scope":"tauceti/analysis/part-02/distribution","status":"fully_proved","summary":"A function locally integrable on an open interval whose pairing with the derivative of every test function on the interval vanishes, `\u222b x, deriv \u03c8 x \u2022 f x = 0`, is almost everywhere equal to a constant on the interval.","title":"The du Bois-Reymond lemma"},{"id":"tauceti/analysis/part-02/distribution/greens-identity-on-schwartz-space","kind":"node","lean":"SchwartzMap.integral_bilinear_laplacian_right_eq_neg_sum","parent":1949,"result":true,"route":"../roadmap/tauceti/analysis/part-02/distribution/index.html#greens-identity-on-schwartz-space","scope":"tauceti/analysis/part-02/distribution","status":"fully_proved","summary":"for a continuous bilinear map `B` and an orthonormal basis `b`, `\u222b B(f, \u0394g) = -\u2211\u1d62 \u222b B(\u2202_{b\u1d62} f, \u2202_{b\u1d62} g)`.","title":"Green's identity on Schwartz space"},{"id":"tauceti/analysis/part-02/distribution/the-local-constancy-theorem","kind":"node","lean":"Orientation.existsUnique_comp_monoCLM_eq_smul_integrationCurrent","parent":1949,"result":true,"route":"../roadmap/tauceti/analysis/part-02/distribution/index.html#the-local-constancy-theorem","scope":"tauceti/analysis/part-02/distribution","status":"fully_proved","summary":"A top-degree current with zero boundary on an arbitrary open subset `\u03a9` of an oriented real vector space of dimension `k + 1` restricts, on every connected open subset `U \u2286 \u03a9`, to a unique real multiple of the integration current `[[U]]`.","title":"The local constancy theorem"},{"id":"tauceti/analysis/part-02/distribution/the-monotone-du-bois-reymond-lemma","kind":"node","lean":"ContinuousOn.monotoneOn_of_integral_deriv_mul_nonpos","parent":1949,"result":true,"route":"../roadmap/tauceti/analysis/part-02/distribution/index.html#the-monotone-du-bois-reymond-lemma","scope":"tauceti/analysis/part-02/distribution","status":"fully_proved","summary":"A real function continuous on an open interval whose distributional derivative is nonnegative, in the sense that `\u222b x, deriv \u03c8 x * f x \u2264 0` for every nonnegative test function `\u03c8` on the interval, is monotone on the interval.","title":"The monotone du Bois-Reymond lemma"},{"id":"tauceti/analysis/part-02/distribution/plancherels-theorem-for-the-inverse-fourier-tran","kind":"node","lean":"SchwartzMap.norm_fourierInv_toL2_eq","parent":1949,"result":true,"route":"../roadmap/tauceti/analysis/part-02/distribution/index.html#plancherels-theorem-for-the-inverse-fourier-tran","scope":"tauceti/analysis/part-02/distribution","status":"fully_proved","summary":"Formalized in Tau Ceti as `SchwartzMap.norm_fourierInv_toL2_eq`.","title":"Plancherel's theorem for the inverse Fourier transform of Schwartz functions"},{"id":"tauceti/analysis/part-02/distribution/the-weighted-plancherel-bound","kind":"node","lean":"SchwartzMap.integral_norm_pow_mul_norm_fourierInv_sq_le","parent":1949,"result":true,"route":"../roadmap/tauceti/analysis/part-02/distribution/index.html#the-weighted-plancherel-bound","scope":"tauceti/analysis/part-02/distribution","status":"fully_proved","summary":"for a Schwartz function `\u03c3`, an orthonormal basis `b` of `V` indexed by `\u03b9`, and `N \u2260 0`,","title":"The weighted Plancherel bound"},{"id":"tauceti/analysis/part-02/distribution/the-weighted-plancherel-identity","kind":"node","lean":"SchwartzMap.integral_inner_pow_mul_norm_fourierInv_sq","parent":1949,"result":true,"route":"../roadmap/tauceti/analysis/part-02/distribution/index.html#the-weighted-plancherel-identity","scope":"tauceti/analysis/part-02/distribution","status":"fully_proved","summary":"for a Schwartz function `\u03c3`,","title":"The weighted Plancherel identity"},{"data_url":"scopes/tauceti/analysis/part-02/fourier.json","id":"tauceti/analysis/part-02/fourier","kind":"scope","parent":1841,"result":false,"route":"../roadmap/tauceti/analysis/part-02/fourier/index.html","scope":"tauceti/analysis/part-02/fourier","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Analysis.Fourier` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Fourier"},{"id":"tauceti/analysis/part-02/fourier/brown-halmos-theorem","kind":"node","lean":"TauCeti.existsUnique_toeplitz_eq_iff_adjoint_comp_comp_eq","parent":1959,"result":true,"route":"../roadmap/tauceti/analysis/part-02/fourier/index.html#brown-halmos-theorem","scope":"tauceti/analysis/part-02/fourier","status":"fully_proved","summary":"A bounded operator `T` on Hardy space is a Toeplitz operator `T_a` for a unique essentially bounded symbol `a` if and only if `S\u2020 T S = T`, where `S = T_z` is the forward shift.","title":"Brown\u2013Halmos theorem"},{"id":"tauceti/analysis/part-02/fourier/classical-toeplitz-product-identity","kind":"node","lean":"TauCeti.toeplitz_comp_sub_toeplitz_eq_neg_classicalHankel_comp_classicalHankel","parent":1959,"result":true,"route":"../roadmap/tauceti/analysis/part-02/fourier/index.html#classical-toeplitz-product-identity","scope":"tauceti/analysis/part-02/fourier","status":"fully_proved","summary":"The product defect is minus the product of the classical Hankel operator of the left symbol and that of the reflected right symbol.","title":"Classical Toeplitz product identity"},{"id":"tauceti/analysis/part-02/fourier/coburns-alternative","kind":"node","lean":"TauCeti.ker_toeplitz_eq_bot_or_ker_adjoint_toeplitz_eq_bot","parent":1959,"result":true,"route":"../roadmap/tauceti/analysis/part-02/fourier/index.html#coburns-alternative","scope":"tauceti/analysis/part-02/fourier","status":"fully_proved","summary":"For a nonzero essentially bounded symbol `a`, either the Toeplitz operator `T_a` or its adjoint has trivial kernel.","title":"Coburn's alternative"},{"id":"tauceti/analysis/part-02/fourier/the-convolution-theorem-for-the-inverse-fourier","kind":"node","lean":"Real.fourierInv_mul_convolution_eq","parent":1959,"result":true,"route":"../roadmap/tauceti/analysis/part-02/fourier/index.html#the-convolution-theorem-for-the-inverse-fourier","scope":"tauceti/analysis/part-02/fourier","status":"fully_proved","summary":"The inverse Fourier transform of the convolution of two integrable functions is the product of their inverse Fourier transforms.","title":"The convolution theorem for the inverse Fourier transform"},{"id":"tauceti/analysis/part-02/fourier/equality-in-wirtingers-inequality","kind":"node","lean":"TauCeti.integral_norm_sub_average_sq_eq_iff","parent":1959,"result":true,"route":"../roadmap/tauceti/analysis/part-02/fourier/index.html#equality-in-wirtingers-inequality","scope":"tauceti/analysis/part-02/fourier","status":"fully_proved","summary":"Equality holds exactly when, on `[a, b]`, the function is its mean plus a first harmonic `cos (2\u03c0 t / (b - a)) \u2022 u + sin (2\u03c0 t / (b - a)) \u2022 v`.","title":"Equality in Wirtinger's inequality"},{"id":"tauceti/analysis/part-02/fourier/the-exterior-reciprocal-fourier-identity","kind":"node","lean":"TauCeti.HaagerupZsido.integral_exp_mul_I_mul_reciprocalKernel","parent":1959,"result":true,"route":"../roadmap/tauceti/analysis/part-02/fourier/index.html#the-exterior-reciprocal-fourier-identity","scope":"tauceti/analysis/part-02/fourier","status":"fully_proved","summary":"For `|x| \u2265 1`, the Fourier integral of the Haagerup\u2013Zsid\u00f3 kernel is `\u222b e^{itx} k(t) dt = 1 / x`.","title":"The exterior reciprocal Fourier identity"},{"id":"tauceti/analysis/part-02/fourier/kroneckers-theorem-for-linear-flows-on-tori","kind":"node","lean":"TauCeti.AddCircle.denseRange_coe_add_mul_iff","parent":1959,"result":true,"route":"../roadmap/tauceti/analysis/part-02/fourier/index.html#kroneckers-theorem-for-linear-flows-on-tori","scope":"tauceti/analysis/part-02/fourier","status":"fully_proved","summary":"A line `t \u21a6 \u03b8 + t \u03bd` in the torus `(\u211d / p\u2124)^\u03b9` is dense exactly when its velocity `\u03bd` is nonresonant: `\u2211 i, k i * \u03bd i \u2260 0` for every nonzero integer vector `k`.","title":"Kronecker's theorem for linear flows on tori"},{"id":"tauceti/analysis/part-02/fourier/parsevals-energy-identity","kind":"node","lean":"TauCeti.hasSum_sq_frequency_mul_sq_fourierCoeffOn","parent":1959,"result":true,"route":"../roadmap/tauceti/analysis/part-02/fourier/index.html#parsevals-energy-identity","scope":"tauceti/analysis/part-02/fourier","status":"fully_proved","summary":"Parseval's energy identity for an absolutely continuous closed complex curve with square-integrable derivative. Each frequency contributes its squared frequency times the squared norm of the corresponding Fourier coefficient.","title":"Parseval's energy identity"},{"id":"tauceti/analysis/part-02/fourier/plancherel-bound-for-fourier-multipliers","kind":"node","lean":"MeasureTheory.Lp.norm_fourierMultiplierCLM_le","parent":1959,"result":true,"route":"../roadmap/tauceti/analysis/part-02/fourier/index.html#plancherel-bound-for-fourier-multipliers","scope":"tauceti/analysis/part-02/fourier","status":"fully_proved","summary":"`\u2016T_m\u2016 \u2264 \u2016m\u2016_\u221e`.","title":"Plancherel bound for Fourier multipliers"},{"id":"tauceti/analysis/part-02/fourier/real-cauchy-poisson-identity","kind":"node","lean":"TauCeti.tsum_exp_neg_mul_abs_int_eq_tsum_div","parent":1959,"result":true,"route":"../roadmap/tauceti/analysis/part-02/fourier/index.html#real-cauchy-poisson-identity","scope":"tauceti/analysis/part-02/fourier","status":"fully_proved","summary":"For `y > 0`, the sum over integers of `exp (-2\u03c0 y |n|)` equals the sum of `y / (\u03c0 (y\u00b2 + n\u00b2))`.","title":"Real Cauchy Poisson identity"},{"id":"tauceti/analysis/part-02/fourier/riemann-lebesgue-on-a-vertical-line","kind":"node","lean":"TauCeti.tendsto_integral_cpow_mul_I_smul_atTop","parent":1959,"result":true,"route":"../roadmap/tauceti/analysis/part-02/fourier/index.html#riemann-lebesgue-on-a-vertical-line","scope":"tauceti/analysis/part-02/fourier","status":"fully_proved","summary":"the integral of `f` against the oscillating factor `x ^ (i t)` tends to `0` as `x \u2192 \u221e`.","title":"Riemann--Lebesgue on a vertical line"},{"id":"tauceti/analysis/part-02/fourier/spectral-theorem-for-strongly-continuous-unitary","kind":"node","lean":"ContRepresentation.exists_pontryaginMeasureTransform_eq_inner","parent":1959,"result":true,"route":"../roadmap/tauceti/analysis/part-02/fourier/index.html#spectral-theorem-for-strongly-continuous-unitary","scope":"tauceti/analysis/part-02/fourier","status":"fully_proved","summary":"For a strongly continuous unitary representation `\u03c0` of a locally compact abelian group `G` on a Hilbert space `H` and a vector `\u03be`, the matrix coefficient `g \u21a6 \u27ea\u03be, \u03c0(g) \u03be\u27eb` is the Fourier\u2013Stieltjes transform of a finite inner regular measure on the Pontryagin dual of `G`.","title":"Spectral theorem for strongly continuous unitary representations, cyclic form"},{"id":"tauceti/analysis/part-02/fourier/weyls-equidistribution-theorem-for-linear-flows","kind":"node","lean":"TauCeti.UnitAddTorus.tendsto_inv_smul_intervalIntegral","parent":1959,"result":true,"route":"../roadmap/tauceti/analysis/part-02/fourier/index.html#weyls-equidistribution-theorem-for-linear-flows","scope":"tauceti/analysis/part-02/fourier","status":"fully_proved","summary":"If the velocity `\u03bd` is nonresonant, that is `\u27eak, \u03bd\u27eb \u2260 0` for every nonzero `k \u2208 \u2124^d`, then along every line `t \u21a6 \u03b8 + t \u03bd` of the unit torus the time averages `T\u207b\u00b9 \u222b\u2080\u1d40 f (\u03b8 + t \u03bd) dt` of a continuous function `f` tend to its space average `\u222b f`.","title":"Weyl's equidistribution theorem for linear flows"},{"id":"tauceti/analysis/part-02/fourier/wirtingers-inequality","kind":"node","lean":"TauCeti.integral_norm_sub_average_sq_le","parent":1959,"result":true,"route":"../roadmap/tauceti/analysis/part-02/fourier/index.html#wirtingers-inequality","scope":"tauceti/analysis/part-02/fourier","status":"fully_proved","summary":"An absolutely continuous function on `[a, b]` with `f a = f b`, valued in a finite-dimensional real inner product space and with square-integrable almost-everywhere derivative, deviates from its mean in `L\u00b2` by at most `(b - a) / (2\u03c0)` times the `L\u00b2` norm of its derivative.","title":"Wirtinger's inequality"},{"data_url":"scopes/tauceti/analysis/part-02/fredholm.json","id":"tauceti/analysis/part-02/fredholm","kind":"scope","parent":1841,"result":false,"route":"../roadmap/tauceti/analysis/part-02/fredholm/index.html","scope":"tauceti/analysis/part-02/fredholm","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Analysis.Fredholm` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Fredholm"},{"id":"tauceti/analysis/part-02/fredholm/atkinsons-theorem","kind":"node","lean":"TauCeti.isFredholm_iff_exists_isCompactOperator_comp_sub_one, ContinuousLinearMap.IsFredholm.of_isCompactOperator_comp_sub_one","parent":1974,"result":true,"route":"../roadmap/tauceti/analysis/part-02/fredholm/index.html#atkinsons-theorem","scope":"tauceti/analysis/part-02/fredholm","status":"fully_proved","summary":"an operator between Banach spaces is Fredholm if and only if it is invertible modulo compact operators.","title":"Atkinson's theorem"},{"id":"tauceti/analysis/part-02/fredholm/riesz-schauder","kind":"node","lean":"TauCeti.isFredholm_one_add, TauCeti.isFredholm_one_sub","parent":1974,"result":true,"route":"../roadmap/tauceti/analysis/part-02/fredholm/index.html#riesz-schauder","scope":"tauceti/analysis/part-02/fredholm","status":"fully_proved","summary":"Riesz--Schauder, in the form in which compact perturbations of the identity arise below.","title":"Riesz--Schauder"},{"id":"tauceti/analysis/part-02/fredholm/schwartzs-lemma","kind":"node","lean":"ContinuousLinearMap.cofg_range_add_of_surjective_of_isCompactOperator","parent":1974,"result":true,"route":"../roadmap/tauceti/analysis/part-02/fredholm/index.html#schwartzs-lemma","scope":"tauceti/analysis/part-02/fredholm","status":"fully_proved","summary":"A compact perturbation of a surjection between real or complex Banach spaces has finite-dimensional cokernel, without any hypothesis on its kernel.","title":"Schwartz's lemma"},{"data_url":"scopes/tauceti/analysis/part-02/gradient-flow.json","id":"tauceti/analysis/part-02/gradient-flow","kind":"scope","parent":1841,"result":false,"route":"../roadmap/tauceti/analysis/part-02/gradient-flow/index.html","scope":"tauceti/analysis/part-02/gradient-flow","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Analysis.GradientFlow` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Gradient Flow"},{"id":"tauceti/analysis/part-02/gradient-flow/de-giorgi-energy-identity","kind":"node","lean":"TauCeti.moreauYosida_add_integral_eq","parent":1978,"result":true,"route":"../roadmap/tauceti/analysis/part-02/gradient-flow/index.html#de-giorgi-energy-identity","scope":"tauceti/analysis/part-02/gradient-flow","status":"fully_proved","summary":"Let `\u03c6` be lower semicontinuous at `x` with `\u03c6(x) \u2260 \u22a4`, and let `y r \u2208 J_r x` be a resolvent step for every time `r \u2208 (0, \u03c3]`, with `\u03c6_\u03c3(x) \u2260 \u22a5`. Then `\u03c6_\u03c3(x) + \u222b\u2080^\u03c3 d(x, y r)\u00b2 / (2r\u00b2) dr = \u03c6(x)`.","title":"De Giorgi energy identity"},{"id":"tauceti/analysis/part-02/gradient-flow/de-giorgi-energy-inequality","kind":"node","lean":"TauCeti.moreauYosida_add_integral_le","parent":1978,"result":true,"route":"../roadmap/tauceti/analysis/part-02/gradient-flow/index.html#de-giorgi-energy-inequality","scope":"tauceti/analysis/part-02/gradient-flow","status":"fully_proved","summary":"Let `y r \u2208 J_r x` be a resolvent step for every time `r \u2208 (0, \u03c3]`, with `\u03c6(x) \u2260 \u22a4` and `\u03c6_\u03c3(x) \u2260 \u22a5`. Then `\u03c6_\u03c3(x) + \u222b\u2080^\u03c3 d(x, y r)\u00b2 / (2r\u00b2) dr \u2264 \u03c6(x)`. Equality holds when `\u03c6` is lower semicontinuous at `x` (`TauCeti.moreauYosida_add_integral_eq`).","title":"De Giorgi energy inequality"},{"id":"tauceti/analysis/part-02/gradient-flow/holder-estimate-for-the-piecewise-constant-inter","kind":"node","lean":"TauCeti.edist_ceil_sq_le_of_forall_mem_moreauYosidaResolvent","parent":1978,"result":true,"route":"../roadmap/tauceti/analysis/part-02/gradient-flow/index.html#holder-estimate-for-the-piecewise-constant-inter","scope":"tauceti/analysis/part-02/gradient-flow","status":"fully_proved","summary":"Under the hypotheses of `TauCeti.edist_sq_le_of_forall_mem_moreauYosidaResolvent`, the piecewise constant interpolant `t \u21a6 u \u2308t / \u03c4\u2309\u208a` of a minimizing movement sequence satisfies, for `0 \u2264 s \u2264 t`, `d(u \u2308s / \u03c4\u2309\u208a, u \u2308t / \u03c4\u2309\u208a)\u00b2 \u2264 2 (t - s + \u03c4) (E - A) (1 + exp (8 (t + \u03c4) / \u03c4'))`. As `\u03c4 \u2192 0` this becomes a `1/2`-H\u00f6lder bound on bounded time intervals.","title":"H\u00f6lder estimate for the piecewise constant interpolant"},{"id":"tauceti/analysis/part-02/gradient-flow/youngs-inequality-along-a-curve","kind":"node","lean":"AbsolutelyContinuousOnInterval.lintegral_mul_metricDerivative_le_dissipation","parent":1978,"result":true,"route":"../roadmap/tauceti/analysis/part-02/gradient-flow/index.html#youngs-inequality-along-a-curve","scope":"tauceti/analysis/part-02/gradient-flow","status":"fully_proved","summary":"the integral of `g (u r) |u'|(r)` is at most the dissipation.","title":"Young's inequality along a curve"},{"data_url":"scopes/tauceti/analysis/part-02/inner-product-space.json","id":"tauceti/analysis/part-02/inner-product-space","kind":"scope","parent":1841,"result":false,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/index.html","scope":"tauceti/analysis/part-02/inner-product-space","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Analysis.InnerProductSpace` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Inner Product Space"},{"data_url":"scopes/tauceti/analysis/part-02/inner-product-space/part-01.json","id":"tauceti/analysis/part-02/inner-product-space/part-01","kind":"scope","navigation":"transparent","parent":1983,"result":false,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-01/index.html","scope":"tauceti/analysis/part-02/inner-product-space/part-01","status":"fully_proved","title":"Bessel's inequality for a\u2026 to The interval/exterior Sylvester\u2026"},{"id":"tauceti/analysis/part-02/inner-product-space/part-01/bessels-inequality-for-a-functional","kind":"node","lean":"Orthonormal.summable_norm_apply_sq, LinearMap.sum_norm_inner_leftSingularVector_sq_le, Orthonormal.norm_analysisMap_le","parent":1984,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-01/index.html#bessels-inequality-for-a-functional","scope":"tauceti/analysis/part-02/inner-product-space/part-01","status":"fully_proved","summary":"The values of a continuous linear functional `\u03c6` on an orthonormal family are square summable. The space need not be complete.","title":"Bessel's inequality for a functional"},{"id":"tauceti/analysis/part-02/inner-product-space/part-01/cauchy-schwarz-for-the-column-series","kind":"node","lean":"HilbertBasis.tsum_norm_inner_mul_norm_le, ContinuousLinearMap.sq_opNorm_sq_mul_re_inner_sub_re_inner_le","parent":1984,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-01/index.html#cauchy-schwarz-for-the-column-series","scope":"tauceti/analysis/part-02/inner-product-space/part-01","status":"fully_proved","summary":"`\u2211' i, \u2016\u27eab\u1d62, x\u27eb\u2016 \u2016f\u1d62\u2016 \u2264 \u2016x\u2016 \u2016f\u2016`.","title":"Cauchy\u2013Schwarz for the column series"},{"id":"tauceti/analysis/part-02/inner-product-space/part-01/the-cotlar-stein-lemma","kind":"node","lean":"TauCeti.exists_hasSum_apply_of_almost_orthogonal, TauCeti.norm_sum_le_of_almost_orthogonal","parent":1984,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-01/index.html#the-cotlar-stein-lemma","scope":"tauceti/analysis/part-02/inner-product-space/part-01","status":"fully_proved","summary":"Let `T i` be operators between Hilbert spaces which are almost orthogonal in the sense that `\u2211 j, \u221a\u2016(T i)\u2020 \u2218L T j\u2016 \u2264 A` and `\u2211 j, \u221a\u2016T i \u2218L (T j)\u2020\u2016 \u2264 A` for every `i`, the sums over `j` being taken over arbitrary finite sets. Then for every `x` the series `\u2211 i, T i x` converges unconditionally, and its sum is a bounded operator of norm at most `A`.","title":"The Cotlar\u2013Stein lemma"},{"id":"tauceti/analysis/part-02/inner-product-space/part-01/courant-fischer-lower-direction","kind":"node","lean":"LinearMap.IsSymmetric.exists_submodule_forall_unit_eigenvalue_le_re_inner, LinearMap.IsSymmetric.exists_unit_vector_re_inner_le_eigenvalue, LinearMap.IsSymmetric.exists_unit_vector_re_inner_le_eigenvalue_of_le_finrank","parent":1984,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-01/index.html#courant-fischer-lower-direction","scope":"tauceti/analysis/part-02/inner-product-space/part-01","status":"fully_proved","summary":"There is a subspace `V` of dimension `k + 1` on which every unit vector `x` satisfies `\u03bb\u2096(T) \u2264 re \u27eaT x, x\u27eb`, where `\u03bb` is the decreasing enumeration `LinearMap.IsSymmetric.eigenvalues` of the eigenvalues of the symmetric operator `T`.","title":"Courant\u2013Fischer, lower direction"},{"id":"tauceti/analysis/part-02/inner-product-space/part-01/daviss-eigenvalue-change-lower-bound","kind":"node","lean":"LinearMap.IsSymmetric.le_sum_sq_eigenvalues_sub","parent":1984,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-01/index.html#daviss-eigenvalue-change-lower-bound","scope":"tauceti/analysis/part-02/inner-product-space/part-01","status":"fully_proved","summary":"Let `T` and `S` be symmetric, let `(v\u1d62)` be the eigenbasis of `T`, and suppose the eigenvalues `\u03bc\u1d62` of `S` are pairwise at distance at least `\u03b3 \u2265 0`. If the diagonal entries `h\u1d62 = Re\u27eav\u1d62, (S - T) v\u1d62\u27eb` of the perturbation satisfy `\u2211\u1d62 h\u1d62\u00b2 \u2264 (\u03b3 / \u221a2)\u00b2`, then the sorted spectra satisfy","title":"Davis's eigenvalue-change lower bound"},{"id":"tauceti/analysis/part-02/inner-product-space/part-01/daviss-sin-2-theorem-angle-form","kind":"node","lean":"LinearMap.IsSymmetric.sin_two_arccos_le, LinearMap.IsSymmetric.sin_two_theta_le_of_mem, LinearMap.IsSymmetric.sin_two_theta_le","parent":1984,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-01/index.html#daviss-sin-2-theorem-angle-form","scope":"tauceti/analysis/part-02/inner-product-space/part-01","status":"fully_proved","summary":"Under the hypotheses of `sin_two_theta_le`, with `\u03b8 = arccos \u2016P x\u2016` the angle between the unit eigenvector `x` and the invariant subspace `U`,","title":"Davis's sin 2\u03b8 theorem, angle form"},{"id":"tauceti/analysis/part-02/inner-product-space/part-01/the-davis-kahan-sin-2-theorem","kind":"node","lean":"TauCeti.UnitarilyInvariantSeminorm.mul_map_sinTwoAngleOperator_le_two_mul_map_sub","parent":1984,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-01/index.html#the-davis-kahan-sin-2-theorem","scope":"tauceti/analysis/part-02/inner-product-space/part-01","status":"fully_proved","summary":"Let `A` and `B` be symmetric operators on a finite-dimensional inner product space, let `U` be invariant under `A` and `V` invariant under `B`. If `re \u27eaA z, z\u27eb \u2264 a \u2016z\u2016\u00b2` on `U\u15ee` and `b \u2016y\u2016\u00b2 \u2264 re \u27eaA y, y\u27eb` on `U`, then the double-angle operator `2 P_{U\u15ee} P_V P_U` satisfies `(b - a) N(2 P_{U\u15ee} P_V P_U) \u2264 2 N(B - A)` for every unitarily invariant seminorm `N`. No spectral hypothesis is placed on `B`.","title":"The Davis\u2013Kahan `sin 2\u0398` theorem"},{"id":"tauceti/analysis/part-02/inner-product-space/part-01/the-dimension-free-davis-kahan-sin-theorem-for-t","kind":"node","lean":"Submodule.mul_projectionGap_le_norm_sub","parent":1984,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-01/index.html#the-dimension-free-davis-kahan-sin-theorem-for-t","scope":"tauceti/analysis/part-02/inner-product-space/part-01","status":"fully_proved","summary":"Let `U` be invariant under the symmetric operator `A` and `V` invariant under the symmetric operator `B`. Suppose that the quadratic form of `A` on `U` is at least `c + g` while that of `B` on `V\u15ee` is at most `c`, and that the quadratic form of `B` on `V` is at least `c' + g` while that of `A` on `U\u15ee` is at most `c'`. Then the gap between `U` and `V` satisfies `g \u2016P_U - P_V\u2016 \u2264 \u2016B - A\u2016`.","title":"The dimension-free Davis\u2013Kahan `sin \u0398` theorem for the gap"},{"id":"tauceti/analysis/part-02/inner-product-space/part-01/the-dimension-free-directed-davis-kahan-sin-theo","kind":"node","lean":"Submodule.mul_norm_starProjection_comp_starProjection_le_norm_sub","parent":1984,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-01/index.html#the-dimension-free-directed-davis-kahan-sin-theo","scope":"tauceti/analysis/part-02/inner-product-space/part-01","status":"fully_proved","summary":"Let `U` be invariant under `A` and `W` invariant under the symmetric operator `B`. If the quadratic form of `A` on `U` is at least `c + g` and that of `B` on `W` is at most `c`, then `g \u2016P_W P_U\u2016 \u2264 \u2016B - A\u2016`. For `g > 0` this bounds `\u2016P_W P_U\u2016` by `\u2016B - A\u2016 / g`. Neither completeness nor finite dimension is assumed.","title":"The dimension-free directed Davis\u2013Kahan `sin \u0398` theorem"},{"id":"tauceti/analysis/part-02/inner-product-space/part-01/the-directed-davis-kahan-sin-theorem-for-the-dir","kind":"node","lean":"Submodule.mul_directedProjectionGap_le_norm_sub","parent":1984,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-01/index.html#the-directed-davis-kahan-sin-theorem-for-the-dir","scope":"tauceti/analysis/part-02/inner-product-space/part-01","status":"fully_proved","summary":"Let `U` be invariant under `A` and `V` invariant under the symmetric operator `B`. If the quadratic form of `A` on `U` is at least `c + g` and that of `B` on `V\u15ee` is at most `c`, then the directed gap `\u2016P_{V\u15ee} P_U\u2016` from `U` to `V` satisfies `g \u2016P_{V\u15ee} P_U\u2016 \u2264 \u2016B - A\u2016`.","title":"The directed Davis\u2013Kahan `sin \u0398` theorem for the directed gap"},{"id":"tauceti/analysis/part-02/inner-product-space/part-01/forward-schur-horn-inequality-karamata-form","kind":"node","lean":"LinearMap.IsSymmetric.sum_map_re_inner_apply_self_le_sum_map_eigenvalues","parent":1984,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-01/index.html#forward-schur-horn-inequality-karamata-form","scope":"tauceti/analysis/part-02/inner-product-space/part-01","status":"fully_proved","summary":"If `\u03c6` is convex on a set `s` containing the eigenvalues of a symmetric operator `T`, then for every orthonormal basis `e`, the sum of `\u03c6` over the diagonal entries `Re\u27eaT (e k), e k\u27eb` is at most the sum of `\u03c6` over the eigenvalues.","title":"Forward Schur\u2013Horn inequality, Karamata form"},{"id":"tauceti/analysis/part-02/inner-product-space/part-01/frobenius-identity-for-the-overlap-operator","kind":"node","lean":"TauCeti.sum_sq_singularValues_overlapOperator","parent":1984,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-01/index.html#frobenius-identity-for-the-overlap-operator","scope":"tauceti/analysis/part-02/inner-product-space/part-01","status":"fully_proved","summary":"the squared singular values of the overlap operator sum to the squared Frobenius norm `\u2211\u2c7c \u2211\u1d62 \u2016\u27eau\u1d62, v\u2c7c\u27eb\u2016\u00b2` of its matrix.","title":"Frobenius identity for the overlap operator"},{"id":"tauceti/analysis/part-02/inner-product-space/part-01/the-frobenius-sylvester-bound","kind":"node","lean":"ContinuousLinearMap.mul_frobenius_le_frobenius_sylvester_apply","parent":1984,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-01/index.html#the-frobenius-sylvester-bound","scope":"tauceti/analysis/part-02/inner-product-space/part-01","status":"fully_proved","summary":"In finite dimension, if `A` and `B` are symmetric and every eigenvalue of `A` is at distance at least `\u03b4` from every eigenvalue of `B`, then `\u03b4 \u2016X\u2016_F \u2264 \u2016A X - X B\u2016_F` for every `X`. In particular every solution of the Sylvester equation `A X - X B = C` satisfies `\u03b4 \u2016X\u2016_F \u2264 \u2016C\u2016_F`.","title":"The Frobenius Sylvester bound"},{"id":"tauceti/analysis/part-02/inner-product-space/part-01/the-frobenius-upper-residual-bound","kind":"node","lean":"LinearMap.IsSymmetric.sum_norm_smul_sub_apply_eigenvectorBasis_sq_le_four_mul_frobenius_sq","parent":1984,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-01/index.html#the-frobenius-upper-residual-bound","scope":"tauceti/analysis/part-02/inner-product-space/part-01","status":"fully_proved","summary":"The residuals `R\u2c7c = \u03bb\u2c7c(T) u\u2c7c - T u\u2c7c` of the eigenvectors `u\u2c7c` of `S` satisfy `\u2211_{j \u2208 s} \u2016R\u2c7c\u2016\u00b2 \u2264 4 \u2016S - T\u2016_F\u00b2` for every index block `s`.","title":"The Frobenius upper residual bound"},{"id":"tauceti/analysis/part-02/inner-product-space/part-01/hadamards-inequality","kind":"node","lean":"OrthonormalBasis.norm_det_le_prod_norm, exteriorPower.norm_\u03b9Multi_le_prod_norm, Matrix.det_gram_le_prod_norm_sq, Matrix.re_det_gram_le_prod_norm_sq, Matrix.re_det_gram_le_pow","parent":1984,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-01/index.html#hadamards-inequality","scope":"tauceti/analysis/part-02/inner-product-space/part-01","status":"fully_proved","summary":"The determinant of a family `v` in the coordinates of an orthonormal basis has norm at most the product of the norms of the vectors. The family may be linearly dependent.","title":"Hadamard's inequality"},{"id":"tauceti/analysis/part-02/inner-product-space/part-01/hilbert-schmidt-norm-identity","kind":"node","lean":"HilbertBasis.hilbertSchmidtNorm_columnOperator","parent":1984,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-01/index.html#hilbert-schmidt-norm-identity","scope":"tauceti/analysis/part-02/inner-product-space/part-01","status":"fully_proved","summary":"The Hilbert\u2013Schmidt norm of `T_f` is the `\u2113\u00b2` norm of `f`.","title":"Hilbert\u2013Schmidt norm identity"},{"id":"tauceti/analysis/part-02/inner-product-space/part-01/the-hoffman-wielandt-inequality","kind":"node","lean":"LinearMap.IsSymmetric.sum_sq_eigenvalues_sub_le_sum_sq_norm_apply","parent":1984,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-01/index.html#the-hoffman-wielandt-inequality","scope":"tauceti/analysis/part-02/inner-product-space/part-01","status":"fully_proved","summary":"For symmetric `T` and `S` with eigenvalues listed in decreasing order and every orthonormal basis `e`, the squared Euclidean distance between the two spectra is at most the squared Frobenius distance `\u2211\u2096 \u2016T e\u2096 - S e\u2096\u2016\u00b2` between the operators.","title":"The Hoffman\u2013Wielandt inequality"},{"id":"tauceti/analysis/part-02/inner-product-space/part-01/hopfs-boundary-point-lemma","kind":"node","lean":"TauCeti.fderiv_pos_of_laplacian_nonneg_of_lt_ball_of_le_sphere, TauCeti.fderiv_pos_of_mul_le_laplacian_add_fderiv_of_lt_ball_of_le_sphere, TauCeti.fderiv_neg_of_laplacian_add_fderiv_le_mul_of_gt_ball_of_ge_sphere, TauCeti.fderiv_pos_of_harmonicOnNhd_of_lt_closedBall, TauCeti.fderiv_neg_of_harmonicOnNhd_of_gt_closedBall, TauCeti.fderiv_pos_of_laplacian_nonneg_of_lt_closedBall, TauCeti.fderiv_neg_of_laplacian_nonpos_of_gt_ball_of_ge_sphere, TauCeti.fderiv_neg_of_laplacian_nonpos_of_gt_closedBall","parent":1984,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-01/index.html#hopfs-boundary-point-lemma","scope":"tauceti/analysis/part-02/inner-product-space/part-01","status":"fully_proved","summary":"Let `B = ball y R` be a ball in a finite-dimensional real inner product space, let `e` be a unit vector, and let `x\u2080 = y + R \u2022 e` be the point where the outward ray in direction `e` meets the sphere `\u2202B`. If `u` is continuous on `closedBall y R`, twice continuously differentiable on `B`, and differentiable at `x\u2080`, is subharmonic on `B` (`0 \u2264 \u0394 u`), stays strictly below the value `u x\u2080` inside `B` and weakly below\u2026","title":"Hopf's boundary-point lemma"},{"id":"tauceti/analysis/part-02/inner-product-space/part-01/horns-theorem-linearmap-issymmetric-exists-ortho","kind":"node","lean":"LinearMap.IsSymmetric.exists_orthonormalBasis_re_inner_apply_self_eq_of_mem_convexHull","parent":1984,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-01/index.html#horns-theorem-linearmap-issymmetric-exists-ortho","scope":"tauceti/analysis/part-02/inner-product-space/part-01","status":"fully_proved","summary":"Every point `d` of the convex hull of the rearrangements of the spectrum of a symmetric operator `T` is the diagonal of `T` in some orthonormal basis: there is an orthonormal basis `e` with `Re\u27eaT e\u2096, e\u2096\u27eb = d\u2096` for every `k`.","title":"Horn's theorem (`LinearMap.IsSymmetric.exists_orthonormalBasis_re_inner_apply_self_eq_of_mem_convexHull`)"},{"id":"tauceti/analysis/part-02/inner-product-space/part-01/the-interval-exterior-sylvester-bound","kind":"node","lean":"TauCeti.UnitarilyInvariantSeminorm.mul_map_le_map_sylvester_apply_of_pointIntervalExteriorGap, TauCeti.UnitarilyInvariantSeminorm.mul_map_le_map_sylvester_apply_of_pointIntervalExteriorGap_of_mem, TauCeti.UnitarilyInvariantSeminorm.mul_map_le_map_sylvester_apply_of_pointIntervalExteriorGap_swap_of_mem, TauCeti.UnitarilyInvariantSeminorm.mul_map_le_map_sylvester_apply_of_pointIntervalExteriorGap_swap","parent":1984,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-01/index.html#the-interval-exterior-sylvester-bound","scope":"tauceti/analysis/part-02/inner-product-space/part-01","status":"fully_proved","summary":"Let `A` and `B` be symmetric operators on finite-dimensional inner product spaces. If the eigenvalues of `A` lie in `[a, b]` and those of `B` outside `(a - \u03b4, b + \u03b4)`, then `\u03b4 N(X) \u2264 N(A X - X B)` for every unitarily invariant seminorm `N`. In particular every solution of the Sylvester equation `A X - X B = C` satisfies `\u03b4 N(X) \u2264 N(C)`.","title":"The interval/exterior Sylvester bound"},{"data_url":"scopes/tauceti/analysis/part-02/inner-product-space/part-02.json","id":"tauceti/analysis/part-02/inner-product-space/part-02","kind":"scope","navigation":"transparent","parent":1983,"result":false,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-02/index.html","scope":"tauceti/analysis/part-02/inner-product-space/part-02","status":"fully_proved","title":"Ky Fan perturbation `sin \u0398`\u2026 to The Schwarz reflection principle"},{"id":"tauceti/analysis/part-02/inner-product-space/part-02/ky-fan-perturbation-sin-bound","kind":"node","lean":"TauCeti.UnitarilyInvariantSeminorm.mul_kyFan_starProjection_comp_starProjection_le_kyFan_sub","parent":2005,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-02/index.html#ky-fan-perturbation-sin-bound","scope":"tauceti/analysis/part-02/inner-product-space/part-02","status":"fully_proved","summary":"For every `k`, interval/exterior spectral separation implies `\u03b4 K\u2096(P_{V\u15ee} P_U) \u2264 K\u2096(B - A)`.","title":"Ky Fan perturbation `sin \u0398` bound"},{"id":"tauceti/analysis/part-02/inner-product-space/part-02/ky-fan-residual-sin-bound","kind":"node","lean":"TauCeti.UnitarilyInvariantSeminorm.mul_kyFan_starProjection_comp_le_kyFan_sylvester_apply","parent":2005,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-02/index.html#ky-fan-residual-sin-bound","scope":"tauceti/analysis/part-02/inner-product-space/part-02","status":"fully_proved","summary":"Under interval/exterior spectral separation, `\u03b4 K\u2096(P_{U\u15ee} X) \u2264 K\u2096(A X - X M)` for every trial map `X` and every `k`.","title":"Ky Fan residual `sin \u0398` bound"},{"id":"tauceti/analysis/part-02/inner-product-space/part-02/ky-fans-maximum-principle-attainment","kind":"node","lean":"LinearMap.exists_orthonormal_sum_inner_eq_kyFanSum, LinearMap.norm_sum_inner_le_kyFanSum, LinearMap.norm_sum_inner_add_inner_le_kyFanSum, LinearMap.IsPositive.exists_orthonormal_sum_re_inner_eq_kyFanSum","parent":2005,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-02/index.html#ky-fans-maximum-principle-attainment","scope":"tauceti/analysis/part-02/inner-product-space/part-02","status":"fully_proved","summary":"the first `k` right singular vectors and the corresponding unit left singular vectors, completed where the singular value vanishes, are orthonormal families with `\u2211\u1d62 \u27eau\u1d62, A v\u1d62\u27eb = \u03c3\u2080(A) + \u22ef + \u03c3\u2096\u208b\u2081(A)`.","title":"Ky Fan's maximum principle, attainment"},{"id":"tauceti/analysis/part-02/inner-product-space/part-02/ky-fans-trace-inequality","kind":"node","lean":"LinearMap.IsSymmetric.sum_re_inner_apply_self_le_sum_eigenvalues","parent":2005,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-02/index.html#ky-fans-trace-inequality","scope":"tauceti/analysis/part-02/inner-product-space/part-02","status":"fully_proved","summary":"for a symmetric `T` and an orthonormal family `(w\u1d62)` with `k` members, `\u2211\u1d62 re \u27eaT w\u1d62, w\u1d62\u27eb` is at most the sum `\u03bb\u2080 + \u22ef + \u03bb\u2096\u208b\u2081` of the `k` largest eigenvalues of `T`.","title":"Ky Fan's trace inequality"},{"id":"tauceti/analysis/part-02/inner-product-space/part-02/lions-representation-theorem","kind":"node","lean":"LinearMap.exists_forall_apply_eq_and_norm_le_of_coercive","parent":2005,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-02/index.html#lions-representation-theorem","scope":"tauceti/analysis/part-02/inner-product-space/part-02","status":"fully_proved","summary":"Let `H` be a real Hilbert space, `\u03a6` a real vector space with a map `j : \u03a6 \u2192 H` and a function `p : \u03a6 \u2192 \u211d` dominating it, `\u2016j \u03c6\u2016 \u2264 p \u03c6`, and let `E` assign to each `\u03c6 \u2208 \u03a6` a continuous linear functional on `H` that is coercive on the diagonal, `c p(\u03c6)\u00b2 \u2264 E \u03c6 (j \u03c6)` with `c > 0`. Then every linear functional `\u2113` on `\u03a6` with `|\u2113 \u03c6| \u2264 M p(\u03c6)` is represented by some `u \u2208 H`, in the sense that `E \u03c6 u = \u2113 \u03c6` for every\u2026","title":"Lions' representation theorem"},{"id":"tauceti/analysis/part-02/inner-product-space/part-02/liouville-theorem-under-the-sign-condition-f-f-0","kind":"node","lean":"TauCeti.eq_zero_of_tendsto_cocompact_of_mul_laplacian_nonneg","parent":2005,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-02/index.html#liouville-theorem-under-the-sign-condition-f-f-0","scope":"tauceti/analysis/part-02/inner-product-space/part-02","status":"fully_proved","summary":"A `C\u00b2` function on a nontrivial finite-dimensional real inner product space which tends to zero at infinity and satisfies `0 \u2264 f x * \u0394 f x` everywhere vanishes identically. The sign condition holds for solutions of `\u0394 f = c f` with `c \u2265 0`, and for differences of solutions of semilinear equations `\u0394 u = F(x, u)` with `F` nondecreasing in `u`.","title":"Liouville theorem under the sign condition `f \u00b7 \u0394 f \u2265 0`"},{"id":"tauceti/analysis/part-02/inner-product-space/part-02/the-lipschitz-bound-for-bounded-harmonic-functio","kind":"node","lean":"InnerProductSpace.HarmonicOnNhd.norm_sub_le_of_norm_le","parent":2005,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-02/index.html#the-lipschitz-bound-for-bounded-harmonic-functio","scope":"tauceti/analysis/part-02/inner-product-space/part-02","status":"fully_proved","summary":"Let `u` be harmonic on neighbourhoods of the closed balls `closedBall y s` and `closedBall z s`, with `0 < s` and `dist y z \u2264 s`, and let `\u2016u\u2016 \u2264 M` on the open balls `ball y s` and `ball z s`. Then","title":"The Lipschitz bound for bounded harmonic functions"},{"id":"tauceti/analysis/part-02/inner-product-space/part-02/the-pairwise-2-davis-kahan-sin-theorem","kind":"node","lean":"TauCeti.UnitarilyInvariantSeminorm.mul_map_starProjection_comp_starProjection_le_pi_div_two_mul_map_sub","parent":2005,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-02/index.html#the-pairwise-2-davis-kahan-sin-theorem","scope":"tauceti/analysis/part-02/inner-product-space/part-02","status":"fully_proved","summary":"Let `A` and `B` be symmetric operators on a finite-dimensional inner product space, let `U` be invariant under `A` and `V` invariant under `B`. If every eigenvalue of `A` on `U` is at distance at least `\u03b4` from every eigenvalue of `B` on `V\u15ee`, then the directed sine map `P_{V\u15ee} P_U` from `U` towards `V` satisfies `\u03b4 N(P_{V\u15ee} P_U) \u2264 (\u03c0 / 2) N(B - A)` for every unitarily invariant seminorm `N`.","title":"The pairwise `\u03c0 / 2` Davis\u2013Kahan `sin \u0398` theorem"},{"id":"tauceti/analysis/part-02/inner-product-space/part-02/parsevals-identity-for-a-functional","kind":"node","lean":"OrthonormalBasis.sum_sq_norm_apply_eq_norm_sq, Orthonormal.norm_starProjection_sq_eq_sum, HilbertBasis.tsum_enorm_sq_inner, HilbertBasis.hasSum_norm_sq_inner, HilbertBasis.tsum_norm_sq_inner","parent":2005,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-02/index.html#parsevals-identity-for-a-functional","scope":"tauceti/analysis/part-02/inner-product-space/part-02","status":"fully_proved","summary":"The squared values of a continuous linear functional on a finite orthonormal basis sum to the square of its operator norm: by the Riesz representation, they are the squared coordinates of the representing vector.","title":"Parseval's identity for a functional"},{"id":"tauceti/analysis/part-02/inner-product-space/part-02/the-pointwise-clarkson-inequality","kind":"node","lean":"TauCeti.norm_add_rpow_add_norm_sub_rpow_le","parent":2005,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-02/index.html#the-pointwise-clarkson-inequality","scope":"tauceti/analysis/part-02/inner-product-space/part-02","status":"fully_proved","summary":"For `2 \u2264 q` and vectors `a`, `b` of a normed group satisfying the parallelogram identity, `\u2016a + b\u2016 ^ q + \u2016a - b\u2016 ^ q \u2264 2 ^ (q - 1) * (\u2016a\u2016 ^ q + \u2016b\u2016 ^ q)`.","title":"The pointwise Clarkson inequality"},{"id":"tauceti/analysis/part-02/inner-product-space/part-02/population-gap-frobenius-sin-theta-theorem","kind":"node","lean":"LinearMap.IsSymmetric.sinThetaFrobenius_eigenvectorSpan_le","parent":2005,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-02/index.html#population-gap-frobenius-sin-theta-theorem","scope":"tauceti/analysis/part-02/inner-product-space/part-02","status":"fully_proved","summary":"For the subspaces selected by the same indices in the canonical eigenbases of symmetric operators `A` (population) and `B` (perturbed), a gap in the population eigenvalues alone gives","title":"Population-gap Frobenius sin-theta theorem"},{"id":"tauceti/analysis/part-02/inner-product-space/part-02/population-gap-sine-theorem-for-arbitrary-ordere","kind":"node","lean":"LinearMap.IsSymmetric.sinThetaFrobenius_orderedEigenframe_le","parent":2005,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-02/index.html#population-gap-sine-theorem-for-arbitrary-ordere","scope":"tauceti/analysis/part-02/inner-product-space/part-02","status":"fully_proved","summary":"The frames select the same sorted indices. Only the population eigenvalues need a gap from the unselected indices; the perturbed frame may be any orthonormal choice inside repeated sample eigenspaces.","title":"Population-gap sine theorem for arbitrary ordered eigenframes"},{"id":"tauceti/analysis/part-02/inner-product-space/part-02/pythagorean-gram-identity-for-complementary-line","kind":"node","lean":"LinearIsometry.adjoint_comp_self_adjoint_comp_eq_id_sub_comp_adjoint","parent":2005,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-02/index.html#pythagorean-gram-identity-for-complementary-line","scope":"tauceti/analysis/part-02/inner-product-space/part-02","status":"fully_proved","summary":"If the ranges of linear maps `F\u2080` and `F\u2081` satisfy `F\u2080F\u2080\u22c6 + F\u2081F\u2081\u22c6 = 1` and `E\u2080` is an isometry, then `(F\u2080\u22c6E\u2080)\u22c6(F\u2080\u22c6E\u2080) = 1 - (E\u2080\u22c6F\u2081)(E\u2080\u22c6F\u2081)\u22c6`. No isometry hypothesis on `F\u2080` or `F\u2081` is needed.","title":"Pythagorean Gram identity for complementary linear blocks"},{"id":"tauceti/analysis/part-02/inner-product-space/part-02/the-rayleigh-lower-bound","kind":"node","lean":"IsCoercive.inv_norm_formSolutionOperator_mul_norm_apply_sq_le","parent":2005,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-02/index.html#the-rayleigh-lower-bound","scope":"tauceti/analysis/part-02/inner-product-space/part-02","status":"fully_proved","summary":"`\u2016S\u2016\u207b\u00b9 \u2016J v\u2016\u00b2 \u2264 B v v` for every `v : V`. For the Dirichlet problem this is the Poincar\u00e9 inequality with the reciprocal solution-operator norm as its constant. Beyond coercivity, only symmetry is needed: when `S = 0` the left-hand side vanishes and the bound is the nonnegativity of the energy.","title":"The Rayleigh lower bound"},{"id":"tauceti/analysis/part-02/inner-product-space/part-02/the-rayleigh-principle","kind":"node","lean":"IsCoercive.isLeast_rayleighQuotient","parent":2005,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-02/index.html#the-rayleigh-principle","scope":"tauceti/analysis/part-02/inner-product-space/part-02","status":"fully_proved","summary":"the least variational eigenvalue `\u2016S\u2016\u207b\u00b9` is the *minimum* of the Rayleigh quotient `B v v / \u2016J v\u2016\u00b2` over the vectors with `J v \u2260 0`. The minimum is attained at an eigenfunction, which is why compactness of `J` is assumed.","title":"The Rayleigh principle"},{"id":"tauceti/analysis/part-02/inner-product-space/part-02/the-rectangular-double-angle-gram-identity","kind":"node","lean":"LinearMap.adjoint_comp_self_comp_eq_sub_of_adjoint_comp_self_eq_id_sub","parent":2005,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-02/index.html#the-rectangular-double-angle-gram-identity","scope":"tauceti/analysis/part-02/inner-product-space/part-02","status":"fully_proved","summary":"If `C\u22c6C = 1 - T T\u22c6` then the Gram operator of the composite `C \u2218 T` is `M - M\u00b2` for `M = T\u22c6T`.","title":"The rectangular double-angle Gram identity"},{"id":"tauceti/analysis/part-02/inner-product-space/part-02/reflected-cross-block-gram-identity-for-linear-b","kind":"node","lean":"LinearIsometry.adjoint_comp_self_adjoint_two_smul_comp_adjoint_sub_id_comp_comp_of_adjoint_comp_eq_zero","parent":2005,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-02/index.html#reflected-cross-block-gram-identity-for-linear-b","scope":"tauceti/analysis/part-02/inner-product-space/part-02","status":"fully_proved","summary":"If `E\u2080` is an isometry and `F\u2080`, `F\u2081` are linear maps with complementary final operators and orthogonal cross block, the reflected cross block has Gram operator `4 (M - M\u00b2)` for `M = T\u22c6T` and `T = E\u2080\u22c6F\u2081`. For arbitrary `F\u2081`, complementarity alone does not imply orthogonality, so `F\u2080\u22c6F\u2081 = 0` is explicitly assumed; it follows if `F\u2081` is a partial isometry.","title":"Reflected cross-block Gram identity for linear blocks"},{"id":"tauceti/analysis/part-02/inner-product-space/part-02/the-reverse-separated-sylvester-bound","kind":"node","lean":"ContinuousLinearMap.mul_norm_le_norm_sylvester_apply_of_re_inner_le_of_le_re_inner","parent":2005,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-02/index.html#the-reverse-separated-sylvester-bound","scope":"tauceti/analysis/part-02/inner-product-space/part-02","status":"fully_proved","summary":"If the real part of the quadratic form of `A` is at most `c` and that of `B` is at least `c + g`, then `g \u2016X\u2016 \u2264 \u2016A X - X B\u2016` for every bounded operator `X`. For `g > 0`, every solution of `A X - X B = Y` therefore satisfies `\u2016X\u2016 \u2264 \u2016Y\u2016 / g`.","title":"The reverse separated Sylvester bound"},{"id":"tauceti/analysis/part-02/inner-product-space/part-02/the-schur-horn-theorem-linearmap-issymmetric-ran","kind":"node","lean":"LinearMap.IsSymmetric.range_re_inner_apply_self_eq_convexHull_eigenvalues_comp_perm, LinearMap.IsSymmetric.exists_orthonormalBasis_re_inner_apply_self_eq_iff","parent":2005,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-02/index.html#the-schur-horn-theorem-linearmap-issymmetric-ran","scope":"tauceti/analysis/part-02/inner-product-space/part-02","status":"fully_proved","summary":"The diagonals `(Re\u27eaT e\u2096, e\u2096\u27eb)\u2096` of a symmetric operator `T` in the orthonormal bases `e` indexed by `Fin n` form exactly the convex hull of the rearrangements of the spectrum of `T`.","title":"The Schur\u2013Horn theorem (`LinearMap.IsSymmetric.range_re_inner_apply_self_eq_convexHull_eigenvalues_comp_perm`)"},{"id":"tauceti/analysis/part-02/inner-product-space/part-02/the-schwarz-reflection-principle","kind":"node","lean":"TauCeti.WeaklyHarmonicOn.exists_harmonicOnNhd_ae_eq_of_inf_normalHalfSpace, TauCeti.WeaklyHarmonicOn.sub_comp_normalReflection","parent":2005,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-02/index.html#the-schwarz-reflection-principle","scope":"tauceti/analysis/part-02/inner-product-space/part-02","status":"fully_proved","summary":"Let `U` be an open set mapped into itself by the reflection `\u03c1` in the hyperplane `{a} \u00d7 E`, and let `w` be weakly harmonic on `U \u2229 H`, where `H = {x | a < x.fst}`, and zero almost everywhere on `U` off `H`. Suppose that `w` is weakly differentiable on all of `U`, with weak gradient `G` vanishing almost everywhere on `U` off `H`, so that `w` vanishes on the flat boundary `U \u2229 {a} \u00d7 E` in the Sobolev sense. Then\u2026","title":"The Schwarz reflection principle"},{"data_url":"scopes/tauceti/analysis/part-02/inner-product-space/part-03.json","id":"tauceti/analysis/part-02/inner-product-space/part-03","kind":"scope","navigation":"transparent","parent":1983,"result":false,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-03/index.html","scope":"tauceti/analysis/part-02/inner-product-space/part-03","status":"fully_proved","title":"The separated Sylvester bound to The Yu\u2013Wang\u2013Samworth overlap\u2026"},{"id":"tauceti/analysis/part-02/inner-product-space/part-03/the-separated-sylvester-bound","kind":"node","lean":"ContinuousLinearMap.mul_norm_le_norm_sylvester_apply_of_le_re_inner_of_re_inner_le","parent":2026,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-03/index.html#the-separated-sylvester-bound","scope":"tauceti/analysis/part-02/inner-product-space/part-03","status":"fully_proved","summary":"If the real part of the quadratic form of `A` is at least `c + g` and that of `B` is at most `c`, then `g \u2016X\u2016 \u2264 \u2016A X - X B\u2016` for every bounded operator `X`. For `g > 0`, every solution of `A X - X B = Y` therefore satisfies `\u2016X\u2016 \u2264 \u2016Y\u2016 / g`.","title":"The separated Sylvester bound"},{"id":"tauceti/analysis/part-02/inner-product-space/part-03/the-singular-value-hoffman-wielandt-inequality","kind":"node","lean":"LinearMap.sum_sq_singularValues_sub_le_sq_frobenius_apply_sub_of_finrank_le, LinearMap.sum_sq_singularValues_sub_le_sq_frobenius_apply_sub_min","parent":2026,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-03/index.html#the-singular-value-hoffman-wielandt-inequality","scope":"tauceti/analysis/part-02/inner-product-space/part-03","status":"fully_proved","summary":"The singular-value Hoffman\u2013Wielandt inequality when the domain dimension does not exceed the codomain dimension. The squared distance between the ordered singular values is at most the squared Frobenius seminorm of the difference.","title":"The singular-value Hoffman\u2013Wielandt inequality"},{"id":"tauceti/analysis/part-02/inner-product-space/part-03/the-spectral-theorem-for-compact-normal-operator","kind":"node","lean":"IsStarNormal.orthogonalComplement_iSup_eigenspaces_eq_bot","parent":2026,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-03/index.html#the-spectral-theorem-for-compact-normal-operator","scope":"tauceti/analysis/part-02/inner-product-space/part-03","status":"fully_proved","summary":"the eigenspaces of a compact normal operator on a complex Hilbert space have trivial mutual orthogonal complement.","title":"The spectral theorem for compact normal operators"},{"id":"tauceti/analysis/part-02/inner-product-space/part-03/spectral-theorem-for-normal-operators","kind":"node","lean":"ContinuousLinearMap.borelCalculus_restrict_id, LinearMap.exists_orthonormalBasis_apply_eq_smul_of_isStarNormal","parent":2026,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-03/index.html#spectral-theorem-for-normal-operators","scope":"tauceti/analysis/part-02/inner-product-space/part-03","status":"fully_proved","summary":"The Borel calculus of the coordinate function on the spectrum is `a` itself, that is, `a = \u222b z dP(z)`.","title":"Spectral theorem for normal operators"},{"id":"tauceti/analysis/part-02/inner-product-space/part-03/the-spectral-theorem-for-the-solution-operator","kind":"node","lean":"IsCoercive.orthogonalComplement_iSup_eigenspaces_formSolutionOperator_eq_bot","parent":2026,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-03/index.html#the-spectral-theorem-for-the-solution-operator","scope":"tauceti/analysis/part-02/inner-product-space/part-03","status":"fully_proved","summary":"its eigenspaces have trivial orthogonal complement in `H`, even when `H` is incomplete.","title":"The spectral theorem for the solution operator"},{"id":"tauceti/analysis/part-02/inner-product-space/part-03/sylvester-bound-from-a-disc-and-the-exterior-of","kind":"node","lean":"TauCeti.UnitarilyInvariantSeminorm.mul_map_le_map_sylvester_apply_of_norm_sub_smul_le_of_le_norm_sub_smul, TauCeti.UnitarilyInvariantSeminorm.mul_map_le_map_sylvester_apply_of_le_norm_sub_smul_of_norm_sub_smul_le","parent":2026,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-03/index.html#sylvester-bound-from-a-disc-and-the-exterior-of","scope":"tauceti/analysis/part-02/inner-product-space/part-03","status":"fully_proved","summary":"If `\u2016A x - c x\u2016 \u2264 r \u2016x\u2016` for every `x` and `(r + \u03b4) \u2016y\u2016 \u2264 \u2016B y - c y\u2016` for every `y`, then `\u03b4 N(X) \u2264 N(A X - X B)` for every unitarily invariant seminorm `N`.","title":"Sylvester bound from a disc and the exterior of a larger disc"},{"id":"tauceti/analysis/part-02/inner-product-space/part-03/the-variational-fredholm-alternative","kind":"node","lean":"IsCoercive.fredholmAlternative_formPerturbation","parent":2026,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-03/index.html#the-variational-fredholm-alternative","scope":"tauceti/analysis/part-02/inner-product-space/part-03","status":"fully_proved","summary":"For a compact continuous linear map `J`, either the homogeneous perturbation of `B` by `\u03ba \u27eaJ \u00b7, J \u00b7\u27eb` has a nonzero solution, or every represented forcing has a unique solution.","title":"The variational Fredholm alternative"},{"id":"tauceti/analysis/part-02/inner-product-space/part-03/the-von-neumann-trace-inequality","kind":"node","lean":"LinearMap.IsSymmetric.re_trace_comp_le_sum_eigenvalues_mul_eigenvalues","parent":2026,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-03/index.html#the-von-neumann-trace-inequality","scope":"tauceti/analysis/part-02/inner-product-space/part-03","status":"fully_proved","summary":"For symmetric `T` and `S` with eigenvalues listed in decreasing order, `Re tr (T S) \u2264 \u2211\u1d62 \u03bb\u1d62(T) \u03bb\u1d62(S)`.","title":"The von Neumann trace inequality"},{"id":"tauceti/analysis/part-02/inner-product-space/part-03/von-neumanns-criterion","kind":"node","lean":"LinearPMap.isSelfAdjoint_of_surjective_addSub","parent":2026,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-03/index.html#von-neumanns-criterion","scope":"tauceti/analysis/part-02/inner-product-space/part-03","status":"fully_proved","summary":"A symmetric partial linear map is self-adjoint if its shifts by `+i` and `-i` are surjective. Density follows from the first shift, so no independent density hypothesis is needed.","title":"Von Neumann's criterion"},{"id":"tauceti/analysis/part-02/inner-product-space/part-03/von-neumanns-theorem","kind":"node","lean":"LinearPMap.dense_adjoint_domain","parent":2026,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-03/index.html#von-neumanns-theorem","scope":"tauceti/analysis/part-02/inner-product-space/part-03","status":"fully_proved","summary":"The adjoint of a closed densely defined partial linear map between complete inner product spaces has dense domain.","title":"von Neumann's theorem"},{"id":"tauceti/analysis/part-02/inner-product-space/part-03/weyls-additive-inequality-upper-form","kind":"node","lean":"LinearMap.IsSymmetric.eigenvalues_add_le, LinearMap.IsSymmetric.le_eigenvalues_add","parent":2026,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-03/index.html#weyls-additive-inequality-upper-form","scope":"tauceti/analysis/part-02/inner-product-space/part-03","status":"fully_proved","summary":"for symmetric operators `T` and `S` and indices with `i + j \u2264 k`, the decreasingly sorted eigenvalues satisfy `\u03bb\u2096(T + S) \u2264 \u03bb\u1d62(T) + \u03bb\u2c7c(S)`. Indices start at `0`, so for `k = i + j` this is the classical `\u03bb_{i+j-1}(T + S) \u2264 \u03bb\u1d62(T) + \u03bb\u2c7c(S)` in one-based indexing.","title":"Weyl's additive inequality, upper form"},{"id":"tauceti/analysis/part-02/inner-product-space/part-03/weyls-inequality","kind":"node","lean":"LinearMap.IsSymmetric.abs_eigenvalue_sub_eigenvalue_le, LinearMap.IsSymmetric.abs_eigenvalue_sub_eigenvalue_le_norm","parent":2026,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-03/index.html#weyls-inequality","scope":"tauceti/analysis/part-02/inner-product-space/part-03","status":"fully_proved","summary":"Weyl's inequality for symmetric operators on a finite-dimensional inner product space over `\ud835\udd5c = \u211d, \u2102`: the `k`-th (decreasingly sorted) eigenvalues of `T` and `S` differ by at most the operator norm of `T \u2212 S`.","title":"Weyl's inequality"},{"id":"tauceti/analysis/part-02/inner-product-space/part-03/weyls-lemma-for-continuous-functions","kind":"node","lean":"TauCeti.harmonicOnNhd_iff_weaklyHarmonicOn, TauCeti.WeaklyHarmonicOn.exists_harmonicOnNhd_ae_eq","parent":2026,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-03/index.html#weyls-lemma-for-continuous-functions","scope":"tauceti/analysis/part-02/inner-product-space/part-03","status":"fully_proved","summary":"A function continuous on the open set `\u03a9` is harmonic on `\u03a9` if and only if it is weakly harmonic there: `\u222b \u0394\u03c6 \u2022 u \u2202\u03bc = 0` for every test function `\u03c6 \u2208 \ud835\udcd3(\u03a9)`.","title":"Weyl's lemma for continuous functions"},{"id":"tauceti/analysis/part-02/inner-product-space/part-03/youngs-inequality-for-real-inner-products","kind":"node","lean":"TauCeti.two_mul_real_inner_le_add_mul_norm_sq","parent":2026,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-03/index.html#youngs-inequality-for-real-inner-products","scope":"tauceti/analysis/part-02/inner-product-space/part-03","status":"fully_proved","summary":"for every weight `\u03b5 > 0`, `2 \u27eax, y\u27eb \u2264 \u03b5 \u2016x\u2016\u00b2 + \u03b5\u207b\u00b9 \u2016y\u2016\u00b2`.","title":"Young's inequality for real inner products"},{"id":"tauceti/analysis/part-02/inner-product-space/part-03/yu-wang-samworth-bound-for-left-singular-vectors","kind":"node","lean":"LinearMap.sq_mul_sum_sum_norm_inner_self_comp_adjoint_eigenvectorBasis_sq_le_four_mul_card_mul_sq, LinearMap.sq_mul_sum_sum_norm_inner_rightSingularBasis_sq_le_four_mul_card_mul_sq","parent":2026,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-03/index.html#yu-wang-samworth-bound-for-left-singular-vectors","scope":"tauceti/analysis/part-02/inner-product-space/part-03","status":"fully_proved","summary":"A population gap in the eigenvalues of the target Gram operator gives the same cross-block bound as for right singular vectors. The eigenbases of `A A\u2020` and `\u00c2 \u00c2\u2020` include complete bases of the kernels, so no positive-rank assumption is needed.","title":"Yu\u2013Wang\u2013Samworth bound for left singular vectors"},{"id":"tauceti/analysis/part-02/inner-product-space/part-03/the-yu-wang-samworth-overlap-bound-frobenius-for","kind":"node","lean":"LinearMap.IsSymmetric.sq_mul_sum_sum_norm_inner_sq_le_four_mul_frobenius_sq, LinearMap.IsSymmetric.sq_mul_sum_sum_norm_inner_sq_le_four_mul_card_mul_sq","parent":2026,"result":true,"route":"../roadmap/tauceti/analysis/part-02/inner-product-space/part-03/index.html#the-yu-wang-samworth-overlap-bound-frobenius-for","scope":"tauceti/analysis/part-02/inner-product-space/part-03","status":"fully_proved","summary":"If `\u0394 \u2265 0` separates the eigenvalues of the population operator `T` indexed by `s` from the remaining ones, then the eigenvectors `u\u2c7c` of the perturbation `S` satisfy `\u0394\u00b2 \u2211_{j \u2208 s} \u2211_{k \u2209 s} |\u27eav\u2096, u\u2c7c\u27eb|\u00b2 \u2264 4 \u2016S - T\u2016_F\u00b2`. No gap is assumed in the spectrum of `S`.","title":"The Yu\u2013Wang\u2013Samworth overlap bound, Frobenius form"},{"data_url":"scopes/tauceti/analysis/part-02/matrix.json","id":"tauceti/analysis/part-02/matrix","kind":"scope","parent":1841,"result":false,"route":"../roadmap/tauceti/analysis/part-02/matrix/index.html","scope":"tauceti/analysis/part-02/matrix","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Analysis.Matrix` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Matrix"},{"id":"tauceti/analysis/part-02/matrix/the-2-2-schur-horn-theorem","kind":"node","lean":"Matrix.exists_isSymm_fin_two_charpoly_eq_iff","parent":2043,"result":true,"route":"../roadmap/tauceti/analysis/part-02/matrix/index.html#the-2-2-schur-horn-theorem","scope":"tauceti/analysis/part-02/matrix","status":"fully_proved","summary":"For `l\u2082 \u2264 l\u2081`, a real symmetric `2 \u00d7 2` matrix with diagonal `(d\u2081, d\u2082)` and characteristic polynomial `(X - l\u2081) (X - l\u2082)` exists exactly when `(d\u2081, d\u2082)` is majorized by `(l\u2081, l\u2082)`, that is, when `l\u2082 \u2264 d\u2081 \u2264 l\u2081` and `d\u2081 + d\u2082 = l\u2081 + l\u2082`. In that case `!![d\u2081, b; b, d\u2082]` with `b = \u221a((l\u2081 - d\u2081) (d\u2081 - l\u2082))` is such a matrix.","title":"The `2 \u00d7 2` Schur\u2013Horn theorem"},{"id":"tauceti/analysis/part-02/matrix/grothendiecks-inequality-over-the-reals-with-con","kind":"node","lean":"Matrix.abs_sum_sum_mul_inner_le_eight_mul, Matrix.innerProductSup_le_eight_mul, Matrix.norm_sum_sum_mul_inner_le_sixteen_mul","parent":2043,"result":true,"route":"../roadmap/tauceti/analysis/part-02/matrix/index.html#grothendiecks-inequality-over-the-reals-with-con","scope":"tauceti/analysis/part-02/matrix","status":"fully_proved","summary":"if `|\u2211 i, \u2211 j, a i j * s i * t j| \u2264 C` for all real scalars of absolute value at most one, then `|\u2211 i, \u2211 j, a i j * \u27eau i, v j\u27eb| \u2264 8 * C` for all families of vectors of norm at most one in any real inner product space.","title":"Grothendieck's inequality over the reals, with constant `\u03ba_\u211d = 8`"},{"id":"tauceti/analysis/part-02/matrix/hadamard-fischer-inequality","kind":"node","lean":"Matrix.PosSemidef.det_fromBlocks_le","parent":2043,"result":true,"route":"../roadmap/tauceti/analysis/part-02/matrix/index.html#hadamard-fischer-inequality","scope":"tauceti/analysis/part-02/matrix","status":"fully_proved","summary":"The determinant of a positive-semidefinite block matrix is at most the product of the diagonal-block determinants.","title":"Hadamard--Fischer inequality"},{"id":"tauceti/analysis/part-02/matrix/the-hoffman-wielandt-inequality-for-hermitian-ma","kind":"node","lean":"Matrix.IsHermitian.sum_sq_eigenvalues\u2080_sub_le_frobenius_norm_sub_sq","parent":2043,"result":true,"route":"../roadmap/tauceti/analysis/part-02/matrix/index.html#the-hoffman-wielandt-inequality-for-hermitian-ma","scope":"tauceti/analysis/part-02/matrix","status":"fully_proved","summary":"the squared Euclidean distance between the decreasingly sorted spectra is at most the squared Frobenius norm `\u2016A - B\u2016_F\u00b2`.","title":"The Hoffman\u2013Wielandt inequality for Hermitian matrices"},{"id":"tauceti/analysis/part-02/matrix/horns-theorem-matrix-exists-mem-unitarygroup-re","kind":"node","lean":"Matrix.exists_mem_unitaryGroup_re_diag_mul_diagonal_mul_star_eq","parent":2043,"result":true,"route":"../roadmap/tauceti/analysis/part-02/matrix/index.html#horns-theorem-matrix-exists-mem-unitarygroup-re","scope":"tauceti/analysis/part-02/matrix","status":"fully_proved","summary":"Every point `d` of the convex hull of the rearrangements of `w` is the diagonal of `U diag(w) U\u1d34` for some unitary `U`.","title":"Horn's theorem (`Matrix.exists_mem_unitaryGroup_re_diag_mul_diagonal_mul_star_eq`)"},{"id":"tauceti/analysis/part-02/matrix/jensens-operator-inequality-for-isometries","kind":"node","lean":"Matrix.conjTranspose_mul_cfc_mul_le_cfc_conjTranspose_mul_mul","parent":2043,"result":true,"route":"../roadmap/tauceti/analysis/part-02/matrix/index.html#jensens-operator-inequality-for-isometries","scope":"tauceti/analysis/part-02/matrix","status":"fully_proved","summary":"Let `f` be operator concave on a set `S` of matrices closed under unitary conjugation, in the sense that `A \u21a6 cfc f A` is concave on `S`. For a Hermitian `A \u2208 S` and an isometry `V`, that is `V\u1d34 * V = 1`, compressing by `V` lowers the functional calculus: `V\u1d34 * cfc f A * V \u2264 cfc f (V\u1d34 * A * V)`.","title":"Jensen's operator inequality for isometries"},{"id":"tauceti/analysis/part-02/matrix/kleins-inequality-for-the-relative-entropy","kind":"node","lean":"Matrix.PosSemidef.re_trace_sub_trace_le_relativeEntropy","parent":2043,"result":true,"route":"../roadmap/tauceti/analysis/part-02/matrix/index.html#kleins-inequality-for-the-relative-entropy","scope":"tauceti/analysis/part-02/matrix","status":"fully_proved","summary":"For positive semidefinite `\u03c1` and `\u03c3`, `Tr \u03c1 - Tr \u03c3 \u2264 D(\u03c1 \u2016 \u03c3)`.","title":"Klein's inequality for the relative entropy"},{"id":"tauceti/analysis/part-02/matrix/koteljanskiis-inequality-for-overlapping-blocks","kind":"node","lean":"Matrix.PosSemidef.det_mul_det_fromBlocks_le","parent":2043,"result":true,"route":"../roadmap/tauceti/analysis/part-02/matrix/index.html#koteljanskiis-inequality-for-overlapping-blocks","scope":"tauceti/analysis/part-02/matrix","status":"fully_proved","summary":"The product of the principal minors on the common block and on all three blocks is bounded by the product of the two intermediate principal minors. No block need be nonsingular.","title":"Koteljanskii's inequality for overlapping blocks"},{"id":"tauceti/analysis/part-02/matrix/minkowskis-determinant-inequality","kind":"node","lean":"Matrix.PosSemidef.det_rpow_add_det_rpow_le, LinearMap.IsPositive.det_rpow_add_det_rpow_le","parent":2043,"result":true,"route":"../roadmap/tauceti/analysis/part-02/matrix/index.html#minkowskis-determinant-inequality","scope":"tauceti/analysis/part-02/matrix","status":"fully_proved","summary":"For positive semidefinite real `n \u00d7 n` matrices `A` and `B` with `n \u2265 1`, `det A ^ (1 / n) + det B ^ (1 / n) \u2264 det (A + B) ^ (1 / n)`.","title":"Minkowski's determinant inequality"},{"id":"tauceti/analysis/part-02/matrix/row-hodge-lower-bound-for-the-principal-minor-se","kind":"node","lean":"Matrix.rowHodgePolygon_le_newtonPolygon_charPowerSeries","parent":2043,"result":true,"route":"../roadmap/tauceti/analysis/part-02/matrix/index.html#row-hodge-lower-bound-for-the-principal-minor-se","scope":"tauceti/analysis/part-02/matrix","status":"fully_proved","summary":"Let `v` be a real additive valuation with `\u2016x\u2016 = b ^ (-v x)` for `x \u2260 0`, where `b > 1`. For a bounded row majorant `\u2016A i j\u2016 \u2264 r i`, the row Hodge polygon of `r` lies below the Newton polygon of the principal-minor series `A.charPowerSeries`, which is the Fredholm determinant `det (1 - T A)` when its principal-minor families are summable.","title":"Row Hodge lower bound for the principal-minor series"},{"id":"tauceti/analysis/part-02/matrix/schurs-theorem-matrix-re-diag-mul-diagonal-mul-s","kind":"node","lean":"Matrix.re_diag_mul_diagonal_mul_star_mem_convexHull","parent":2043,"result":true,"route":"../roadmap/tauceti/analysis/part-02/matrix/index.html#schurs-theorem-matrix-re-diag-mul-diagonal-mul-s","scope":"tauceti/analysis/part-02/matrix","status":"fully_proved","summary":"For a unitary `U`, the diagonal of `U diag(w) U\u1d34` lies in the convex hull of the rearrangements `w \u2218 \u03c3` of `w`.","title":"Schur's theorem (`Matrix.re_diag_mul_diagonal_mul_star_mem_convexHull`)"},{"id":"tauceti/analysis/part-02/matrix/the-schur-horn-theorem-matrix-range-re-diag-unit","kind":"node","lean":"Matrix.range_re_diag_unitary_mul_diagonal_mul_star, Matrix.IsHermitian.range_re_diag_unitary_mul_mul_star, Matrix.range_re_diag_unitary_mul_diagonal_single_mul_star","parent":2043,"result":true,"route":"../roadmap/tauceti/analysis/part-02/matrix/index.html#the-schur-horn-theorem-matrix-range-re-diag-unit","scope":"tauceti/analysis/part-02/matrix","status":"fully_proved","summary":"The diagonals of the matrices `U diag(w) U\u1d34`, for `U` unitary, form exactly the convex hull of the rearrangements of `w`. In the language of moment maps: the image of the unitary orbit of `diag(w)` under the moment map of the diagonal torus is the permutohedron of `w`.","title":"The Schur\u2013Horn theorem (`Matrix.range_re_diag_unitary_mul_diagonal_mul_star`)"},{"id":"tauceti/analysis/part-02/matrix/the-singular-value-hoffman-wielandt-inequality-f","kind":"node","lean":"Matrix.sum_sq_singularValues_toEuclideanLin_sub_le_frobenius_norm_sub_sq","parent":2043,"result":true,"route":"../roadmap/tauceti/analysis/part-02/matrix/index.html#the-singular-value-hoffman-wielandt-inequality-f","scope":"tauceti/analysis/part-02/matrix","status":"fully_proved","summary":"the squared distance between their ordered singular values, summed up to the smaller dimension, is bounded by the squared Frobenius norm of their difference.","title":"The singular-value Hoffman\u2013Wielandt inequality for same-shaped rectangular matrices"},{"id":"tauceti/analysis/part-02/matrix/the-spectral-theorem-for-normal-matrices","kind":"node","lean":"Matrix.exists_mem_unitaryGroup_star_mul_mul_eq_diagonal","parent":2043,"result":true,"route":"../roadmap/tauceti/analysis/part-02/matrix/index.html#the-spectral-theorem-for-normal-matrices","scope":"tauceti/analysis/part-02/matrix","status":"fully_proved","summary":"A normal complex matrix is unitarily diagonalizable: there is a unitary matrix `U` with `star U * A * U` diagonal.","title":"The spectral theorem for normal matrices"},{"id":"tauceti/analysis/part-02/matrix/spectral-wasserstein-bound-squared-form","kind":"node","lean":"Matrix.wassersteinEDist_two_spectralMeasure_sq_le","parent":2043,"result":true,"route":"../roadmap/tauceti/analysis/part-02/matrix/index.html#spectral-wasserstein-bound-squared-form","scope":"tauceti/analysis/part-02/matrix","status":"fully_proved","summary":"The squared `W\u2082` distance between the empirical spectral measures is at most the normalized squared Frobenius distance.","title":"Spectral Wasserstein bound, squared form"},{"id":"tauceti/analysis/part-02/matrix/sylvesters-law-of-inertia-for-hermitian-matrices","kind":"node","lean":"Matrix.IsHermitian.signature_congr","parent":2043,"result":true,"route":"../roadmap/tauceti/analysis/part-02/matrix/index.html#sylvesters-law-of-inertia-for-hermitian-matrices","scope":"tauceti/analysis/part-02/matrix","status":"fully_proved","summary":"The signature is unchanged by `*`-congruence `A \u21a6 P * A * P\u1d34` with `P` invertible.","title":"Sylvester's law of inertia for Hermitian matrices"},{"id":"tauceti/analysis/part-02/matrix/weyls-additive-inequality-for-hermitian-matrices","kind":"node","lean":"Matrix.IsHermitian.eigenvalues\u2080_add_le, Matrix.IsHermitian.le_eigenvalues\u2080_add","parent":2043,"result":true,"route":"../roadmap/tauceti/analysis/part-02/matrix/index.html#weyls-additive-inequality-for-hermitian-matrices","scope":"tauceti/analysis/part-02/matrix","status":"fully_proved","summary":"if `i + j \u2264 k`, then `\u03bb\u2096(A + B) \u2264 \u03bb\u1d62(A) + \u03bb\u2c7c(B)` for the decreasing enumerations `eigenvalues\u2080`. For `k = i + j` this is the classical `\u03bb_{i+j-1}(A + B) \u2264 \u03bb\u1d62(A) + \u03bb\u2c7c(B)` in one-based indexing.","title":"Weyl's additive inequality for Hermitian matrices, upper form"},{"id":"tauceti/analysis/part-02/matrix/weyls-perturbation-bound-for-ordered-eigenvalues","kind":"node","lean":"Matrix.IsHermitian.abs_eigenvalues\u2080_sub_le_norm","parent":2043,"result":true,"route":"../roadmap/tauceti/analysis/part-02/matrix/index.html#weyls-perturbation-bound-for-ordered-eigenvalues","scope":"tauceti/analysis/part-02/matrix","status":"fully_proved","summary":"corresponding eigenvalues differ by at most the Euclidean operator norm of the matrix difference.","title":"Weyl's perturbation bound for ordered eigenvalues of Hermitian matrices"},{"data_url":"scopes/tauceti/analysis/part-02/normed.json","id":"tauceti/analysis/part-02/normed","kind":"scope","parent":1841,"result":false,"route":"../roadmap/tauceti/analysis/part-02/normed/index.html","scope":"tauceti/analysis/part-02/normed","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Analysis.Normed` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Normed"},{"data_url":"scopes/tauceti/analysis/part-02/normed/part-01.json","id":"tauceti/analysis/part-02/normed/part-01","kind":"scope","navigation":"transparent","parent":2062,"result":false,"route":"../roadmap/tauceti/analysis/part-02/normed/part-01/index.html","scope":"tauceti/analysis/part-02/normed/part-01","status":"fully_proved","title":"Banach's lemma to Neumann series criterion\u2026"},{"id":"tauceti/analysis/part-02/normed/part-01/banachs-lemma","kind":"node","lean":"ContinuousLinearMap.exists_preimage_norm_le_of_mem_closure","parent":2063,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-01/index.html#banachs-lemma","scope":"tauceti/analysis/part-02/normed/part-01","status":"fully_proved","summary":"If every `y` lies in the closure of the image of the closed ball of radius `C * \u2016y\u2016`, then every `y` has a preimage of norm at most `2 * C * \u2016y\u2016`. In particular, the map is surjective.","title":"Banach's lemma"},{"id":"tauceti/analysis/part-02/normed/part-01/banach-steinhaus-theorem-uniform-boundedness-pri","kind":"node","lean":"TauCeti.NormedRing.IsTate.banach_steinhaus","parent":2063,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-01/index.html#banach-steinhaus-theorem-uniform-boundedness-pri","scope":"tauceti/analysis/part-02/normed/part-01","status":"fully_proved","summary":"If a family `g : \u03b9 \u2192 E \u2192L[R] F` of continuous linear maps from a complete seminormed module `E` over a Tate ring `R` is pointwise bounded, then the operator norms `\u2016g i\u2016` are uniformly bounded.","title":"Banach\u2013Steinhaus theorem (Uniform Boundedness Principle) over a Tate ring"},{"id":"tauceti/analysis/part-02/normed/part-01/bounded-inverse-theorem","kind":"node","lean":"ContinuousLinearEquiv.ofBijectiveOfIsTate, LinearEquiv.continuous_symm_of_isTate","parent":2063,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-01/index.html#bounded-inverse-theorem","scope":"tauceti/analysis/part-02/normed/part-01","status":"fully_proved","summary":"a bijective continuous linear map from a complete ultrametric seminormed module to a complete normed module over a Tate ring is a continuous linear equivalence.","title":"Bounded Inverse Theorem"},{"id":"tauceti/analysis/part-02/normed/part-01/closed-graph-theorem-for-normed-modules-over-a-t","kind":"node","lean":"LinearMap.continuous_of_isClosed_graph_of_isTate","parent":2063,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-01/index.html#closed-graph-theorem-for-normed-modules-over-a-t","scope":"tauceti/analysis/part-02/normed/part-01","status":"fully_proved","summary":"a linear map from a complete ultrametric normed module to a complete ultrametric seminormed module whose graph is closed is continuous.","title":"Closed Graph Theorem for normed modules over a Tate ring"},{"id":"tauceti/analysis/part-02/normed/part-01/the-continuous-first-isomorphism-theorem-for-ban","kind":"node","lean":"ContinuousAlgHom.quotientKerEquivOfSurjective","parent":2063,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-01/index.html#the-continuous-first-isomorphism-theorem-for-ban","scope":"tauceti/analysis/part-02/normed/part-01","status":"fully_proved","summary":"a continuous surjective algebra homomorphism `f : A \u2192 B` induces an isomorphism of topological algebras `A \u29f8 ker f \u2243 B`. Its inverse is continuous by the open mapping theorem; it need not be an isometry.","title":"The continuous first isomorphism theorem for Banach algebras"},{"id":"tauceti/analysis/part-02/normed/part-01/cramers-rule-bound","kind":"node","lean":"TauCeti.norm_det_mul_max_norm_le","parent":2063,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-01/index.html#cramers-rule-bound","scope":"tauceti/analysis/part-02/normed/part-01","status":"fully_proved","summary":"Cramer's rule bound for two linear forms `L = a x + b y` and `L' = a' x + b' y` in two variables: the determinant `D = a b' - b a'` satisfies `\u2016D\u2016 max(\u2016x\u2016, \u2016y\u2016) \u2264 (\u2016a\u2016 + \u2016b\u2016 + \u2016a'\u2016 + \u2016b'\u2016) (\u2016L\u2016 + \u2016L'\u2016)`.","title":"Cramer's rule bound"},{"id":"tauceti/analysis/part-02/normed/part-01/the-duhamel-identity-for-the-exponential-of-a-bo","kind":"node","lean":"ContinuousLinearMap.exp_smul_apply_sub_eq_intervalIntegral","parent":2063,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-01/index.html#the-duhamel-identity-for-the-exponential-of-a-bo","scope":"tauceti/analysis/part-02/normed/part-01","status":"fully_proved","summary":"For `B : X \u2192L[\u211d] X`, `exp (t B) x - x = \u222b\u2080\u1d57 exp (u B) (B x) du`.","title":"The Duhamel identity for the exponential of a bounded operator"},{"id":"tauceti/analysis/part-02/normed/part-01/ehrlings-inequality","kind":"node","lean":"IsCompactOperator.exists_norm_le_mul_norm_add_mul_norm","parent":2063,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-01/index.html#ehrlings-inequality","scope":"tauceti/analysis/part-02/normed/part-01","status":"fully_proved","summary":"If `i : X \u2192 Y` is a compact operator and `j : Y \u2192 Z` is an injective continuous linear map, then for every `\u03b5 > 0` there is `C \u2265 0` with `\u2016i x\u2016 \u2264 \u03b5 * \u2016x\u2016 + C * \u2016j (i x)\u2016` for all `x`.","title":"Ehrling's inequality"},{"id":"tauceti/analysis/part-02/normed/part-01/gantmachers-criterion","kind":"node","lean":"ContinuousLinearMap.isWeaklyCompact_iff_range_subset, ContinuousLinearMap.IsWeaklyCompact.of_range_precomp_precomp_subset, ContinuousLinearMap.IsWeaklyCompact.range_precomp_precomp_subset","parent":2063,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-01/index.html#gantmachers-criterion","scope":"tauceti/analysis/part-02/normed/part-01","status":"fully_proved","summary":"A bounded operator is weakly compact iff its bidual adjoint `T**` takes values in the canonical image of the codomain in its bidual. No completeness is assumed.","title":"Gantmacher's criterion"},{"id":"tauceti/analysis/part-02/normed/part-01/gantmachers-theorem","kind":"node","lean":"ContinuousLinearMap.isWeaklyCompact_precomp_iff, ContinuousLinearMap.IsWeaklyCompact.of_precomp, ContinuousLinearMap.IsWeaklyCompact.precomp","parent":2063,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-01/index.html#gantmachers-theorem","scope":"tauceti/analysis/part-02/normed/part-01","status":"fully_proved","summary":"An operator into a Banach space is weakly compact iff its adjoint is weakly compact.","title":"Gantmacher's theorem"},{"id":"tauceti/analysis/part-02/normed/part-01/gelfands-formula-in-a-nonunital-complex-banach-a","kind":"node","lean":"TauCeti.spectralRadius_eq_gelfandRadius_toLp_inr","parent":2063,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-01/index.html#gelfands-formula-in-a-nonunital-complex-banach-a","scope":"tauceti/analysis/part-02/normed/part-01","status":"fully_proved","summary":"the spectral radius of `a`, computed from its quasispectrum, is the Gelfand radius of its image in the \u2113\u00b9 unitization.","title":"Gelfand's formula in a nonunital complex Banach algebra"},{"id":"tauceti/analysis/part-02/normed/part-01/grothendiecks-lemma","kind":"node","lean":"TauCeti.isCompact_closure_toWeakSpace_of_forall_exists_isCompact","parent":2063,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-01/index.html#grothendiecks-lemma","scope":"tauceti/analysis/part-02/normed/part-01","status":"fully_proved","summary":"In a real or complex Banach space, a set has weakly compact closure if, for every `\u03b5 > 0`, each of its points lies within norm distance `\u03b5` of a fixed weakly compact set.","title":"Grothendieck's lemma"},{"id":"tauceti/analysis/part-02/normed/part-01/holders-inequality-for-weak-norms","kind":"node","lean":"TauCeti.weakLpNorm_smul_le","parent":2063,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-01/index.html#holders-inequality-for-weak-norms","scope":"tauceti/analysis/part-02/normed/part-01","status":"fully_proved","summary":"Multiplying a finite family by scalar weights `c` multiplies its weak norm by at most the `\u2113\u02e2` norm of the weights, when `1 / p = 1 / s + 1 / q` with finite `p` and `q`.","title":"H\u00f6lder's inequality for weak norms"},{"id":"tauceti/analysis/part-02/normed/part-01/johnsons-theorem","kind":"node","lean":"NonUnitalAlgHom.continuous_of_surjective_of_jacobson_eq_bot, NonUnitalAlgHom.continuous_of_surjective_of_nonUnitalJacobson_eq_bot","parent":2063,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-01/index.html#johnsons-theorem","scope":"tauceti/analysis/part-02/normed/part-01","status":"fully_proved","summary":"A surjective homomorphism `\u03b8 : A \u2192 B` of complex Banach algebras, not assumed continuous, is continuous when `B` is unital with zero Jacobson radical.","title":"Johnson's theorem"},{"id":"tauceti/analysis/part-02/normed/part-01/johnsons-uniqueness-of-norm-theorem","kind":"node","lean":"AlgebraNorm.exists_le_mul_and_le_mul_of_jacobson_eq_bot, RingNorm.exists_le_mul_and_le_mul_of_nonUnitalJacobson_eq_bot","parent":2063,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-01/index.html#johnsons-uniqueness-of-norm-theorem","scope":"tauceti/analysis/part-02/normed/part-01","status":"fully_proved","summary":"On a unital complex algebra `A` with zero Jacobson radical, any two algebra norms under which `A` is complete are equivalent.","title":"Johnson's uniqueness-of-norm theorem"},{"id":"tauceti/analysis/part-02/normed/part-01/johnson-sinclair-theorem-nonunital-form","kind":"node","lean":"LinearMap.continuous_of_derivation_of_nonUnitalJacobson_eq_bot, LinearMap.continuous_of_derivation_of_jacobson_eq_bot","parent":2063,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-01/index.html#johnson-sinclair-theorem-nonunital-form","scope":"tauceti/analysis/part-02/normed/part-01","status":"fully_proved","summary":"Let `B` be a complex Banach algebra, not necessarily unital, with zero Jacobson radical. Every linear map `D : B \u2192 B` satisfying the Leibniz rule `D (x * y) = x * D y + D x * y` is continuous.","title":"Johnson\u2013Sinclair theorem, nonunital form"},{"id":"tauceti/analysis/part-02/normed/part-01/local-neumann-formula-for-the-resolvent","kind":"node","lean":"TauCeti.LinearPMap.resolvent_eq_mul_inverse_one_sub","parent":2063,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-01/index.html#local-neumann-formula-for-the-resolvent","scope":"tauceti/analysis/part-02/normed/part-01","status":"fully_proved","summary":"Inside the ball `\u2016mu - lambda\u2016 * \u2016R(lambda)\u2016 < 1`, the resolvent at `mu` is obtained by multiplying `R(lambda)` by the ring inverse of `1 - (lambda - mu) R(lambda)`.","title":"Local Neumann formula for the resolvent"},{"id":"tauceti/analysis/part-02/normed/part-01/mazurs-lemma-in-sequential-form","kind":"node","lean":"TauCeti.exists_tendsto_sum_smul_of_tendsto_weak, TauCeti.exists_sum_smul_norm_sub_lt_of_tendsto_weak","parent":2063,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-01/index.html#mazurs-lemma-in-sequential-form","scope":"tauceti/analysis/part-02/normed/part-01","status":"fully_proved","summary":"choose a finite real convex combination from the `n`th tail for every `n`, so that the combinations converge in norm to the weak limit.","title":"Mazur's lemma, in sequential form"},{"id":"tauceti/analysis/part-02/normed/part-01/the-method-of-continuity-gilbarg-trudinger-theor","kind":"node","lean":"ContinuousLinearMap.surjective_iff_of_forall_norm_le_lineMap","parent":2063,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-01/index.html#the-method-of-continuity-gilbarg-trudinger-theor","scope":"tauceti/analysis/part-02/normed/part-01","status":"fully_proved","summary":"Let `E` be a Banach space and `L\u2080, L\u2081 : E \u2192L[\ud835\udd5c] F`. If the operators `L t = (1 - t) L\u2080 + t L\u2081` satisfy the a priori estimate `\u2016x\u2016 \u2264 C * \u2016L t x\u2016` for every `t \u2208 [0, 1]`, then `L\u2080` is surjective if and only if `L\u2081` is.","title":"The method of continuity (Gilbarg\u2013Trudinger, Theorem 5.2)"},{"id":"tauceti/analysis/part-02/normed/part-01/neumann-series-criterion-tauceti-isnormmulunit-o","kind":"node","lean":"TauCeti.IsNormMulUnit.of_norm_sub_lt","parent":2063,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-01/index.html#neumann-series-criterion-tauceti-isnormmulunit-o","scope":"tauceti/analysis/part-02/normed/part-01","status":"fully_proved","summary":"In an ultrametric norm-one ring with summable geometric series, an element whose distance from a multiplicative unit `u` is less than `\u2016u\u2016` is a multiplicative unit.","title":"Neumann series criterion (`TauCeti.IsNormMulUnit.of_norm_sub_lt`)"},{"data_url":"scopes/tauceti/analysis/part-02/normed/part-02.json","id":"tauceti/analysis/part-02/normed/part-02","kind":"scope","navigation":"transparent","parent":2062,"result":false,"route":"../roadmap/tauceti/analysis/part-02/normed/part-02/index.html","scope":"tauceti/analysis/part-02/normed/part-02","status":"fully_proved","title":"Norm double-annihilator theorem to \u0160mulian's theorem"},{"id":"tauceti/analysis/part-02/normed/part-02/norm-double-annihilator-theorem","kind":"node","lean":"Submodule.orthogonalBilin_orthogonalBilin_topDualPairing_eq_topologicalClosure","parent":2084,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-02/index.html#norm-double-annihilator-theorem","scope":"tauceti/analysis/part-02/normed/part-02","status":"fully_proved","summary":"The preannihilator of the continuous annihilator of a subspace is its norm closure.","title":"Norm double-annihilator theorem"},{"id":"tauceti/analysis/part-02/normed/part-02/open-mapping-theorem-for-normed-modules-over-a-t","kind":"node","lean":"LinearMap.isOpenMap_of_isTate","parent":2084,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-02/index.html#open-mapping-theorem-for-normed-modules-over-a-t","scope":"tauceti/analysis/part-02/normed/part-02","status":"fully_proved","summary":"A surjective continuous linear map from a complete ultrametric seminormed module to a complete normed module over a Tate ring is an open map.","title":"Open Mapping Theorem for normed modules over a Tate ring"},{"id":"tauceti/analysis/part-02/normed/part-02/the-quadratic-taylor-estimate-for-the-exponentia","kind":"node","lean":"NormedSpace.isBigO_exp_sub_quadratic, NormedSpace.isBigO_exp_mul_exp_sub_one_sub_quadratic, NormedSpace.isBigO_log_one_add_sub_quadratic","parent":2084,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-02/index.html#the-quadratic-taylor-estimate-for-the-exponentia","scope":"tauceti/analysis/part-02/normed/part-02","status":"fully_proved","summary":"In a complete normed algebra, `exp x - (1 + x + 2\u207b\u00b9 \u2022 x ^ 2)` is `O(\u2016x\u2016 ^ 3)` as `x \u2192 0`.","title":"The quadratic Taylor estimate for the exponential"},{"id":"tauceti/analysis/part-02/normed/part-02/quantitative-open-mapping-theorem","kind":"node","lean":"ContinuousLinearMap.exists_preimage_norm_le_of_isTate, LinearMap.exists_preimage_norm_le_of_surjective","parent":2084,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-02/index.html#quantitative-open-mapping-theorem","scope":"tauceti/analysis/part-02/normed/part-02","status":"fully_proved","summary":"for a surjective continuous linear map `f : M \u2192L[R] N` from a complete ultrametric seminormed module to a complete normed module over a Tate ring, there exists `C > 0` such that every `y : N` lifts to `x : M` with `\u2016x\u2016 \u2264 C * \u2016y\u2016`.","title":"Quantitative Open Mapping Theorem"},{"id":"tauceti/analysis/part-02/normed/part-02/rickarts-scalar-commutant-theorem","kind":"node","lean":"IsSimpleModule.algebraMap_end_bijective_of_completeSpace, LinearMap.existsUnique_eq_smul_of_isSimpleModule","parent":2084,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-02/index.html#rickarts-scalar-commutant-theorem","scope":"tauceti/analysis/part-02/normed/part-02","status":"fully_proved","summary":"the scalar map from the complex numbers to the algebraic endomorphism ring of a simple Banach module is bijective.","title":"Rickart's scalar-commutant theorem"},{"id":"tauceti/analysis/part-02/normed/part-02/rieszs-lemma-translated-form","kind":"node","lean":"TauCeti.riesz_lemma_translate, TauCeti.riesz_lemma_of_norm_lt_of_lt","parent":2084,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-02/index.html#rieszs-lemma-translated-form","scope":"tauceti/analysis/part-02/normed/part-02","status":"fully_proved","summary":"for a closed submodule `F`, a vector `x` and `r < 1`, some `y \u2208 F` satisfies `r * \u2016x - y\u2016 \u2264 \u2016x - y - z\u2016` for every `z \u2208 F`, so `x - y` is almost as far from `F` as its norm. For `x \u2208 F` one may take `y = x`.","title":"Riesz's lemma, translated form"},{"id":"tauceti/analysis/part-02/normed/part-02/schauders-theorem","kind":"node","lean":"ContinuousLinearMap.isCompactOperator_precomp_iff, IsCompactOperator.of_precomp, IsCompactOperator.precomp","parent":2084,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-02/index.html#schauders-theorem","scope":"tauceti/analysis/part-02/normed/part-02","status":"fully_proved","summary":"an operator into a Banach space over `\u211d` or `\u2102` is compact if and only if its adjoint is compact.","title":"Schauder's theorem"},{"id":"tauceti/analysis/part-02/normed/part-02/schurs-theorem-lp-tendsto-of-tendsto-toweakspace","kind":"node","lean":"lp.tendsto_of_tendsto_toWeakSpace_one","parent":2084,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-02/index.html#schurs-theorem-lp-tendsto-of-tendsto-toweakspace","scope":"tauceti/analysis/part-02/normed/part-02","status":"fully_proved","summary":"along a countably generated filter, in particular for sequences, weak convergence in \u2113\u00b9 implies norm convergence.","title":"Schur's theorem (`lp.tendsto_of_tendsto_toWeakSpace_one`)"},{"id":"tauceti/analysis/part-02/normed/part-02/sequential-closed-graph-theorem-for-normed-modul","kind":"node","lean":"LinearMap.continuous_of_seq_closed_graph_of_isTate","parent":2084,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-02/index.html#sequential-closed-graph-theorem-for-normed-modul","scope":"tauceti/analysis/part-02/normed/part-02","status":"fully_proved","summary":"a linear map from a complete ultrametric normed module to a complete ultrametric seminormed module whose graph is sequentially closed is continuous.","title":"Sequential Closed Graph Theorem for normed modules over a Tate ring"},{"id":"tauceti/analysis/part-02/normed/part-02/serres-potential-orthonormalisation-theorem","kind":"node","lean":"TauCeti.isPotentiallyONable_of_discrete_norm","parent":2084,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-02/index.html#serres-potential-orthonormalisation-theorem","scope":"tauceti/analysis/part-02/normed/part-02","status":"fully_proved","summary":"Every Banach module over a complete ultrametric norm-one commutative ring with discrete scalar norms and field residue ring is continuously linearly equivalent to a native model space `C\u2080(I, R)`. In particular this applies to complete discretely normed fields. The index type need not be countable.","title":"Serre's potential orthonormalisation theorem"},{"id":"tauceti/analysis/part-02/normed/part-02/the-singer-wermer-theorem","kind":"node","lean":"ContinuousLinearMap.apply_mem_nonUnitalJacobson_of_derivation, ContinuousLinearMap.apply_mem_jacobson_of_derivation","parent":2084,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-02/index.html#the-singer-wermer-theorem","scope":"tauceti/analysis/part-02/normed/part-02","status":"fully_proved","summary":"A bounded derivation `D` of a commutative complex Banach algebra, possibly without unit, takes values in the Jacobson radical.","title":"The Singer\u2013Wermer theorem"},{"id":"tauceti/analysis/part-02/normed/part-02/ultrametric-neumann-estimate","kind":"node","lean":"TauCeti.norm_inverse_one_sub, TauCeti.norm_inverse_one_sub_sub_one","parent":2084,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-02/index.html#ultrametric-neumann-estimate","scope":"tauceti/analysis/part-02/normed/part-02","status":"fully_proved","summary":"In a normed ring with `\u20161\u2016 = 1`, convergent geometric series and an ultrametric norm, the inverse of `1 - x` has norm one whenever `\u2016x\u2016 < 1`.","title":"Ultrametric Neumann estimate"},{"id":"tauceti/analysis/part-02/normed/part-02/youngs-inequality","kind":"node","lean":"DiscreteConvolution.tsum_addRingConvolution_sq_le, DiscreteConvolution.tsum_addRingConvolution_sq_le'","parent":2084,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-02/index.html#youngs-inequality","scope":"tauceti/analysis/part-02/normed/part-02","status":"fully_proved","summary":"Young's inequality `\u2016f \u22c6 g\u2016\u2082 \u2264 \u2016g\u2016\u2081 \u2016f\u2016\u2082` for families in `\u211d\u22650\u221e` on an additive group, in squared form. No summability hypotheses are needed.","title":"Young's inequality"},{"id":"tauceti/analysis/part-02/normed/part-02/silovs-theorem","kind":"node","lean":"NonUnitalAlgHom.continuous_of_nonUnitalJacobson_eq_bot, NonUnitalAlgHom.continuous_of_jacobson_eq_bot","parent":2084,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-02/index.html#silovs-theorem","scope":"tauceti/analysis/part-02/normed/part-02","status":"fully_proved","summary":"Every homomorphism from a complex Banach algebra into a commutative semisimple complex Banach algebra is continuous. Neither algebra needs a unit.","title":"\u0160ilov's theorem"},{"id":"tauceti/analysis/part-02/normed/part-02/smulians-theorem","kind":"node","lean":"TauCeti.isSeqCompact_toWeakSpace_of_isCompact, TauCeti.tendsto_subseq_toWeakSpace_of_isCompact_closure","parent":2084,"result":true,"route":"../roadmap/tauceti/analysis/part-02/normed/part-02/index.html#smulians-theorem","scope":"tauceti/analysis/part-02/normed/part-02","status":"fully_proved","summary":"in a real or complex normed space, a weakly compact set is weakly sequentially compact. Neither completeness nor separability of the space is assumed.","title":"\u0160mulian's theorem"},{"data_url":"scopes/tauceti/analysis/part-02/ode.json","id":"tauceti/analysis/part-02/ode","kind":"scope","parent":1841,"result":false,"route":"../roadmap/tauceti/analysis/part-02/ode/index.html","scope":"tauceti/analysis/part-02/ode","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Analysis.ODE` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"ODE"},{"id":"tauceti/analysis/part-02/ode/caratheodorys-existence-theorem-for-linear-equat","kind":"node","lean":"TauCeti.exists_continuousOn_eq_forall_mem_Icc_eq_picard","parent":2100,"result":true,"route":"../roadmap/tauceti/analysis/part-02/ode/index.html#caratheodorys-existence-theorem-for-linear-equat","scope":"tauceti/analysis/part-02/ode","status":"fully_proved","summary":"Let `M : \u211d \u2192 E \u2192L[\u211d] E` be almost everywhere strongly measurable and essentially bounded on `(a, b)`, and let `F : \u211d \u2192 E` be integrable on `(a, b)`. Then the linear equation `x' = M(t) x + F(t)` with `x(a) = x\u2080` has a solution in Carath\u00e9odory's sense: a curve `x` continuous on `[a, b]` with `x a = x\u2080` and `x t = x\u2080 + \u222b s in a..t, (M s (x s) + F s)` for every `t \u2208 [a, b]`.","title":"Carath\u00e9odory's existence theorem for linear equations"},{"id":"tauceti/analysis/part-02/ode/the-frobenius-theorem-for-total-differential-equ","kind":"node","lean":"TauCeti.eventually_exists_hasFDerivAt_iff_eventually_isFrobeniusIntegrableAt","parent":2100,"result":true,"route":"../roadmap/tauceti/analysis/part-02/ode/index.html#the-frobenius-theorem-for-total-differential-equ","scope":"tauceti/analysis/part-02/ode","status":"fully_proved","summary":"Let `E` and `F` be finite-dimensional real normed spaces and let `f : E \u00d7 F \u2192 (E \u2192L[\u211d] F)` be `C^(n+1)` near `p\u2080`, for instance `C\u00b9`. The total differential equation `D u x = f (x, u x)` has a local solution through every point near `p\u2080` if and only if `f` satisfies the Frobenius integrability condition near `p\u2080`.","title":"The Frobenius theorem for total differential equations"},{"id":"tauceti/analysis/part-02/ode/gronwalls-inequality-integral-form","kind":"node","lean":"TauCeti.le_mul_exp_of_le_add_mul_integral, TauCeti.mul_exp_integral_le_of_mul_le_deriv","parent":2100,"result":true,"route":"../roadmap/tauceti/analysis/part-02/ode/index.html#gronwalls-inequality-integral-form","scope":"tauceti/analysis/part-02/ode","status":"fully_proved","summary":"If `\u03c6` is continuous on `[a, b]`, `0 \u2264 K` and `\u03c6 t \u2264 c + K \u222b\u2090\u1d57 \u03c6` for every `t \u2208 [a, b]`, then `\u03c6 t \u2264 c exp (K (t - a))` on `[a, b]`.","title":"Gr\u00f6nwall's inequality, integral form"},{"id":"tauceti/analysis/part-02/ode/the-local-frobenius-theorem-tauceti-exists-event","kind":"node","lean":"TauCeti.exists_eventually_hasFDerivAt_of_isFrobeniusIntegrableAt","parent":2100,"result":true,"route":"../roadmap/tauceti/analysis/part-02/ode/index.html#the-local-frobenius-theorem-tauceti-exists-event","scope":"tauceti/analysis/part-02/ode","status":"fully_proved","summary":"Let `E` and `F` be finite-dimensional real normed spaces and let `f : E \u00d7 F \u2192 (E \u2192L[\u211d] F)` be `C^(n+1)` near `(x\u2080, y\u2080)`, for instance `C\u00b9`. If `f` satisfies the Frobenius integrability condition near `(x\u2080, y\u2080)`, then the total differential equation `D u x = f (x, u x)` has a local solution `u` with `u x\u2080 = y\u2080`, which is `C^(n+1)` at `x\u2080`.","title":"The local Frobenius theorem (`TauCeti.exists_eventually_hasFDerivAt_of_isFrobeniusIntegrableAt`)"},{"id":"tauceti/analysis/part-02/ode/the-local-stable-manifold-theorem-lipschitz-form","kind":"node","lean":"ContinuousLinearMap.exists_setOf_exists_isIntegralCurveOn_mapsTo_closedBall_eq_image","parent":2100,"result":true,"route":"../roadmap/tauceti/analysis/part-02/ode/index.html#the-local-stable-manifold-theorem-lipschitz-form","scope":"tauceti/analysis/part-02/ode","status":"fully_proved","summary":"Near a hyperbolic equilibrium the initial values of the forward solutions that stay in a fixed small ball, truncated by the condition `\u2016P x\u2016 \u2264 \u03c1`, form the graph of a Lipschitz map over a ball in the stable subspace `range P`.","title":"The local stable-manifold theorem, Lipschitz form"},{"id":"tauceti/analysis/part-02/ode/the-local-unstable-manifold-theorem-lipschitz-fo","kind":"node","lean":"ContinuousLinearMap.exists_setOf_exists_isIntegralCurveOn_Iic_mapsTo_closedBall_eq_image","parent":2100,"result":true,"route":"../roadmap/tauceti/analysis/part-02/ode/index.html#the-local-unstable-manifold-theorem-lipschitz-fo","scope":"tauceti/analysis/part-02/ode","status":"fully_proved","summary":"Near a hyperbolic equilibrium the initial values of backward solutions confined to a fixed small ball, truncated by the norm of their `1 - P` component, form a graph over a ball in `range (1 - P)`.","title":"The local unstable-manifold theorem, Lipschitz form"},{"id":"tauceti/analysis/part-02/ode/the-straightening-theorem-flow-box-theorem","kind":"node","lean":"TauCeti.exists_openPartialHomeomorph_fderiv_eq_const","parent":2100,"result":true,"route":"../roadmap/tauceti/analysis/part-02/ode/index.html#the-straightening-theorem-flow-box-theorem","scope":"tauceti/analysis/part-02/ode","status":"fully_proved","summary":"A `C^n` vector field `w`, `1 \u2264 n`, defined near a point `a` of a finite-dimensional real normed space, with `w a \u2260 0`, is constant in suitable `C^n` coordinates: there is an open partial homeomorphism `\u03c6`, `C^n` on its source with `C^n` inverse on its target, such that `a \u2208 \u03c6.source` and `d\u03c6_y (w y) = w a` for every `y \u2208 \u03c6.source`.","title":"The straightening theorem (flow-box theorem)"},{"id":"tauceti/analysis/part-02/ode/uniqueness-in-the-local-frobenius-theorem","kind":"node","lean":"TauCeti.eventuallyEq_of_eventually_hasFDerivAt","parent":2100,"result":true,"route":"../roadmap/tauceti/analysis/part-02/ode/index.html#uniqueness-in-the-local-frobenius-theorem","scope":"tauceti/analysis/part-02/ode","status":"fully_proved","summary":"Two solutions of the total differential equation `D u x = f (x, u x)` near `x\u2080` taking the same value at `x\u2080` agree near `x\u2080`, as soon as `f` is Lipschitz near `(x\u2080, u\u2081 x\u2080)`. Neither integrability nor finite dimensionality is needed: along each ray from `x\u2080` both solutions solve the same ordinary differential equation.","title":"Uniqueness in the local Frobenius theorem"},{"data_url":"scopes/tauceti/analysis/part-02/operator-ideal.json","id":"tauceti/analysis/part-02/operator-ideal","kind":"scope","parent":1841,"result":false,"route":"../roadmap/tauceti/analysis/part-02/operator-ideal/index.html","scope":"tauceti/analysis/part-02/operator-ideal","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Analysis.OperatorIdeal` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Operator Ideal"},{"id":"tauceti/analysis/part-02/operator-ideal/eckart-young-for-ky-fan-gauges","kind":"node","lean":"ContinuousLinearMap.kyFanGauge_eq_kyFanSum, ContinuousLinearMap.singularValue_eq_linearMap_singularValues","parent":2109,"result":true,"route":"../roadmap/tauceti/analysis/part-02/operator-ideal/index.html#eckart-young-for-ky-fan-gauges","scope":"tauceti/analysis/part-02/operator-ideal","status":"fully_proved","summary":"Between finite-dimensional inner product spaces, the Ky Fan gauge `K\u2096(T)` of a bounded linear map is the Ky Fan `k`-sum `\u03c3\u2080(T) + \u22ef + \u03c3\u2096\u208b\u2081(T)` of its singular values.","title":"Eckart\u2013Young for Ky Fan gauges"},{"id":"tauceti/analysis/part-02/operator-ideal/fatou-inequality-for-the-trace-class-gauge","kind":"node","lean":"ContinuousLinearMap.nuclearENorm_le_tsum_of_hasSum","parent":2109,"result":true,"route":"../roadmap/tauceti/analysis/part-02/operator-ideal/index.html#fatou-inequality-for-the-trace-class-gauge","scope":"tauceti/analysis/part-02/operator-ideal","status":"fully_proved","summary":"For a series converging in operator norm, the gauge of its sum is at most the sum of the gauges, even when that sum is infinite.","title":"Fatou inequality for the trace-class gauge"},{"id":"tauceti/analysis/part-02/operator-ideal/ky-fan-triangle-inequality","kind":"node","lean":"ContinuousLinearMap.kyFanGauge_add_le, TauCeti.HasMinMaxLowerBound.kyFanGauge_add_le","parent":2109,"result":true,"route":"../roadmap/tauceti/analysis/part-02/operator-ideal/index.html#ky-fan-triangle-inequality","scope":"tauceti/analysis/part-02/operator-ideal","status":"fully_proved","summary":"For bounded linear maps from a complete inner product space to an inner product space, the Ky Fan gauges are subadditive: `K\u2096(S + T) \u2264 K\u2096(S) + K\u2096(T)`.","title":"Ky Fan triangle inequality"},{"id":"tauceti/analysis/part-02/operator-ideal/ky-fans-maximum-principle","kind":"node","lean":"ContinuousLinearMap.norm_sum_inner_le_kyFanGauge, ContinuousLinearMap.IsPositive.exists_orthonormal_kyFanGauge_le_sum_re_inner_add, ContinuousLinearMap.IsPositive.kyFanGauge_le_iff","parent":2109,"result":true,"route":"../roadmap/tauceti/analysis/part-02/operator-ideal/index.html#ky-fans-maximum-principle","scope":"tauceti/analysis/part-02/operator-ideal","status":"fully_proved","summary":"For a bounded linear map `T` between inner product spaces and orthonormal families `(u\u1d62)` in `F` and `(v\u1d62)` in `E` with `k` members, `\u2016\u2211\u1d62 \u27eau\u1d62, T v\u1d62\u27eb\u2016 \u2264 a\u2080(T) + \u22ef + a\u2096\u208b\u2081(T)`.","title":"Ky Fan's maximum principle"},{"id":"tauceti/analysis/part-02/operator-ideal/weyls-perturbation-bound-for-singular-values","kind":"node","lean":"LinearMap.abs_singularValues_sub_singularValues_le","parent":2109,"result":true,"route":"../roadmap/tauceti/analysis/part-02/operator-ideal/index.html#weyls-perturbation-bound-for-singular-values","scope":"tauceti/analysis/part-02/operator-ideal","status":"fully_proved","summary":"Each singular value of a linear map between finite-dimensional inner product spaces is `1`-Lipschitz in the operator norm: `|\u03c3\u2099(A) - \u03c3\u2099(B)| \u2264 \u2016A - B\u2016`.","title":"Weyl's perturbation bound for singular values"},{"data_url":"scopes/tauceti/analysis/part-02/pde.json","id":"tauceti/analysis/part-02/pde","kind":"scope","parent":1841,"result":false,"route":"../roadmap/tauceti/analysis/part-02/pde/index.html","scope":"tauceti/analysis/part-02/pde","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Analysis.PDE` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"PDE"},{"data_url":"scopes/tauceti/analysis/part-02/pde/part-01.json","id":"tauceti/analysis/part-02/pde/part-01","kind":"scope","navigation":"transparent","parent":2115,"result":false,"route":"../roadmap/tauceti/analysis/part-02/pde/part-01/index.html","scope":"tauceti/analysis/part-02/pde/part-01","status":"fully_proved","title":"The Caccioppoli inequality to The local Harnack inequality"},{"id":"tauceti/analysis/part-02/pde/part-01/the-caccioppoli-inequality","kind":"node","lean":"TauCeti.PDE.UniformlyEllipticOn.setIntegral_sq_mul_norm_gradient_sq_le, TauCeti.PDE.UniformlyEllipticOn.setIntegral_sq_mul_norm_gradient_posPartAbove_sq_le, TauCeti.PDE.UniformlyEllipticOn.setIntegral_sq_mul_norm_gradient_posPartAbove_sq_le_lowerOrder, TauCeti.PDE.UniformlyEllipticOn.setIntegral_sq_mul_norm_gradient_posPartAbove_sq_le_of_memLp, TauCeti.PDE.UniformlyEllipticOn.setIntegral_sq_mul_norm_gradient_posPartAbove_sq_le_lowerOrder_of_memLp, TauCeti.PDE.UniformlyEllipticOn.setIntegral_sq_mul_rpow_mul_norm_gradient_sq_le, TauCeti.PDE.UniformlyEllipticOn.setIntegral_sq_mul_norm_gradient_sq_le_of_ae_eq_rpow, TauCeti.PDE.exists_setIntegral_ball_norm_gradient_posPartAbove_sq_le, TauCeti.PDE.exists_setIntegral_ball_norm_gradient_posPartAbove_sq_le_lowerOrder","parent":2116,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-01/index.html#the-caccioppoli-inequality","scope":"tauceti/analysis/part-02/pde/part-01","status":"fully_proved","summary":"Let `a` be measurable and uniformly elliptic on `\u03a9` with constants `0 < \u03bb \u2264 \u039b`, and let `u \u2208 H\u00b9(\u03a9)` be a weak solution of `-\u2202\u1d62(a\u2071\u02b2 \u2202\u2c7cu) = f` in `\u03a9`, in the sense that `a(u, v) = \u222b_\u03a9 f v` for every `v \u2208 H\u00b9\u2080(\u03a9)`, with no boundary condition on `u`. Then for every smooth `\u03c8` compactly supported in `\u03a9`,","title":"The Caccioppoli inequality"},{"id":"tauceti/analysis/part-02/pde/part-01/cauchy-schwarz-for-jet-fields","kind":"node","lean":"TauCeti.PDE.integral_norm_jetField_mul_le","parent":2116,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-01/index.html#cauchy-schwarz-for-jet-fields","scope":"tauceti/analysis/part-02/pde/part-01","status":"fully_proved","summary":"The integral of the product of the jet norms of two Sobolev functions is at most the product of their `W^{1,2}` norms. This is the estimate that turns the pointwise operator-norm bound on the energy integrand into boundedness of the energy form.","title":"Cauchy--Schwarz for jet fields"},{"id":"tauceti/analysis/part-02/pde/part-01/the-centered-harnack-inequality-on-a-planar-disk","kind":"node","lean":"InnerProductSpace.HarmonicContOnCl.harnack_inequality_center_of_nonneg_on_sphere, InnerProductSpace.HarmonicOnNhd.harnack_inequality_center","parent":2116,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-01/index.html#the-centered-harnack-inequality-on-a-planar-disk","scope":"tauceti/analysis/part-02/pde/part-01","status":"fully_proved","summary":"If `f` is harmonic on the open disk of radius `R` about `c`, continuous on its closure and nonnegative on its boundary circle, then every `w` in the open disk satisfies","title":"The centered Harnack inequality on a planar disk"},{"id":"tauceti/analysis/part-02/pde/part-01/duhamels-formula-for-the-heat-equation","kind":"node","lean":"TauCeti.heatKernel_convolution_sub_eq_integral, TauCeti.heatSemigroup_sub_eq_integral","parent":2116,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-01/index.html#duhamels-formula-for-the-heat-equation","scope":"tauceti/analysis/part-02/pde/part-01","status":"fully_proved","summary":"For `\u03c6` of class `C\u00b2` with compact support and `t > 0`, `(K_t \u22c6 \u03c6)(x) - \u03c6(x) = \u222b\u2080\u1d57 (K_s \u22c6 \u0394 \u03c6)(x) ds` at every point `x`.","title":"Duhamel's formula for the heat equation"},{"id":"tauceti/analysis/part-02/pde/part-01/duhamels-identity-in-supremum-norm","kind":"node","lean":"TauCeti.heatSemigroupBUC_sub_eq_integral","parent":2116,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-01/index.html#duhamels-identity-in-supremum-norm","scope":"tauceti/analysis/part-02/pde/part-01","status":"fully_proved","summary":"For bounded uniformly continuous `C\u00b2` data `f` with bounded first and second derivatives and bounded uniformly continuous Laplacian `g`, `e^{t\u0394} f - f = \u222b\u2080\u1d57 e^{s\u0394} g ds` as a Banach-valued integral.","title":"Duhamel's identity in supremum norm"},{"id":"tauceti/analysis/part-02/pde/part-01/duhamels-principle-in-integrated-form","kind":"node","lean":"TauCeti.heatDuhamelPotential_eq_integral","parent":2116,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-01/index.html#duhamels-principle-in-integrated-form","scope":"tauceti/analysis/part-02/pde/part-01","status":"fully_proved","summary":"Let `f` be jointly measurable and essentially bounded on `(0, t) \u00d7 E`, with `f(s, \u00b7)` `\u03b1`-H\u00f6lder for almost every `0 < s < t`, where `\u03b1 > 0`. Then the Duhamel potential `w` satisfies `w(t, x) = \u222b\u2080\u1d57 (\u0394w(\u03c3, x) + f(\u03c3, x)) d\u03c3`.","title":"Duhamel's principle in integrated form"},{"id":"tauceti/analysis/part-02/pde/part-01/the-fredholm-alternative-in-eigenvalue-language","kind":"node","lean":"TauCeti.PDE.existsUnique_isWeakSolutionDirichlet_sub_const_of_not_isDirichletEigenvalue, TauCeti.PDE.fredholmAlternative_isWeakSolutionDirichlet_sub_const","parent":2116,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-01/index.html#the-fredholm-alternative-in-eigenvalue-language","scope":"tauceti/analysis/part-02/pde/part-01","status":"fully_proved","summary":"On a bounded domain, if `\u03ba` is not a Dirichlet eigenvalue then `L u - \u03ba u = f` in `\u03a9`, `u = 0` on `\u2202\u03a9`, has exactly one weak solution for every `f \u2208 L\u00b2(\u03a9)`.","title":"The Fredholm alternative in eigenvalue language"},{"id":"tauceti/analysis/part-02/pde/part-01/the-global-schauder-estimate-on-the-whole-space","kind":"node","lean":"TauCeti.PDE.exists_holderWith_iteratedFDeriv_two_and_norm_le_of_sum_smul_fderiv_fderiv, TauCeti.PDE.exists_norm_le_of_sum_smul_fderiv_fderiv, TauCeti.PDE.exists_holderWith_iteratedFDeriv_two_of_sum_smul_fderiv_fderiv_of_holderWith, TauCeti.PDE.exists_holderWith_iteratedFDeriv_two_and_norm_le_of_sum_smul_fderiv_fderiv_add_fderiv_add_mul","parent":2116,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-01/index.html#the-global-schauder-estimate-on-the-whole-space","scope":"tauceti/analysis/part-02/pde/part-01","status":"fully_proved","summary":"Let `0 < \u03b1 < 1`, and let `\u03bb`, `\u039b` and `L` be given. Then there is a constant `C`, depending only on `n`, `\u03b1`, `\u03bb`, `\u039b` and `L`, with the following property. Let `a` be a matrix-valued coefficient on `\u211d^{n+1}` which is uniformly elliptic and bounded with constants `0 < \u03bb \u2264 \u039b` (`UniformlyEllipticOn univ a \u03bb \u039b`) and globally `\u03b1`-H\u00f6lder entrywise with constant `L`, and put `L u = \u2211\u1d62\u2c7c a\u1d62\u2c7c \u2202\u1d62\u2202\u2c7cu`. Let `u` be `C\u00b2` with\u2026","title":"The global Schauder estimate on the whole space"},{"id":"tauceti/analysis/part-02/pde/part-01/greens-first-identity-against-h10","kind":"node","lean":"TauCeti.PDE.energyFormH1_one_zero_zero_eq_setIntegral_neg_laplacian_mul","parent":2116,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-01/index.html#greens-first-identity-against-h10","scope":"tauceti/analysis/part-02/pde/part-01","status":"fully_proved","summary":"Let `u \u2208 H\u00b9(\u03a9)` agree almost everywhere on `\u03a9` with a function `w` of class `C\u00b2` on `\u03a9` whose Laplacian lies in `L\u00b2(\u03a9)`. Then `\u222b_\u03a9 \u2207v \u00b7 \u2207u = -\u222b_\u03a9 \u0394w v` for every `v \u2208 H\u00b9\u2080(\u03a9)`: `u` solves `-\u0394u = -\u0394w` weakly. No boundary condition is imposed on `u`.","title":"Green's first identity against `H\u00b9\u2080(\u03a9)`"},{"id":"tauceti/analysis/part-02/pde/part-01/gardings-inequality-on-h1","kind":"node","lean":"TauCeti.PDE.UniformlyEllipticOn.garding_energyFormH1_self, TauCeti.PDE.UniformlyEllipticOn.garding_energyFormH1_self_norm_of_norm_le, TauCeti.PDE.UniformlyEllipticOn.garding_energyFormH1_self_norm","parent":2116,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-01/index.html#gardings-inequality-on-h1","scope":"tauceti/analysis/part-02/pde/part-01","status":"fully_proved","summary":"For a uniformly elliptic principal coefficient with lower constant `\u03bb`, a drift bounded by `\u03b2` and a nonnegative mass coefficient,","title":"G\u00e5rding's inequality on `H\u00b9(\u03a9)`"},{"id":"tauceti/analysis/part-02/pde/part-01/harnacks-convergence-theorem","kind":"node","lean":"TauCeti.tendstoLocallyUniformlyOn_iSup_of_monotone","parent":2116,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-01/index.html#harnacks-convergence-theorem","scope":"tauceti/analysis/part-02/pde/part-01","status":"fully_proved","summary":"Let `F` be a family of functions harmonic on a preconnected open set `U`, monotone at each point of `U` and bounded above at some point `x\u2080 \u2208 U`. Then `F` converges locally uniformly on `U` to its pointwise supremum.","title":"Harnack's convergence theorem"},{"id":"tauceti/analysis/part-02/pde/part-01/harnacks-inequality","kind":"node","lean":"IsCompact.harnack_inequality, InnerProductSpace.HarmonicOnNhd.harnack_inequality, InnerProductSpace.HarmonicContOnCl.harnack_inequality_of_nonneg_on_sphere","parent":2116,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-01/index.html#harnacks-inequality","scope":"tauceti/analysis/part-02/pde/part-01","status":"fully_proved","summary":"Let `K` be a compact subset of a preconnected open set `U`. There is a constant `C`, depending only on `K` and `U`, such that every function `u` harmonic and nonnegative on `U` satisfies `u x \u2264 C * u y` for all `x, y \u2208 K`; that is, `sup_K u \u2264 C * inf_K u`.","title":"Harnack's inequality"},{"id":"tauceti/analysis/part-02/pde/part-01/the-hopf-lax-subsolution-inequality","kind":"node","lean":"TauCeti.hopfLax_add_add_smul_le, TauCeti.hopfLax_norm_rpow_add_add_smul_le","parent":2116,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-01/index.html#the-hopf-lax-subsolution-inequality","scope":"tauceti/analysis/part-02/pde/part-01","status":"fully_proved","summary":"For a convex Lagrangian `L`, moving from `(t, x)` at velocity `w` for time `h \u2265 0` raises the Hopf\u2013Lax value by at most `h L w`: `u (t + h, x + h w) \u2264 u (t, x) + h L w` for `t > 0`.","title":"The Hopf\u2013Lax subsolution inequality"},{"id":"tauceti/analysis/part-02/pde/part-01/holder-bound-on-the-laplacian-of-a-heat-convolut","kind":"node","lean":"TauCeti.exists_norm_laplacian_heatKernel_convolution_le","parent":2116,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-01/index.html#holder-bound-on-the-laplacian-of-a-heat-convolut","scope":"tauceti/analysis/part-02/pde/part-01","status":"fully_proved","summary":"The Laplacian of the heat convolution of a bounded `\u03b1`-H\u00f6lder function with constant `H` is `O(H \u03c4^((\u03b1 - 2)/2))`, uniformly in the function.","title":"H\u00f6lder bound on the Laplacian of a heat convolution"},{"id":"tauceti/analysis/part-02/pde/part-01/holder-bounds-on-the-derivatives-of-a-heat-convo","kind":"node","lean":"TauCeti.norm_iteratedFDeriv_heatKernel_convolution_le_of_holderWith","parent":2116,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-01/index.html#holder-bounds-on-the-derivatives-of-a-heat-convo","scope":"tauceti/analysis/part-02/pde/part-01","status":"fully_proved","summary":"If `g` is bounded and `\u03b1`-H\u00f6lder with constant `H`, then for `m \u2265 1` and `t > 0`, `\u2016D^m (K_t \u22c6 g)(x)\u2016 \u2264 H \u222b \u2016D^m K_t(z)\u2016 \u2016z\u2016^\u03b1 dz`. By `TauCeti.exists_integral_norm_iteratedFDeriv_heatKernel_mul_rpow_le` the right side is `O(H t^((\u03b1 - m) / 2))`, which for `m = 2` is integrable in time near `t = 0` when `\u03b1 > 0`.","title":"H\u00f6lder bounds on the derivatives of a heat convolution"},{"id":"tauceti/analysis/part-02/pde/part-01/the-holder-estimate-for-the-hessian-of-the-duham","kind":"node","lean":"TauCeti.exists_holderWith_iteratedFDeriv_two_heatDuhamelPotential","parent":2116,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-01/index.html#the-holder-estimate-for-the-hessian-of-the-duham","scope":"tauceti/analysis/part-02/pde/part-01","status":"fully_proved","summary":"For `0 < \u03b1 < 1` there is a constant `C`, depending only on `\u03b1` and the dimension, such that the Hessian `D\u00b2 w(t, \u00b7)` is `\u03b1`-H\u00f6lder with constant `C H` whenever `f` is jointly measurable and essentially bounded on `(0, t) \u00d7 E` and `f(s, \u00b7)` is `\u03b1`-H\u00f6lder with constant `H` for almost every `0 < s < t`. Neither the bound on `f` nor the time `t` enters the estimate.","title":"The H\u00f6lder estimate for the Hessian of the Duhamel potential"},{"id":"tauceti/analysis/part-02/pde/part-01/the-interior-calderon-zygmund-estimate-for-conti","kind":"node","lean":"TauCeti.PDE.exists_lpNorm_iteratedGradient_le_of_sum_mul_inner_add_inner_add_mul_eq_of_continuousAt, TauCeti.PDE.exists_mul_lpNorm_iteratedGradient_ball_le_of_sum_mul_inner_add_inner_add_mul_eq","parent":2116,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-01/index.html#the-interior-calderon-zygmund-estimate-for-conti","scope":"tauceti/analysis/part-02/pde/part-01","status":"fully_proved","summary":"Let `0 < n` and `1 < p < \u221e`, let `\u03a9 \u2286 \u211d\u207f` be open and `K \u2286 \u03a9` compact, and let `a` be a matrix-valued coefficient, measurable on `\u03a9` and continuous at each `x \u2208 K`, such that `a x` is elliptic at each `x \u2208 K`, that is, `\u03bb\u2093 \u2016\u03be\u2016\u00b2 \u2264 \u03be\u1d40 a(x) \u03be` for some `\u03bb\u2093 > 0`. Let `b` and `c` be measurable lower-order coefficients with `\u2016b(x)\u2016 \u2264 \u03b2` and `|c(x)| \u2264 \u03b3` for almost every `x \u2208 \u03a9`. Then there is a constant `C`, depending on\u2026","title":"The interior Calder\u00f3n\u2013Zygmund estimate for continuous coefficients"},{"id":"tauceti/analysis/part-02/pde/part-01/the-interior-schauder-estimate-on-small-balls","kind":"node","lean":"TauCeti.PDE.exists_holderOnWith_iteratedFDeriv_two_closedBall_of_sum_smul_fderiv_fderiv, TauCeti.PDE.exists_holderOnWith_iteratedFDeriv_two_of_sum_smul_fderiv_fderiv, TauCeti.PDE.exists_holderOnWith_iteratedFDeriv_two_of_sum_smul_fderiv_fderiv_of_holderOnWith, TauCeti.PDE.exists_holderOnWith_iteratedFDeriv_two_of_laplacian, TauCeti.PDE.exists_holderOnWith_iteratedFDeriv_two_of_sum_smul_fderiv_fderiv_add_fderiv_add_mul","parent":2116,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-01/index.html#the-interior-schauder-estimate-on-small-balls","scope":"tauceti/analysis/part-02/pde/part-01","status":"fully_proved","summary":"Let `0 < \u03b1 < 1`. There is `C\u2080 > 0`, depending only on dimension and `\u03b1`, with the following property. Let `0 < \u03bb`, `\u039b`, `L`, `M` and `0 < R` be given, put `\u03ba = C\u2080 (\u039b / \u03bb)^(1 + (n+1)/4 + \u03b1/2)`, and assume the scale condition","title":"The interior Schauder estimate on small balls"},{"id":"tauceti/analysis/part-02/pde/part-01/the-limit-in-harnacks-convergence-theorem-is-har","kind":"node","lean":"TauCeti.harmonicOnNhd_iSup_of_monotone","parent":2116,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-01/index.html#the-limit-in-harnacks-convergence-theorem-is-har","scope":"tauceti/analysis/part-02/pde/part-01","status":"fully_proved","summary":"Let `F` be a family of functions harmonic on a preconnected open set `U`, monotone at each point of `U` and bounded above at some point `x\u2080 \u2208 U`. Then its pointwise supremum is harmonic on `U`.","title":"The limit in Harnack's convergence theorem is harmonic"},{"id":"tauceti/analysis/part-02/pde/part-01/the-local-harnack-inequality","kind":"node","lean":"InnerProductSpace.HarmonicOnNhd.le_three_pow_mul_of_nonneg","parent":2116,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-01/index.html#the-local-harnack-inequality","scope":"tauceti/analysis/part-02/pde/part-01","status":"fully_proved","summary":"If `u` is harmonic and nonnegative on `ball c (4 * r)`, then any two of its values on `ball c r` are within the factor `3 ^ n` of each other, where `n` is the dimension of `E`.","title":"The local Harnack inequality"},{"data_url":"scopes/tauceti/analysis/part-02/pde/part-02.json","id":"tauceti/analysis/part-02/pde/part-02","kind":"scope","navigation":"transparent","parent":2115,"result":false,"route":"../roadmap/tauceti/analysis/part-02/pde/part-02/index.html","scope":"tauceti/analysis/part-02/pde/part-02","status":"fully_proved","title":"The localized Calder\u00f3n\u2013Zygmund\u2026 to The zero-forcing Caccioppoli\u2026"},{"id":"tauceti/analysis/part-02/pde/part-02/the-localized-calderon-zygmund-estimate-for-coef","kind":"node","lean":"TauCeti.PDE.exists_lpNorm_smul_iteratedGradient_le_of_sum_mul_inner_eq_of_abs_sub_le","parent":2137,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-02/index.html#the-localized-calderon-zygmund-estimate-for-coef","scope":"tauceti/analysis/part-02/pde/part-02","status":"fully_proved","summary":"Let `0 < n` and `1 < p < \u221e`. There are constants `C` and `\u03b4 > 0`, depending only on `n` and `p`, with the following property. Let `A` be a matrix with `\u03bb \u2016\u03be\u2016\u00b2 \u2264 \u03be\u1d40 A \u03be` and `0 < \u03bb`, let `\u03a9` be open, and let `\u03b6` be smooth with compact support in `\u03a9`, `\u2016\u2207\u03b6\u2016 \u2264 M\u2081` and `\u2016D\u00b2\u03b6\u2016 \u2264 M\u2082`. Let `a` be a measurable matrix-valued coefficient on `\u03a9` with `|a\u1d62\u2c7c(x) - A\u1d62\u2c7c| \u2264 \u03b5` and `|a\u1d62\u2c7c(x)| \u2264 \u039b` for almost every `x` in the support\u2026","title":"The localized Calder\u00f3n\u2013Zygmund estimate for coefficients of small oscillation"},{"id":"tauceti/analysis/part-02/pde/part-02/mosers-lower-bound-from-small-negative-powers","kind":"node","lean":"TauCeti.PDE.exists_ae_inv_value_le_mul_rpow_mul_setIntegral_rpow","parent":2137,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-02/index.html#mosers-lower-bound-from-small-negative-powers","scope":"tauceti/analysis/part-02/pde/part-02","status":"fully_proved","summary":"For every `0 < p \u2264 2` and every positive weak supersolution in any dimension `n \u2265 1`, `u\u207b\u00b9 \u2264 D R^{-n/p} (\u222b_{B(x\u2080, R)} u\u207b\u1d56)^{1/p}` almost everywhere on `B(x\u2080, R/2)`. The constant depends only on `\u03bb`, `\u039b`, `p`, the dimension and Haar normalization, not on the positive lower bound for `u`. No boundary regularity is required.","title":"Moser's lower bound from small negative powers"},{"id":"tauceti/analysis/part-02/pde/part-02/the-parabolic-holder-estimate-of-the-cauchy-heat","kind":"node","lean":"TauCeti.holderWith_heatCauchyPotential","parent":2137,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-02/index.html#the-parabolic-holder-estimate-of-the-cauchy-heat","scope":"tauceti/analysis/part-02/pde/part-02","status":"fully_proved","summary":"If `g` is continuous and `\u03b1`-H\u00f6lder with constant `H`, its Cauchy potential is `\u03b1`-H\u00f6lder on parabolic space-time, whose distance is `\u2016x - x'\u2016 + \u221a|t - s|`, with constant `H (1 + M_\u03b1)`. Here `M_\u03b1 = \u222b \u2016y\u2016^\u03b1 K_1(y) dy` is `TauCeti.heatKernelMoment`, which depends only on `\u03b1` and the dimension. Continuity is automatic when `\u03b1 > 0` (`HolderWith.continuous`).","title":"The parabolic H\u00f6lder estimate of the Cauchy heat potential"},{"id":"tauceti/analysis/part-02/pde/part-02/perrons-theorem","kind":"node","lean":"TauCeti.harmonicOnNhd_perronSolution","parent":2137,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-02/index.html#perrons-theorem","scope":"tauceti/analysis/part-02/pde/part-02","status":"fully_proved","summary":"Let `\u03a9` be a bounded open set, let `g` be bounded above on `frontier \u03a9`, and suppose the Perron family of `g` is nonempty. Then the Perron solution `TauCeti.perronSolution \u03a9 g` is harmonic in `\u03a9`.","title":"Perron's theorem"},{"id":"tauceti/analysis/part-02/pde/part-02/pointwise-garding-inequality","kind":"node","lean":"TauCeti.PDE.garding_energyIntegrand_self_of_bounds","parent":2137,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-02/index.html#pointwise-garding-inequality","scope":"tauceti/analysis/part-02/pde/part-02","status":"fully_proved","summary":"With a nonnegative mass coefficient (`c \u2265 0`), the diagonal of the jet form is bounded below by `(\u03bb/2)\u2016\u2207u\u2016\u00b2 \u2212 (\u03b2\u00b2/2\u03bb)|u|\u00b2`. The ellipticity floor `\u03bb\u2016\u2207u\u2016\u00b2` pays for the drift term via Young's inequality, leaving half the floor and a mass defect proportional to `\u03b2\u00b2/\u03bb`. Integrating over `\u03a9` this is G\u00e5rding's inequality `a(u, u) \u2265 (\u03bb/2)\u2016\u2207u\u2016\u00b2_{L\u00b2} \u2212 (\u03b2\u00b2/2\u03bb)\u2016u\u2016\u00b2_{L\u00b2}`.","title":"Pointwise G\u00e5rding inequality"},{"id":"tauceti/analysis/part-02/pde/part-02/the-rayleigh-principle-for-the-dirichlet-problem","kind":"node","lean":"TauCeti.PDE.isLeast_rayleighQuotient_firstDirichletEigenvalue","parent":2137,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-02/index.html#the-rayleigh-principle-for-the-dirichlet-problem","scope":"tauceti/analysis/part-02/pde/part-02","status":"fully_proved","summary":"On a nonempty bounded domain with a symmetric energy form, the first Dirichlet eigenvalue is the *minimum* of the Rayleigh quotient","title":"The Rayleigh principle for the Dirichlet problem"},{"id":"tauceti/analysis/part-02/pde/part-02/schauder-estimate-for-a-hessian-supported-in-a-b","kind":"node","lean":"TauCeti.PDE.exists_holderWith_iteratedFDeriv_two_of_support_subset_closedBall","parent":2137,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-02/index.html#schauder-estimate-for-a-hessian-supported-in-a-b","scope":"tauceti/analysis/part-02/pde/part-02","status":"fully_proved","summary":"Let `0 < \u03b1 < 1`. There is `C > 0`, depending only on dimension and `\u03b1`, with the following property. Let `a` be `\u03b1`-H\u00f6lder entrywise with constant `L` on the support ball, and suppose `a x\u2080` has ellipticity bounds `\u03bb, \u039b`, with `\u03bb > 0`. Put","title":"Schauder estimate for a Hessian supported in a ball"},{"id":"tauceti/analysis/part-02/pde/part-02/the-sharp-poincare-inequality-on-an-interval","kind":"node","lean":"TauCeti.PDE.isGreatest_mul_norm_value_sq_le_energyFormH1_intervalDomain","parent":2137,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-02/index.html#the-sharp-poincare-inequality-on-an-interval","scope":"tauceti/analysis/part-02/pde/part-02","status":"fully_proved","summary":"For every `u \u2208 H\u00b9\u2080(a, b)`,","title":"The sharp Poincar\u00e9 inequality on an interval"},{"id":"tauceti/analysis/part-02/pde/part-02/the-spatial-holder-estimate-of-the-cauchy-heat-p","kind":"node","lean":"HolderWith.heatCauchyPotential","parent":2137,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-02/index.html#the-spatial-holder-estimate-of-the-cauchy-heat-p","scope":"tauceti/analysis/part-02/pde/part-02","status":"fully_proved","summary":"If `g` is continuous and `\u03b1`-H\u00f6lder with constant `H`, then so is every time slice of its Cauchy potential, with the same constant.","title":"The spatial H\u00f6lder estimate of the Cauchy heat potential"},{"id":"tauceti/analysis/part-02/pde/part-02/the-spectral-theorem-for-the-dirichlet-problem","kind":"node","lean":"TauCeti.PDE.orthogonalComplement_iSup_eigenspaces_dirichletSolutionOperator_eq_bot","parent":2137,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-02/index.html#the-spectral-theorem-for-the-dirichlet-problem","scope":"tauceti/analysis/part-02/pde/part-02","status":"fully_proved","summary":"on a bounded domain and for a symmetric energy form, the eigenvectors of the solution operator span a dense subspace of `L\u00b2(\u03a9)`.","title":"The spectral theorem for the Dirichlet problem"},{"id":"tauceti/analysis/part-02/pde/part-02/stampacchias-global-bound","kind":"node","lean":"TauCeti.PDE.UniformlyEllipticOn.ae_value_le_add_of_forall_eLpNorm_le, TauCeti.PDE.exists_ae_value_le_add_mul_div_mul_rpow","parent":2137,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-02/index.html#stampacchias-global-bound","scope":"tauceti/analysis/part-02/pde/part-02","status":"fully_proved","summary":"Let `\u03a9` have finite measure, let `a` be measurable and uniformly elliptic on `\u03a9` with constants `0 < \u03bb \u2264 \u039b`, and let `c` be bounded, measurable and nonnegative. Suppose that `W^{1,2}_0(\u03a9)` satisfies a Sobolev inequality `\u2016v\u2016_{L\u02e2} \u2264 S \u2016\u2207v\u2016_{L\u00b2}`, and let `f \u2208 L\u02b3(\u03a9)` with `s (1 - 1/r) > 2`. If `u \u2208 H\u00b9(\u03a9)` is a weak subsolution of","title":"Stampacchia's global bound"},{"id":"tauceti/analysis/part-02/pde/part-02/the-temporal-holder-estimate-of-the-cauchy-heat","kind":"node","lean":"TauCeti.norm_heatCauchyPotential_sub_heatCauchyPotential_le, TauCeti.exists_holderOnWith_iteratedFDeriv_two_heatDuhamelPotential","parent":2137,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-02/index.html#the-temporal-holder-estimate-of-the-cauchy-heat","scope":"tauceti/analysis/part-02/pde/part-02","status":"fully_proved","summary":"If `g` is continuous and `\u03b1`-H\u00f6lder with constant `H`, its Cauchy potential `u` satisfies `\u2016u(t, x) - u(s, x)\u2016 \u2264 H M_\u03b1 |t - s|^(\u03b1/2)`, where `M_\u03b1 = \u222b \u2016y\u2016^\u03b1 K_1(y) dy` is `TauCeti.heatKernelMoment`.","title":"The temporal H\u00f6lder estimate of the Cauchy heat potential"},{"id":"tauceti/analysis/part-02/pde/part-02/the-value-field-parabolic-holder-estimate-for-th","kind":"node","lean":"TauCeti.holderWith_heatDuhamelPotential","parent":2137,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-02/index.html#the-value-field-parabolic-holder-estimate-for-th","scope":"tauceti/analysis/part-02/pde/part-02","status":"fully_proved","summary":"If the forcing is jointly measurable and its spatial slices on `[0,T]` are bounded by `M` and `\u03b1`-H\u00f6lder with constant `H`, where `0 < \u03b1 \u2264 2`, then its Duhamel potential is parabolically `\u03b1`-H\u00f6lder on the closed slab, with constant `T H (1 + M_\u03b1) + M max(1,T)`. Here `M_\u03b1` is the heat-kernel moment. No temporal continuity of the forcing is assumed.","title":"The value-field parabolic H\u00f6lder estimate for the Duhamel potential"},{"id":"tauceti/analysis/part-02/pde/part-02/the-weitzenbock-formula-at-a-vortex","kind":"node","lean":"TauCeti.Vortex.gaugeFixedOperator_gaugeFixedAdjoint_of_equation_eq_zero, TauCeti.Vortex.gaugeFixedOperator_gaugeFixedAdjoint, TauCeti.Vortex.gaugeFixedAdjoint_gaugeFixedOperator, TauCeti.Vortex.gaugeFixedAdjoint_gaugeFixedOperator_of_equation_eq_zero","parent":2137,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-02/index.html#the-weitzenbock-formula-at-a-vortex","scope":"tauceti/analysis/part-02/pde/part-02","status":"fully_proved","summary":"where the vortex equation holds, `L_x L_x* = (-\u0394 + P) \u2295 (-\u0394_a + 1 / 8 + P / 2)` with `P = p\u00b2 + q\u00b2 = |\u03b1|\u00b2 / 4`. The section part is displayed in real coordinates as in `gaugeFixedOperator_gaugeFixedAdjoint`.","title":"The Weitzenb\u00f6ck formula at a vortex"},{"id":"tauceti/analysis/part-02/pde/part-02/the-zero-forcing-caccioppoli-inequality-for-a-po","kind":"node","lean":"TauCeti.PDE.UniformlyEllipticOn.setIntegral_sq_mul_norm_gradient_posPartAbove_sq_le_of_nonpos, TauCeti.PDE.UniformlyEllipticOn.setIntegral_sq_mul_norm_gradient_posPartAbove_sq_le_lowerOrder_of_nonpos","parent":2137,"result":true,"route":"../roadmap/tauceti/analysis/part-02/pde/part-02/index.html#the-zero-forcing-caccioppoli-inequality-for-a-po","scope":"tauceti/analysis/part-02/pde/part-02","status":"fully_proved","summary":"This is the specialization of `UniformlyEllipticOn.setIntegral_sq_mul_norm_gradient_posPartAbove_sq_le_of_memLp` to `-\u2202\u1d62(a\u2071\u02b2 \u2202\u2c7cu) \u2264 0`.","title":"The zero-forcing Caccioppoli inequality for a positive truncation of a weak subsolution"},{"data_url":"scopes/tauceti/analysis/part-02/polynomial.json","id":"tauceti/analysis/part-02/polynomial","kind":"scope","parent":1841,"result":false,"route":"../roadmap/tauceti/analysis/part-02/polynomial/index.html","scope":"tauceti/analysis/part-02/polynomial","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Analysis.Polynomial` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Polynomial"},{"id":"tauceti/analysis/part-02/polynomial/cauchys-bound-on-the-symmetric-power","kind":"node","lean":"TauCeti.Sym.norm_le_norm_coeffEquiv_ofFn_add_one","parent":2153,"result":true,"route":"../roadmap/tauceti/analysis/part-02/polynomial/index.html#cauchys-bound-on-the-symmetric-power","scope":"tauceti/analysis/part-02/polynomial","status":"fully_proved","summary":"the points of an unordered tuple are bounded by one more than the sup-norm of its elementary symmetric chart, because each of them is a root of the monic polynomial those coordinates present.","title":"Cauchy's bound on the symmetric power"},{"id":"tauceti/analysis/part-02/polynomial/colemans-bound-for-truncations","kind":"node","lean":"PowerSeries.norm_coeff_colemanDBounded_trunc_le","parent":2153,"result":true,"route":"../roadmap/tauceti/analysis/part-02/polynomial/index.html#colemans-bound-for-truncations","scope":"tauceti/analysis/part-02/polynomial","status":"fully_proved","summary":"Let `\u03c1` be a nonnegative antitone Newton majorant of a power series `F`, `\u2016coeff j F\u2016 \u2264 \u220f i < j, \u03c1 i`, and let `B` be restricted at every radius `\u03c1 i` with `B(0) = 0`. Then the Coleman transforms of the truncations of `F` along the truncations of `B` satisfy `\u2016coeff k D_(n,m)(trunc a B, trunc b F)\u2016 \u2264 \u220f i < k, \u03b2_B(\u03c1 i)`, uniformly in the ranks, degree bounds and truncation orders. In particular this bounds\u2026","title":"Coleman's bound for truncations"},{"id":"tauceti/analysis/part-02/polynomial/colemans-coefficient-bound","kind":"node","lean":"Polynomial.norm_coeff_colemanDBounded_le","parent":2153,"result":true,"route":"../roadmap/tauceti/analysis/part-02/polynomial/index.html#colemans-coefficient-bound","scope":"tauceti/analysis/part-02/polynomial","status":"fully_proved","summary":"Let `P` and `B` be polynomials over an ultrametric seminormed ring with `\u20161\u2016 = 1` and `B(0) = 0`. If `\u03c1` is a nonnegative antitone sequence with `\u2016coeff j P\u2016 \u2264 \u220f i < j, \u03c1 i` for `j \u2264 n`, and `\u2016coeff l B\u2016 * \u03c1 t ^ l \u2264 \u03b2 t` for every `t` and every `l \u2265 1`, then the `k`-th coefficient of Coleman's transform `D_(n,m)(B, P)` has norm at most `\u220f t < k, \u03b2 t`, for all ranks `n` and degree bounds `m`.","title":"Coleman's coefficient bound"},{"id":"tauceti/analysis/part-02/polynomial/colemans-perturbation-bound","kind":"node","lean":"Polynomial.dist_coeff_colemanDBounded_mul_le","parent":2153,"result":true,"route":"../roadmap/tauceti/analysis/part-02/polynomial/index.html#colemans-perturbation-bound","scope":"tauceti/analysis/part-02/polynomial","status":"fully_proved","summary":"Let `\u03c1` be a nonnegative antitone sequence and `r \u2265 1`. If `P\u2081` has Newton majorant `\u03c1` up to degree `n`, the perturbation `P\u2082 - P\u2081` satisfies the same bound after multiplication by `r`, and the coefficients of positive degree of `B\u2081` and of `r (B\u2082 - B\u2081)` are bounded by `\u03b2` at every radius `\u03c1 t`, where `B\u2081(0) = B\u2082(0) = 0`, then the coefficients of the bounded Coleman transforms satisfy\u2026","title":"Coleman's perturbation bound"},{"id":"tauceti/analysis/part-02/polynomial/northcotts-theorem-for-integer-polynomials","kind":"node","lean":"Polynomial.finite_setOf_natDegree_le_supNorm_le","parent":2153,"result":true,"route":"../roadmap/tauceti/analysis/part-02/polynomial/index.html#northcotts-theorem-for-integer-polynomials","scope":"tauceti/analysis/part-02/polynomial","status":"fully_proved","summary":"Only finitely many integer polynomials have degree at most `n` and coefficient sup norm (naive height) at most `B`.","title":"Northcott's theorem for integer polynomials"},{"data_url":"scopes/tauceti/analysis/part-02/semigroups.json","id":"tauceti/analysis/part-02/semigroups","kind":"scope","parent":1841,"result":false,"route":"../roadmap/tauceti/analysis/part-02/semigroups/index.html","scope":"tauceti/analysis/part-02/semigroups","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Analysis.Semigroups` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Semigroups"},{"id":"tauceti/analysis/part-02/semigroups/the-bounded-perturbation-theorem","kind":"node","lean":"TauCeti.Semigroups.StronglyContinuousSemigroup.exists_generator_eq_vadd, TauCeti.Semigroups.StronglyContinuousSemigroup.exists_generator_eq_vadd_of_hasGrowthBound_one, TauCeti.Semigroups.ContractionSemigroup.exists_generator_eq_vadd, TauCeti.Semigroups.IsMDissipative.exists_stronglyContinuousSemigroup_generator_eq_vadd","parent":2159,"result":true,"route":"../roadmap/tauceti/analysis/part-02/semigroups/index.html#the-bounded-perturbation-theorem","scope":"tauceti/analysis/part-02/semigroups","status":"fully_proved","summary":"If a strongly continuous semigroup `S` has growth bound `(omega, M)` and `B` is bounded, then `B +\u1d65 S.generator` generates a strongly continuous semigroup with growth bound `(omega + M * \u2016B\u2016, M)`.","title":"The bounded perturbation theorem"},{"id":"tauceti/analysis/part-02/semigroups/converse-of-the-lumer-phillips-theorem","kind":"node","lean":"TauCeti.Semigroups.ContractionSemigroup.isDissipative_generator, TauCeti.Semigroups.ContractionSemigroup.real_inner_generator_nonpos","parent":2159,"result":true,"route":"../roadmap/tauceti/analysis/part-02/semigroups/index.html#converse-of-the-lumer-phillips-theorem","scope":"tauceti/analysis/part-02/semigroups","status":"fully_proved","summary":"the generator of a contraction semigroup is dissipative.","title":"Converse of the Lumer--Phillips theorem"},{"id":"tauceti/analysis/part-02/semigroups/the-hille-yosida-bound-at-a-complex-spectral-par","kind":"node","lean":"TauCeti.Semigroups.StronglyContinuousSemigroup.norm_resolvent_complexGenerator_le","parent":2159,"result":true,"route":"../roadmap/tauceti/analysis/part-02/semigroups/index.html#the-hille-yosida-bound-at-a-complex-spectral-par","scope":"tauceti/analysis/part-02/semigroups","status":"fully_proved","summary":"on the half-plane of a growth bound `(omega, M)`, `\u2016R(lambda, A)\u2016 \u2264 M / (re lambda - omega)`.","title":"The Hille--Yosida bound at a complex spectral parameter"},{"id":"tauceti/analysis/part-02/semigroups/the-hille-yosida-power-bounds-at-a-complex-spect","kind":"node","lean":"TauCeti.Semigroups.StronglyContinuousSemigroup.norm_resolvent_complexGenerator_pow_le","parent":2159,"result":true,"route":"../roadmap/tauceti/analysis/part-02/semigroups/index.html#the-hille-yosida-power-bounds-at-a-complex-spect","scope":"tauceti/analysis/part-02/semigroups","status":"fully_proved","summary":"on the half-plane of a growth bound `(omega, M)`, `\u2016R(lambda, A)\u207f\u2016 \u2264 M / (re lambda - omega)\u207f` for every `n`.","title":"The Hille--Yosida power bounds at a complex spectral parameter"},{"id":"tauceti/analysis/part-02/semigroups/hille-yosida-resolvent-bound","kind":"node","lean":"TauCeti.Semigroups.StronglyContinuousSemigroup.resolvent_norm_le, TauCeti.Semigroups.StronglyContinuousSemigroup.norm_generator_resolvent_le","parent":2159,"result":true,"route":"../roadmap/tauceti/analysis/part-02/semigroups/index.html#hille-yosida-resolvent-bound","scope":"tauceti/analysis/part-02/semigroups","status":"fully_proved","summary":"`\u2016R \u03bb\u2016 \u2264 M/(\u03bb-\u03c9)` for a C\u2080 semigroup with growth bound `(\u03c9, M)` and `\u03bb > \u03c9` (Hille 1948, Yosida 1948; Engel\u2013Nagel Ch. II).","title":"Hille\u2013Yosida resolvent bound"},{"id":"tauceti/analysis/part-02/semigroups/lumer-phillips-in-duality-map-form","kind":"node","lean":"TauCeti.Semigroups.exists_contractionSemigroup_generator_eq_iff_exists_mem_dualitySet_apply_nonpos, TauCeti.Semigroups.exists_contractionSemigroup_generator_eq_iff, TauCeti.Semigroups.IsMDissipative.exists_contractionSemigroup_generator_eq, TauCeti.Semigroups.exists_contractionSemigroup_generator_eq_iff_forall_mem_dualitySet_apply_nonpos","parent":2159,"result":true,"route":"../roadmap/tauceti/analysis/part-02/semigroups/index.html#lumer-phillips-in-duality-map-form","scope":"tauceti/analysis/part-02/semigroups","status":"fully_proved","summary":"An unbounded operator on a real Banach space generates a contraction semigroup exactly when it is densely defined, each `x \u2208 D(A)` has a duality-set member `f` with `f (A x) \u2264 0`, and `lambda \u2022 I - A` maps `D(A)` onto `X` for some `lambda > 0`.","title":"Lumer--Phillips in duality-map form"},{"id":"tauceti/analysis/part-02/semigroups/sharp-hille-yosida-power-bound-for-the-generator","kind":"node","lean":"TauCeti.Semigroups.StronglyContinuousSemigroup.norm_generator_resolvent_pow_le","parent":2159,"result":true,"route":"../roadmap/tauceti/analysis/part-02/semigroups/index.html#sharp-hille-yosida-power-bound-for-the-generator","scope":"tauceti/analysis/part-02/semigroups","status":"fully_proved","summary":"For a C\u2080-semigroup with growth bound `(omega, M)` and `lambda > omega`, `\u2016R(lambda, generator S) ^ n\u2016 \u2264 M / (lambda - omega) ^ n`.","title":"Sharp Hille--Yosida power bound for the generator resolvent"},{"data_url":"scopes/tauceti/analysis/part-02/sobolev.json","id":"tauceti/analysis/part-02/sobolev","kind":"scope","parent":1841,"result":false,"route":"../roadmap/tauceti/analysis/part-02/sobolev/index.html","scope":"tauceti/analysis/part-02/sobolev","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Analysis.Sobolev` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Sobolev"},{"data_url":"scopes/tauceti/analysis/part-02/sobolev/part-01.json","id":"tauceti/analysis/part-02/sobolev/part-01","kind":"scope","navigation":"transparent","parent":2167,"result":false,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-01/index.html","scope":"tauceti/analysis/part-02/sobolev/part-01","status":"fully_proved","title":"The `L\u00b2` Leibniz estimate to No Poincar\u00e9 inequality holds on\u2026"},{"id":"tauceti/analysis/part-02/sobolev/part-01/the-l2-leibniz-estimate","kind":"node","lean":"TauCeti.W1p.norm_gradient_contDiffSMul_sq_le","parent":2168,"result":true,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-01/index.html#the-l2-leibniz-estimate","scope":"tauceti/analysis/part-02/sobolev/part-01","status":"fully_proved","summary":"At exponent two, the gradient `\u03c8 \u2207u + u \u2207\u03c8` of `\u03c8 u` satisfies","title":"The `L\u00b2` Leibniz estimate"},{"id":"tauceti/analysis/part-02/sobolev/part-01/the-analytic-content-of-the-extension-theorem","kind":"node","lean":"TauCeti.W1p0.hasWeakFDerivOn_indicator","parent":2168,"result":true,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-01/index.html#the-analytic-content-of-the-extension-theorem","scope":"tauceti/analysis/part-02/sobolev/part-01","status":"fully_proved","summary":"The zero-extension of `u \u2208 W^{1,p}_0(\u03a9)` is weakly differentiable on the larger open set `\u03a9'`, with weak gradient the zero-extension of the weak gradient of `u`. This is the statement that can *fail* for a general `u \u2208 W^{1,p}(\u03a9)`: with no condition forcing `u` to vanish towards `\u2202\u03a9`, its zero-extension need not be weakly differentiable on `\u03a9'` at all.","title":"The analytic content of the extension theorem"},{"id":"tauceti/analysis/part-02/sobolev/part-01/the-borderline-sobolev-inequality-on-w-1-n-0","kind":"node","lean":"TauCeti.W1p.eLpNorm_value_le_mul_measure_rpow_mul_enorm_gradient, TauCeti.eLpNorm_le_mul_measure_rpow_mul_eLpNorm_fderiv","parent":2168,"result":true,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-01/index.html#the-borderline-sobolev-inequality-on-w-1-n-0","scope":"tauceti/analysis/part-02/sobolev/part-01","status":"fully_proved","summary":"Let `n` be the dimension and `\u03c9 = \u03bc(B(0, 1))`. If `\u03a9` has finite measure, then every `u \u2208 W^{1,n}_0(\u03a9)` lies in `L^q(\u03a9)` for every finite `q \u2265 n`, with","title":"The borderline Sobolev inequality on `W^{1,n}_0(\u03a9)`"},{"id":"tauceti/analysis/part-02/sobolev/part-01/de-giorgis-isoperimetric-inequality","kind":"node","lean":"TauCeti.W1p.sub_mul_measureReal_mul_measureReal_le_of_convex, TauCeti.W1p.sub_mul_measureReal_mul_measureReal_le_of_eq_ball, TauCeti.W1p.sub_mul_measureReal_mul_measureReal_le_of_ball_subset, TauCeti.W1p.sq_sub_mul_measureReal_mul_measureReal_le_of_ball_subset","parent":2168,"result":true,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-01/index.html#de-giorgis-isoperimetric-inequality","scope":"tauceti/analysis/part-02/sobolev/part-01","status":"fully_proved","summary":"Let `\u03a9` be a bounded convex open set in a finite-dimensional real inner product space of dimension `n`, let `u \u2208 W^{1,p}(\u03a9)` with `1 \u2264 p \u2264 \u221e`, and let `k < l`. Then, all sets being taken inside `\u03a9`,","title":"De Giorgi's isoperimetric inequality"},{"id":"tauceti/analysis/part-02/sobolev/part-01/the-divergence-theorem-on-a-half-space-tauceti-s","kind":"node","lean":"TauCeti.setIntegral_normalHalfSpace_fderiv_apply","parent":2168,"result":true,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-01/index.html#the-divergence-theorem-on-a-half-space-tauceti-s","scope":"tauceti/analysis/part-02/sobolev/part-01","status":"fully_proved","summary":"For a compactly supported `C\u00b9` function `g` on `\u211d \u00d7 E` and a direction `v`, the integral of the directional derivative `\u2202_v g` over the half-space `{x | a < x.fst}` is the boundary integral of `g` over `{a} \u00d7 E`, weighted by `\u27eav, -e\u2081\u27eb = -v.fst`, the normal component of `v` along the outward unit normal.","title":"The divergence theorem on a half-space (`TauCeti.setIntegral_normalHalfSpace_fderiv_apply`)"},{"id":"tauceti/analysis/part-02/sobolev/part-01/the-du-bois-reymond-lemma-for-weak-derivatives","kind":"node","lean":"TauCeti.HasWeakLineDerivOn.exists_ae_eq_const","parent":2168,"result":true,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-01/index.html#the-du-bois-reymond-lemma-for-weak-derivatives","scope":"tauceti/analysis/part-02/sobolev/part-01","status":"fully_proved","summary":"A function whose weak derivative on `Ioo a b` is `0` is almost everywhere equal to a constant on `Ioo a b`.","title":"The du Bois-Reymond lemma for weak derivatives"},{"id":"tauceti/analysis/part-02/sobolev/part-01/extension-theorem-for-bounded-lipschitz-domains","kind":"node","lean":"TauCeti.W1p.exists_extensionL_of_isLipschitzDomainWith","parent":2168,"result":true,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-01/index.html#extension-theorem-for-bounded-lipschitz-domains","scope":"tauceti/analysis/part-02/sobolev/part-01","status":"fully_proved","summary":"There is a bounded linear extension from `W^{1,p}(\u03a9)` to the whole space, and restriction to `\u03a9` recovers the original function. Both endpoint exponents `p = 1` and `p = \u221e` are included.","title":"Extension theorem for bounded Lipschitz domains"},{"id":"tauceti/analysis/part-02/sobolev/part-01/the-fundamental-theorem-of-calculus-for-a-contin","kind":"node","lean":"TauCeti.HasWeakLineDerivOn.integral_eq_sub","parent":2168,"result":true,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-01/index.html#the-fundamental-theorem-of-calculus-for-a-contin","scope":"tauceti/analysis/part-02/sobolev/part-01","status":"fully_proved","summary":"Let `u'` be the weak derivative of `u` on `Ioo a b`, integrable on `Ioo a b`, and let `v` be continuous on `Icc a b` and equal to `u` almost everywhere on `Ioo a b`. Then `\u222b r in s..t, u' r = v t - v s` for all `s, t \u2208 Icc a b`.","title":"The fundamental theorem of calculus for a continuous representative"},{"id":"tauceti/analysis/part-02/sobolev/part-01/the-fundamental-theorem-of-calculus-for-weak-der","kind":"node","lean":"TauCeti.HasWeakLineDerivOn.exists_ae_eq_add_intervalIntegral, TauCeti.HasWeakLineDerivOn.exists_ae_eq_add_intervalIntegral_left","parent":2168,"result":true,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-01/index.html#the-fundamental-theorem-of-calculus-for-weak-der","scope":"tauceti/analysis/part-02/sobolev/part-01","status":"fully_proved","summary":"If `u'` is the weak derivative of `u` on `Ioo a b` and `t\u2080 \u2208 Ioo a b`, then `u` is almost everywhere on `Ioo a b` equal to a constant plus the primitive `t \u21a6 \u222b s in t\u2080..t, u' s`.","title":"The fundamental theorem of calculus for weak derivatives"},{"id":"tauceti/analysis/part-02/sobolev/part-01/the-fundamental-theorem-of-calculus-for-weak-der-2","kind":"node","lean":"TauCeti.HasWeakLineDerivOn.exists_ae_eq_add_intervalIntegral_top","parent":2168,"result":true,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-01/index.html#the-fundamental-theorem-of-calculus-for-weak-der-2","scope":"tauceti/analysis/part-02/sobolev/part-01","status":"fully_proved","summary":"If `u'` is the weak derivative of `u` on all of `\u211d`, then `u` is almost everywhere equal to a constant plus the primitive `t \u21a6 \u222b s in t\u2080..t, u' s`, for any base point `t\u2080`.","title":"The fundamental theorem of calculus for weak derivatives on the line"},{"id":"tauceti/analysis/part-02/sobolev/part-01/the-gagliardo-nirenberg-sobolev-inequality-on-w","kind":"node","lean":"TauCeti.W1p.norm_sobolevEmbeddingTopL_le, TauCeti.norm_testFunctionLp_le_mul_norm_gradientTestFunctionLp, TauCeti.D1p0.norm_sobolevEmbeddingL_le, TauCeti.W1p.eLpNorm_value_le_mul_enorm_gradient","parent":2168,"result":true,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-01/index.html#the-gagliardo-nirenberg-sobolev-inequality-on-w","scope":"tauceti/analysis/part-02/sobolev/part-01","status":"fully_proved","summary":"The image of `u` under the whole-space Sobolev embedding is bounded in `L^{p\u22c6}` by the `L\u1d56` norm of the gradient of `u` alone, with Mathlib's constant `MeasureTheory.SNormLESNormFDerivOfEqConst`.","title":"The Gagliardo--Nirenberg--Sobolev inequality on `W^{1,p}(E)`"},{"id":"tauceti/analysis/part-02/sobolev/part-01/greens-second-identity-against-a-test-function","kind":"node","lean":"ContDiffOn.integral_laplacian_smul_eq_integral_smul_laplacian","parent":2168,"result":true,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-01/index.html#greens-second-identity-against-a-test-function","scope":"tauceti/analysis/part-02/sobolev/part-01","status":"fully_proved","summary":"For `u` of class `C\u00b2` on the open set `\u03a9` and a test function `\u03c6` on `\u03a9`, `\u222b \u0394\u03c6 \u2022 u \u2202\u03bc = \u222b \u03c6 \u2022 \u0394u \u2202\u03bc`: the classical Laplacian of a `C\u00b2` function is its distributional Laplacian.","title":"Green's second identity against a test function"},{"id":"tauceti/analysis/part-02/sobolev/part-01/the-interior-sobolev-inequality","kind":"node","lean":"TauCeti.W1p.exists_eLpNorm_value_le_of_isCompact_closure","parent":2168,"result":true,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-01/index.html#the-interior-sobolev-inequality","scope":"tauceti/analysis/part-02/sobolev/part-01","status":"fully_proved","summary":"If `V` has compact closure contained in the open set `\u03a9`, `1 \u2264 p` and `1/p\u22c6 + 1/n = 1/p` with `p\u22c6 < \u221e`, then for every `q \u2264 p\u22c6` there is a constant `C` with `\u2016u\u2016_{L^q(V)} \u2264 C \u2016u\u2016_{W^{1,p}(\u03a9)}` for all `u \u2208 W^{1,p}(\u03a9)`. The full Sobolev norm is needed on the right: on a bounded `\u03a9`, a nonzero constant has vanishing gradient.","title":"The interior Sobolev inequality"},{"id":"tauceti/analysis/part-02/sobolev/part-01/jensens-bound-for-the-gradient-of-a-mollificatio","kind":"node","lean":"TauCeti.HasWeakFDerivOn.lintegral_ofReal_map_norm_fderiv_convolution_le","parent":2168,"result":true,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-01/index.html#jensens-bound-for-the-gradient-of-a-mollificatio","scope":"tauceti/analysis/part-02/sobolev/part-01","status":"fully_proved","summary":"If `U` is a weak derivative field of `u` on the whole space and `rho` is a smooth compactly supported probability kernel, then `\u222b \u03a6(\u2016D(u \u22c6 rho)\u2016) \u2264 \u222b \u03a6(\u2016U\u2016)` for every `\u03a6` convex and monotone on `[0, \u221e)`, in extended nonnegative form.","title":"Jensen's bound for the gradient of a mollification"},{"id":"tauceti/analysis/part-02/sobolev/part-01/the-local-morrey-inequality","kind":"node","lean":"TauCeti.W1p.exists_dist_preciseRepresentative_le","parent":2168,"result":true,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-01/index.html#the-local-morrey-inequality","scope":"tauceti/analysis/part-02/sobolev/part-01","status":"fully_proved","summary":"Let `n < p \u2264 \u221e`, where `n` is the dimension, and let `\u03b1 = 1 - n / p`. There is a constant `C > 0`, depending only on `n`, `p` and the Haar measure `\u03bc`, such that the following holds. Let `u \u2208 W^{1,p}(\u03a9)` and let `x, y` be points with `closedBall x (2 |x - y|) \u2286 \u03a9`. Then the precise representative `u*` of `u` satisfies","title":"The local Morrey inequality"},{"id":"tauceti/analysis/part-02/sobolev/part-01/meyers-serrin-h-w-in-w-k-p","kind":"node","lean":"TauCeti.Wkp.dense_contDiffOn_representatives","parent":2168,"result":true,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-01/index.html#meyers-serrin-h-w-in-w-k-p","scope":"tauceti/analysis/part-02/sobolev/part-01","status":"fully_proved","summary":"For `1 \u2264 p < \u221e`, every order `k` and an arbitrary open set `\u03a9`, the elements of `W^{k,p}(\u03a9)` represented by functions smooth on `\u03a9` are dense in `W^{k,p}(\u03a9)`, in the full Sobolev norm.","title":"Meyers\u2013Serrin: `H = W` in `W^{k,p}(\u03a9)`"},{"id":"tauceti/analysis/part-02/sobolev/part-01/the-morrey-potential-estimate-for-the-mean","kind":"node","lean":"TauCeti.enorm_sub_setAverage_le_of_starConvex_of_finrank_lt, TauCeti.enorm_sub_setAverage_le_of_convex_of_finrank_lt","parent":2168,"result":true,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-01/index.html#the-morrey-potential-estimate-for-the-mean","scope":"tauceti/analysis/part-02/sobolev/part-01","status":"fully_proved","summary":"If `u` is `C\u00b9` on an open set `\u03a9` which is star-convex about `x` and contained in `closedBall x D`, and `p` exceeds the dimension `n` of the space, then for every `S \u2286 \u03a9` of positive measure the deviation of `u x` from the mean of `u` over `S` is at most `D ^ n / (n \u03bc(S)) * K ^ (1 - 1 / p) * D ^ (1 - n / p)` times the `L\u1d56(\u03a9)` norm of the derivative of `u`, where `K = n \u03bc(B(0, 1)) (p - 1) / (p - n)`.","title":"The Morrey potential estimate for the mean"},{"id":"tauceti/analysis/part-02/sobolev/part-01/morreys-dirichlet-growth-theorem","kind":"node","lean":"TauCeti.W1p.holderOnWith_preciseRepresentative_of_setIntegral_norm_gradient_le, TauCeti.W1p.holderWith_preciseRepresentative_of_setIntegral_norm_gradient_le","parent":2168,"result":true,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-01/index.html#morreys-dirichlet-growth-theorem","scope":"tauceti/analysis/part-02/sobolev/part-01","status":"fully_proved","summary":"Let `u \u2208 W^{1,p}(\u03a9)` and `\u03b1 > 0`, and let `K` be a set of diameter at most `R` whose closed `2R`-thickening lies in `\u03a9`. If the gradient of `u` satisfies the growth condition","title":"Morrey's Dirichlet growth theorem"},{"id":"tauceti/analysis/part-02/sobolev/part-01/morreys-inequality","kind":"node","lean":"TauCeti.enorm_sub_le_of_contDiff_of_finrank_lt, TauCeti.enorm_sub_le_of_mem_ball_of_finrank_lt, TauCeti.holderWith_of_contDiff_of_finrank_lt","parent":2168,"result":true,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-01/index.html#morreys-inequality","scope":"tauceti/analysis/part-02/sobolev/part-01","status":"fully_proved","summary":"If `u` is `C\u00b9` on the whole space and `p` exceeds the dimension `n` of the space, then `\u2016u x - u y\u2016` is at most `2 ^ (n + 1) / (n \u03c9) * K ^ (1 - 1 / p) * (2 \u2016x - y\u2016) ^ (1 - n / p)` times the `L\u1d56` norm of the derivative of `u`, where `\u03c9 = \u03bc(B(0, 1))` and `K = n \u03c9 (p - 1) / (p - n)`.","title":"Morrey's inequality"},{"id":"tauceti/analysis/part-02/sobolev/part-01/no-poincare-inequality-holds-on-the-whole-space","kind":"node","lean":"TauCeti.not_exists_eLpNorm_le_const_mul_eLpNorm_fderiv","parent":2168,"result":true,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-01/index.html#no-poincare-inequality-holds-on-the-whole-space","scope":"tauceti/analysis/part-02/sobolev/part-01","status":"fully_proved","summary":"There is no constant `C` with `\u2016u\u2016_p \u2264 C \u2016Du\u2016_p` for every compactly supported `C\u1d4f` function `u`, for any order `k \u2264 \u221e` and any `p \u2260 0`, including `p = \u221e`. In particular, the inequality fails even when restricted to smooth test functions.","title":"No Poincar\u00e9 inequality holds on the whole space"},{"data_url":"scopes/tauceti/analysis/part-02/sobolev/part-02.json","id":"tauceti/analysis/part-02/sobolev/part-02","kind":"scope","navigation":"transparent","parent":2167,"result":false,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-02/index.html","scope":"tauceti/analysis/part-02/sobolev/part-02","status":"fully_proved","title":"The one-dimensional Poincar\u00e9\u2026 to Trudinger's inequality on\u2026"},{"id":"tauceti/analysis/part-02/sobolev/part-02/the-one-dimensional-poincare-inequality","kind":"node","lean":"TauCeti.eLpNorm_le_eLpNorm_deriv_of_support_subset_Icc, TauCeti.lintegral_enorm_rpow_le_of_support_subset_Icc","parent":2189,"result":true,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-02/index.html#the-one-dimensional-poincare-inequality","scope":"tauceti/analysis/part-02/sobolev/part-02","status":"fully_proved","summary":"a `C\u00b9` function on `\u211d` supported in an interval of length `b - a` obeys `\u2016g\u2016_p \u2264 (b - a) \u2016g'\u2016_p` for every `1 \u2264 p \u2264 \u221e`. The endpoints may be in either order: an empty or degenerate interval carries only the zero function.","title":"The one-dimensional Poincar\u00e9 inequality"},{"id":"tauceti/analysis/part-02/sobolev/part-02/the-poincare-inequality-on-a-ball","kind":"node","lean":"TauCeti.eLpNorm_le_eLpNorm_fderiv_of_support_subset_ball, TauCeti.W1p.norm_value_le_mul_norm_gradient_of_subset_slab, TauCeti.W1p.norm_value_le_mul_norm_gradient_of_subset_ball, TauCeti.eLpNorm_le_eLpNorm_fderiv_of_support_subset_slab, TauCeti.W1p.setAverage_norm_value_sub_setAverage_closedBall_le","parent":2189,"result":true,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-02/index.html#the-poincare-inequality-on-a-ball","scope":"tauceti/analysis/part-02/sobolev/part-02","status":"fully_proved","summary":"A `C\u00b9` function on `\u211d^{n+1}` supported in a ball of radius `R` satisfies `\u2016u\u2016_p \u2264 2R \u2016Du\u2016_p` for every `1 \u2264 p \u2264 \u221e`.","title":"The Poincar\u00e9 inequality on a ball"},{"id":"tauceti/analysis/part-02/sobolev/part-02/the-poincare-wirtinger-inequality-on-a-ball","kind":"node","lean":"TauCeti.W1p.eLpNorm_value_sub_setAverage_le_of_eq_ball, TauCeti.W1p.exists_eLpNorm_value_sub_setAverage_le_of_isCompactOperator, TauCeti.W1p.eLpNorm_value_sub_setAverage_le_of_convex, TauCeti.W1p.setIntegral_value_sub_setAverage_sq_le_of_ball_subset, TauCeti.W1p.setIntegral_abs_value_sub_setAverage_le_of_ball_subset, TauCeti.eLpNorm_sub_setAverage_le_of_convex, TauCeti.lintegral_enorm_sub_setAverage_rpow_le_of_convex, TauCeti.setLIntegral_enorm_sub_setAverage_le_mul_measure_rpow_mul_of_convex, TauCeti.W1p.exists_eLpNorm_value_sub_setAverage_le_of_isLipschitzDomainWith","parent":2189,"result":true,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-02/index.html#the-poincare-wirtinger-inequality-on-a-ball","scope":"tauceti/analysis/part-02/sobolev/part-02","status":"fully_proved","summary":"On a ball of radius `R` the constant of `TauCeti.W1p.eLpNorm_value_sub_setAverage_le_of_convex` is `2 ^ (n + 1) * R`: the deviation of `u \u2208 W^{1,p}(B(c, R))` from its mean over the ball is at most `2 ^ (n + 1) * R` times the `L\u1d56` norm of its weak gradient. The constant is proportional to `R`, as the scaling of both sides forces.","title":"The Poincar\u00e9\u2013Wirtinger inequality on a ball"},{"id":"tauceti/analysis/part-02/sobolev/part-02/the-pointwise-morrey-estimate","kind":"node","lean":"TauCeti.W1p.dist_preciseRepresentative_le_of_setIntegral_norm_gradient_le","parent":2189,"result":true,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-02/index.html#the-pointwise-morrey-estimate","scope":"tauceti/analysis/part-02/sobolev/part-02","status":"fully_proved","summary":"Let `u \u2208 W^{1,p}(\u03a9)`, `\u03b1 > 0`, `d = dist x y`, and suppose that `closedBall x (2 d) \u2286 \u03a9` and that the gradient of `u` satisfies the growth condition `\u222b_{B(z, r)} \u2016\u2207u\u2016 \u2264 M r^(n - 1 + \u03b1)` for `z = x`, `0 < r \u2264 2 d` and for `z = y`, `0 < r \u2264 d`. Then the precise representative `u*` of `u` satisfies","title":"The pointwise Morrey estimate"},{"id":"tauceti/analysis/part-02/sobolev/part-02/rellich-kondrachov-on-bounded-lipschitz-domains","kind":"node","lean":"TauCeti.W1p.isCompactOperator_valueL_of_isLipschitzDomainWith, TauCeti.W1p0.isCompactOperator_valueL","parent":2189,"result":true,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-02/index.html#rellich-kondrachov-on-bounded-lipschitz-domains","scope":"tauceti/analysis/part-02/sobolev/part-02","status":"fully_proved","summary":"If `\u03a9` is a bounded Lipschitz domain and `1 \u2264 p < \u221e`, the value map `W^{1,p}(\u03a9) \u2192 L\u1d56(\u03a9)` is a compact operator.","title":"Rellich\u2013Kondrachov on bounded Lipschitz domains"},{"id":"tauceti/analysis/part-02/sobolev/part-02/the-sobolev-inequality-on-w-1-p-0-of-a-ball","kind":"node","lean":"TauCeti.W1p.eLpNorm_value_le_mul_rpow_mul_enorm_gradient_of_eq_ball, TauCeti.W1p.eLpNorm_value_le_mul_measure_rpow_mul_enorm_gradient_of_lt, TauCeti.W1p.exists_eLpNorm_value_le_of_isLipschitzDomainWith, TauCeti.eLpNorm_le_mul_measure_rpow_mul_eLpNorm_fderiv_of_lt","parent":2189,"result":true,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-02/index.html#the-sobolev-inequality-on-w-1-p-0-of-a-ball","scope":"tauceti/analysis/part-02/sobolev/part-02","status":"fully_proved","summary":"Let `n \u2265 1` be the dimension and `\u03c9 = \u03bc(B(0, 1))`, and let `p \u2264 q < \u221e` with `\u03b4 = 1/p - 1/q < 1/n`. If `\u03a9 = B(x\u2080, R)`, then every `u \u2208 W^{1,p}_0(\u03a9)` satisfies","title":"The Sobolev inequality on `W^{1,p}_0` of a ball"},{"id":"tauceti/analysis/part-02/sobolev/part-02/the-supercritical-sobolev-multiplication-estimat","kind":"node","lean":"TauCeti.W1p.norm_mul_le","parent":2189,"result":true,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-02/index.html#the-supercritical-sobolev-multiplication-estimat","scope":"tauceti/analysis/part-02/sobolev/part-02","status":"fully_proved","summary":"The `W^{1,p}` norm of a product is at most three times the operator norm of Morrey's embedding times the product of the norms of the factors. The embedding norm is what enters because the only control on a factor outside `L^p` is the supremum norm of its Morrey representative.","title":"The supercritical Sobolev multiplication estimate"},{"id":"tauceti/analysis/part-02/sobolev/part-02/trudingers-inequality-on-w-1-n-0","kind":"node","lean":"TauCeti.W1p.setLIntegral_exp_abs_value_div_rpow_le, TauCeti.setLIntegral_exp_norm_div_rpow_le, TauCeti.setLIntegral_exp_norm_div_rpow_le_of_eLpNorm_fderiv_le","parent":2189,"result":true,"route":"../roadmap/tauceti/analysis/part-02/sobolev/part-02/index.html#trudingers-inequality-on-w-1-n-0","scope":"tauceti/analysis/part-02/sobolev/part-02","status":"fully_proved","summary":"Let `n \u2265 2` be the dimension and `\u03c9 = \u03bc(B(0, 1))`. Every `u \u2208 W^{1,n}_0(\u03a9)` satisfies","title":"Trudinger's inequality on `W^{1,n}_0(\u03a9)`"},{"data_url":"scopes/tauceti/analysis/part-02/special-functions.json","id":"tauceti/analysis/part-02/special-functions","kind":"scope","parent":1841,"result":false,"route":"../roadmap/tauceti/analysis/part-02/special-functions/index.html","scope":"tauceti/analysis/part-02/special-functions","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Analysis.SpecialFunctions` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Special Functions"},{"id":"tauceti/analysis/part-02/special-functions/the-addition-theorem-for-cn-dlmf-22-8-2","kind":"node","lean":"TauCeti.Real.jacobiCn_add","parent":2198,"result":true,"route":"../roadmap/tauceti/analysis/part-02/special-functions/index.html#the-addition-theorem-for-cn-dlmf-22-8-2","scope":"tauceti/analysis/part-02/special-functions","status":"fully_proved","summary":"`cn(u + v) = (cn u \u00b7 cn v - sn u \u00b7 dn u \u00b7 sn v \u00b7 dn v) / (1 - k\u00b2 sn\u00b2(u) sn\u00b2(v))`.","title":"The addition theorem for `cn` (DLMF 22.8.2)"},{"id":"tauceti/analysis/part-02/special-functions/the-addition-theorem-for-dn-dlmf-22-8-3","kind":"node","lean":"TauCeti.Real.jacobiDn_add","parent":2198,"result":true,"route":"../roadmap/tauceti/analysis/part-02/special-functions/index.html#the-addition-theorem-for-dn-dlmf-22-8-3","scope":"tauceti/analysis/part-02/special-functions","status":"fully_proved","summary":"`dn(u + v) = (dn u \u00b7 dn v - k\u00b2 \u00b7 sn u \u00b7 cn u \u00b7 sn v \u00b7 cn v) / (1 - k\u00b2 sn\u00b2(u) sn\u00b2(v))`.","title":"The addition theorem for `dn` (DLMF 22.8.3)"},{"id":"tauceti/analysis/part-02/special-functions/the-addition-theorem-for-sn-dlmf-22-8-1","kind":"node","lean":"TauCeti.Real.jacobiSn_add","parent":2198,"result":true,"route":"../roadmap/tauceti/analysis/part-02/special-functions/index.html#the-addition-theorem-for-sn-dlmf-22-8-1","scope":"tauceti/analysis/part-02/special-functions","status":"fully_proved","summary":"`sn(u + v) = (sn u \u00b7 cn v \u00b7 dn v + sn v \u00b7 cn u \u00b7 dn u) / (1 - k\u00b2 sn\u00b2(u) sn\u00b2(v))`.","title":"The addition theorem for `sn` (DLMF 22.8.1)"},{"id":"tauceti/analysis/part-02/special-functions/the-addition-theorem-for","kind":"node","lean":"PeriodPair.derivWeierstrassP_add","parent":2198,"result":true,"route":"../roadmap/tauceti/analysis/part-02/special-functions/index.html#the-addition-theorem-for","scope":"tauceti/analysis/part-02/special-functions","status":"fully_proved","summary":"If `z, w \u2209 \u039b` and `\u2118(z) \u2260 \u2118(w)`, then `\u2118'(z + w) = -m (\u2118(z + w) - \u2118(z)) - \u2118'(z)` with `m = (\u2118'(z) - \u2118'(w)) / (\u2118(z) - \u2118(w))`: the points `(\u2118(z), \u2118'(z))`, `(\u2118(w), \u2118'(w))` and `(\u2118(z + w), -\u2118'(z + w))` lie on a line of slope `m`.","title":"The addition theorem for `\u2118'`"},{"id":"tauceti/analysis/part-02/special-functions/the-addition-theorem-for-2","kind":"node","lean":"PeriodPair.weierstrassP_add","parent":2198,"result":true,"route":"../roadmap/tauceti/analysis/part-02/special-functions/index.html#the-addition-theorem-for-2","scope":"tauceti/analysis/part-02/special-functions","status":"fully_proved","summary":"If `z, w \u2209 \u039b` and `\u2118(z) \u2260 \u2118(w)`, then `\u2118(z + w) = ((\u2118'(z) - \u2118'(w)) / (\u2118(z) - \u2118(w)))\u00b2 / 4 - \u2118(z) - \u2118(w)`.","title":"The addition theorem for `\u2118`"},{"id":"tauceti/analysis/part-02/special-functions/bernsteins-exponential-bound","kind":"node","lean":"Real.exp_le_one_add_add_sq_div_of_le","parent":2198,"result":true,"route":"../roadmap/tauceti/analysis/part-02/special-functions/index.html#bernsteins-exponential-bound","scope":"tauceti/analysis/part-02/special-functions","status":"fully_proved","summary":"If `x \u2264 c` with `0 \u2264 c < 3`, then `exp x \u2264 1 + x + x ^ 2 / (2 * (1 - c / 3))`. Only an upper bound on `x` is needed: negative `x` are covered by `Real.exp_le_one_add_add_sq_div_two_of_nonpos`.","title":"Bernstein's exponential bound"},{"id":"tauceti/analysis/part-02/special-functions/the-duplication-theorem-for-r-f-dlmf-19-26-18-at","kind":"node","lean":"Complex.carlsonRF_ofReal_duplication","parent":2198,"result":true,"route":"../roadmap/tauceti/analysis/part-02/special-functions/index.html#the-duplication-theorem-for-r-f-dlmf-19-26-18-at","scope":"tauceti/analysis/part-02/special-functions","status":"fully_proved","summary":"for real `x, y, z \u2265 0`, with `\u03bb = \u221ax \u221ay + \u221ay \u221az + \u221az \u221ax`, `R_F(x, y, z) = R_F((x + \u03bb)/4, (y + \u03bb)/4, (z + \u03bb)/4)`.","title":"The duplication theorem for `R_F` (DLMF 19.26.18) at nonnegative real arguments"},{"id":"tauceti/analysis/part-02/special-functions/eulers-limit-formula-in-ratio-form","kind":"node","lean":"Real.isEquivalent_Gamma_natCast_add_one_add","parent":2198,"result":true,"route":"../roadmap/tauceti/analysis/part-02/special-functions/index.html#eulers-limit-formula-in-ratio-form","scope":"tauceti/analysis/part-02/special-functions","status":"fully_proved","summary":"For every real `s`, `\u0393(n + 1 + s)` is asymptotic to `n ^ s * n!` as the natural number `n` tends to infinity.","title":"Euler's limit formula in ratio form"},{"id":"tauceti/analysis/part-02/special-functions/fatous-lemma-on-circles","kind":"node","lean":"TauCeti.volume_setOf_tendsto_circleMap_nhdsNE_zero","parent":2198,"result":true,"route":"../roadmap/tauceti/analysis/part-02/special-functions/index.html#fatous-lemma-on-circles","scope":"tauceti/analysis/part-02/special-functions","status":"fully_proved","summary":"If `log\u207a \u2016f\u2016\u207b\u00b9` is integrable on the circles about `c` of radii `\u03c1 n`, and its circle averages there are at most `B`, then the angles in one period along which `f (circleMap c (\u03c1 n) \u03b8)` tends to `0` through nonzero values form a null set.","title":"Fatou's lemma on circles"},{"id":"tauceti/analysis/part-02/special-functions/gausss-identity","kind":"node","lean":"TauCeti.Real.lemniscateConstant_eq_pi_div_agm","parent":2198,"result":true,"route":"../roadmap/tauceti/analysis/part-02/special-functions/index.html#gausss-identity","scope":"tauceti/analysis/part-02/special-functions","status":"fully_proved","summary":"the lemniscate constant is `\u03d6 = \u03c0 / M(1, \u221a2)`, where `M` is the arithmetic-geometric mean `NNReal.agm`.","title":"Gauss's identity"},{"id":"tauceti/analysis/part-02/special-functions/gausss-theorem","kind":"node","lean":"TauCeti.Real.legendreK_eq_pi_div_two_mul_agm","parent":2198,"result":true,"route":"../roadmap/tauceti/analysis/part-02/special-functions/index.html#gausss-theorem","scope":"tauceti/analysis/part-02/special-functions","status":"fully_proved","summary":"for `k ^ 2 < 1`, the complete elliptic integral of the first kind is `K(k) = \u03c0 / (2 M(1, k'))`, where `M` is the arithmetic-geometric mean and `k' = \u221a(1 - k\u00b2)` is the complementary modulus.","title":"Gauss's theorem"},{"id":"tauceti/analysis/part-02/special-functions/the-gauss-brent-salamin-formula","kind":"node","lean":"TauCeti.Real.pi_eq_four_mul_agm_sq_div_one_sub_tsum","parent":2198,"result":true,"route":"../roadmap/tauceti/analysis/part-02/special-functions/index.html#the-gauss-brent-salamin-formula","scope":"tauceti/analysis/part-02/special-functions","status":"fully_proved","summary":"with the AGM iteration `a\u2080 = 1`, `g\u2080 = 1 / \u221a2` and `c\u2c7c\u00b2 = a\u2c7c\u00b2 - g\u2c7c\u00b2`, `\u03c0 = 4 M(1, 1/\u221a2)\u00b2 / (1 - \u2211_{j \u2265 1} 2^(j+1) c\u2c7c\u00b2)`. Mathlib's `agmSequences 1 (\u221a2)\u207b\u00b9 n` is the pair `(g\u2099\u208a\u2081, a\u2099\u208a\u2081)`, so the series index is `n = j - 1`.","title":"The Gauss\u2013Brent\u2013Salamin formula"},{"id":"tauceti/analysis/part-02/special-functions/jordans-integral-bound","kind":"node","lean":"TauCeti.integral_exp_neg_mul_sin_lt","parent":2198,"result":true,"route":"../roadmap/tauceti/analysis/part-02/special-functions/index.html#jordans-integral-bound","scope":"tauceti/analysis/part-02/special-functions","status":"fully_proved","summary":"For `c > 0`, `\u222b\u2080^\u03c0 e^{-c sin \u03b8} d\u03b8 < \u03c0 / c`.","title":"Jordan's integral bound"},{"id":"tauceti/analysis/part-02/special-functions/peetres-inequality-tauceti-one-add-norm-add-sq-r","kind":"node","lean":"TauCeti.one_add_norm_add_sq_rpow_le_mul, TauCeti.one_add_norm_sq_rpow_le_mul_sub, TauCeti.one_add_norm_add_sq_rpow_le","parent":2198,"result":true,"route":"../roadmap/tauceti/analysis/part-02/special-functions/index.html#peetres-inequality-tauceti-one-add-norm-add-sq-r","scope":"tauceti/analysis/part-02/special-functions","status":"fully_proved","summary":"For every real `s`, the Japanese bracket `\u27e8x\u27e9 = (1 + \u2016x\u2016 ^ 2) ^ (1/2)` satisfies `\u27e8x + y\u27e9 ^ s \u2264 2 ^ (|s| / 2) \u27e8x\u27e9 ^ s \u27e8y\u27e9 ^ |s|`.","title":"Peetre's inequality (`TauCeti.one_add_norm_add_sq_rpow_le_mul`)"},{"id":"tauceti/analysis/part-02/special-functions/salamins-formula-for-complementary-moduli","kind":"node","lean":"TauCeti.Real.pi_eq_four_mul_agm_mul_agm_div_one_sub_tsum","parent":2198,"result":true,"route":"../roadmap/tauceti/analysis/part-02/special-functions/index.html#salamins-formula-for-complementary-moduli","scope":"tauceti/analysis/part-02/special-functions","status":"fully_proved","summary":"if `x, y > 0` and `x\u00b2 + y\u00b2 = 1`, then `\u03c0 = 4 M(1, x) M(1, y) / (1 - \u2211\u2099 2\u207f\u207a\u00b9 (c\u2099\u208a\u2081(x)\u00b2 + c\u2099\u208a\u2081(y)\u00b2))`, where `M` is the AGM and `c\u2099\u208a\u2081(x)\u00b2 = a\u2099\u208a\u2081\u00b2 - g\u2099\u208a\u2081\u00b2` is the squared correction of the AGM iteration started at `(1, x)`. Mathlib's `agmSequences 1 x n` is the pair `(g\u2099\u208a\u2081, a\u2099\u208a\u2081)`.","title":"Salamin's formula for complementary moduli"},{"id":"tauceti/analysis/part-02/special-functions/the-scalar-bellman-majorization-estimate","kind":"node","lean":"TauCeti.mul_burkholderPhi_le","parent":2198,"result":true,"route":"../roadmap/tauceti/analysis/part-02/special-functions/index.html#the-scalar-bellman-majorization-estimate","scope":"tauceti/analysis/part-02/special-functions","status":"fully_proved","summary":"For `2 \u2264 p` and `x, y \u2265 0`, `((p - 1) ^ (p - 1) / p ^ (p - 2)) \u03a6_p(x, y) \u2264 (p - 1) ^ p x ^ p - y ^ p`.","title":"The scalar Bellman majorization estimate"},{"id":"tauceti/analysis/part-02/special-functions/the-scalar-bellman-midpoint-estimate","kind":"node","lean":"TauCeti.two_mul_burkholderPhi_abs_le","parent":2198,"result":true,"route":"../roadmap/tauceti/analysis/part-02/special-functions/index.html#the-scalar-bellman-midpoint-estimate","scope":"tauceti/analysis/part-02/special-functions","status":"fully_proved","summary":"For `2 \u2264 p`, real `x`, `y`, `a` and `\u03b5` with `|\u03b5| = 1`, `2 \u03a6_p(|x|, |y|) \u2264 \u03a6_p(|x + a|, |y + \u03b5 a|) + \u03a6_p(|x - a|, |y - \u03b5 a|)`.","title":"The scalar Bellman midpoint estimate"},{"id":"tauceti/analysis/part-02/special-functions/the-uniformization-theorem","kind":"node","lean":"PeriodPair.exists_g\u2082_g\u2083_eq","parent":2198,"result":true,"route":"../roadmap/tauceti/analysis/part-02/special-functions/index.html#the-uniformization-theorem","scope":"tauceti/analysis/part-02/special-functions","status":"fully_proved","summary":"for all `a, b \u2208 \u2102` with `a\u00b3 - 27 b\u00b2 \u2260 0` there is a lattice `\u039b` with `g\u2082(\u039b) = a` and `g\u2083(\u039b) = b`.","title":"The uniformization theorem"},{"data_url":"scopes/tauceti/analysis/part-02/other-results.json","id":"tauceti/analysis/part-02/other-results","kind":"scope","parent":1841,"result":false,"route":"../roadmap/tauceti/analysis/part-02/other-results/index.html","scope":"tauceti/analysis/part-02/other-results","status":"fully_proved","summary":"Results named in the rest of Tau Ceti's `TauCeti.Analysis` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Other results"},{"id":"tauceti/analysis/part-02/other-results/the-bicommutant-theorem","kind":"node","lean":"StarSubalgebra.double_commutant_iff_isClosed_image_ofCLM","parent":2217,"result":true,"route":"../roadmap/tauceti/analysis/part-02/other-results/index.html#the-bicommutant-theorem","scope":"tauceti/analysis/part-02/other-results","status":"fully_proved","summary":"a unital star subalgebra equals its double commutant if and only if it is weak operator closed.","title":"The bicommutant theorem"},{"id":"tauceti/analysis/part-02/other-results/the-bipolar-theorem","kind":"node","lean":"AbsConvex.bipolar_eq_closure","parent":2217,"result":true,"route":"../roadmap/tauceti/analysis/part-02/other-results/index.html#the-bipolar-theorem","scope":"tauceti/analysis/part-02/other-results","status":"fully_proved","summary":"the bipolar of a nonempty absolutely convex set equals its closure in the original locally convex topology.","title":"The bipolar theorem"},{"id":"tauceti/analysis/part-02/other-results/the-bipolar-theorem-for-an-arbitrary-set","kind":"node","lean":"StrongDual.bipolar_eq_closedAbsConvexHull","parent":2217,"result":true,"route":"../roadmap/tauceti/analysis/part-02/other-results/index.html#the-bipolar-theorem-for-an-arbitrary-set","scope":"tauceti/analysis/part-02/other-results","status":"fully_proved","summary":"Its bipolar is the closed absolutely convex hull of the set together with zero, including when the set is empty.","title":"The bipolar theorem for an arbitrary set"},{"id":"tauceti/analysis/part-02/other-results/discrete-gronwall-bound-under-coercivity","kind":"node","lean":"TauCeti.sq_le_mul_one_add_div_pow_of_sq_le_mul_sub","parent":2217,"result":true,"route":"../roadmap/tauceti/analysis/part-02/other-results/index.html#discrete-gronwall-bound-under-coercivity","scope":"tauceti/analysis/part-02/other-results","status":"fully_proved","summary":"Let `D n \u2265 0` with `D 0 = 0` grow by steps `D (n + 1) \u2264 D n + d n` whose squared lengths are paid for by an energy `e`, `d n \u00b2 \u2264 2\u03c4 (e n - e (n + 1))`, and let the energy satisfy the coercivity bound `2\u03c4' (A - e n) \u2264 D n \u00b2`, where `4\u03c4 \u2264 \u03c4'`. Then `D n \u00b2 \u2264 2\u03c4' (e 0 - A) (1 + 8\u03c4 / \u03c4')\u207f`.","title":"Discrete Gr\u00f6nwall bound under coercivity"},{"id":"tauceti/analysis/part-02/other-results/evertses-lemma","kind":"node","lean":"AbsoluteValue.exists_evertse_of_approx","parent":2217,"result":true,"route":"../roadmap/tauceti/analysis/part-02/other-results/index.html#evertses-lemma","scope":"tauceti/analysis/part-02/other-results","status":"fully_proved","summary":"Evertse's lemma (Evertse; Bombieri\u2013Gubler, Lemma 7.5.29), for a field `K` with a set `T` of absolute values at which a subset `R` approximates every family of targets simultaneously, `v (\u03be + \u03b3 v) \u2264 A v`. Let `x 0, \u2026, x (m - 1)` be linearly independent in `V`, and at each `v \u2208 T` let the forms `L v k` separate the points of `V`, with weights `\u03bd v k > 0` and a nondecreasing `\u03bc v` bounding\u2026","title":"Evertse's lemma"},{"id":"tauceti/analysis/part-02/other-results/first-order-taylor-bound-for-a-holder-derivative","kind":"node","lean":"HolderOnWith.norm_image_sub_sub_le_of_hasFDerivWithinAt","parent":2217,"result":true,"route":"../roadmap/tauceti/analysis/part-02/other-results/index.html#first-order-taylor-bound-for-a-holder-derivative","scope":"tauceti/analysis/part-02/other-results","status":"fully_proved","summary":"If `g` has derivative `g'` within `s` and `g'` is `\u03b1`-H\u00f6lder with constant `H` on `s`, then for a segment from `x` to `x + w` contained in `s`, the error of the linear approximation is at most `H \u2016w\u2016^(1 + \u03b1)`.","title":"First-order Taylor bound for a H\u00f6lder derivative"},{"id":"tauceti/analysis/part-02/other-results/flux-divergence-form-of-greens-theorem-for-a-sim","kind":"node","lean":"TauCeti.Green.green_ac_flux_jordan","parent":2217,"result":true,"route":"../roadmap/tauceti/analysis/part-02/other-results/index.html#flux-divergence-form-of-greens-theorem-for-a-sim","scope":"tauceti/analysis/part-02/other-results","status":"fully_proved","summary":"the boundary integral of `X dy - Y dx` equals `Contour.jordanSign \u03b3 a b \u2208 {1, -1}` times the divergence integral over `filledHull (\u03b3 '' uIcc a b) \\ (\u03b3 '' uIcc a b)`.","title":"Flux-divergence form of Green's theorem for a simple closed absolutely continuous curve"},{"id":"tauceti/analysis/part-02/other-results/the-gagliardo-nirenberg-sobolev-inequality-for-i","kind":"node","lean":"TauCeti.eLpNorm_le_eLpNorm_fderiv_one_of_integrable","parent":2217,"result":true,"route":"../roadmap/tauceti/analysis/part-02/other-results/index.html#the-gagliardo-nirenberg-sobolev-inequality-for-i","scope":"tauceti/analysis/part-02/other-results","status":"fully_proved","summary":"Let `E` have dimension `n \u2265 2` and let `p = n / (n - 1)`. Every integrable `C\u00b9` function `u` satisfies `\u2016u\u2016_p \u2264 C \u2016Du\u2016\u2081`, with `\u2016Du\u2016\u2081` finite or infinite, where `C` is the constant `MeasureTheory.eLpNormLESNormFDerivOneConst \u03bc p` of the compactly supported case `MeasureTheory.eLpNorm_le_eLpNorm_fderiv_one`.","title":"The Gagliardo\u2013Nirenberg\u2013Sobolev inequality for integrable functions"},{"id":"tauceti/analysis/part-02/other-results/greens-theorem-for-a-simple-closed-absolutely-co","kind":"node","lean":"TauCeti.Green.green_ac_jordan","parent":2217,"result":true,"route":"../roadmap/tauceti/analysis/part-02/other-results/index.html#greens-theorem-for-a-simple-closed-absolutely-co","scope":"tauceti/analysis/part-02/other-results","status":"fully_proved","summary":"with no supplied index hypothesis, both the boundary circulation and the interior curl are integrable and the circulation equals `Contour.jordanSign \u03b3 a b \u2208 {1, -1}` times the integral of `curl P Q` over the bounded inside `filledHull (\u03b3 '' uIcc a b) \\ (\u03b3 '' uIcc a b)`.","title":"Green's theorem for a simple closed absolutely continuous curve"},{"id":"tauceti/analysis/part-02/other-results/holders-inequality-for-unconditional-sums-in-r0","kind":"node","lean":"ENNReal.inner_le_Lp_mul_Lq_tsum","parent":2217,"result":true,"route":"../roadmap/tauceti/analysis/part-02/other-results/index.html#holders-inequality-for-unconditional-sums-in-r0","scope":"tauceti/analysis/part-02/other-results","status":"fully_proved","summary":"for conjugate exponents `p` and `q`, `\u2211' i, f i * g i \u2264 (\u2211' i, f i ^ p) ^ (1 / p) * (\u2211' i, g i ^ q) ^ (1 / q)`. No summability hypotheses are needed; both sides may be infinite.","title":"H\u00f6lder's inequality for unconditional sums in `\u211d\u22650\u221e`"},{"id":"tauceti/analysis/part-02/other-results/jensens-inequality-for-convolution","kind":"node","lean":"ConvexOn.ofReal_map_norm_convolution_le_lintegral","parent":2217,"result":true,"route":"../roadmap/tauceti/analysis/part-02/other-results/index.html#jensens-inequality-for-convolution","scope":"tauceti/analysis/part-02/other-results","status":"fully_proved","summary":"If `\u03a6` is convex and monotone on `[0, \u221e)` and `\u03c1` is a probability kernel, then at every point `\u03a6(\u2016(\u03c1 \u22c6 g)(x)\u2016) \u2264 \u222b \u03c1(t) \u03a6(\u2016g(x - t)\u2016) dt`, in extended nonnegative form.","title":"Jensen's inequality for convolution"},{"id":"tauceti/analysis/part-02/other-results/minkowskis-inequality-for-unconditional-sums-in","kind":"node","lean":"ENNReal.Lp_add_le_tsum","parent":2217,"result":true,"route":"../roadmap/tauceti/analysis/part-02/other-results/index.html#minkowskis-inequality-for-unconditional-sums-in","scope":"tauceti/analysis/part-02/other-results","status":"fully_proved","summary":"for `1 \u2264 p`, `(\u2211' i, (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211' i, f i ^ p) ^ (1 / p) + (\u2211' i, g i ^ p) ^ (1 / p)`. No summability hypotheses are needed; both sides may be infinite.","title":"Minkowski's inequality for unconditional sums in `\u211d\u22650\u221e`"},{"id":"tauceti/analysis/part-02/other-results/stampacchias-lemma","kind":"node","lean":"TauCeti.apply_add_eq_zero_of_le_div_sub_rpow_mul_rpow","parent":2217,"result":true,"route":"../roadmap/tauceti/analysis/part-02/other-results/index.html#stampacchias-lemma","scope":"tauceti/analysis/part-02/other-results","status":"fully_proved","summary":"Let `\u03c6` be nonnegative and antitone on `[k\u2080, \u221e)`, with","title":"Stampacchia's lemma"},{"id":"tauceti/analysis/part-02/other-results/the-von-neumann-density-theorem","kind":"node","lean":"StarSubalgebra.exists_forall_norm_sub_apply_lt_of_mem_centralizer_centralizer, VonNeumannAlgebra.exists_forall_norm_sub_apply_lt_of_mem_adjoin, StarSubalgebra.apply_mem_closure_of_mem_centralizer_centralizer","parent":2217,"result":true,"route":"../roadmap/tauceti/analysis/part-02/other-results/index.html#the-von-neumann-density-theorem","scope":"tauceti/analysis/part-02/other-results","status":"fully_proved","summary":"An operator `x` in the double commutant of a unital star subalgebra `S` of the bounded operators on a Hilbert space is a strong-operator limit of elements of `S`: for every finite family of vectors `v i` and every `\u03b5 > 0` there is `s \u2208 S` with `\u2016x (v i) - s (v i)\u2016 < \u03b5` for all `i`.","title":"The von Neumann density theorem"},{"id":"tauceti/analysis/part-02/other-results/youngs-inequality-l1-l2-l2","kind":"node","lean":"MeasureTheory.lintegral_lconvolution_sq_le","parent":2217,"result":true,"route":"../roadmap/tauceti/analysis/part-02/other-results/index.html#youngs-inequality-l1-l2-l2","scope":"tauceti/analysis/part-02/other-results","status":"fully_proved","summary":"For extended nonnegative functions on a group with a left-invariant measure, `\u222b\u207b (f \u22c6\u2097[\u03bc] g) ^ 2 \u2264 (\u222b\u207b f) ^ 2 * \u222b\u207b g ^ 2`.","title":"Young's inequality `L\u00b9 \u22c6 L\u00b2 \u2286 L\u00b2`"},{"id":"tauceti/analysis/part-02/other-results/the-zero-spatial-bcr-theorem-is-the-hausdorff-be","kind":"node","lean":"TauCeti.bcr_zero_spatial_iff_hausdorff_bernstein_widder","parent":2217,"result":true,"route":"../roadmap/tauceti/analysis/part-02/other-results/index.html#the-zero-spatial-bcr-theorem-is-the-hausdorff-be","scope":"tauceti/analysis/part-02/other-results","status":"fully_proved","summary":"For a real function `f`, the bounded, continuous positive-definite condition on `(t, _) \u21a6 f t : \u211d\u22650 \u00d7 PUnit \u2192 \u2102` holds exactly when `f` is continuous on `[0, \u221e)` and completely monotone on `(0, \u221e)`.","title":"The zero-spatial BCR theorem is the Hausdorff--Bernstein--Widder theorem"},{"data_url":"scopes/tauceti/category-theory.json","id":"tauceti/category-theory","kind":"scope","parent":1095,"result":false,"route":"../roadmap/tauceti/category-theory/index.html","scope":"tauceti/category-theory","status":"fully_proved","summary":"Results in Tau Ceti's `TauCeti.CategoryTheory` modules, and the roadmaps whose articles are mostly there: by roadmap, then by area.","title":"Category Theory"},{"data_url":"scopes/tauceti/category-theory/geometric-representation-categories-and-coherent.json","id":"tauceti/category-theory/geometric-representation-categories-and-coherent","kind":"scope","parent":2234,"result":false,"route":"../roadmap/tauceti/category-theory/geometric-representation-categories-and-coherent/index.html","scope":"tauceti/category-theory/geometric-representation-categories-and-coherent","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Geometric representation categories and coherent local transport"},{"id":"tauceti/category-theory/geometric-representation-categories-and-coherent/categorytheory-monoidalcategory-monoidalleftacti","kind":"node","lean":"CategoryTheory.MonoidalCategory.MonoidalLeftAction.pseudofunctorEquiv","parent":2235,"result":true,"route":"../roadmap/tauceti/category-theory/geometric-representation-categories-and-coherent/index.html#categorytheory-monoidalcategory-monoidalleftacti","scope":"tauceti/category-theory/geometric-representation-categories-and-coherent","status":"fully_proved","summary":"Left actions of `Discrete M` on categories correspond bijectively to pseudofunctors `LocallyDiscrete (SingleObj M) \u2964\u1d56 Cat`: an action on `C` is sent to `pseudofunctor M C`, and a pseudofunctor `F` to the action `F.monoidalLeftAction` on its value at the unique object.","title":"CategoryTheory.MonoidalCategory.MonoidalLeftAction.pseudofunctorEquiv"},{"id":"tauceti/category-theory/geometric-representation-categories-and-coherent/categorytheory-monoidalcategory-monoidalleftacti-2","kind":"node","lean":"CategoryTheory.MonoidalCategory.MonoidalLeftAction.strongTransEquiv","parent":2235,"result":true,"route":"../roadmap/tauceti/category-theory/geometric-representation-categories-and-coherent/index.html#categorytheory-monoidalcategory-monoidalleftacti-2","scope":"tauceti/category-theory/geometric-representation-categories-and-coherent","status":"fully_proved","summary":"Equivariant functors are the strong transformations between the pseudofunctors attached to the actions: functors `C \u2964 D` equipped with a left-linear structure for the actions of `Discrete M` correspond to strong transformations `pseudofunctor M C \u27f6 pseudofunctor M D`. The correspondence keeps the lineators: the naturality isomorphisms of `strongTrans M F` have components the lineators of `F`.","title":"CategoryTheory.MonoidalCategory.MonoidalLeftAction.strongTransEquiv"},{"id":"tauceti/category-theory/geometric-representation-categories-and-coherent/categorytheory-monoidalcategory-monoidalleftacti-3","kind":"node","lean":"CategoryTheory.MonoidalCategory.MonoidalLeftAction.modificationEquiv","parent":2235,"result":true,"route":"../roadmap/tauceti/category-theory/geometric-representation-categories-and-coherent/index.html#categorytheory-monoidalcategory-monoidalleftacti-3","scope":"tauceti/category-theory/geometric-representation-categories-and-coherent","status":"fully_proved","summary":"Equivariant natural transformations `F\u2081 \u27f6 F\u2082` correspond to modifications `strongTrans M F\u2081 \u27f6 strongTrans M F\u2082`.","title":"CategoryTheory.MonoidalCategory.MonoidalLeftAction.modificationEquiv"},{"id":"tauceti/category-theory/geometric-representation-categories-and-coherent/tauceti-centralunitiso-associativity-iff","kind":"node","lean":"TauCeti.centralUnitIso_associativity_iff","parent":2235,"result":true,"route":"../roadmap/tauceti/category-theory/geometric-representation-categories-and-coherent/index.html#tauceti-centralunitiso-associativity-iff","scope":"tauceti/category-theory/geometric-representation-categories-and-coherent","status":"fully_proved","summary":"The associativity condition of a pseudofunctor out of `LocallyDiscrete (SingleObj G)` with identity `1`-cells and central compositors `centralUnitIso C (c (h, g))` at `g \u226b h = h * g`, evaluated at `(g, h, j)`, is the `2`-cocycle identity `c (j * h, g) * c (j, h) = c (h, g) * c (j, h * g)` of `c` at `(j, h, g)`.","title":"TauCeti.centralUnitIso_associativity_iff"},{"id":"tauceti/category-theory/geometric-representation-categories-and-coherent/tauceti-centralunitiso-naturality-comp-iff","kind":"node","lean":"TauCeti.centralUnitIso_naturality_comp_iff","parent":2235,"result":true,"route":"../roadmap/tauceti/category-theory/geometric-representation-categories-and-coherent/index.html#tauceti-centralunitiso-naturality-comp-iff","scope":"tauceti/category-theory/geometric-representation-categories-and-coherent","status":"fully_proved","summary":"The composition condition of a strong transformation with identity components and naturality `2`-cells `centralUnitIso C (b g)`, between pseudofunctors with identity `1`-cells and central compositors `centralUnitIso C (c (h, g))` and `centralUnitIso C (c' (h, g))` at `g \u226b h = h * g`, evaluated at `(g, h)`, states that `c'` is `c` multiplied by the coboundary of `b` at `(h, g)`.","title":"TauCeti.centralUnitIso_naturality_comp_iff"},{"id":"tauceti/category-theory/geometric-representation-categories-and-coherent/tauceti-derivationextension-innerequiv","kind":"node","lean":"TauCeti.DerivationExtension.innerEquiv","parent":2235,"result":true,"route":"../roadmap/tauceti/category-theory/geometric-representation-categories-and-coherent/index.html#tauceti-derivationextension-innerequiv","scope":"tauceti/category-theory/geometric-representation-categories-and-coherent","status":"fully_proved","summary":"Changing a derivation by an inner derivation does not change its self-extension: when `\u03b4' = \u03b4 + ad b`, the map `(u, v) \u21a6 (u + b \u2022 v, v)` is an isomorphism `E_\u03b4(M) \u2243 E_\u03b4'(M)`.","title":"TauCeti.DerivationExtension.innerEquiv"},{"id":"tauceti/category-theory/geometric-representation-categories-and-coherent/tauceti-isobjectwisecongruenceinvariant-iff-of-e","kind":"node","lean":"TauCeti.isObjectwiseCongruenceInvariant_iff_of_equivalence","parent":2235,"result":true,"route":"../roadmap/tauceti/category-theory/geometric-representation-categories-and-coherent/index.html#tauceti-isobjectwisecongruenceinvariant-iff-of-e","scope":"tauceti/category-theory/geometric-representation-categories-and-coherent","status":"fully_proved","summary":"Objectwise congruence invariance is independent of an equivariant presentation of the category. Only the forward functor of the equivalence needs a specified lineator.","title":"TauCeti.isObjectwiseCongruenceInvariant_iff_of_equivalence"},{"id":"tauceti/category-theory/geometric-representation-categories-and-coherent/monoidhom-restrictscalarsmonoidalleftaction","kind":"node","lean":"MonoidHom.restrictScalarsMonoidalLeftAction","parent":2235,"result":true,"route":"../roadmap/tauceti/category-theory/geometric-representation-categories-and-coherent/index.html#monoidhom-restrictscalarsmonoidalleftaction","scope":"tauceti/category-theory/geometric-representation-categories-and-coherent","status":"fully_proved","summary":"The left action of a monoid `H` on `ModuleCat A` by twisting along a homomorphism `a : H \u2192* RingAut A`: an element `h` acts by restricting scalars along `(a h)\u207b\u00b9`. The associator `(g * h) \u2299\u2097 M \u2245 g \u2299\u2097 h \u2299\u2097 M` and the unitor `1 \u2299\u2097 M \u2245 M` are the comparison isomorphisms of restriction of scalars, with identity underlying linear maps.","title":"MonoidHom.restrictScalarsMonoidalLeftAction"},{"id":"tauceti/category-theory/geometric-representation-categories-and-coherent/tauceti-derivationextension","kind":"node","lean":"TauCeti.DerivationExtension","parent":2235,"result":true,"route":"../roadmap/tauceti/category-theory/geometric-representation-categories-and-coherent/index.html#tauceti-derivationextension","scope":"tauceti/category-theory/geometric-representation-categories-and-coherent","status":"fully_proved","summary":"The self-extension `E_\u03b4(M)` of a module `M` attached to a derivation `\u03b4`: pairs `(u, v)` of elements of `M`, on which `a` acts by `a \u2022 (u, v) = (a \u2022 u + \u03b4 a \u2022 v, a \u2022 v)`. The first coordinate carries the submodule `M` and the second the quotient `M`.","title":"TauCeti.DerivationExtension"},{"data_url":"scopes/tauceti/category-theory/pivotal-and-spherical-categories.json","id":"tauceti/category-theory/pivotal-and-spherical-categories","kind":"scope","parent":2234,"result":false,"route":"../roadmap/tauceti/category-theory/pivotal-and-spherical-categories/index.html","scope":"tauceti/category-theory/pivotal-and-spherical-categories","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Pivotal and spherical categories"},{"id":"tauceti/category-theory/pivotal-and-spherical-categories/the-global-dimension-of-a-pivotal-fusion-categor","kind":"node","lean":"TauCeti.globalDim_ne_zero","parent":2245,"result":true,"route":"../roadmap/tauceti/category-theory/pivotal-and-spherical-categories/index.html#the-global-dimension-of-a-pivotal-fusion-categor","scope":"tauceti/category-theory/pivotal-and-spherical-categories","status":"fully_proved","summary":"Here `C` is pivotal, `k`-linear abelian and finite semisimple over an algebraically closed field `k` of characteristic zero, with finite-dimensional morphism spaces, simple tensor unit, and finitely many simple classes.","title":"The global dimension of a pivotal fusion category over a field of characteristic zero is nonzero (Etingof\u2013Nikshych\u2013Ostrik)"},{"id":"tauceti/category-theory/pivotal-and-spherical-categories/characters-of-the-grothendieck-ring-are-characte","kind":"node","lean":"TauCeti.SplitK0.ringHom_apply_dual","parent":2245,"result":true,"route":"../roadmap/tauceti/category-theory/pivotal-and-spherical-categories/index.html#characters-of-the-grothendieck-ring-are-characte","scope":"tauceti/category-theory/pivotal-and-spherical-categories","status":"fully_proved","summary":"A ring homomorphism `\u03c8` from the Grothendieck ring of a fusion category to `\u211d` or `\u2102` intertwines the duality `[X] \u21a6 [X\u1601]` with complex conjugation.","title":"Characters of the Grothendieck ring are \u22c6-characters"},{"id":"tauceti/category-theory/pivotal-and-spherical-categories/the-frobenius-perron-character","kind":"node","lean":"Module.Basis.existsUnique_ringHom_nonneg","parent":2245,"result":true,"route":"../roadmap/tauceti/category-theory/pivotal-and-spherical-categories/index.html#the-frobenius-perron-character","scope":"tauceti/category-theory/pivotal-and-spherical-categories","status":"fully_proved","summary":"A ring with a finite nonempty `\u2124`-basis with nonnegative structure constants, such that every basis element occurs in `b j * \u2211 i, b i` for every `j`, has exactly one character to `\u211d` which is nonnegative on the basis.","title":"The Frobenius\u2013Perron character"},{"id":"tauceti/category-theory/pivotal-and-spherical-categories/tauceti-splitk0-spectralradius-fusionmatrix","kind":"node","lean":"TauCeti.SplitK0.spectralRadius_fusionMatrix","parent":2245,"result":true,"route":"../roadmap/tauceti/category-theory/pivotal-and-spherical-categories/index.html#tauceti-splitk0-spectralradius-fusionmatrix","scope":"tauceti/category-theory/pivotal-and-spherical-categories","status":"fully_proved","summary":"The value of the Frobenius\u2013Perron character on a simple class `S` is the spectral radius of the fusion matrix of `S`.","title":"TauCeti.SplitK0.spectralRadius_fusionMatrix"},{"id":"tauceti/category-theory/pivotal-and-spherical-categories/the-global-dimension-does-not-depend-on-the-pivo","kind":"node","lean":"TauCeti.globalDim_pivotalCategory_eq","parent":2245,"result":true,"route":"../roadmap/tauceti/category-theory/pivotal-and-spherical-categories/index.html#the-global-dimension-does-not-depend-on-the-pivo","scope":"tauceti/category-theory/pivotal-and-spherical-categories","status":"fully_proved","summary":"over an algebraically closed field `k`, if the endomorphisms of each simple object are finite-dimensional, then any two pivotal structures `P` and `Q` on `C` give the same global dimension.","title":"The global dimension does not depend on the pivotal structure"},{"id":"tauceti/category-theory/pivotal-and-spherical-categories/tauceti-symmetricsphericalcategory","kind":"node","lean":"TauCeti.symmetricSphericalCategory","parent":2245,"result":true,"route":"../roadmap/tauceti/category-theory/pivotal-and-spherical-categories/index.html#tauceti-symmetricsphericalcategory","scope":"tauceti/category-theory/pivotal-and-spherical-categories","status":"fully_proved","summary":"The canonical pivotal structure on a symmetric rigid category is spherical.","title":"TauCeti.symmetricSphericalCategory"},{"id":"tauceti/category-theory/pivotal-and-spherical-categories/fdrep-trace-eq-smul","kind":"node","lean":"FDRep.trace_eq_smul","parent":2245,"result":true,"route":"../roadmap/tauceti/category-theory/pivotal-and-spherical-categories/index.html#fdrep-trace-eq-smul","scope":"tauceti/category-theory/pivotal-and-spherical-categories","status":"fully_proved","summary":"The categorical trace of an equivariant endomorphism of a finite-dimensional representation is its linear trace, acting on the trivial representation by scalar multiplication.","title":"FDRep.trace_eq_smul"},{"id":"tauceti/category-theory/pivotal-and-spherical-categories/tauceti-splitk0-simplebasis","kind":"node","lean":"TauCeti.SplitK0.simpleBasis","parent":2245,"result":true,"route":"../roadmap/tauceti/category-theory/pivotal-and-spherical-categories/index.html#tauceti-splitk0-simplebasis","scope":"tauceti/category-theory/pivotal-and-spherical-categories","status":"fully_proved","summary":"The classes of the simple representatives form a `\u2124`-basis of split `K\u2080`.","title":"TauCeti.SplitK0.simpleBasis"},{"id":"tauceti/category-theory/pivotal-and-spherical-categories/tauceti-fusioncoeff","kind":"node","lean":"TauCeti.fusionCoeff","parent":2245,"result":true,"route":"../roadmap/tauceti/category-theory/pivotal-and-spherical-categories/index.html#tauceti-fusioncoeff","scope":"tauceti/category-theory/pivotal-and-spherical-categories","status":"fully_proved","summary":"The fusion coefficient `N_{ST}^U` of three simple classes: the dimension over `k` of the morphisms from the tensor product of the representatives of `S` and `T` to the representative of `U`. Over an algebraically closed field, with finite-dimensional morphism spaces, it is the multiplicity of `U` in `S \u2297 T` (`TauCeti.fusionCoeff_eq_simpleMultiplicities`).","title":"TauCeti.fusionCoeff"},{"id":"tauceti/category-theory/pivotal-and-spherical-categories/tauceti-frobeniusperrondim","kind":"node","lean":"TauCeti.frobeniusPerronDim","parent":2245,"result":true,"route":"../roadmap/tauceti/category-theory/pivotal-and-spherical-categories/index.html#tauceti-frobeniusperrondim","scope":"tauceti/category-theory/pivotal-and-spherical-categories","status":"fully_proved","summary":"The Frobenius\u2013Perron dimension of an object of a fusion category: the value of the Frobenius\u2013Perron character `TauCeti.SplitK0.frobeniusPerronChar` at the class of the object.","title":"TauCeti.frobeniusPerronDim"},{"id":"tauceti/category-theory/pivotal-and-spherical-categories/tauceti-balancedcategory","kind":"node","lean":"TauCeti.BalancedCategory","parent":2245,"result":true,"route":"../roadmap/tauceti/category-theory/pivotal-and-spherical-categories/index.html#tauceti-balancedcategory","scope":"tauceti/category-theory/pivotal-and-spherical-categories","status":"fully_proved","summary":"A *balanced* structure on a braided monoidal category: a twist `\u03b8 : \ud835\udfed C \u2245 \ud835\udfed C`, a natural automorphism of the identity functor, satisfying the balancing axiom `\u03b8_{X \u2297 Y} = (\u03b8_X \u2297 \u03b8_Y) \u226b \u03b2_{X,Y} \u226b \u03b2_{Y,X}`.","title":"TauCeti.BalancedCategory"},{"id":"tauceti/category-theory/pivotal-and-spherical-categories/tauceti-pivotalbalancedequiv","kind":"node","lean":"TauCeti.pivotalBalancedEquiv","parent":2245,"result":true,"route":"../roadmap/tauceti/category-theory/pivotal-and-spherical-categories/index.html#tauceti-pivotalbalancedequiv","scope":"tauceti/category-theory/pivotal-and-spherical-categories","status":"fully_proved","summary":"Pivotal structures on a braided rigid category correspond to balanced structures via `\u03b8_X = \u03c6_X \u226b u_X\u207b\u00b9`, with inverse `\u03c6_X = \u03b8_X \u226b u_X`.","title":"TauCeti.pivotalBalancedEquiv"},{"id":"tauceti/category-theory/pivotal-and-spherical-categories/the-freyd-yetter-identity","kind":"node","lean":"TauCeti.pivotalIsoApp_rightDual","parent":2245,"result":true,"route":"../roadmap/tauceti/category-theory/pivotal-and-spherical-categories/index.html#the-freyd-yetter-identity","scope":"tauceti/category-theory/pivotal-and-spherical-categories","status":"fully_proved","summary":"the component of the pivotal isomorphism at `X\u1601` is the mate of the inverse of its component at `X`.","title":"The Freyd\u2013Yetter identity"},{"data_url":"scopes/tauceti/category-theory/exact.json","id":"tauceti/category-theory/exact","kind":"scope","parent":2234,"result":false,"route":"../roadmap/tauceti/category-theory/exact/index.html","scope":"tauceti/category-theory/exact","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.CategoryTheory.Exact` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Exact"},{"id":"tauceti/category-theory/exact/the-dual-noether-isomorphism","kind":"node","lean":"TauCeti.ExactStructure.exists_conflation_comp'","parent":2259,"result":true,"route":"../roadmap/tauceti/category-theory/exact/index.html#the-dual-noether-isomorphism","scope":"tauceti/category-theory/exact","status":"fully_proved","summary":"Given conflations `X \u27f6 Y \u27f6 Z` and `V \u27f6 W \u27f6 Y`, the pullback `Q` of `W \u27f6 Y` along `X \u27f6 Y` is at once the kernel of the composite deflation `W \u27f6 Z` and an extension of `X` by `V`.","title":"The dual Noether isomorphism"},{"id":"tauceti/category-theory/exact/the-graded-comparison-theorem","kind":"node","lean":"TauCeti.ExactStructure.FiniteResolution.liftShiftHomotopy","parent":2259,"result":true,"route":"../roadmap/tauceti/category-theory/exact/index.html#the-graded-comparison-theorem","scope":"tauceti/category-theory/exact","status":"fully_proved","summary":"The comparison map between two finite projective resolutions commutes with the grading shift up to homotopy: the lift of `f{1}` between the shifted resolutions is homotopic to the shift of the lift of `f`.","title":"The graded comparison theorem"},{"id":"tauceti/category-theory/exact/the-noether-isomorphism-for-exact-categories","kind":"node","lean":"TauCeti.ExactStructure.exists_conflation_comp","parent":2259,"result":true,"route":"../roadmap/tauceti/category-theory/exact/index.html#the-noether-isomorphism-for-exact-categories","scope":"tauceti/category-theory/exact","status":"fully_proved","summary":"Given conflations `X \u27f6 Y \u27f6 Z` and `Y \u27f6 W \u27f6 V`, there is an object `Q` \u2014 the pushout of `Y \u27f6 W` along `Y \u27f6 Z` \u2014 which is at once the cokernel of the composite inflation `X \u27f6 W` and an extension of `V` by `Z`.","title":"The Noether isomorphism for exact categories"},{"id":"tauceti/category-theory/exact/schanuels-lemma","kind":"node","lean":"TauCeti.ExactStructure.nonempty_iso_biprod_of_projective","parent":2259,"result":true,"route":"../roadmap/tauceti/category-theory/exact/index.html#schanuels-lemma","scope":"tauceti/category-theory/exact","status":"fully_proved","summary":"Two conflations `K \u21aa Q \u21a0 X` and `K' \u21aa Q' \u21a0 X` with projective middle terms have stably isomorphic kernels: `K \u229e Q' \u2245 K' \u229e Q`.","title":"Schanuel's lemma"},{"data_url":"scopes/tauceti/category-theory/grothendieck-group.json","id":"tauceti/category-theory/grothendieck-group","kind":"scope","parent":2234,"result":false,"route":"../roadmap/tauceti/category-theory/grothendieck-group/index.html","scope":"tauceti/category-theory/grothendieck-group","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.CategoryTheory.GrothendieckGroup` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Grothendieck Group"},{"id":"tauceti/category-theory/grothendieck-group/the-euler-poincare-theorem-in-abelian-k0","kind":"node","lean":"TauCeti.AbelianK0.eulerChar_eq_homologyEulerChar, TauCeti.AbelianK0.AdditiveInvariant.sum_negOnePow_obj_X_eq_sum_negOnePow_obj_homology, TauCeti.ExactK0.AdditiveInvariant.sum_negOnePow_obj_X_eq_sum_negOnePow_obj_homology, TauCeti.ExactK0.sum_negOnePow_of_X_eq_sum_negOnePow_of_homology","parent":2264,"result":true,"route":"../roadmap/tauceti/category-theory/grothendieck-group/index.html#the-euler-poincare-theorem-in-abelian-k0","scope":"tauceti/category-theory/grothendieck-group","status":"fully_proved","summary":"For a cochain complex that is strictly supported in degrees `a` to `b`, and any finite range of degrees containing `[a, b]`, the alternating sum of the classes of the terms equals the alternating sum of the classes of the cohomology objects.","title":"The Euler\u2013Poincar\u00e9 theorem in abelian `K\u2080`"},{"id":"tauceti/category-theory/grothendieck-group/the-generating-half-of-the-resolution-theorem","kind":"node","lean":"TauCeti.ExactK0.closure_propClasses_eq_top","parent":2264,"result":true,"route":"../roadmap/tauceti/category-theory/grothendieck-group/index.html#the-generating-half-of-the-resolution-theorem","scope":"tauceti/category-theory/grothendieck-group","status":"fully_proved","summary":"If every object of `C` admits a finite `P`-resolution, then the classes of the objects satisfying `P` generate exact `K\u2080`.","title":"The generating half of the resolution theorem"},{"id":"tauceti/category-theory/grothendieck-group/the-graded-resolution-theorem","kind":"node","lean":"TauCeti.GradedExactStructure.laurentResolutionEquiv, TauCeti.GradedExactStructure.IsResolving.laurentResolutionEquiv","parent":2264,"result":true,"route":"../roadmap/tauceti/category-theory/grothendieck-group/index.html#the-graded-resolution-theorem","scope":"tauceti/category-theory/grothendieck-group","status":"fully_proved","summary":"Let `P` be a class of `E`-projectives containing a zero object, closed under binary biproducts, and stable under the grading shift. Then the inclusion of `P` into the objects of finite `P`-dimension induces an isomorphism of graded Grothendieck groups as `\u2124[q,q\u207b\u00b9]`-modules. Its inverse sends the class of an object to its Euler class, by `TauCeti.GradedExactStructure.laurentResolutionEquiv_symm_of`.","title":"The graded resolution theorem"},{"id":"tauceti/category-theory/grothendieck-group/the-resolution-theorem-for-finite-projective-res","kind":"node","lean":"TauCeti.ExactStructure.resolutionEquiv","parent":2264,"result":true,"route":"../roadmap/tauceti/category-theory/grothendieck-group/index.html#the-resolution-theorem-for-finite-projective-res","scope":"tauceti/category-theory/grothendieck-group","status":"fully_proved","summary":"When every object satisfying `P` is `E`-projective, the inclusion of the full subcategory on `P` into the full subcategory of objects admitting a finite `P`-resolution induces an isomorphism of exact Grothendieck groups. Its inverse sends the class of an object to the alternating class of any finite `P`-resolution of it, by `TauCeti.ExactStructure.resolutionEquiv_symm_of`.","title":"The resolution theorem for finite projective resolutions"},{"data_url":"scopes/tauceti/category-theory/preadditive.json","id":"tauceti/category-theory/preadditive","kind":"scope","parent":2234,"result":false,"route":"../roadmap/tauceti/category-theory/preadditive/index.html","scope":"tauceti/category-theory/preadditive","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.CategoryTheory.Preadditive` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Preadditive"},{"id":"tauceti/category-theory/preadditive/jacobsons-lemma","kind":"node","lean":"CategoryTheory.isIso_id_sub_comp_comm","parent":2269,"result":true,"route":"../roadmap/tauceti/category-theory/preadditive/index.html#jacobsons-lemma","scope":"tauceti/category-theory/preadditive","status":"fully_proved","summary":"`\ud835\udfd9 X - a \u226b b` is invertible exactly when `\ud835\udfd9 Y - b \u226b a` is.","title":"Jacobson's lemma"},{"id":"tauceti/category-theory/preadditive/schurs-lemma-as-an-identification-of-algebras","kind":"node","lean":"TauCeti.endAlgEquivOfSimple","parent":2269,"result":true,"route":"../roadmap/tauceti/category-theory/preadditive/index.html#schurs-lemma-as-an-identification-of-algebras","scope":"tauceti/category-theory/preadditive","status":"fully_proved","summary":"the endomorphisms of a simple object `X` with finite-dimensional endomorphisms over an algebraically closed field `k` are the scalars `k`. This is the inverse of the structure map `algebraMap k (End X)`, `c \u21a6 c \u2022 \ud835\udfd9 X`.","title":"Schur's lemma as an identification of algebras"},{"id":"tauceti/category-theory/preadditive/tauceti-islocalring-end-of-indecomposable","kind":"node","lean":"TauCeti.isLocalRing_end_of_indecomposable","parent":2269,"result":true,"route":"../roadmap/tauceti/category-theory/preadditive/index.html#tauceti-islocalring-end-of-indecomposable","scope":"tauceti/category-theory/preadditive","status":"fully_proved","summary":"Fitting's lemma: an indecomposable object has a local endomorphism ring, in an idempotent-complete linear category over a commutative Artinian ring `R` when the endomorphisms of the object are finitely generated over `R`, for instance in a Hom-finite linear category over a field. Indecomposability leaves `0` and `1` as the only idempotents of `End X` (`TauCeti.idempotent_eq_zero_or_id_of_indecomposable`).","title":"TauCeti.isLocalRing_end_of_indecomposable"},{"data_url":"scopes/tauceti/category-theory/other-results.json","id":"tauceti/category-theory/other-results","kind":"scope","parent":2234,"result":false,"route":"../roadmap/tauceti/category-theory/other-results/index.html","scope":"tauceti/category-theory/other-results","status":"fully_proved","summary":"Results named in the rest of Tau Ceti's `TauCeti.CategoryTheory` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Other results"},{"id":"tauceti/category-theory/other-results/the-dg-yoneda-lemma","kind":"node","lean":"TauCeti.dgRepresentableHomIso, TauCeti.dgRepresentableHomEquiv","parent":2273,"result":true,"route":"../roadmap/tauceti/category-theory/other-results/index.html#the-dg-yoneda-lemma","scope":"tauceti/category-theory/other-results","status":"fully_proved","summary":"evaluation at the identity of `X` is an isomorphism from the Hom complex of right DG modules `Hom(Hom(-, X), F)` to the complex `F(X)`.","title":"The DG Yoneda lemma"},{"id":"tauceti/category-theory/other-results/frobenius-reciprocity-in-multiplicity-form","kind":"node","lean":"TauCeti.fusionCoeff_eq_simpleMultiplicities_rightDual_tensor, TauCeti.fusionCoeff_dual_left, TauCeti.fusionCoeff_dual_right, TauCeti.fusionCoeff_eq_simpleMultiplicities_tensor_rightDual","parent":2273,"result":true,"route":"../roadmap/tauceti/category-theory/other-results/index.html#frobenius-reciprocity-in-multiplicity-form","scope":"tauceti/category-theory/other-results","status":"fully_proved","summary":"`N_{S T}^U` is the multiplicity of `T` in `S\u1601 \u2297 U`, coming from `(S\u1601 \u2297 U \u27f6 T) \u2243 (U \u27f6 S \u2297 T)`.","title":"Frobenius reciprocity in multiplicity form"},{"id":"tauceti/category-theory/other-results/ng-schauenburg","kind":"node","lean":"TauCeti.iterate_frobeniusSchurRotate_eq_id","parent":2273,"result":true,"route":"../roadmap/tauceti/category-theory/other-results/index.html#ng-schauenburg","scope":"tauceti/category-theory/other-results","status":"fully_proved","summary":"the `(n + 1)`-st iterate of `E_V^{(n+1)}` is the identity.","title":"Ng\u2013Schauenburg"},{"id":"tauceti/category-theory/other-results/ng-schauenburg-trichotomy","kind":"node","lean":"TauCeti.frobeniusSchurIndicator_two_eq_one_or_eq_zero_or_eq_neg_one","parent":2273,"result":true,"route":"../roadmap/tauceti/category-theory/other-results/index.html#ng-schauenburg-trichotomy","scope":"tauceti/category-theory/other-results","status":"fully_proved","summary":"for a simple object `V` (with `V\u1601` simple and `V \u27f6 V` finite-dimensional over an algebraically closed field `k`), the second Frobenius\u2013Schur indicator takes values in `{1, 0, -1}`.","title":"Ng\u2013Schauenburg trichotomy"},{"data_url":"scopes/tauceti/combinatorics.json","id":"tauceti/combinatorics","kind":"scope","parent":1095,"result":false,"route":"../roadmap/tauceti/combinatorics/index.html","scope":"tauceti/combinatorics","status":"fully_proved","summary":"Results in Tau Ceti's `TauCeti.Combinatorics` modules, and the roadmaps whose articles are mostly there: by roadmap, then by area.","title":"Combinatorics"},{"data_url":"scopes/tauceti/combinatorics/dense-graph-limits-and-graphons.json","id":"tauceti/combinatorics/dense-graph-limits-and-graphons","kind":"scope","parent":2278,"result":false,"route":"../roadmap/tauceti/combinatorics/dense-graph-limits-and-graphons/index.html","scope":"tauceti/combinatorics/dense-graph-limits-and-graphons","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Dense graph limits and graphons"},{"id":"tauceti/combinatorics/dense-graph-limits-and-graphons/the-lovasz-szegedy-characterization-of-homomorph","kind":"node","lean":"TauCeti.DenseGraphLimits.lovasz_szegedy_representability","parent":2279,"result":true,"route":"../roadmap/tauceti/combinatorics/dense-graph-limits-and-graphons/index.html#the-lovasz-szegedy-characterization-of-homomorph","scope":"tauceti/combinatorics/dense-graph-limits-and-graphons","status":"fully_proved","summary":"A graph parameter is the homomorphism density of a graphon on an atomless standard Borel carrier `(\u03a9, \u03bc)`, such as the unit interval, if and only if it is isomorphism invariant, multiplicative, normalized and reflection positive.","title":"The Lov\u00e1sz\u2013Szegedy characterization of homomorphism densities"},{"id":"tauceti/combinatorics/dense-graph-limits-and-graphons/every-exchangeable-law-on-infinite-graphs-is-a-g","kind":"node","lean":"TauCeti.DenseGraphLimits.InfiniteExchangeableGraphLaw.existsUnique_bind_infiniteSampleLawOnSpace","parent":2279,"result":true,"route":"../roadmap/tauceti/combinatorics/dense-graph-limits-and-graphons/index.html#every-exchangeable-law-on-infinite-graphs-is-a-g","scope":"tauceti/combinatorics/dense-graph-limits-and-graphons","status":"fully_proved","summary":"The integral form of the Diaconis\u2013Janson correspondence: the law is the bind of a unique probability measure on graphon space against the joint sampling laws.","title":"Every exchangeable law on infinite graphs is a graphon mixture, for exactly one mixing measure"},{"id":"tauceti/combinatorics/dense-graph-limits-and-graphons/dissociated-relational-laws-of-graphs-are-grapho","kind":"node","lean":"TauCeti.DenseGraphLimits.isDissociated_iff_exists_toRelationalLaw_sampleExchangeableLaw","parent":2279,"result":true,"route":"../roadmap/tauceti/combinatorics/dense-graph-limits-and-graphons/index.html#dissociated-relational-laws-of-graphs-are-grapho","scope":"tauceti/combinatorics/dense-graph-limits-and-graphons","status":"fully_proved","summary":"An exchangeable relational law of the language of graphs whose marginals are concentrated on simple graphs is dissociated exactly when it is the relational law of the sampling law of a graphon, which may be taken on any atomless standard Borel carrier `(\u03a9, \u03bc)`, such as the unit interval.","title":"Dissociated relational laws of graphs are graphon sampling laws"},{"id":"tauceti/combinatorics/dense-graph-limits-and-graphons/cut-distance-convergence-is-convergence-of-all-h","kind":"node","lean":"TauCeti.DenseGraphLimits.tendsto_cutDist_iff_forall_homDensity_tendsto","parent":2279,"result":true,"route":"../roadmap/tauceti/combinatorics/dense-graph-limits-and-graphons/index.html#cut-distance-convergence-is-convergence-of-all-h","scope":"tauceti/combinatorics/dense-graph-limits-and-graphons","status":"fully_proved","summary":"Cut-distance convergence is convergence of all homomorphism densities, for graphons on arbitrary, possibly different, probability carriers, indexed by any type and taken along any filter: `\u03b4\u25a1(W\u1d62, W) \u2192 0` if and only if `t(F, W\u1d62) \u2192 t(F, W)` for every finite simple graph `F`. Sequences are the case of `atTop` on `\u2115`.","title":"Cut-distance convergence is convergence of all homomorphism densities"},{"id":"tauceti/combinatorics/dense-graph-limits-and-graphons/every-convergent-graph-sequence-has-a-limit-grap","kind":"node","lean":"TauCeti.DenseGraphLimits.exists_graphon_tendsto_cutDist_finiteGraphGraphon_iff_forall_homDensityFin_cauchySeq","parent":2279,"result":true,"route":"../roadmap/tauceti/combinatorics/dense-graph-limits-and-graphons/index.html#every-convergent-graph-sequence-has-a-limit-grap","scope":"tauceti/combinatorics/dense-graph-limits-and-graphons","status":"fully_proved","summary":"Every convergent graph sequence has a limit graphon (Lov\u00e1sz--Szegedy), on every atomless standard Borel carrier `(\u03a9, \u03bc)`, such as the unit interval. A sequence of finite graphs on nonempty vertex sets converges in the cut distance of step graphons to some graphon on `(\u03a9, \u03bc)` if and only if every finite homomorphism density `t(F, G\u2099)` is a Cauchy, hence convergent, sequence.","title":"Every convergent graph sequence has a limit graphon"},{"id":"tauceti/combinatorics/dense-graph-limits-and-graphons/lovasz-szegedy-compactness-over-atomless-standar","kind":"node","lean":"TauCeti.DenseGraphLimits.GraphonSpace.instCompactSpace","parent":2279,"result":true,"route":"../roadmap/tauceti/combinatorics/dense-graph-limits-and-graphons/index.html#lovasz-szegedy-compactness-over-atomless-standar","scope":"tauceti/combinatorics/dense-graph-limits-and-graphons","status":"fully_proved","summary":"The cut-distance quotient of graphons on a standard Borel space with an atomless probability measure is a compact metric space, being isometric to `GraphonSpaceI`.","title":"Lov\u00e1sz--Szegedy compactness over atomless standard Borel carriers"},{"id":"tauceti/combinatorics/dense-graph-limits-and-graphons/frieze-kannan-weak-regularity","kind":"node","lean":"TauCeti.DenseGraphLimits.weak_regularity_frieze_kannan","parent":2279,"result":true,"route":"../roadmap/tauceti/combinatorics/dense-graph-limits-and-graphons/index.html#frieze-kannan-weak-regularity","scope":"tauceti/combinatorics/dense-graph-limits-and-graphons","status":"fully_proved","summary":"Every graphon has a measurable block-average step graphon within `\u03b5` in cut norm, on a partition with at most `4 ^ \u23081 / \u03b5\u00b2\u2309` parts.","title":"Frieze--Kannan weak regularity"},{"id":"tauceti/combinatorics/dense-graph-limits-and-graphons/tauceti-densegraphlimits-cutdist","kind":"node","lean":"TauCeti.DenseGraphLimits.cutDist","parent":2279,"result":true,"route":"../roadmap/tauceti/combinatorics/dense-graph-limits-and-graphons/index.html#tauceti-densegraphlimits-cutdist","scope":"tauceti/combinatorics/dense-graph-limits-and-graphons","status":"fully_proved","summary":"The cut distance of two graphons: the infimum, over couplings of their carriers, of the cut norm of the overlaid difference.","title":"TauCeti.DenseGraphLimits.cutDist"},{"id":"tauceti/combinatorics/dense-graph-limits-and-graphons/tauceti-densegraphlimits-cutdist-triangle","kind":"node","lean":"TauCeti.DenseGraphLimits.cutDist_triangle","parent":2279,"result":true,"route":"../roadmap/tauceti/combinatorics/dense-graph-limits-and-graphons/index.html#tauceti-densegraphlimits-cutdist-triangle","scope":"tauceti/combinatorics/dense-graph-limits-and-graphons","status":"fully_proved","summary":"The coupling-primary graphon cut distance satisfies the triangle inequality on arbitrary probability carriers.","title":"TauCeti.DenseGraphLimits.cutDist_triangle"},{"id":"tauceti/combinatorics/dense-graph-limits-and-graphons/tauceti-densegraphlimits-graphon-unitintervalrep","kind":"node","lean":"TauCeti.DenseGraphLimits.Graphon.unitIntervalRepr","parent":2279,"result":true,"route":"../roadmap/tauceti/combinatorics/dense-graph-limits-and-graphons/index.html#tauceti-densegraphlimits-graphon-unitintervalrep","scope":"tauceti/combinatorics/dense-graph-limits-and-graphons","status":"fully_proved","summary":"A graphon on the unit interval at cut distance zero from `W`, for `W` on an arbitrary probability carrier.","title":"TauCeti.DenseGraphLimits.Graphon.unitIntervalRepr"},{"id":"tauceti/combinatorics/dense-graph-limits-and-graphons/tauceti-probability-exchangeablerelationallaw","kind":"node","lean":"TauCeti.Probability.ExchangeableRelationalLaw","parent":2279,"result":true,"route":"../roadmap/tauceti/combinatorics/dense-graph-limits-and-graphons/index.html#tauceti-probability-exchangeablerelationallaw","scope":"tauceti/combinatorics/dense-graph-limits-and-graphons","status":"fully_proved","summary":"An exchangeable random structure of `L`, presented by its consistent finite marginals: a probability law on the configurations on `Fin k` for every `k`, consistent under restriction along every injection of vertices.","title":"TauCeti.Probability.ExchangeableRelationalLaw"},{"id":"tauceti/combinatorics/dense-graph-limits-and-graphons/exchangeable-laws-on-infinite-graphs-are-the-exc","kind":"node","lean":"TauCeti.DenseGraphLimits.InfiniteExchangeableGraphLaw.relationalLawEquiv","parent":2279,"result":true,"route":"../roadmap/tauceti/combinatorics/dense-graph-limits-and-graphons/index.html#exchangeable-laws-on-infinite-graphs-are-the-exc","scope":"tauceti/combinatorics/dense-graph-limits-and-graphons","status":"fully_proved","summary":"The inverse pulls the law back along the encoding.","title":"Exchangeable laws on infinite graphs are the exchangeable relational laws of the language of graphs on `\u2115` concentrated on simple graphs, that is, on symmetric irreflexive configurations"},{"data_url":"scopes/tauceti/combinatorics/finite-graph-connectivity-and-network-flows.json","id":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows","kind":"scope","parent":2278,"result":false,"route":"../roadmap/tauceti/combinatorics/finite-graph-connectivity-and-network-flows/index.html","scope":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Finite graph connectivity and network flows"},{"id":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows/max-flow-min-cut-theorem","kind":"node","lean":"TauCeti.Flow.exists_val_eq_arrowSum","parent":2292,"result":true,"route":"../roadmap/tauceti/combinatorics/finite-graph-connectivity-and-network-flows/index.html#max-flow-min-cut-theorem","scope":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows","status":"fully_proved","summary":"For nonnegative capacities and distinct terminals, some ordinary flow has value equal to the capacity of the arrows leaving some vertex set containing `s` but not `t`. By weak duality (`TauCeti.Flow.val_le_arrowSum`) this flow is maximum and this cut is minimum.","title":"Max-flow/min-cut theorem"},{"id":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows/max-flow-min-cut-theorem-for-bounded-assignments","kind":"node","lean":"TauCeti.BoundedFlow.exists_val_eq_cutBound","parent":2292,"result":true,"route":"../roadmap/tauceti/combinatorics/finite-graph-connectivity-and-network-flows/index.html#max-flow-min-cut-theorem-for-bounded-assignments","scope":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows","status":"fully_proved","summary":"If some bounded `s`\u2013`t` assignment between distinct terminals exists, then some bounded `s`\u2013`t` assignment has value equal to the cut bound of some vertex set containing `s` but not `t`. By weak duality (`TauCeti.BoundedFlow.val_le_cutBound`) this assignment is maximum and this cut bound is minimum.","title":"Max-flow/min-cut theorem for bounded assignments"},{"id":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows/integrality-of-maximum-flows","kind":"node","lean":"TauCeti.Flow.exists_isMaximum_mem","parent":2292,"result":true,"route":"../roadmap/tauceti/combinatorics/finite-graph-connectivity-and-network-flows/index.html#integrality-of-maximum-flows","scope":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows","status":"fully_proved","summary":"For nonnegative capacities in an additive subgroup `H`, some maximum ordinary flow takes values in `H`. For distinct terminals, its value is the capacity of some cut by `TauCeti.Flow.IsMaximum.exists_val_eq_arrowSum`.","title":"Integrality of maximum flows"},{"id":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows/path-cycle-decomposition","kind":"node","lean":"TauCeti.exists_pathCycleDecomposition","parent":2292,"result":true,"route":"../roadmap/tauceti/combinatorics/finite-graph-connectivity-and-network-flows/index.html#path-cycle-decomposition","scope":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows","status":"fully_proved","summary":"Every nonnegative arrow assignment `f` on a finite arrow family is a sum of positively weighted directed paths, each from a vertex of negative excess to a vertex of positive excess, and positively weighted directed cycles. The number of components is at most the number of arrows on which `f` is positive, and if the values of `f` lie in an additive subgroup `H`, then so do all weights.","title":"Path\u2013cycle decomposition"},{"id":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows/acyclic-flows","kind":"node","lean":"TauCeti.Flow.exists_isAcyclic","parent":2292,"result":true,"route":"../roadmap/tauceti/combinatorics/finite-graph-connectivity-and-network-flows/index.html#acyclic-flows","scope":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows","status":"fully_proved","summary":"Every ordinary `s`\u2013`t` flow `f` has an acyclic ordinary `s`\u2013`t` flow of the same value that is arrowwise at most `f`.","title":"Acyclic flows"},{"id":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows/the-poset-of-minimum-cuts","kind":"node","lean":"TauCeti.BoundedFlow.IsMaximum.minimizersOrderIso","parent":2292,"result":true,"route":"../roadmap/tauceti/combinatorics/finite-graph-connectivity-and-network-flows/index.html#the-poset-of-minimum-cuts","scope":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows","status":"fully_proved","summary":"For a maximum bounded `s`\u2013`t` assignment `f` between distinct terminals, the minimum cuts, ordered by inclusion, are order isomorphic to the upper sets of the poset of free residual components, ordered by inclusion. A minimum cut corresponds to the free components contained in it, and an upper set to the vertices of its components together with the vertices forced onto the source side.","title":"The poset of minimum cuts"},{"id":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows/multi-cut-non-crossing-lemma","kind":"node","lean":"TauCeti.TerminalCut.exists_isMinimum_forall_not_crosses","parent":2292,"result":true,"route":"../roadmap/tauceti/combinatorics/finite-graph-connectivity-and-network-flows/index.html#multi-cut-non-crossing-lemma","scope":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows","status":"fully_proved","summary":"Let `\ud835\udc9e` be a pairwise non-crossing family of cuts, each a minimum cut of a symmetric submodular `f` between some pair of vertices. If none of them separates the distinct vertices `s` and `t`, then some minimum `s`\u2013`t` cut crosses none of them.","title":"Multi-cut non-crossing lemma"},{"id":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows/ultrametric-inequality-for-minimum-cut-values","kind":"node","lean":"TauCeti.TerminalCut.min_pairMinimumValue_le","parent":2292,"result":true,"route":"../roadmap/tauceti/combinatorics/finite-graph-connectivity-and-network-flows/index.html#ultrametric-inequality-for-minimum-cut-values","scope":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows","status":"fully_proved","summary":"every cut separating `s` from `t` separates `s` from `v` or `v` from `t`. The intermediate vertex `v` is arbitrary.","title":"Ultrametric inequality for minimum cut values"},{"id":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows/graph-isbridge-iff-forall-cycle-notmem","kind":"node","lean":"Graph.isBridge_iff_forall_cycle_notMem","parent":2292,"result":true,"route":"../roadmap/tauceti/combinatorics/finite-graph-connectivity-and-network-flows/index.html#graph-isbridge-iff-forall-cycle-notmem","scope":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows","status":"fully_proved","summary":"An actual edge is a bridge exactly when it lies on no cycle.","title":"Graph.isBridge_iff_forall_cycle_notMem"},{"id":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows/bridges-by-component-counts","kind":"node","lean":"Graph.isBridge_iff_componentCount_deleteEdges_eq_add_one","parent":2292,"result":true,"route":"../roadmap/tauceti/combinatorics/finite-graph-connectivity-and-network-flows/index.html#bridges-by-component-counts","scope":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows","status":"fully_proved","summary":"With finitely many connected components, an edge is a bridge exactly when deleting it increases the number of connected components by one.","title":"Bridges by component counts"},{"id":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows/tauceti-residualupdate","kind":"node","lean":"TauCeti.residualUpdate","parent":2292,"result":true,"route":"../roadmap/tauceti/combinatorics/finite-graph-connectivity-and-network-flows/index.html#tauceti-residualupdate","scope":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows","status":"fully_proved","summary":"The *residual update* of `f` by a residual assignment `r`: the value on each arrow `e` is `f e + r e\u207a - r e\u207b`, where `e\u207a = Sum.inl e` and `e\u207b = Sum.inr e` are its residual copies.","title":"TauCeti.residualUpdate"},{"id":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows/tauceti-arrowwalk-splitequiv","kind":"node","lean":"TauCeti.ArrowWalk.splitEquiv","parent":2292,"result":true,"route":"../roadmap/tauceti/combinatorics/finite-graph-connectivity-and-network-flows/index.html#tauceti-arrowwalk-splitequiv","scope":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows","status":"fully_proved","summary":"Lifting identifies the walks from `s` to `t` with the walks in the split network from the entrance of `s` to the entrance (`b = false`) or exit (`b = true`) of `t`; the inverse is projection.","title":"TauCeti.ArrowWalk.splitEquiv"},{"id":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows/tauceti-auxterminalhom","kind":"node","lean":"TauCeti.AuxTerminalHom","parent":2292,"result":true,"route":"../roadmap/tauceti/combinatorics/finite-graph-connectivity-and-network-flows/index.html#tauceti-auxterminalhom","scope":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows","status":"fully_proved","summary":"The arrow family of `Q` with auxiliary terminals for the vertex sets `A` and `B`, on the vertex type `V \u2295 Bool`. The arrows between original vertices `Sum.inl v` and `Sum.inl w` are those of `Q v w`; the fresh source `\u03c3 = Sum.inr false` has one arrow to each `Sum.inl a` with `a \u2208 A`; the fresh sink `\u03c4 = Sum.inr true` has one arrow from each `Sum.inl b` with `b \u2208 B`; and there are no other arrows.","title":"TauCeti.AuxTerminalHom"},{"id":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows/graph-walk","kind":"node","lean":"Graph.Walk","parent":2292,"result":true,"route":"../roadmap/tauceti/combinatorics/finite-graph-connectivity-and-network-flows/index.html#graph-walk","scope":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows","status":"fully_proved","summary":"An edge-labelled walk in a multigraph. Each step records an edge together with its incidence; a zero-length walk may be based at any ambient point.","title":"Graph.Walk"},{"id":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows/simplegraph-walkequivofsimplegraph","kind":"node","lean":"SimpleGraph.walkEquivOfSimpleGraph","parent":2292,"result":true,"route":"../roadmap/tauceti/combinatorics/finite-graph-connectivity-and-network-flows/index.html#simplegraph-walkequivofsimplegraph","scope":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows","status":"fully_proved","summary":"Walks in a simple graph are equivalent to edge-labelled walks in its associated `Graph`.","title":"SimpleGraph.walkEquivOfSimpleGraph"},{"id":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows/gomory-hu-theorem","kind":"node","lean":"TauCeti.TerminalCut.exists_isCutTree","parent":2292,"result":true,"route":"../roadmap/tauceti/combinatorics/finite-graph-connectivity-and-network-flows/index.html#gomory-hu-theorem","scope":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows","status":"fully_proved","summary":"Every symmetric submodular set function on a nonempty finite vertex type has a cut tree: a weighted tree on the vertices in which the minimum cut value between any two distinct vertices is the minimum weight on the tree path between them, and deleting any tree edge leaves a minimum cut between its ends.","title":"Gomory\u2013Hu theorem"},{"id":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows/query-theorem","kind":"node","lean":"TauCeti.TerminalCut.IsCutTree.isMinimum_edgeSide_of_mem_edges","parent":2292,"result":true,"route":"../roadmap/tauceti/combinatorics/finite-graph-connectivity-and-network-flows/index.html#query-theorem","scope":"tauceti/combinatorics/finite-graph-connectivity-and-network-flows","status":"fully_proved","summary":"If `s(u, v)` has minimum weight on the tree path from `s` to `t`, and `s` lies on the side of `u`, then that side is a minimum `s`\u2013`t` cut. Since the two sides of a tree edge cover the vertices, one of the orientations `(u, v)` and `(v, u)` applies.","title":"Query theorem"},{"data_url":"scopes/tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit.json","id":"tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit","kind":"scope","parent":2278,"result":false,"route":"../roadmap/tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit/index.html","scope":"tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Graph regularity, finite weak regularity, and arity-3 hypergraph complexes"},{"id":"tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit/finite-frieze-kannan-weak-regularity","kind":"node","lean":"SimpleGraph.frieze_kannan","parent":2310,"result":true,"route":"../roadmap/tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit/index.html#finite-frieze-kannan-weak-regularity","scope":"tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit","status":"fully_proved","summary":"For `\u03b5 > 0`, the vertices of `G` have a partition `P` into at most `4 ^ \u23081 / \u03b5\u00b2\u2309` parts such that, for all vertex sets `A` and `B`, the number of edges from `A` to `B` is within `\u03b5 * (card V) ^ 2` of its stepped prediction `G.steppedCount P A B`.","title":"Finite Frieze\u2013Kannan weak regularity"},{"id":"tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit/strong-graph-regularity","kind":"node","lean":"SimpleGraph.exists_strong_regular","parent":2310,"result":true,"route":"../roadmap/tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit/index.html#strong-graph-regularity","scope":"tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit","status":"fully_proved","summary":"Let `\u03b5 > 0` and let `F : \u2115 \u2192 \u211d` be a positive schedule. Every equipartition `P\u2080` of a finite graph `G` is almost refined, up to `\u03b5`, by the coarse partition `P` of a strong regularity witness `(P, Q)`, where `P` has at least `l` parts for any requested `l \u2264 card \u03b1`, and `Q` has at most `strongGraphRegularityBound \u03b5 F (max l #P\u2080.parts)` parts.","title":"Strong graph regularity"},{"id":"tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit/szemeredis-regularity-lemma-relative-to-a-seed","kind":"node","lean":"SimpleGraph.exists_regular_equipartition_almost_refining, SimpleGraph.exists_regular_almostRefines_le_card","parent":2310,"result":true,"route":"../roadmap/tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit/index.html#szemeredis-regularity-lemma-relative-to-a-seed","scope":"tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit","status":"fully_proved","summary":"Every equipartition `P\u2080` of a finite graph is almost refined, up to `\u03b5`, by an `\u03b5`-uniform equipartition with at most `refiningRegularityBound \u03b5 #P\u2080.parts` parts.","title":"Szemer\u00e9di's regularity lemma relative to a seed"},{"id":"tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit/induced-counting-lemma-for-three-cells","kind":"node","lean":"SimpleGraph.abs_card_induced_fin_three_sub_le","parent":2310,"result":true,"route":"../roadmap/tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit/index.html#induced-counting-lemma-for-three-cells","scope":"tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit","status":"fully_proved","summary":"Let `A 0`, `A 1`, `A 2` be pairwise `\u03b5`-uniform cells of `G`, and `J` a graph on `Fin 3`. Then the number of maps `g` with `g i \u2208 A i` such that `g i` and `g j` are adjacent in `G` exactly when `i` and `j` are adjacent in `J` differs from `|A 0| |A 1| |A 2|` times the product, over the pairs `i < j`, of the edge density of `(A i, A j)` (if `i` and `j` are adjacent in `J`) or of its nonedge density `1 - d(A i, A j)`\u2026","title":"Induced counting lemma for three cells"},{"id":"tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit/finite-partition-energy-is-graphon-partition-ene","kind":"node","lean":"SimpleGraph.graphonPartitionEnergy_finiteGraphGraphon","parent":2310,"result":true,"route":"../roadmap/tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit/index.html#finite-partition-energy-is-graphon-partition-ene","scope":"tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit","status":"fully_proved","summary":"For a graph `G` on `Fin m` with `0 < m` and a partition `P` of its vertices, the size-weighted energy of `P` is the partition energy of the graphon of `G` over the partition of the unit interval grouping the vertex cells by the parts of `P`.","title":"Finite partition energy is graphon partition energy"},{"id":"tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit/simplegraph-cutdiscrepancy","kind":"node","lean":"SimpleGraph.cutDiscrepancy","parent":2310,"result":true,"route":"../roadmap/tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit/index.html#simplegraph-cutdiscrepancy","scope":"tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit","status":"fully_proved","summary":"The cut discrepancy of `G` with respect to `P`: the largest deviation, over all rectangles `(A, B)`, between the number of edges from `A` to `B` and its stepped prediction. It is count-scaled, so it lies between `0` and `(card V) ^ 2`.","title":"SimpleGraph.cutDiscrepancy"},{"id":"tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit/tauceti-triadiccomplex3","kind":"node","lean":"TauCeti.TriadicComplex3","parent":2310,"result":true,"route":"../roadmap/tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit/index.html#tauceti-triadiccomplex3","scope":"tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit","status":"fully_proved","summary":"A triadic complex on `V`: a finite palette of `pairColorCount` pair colors chosen by the complex, a lower skeleton over that palette, and a finite family of polyads over that skeleton.","title":"TauCeti.TriadicComplex3"},{"id":"tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit/tauceti-colored3graph-topregularovermostpolyads","kind":"node","lean":"TauCeti.Colored3Graph.TopRegularOverMostPolyads","parent":2310,"result":true,"route":"../roadmap/tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit/index.html#tauceti-colored3graph-topregularovermostpolyads","scope":"tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit","status":"fully_proved","summary":"`H` is `(\u03b7, r)`-top-regular over most polyads of `C`: the polyads over which `H` is not `(\u03b7, r)`-top-regular carry at most an `\u03b5`-proportion of the selected support.","title":"TauCeti.Colored3Graph.TopRegularOverMostPolyads"},{"id":"tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit/tauceti-colored3graph-istopregularoverpolyad","kind":"node","lean":"TauCeti.Colored3Graph.IsTopRegularOverPolyad","parent":2310,"result":true,"route":"../roadmap/tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit/index.html#tauceti-colored3graph-istopregularoverpolyad","scope":"tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit","status":"fully_proved","summary":"`H` is `(\u03b4, r)`-top-regular over the polyad `P` when, for every top color `c` and every family of `r` subpolyads of `P` whose union support has at least `\u03b4` times the size of the support of `P`, the relative density of `c` on that union is within `\u03b4` of the relative density of `c` over `P`. Members of the family may repeat, so this tests unions of at most `r` subpolyads.","title":"TauCeti.Colored3Graph.IsTopRegularOverPolyad"},{"id":"tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit/slicing-lemma","kind":"node","lean":"TauCeti.Colored3Graph.IsTopRegularOverPolyad.of_pairSupport_subset","parent":2310,"result":true,"route":"../roadmap/tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit/index.html#slicing-lemma","scope":"tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit","status":"fully_proved","summary":"If `H` is `(\u03b4, r)`-top-regular over `P` at a positive rank, then it is `(max (2\u03b4) (\u03b4 / \u03b1), r)`-top-regular over every polyad `P'` whose pair graphs lie in those of `P` and whose support is at least an `\u03b1`-fraction of the support of `P`. The polyad `P'` may live over a different skeleton, so this is how top regularity passes to a refined lower skeleton.","title":"Slicing lemma"},{"id":"tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit/cauchy-schwarz-on-a-resolved-witness","kind":"node","lean":"TauCeti.Colored3Graph.card_mul_sq_relDensityOn_sub_le_sum, TauCeti.PairColorSystem.card_mul_sq_pairColorDensity_sub_le_sum","parent":2310,"result":true,"route":"../roadmap/tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit/index.html#cauchy-schwarz-on-a-resolved-witness","scope":"tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit","status":"fully_proved","summary":"The squared deviation of its density from any real number `a`, multiplied by its size, is bounded by the sum of the squared deviations of the fine densities from `a`. The empty witness is allowed.","title":"Cauchy\u2013Schwarz on a resolved witness"},{"id":"tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit/szemeredis-regularity-lemma-below-a-starting-equ","kind":"node","lean":"SimpleGraph.szemeredi_regularity_le","parent":2310,"result":true,"route":"../roadmap/tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit/index.html#szemeredis-regularity-lemma-below-a-starting-equ","scope":"tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit","status":"fully_proved","summary":"If `P\u2081` is an equipartition with at least `7` parts and `100 \u2264 4 ^ #P\u2081.parts * \u03b5 ^ 5`, and the graph is large enough, then some `\u03b5`-uniform equipartition refines `P\u2081` and has at most `stepBound^[\u230a4 / \u03b5 ^ 5\u230b\u208a + 1] #P\u2081.parts` parts.","title":"Szemer\u00e9di's regularity lemma below a starting equipartition"},{"id":"tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit/szemeredis-regularity-lemma-refining-an-equipart","kind":"node","lean":"SimpleGraph.exists_regular_exact_refining_equipartition","parent":2310,"result":true,"route":"../roadmap/tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit/index.html#szemeredis-regularity-lemma-refining-an-equipart","scope":"tauceti/combinatorics/graph-regularity-finite-weak-regularity-and-arit","status":"fully_proved","summary":"Every equipartition `P` of a finite graph is refined exactly by a `\u03b4`-uniform equipartition with at most `nestedRefinementBound \u03b4 #P.parts` parts.","title":"Szemer\u00e9di's regularity lemma refining an equipartition exactly"},{"data_url":"scopes/tauceti/combinatorics/dense-graph-limits.json","id":"tauceti/combinatorics/dense-graph-limits","kind":"scope","parent":2278,"result":false,"route":"../roadmap/tauceti/combinatorics/dense-graph-limits/index.html","scope":"tauceti/combinatorics/dense-graph-limits","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Combinatorics.DenseGraphLimits` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Dense Graph Limits"},{"id":"tauceti/combinatorics/dense-graph-limits/goodmans-inequality","kind":"node","lean":"TauCeti.DenseGraphLimits.goodman_triangle_density","parent":2324,"result":true,"route":"../roadmap/tauceti/combinatorics/dense-graph-limits/index.html#goodmans-inequality","scope":"tauceti/combinatorics/dense-graph-limits","status":"fully_proved","summary":"For every graphon, triangle density is at least `2 * t(K\u2082, W)^2 - t(K\u2082, W)`.","title":"Goodman's inequality"},{"id":"tauceti/combinatorics/dense-graph-limits/mantels-inequality-for-graphons","kind":"node","lean":"TauCeti.DenseGraphLimits.mantel_triangle_free","parent":2324,"result":true,"route":"../roadmap/tauceti/combinatorics/dense-graph-limits/index.html#mantels-inequality-for-graphons","scope":"tauceti/combinatorics/dense-graph-limits","status":"fully_proved","summary":"A graphon with zero triangle density has edge density at most `1 / 2`.","title":"Mantel's inequality for graphons"},{"id":"tauceti/combinatorics/dense-graph-limits/sidorenkos-inequality-for-the-4-cycle","kind":"node","lean":"TauCeti.DenseGraphLimits.sidorenko_cycleGraph_four","parent":2324,"result":true,"route":"../roadmap/tauceti/combinatorics/dense-graph-limits/index.html#sidorenkos-inequality-for-the-4-cycle","scope":"tauceti/combinatorics/dense-graph-limits","status":"fully_proved","summary":"The 4-cycle homomorphism density is at least the fourth power of the edge density, on every probability carrier. This is the graphon form of the `K_{2,2}` case in Y. Zhao, *Graph Theory and Additive Combinatorics*, Theorem 5.2.1.","title":"Sidorenko's inequality for the 4-cycle"},{"data_url":"scopes/tauceti/combinatorics/graph.json","id":"tauceti/combinatorics/graph","kind":"scope","parent":2278,"result":false,"route":"../roadmap/tauceti/combinatorics/graph/index.html","scope":"tauceti/combinatorics/graph","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Combinatorics.Graph` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Graph"},{"id":"tauceti/combinatorics/graph/hakimis-theorem","kind":"node","lean":"Graph.exists_orientation_indegree_eq_iff","parent":2328,"result":true,"route":"../roadmap/tauceti/combinatorics/graph/index.html#hakimis-theorem","scope":"tauceti/combinatorics/graph","status":"fully_proved","summary":"A finite multigraph has an orientation with indegree `d` exactly when `d` sums to the number of actual edges and every vertex set `S` has at most `d(S)` internal edges.","title":"Hakimi's theorem"},{"id":"tauceti/combinatorics/graph/konigs-theorem-for-multiset-capacitated-matching","kind":"node","lean":"Graph.IsBipartition.exists_isCapacitatedMatching_isVertexCover_sum_eq","parent":2328,"result":true,"route":"../roadmap/tauceti/combinatorics/graph/index.html#konigs-theorem-for-multiset-capacitated-matching","scope":"tauceti/combinatorics/graph","status":"fully_proved","summary":"Let `L` and `R` form a bipartition of a multigraph with finitely many actual edges, and let `b v : \u2115` be a capacity on every actual vertex. Some multiset capacitated matching `x` and some vertex cover `C` satisfy `\u2211 e, x e = \u2211 v \u2208 C, b v`. By `Graph.IsCapacitatedMatching.sum_le_sum_of_isVertexCover`, the matching is then largest and the cover of least weight.","title":"K\u0151nig's theorem for multiset capacitated matchings"},{"id":"tauceti/combinatorics/graph/mengers-theorem-for-adjacent-vertices-predicate","kind":"node","lean":"Graph.forall_reachable_deleteEdges_deleteVerts_iff_exists_paths, Graph.exists_paths_setEdgeSeparator_card_eq, Graph.forall_exists_reachable_deleteEdges_iff_exists_paths, SimpleGraph.exists_paths_setEdgeSeparator_card_eq, SimpleGraph.forall_exists_reachable_deleteEdges_iff_exists_paths, Graph.exists_paths_edgeSeparator_card_eq, Graph.isEdgeReachable_iff_exists_paths, SimpleGraph.exists_paths_edgeSeparator_card_eq, SimpleGraph.isEdgeReachable_iff_exists_paths, Graph.exists_paths_vertexSeparator_card_add_card_eq, Graph.exists_paths_vertexSeparator_card_eq, Graph.forall_reachable_deleteVerts_iff_exists_paths, Graph.isVertexReachable_iff_exists_paths, Graph.exists_paths_setSeparator_card_eq, Graph.forall_exists_reachable_deleteVerts_iff_exists_paths","parent":2328,"result":true,"route":"../roadmap/tauceti/combinatorics/graph/index.html#mengers-theorem-for-adjacent-vertices-predicate","scope":"tauceti/combinatorics/graph","status":"fully_proved","summary":"For distinct points, after deleting every direct edge, every set of fewer than `k` nonterminal points leaves the terminals reachable exactly when there are distinct internally disjoint paths indexed by `Fin k \u2295 {e // G.IsLink e s t}`.","title":"Menger's theorem for adjacent vertices, predicate form"},{"id":"tauceti/combinatorics/graph/undirected-max-flow-min-cut-theorem","kind":"node","lean":"Graph.exists_flow_val_eq_edgeCutCapacity, Graph.exists_flow_val_eq_edgeCutCapacity_mem","parent":2328,"result":true,"route":"../roadmap/tauceti/combinatorics/graph/index.html#undirected-max-flow-min-cut-theorem","scope":"tauceti/combinatorics/graph","status":"fully_proved","summary":"For nonnegative capacities on the edges of a multigraph and distinct actual vertices `s` and `t`, some acyclic `s`\u2013`t` flow has value equal to the capacity of some set of actual vertices containing `s` but not `t`. By `TauCeti.Flow.val_le_edgeCutCapacity` this flow is maximum and this cut is minimum.","title":"Undirected max-flow/min-cut theorem"},{"id":"tauceti/combinatorics/graph/vertex-capacitated-mengers-theorem-for-multigrap","kind":"node","lean":"Graph.exists_paths_vertexSeparator_sum_eq, Graph.exists_paths_setSeparator_sum_eq","parent":2328,"result":true,"route":"../roadmap/tauceti/combinatorics/graph/index.html#vertex-capacitated-mengers-theorem-for-multigrap","scope":"tauceti/combinatorics/graph","status":"fully_proved","summary":"Let `s \u2260 t` be points of a multigraph with finitely many actual vertices, joined by no edge, and let `c v` be a capacity on every vertex. Some family of `k` paths from `s` to `t`, in which every vertex `v` other than `s` and `t` lies on at most `c v` members, and some set `X` of actual vertices avoiding `s` and `t`, whose deletion leaves `s` and `t` unreachable, satisfy `\u2211 v \u2208 X, c v = k`. By\u2026","title":"Vertex-capacitated Menger's theorem for multigraphs"},{"id":"tauceti/combinatorics/graph/whitneys-inequality-for-multigraphs","kind":"node","lean":"Graph.vertexConnectivity_le_edgeConnectivity, Graph.IsVertexConnected.isEdgeConnected","parent":2328,"result":true,"route":"../roadmap/tauceti/combinatorics/graph/index.html#whitneys-inequality-for-multigraphs","scope":"tauceti/combinatorics/graph","status":"fully_proved","summary":"vertex connectivity is at most edge connectivity.","title":"Whitney's inequality for multigraphs"},{"data_url":"scopes/tauceti/combinatorics/network.json","id":"tauceti/combinatorics/network","kind":"scope","parent":2278,"result":false,"route":"../roadmap/tauceti/combinatorics/network/index.html","scope":"tauceti/combinatorics/network","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Combinatorics.Network` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Network"},{"id":"tauceti/combinatorics/network/the-directed-whitney-inequality","kind":"node","lean":"TauCeti.IsVertexStrong.isArcStrong, TauCeti.vertexStrongConnectivity_le_arcConnectivity","parent":2335,"result":true,"route":"../roadmap/tauceti/combinatorics/network/index.html#the-directed-whitney-inequality","scope":"tauceti/combinatorics/network","status":"fully_proved","summary":"A `k`-vertex-strong arrow family is `k`-arc-strong.","title":"The directed Whitney inequality"},{"id":"tauceti/combinatorics/network/generalized-hoffmans-theorem","kind":"node","lean":"TauCeti.ExcessFlow.nonempty_iff","parent":2335,"result":true,"route":"../roadmap/tauceti/combinatorics/network/index.html#generalized-hoffmans-theorem","scope":"tauceti/combinatorics/network","status":"fully_proved","summary":"Ordered signed bounds admit an assignment of prescribed excess exactly when that excess is balanced and every cut can supply its demand.","title":"Generalized Hoffman's theorem"},{"id":"tauceti/combinatorics/network/hoffmans-circulation-theorem","kind":"node","lean":"TauCeti.exists_isCirculation_iff","parent":2335,"result":true,"route":"../roadmap/tauceti/combinatorics/network/index.html#hoffmans-circulation-theorem","scope":"tauceti/combinatorics/network","status":"fully_proved","summary":"A bounded circulation exists exactly when every vertex set has total lower bound entering it at most the total upper bound leaving it.","title":"Hoffman's circulation theorem"},{"id":"tauceti/combinatorics/network/max-flow-min-cut-theorem-between-vertex-sets","kind":"node","lean":"TauCeti.SetFlow.exists_val_eq_arrowSum","parent":2335,"result":true,"route":"../roadmap/tauceti/combinatorics/network/index.html#max-flow-min-cut-theorem-between-vertex-sets","scope":"tauceti/combinatorics/network","status":"fully_proved","summary":"For disjoint `A` and `B` and nonnegative capacities, some `A`\u2013`B` flow has value equal to the capacity of some `A`\u2013`B` cut. By weak duality (`TauCeti.SetFlow.val_le_arrowSum`) this flow is maximum and this cut is minimum.","title":"Max-flow/min-cut theorem between vertex sets"},{"id":"tauceti/combinatorics/network/mengers-theorem-for-vertex-strong-arrow-families","kind":"node","lean":"TauCeti.isVertexStrong_iff_exists_paths, TauCeti.exists_paths_arrowSeparator_card_eq, TauCeti.exists_paths_setArrowSeparator_card_eq, TauCeti.forall_exists_reachableWithout_iff_exists_paths, TauCeti.forall_reachableWithout_iff_exists_paths, TauCeti.exists_paths_vertexSeparator_card_eq, TauCeti.exists_paths_isMixedSeparator_card_eq, TauCeti.forall_not_isMixedSeparator_iff_exists_paths, TauCeti.forall_reachableAvoiding_iff_exists_paths, TauCeti.exists_paths_setSeparator_card_eq, TauCeti.forall_exists_reachableAvoiding_iff_exists_paths","parent":2335,"result":true,"route":"../roadmap/tauceti/combinatorics/network/index.html#mengers-theorem-for-vertex-strong-arrow-families","scope":"tauceti/combinatorics/network","status":"fully_proved","summary":"An arrow family on a finite vertex type is `k`-vertex-strong exactly when it has more than `k` vertices and any two distinct vertices `s` and `t` are joined by `k` distinct directed paths pairwise sharing no vertex other than `s` and `t`. When there are arrows from `s` to `t`, one of these paths may be a single arrow.","title":"Menger's theorem for vertex-strong arrow families"},{"id":"tauceti/combinatorics/network/vertex-capacitated-mengers-theorem-for-directed","kind":"node","lean":"TauCeti.exists_paths_vertexSeparator_sum_eq, TauCeti.exists_paths_setSeparator_sum_eq","parent":2335,"result":true,"route":"../roadmap/tauceti/combinatorics/network/index.html#vertex-capacitated-mengers-theorem-for-directed","scope":"tauceti/combinatorics/network","status":"fully_proved","summary":"Let `s \u2260 t` be vertices of an arrow family on a finite vertex type, with no arrow from `s` to `t`, and let `c v` be a capacity on every vertex. Some family of `k` directed `s`\u2013`t` paths, in which every vertex `v` other than `s` and `t` lies on at most `c v` members, and some set `X` of vertices avoiding `s` and `t`, whose deletion leaves `t` unreachable from `s`, satisfy `\u2211 v \u2208 X, c v = k`. By\u2026","title":"Vertex-capacitated Menger's theorem for directed paths"},{"data_url":"scopes/tauceti/combinatorics/simple-graph.json","id":"tauceti/combinatorics/simple-graph","kind":"scope","parent":2278,"result":false,"route":"../roadmap/tauceti/combinatorics/simple-graph/index.html","scope":"tauceti/combinatorics/simple-graph","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Combinatorics.SimpleGraph` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Simple Graph"},{"id":"tauceti/combinatorics/simple-graph/diracs-theorem","kind":"node","lean":"SimpleGraph.IsVertexConnected.exists_isCycle_forall_mem_support","parent":2342,"result":true,"route":"../roadmap/tauceti/combinatorics/simple-graph/index.html#diracs-theorem","scope":"tauceti/combinatorics/simple-graph","status":"fully_proved","summary":"For `2 \u2264 k`, every set of at most `k` vertices of a `k`-vertex-connected simple graph on a finite vertex type lies on a common cycle.","title":"Dirac's theorem"},{"id":"tauceti/combinatorics/simple-graph/the-global-form-of-mengers-theorem","kind":"node","lean":"SimpleGraph.isVertexConnected_iff_forall_exists_paths","parent":2342,"result":true,"route":"../roadmap/tauceti/combinatorics/simple-graph/index.html#the-global-form-of-mengers-theorem","scope":"tauceti/combinatorics/simple-graph","status":"fully_proved","summary":"A simple graph on a finite vertex type with more than `k` vertices is `k`-vertex-connected exactly when every two distinct vertices `s` and `t`, adjacent or not, are joined by `k` distinct paths pairwise sharing no vertex other than `s` and `t`.","title":"The global form of Menger's theorem"},{"id":"tauceti/combinatorics/simple-graph/lempel-even-cederbaum-theorem","kind":"node","lean":"SimpleGraph.nonempty_stNumbering_iff_isVertexConnected_two_sup_edge, SimpleGraph.nonempty_stNumbering_of_isVertexConnected_two_sup_edge, SimpleGraph.nonempty_stNumbering_iff_isVertexConnected_two","parent":2342,"result":true,"route":"../roadmap/tauceti/combinatorics/simple-graph/index.html#lempel-even-cederbaum-theorem","scope":"tauceti/combinatorics/simple-graph","status":"fully_proved","summary":"On a finite vertex type with at least three vertices, a simple graph `G` admits an `st`-numbering for distinct vertices `s \u2260 t` if and only if `G \u2294 SimpleGraph.edge s t` is `2`-vertex-connected.","title":"Lempel\u2013Even\u2013Cederbaum theorem"},{"id":"tauceti/combinatorics/simple-graph/mengers-theorem-for-fans","kind":"node","lean":"SimpleGraph.exists_fan_separator_card_eq, SimpleGraph.forall_exists_walk_iff_exists_fan, SimpleGraph.exists_paths_vertexSeparator_card_eq, SimpleGraph.exists_paths_vertexSeparator_card_add_one_eq, SimpleGraph.forall_reachable_induce_iff_exists_paths, SimpleGraph.isVertexReachable_iff_exists_paths, SimpleGraph.exists_paths_setSeparator_card_eq, SimpleGraph.forall_exists_walk_iff_exists_paths","parent":2342,"result":true,"route":"../roadmap/tauceti/combinatorics/simple-graph/index.html#mengers-theorem-for-fans","scope":"tauceti/combinatorics/simple-graph","status":"fully_proved","summary":"Let `x` be a vertex outside a set `U` of vertices of a simple graph on a finite vertex type. Some family of `k` paths, the `i`-th from `x` to `u i \u2208 U` with its other vertices outside `U`, any two of which share only the vertex `x`, and some set of `k` vertices other than `x` meeting every walk from `x` to a vertex of `U` exist. By `SimpleGraph.card_le_card_of_fan`, the fan is then largest and the separator smallest.","title":"Menger's theorem for fans"},{"id":"tauceti/combinatorics/simple-graph/vertex-capacitated-mengers-theorem","kind":"node","lean":"SimpleGraph.exists_paths_vertexSeparator_sum_eq, SimpleGraph.exists_paths_setSeparator_sum_eq","parent":2342,"result":true,"route":"../roadmap/tauceti/combinatorics/simple-graph/index.html#vertex-capacitated-mengers-theorem","scope":"tauceti/combinatorics/simple-graph","status":"fully_proved","summary":"Let `s` and `t` be distinct nonadjacent vertices of a simple graph on a finite vertex type, and let `c v` be a capacity on every vertex. Some family of `k` paths from `s` to `t`, in which every vertex `v` other than `s` and `t` lies on at most `c v` members, and some set `X` of vertices other than `s` and `t`, whose deletion separates `s` from `t`, satisfy `\u2211 v \u2208 X, c v = k`. By\u2026","title":"Vertex-capacitated Menger's theorem"},{"id":"tauceti/combinatorics/simple-graph/whitneys-inequality","kind":"node","lean":"SimpleGraph.vertexConnectivity_le_edgeConnectivity, SimpleGraph.IsVertexConnected.isEdgeConnected","parent":2342,"result":true,"route":"../roadmap/tauceti/combinatorics/simple-graph/index.html#whitneys-inequality","scope":"tauceti/combinatorics/simple-graph","status":"fully_proved","summary":"the vertex connectivity of a simple graph is at most its edge connectivity.","title":"Whitney's inequality"},{"id":"tauceti/combinatorics/simple-graph/whitneys-theorem-with-prescribed-initial-cycle","kind":"node","lean":"SimpleGraph.IsVertexConnected.exists_isOpen_earDecomposition_initial_eq","parent":2342,"result":true,"route":"../roadmap/tauceti/combinatorics/simple-graph/index.html#whitneys-theorem-with-prescribed-initial-cycle","scope":"tauceti/combinatorics/simple-graph","status":"fully_proved","summary":"For any cycle `c` in a `2`-vertex-connected simple graph on a finite vertex type, there is an open ear decomposition of `\u22a4` whose initial subgraph is `c.toSubgraph`.","title":"Whitney's theorem with prescribed initial cycle"},{"data_url":"scopes/tauceti/combinatorics/other-results.json","id":"tauceti/combinatorics/other-results","kind":"scope","parent":2278,"result":false,"route":"../roadmap/tauceti/combinatorics/other-results/index.html","scope":"tauceti/combinatorics/other-results","status":"fully_proved","summary":"Results named in the rest of Tau Ceti's `TauCeti.Combinatorics` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Other results"},{"id":"tauceti/combinatorics/other-results/the-frobenius-determinant-formula","kind":"node","lean":"TauCeti.standardCount_mul_prod_factorial_betaNumber","parent":2350,"result":true,"route":"../roadmap/tauceti/combinatorics/other-results/index.html#the-frobenius-determinant-formula","scope":"tauceti/combinatorics/other-results","status":"fully_proved","summary":"The Frobenius determinant formula for the number `f^\u03bc` of standard Young tableaux of shape `\u03bc`: for any bound `r` on the number of rows, `f^\u03bc` times the product of the factorials of the beta-numbers is `\u03bc.card !` times the product of their differences over the ordered pairs of rows.","title":"The Frobenius determinant formula"},{"id":"tauceti/combinatorics/other-results/halls-theorem-for-expanding-sets","kind":"node","lean":"Finset.exists_injective_of_mul_card_le_card_smul","parent":2350,"result":true,"route":"../roadmap/tauceti/combinatorics/other-results/index.html#halls-theorem-for-expanding-sets","scope":"tauceti/combinatorics/other-results","status":"fully_proved","summary":"If a finite set `D` acting on `\u03b2` satisfies `k \u00b7 |F| \u2264 |D \u2022 F|` for every finite `F \u2286 \u03b2`, then there is an injective map `j : \u03b2 \u00d7 Fin k \u2192 \u03b2` moving each point by an element of `D`: `j (x, i) = d \u2022 x` for some `d \u2208 D`.","title":"Hall's theorem for expanding sets"},{"id":"tauceti/combinatorics/other-results/hoeffdings-inequality-for-a-color-count","kind":"node","lean":"TauCeti.measureReal_le_abs_card_filter_sub_le","parent":2350,"result":true,"route":"../roadmap/tauceti/combinatorics/other-results/index.html#hoeffdings-inequality-for-a-color-count","scope":"tauceti/combinatorics/other-results","status":"fully_proved","summary":"Color the points of a finite type `\u03b9` independently according to a `PMF` `q` on `\u03ba`. The number of points of `T` with color `c` deviates from its mean `q c * #T` by at least `a \u2265 0` with probability at most `2 * exp (-2 a\u00b2 / #T)`.","title":"Hoeffding's inequality for a color count"},{"id":"tauceti/combinatorics/other-results/hoeffdings-lemma-on-the-grid-0-1-r-1","kind":"node","lean":"TauCeti.sum_range_exp_mul_div_sub_half_le","parent":2350,"result":true,"route":"../roadmap/tauceti/combinatorics/other-results/index.html#hoeffdings-lemma-on-the-grid-0-1-r-1","scope":"tauceti/combinatorics/other-results","status":"fully_proved","summary":"For `r \u2260 0` and every real `t`, the exponential moment of the uniformly distributed centred point `k / r - 1 / 2` satisfies `\u2211 k \u2208 range (r + 1), exp (t (k / r - 1 / 2)) \u2264 (r + 1) exp (t\u00b2 / 8)`.","title":"Hoeffding's lemma on the grid `{0, 1 / r, \u2026, 1}`"},{"id":"tauceti/combinatorics/other-results/mengers-theorem-for-vertex-deletion-in-a-digraph","kind":"node","lean":"Digraph.forall_reachableAvoiding_iff_exists_paths, Digraph.exists_paths_vertexSeparator_card_eq","parent":2350,"result":true,"route":"../roadmap/tauceti/combinatorics/other-results/index.html#mengers-theorem-for-vertex-deletion-in-a-digraph","scope":"tauceti/combinatorics/other-results","status":"fully_proved","summary":"Distinct nonadjacent vertices remain joined after deleting any fewer than `k` other vertices exactly when there are `k` pairwise internally vertex-disjoint directed paths between them.","title":"Menger's theorem for vertex deletion in a digraph, predicate form"},{"id":"tauceti/combinatorics/other-results/the-riemann-hurwitz-inequality-for-a-block-quoti","kind":"node","lean":"TauCeti.PermutationTriple.eulerChar_le_ncard_mul_eulerChar_blockQuotient","parent":2350,"result":true,"route":"../roadmap/tauceti/combinatorics/other-results/index.html#the-riemann-hurwitz-inequality-for-a-block-quoti","scope":"tauceti/combinatorics/other-results","status":"fully_proved","summary":"The Euler characteristic of a triple with transitive monodromy is at most `|B|` times that of its quotient by a nonempty block `B`.","title":"The Riemann\u2013Hurwitz inequality for a block quotient"},{"id":"tauceti/combinatorics/other-results/schutzenbergers-symmetry-theorem","kind":"node","lean":"Equiv.Perm.rsk_inv","parent":2350,"result":true,"route":"../roadmap/tauceti/combinatorics/other-results/index.html#schutzenbergers-symmetry-theorem","scope":"tauceti/combinatorics/other-results","status":"fully_proved","summary":"The Robinson--Schensted correspondence of the inverse permutation exchanges the insertion and recording tableaux.","title":"Sch\u00fctzenberger's symmetry theorem"},{"id":"tauceti/combinatorics/other-results/vertex-capacitated-mengers-theorem-for-a-digraph","kind":"node","lean":"Digraph.exists_paths_vertexSeparator_sum_eq","parent":2350,"result":true,"route":"../roadmap/tauceti/combinatorics/other-results/index.html#vertex-capacitated-mengers-theorem-for-a-digraph","scope":"tauceti/combinatorics/other-results","status":"fully_proved","summary":"Let `s` and `t` be distinct vertices with no arrow from `s` to `t`, and let `c v` be the capacity of vertex `v`. Some family of `k` directed `s`-to-`t` paths respects every nonterminal vertex capacity, and some terminal-excluding vertex separator has total capacity `k`.","title":"Vertex-capacitated Menger's theorem for a digraph"},{"data_url":"scopes/tauceti/field-theory.json","id":"tauceti/field-theory","kind":"scope","parent":1095,"result":false,"route":"../roadmap/tauceti/field-theory/index.html","scope":"tauceti/field-theory","status":"fully_proved","summary":"Results in Tau Ceti's `TauCeti.FieldTheory` modules, and the roadmaps whose articles are mostly there: by roadmap, then by area.","title":"Field Theory"},{"data_url":"scopes/tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri.json","id":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri","kind":"scope","parent":2359,"result":false,"route":"../roadmap/tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/index.html","scope":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Algebraic curves \u2014 function fields, divisors, and Riemann\u2013Roch"},{"id":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/the-riemann-roch-theorem-stichtenoth-theorem-1-5","kind":"node","lean":"TauCeti.isRiemannRochDivisor_weilDifferentialDivisor, TauCeti.Divisor.dim_eq_degree_add_one_sub_genus_of_two_mul_genus_sub_one_le_degree, TauCeti.isRiemannRochDivisor_of_isGreatest_mem_weilDifferentialFiltration, TauCeti.exists_isRiemannRochDivisor","parent":2360,"result":true,"route":"../roadmap/tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/index.html#the-riemann-roch-theorem-stichtenoth-theorem-1-5","scope":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri","status":"fully_proved","summary":"the divisor of a nonzero Weil differential is a Riemann\u2013Roch divisor for the genus.","title":"The Riemann\u2013Roch theorem (Stichtenoth, Theorem 1.5.15), for the divisor `(\u03c9)` itself"},{"id":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/the-hurwitz-genus-formula-stichtenoth-theorem-3","kind":"node","lean":"TauCeti.hurwitz_genus_formula, TauCeti.hurwitz_genus_formula_ratFunc, TauCeti.hurwitz_genus_formula_geometricDegree, TauCeti.two_mul_genus_sub_two_eq_iff_forall_isTame, TauCeti.geometricDegree_mul_add_degree_tameDifferent_le","parent":2360,"result":true,"route":"../roadmap/tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/index.html#the-hurwitz-genus-formula-stichtenoth-theorem-3","scope":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri","status":"fully_proved","summary":"for a finite separable extension `F' / k'` of the function field `F / k`, both with exact constants and with `k' / k` finite separable, `[k' : k] \u00b7 (2g' - 2) = [F' : F] \u00b7 (2g - 2) + [k' : k] \u00b7 deg Diff(F'/F)`.","title":"The Hurwitz genus formula (Stichtenoth, Theorem 3.4.13)"},{"id":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/the-anti-equivalence-between-regular-proper-curv","kind":"node","lean":"TauCeti.AlgebraicGeometry.RegularProperCurveCat.equivFunctionFieldCat","parent":2360,"result":true,"route":"../roadmap/tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/index.html#the-anti-equivalence-between-regular-proper-curv","scope":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri","status":"fully_proved","summary":"the function-field functor `X \u21a6 k(X)` is an equivalence `RegularProperCurveCat k \u224c (FunctionFieldCat k)\u1d52\u1d56` with inverse the model functor `F \u21a6 twoChartModel F`.","title":"The anti-equivalence between regular proper curves and algebraic function fields over a field `k`"},{"id":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/abhyankars-lemma-stichtenoth-theorem-3-9-1","kind":"node","lean":"TauCeti.Place.ramificationIdx_restrict_sup, TauCeti.Place.ramificationIdx_restrict_sup_of_isGalois","parent":2360,"result":true,"route":"../roadmap/tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/index.html#abhyankars-lemma-stichtenoth-theorem-3-9-1","scope":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri","status":"fully_proved","summary":"Let `E\u2081` and `E\u2082` be intermediate fields of a finite separable extension `F' / F` of function fields, and `P'` a place of `F'` such that the residue field of the place `P` below it in `F` is perfect, for instance because `k` is perfect (`TauCeti.Place.perfectField_residueField`). If the ramification index of `P' \u2229 E\u2081` over `P` is nonzero in `k`, then the ramification index of `P' \u2229 E\u2081 E\u2082` over `P` is the least\u2026","title":"Abhyankar's lemma (Stichtenoth, Theorem 3.9.1)"},{"id":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/the-genus-of-a-smooth-projective-plane-curve-ful","kind":"node","lean":"TauCeti.genus_eq_of_isHomogeneous","parent":2360,"result":true,"route":"../roadmap/tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/index.html#the-genus-of-a-smooth-projective-plane-curve-ful","scope":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri","status":"fully_proved","summary":"let `G \u2208 k[X\u2080, X\u2081, X\u2082]` be homogeneous of degree `d`, with every point of `G = 0` over an algebraically closed extension `K` of `k` nonsingular. Let `F / k` be generated by a transcendental `x` and an element `y` with `G(x, y, 1) = 0`. Then `F` has genus `(d - 1)(d - 2) / 2`.","title":"The genus of a smooth projective plane curve (Fulton, *Algebraic Curves*, Section 8.3)"},{"id":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/tauceti-divisor-sectionring","kind":"node","lean":"TauCeti.Divisor.sectionRing","parent":2360,"result":true,"route":"../roadmap/tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/index.html#tauceti-divisor-sectionring","scope":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri","status":"fully_proved","summary":"The section ring `\u2a01\u2099 L(nD)` of a divisor `D`: the `k`-subalgebra of `F[X]` of polynomials `\u2211 f\u2099 X\u207f` with `f\u2099 \u2208 L(nD)` for every `n`.","title":"TauCeti.Divisor.sectionRing"},{"id":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/high-powers-of-a-transcendental-function-have-po","kind":"node","lean":"TauCeti.riemannRochSpace_nsmul_poles_pow_le_pow","parent":2360,"result":true,"route":"../roadmap/tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/index.html#high-powers-of-a-transcendental-function-have-po","scope":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri","status":"fully_proved","summary":"over an exact constant field, if `z` is transcendental over `k`, `e \u2265 1` and `e \u2265 2g`, then every section of `n (z\u1d49)\u221e` is a sum of products of `n` sections of `(z\u1d49)\u221e`.","title":"High powers of a transcendental function have pole divisors whose section ring is generated in degree one"},{"id":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/the-automorphism-group-acting-on-the-weil-differ","kind":"node","lean":"TauCeti.weilDifferentialRepresentation","parent":2360,"result":true,"route":"../roadmap/tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/index.html#the-automorphism-group-acting-on-the-weil-differ","scope":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri","status":"fully_proved","summary":"the restriction of the dual of `TauCeti.repartitionRepresentation` to the stable subspace `\u03a9_F`, so that `(\u03c3 \u2022 \u03c9) a = \u03c9 (\u03c3\u207b\u00b9 \u2022 a)`.","title":"The automorphism group acting on the Weil differentials of `F' / k`"},{"id":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/at-every-place-the-completion-is-f-p-t-separable","kind":"node","lean":"TauCeti.Place.exists_completion_ringEquiv_laurentSeries","parent":2360,"result":true,"route":"../roadmap/tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/index.html#at-every-place-the-completion-is-f-p-t-separable","scope":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri","status":"fully_proved","summary":"some isomorphism of the completion with the Laurent-series field over the residue field carries the valuation of `P` to the `T`-adic valuation. When `F_P` is inseparable over `k` there may be no `k`-linear such isomorphism (`TauCeti.Place.isEmpty_completion_algEquiv_laurentSeries`).","title":"At every place the completion is `F_P((T))`, separable residue field or not"},{"id":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/the-abstract-residue-theorem-stichtenoth-1-45","kind":"node","lean":"TauCeti.finsum_repartitionDualComponent_eq_zero","parent":2360,"result":true,"route":"../roadmap/tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/index.html#the-abstract-residue-theorem-stichtenoth-1-45","scope":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri","status":"fully_proved","summary":"the local components of a Weil differential sum to zero on every function of `F`.","title":"The abstract residue theorem (Stichtenoth, (1.45))"},{"id":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/the-duality-theorem-in-dimensions","kind":"node","lean":"TauCeti.dim_sub_eq_finrank_weilDifferentialFiltration, TauCeti.riemannRochSpaceEquivWeilDifferentialFiltration","parent":2360,"result":true,"route":"../roadmap/tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/index.html#the-duality-theorem-in-dimensions","scope":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri","status":"fully_proved","summary":"`\u2113(W - D) = dim_k \u03a9_F(D)`.","title":"The duality theorem, in dimensions"},{"id":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/the-residue-theorem-stichtenoth-corollary-4-3-3","kind":"node","lean":"TauCeti.finsum_kaehlerResidueOfPerfectField_eq_zero","parent":2360,"result":true,"route":"../roadmap/tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/index.html#the-residue-theorem-stichtenoth-corollary-4-3-3","scope":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri","status":"fully_proved","summary":"over an algebraically closed field, the residues of a K\u00e4hler differential `\u03c9` at the places of `F` sum to zero, `\u2211_P res_P (\u03c9) = 0`. Every place is rational here, and only finitely many residues are nonzero (`TauCeti.finite_support_kaehlerResidueOfPerfectField`).","title":"The residue theorem (Stichtenoth, Corollary 4.3.3)"},{"id":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/the-residue-theorem-over-a-perfect-field-stichte","kind":"node","lean":"TauCeti.finsum_traceResidue_kaehlerBasisOfSeparating_coord_eq_zero","parent":2360,"result":true,"route":"../roadmap/tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/index.html#the-residue-theorem-over-a-perfect-field-stichte","scope":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri","status":"fully_proved","summary":"over a perfect exact constant field, choose a prime element `t_P` at each place `P`. Then the trace residues of a K\u00e4hler differential `\u03c9` sum to zero, `\u2211_P Tr_{F_P/k} res_{P,t_P} (\u03c9 / dt_P) = 0`, a sum with finitely many nonzero terms (`TauCeti.finite_support_traceResidue_kaehlerBasisOfSeparating_coord`). This is the residue theorem for Weil differentials `TauCeti.finsum_traceResidue_coord_eq_zero` at `u = 1`,\u2026","title":"The residue theorem over a perfect field (Stichtenoth, Corollary 4.3.3 and Remark 4.3.7(d))"},{"id":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/riemanns-theorem","kind":"node","lean":"TauCeti.Divisor.degree_add_one_sub_dim_le_genus, TauCeti.Divisor.degree_add_one_sub_genus_le_dim, TauCeti.Divisor.degree_add_one_le_dim_ratFunc","parent":2360,"result":true,"route":"../roadmap/tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/index.html#riemanns-theorem","scope":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri","status":"fully_proved","summary":"Riemann's theorem (Stichtenoth, Theorem 1.4.17), in the form that unwinds the definition of the genus: `deg D + 1 - \u2113(D) \u2264 g` for every divisor `D`.","title":"Riemann's theorem"},{"id":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/the-riemann-roch-identity-read-as-duality","kind":"node","lean":"TauCeti.Divisor.IsRiemannRochDivisor.indexOfSpecialty_eq","parent":2360,"result":true,"route":"../roadmap/tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri/index.html#the-riemann-roch-identity-read-as-duality","scope":"tauceti/field-theory/algebraic-curves-function-fields-divisors-and-ri","status":"fully_proved","summary":"for a Riemann\u2013Roch divisor `W`, the index of specialty of `D` is `\u2113(W - D)` (Stichtenoth, Theorem 1.5.14 in the presence of Theorem 1.5.15).","title":"The Riemann\u2013Roch identity, read as duality"},{"data_url":"scopes/tauceti/field-theory/galois-groups-of-polynomials.json","id":"tauceti/field-theory/galois-groups-of-polynomials","kind":"scope","parent":2359,"result":false,"route":"../roadmap/tauceti/field-theory/galois-groups-of-polynomials/index.html","scope":"tauceti/field-theory/galois-groups-of-polynomials","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Galois groups of polynomials"},{"id":"tauceti/field-theory/galois-groups-of-polynomials/tauceti-exists-monic-int-polynomial-hasfullsymme","kind":"node","lean":"TauCeti.exists_monic_int_polynomial_hasFullSymmetricGaloisGroup","parent":2376,"result":true,"route":"../roadmap/tauceti/field-theory/galois-groups-of-polynomials/index.html#tauceti-exists-monic-int-polynomial-hasfullsymme","scope":"tauceti/field-theory/galois-groups-of-polynomials","status":"fully_proved","summary":"Every positive degree occurs for a monic integral polynomial irreducible over `\u211a` with full symmetric Galois group.","title":"TauCeti.exists_monic_int_polynomial_hasFullSymmetricGaloisGroup"},{"id":"tauceti/field-theory/galois-groups-of-polynomials/tauceti-existsunique-transitivegrouplabel-five","kind":"node","lean":"TauCeti.existsUnique_transitiveGroupLabel_five","parent":2376,"result":true,"route":"../roadmap/tauceti/field-theory/galois-groups-of-polynomials/index.html#tauceti-existsunique-transitivegrouplabel-five","scope":"tauceti/field-theory/galois-groups-of-polynomials","status":"fully_proved","summary":"A transitive subgroup of the symmetric group on five points carries exactly one label, one of `5T1`, \u2026, `5T5`.","title":"TauCeti.existsUnique_transitiveGroupLabel_five"},{"id":"tauceti/field-theory/galois-groups-of-polynomials/the-factorization-theorem-for-a-separable-resolv","kind":"node","lean":"TauCeti.ResolventSpec.orbitQuotientEquivFactors","parent":2376,"result":true,"route":"../roadmap/tauceti/field-theory/galois-groups-of-polynomials/index.html#the-factorization-theorem-for-a-separable-resolv","scope":"tauceti/field-theory/galois-groups-of-polynomials","status":"fully_proved","summary":"Let `f` be monic of degree `n`, let `E` be a normal splitting extension, number the roots of `f` in `E` by `e`, and let the resolvent of `f` for the specification be separable. The orbits of the Galois image, read through `e`, on the cosets of `H` are then in bijection with the monic irreducible factors of the resolvent over `F`: the orbit of the coset of `\u03c4` goes to the minimal polynomial of the value at the roots\u2026","title":"The factorization theorem for a separable resolvent"},{"id":"tauceti/field-theory/galois-groups-of-polynomials/the-quintic-resolvent-solvability-criterion","kind":"node","lean":"TauCeti.isSolvable_gal_iff_exists_isRoot_specialize_quinticF20Spec","parent":2376,"result":true,"route":"../roadmap/tauceti/field-theory/galois-groups-of-polynomials/index.html#the-quintic-resolvent-solvability-criterion","scope":"tauceti/field-theory/galois-groups-of-polynomials","status":"fully_proved","summary":"Let `f` be a monic irreducible separable quintic over a field. If the specialization of Dummit's `F\u2082\u2080` resolvent is separable, then the polynomial Galois group is solvable if and only if that resolvent has a root in the base field.","title":"The quintic resolvent solvability criterion"},{"id":"tauceti/field-theory/galois-groups-of-polynomials/tauceti-resolventspec","kind":"node","lean":"TauCeti.ResolventSpec","parent":2376,"result":true,"route":"../roadmap/tauceti/field-theory/galois-groups-of-polynomials/index.html#tauceti-resolventspec","scope":"tauceti/field-theory/galois-groups-of-polynomials","status":"fully_proved","summary":"A resolvent specification in degree `n`: a subgroup `H` of `Equiv.Perm (Fin n)`, an integral invariant `\u03a6` in `n` formal roots whose stabilizer under renaming is exactly `H`, and the integral expression of the orbit product of `\u03a6` in the elementary symmetric polynomials. No polynomial and no coefficient ring appears; specialization is `ResolventSpec.specialize`.","title":"TauCeti.ResolventSpec"},{"id":"tauceti/field-theory/galois-groups-of-polynomials/separation-evidence-for-a-specialized-resolvent","kind":"node","lean":"TauCeti.ResolventSeparationEvidence","parent":2376,"result":true,"route":"../roadmap/tauceti/field-theory/galois-groups-of-polynomials/index.html#separation-evidence-for-a-specialized-resolvent","scope":"tauceti/field-theory/galois-groups-of-polynomials","status":"fully_proved","summary":"This records that the specialized resolvent is separable. Under the usual monic degree-`n` splitting description, it says that distinct elements of the symbolic orbit take distinct values at the roots of `f`. This is the hypothesis that lets a root or a factorization of the resolvent constrain the Galois image; specialization never loses degree, but its values can collide.","title":"Separation evidence for a specialized resolvent"},{"id":"tauceti/field-theory/galois-groups-of-polynomials/tauceti-hasgaloislabel","kind":"node","lean":"TauCeti.HasGaloisLabel","parent":2376,"result":true,"route":"../roadmap/tauceti/field-theory/galois-groups-of-polynomials/index.html#tauceti-hasgaloislabel","scope":"tauceti/field-theory/galois-groups-of-polynomials","status":"fully_proved","summary":"The Galois group of `f` carries the transitive-group label `j` of degree `n`: `f` is separable of degree `n`, and some numbering of its roots in the splitting field by `Fin n` carries the image of the Galois action on the roots to a conjugate of the reference subgroup `referenceSubgroup n j`. By `TauCeti.hasGaloisLabel_iff_forall`, every numbering then does.","title":"TauCeti.HasGaloisLabel"},{"id":"tauceti/field-theory/galois-groups-of-polynomials/polynomial-gal-sign","kind":"node","lean":"Polynomial.Gal.sign","parent":2376,"result":true,"route":"../roadmap/tauceti/field-theory/galois-groups-of-polynomials/index.html#polynomial-gal-sign","scope":"tauceti/field-theory/galois-groups-of-polynomials","status":"fully_proved","summary":"The parity character of the action of `p.Gal` on the distinct roots of `p` in its splitting field. It takes values in `\u2124\u02e3`, independently of the characteristic of the base field.","title":"Polynomial.Gal.sign"},{"id":"tauceti/field-theory/galois-groups-of-polynomials/vietas-formulas-in-the-form-the-symmetric-descen","kind":"node","lean":"TauCeti.vietaHom_eq_comp_esymmSubst","parent":2376,"result":true,"route":"../roadmap/tauceti/field-theory/galois-groups-of-polynomials/index.html#vietas-formulas-in-the-form-the-symmetric-descen","scope":"tauceti/field-theory/galois-groups-of-polynomials","status":"fully_proved","summary":"if `f` is the product of the linear factors attached to `x`, then substituting the signed coefficients of `f` is the same as substituting the elementary symmetric polynomials and evaluating at `x`.","title":"Vieta's formulas, in the form the symmetric descent uses"},{"data_url":"scopes/tauceti/field-theory/function-field.json","id":"tauceti/field-theory/function-field","kind":"scope","parent":2359,"result":false,"route":"../roadmap/tauceti/field-theory/function-field/index.html","scope":"tauceti/field-theory/function-field","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.FieldTheory.FunctionField` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Function Field"},{"data_url":"scopes/tauceti/field-theory/function-field/part-01.json","id":"tauceti/field-theory/function-field/part-01","kind":"scope","navigation":"transparent","parent":2386,"result":false,"route":"../roadmap/tauceti/field-theory/function-field/part-01/index.html","scope":"tauceti/field-theory/function-field/part-01","status":"fully_proved","title":"The Artin--Schreier genus\u2026 to The tame different theorem\u2026"},{"id":"tauceti/field-theory/function-field/part-01/the-artin-schreier-genus-formula-cross-multiplie","kind":"node","lean":"TauCeti.artinSchreier_genus_formula","parent":2387,"result":true,"route":"../roadmap/tauceti/field-theory/function-field/part-01/index.html#the-artin-schreier-genus-formula-cross-multiplie","scope":"tauceti/field-theory/function-field/part-01","status":"fully_proved","summary":"The representatives may vary with the place. Pole orders are weighted by residue field degrees, not by the number of poles.","title":"The Artin--Schreier genus formula, cross-multiplied for a possible constant-field extension"},{"id":"tauceti/field-theory/function-field/part-01/castelnuovos-inequality-stichtenoth-theorem-3-11","kind":"node","lean":"TauCeti.genus_le_finrank_mul_genus_add_finrank_mul_genus_add, TauCeti.genus_le_finrank_mul_genus_add_finrank_mul_genus_add_of_isSplitCompletely, TauCeti.genus_le_finrank_mul_genus_add_finrank_mul_genus_add_of_perfectField","parent":2387,"result":true,"route":"../roadmap/tauceti/field-theory/function-field/part-01/index.html#castelnuovos-inequality-stichtenoth-theorem-3-11","scope":"tauceti/field-theory/function-field/part-01","status":"fully_proved","summary":"Let `F / k` be a function field with exact constants and a separating element `x`, and let `F\u2081`, `F\u2082` be subfields of `F` containing `k` with `F = F\u2081 F\u2082` and `F` finite over both. Then `g(F) \u2264 [F : F\u2081] g(F\u2081) + [F : F\u2082] g(F\u2082) + ([F : F\u2081] - 1) ([F : F\u2082] - 1)`. The constant field `k` may be imperfect, and neither subextension is assumed separable.","title":"Castelnuovo's inequality (Stichtenoth, Theorem 3.11.3)"},{"id":"tauceti/field-theory/function-field/part-01/cliffords-theorem","kind":"node","lean":"TauCeti.Divisor.two_mul_dim_le_degree_add_two","parent":2387,"result":true,"route":"../roadmap/tauceti/field-theory/function-field/part-01/index.html#cliffords-theorem","scope":"tauceti/field-theory/function-field/part-01","status":"fully_proved","summary":"Clifford's theorem (Stichtenoth, Theorem 1.6.13), over an arbitrary exact field of constants: a divisor of degree between `0` and `2g - 2` satisfies","title":"Clifford's theorem"},{"id":"tauceti/field-theory/function-field/part-01/dedekinds-different-theorem-first-part-stichteno","kind":"node","lean":"TauCeti.Place.ramificationIdx_le_differentExponent_add_one, TauCeti.Divisor.ramificationIdx_le_coeff_different_add_one, TauCeti.Divisor.tameDifferent_le_different, TauCeti.Place.ramificationIdx_eq_differentExponent_add_one_iff","parent":2387,"result":true,"route":"../roadmap/tauceti/field-theory/function-field/part-01/index.html#dedekinds-different-theorem-first-part-stichteno","scope":"tauceti/field-theory/function-field/part-01","status":"fully_proved","summary":"the different exponent of a place is at least one less than its ramification index. It is stated as `e(P' \u2223 P) \u2264 d(P' \u2223 P) + 1` so that no truncated subtraction of natural numbers appears.","title":"Dedekind's different theorem, first part (Stichtenoth, Theorem 3.5.1(a))"},{"id":"tauceti/field-theory/function-field/part-01/the-eisenstein-criterion-stichtenoth-proposition","kind":"node","lean":"TauCeti.Place.isTotallyRamified_of_isEisensteinAt","parent":2387,"result":true,"route":"../roadmap/tauceti/field-theory/function-field/part-01/index.html#the-eisenstein-criterion-stichtenoth-proposition","scope":"tauceti/field-theory/function-field/part-01","status":"fully_proved","summary":"if `F' = F(y)` and the minimal polynomial of `y` over `F` is Eisenstein at the place of `F / k` below `P'`, then `P'` is totally ramified over `F`. Together with `TauCeti.Place.setOf_restrict_eq_eq_singleton_of_isTotallyRamified` this says that the place below `P'` is totally ramified in `F' / F`: it has exactly one extension, of ramification index `[F' : F]` and relative degree `1`.","title":"The Eisenstein criterion (Stichtenoth, Proposition 3.1.15)"},{"id":"tauceti/field-theory/function-field/part-01/the-fundamental-identity","kind":"node","lean":"TauCeti.Place.sum_ramificationIdx_mul_relativeDegree_eq_finrank, TauCeti.Place.sum_ramificationIdx_mul_relativeDegree_eq_finrank_of_isFunctionField, TauCeti.Place.sum_ramificationIdx_mul_relativeDegree_eq_finrank_of_isSeparable","parent":2387,"result":true,"route":"../roadmap/tauceti/field-theory/function-field/part-01/index.html#the-fundamental-identity","scope":"tauceti/field-theory/function-field/part-01","status":"fully_proved","summary":"The fundamental identity (Stichtenoth, Theorem 3.1.11) at a place of the finite chart of a model: the ramification indices and relative degrees of the places of `F' / k'` lying over the place of a height one prime `\ud835\udd2d` of `R` satisfy `\u2211 e(P' \u2223 P) \u00b7 f(P' \u2223 P) = [F' : F]`. Together with `TauCeti.Place.sum_ramificationIdx_mul_relativeDegree_le_finrank` this settles the fibre over every place that lies on some affine\u2026","title":"The fundamental identity"},{"id":"tauceti/field-theory/function-field/part-01/the-fundamental-inequality-stichtenoth-theorem-3","kind":"node","lean":"TauCeti.Place.sum_ramificationIdx_mul_relativeDegree_le_finrank","parent":2387,"result":true,"route":"../roadmap/tauceti/field-theory/function-field/part-01/index.html#the-fundamental-inequality-stichtenoth-theorem-3","scope":"tauceti/field-theory/function-field/part-01","status":"fully_proved","summary":"the ramification indices and relative degrees of finitely many distinct places of `F' / k'` lying over one place `P` of `F / k` satisfy `\u2211 e(P' \u2223 P) \u00b7 f(P' \u2223 P) \u2264 [F' : F]`.","title":"The fundamental inequality (Stichtenoth, Theorem 3.1.11)"},{"id":"tauceti/field-theory/function-field/part-01/the-galois-correspondence-for-the-inertia-field","kind":"node","lean":"TauCeti.Place.fixingSubgroup_inertiaField, TauCeti.Place.fixingSubgroup_decompositionField","parent":2387,"result":true,"route":"../roadmap/tauceti/field-theory/function-field/part-01/index.html#the-galois-correspondence-for-the-inertia-field","scope":"tauceti/field-theory/function-field/part-01","status":"fully_proved","summary":"the automorphisms of `F'` fixing the inertia field of `P` pointwise are exactly the inertia group of `P`.","title":"The Galois correspondence for the inertia field"},{"id":"tauceti/field-theory/function-field/part-01/hilberts-different-formula","kind":"node","lean":"TauCeti.Place.differentExponent_eq_sum_range_card_ramificationGroup_sub_one, TauCeti.Place.differentExponent_eq_finsum_card_ramificationGroup_sub_one","parent":2387,"result":true,"route":"../roadmap/tauceti/field-theory/function-field/part-01/index.html#hilberts-different-formula","scope":"tauceti/field-theory/function-field/part-01","status":"fully_proved","summary":"Hilbert's different formula (Stichtenoth, Theorem 3.8.7), with the sum cut off at an index `N` from which the ramification groups are trivial: `d(P \u2223 P \u2229 F) = \u2211_{i < N} (|G_i(P)| - 1)`, when the residue extension is separable.","title":"Hilbert's different formula"},{"id":"tauceti/field-theory/function-field/part-01/the-hurwitz-bound-stichtenoth-exercise-3-18","kind":"node","lean":"TauCeti.natCard_le_eighty_four_mul_genus_sub_one, TauCeti.natCard_le_eighty_four_mul_genus_sub_one_of_not_dvd, TauCeti.card_algEquiv_le_eighty_four_mul_genus_sub_one_of_charZero, TauCeti.natCard_le_eighty_four_mul_genus_sub_one_of_charZero","parent":2387,"result":true,"route":"../roadmap/tauceti/field-theory/function-field/part-01/index.html#the-hurwitz-bound-stichtenoth-exercise-3-18","scope":"tauceti/field-theory/function-field/part-01","status":"fully_proved","summary":"a finite group `G` of automorphisms of a function field of genus `g \u2265 2`, every place of which is tame over the fixed field, has order at most `84 (g - 1)`.","title":"The Hurwitz bound (Stichtenoth, Exercise 3.18)"},{"id":"tauceti/field-theory/function-field/part-01/kummers-theorem-stichtenoth-corollary-3-3-8","kind":"node","lean":"TauCeti.Place.restrictOfPrimeEquivNormalizedFactors, TauCeti.Place.exists_injective_restrict_eq, TauCeti.Place.exists_restrict_eq_of_irreducible_map_residue, TauCeti.Place.exists_restrict_eq_relativeDegree_eq_and_isSeparable, TauCeti.Place.exists_finset_card_eq_forall_exists_restrict_eq_valuation_sub_lt_one, TauCeti.Place.exists_equiv_restrict_eq_of_prod_eq_map_residue, TauCeti.Place.restrictEquivNormalizedFactors","parent":2387,"result":true,"route":"../roadmap/tauceti/field-theory/function-field/part-01/index.html#kummers-theorem-stichtenoth-corollary-3-3-8","scope":"tauceti/field-theory/function-field/part-01","status":"fully_proved","summary":"when the height one prime `\ud835\udd2d` of the model `R` is prime to the conductor of `R[y]` in the model `S`, the places of `F' / k'` lying over the place of `\ud835\udd2d` are exactly the normalized irreducible factors of `minpoly R y` modulo `\ud835\udd2d`.","title":"Kummer's theorem (Stichtenoth, Corollary 3.3.8)"},{"id":"tauceti/field-theory/function-field/part-01/kummers-theorem-sees-one-factor-per-place","kind":"node","lean":"TauCeti.Place.map_residue_eq_of_valuation_lt_one","parent":2387,"result":true,"route":"../roadmap/tauceti/field-theory/function-field/part-01/index.html#kummers-theorem-sees-one-factor-per-place","scope":"tauceti/field-theory/function-field/part-01","status":"fully_proved","summary":"a place `P'` over `P` at which two monic polynomials with irreducible reductions both vanish reduces them to the same irreducible polynomial, namely the minimal polynomial of the residue of `y`. This is what makes the places attached to distinct irreducible factors of the reduction of `\u03c6` distinct (Stichtenoth, Theorem 3.3.7).","title":"Kummer's theorem sees one factor per place"},{"id":"tauceti/field-theory/function-field/part-01/the-relative-degree-bound-of-kummers-theorem-sti","kind":"node","lean":"TauCeti.Place.natDegree_map_residue_le_relativeDegree","parent":2387,"result":true,"route":"../roadmap/tauceti/field-theory/function-field/part-01/index.html#the-relative-degree-bound-of-kummers-theorem-sti","scope":"tauceti/field-theory/function-field/part-01","status":"fully_proved","summary":"a place `P'` over `P` at which a monic `g` with irreducible reduction `\u03b3` vanishes has relative degree at least `deg \u03b3`, because `\u03b3` is the minimal polynomial of the residue of `y` at `P'`.","title":"The relative degree bound of Kummer's theorem (Stichtenoth, Theorem 3.3.7)"},{"id":"tauceti/field-theory/function-field/part-01/the-residue-action-on-residues-stichtenoth-theor","kind":"node","lean":"TauCeti.Place.residueAut_residue","parent":2387,"result":true,"route":"../roadmap/tauceti/field-theory/function-field/part-01/index.html#the-residue-action-on-residues-stichtenoth-theor","scope":"tauceti/field-theory/function-field/part-01","status":"fully_proved","summary":"the automorphism of `F'_P` induced by `g` sends the residue of an element `x` of `\ud835\udcaa_P` to the residue of `g x`.","title":"The residue action, on residues (Stichtenoth, Theorem 3.8.2)"},{"id":"tauceti/field-theory/function-field/part-01/the-residue-theorem-for-k-x-stichtenoth-corollar","kind":"node","lean":"TauCeti.finsum_residue_adicOfIrreducible_X_sub_C","parent":2387,"result":true,"route":"../roadmap/tauceti/field-theory/function-field/part-01/index.html#the-residue-theorem-for-k-x-stichtenoth-corollar","scope":"tauceti/field-theory/function-field/part-01","status":"fully_proved","summary":"if every pole of `z` is a rational place, then the residues of `z dx` over `\u2119\u00b9(k)` sum to zero, in the form `\u2211_{a \u2208 k} res_{P_a, x - a} (z) = res_{P_\u221e, x\u207b\u00b9} (x\u00b2 z)`.","title":"The residue theorem for `k(x)` (Stichtenoth, Corollary 4.3.3, for `k(x)`)"},{"id":"tauceti/field-theory/function-field/part-01/the-residue-theorem-for-k-x-over-an-algebraicall","kind":"node","lean":"TauCeti.finsum_residue_adicOfIrreducible_X_sub_C_of_isAlgClosed","parent":2387,"result":true,"route":"../roadmap/tauceti/field-theory/function-field/part-01/index.html#the-residue-theorem-for-k-x-over-an-algebraicall","scope":"tauceti/field-theory/function-field/part-01","status":"fully_proved","summary":"`\u2211_{a \u2208 k} res_{P_a, x - a} (z) = res_{P_\u221e, x\u207b\u00b9} (x\u00b2 z)` for every rational function `z`.","title":"The residue theorem for `k(x)` over an algebraically closed field (Stichtenoth, Corollary 4.3.3, for `k(x)`)"},{"id":"tauceti/field-theory/function-field/part-01/riemanns-inequality-stichtenoth-corollary-3-11-4","kind":"node","lean":"TauCeti.genus_le_finrank_sub_one_mul_finrank_sub_one","parent":2387,"result":true,"route":"../roadmap/tauceti/field-theory/function-field/part-01/index.html#riemanns-inequality-stichtenoth-corollary-3-11-4","scope":"tauceti/field-theory/function-field/part-01","status":"fully_proved","summary":"if `F / k` has exact constants and is generated by two transcendental elements `x` and `y`, then `g \u2264 ([F : k(x)] - 1) ([F : k(y)] - 1)`.","title":"Riemann's inequality (Stichtenoth, Corollary 3.11.4)"},{"id":"tauceti/field-theory/function-field/part-01/stichtenoths-transformation-formula","kind":"node","lean":"TauCeti.Place.residueResidueField_eq_residueResidueField_mul_derivativeOfSeparating, TauCeti.Place.residue_eq_residue_mul, TauCeti.Place.residue_eq_residue_mul_derivativeOfSeparating","parent":2387,"result":true,"route":"../roadmap/tauceti/field-theory/function-field/part-01/index.html#stichtenoths-transformation-formula","scope":"tauceti/field-theory/function-field/part-01","status":"fully_proved","summary":"Stichtenoth's transformation formula `res_{P,s}(z) = res_{P,t}(z \u00b7 ds/dt)` at a place with separable residue field, for two uniformizers `s` and `t` with `t` separating, the residues taking values in `F_P` (Proposition 4.2.9).","title":"Stichtenoth's transformation formula"},{"id":"tauceti/field-theory/function-field/part-01/strong-approximation-stichtenoth-theorem-1-6-5","kind":"node","lean":"TauCeti.Place.exists_forall_mem_ord_sub_eq_and_forall_mem_integers, TauCeti.Place.exists_forall_mem_valuation_sub_le_and_forall_mem_integers","parent":2387,"result":true,"route":"../roadmap/tauceti/field-theory/function-field/part-01/index.html#strong-approximation-stichtenoth-theorem-1-6-5","scope":"tauceti/field-theory/function-field/part-01","status":"fully_proved","summary":"for a set `S` of places of `F / k` that is not all of them and a finite subset `s \u2286 S`, some `x : F` satisfies `ord_P (x - f P) = r P` at every `P \u2208 s` and is regular at every place of `S` outside `s`.","title":"Strong approximation (Stichtenoth, Theorem 1.6.5)"},{"id":"tauceti/field-theory/function-field/part-01/the-tame-different-theorem-fails-at-infinity","kind":"node","lean":"TauCeti.ArtinSchreierMonomial.ramificationIdx_le_differentExponent_of_restrict_eq_infty","parent":2387,"result":true,"route":"../roadmap/tauceti/field-theory/function-field/part-01/index.html#the-tame-different-theorem-fails-at-infinity","scope":"tauceti/field-theory/function-field/part-01","status":"fully_proved","summary":"the different exponent `d` is at least the ramification index `e`, so `e \u2260 d + 1`, unlike at every tame place.","title":"The tame different theorem fails at infinity"},{"data_url":"scopes/tauceti/field-theory/function-field/part-02.json","id":"tauceti/field-theory/function-field/part-02","kind":"scope","navigation":"transparent","parent":2386,"result":false,"route":"../roadmap/tauceti/field-theory/function-field/part-02/index.html","scope":"tauceti/field-theory/function-field/part-02","status":"fully_proved","title":"The weak approximation theorem to Weierstrass gap theorem\u2026"},{"id":"tauceti/field-theory/function-field/part-02/the-weak-approximation-theorem","kind":"node","lean":"TauCeti.Place.exists_forall_ord_sub_eq","parent":2408,"result":true,"route":"../roadmap/tauceti/field-theory/function-field/part-02/index.html#the-weak-approximation-theorem","scope":"tauceti/field-theory/function-field/part-02","status":"fully_proved","summary":"The weak approximation theorem for the places of an algebraic function field (Stichtenoth, Theorem 1.3.1), in the equality form: for finitely many pairwise distinct places `P i`, arbitrary target functions `f i` and arbitrary prescribed integers `r i`, some single `g : F` satisfies `ord_{P i} (g - f i) = r i` for every `i`.","title":"The weak approximation theorem"},{"id":"tauceti/field-theory/function-field/part-02/weierstrass-gap-theorem-stichtenoth-theorem-1-6","kind":"node","lean":"TauCeti.Place.card_weierstrassGaps","parent":2408,"result":true,"route":"../roadmap/tauceti/field-theory/function-field/part-02/index.html#weierstrass-gap-theorem-stichtenoth-theorem-1-6","scope":"tauceti/field-theory/function-field/part-02","status":"fully_proved","summary":"at a rational place of a function field with integrally closed constants and genus `g`, there are exactly `g` gaps.","title":"Weierstrass gap theorem (Stichtenoth, Theorem 1.6.8)"},{"data_url":"scopes/tauceti/field-theory/galois.json","id":"tauceti/field-theory/galois","kind":"scope","parent":2359,"result":false,"route":"../roadmap/tauceti/field-theory/galois/index.html","scope":"tauceti/field-theory/galois","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.FieldTheory.Galois` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Galois"},{"id":"tauceti/field-theory/galois/artins-theorem","kind":"node","lean":"IntermediateField.finiteDimensional_fixedField, IntermediateField.isGalois_fixedField, IntermediateField.finrank_fixedField_eq_natCard","parent":2411,"result":true,"route":"../roadmap/tauceti/field-theory/galois/index.html#artins-theorem","scope":"tauceti/field-theory/galois","status":"fully_proved","summary":"`M` is finite over the field fixed by a finite group of automorphisms.","title":"Artin's theorem"},{"id":"tauceti/field-theory/galois/the-fundamental-theorem-of-galois-theory-in-the","kind":"node","lean":"TauCeti.intermediateFieldEquivClosedSubgroup","parent":2411,"result":true,"route":"../roadmap/tauceti/field-theory/galois/index.html#the-fundamental-theorem-of-galois-theory-in-the","scope":"tauceti/field-theory/galois","status":"fully_proved","summary":"For a normal extension `E/F` the closed subgroups of `Gal(E/F)` correspond, inclusion-reversingly, to the intermediate fields of the separable closure `separableClosure F E / F`; an intermediate field of `E` outside the separable closure has the same fixing subgroup as its separable part.","title":"The fundamental theorem of Galois theory in the ambient model"},{"id":"tauceti/field-theory/galois/hilberts-theorem-90-for-cyclic-extensions","kind":"node","lean":"TauCeti.exists_smul_div_eq_of_norm_eq_one","parent":2411,"result":true,"route":"../roadmap/tauceti/field-theory/galois/index.html#hilberts-theorem-90-for-cyclic-extensions","scope":"tauceti/field-theory/galois","status":"fully_proved","summary":"If `g` generates `Gal(L/K)` and the unit `x` of `L` has norm one, then `x = g y / y` for a unit `y` of `L`.","title":"Hilbert's Theorem 90 for cyclic extensions"},{"data_url":"scopes/tauceti/field-theory/other-results.json","id":"tauceti/field-theory/other-results","kind":"scope","parent":2359,"result":false,"route":"../roadmap/tauceti/field-theory/other-results/index.html","scope":"tauceti/field-theory/other-results","status":"fully_proved","summary":"Results named in the rest of Tau Ceti's `TauCeti.FieldTheory` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Other results"},{"id":"tauceti/field-theory/other-results/capellis-theorem","kind":"node","lean":"TauCeti.X_pow_sub_C_irreducible_iff_forall_prime, TauCeti.X_pow_sub_C_irreducible_of_forall_prime","parent":2415,"result":true,"route":"../roadmap/tauceti/field-theory/other-results/index.html#capellis-theorem","scope":"tauceti/field-theory/other-results","status":"fully_proved","summary":"Over any field `K` and for `n \u2260 0`, the binomial `X\u207f - a` is irreducible if and only if `a` is not a `p`-th power for any prime `p \u2223 n` and, when `4 \u2223 n`, `a` is not of the form `-4b\u2074`. For odd `n` this is Mathlib's `X_pow_sub_C_irreducible_iff_forall_prime_of_odd`.","title":"Capelli's theorem"},{"id":"tauceti/field-theory/other-results/the-comparison-theorem-tauceti-fixedfield-evenau","kind":"node","lean":"TauCeti.fixedField_evenAutSubgroup, TauCeti.fixingSubgroup_discrField","parent":2415,"result":true,"route":"../roadmap/tauceti/field-theory/other-results/index.html#the-comparison-theorem-tauceti-fixedfield-evenau","scope":"tauceti/field-theory/other-results","status":"fully_proved","summary":"In a Galois splitting extension, and away from characteristic `2`, the discriminant field of a separable polynomial is the field fixed by the automorphisms acting on the roots by an even permutation.","title":"The comparison theorem (`TauCeti.fixedField_evenAutSubgroup`)"},{"id":"tauceti/field-theory/other-results/the-cup-norm-theorem","kind":"node","lean":"TauCeti.cup_kummerClass_eq_zero_iff","parent":2415,"result":true,"route":"../roadmap/tauceti/field-theory/other-results/index.html#the-cup-norm-theorem","scope":"tauceti/field-theory/other-results","status":"fully_proved","summary":"For units `a` and `b` of a field in which `2` is invertible, the cup product `(a) \u222a (b) \u2208 H\u00b2(G_K, \ud835\udd3d\u2082)` of their Kummer classes vanishes if and only if the norm equation `b = x\u00b2 - a y\u00b2` has a solution in `K`.","title":"The cup-norm theorem"},{"id":"tauceti/field-theory/other-results/dummits-formula-for-the-resolvent-of-a-quintic-t","kind":"node","lean":"TauCeti.quinticF20Spec_specialize_X_pow_five_add_C_mul_X_add_C, TauCeti.resolventSextic_X_pow_five_add_C_mul_X_add_C","parent":2415,"result":true,"route":"../roadmap/tauceti/field-theory/other-results/index.html#dummits-formula-for-the-resolvent-of-a-quintic-t","scope":"tauceti/field-theory/other-results","status":"fully_proved","summary":"Over every commutative ring, the specialization of the quintic `F\u2082\u2080` specification at `X\u2075 + aX + b` is `X\u2076 + 8aX\u2075 + 40a\u00b2X\u2074 + 160a\u00b3X\u00b3 + 400a\u2074X\u00b2 + (512a\u2075 - 3125b\u2074)X + (256a\u2076 - 9375ab\u2074)`.","title":"Dummit's formula for the resolvent of a quintic trinomial"},{"id":"tauceti/field-theory/other-results/the-equivariant-kummer-isomorphism-on-kummer-cla","kind":"node","lean":"TauCeti.dualTensorHom_kummerH1FiniteRepresentationEquiv","parent":2415,"result":true,"route":"../roadmap/tauceti/field-theory/other-results/index.html#the-equivariant-kummer-isomorphism-on-kummer-cla","scope":"tauceti/field-theory/other-results","status":"fully_proved","summary":"Under `kummerH1FiniteRepresentationEquiv`, the Kummer class of a power class `x` corresponds to the homomorphism `\u03bc\u2099 \u2192 L\u02e3 \u29f8 (L\u02e3)\u207f` sending `\u03be` to `log \u03be \u2022 x`, with `log : \u03bc\u2099 \u2243 \u2124/n` the identification `kummerCoeffAddEquivZMod`.","title":"The equivariant Kummer isomorphism on Kummer classes (`TauCeti.dualTensorHom_kummerH1FiniteRepresentationEquiv`)"},{"id":"tauceti/field-theory/other-results/eulers-criterion-in-a-finite-field","kind":"node","lean":"TauCeti.FiniteField.exists_pow_eq_iff_pow_card_sub_one_div_eq_one","parent":2415,"result":true,"route":"../roadmap/tauceti/field-theory/other-results/index.html#eulers-criterion-in-a-finite-field","scope":"tauceti/field-theory/other-results","status":"fully_proved","summary":"If `n \u2223 #F - 1`, then a nonzero `a \u2208 F` is an `n`-th power exactly when `a ^ ((#F - 1) / n) = 1`.","title":"Euler's criterion in a finite field"},{"id":"tauceti/field-theory/other-results/serres-formula-for-the-evens-norm-of-a-kummer-cl","kind":"node","lean":"TauCeti.galoisEvens2_kummerClass_eq_cup_add_cup","parent":2415,"result":true,"route":"../roadmap/tauceti/field-theory/other-results/index.html#serres-formula-for-the-evens-norm-of-a-kummer-cl","scope":"tauceti/field-theory/other-results","status":"fully_proved","summary":"for an orthogonal basis `y` of the twisted trace form `Tr_*\u27e8a\u27e9 : z \u21a6 Tr (a z\u00b2)` whose values `w_j = Tr (a y_j\u00b2)` are units,","title":"Serre's formula for the Evens norm of a Kummer class (Serre, Th\u00e9or\u00e8me 1\u2032 at `n = 2`)"},{"id":"tauceti/field-theory/other-results/the-two-power-case-of-capellis-theorem","kind":"node","lean":"TauCeti.X_pow_two_pow_sub_C_irreducible","parent":2415,"result":true,"route":"../roadmap/tauceti/field-theory/other-results/index.html#the-two-power-case-of-capellis-theorem","scope":"tauceti/field-theory/other-results","status":"fully_proved","summary":"Over any field `K`, if `a` is not a square in `K` and, when `k \u2265 2`, `a` is not of the form `-4b\u2074`, then `X^(2\u1d4f) - a` is irreducible. The second condition cannot be dropped: `X\u2074 + 4` is reducible over `\u211a`.","title":"The two-power case of Capelli's theorem"},{"data_url":"scopes/tauceti/geometry.json","id":"tauceti/geometry","kind":"scope","parent":1095,"result":false,"route":"../roadmap/tauceti/geometry/index.html","scope":"tauceti/geometry","status":"fully_proved","summary":"Results in Tau Ceti's `TauCeti.Geometry` modules, and the roadmaps whose articles are mostly there: by roadmap, then by area.","title":"Geometry"},{"data_url":"scopes/tauceti/geometry/part-01.json","id":"tauceti/geometry/part-01","kind":"scope","navigation":"transparent","parent":2424,"result":false,"route":"../roadmap/tauceti/geometry/part-01/index.html","scope":"tauceti/geometry/part-01","status":"fully_proved","title":"Analytic toric geometry to Real Algebraic"},{"data_url":"scopes/tauceti/geometry/part-01/analytic-toric-geometry.json","id":"tauceti/geometry/part-01/analytic-toric-geometry","kind":"scope","parent":2425,"result":false,"route":"../roadmap/tauceti/geometry/part-01/analytic-toric-geometry/index.html","scope":"tauceti/geometry/part-01/analytic-toric-geometry","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Analytic toric geometry"},{"id":"tauceti/geometry/part-01/analytic-toric-geometry/tauceti-toric-fan-exists-partialdiffeomorph-anal","kind":"node","lean":"TauCeti.Toric.Fan.exists_partialDiffeomorph_analyticBoundaryComponent_normalForm","parent":2426,"result":true,"route":"../roadmap/tauceti/geometry/part-01/analytic-toric-geometry/index.html#tauceti-toric-fan-exists-partialdiffeomorph-anal","scope":"tauceti/geometry/part-01/analytic-toric-geometry","status":"fully_proved","summary":"The toric boundary has the holomorphic coordinate-hyperplane normal form near every point `x` of the realization. There are a biholomorphism `e` from an open neighbourhood of `x` onto an open subset of `\u2102 ^ n`, for `n` the rank of the lattice, and an injection `j` from the set `s` of boundary components through `x` to the coordinate indices, such that a point of the neighbourhood lies in a boundary component\u2026","title":"TauCeti.Toric.Fan.exists_partialDiffeomorph_analyticBoundaryComponent_normalForm"},{"id":"tauceti/geometry/part-01/analytic-toric-geometry/the-cone-by-cone-properness-criterion","kind":"node","lean":"TauCeti.Toric.FanHom.isProperMap_analyticMap_iff","parent":2426,"result":true,"route":"../roadmap/tauceti/geometry/part-01/analytic-toric-geometry/index.html#the-cone-by-cone-properness-criterion","scope":"tauceti/geometry/part-01/analytic-toric-geometry","status":"fully_proved","summary":"The analytic map of a morphism of regular fans with nonempty source is proper exactly when, for every target cone `\u03c4`, the inverse image of `\u03c4` under the real-linear map is the union of the source cones mapped into `\u03c4`. The nonemptiness hypothesis is necessary: the inclusion of the empty subfan is proper, but `0` lies in the inverse image of every cone and in no cone of the empty fan. See\u2026","title":"The cone-by-cone properness criterion"},{"id":"tauceti/geometry/part-01/analytic-toric-geometry/tauceti-toric-fan-ispropermap-analyticmap-starsu","kind":"node","lean":"TauCeti.Toric.Fan.isProperMap_analyticMap_starSubdivisionAlong","parent":2426,"result":true,"route":"../roadmap/tauceti/geometry/part-01/analytic-toric-geometry/index.html#tauceti-toric-fan-ispropermap-analyticmap-starsu","scope":"tauceti/geometry/part-01/analytic-toric-geometry","status":"fully_proved","summary":"The star subdivision of a regular fan `\u03a6` along a nonzero cone `\u03c3` of `\u03a6` induces a proper analytic map to the realization of `\u03a6`, because the two fans have the same support.","title":"TauCeti.Toric.Fan.isProperMap_analyticMap_starSubdivisionAlong"},{"id":"tauceti/geometry/part-01/analytic-toric-geometry/tauceti-toric-fan-compactspace-analyticrealizati","kind":"node","lean":"TauCeti.Toric.Fan.compactSpace_analyticRealization_iff_isComplete","parent":2426,"result":true,"route":"../roadmap/tauceti/geometry/part-01/analytic-toric-geometry/index.html#tauceti-toric-fan-compactspace-analyticrealizati","scope":"tauceti/geometry/part-01/analytic-toric-geometry","status":"fully_proved","summary":"A nonempty regular fan has compact analytic realization exactly when it is complete. The nonemptiness hypothesis is necessary: the realization of the empty fan is empty, hence compact, while the empty fan is not complete. See `TauCeti.Toric.Fan.compactSpace_analyticRealization_iff_isEmpty_or_isComplete` for the form without it.","title":"TauCeti.Toric.Fan.compactSpace_analyticRealization_iff_isComplete"},{"id":"tauceti/geometry/part-01/analytic-toric-geometry/tauceti-toric-fan-algebraicanalyticequiv-smul","kind":"node","lean":"TauCeti.Toric.Fan.algebraicAnalyticEquiv_smul","parent":2426,"result":true,"route":"../roadmap/tauceti/geometry/part-01/analytic-toric-geometry/index.html#tauceti-toric-fan-algebraicanalyticequiv-smul","scope":"tauceti/geometry/part-01/analytic-toric-geometry","status":"fully_proved","summary":"The algebraic\u2013analytic comparison is equivariant for the coordinate-free complex torus.","title":"TauCeti.Toric.Fan.algebraicAnalyticEquiv_smul"},{"id":"tauceti/geometry/part-01/analytic-toric-geometry/tauceti-toric-fan-algebraicrealizationmodobj","kind":"node","lean":"TauCeti.Toric.Fan.algebraicRealizationModObj","parent":2426,"result":true,"route":"../roadmap/tauceti/geometry/part-01/analytic-toric-geometry/index.html#tauceti-toric-fan-algebraicrealizationmodobj","scope":"tauceti/geometry/part-01/analytic-toric-geometry","status":"fully_proved","summary":"The dense torus acts on the fan scheme over `Spec \u2102` by the glued affine torus actions.","title":"TauCeti.Toric.Fan.algebraicRealizationModObj"},{"id":"tauceti/geometry/part-01/analytic-toric-geometry/tauceti-toric-fan-projectivespace","kind":"node","lean":"TauCeti.Toric.Fan.projectiveSpace","parent":2426,"result":true,"route":"../roadmap/tauceti/geometry/part-01/analytic-toric-geometry/index.html#tauceti-toric-fan-projectivespace","scope":"tauceti/geometry/part-01/analytic-toric-geometry","status":"fully_proved","summary":"The fan of projective space attached to an integral basis `b` of an integral lattice: its cones are spanned by the families of ray generators `projectiveSpaceGenerator b` omitting at least one of them. For a basis indexed by a type with `n` elements it is the fan of `n`-dimensional complex projective space.","title":"TauCeti.Toric.Fan.projectiveSpace"},{"id":"tauceti/geometry/part-01/analytic-toric-geometry/tauceti-toric-projectivespacediffeomorph","kind":"node","lean":"TauCeti.Toric.projectiveSpaceDiffeomorph","parent":2426,"result":true,"route":"../roadmap/tauceti/geometry/part-01/analytic-toric-geometry/index.html#tauceti-toric-projectivespacediffeomorph","scope":"tauceti/geometry/part-01/analytic-toric-geometry","status":"fully_proved","summary":"Homogeneous coordinates identify the analytic realization of the standard projective-space fan biholomorphically with complex projective space, the circle quotient of the unit sphere with its standard homogeneous-ratio atlas.","title":"TauCeti.Toric.projectiveSpaceDiffeomorph"},{"data_url":"scopes/tauceti/geometry/part-01/complex-manifolds-dolbeault-theory-and-kahler-ge.json","id":"tauceti/geometry/part-01/complex-manifolds-dolbeault-theory-and-kahler-ge","kind":"scope","parent":2425,"result":false,"route":"../roadmap/tauceti/geometry/part-01/complex-manifolds-dolbeault-theory-and-kahler-ge/index.html","scope":"tauceti/geometry/part-01/complex-manifolds-dolbeault-theory-and-kahler-ge","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Complex manifolds, Dolbeault theory, and K\u00e4hler geometry"},{"id":"tauceti/geometry/part-01/complex-manifolds-dolbeault-theory-and-kahler-ge/osgoods-lemma","kind":"node","lean":"ContinuousOn.analyticOnNhd_of_differentiableAt_update, ContinuousOn.analyticAt_of_differentiableAt_update","parent":2435,"result":true,"route":"../roadmap/tauceti/geometry/part-01/complex-manifolds-dolbeault-theory-and-kahler-ge/index.html#osgoods-lemma","scope":"tauceti/geometry/part-01/complex-manifolds-dolbeault-theory-and-kahler-ge","status":"fully_proved","summary":"A function of finitely many complex variables which is continuous on an open set and complex differentiable in each variable separately on it is analytic on it.","title":"Osgood's lemma"},{"id":"tauceti/geometry/part-01/complex-manifolds-dolbeault-theory-and-kahler-ge/tauceti-contmdiff-iff-mdifferentiable","kind":"node","lean":"TauCeti.contMDiff_iff_mdifferentiable","parent":2435,"result":true,"route":"../roadmap/tauceti/geometry/part-01/complex-manifolds-dolbeault-theory-and-kahler-ge/index.html#tauceti-contmdiff-iff-mdifferentiable","scope":"tauceti/geometry/part-01/complex-manifolds-dolbeault-theory-and-kahler-ge","status":"fully_proved","summary":"A map between complex manifolds is `C^n`, for any `n \u2260 0`, if and only if it is holomorphic.","title":"TauCeti.contMDiff_iff_mdifferentiable"},{"id":"tauceti/geometry/part-01/complex-manifolds-dolbeault-theory-and-kahler-ge/a-complex-manifold-is-a-real-analytic-manifold","kind":"node","lean":"IsManifold.restrictScalars_of_ne_zero","parent":2435,"result":true,"route":"../roadmap/tauceti/geometry/part-01/complex-manifolds-dolbeault-theory-and-kahler-ge/index.html#a-complex-manifold-is-a-real-analytic-manifold","scope":"tauceti/geometry/part-01/complex-manifolds-dolbeault-theory-and-kahler-ge","status":"fully_proved","summary":"A complex manifold modelled on `\ud835\udcd8(\u2102, E)`, of class `C^m` for some `m \u2260 0` with `E` finite-dimensional, is with the same charts a real `C^n` manifold modelled on `\ud835\udcd8(\u211d, E)` for every `n`; for `n = \u03c9` it is real-analytic.","title":"A complex manifold is a real-analytic manifold"},{"id":"tauceti/geometry/part-01/complex-manifolds-dolbeault-theory-and-kahler-ge/tauceti-holomorphicstructuresheaf-isfractionring","kind":"node","lean":"TauCeti.holomorphicStructureSheaf.isFractionRing_meromorphicGerms","parent":2435,"result":true,"route":"../roadmap/tauceti/geometry/part-01/complex-manifolds-dolbeault-theory-and-kahler-ge/index.html#tauceti-holomorphicstructuresheaf-isfractionring","scope":"tauceti/geometry/part-01/complex-manifolds-dolbeault-theory-and-kahler-ge","status":"fully_proved","summary":"The field of meromorphic germs at `a : \u2102` is the fraction field of the ring of holomorphic germs at `a`.","title":"TauCeti.holomorphicStructureSheaf.isFractionRing_meromorphicGerms"},{"id":"tauceti/geometry/part-01/complex-manifolds-dolbeault-theory-and-kahler-ge/projectivization-instismanifold","kind":"node","lean":"Projectivization.instIsManifold","parent":2435,"result":true,"route":"../roadmap/tauceti/geometry/part-01/complex-manifolds-dolbeault-theory-and-kahler-ge/index.html#projectivization-instismanifold","scope":"tauceti/geometry/part-01/complex-manifolds-dolbeault-theory-and-kahler-ge","status":"fully_proved","summary":"Projective space over a nontrivially normed field, with its standard affine charts, is a `C^r` manifold for every `r`; for `r = \u03c9` it is an analytic manifold. Over `\u2102` this is the standard complex manifold structure on `\u2102\u2119\u207f`.","title":"Projectivization.instIsManifold"},{"id":"tauceti/geometry/part-01/complex-manifolds-dolbeault-theory-and-kahler-ge/tauceti-smoothform-instmodule","kind":"node","lean":"TauCeti.SmoothForm.instModule","parent":2435,"result":true,"route":"../roadmap/tauceti/geometry/part-01/complex-manifolds-dolbeault-theory-and-kahler-ge/index.html#tauceti-smoothform-instmodule","scope":"tauceti/geometry/part-01/complex-manifolds-dolbeault-theory-and-kahler-ge","status":"fully_proved","summary":"Scalars acting on the coefficients act on smooth forms, making them a module. For complex coefficients this is the complex vector space of complex-valued smooth forms.","title":"TauCeti.SmoothForm.instModule"},{"id":"tauceti/geometry/part-01/complex-manifolds-dolbeault-theory-and-kahler-ge/tauceti-contdiffgroupoid-eq-of-ne-zero","kind":"node","lean":"TauCeti.contDiffGroupoid_eq_of_ne_zero","parent":2435,"result":true,"route":"../roadmap/tauceti/geometry/part-01/complex-manifolds-dolbeault-theory-and-kahler-ge/index.html#tauceti-contdiffgroupoid-eq-of-ne-zero","scope":"tauceti/geometry/part-01/complex-manifolds-dolbeault-theory-and-kahler-ge","status":"fully_proved","summary":"For a boundaryless model on a finite-dimensional complex space, the `C^m` and `C^n` groupoids agree whenever `m` and `n` are nonzero.","title":"TauCeti.contDiffGroupoid_eq_of_ne_zero"},{"id":"tauceti/geometry/part-01/complex-manifolds-dolbeault-theory-and-kahler-ge/the-cauchy-integral-formula-on-a-polydisc","kind":"node","lean":"DifferentiableOn.torusIntegral_prod_sub_inv_smul, ContinuousOn.torusIntegral_prod_sub_inv_smul","parent":2435,"result":true,"route":"../roadmap/tauceti/geometry/part-01/complex-manifolds-dolbeault-theory-and-kahler-ge/index.html#the-cauchy-integral-formula-on-a-polydisc","scope":"tauceti/geometry/part-01/complex-manifolds-dolbeault-theory-and-kahler-ge","status":"fully_proved","summary":"If `f` is complex differentiable on the closed polydisc with center `c` and polyradius `R`, then for every `w` in the open polydisc, the integral of `(\u220f i, (z i - w i))\u207b\u00b9 \u2022 f z` over the distinguished boundary `T(c, R)` is `(2\u03c0i)\u207f \u2022 f w`.","title":"The Cauchy integral formula on a polydisc"},{"data_url":"scopes/tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an.json","id":"tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an","kind":"scope","parent":2425,"result":false,"route":"../roadmap/tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an/index.html","scope":"tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Complex manifolds: transport, quotients, bundles, and gluing"},{"id":"tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an/mobius-transformations-of-the-projective-line-ar","kind":"node","lean":"OnePoint.instContMDiffConstSMul","parent":2444,"result":true,"route":"../roadmap/tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an/index.html#mobius-transformations-of-the-projective-line-ar","scope":"tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an","status":"fully_proved","summary":"For `\ud835\udd5c = \u2102`, every transformation `z \u21a6 (a * z + b) / (c * z + d)` of the Riemann sphere is holomorphic, and hence a biholomorphism `Diffeomorph.constSmul \ud835\udcd8(\u2102) \u03c9 g` with inverse the transformation of `g\u207b\u00b9`.","title":"M\u00f6bius transformations of the projective line are analytic"},{"id":"tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an/tauceti-contmdiff-orbitrelquotientlift-iff","kind":"node","lean":"TauCeti.contMDiff_orbitRelQuotientLift_iff","parent":2444,"result":true,"route":"../roadmap/tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an/index.html#tauceti-contmdiff-orbitrelquotientlift-iff","scope":"tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an","status":"fully_proved","summary":"For a free properly discontinuous action by `C^n` maps on a `C^n` manifold and `m \u2264 n`, the descent of a `G`-invariant map is `C^m` exactly when the map is. For holomorphic actions on complex manifolds, this is the holomorphic descent of invariant holomorphic maps.","title":"TauCeti.contMDiff_orbitRelQuotientLift_iff"},{"id":"tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an/tauceti-contmdiff-orbitrelquotientmap","kind":"node","lean":"TauCeti.contMDiff_orbitRelQuotientMap","parent":2444,"result":true,"route":"../roadmap/tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an/index.html#tauceti-contmdiff-orbitrelquotientmap","scope":"tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an","status":"fully_proved","summary":"For free properly discontinuous actions by `C^n` maps on `C^n` manifolds and `m \u2264 n`, a `C^m` map carrying one action to the other along `\u03c3` induces a `C^m` map of orbit spaces.","title":"TauCeti.contMDiff_orbitRelQuotientMap"},{"id":"tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an/mdifferentiableat-mfderiv-restrictscalars","kind":"node","lean":"MDifferentiableAt.mfderiv_restrictScalars","parent":2444,"result":true,"route":"../roadmap/tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an/index.html#mdifferentiableat-mfderiv-restrictscalars","scope":"tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an","status":"fully_proved","summary":"At a point where `f` is differentiable for the models over `\ud835\udd5c'`, the derivative of `f` over the smaller field `\ud835\udd5c` is its derivative over `\ud835\udd5c'` with scalars restricted to `\ud835\udd5c`, both read in the model spaces. For instance, the real derivative of a holomorphic map is its complex derivative.","title":"MDifferentiableAt.mfderiv_restrictScalars"},{"id":"tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an/contmdiff-restrictscalars","kind":"node","lean":"ContMDiff.restrictScalars","parent":2444,"result":true,"route":"../roadmap/tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an/index.html#contmdiff-restrictscalars","scope":"tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an","status":"fully_proved","summary":"A `C^n` map for boundaryless models over `\ud835\udd5c'` is `C^n` for their scalar restrictions to `\ud835\udd5c`. For instance, a holomorphic map between complex manifolds is smooth.","title":"ContMDiff.restrictScalars"},{"id":"tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an/homeomorph-pullbackchartedspace","kind":"node","lean":"Homeomorph.pullbackChartedSpace","parent":2444,"result":true,"route":"../roadmap/tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an/index.html#homeomorph-pullbackchartedspace","scope":"tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an","status":"fully_proved","summary":"The charted-space structure on `N` pulled back along a homeomorphism `e : N \u2243\u209c M`: the atlas is `{e.transOpenPartialHomeomorph c | c \u2208 atlas H M}` and the chart at `x` is the chart of `M` at `e x` precomposed with `e`.","title":"Homeomorph.pullbackChartedSpace"},{"id":"tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an/homeomorph-pullbackdiffeomorph","kind":"node","lean":"Homeomorph.pullbackDiffeomorph","parent":2444,"result":true,"route":"../roadmap/tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an/index.html#homeomorph-pullbackdiffeomorph","scope":"tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an","status":"fully_proved","summary":"A homeomorphism onto a charted space is a `C^n` diffeomorphism for the pulled-back structure: read in a pulled-back chart and in the corresponding chart of `M`, both it and its inverse are the identity.","title":"Homeomorph.pullbackDiffeomorph"},{"id":"tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an/diffeomorph-restrictscalars","kind":"node","lean":"Diffeomorph.restrictScalars","parent":2444,"result":true,"route":"../roadmap/tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an/index.html#diffeomorph-restrictscalars","scope":"tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an","status":"fully_proved","summary":"A `C^n` diffeomorphism for boundaryless models over `\ud835\udd5c'` is a `C^n` diffeomorphism for their scalar restrictions to `\ud835\udd5c`, with the same underlying equivalence. For instance, a biholomorphism between complex manifolds is a diffeomorphism of the underlying real manifolds.","title":"Diffeomorph.restrictScalars"},{"id":"tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an/descent-of-an-invariant-map-to-the-orbit-space","kind":"node","lean":"TauCeti.MulAction.orbitRelQuotientLift","parent":2444,"result":true,"route":"../roadmap/tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an/index.html#descent-of-an-invariant-map-to-the-orbit-space","scope":"tauceti/geometry/part-01/complex-manifolds-transport-quotients-bundles-an","status":"fully_proved","summary":"A map `f : X \u2192 Y` that is invariant under the action of `G` factors through the orbit projection `X \u2192 MulAction.orbitRel.Quotient G X`.","title":"Descent of an invariant map to the orbit space"},{"data_url":"scopes/tauceti/geometry/part-01/curvature-jacobi-fields-submanifolds-and-bochner.json","id":"tauceti/geometry/part-01/curvature-jacobi-fields-submanifolds-and-bochner","kind":"scope","parent":2425,"result":false,"route":"../roadmap/tauceti/geometry/part-01/curvature-jacobi-fields-submanifolds-and-bochner/index.html","scope":"tauceti/geometry/part-01/curvature-jacobi-fields-submanifolds-and-bochner","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Curvature, Jacobi fields, submanifolds, and Bochner geometry"},{"id":"tauceti/geometry/part-01/curvature-jacobi-fields-submanifolds-and-bochner/the-second-bianchi-identity","kind":"node","lean":"CovariantDerivative.covariantDerivative_curvatureOperator_cyclic_eq_zero, CovariantDerivative.derivCurvatureTensor_cyclic_eq_zero, CovariantDerivative.derivCurvatureTensor_cyclic_eq_neg_torsion","parent":2454,"result":true,"route":"../roadmap/tauceti/geometry/part-01/curvature-jacobi-fields-submanifolds-and-bochner/index.html#the-second-bianchi-identity","scope":"tauceti/geometry/part-01/curvature-jacobi-fields-submanifolds-and-bochner","status":"fully_proved","summary":"Let `\u2207` be a smooth connection on a vector bundle and `\u2207\u1d40` a smooth torsion-free connection on the tangent bundle. Writing `(\u2207_X R)(Y, Z) \u03c3 = \u2207_X (R(Y, Z) \u03c3) - R(\u2207\u1d40_X Y, Z) \u03c3 - R(Y, \u2207\u1d40_X Z) \u03c3 - R(Y, Z) (\u2207_X \u03c3)`, the cyclic sum `(\u2207_X R)(Y, Z) \u03c3 + (\u2207_Y R)(Z, X) \u03c3 + (\u2207_Z R)(X, Y) \u03c3` vanishes.","title":"The second Bianchi identity"},{"id":"tauceti/geometry/part-01/curvature-jacobi-fields-submanifolds-and-bochner/covariantderivative-ismetriccompatible-riemanncu","kind":"node","lean":"CovariantDerivative.IsMetricCompatible.riemannCurvatureTensor_eq_of_forall_linearIndependent","parent":2454,"result":true,"route":"../roadmap/tauceti/geometry/part-01/curvature-jacobi-fields-submanifolds-and-bochner/index.html#covariantderivative-ismetriccompatible-riemanncu","scope":"tauceti/geometry/part-01/curvature-jacobi-fields-submanifolds-and-bochner","status":"fully_proved","summary":"For a metric-compatible torsion-free connection, the Riemann curvature tensor at `x` is the only quadrilinear form with the algebraic symmetries of a curvature tensor whose values `T(u, v, v, u)` on linearly independent pairs are the sectional-curvature numerators `\u27e8R(u, v)v, u\u27e9`.","title":"CovariantDerivative.IsMetricCompatible.riemannCurvatureTensor_eq_of_forall_linearIndependent"},{"id":"tauceti/geometry/part-01/curvature-jacobi-fields-submanifolds-and-bochner/covariantderivative-hasconstantsectionalcurvatur","kind":"node","lean":"CovariantDerivative.hasConstantSectionalCurvature_iff_isConstantCurvatureTensor","parent":2454,"result":true,"route":"../roadmap/tauceti/geometry/part-01/curvature-jacobi-fields-submanifolds-and-bochner/index.html#covariantderivative-hasconstantsectionalcurvatur","scope":"tauceti/geometry/part-01/curvature-jacobi-fields-submanifolds-and-bochner","status":"fully_proved","summary":"A metric-compatible torsion-free connection has constant sectional curvature `\u03ba` if and only if its curvature tensor is the constant-curvature tensor `R(w, u)v = \u03ba (\u27e8u, v\u27e9 w - \u27e8w, v\u27e9 u)` at every point. Applied to the Levi-Civita connection, this identifies constant sectional curvature of a Riemannian manifold with the constant-curvature equation.","title":"CovariantDerivative.hasConstantSectionalCurvature_iff_isConstantCurvatureTensor"},{"id":"tauceti/geometry/part-01/curvature-jacobi-fields-submanifolds-and-bochner/curvature-under-conformal-change","kind":"node","lean":"Bundle.ContMDiffRiemannianMetric.curvatureOperator_leviCivitaConnection_rescale","parent":2454,"result":true,"route":"../roadmap/tauceti/geometry/part-01/curvature-jacobi-fields-submanifolds-and-bochner/index.html#curvature-under-conformal-change","scope":"tauceti/geometry/part-01/curvature-jacobi-fields-submanifolds-and-bochner","status":"fully_proved","summary":"Let `\u2207` be a Levi-Civita connection of a `C\u00b9` metric `g` and `A` the conformal connection tensor of a positive function `f`. Then the curvature of the Levi-Civita connection of `f g` is","title":"Curvature under conformal change"},{"id":"tauceti/geometry/part-01/curvature-jacobi-fields-submanifolds-and-bochner/curvature-of-a-connection-plus-a-one-form","kind":"node","lean":"CovariantDerivative.curvatureOperator_addOneForm","parent":2454,"result":true,"route":"../roadmap/tauceti/geometry/part-01/curvature-jacobi-fields-submanifolds-and-bochner/index.html#curvature-of-a-connection-plus-a-one-form","scope":"tauceti/geometry/part-01/curvature-jacobi-fields-submanifolds-and-bochner","status":"fully_proved","summary":"If `A` is a one-form with values in the endomorphisms of `V`, then","title":"Curvature of a connection plus a one-form"},{"id":"tauceti/geometry/part-01/curvature-jacobi-fields-submanifolds-and-bochner/linearmap-six-mul-apply-eq-of-cyclic-eq-zero","kind":"node","lean":"LinearMap.six_mul_apply_eq_of_cyclic_eq_zero","parent":2454,"result":true,"route":"../roadmap/tauceti/geometry/part-01/curvature-jacobi-fields-submanifolds-and-bochner/index.html#linearmap-six-mul-apply-eq-of-cyclic-eq-zero","scope":"tauceti/geometry/part-01/curvature-jacobi-fields-submanifolds-and-bochner","status":"fully_proved","summary":"The polarization identity for curvature-type tensors: for a quadrilinear form `T` antisymmetric in its first and in its last pair of arguments and satisfying the first Bianchi identity, `6 T(x, y, z, w)` is an explicit combination of values `T(a, b, b, a)`. This recovers `T` itself only when multiplication by `6` is injective, see `LinearMap.eq_of_cyclic_eq_zero`.","title":"LinearMap.six_mul_apply_eq_of_cyclic_eq_zero"},{"id":"tauceti/geometry/part-01/curvature-jacobi-fields-submanifolds-and-bochner/vectorfield-jacobi-identity-mliebracket-apply","kind":"node","lean":"VectorField.jacobi_identity_mlieBracket_apply","parent":2454,"result":true,"route":"../roadmap/tauceti/geometry/part-01/curvature-jacobi-fields-submanifolds-and-bochner/index.html#vectorfield-jacobi-identity-mliebracket-apply","scope":"tauceti/geometry/part-01/curvature-jacobi-fields-submanifolds-and-bochner","status":"fully_proved","summary":"The Jacobi identity for the manifold Lie bracket, in cyclic form: `[U, [V, W]] + [V, [W, U]] + [W, [U, V]] = 0`. This rearranges Mathlib's Leibniz form `leibniz_identity_mlieBracket_apply`.","title":"VectorField.jacobi_identity_mlieBracket_apply"},{"data_url":"scopes/tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f.json","id":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f","kind":"scope","parent":2425,"result":false,"route":"../roadmap/tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/index.html","scope":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Differential geometry \u2014 forms, de Rham cohomology, flows, and degree"},{"data_url":"scopes/tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01.json","id":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01","kind":"scope","navigation":"transparent","parent":2462,"result":false,"route":"../roadmap/tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/index.html","scope":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01","status":"fully_proved","title":"TauCeti.integralTopForm_mextDeri\u2026 to Stokes' formula for a homotopy\u2026"},{"id":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/tauceti-integraltopform-mextderiv","kind":"node","lean":"TauCeti.integralTopForm_mextDeriv","parent":2463,"result":true,"route":"../roadmap/tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/index.html#tauceti-integraltopform-mextderiv","scope":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01","status":"fully_proved","summary":"Stokes' theorem for a compactly supported C1 form on an oriented smooth manifold with boundary. The boundary carries its outward-first orientation.","title":"TauCeti.integralTopForm_mextDeriv"},{"id":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/tauceti-green-green-ac-winding","kind":"node","lean":"TauCeti.Green.green_ac_winding","parent":2463,"result":true,"route":"../roadmap/tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/index.html#tauceti-green-green-ac-winding","scope":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01","status":"fully_proved","summary":"Green's theorem for a closed absolutely continuous curve with index vanishing outside the open field domain. The winding numbers may be unbounded and the curve may intersect itself.","title":"TauCeti.Green.green_ac_winding"},{"id":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/the-mayer-vietoris-short-exact-sequence-of-de-rh","kind":"node","lean":"TauCeti.mayerVietorisShortComplex_shortExact","parent":2463,"result":true,"route":"../roadmap/tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/index.html#the-mayer-vietoris-short-exact-sequence-of-de-rh","scope":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01","status":"fully_proved","summary":"On a Hausdorff \u03c3-compact manifold modelled on a finite-dimensional space, the Mayer\u2013Vietoris short complex of two open subsets is short exact.","title":"The Mayer\u2013Vietoris short exact sequence of de Rham complexes"},{"id":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/the-mayer-vietoris-connecting-map-through-a-cut","kind":"node","lean":"TauCeti.deRhamCohomology.mayerVietoris\u03b4_mk_eq_mk_extendByZero","parent":2463,"result":true,"route":"../roadmap/tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/index.html#the-mayer-vietoris-connecting-map-through-a-cut","scope":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01","status":"fully_proved","summary":"Let `\u03c1` be a smooth function on `U \u2294 V` supported in `V` such that `1 - \u03c1` is supported in `U`, and let `\u03c6` be a closed form on `U \u2293 V`. If `\u03c3` is the closed form on `U \u2294 V` extending `d(\u03c1 \u03c6)` by zero (`TauCeti.exists_closedForms_coe_eq_extendByZero_mextDeriv`), then the connecting map sends the class of `\u03c6` to the class of `\u03c3`.","title":"The Mayer\u2013Vietoris connecting map through a cut-off function"},{"id":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/the-compactly-supported-mayer-vietoris-short-exa","kind":"node","lean":"TauCeti.compactlySupportedMayerVietorisShortComplex_shortExact","parent":2463,"result":true,"route":"../roadmap/tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/index.html#the-compactly-supported-mayer-vietoris-short-exa","scope":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01","status":"fully_proved","summary":"On a Hausdorff \u03c3-compact manifold modelled on a finite-dimensional space, the compactly supported Mayer\u2013Vietoris short complex of two open subsets is short exact.","title":"The compactly supported Mayer\u2013Vietoris short exact sequence"},{"id":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/homotopy-invariance-of-de-rham-cohomology","kind":"node","lean":"TauCeti.deRhamCohomology.map_eq_of_homotopy","parent":2463,"result":true,"route":"../roadmap/tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/index.html#homotopy-invariance-of-de-rham-cohomology","scope":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01","status":"fully_proved","summary":"If `F : M \u00d7 [0, 1] \u2192 M'` is a smooth homotopy from `f` to `g`, then `f` and `g` induce the same map on de Rham cohomology. The source `M` may have boundary or corners.","title":"Homotopy invariance of de Rham cohomology"},{"id":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/the-de-rham-cohomology-of-a-normed-space-vanishe","kind":"node","lean":"TauCeti.subsingleton_deRhamCohomology_normedSpace","parent":2463,"result":true,"route":"../roadmap/tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/index.html#the-de-rham-cohomology-of-a-normed-space-vanishe","scope":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.subsingleton_deRhamCohomology_normedSpace`.","title":"The de Rham cohomology of a normed space vanishes in positive degrees"},{"id":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/tauceti-nontrivial-derhamcohomology-circle-one","kind":"node","lean":"TauCeti.nontrivial_deRhamCohomology_circle_one","parent":2463,"result":true,"route":"../roadmap/tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/index.html#tauceti-nontrivial-derhamcohomology-circle-one","scope":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01","status":"fully_proved","summary":"The first de Rham cohomology of the circle is nontrivial.","title":"TauCeti.nontrivial_deRhamCohomology_circle_one"},{"id":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/tauceti-homotopyoperator","kind":"node","lean":"TauCeti.homotopyOperator","parent":2463,"result":true,"route":"../roadmap/tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/index.html#tauceti-homotopyoperator","scope":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01","status":"fully_proved","summary":"The homotopy operator on forms on the cylinder `M \u00d7 [0, 1]`: the `k`-form `x \u21a6 \u222b\u2080\u00b9 \u03c6(x, t)(\u2202\u209c, (\u00b7, 0)) dt` on `M` built from a `(k + 1)`-form `\u03c6` on `M \u00d7 [0, 1]`, where `\u2202\u209c = (0, 1)` is the unit time vector and tangent vectors of `M` enter through `ContinuousLinearMap.inl`. Equivalently `h \u03c6 = \u222b\u2080\u00b9 \u03b9\u209c^*(\u2202\u209c \u231f \u03c6) dt` for the slices `\u03b9\u209c x = (x, t)`. For a `C\u00b9` form with values in a Banach space on a `C\u00b2` manifold,\u2026","title":"TauCeti.homotopyOperator"},{"id":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/the-cochain-homotopy-identity-for-the-homotopy-o","kind":"node","lean":"TauCeti.mextDeriv_homotopyOperator_add_homotopyOperator_mextDeriv","parent":2463,"result":true,"route":"../roadmap/tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/index.html#the-cochain-homotopy-identity-for-the-homotopy-o","scope":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01","status":"fully_proved","summary":"for a `C\u00b9` form `\u03c6` with values in a Banach space on a `C\u00b2` manifold, possibly with boundary or corners, `d (h \u03c6) + h (d \u03c6) = \u03b9\u2081^* \u03c6 - \u03b9\u2080^* \u03c6`, where `\u03b9\u209c x = (x, t)`.","title":"The cochain-homotopy identity for the homotopy operator on the cylinder `M \u00d7 [0, 1]`"},{"id":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/tauceti-compactlysupportedderhamcohomology","kind":"node","lean":"TauCeti.compactlySupportedDeRhamCohomology","parent":2463,"result":true,"route":"../roadmap/tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/index.html#tauceti-compactlysupportedderhamcohomology","scope":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01","status":"fully_proved","summary":"Compactly supported de Rham cohomology is the categorical homology of the compactly supported de Rham complex.","title":"TauCeti.compactlySupportedDeRhamCohomology"},{"id":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/tauceti-compactlysupportedderhamcohomology-exten","kind":"node","lean":"TauCeti.compactlySupportedDeRhamCohomology.extendByZero","parent":2463,"result":true,"route":"../roadmap/tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/index.html#tauceti-compactlysupportedderhamcohomology-exten","scope":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01","status":"fully_proved","summary":"The map on compactly supported de Rham cohomology induced by extension by zero along an inclusion `U \u2264 W` of open subsets of a Hausdorff manifold.","title":"TauCeti.compactlySupportedDeRhamCohomology.extendByZero"},{"id":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/tauceti-compactlysupportedderhamcohomology-integ","kind":"node","lean":"TauCeti.compactlySupportedDeRhamCohomology.integral","parent":2463,"result":true,"route":"../roadmap/tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/index.html#tauceti-compactlysupportedderhamcohomology-integ","scope":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01","status":"fully_proved","summary":"Integration `H^n_c(M) \u2192 \u211d` over an oriented boundaryless manifold, induced by integrating compactly supported closed top forms.","title":"TauCeti.compactlySupportedDeRhamCohomology.integral"},{"id":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/tauceti-poincarepairing","kind":"node","lean":"TauCeti.poincarePairing","parent":2463,"result":true,"route":"../roadmap/tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/index.html#tauceti-poincarepairing","scope":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01","status":"fully_proved","summary":"The Poincar\u00e9 pairing `H^k_dR(M) \u2192 H^l_c(M) \u2192 \u211d` on an oriented boundaryless manifold of dimension `k + l`, sending classes `[\u03c6]` and `[\u03c8]` to `\u222b_M \u03c6 \u2227 \u03c8`. It is the wedge product `H^k_dR(M) \u2192 H^l_c(M) \u2192 H^(k+l)_c(M)` followed by integration.","title":"TauCeti.poincarePairing"},{"id":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/tauceti-pathintegral","kind":"node","lean":"TauCeti.pathIntegral","parent":2463,"result":true,"route":"../roadmap/tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/index.html#tauceti-pathintegral","scope":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01","status":"fully_proved","summary":"The integral of a `1`-form `\u03c6` along a curve `\u03b3 : \u211d \u2192 M` over the oriented interval `a..b`: the interval integral of `\u03c6` evaluated on the velocity of `\u03b3`.","title":"TauCeti.pathIntegral"},{"id":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/the-fundamental-theorem-of-calculus-for-line-int","kind":"node","lean":"TauCeti.pathIntegral_mextDeriv, TauCeti.pathIntegral_mextDeriv_ofFunction","parent":2463,"result":true,"route":"../roadmap/tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/index.html#the-fundamental-theorem-of-calculus-for-line-int","scope":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01","status":"fully_proved","summary":"the line integral of `d\u03c8` along a `C\u00b9` curve is the difference of the values of `\u03c8` at the endpoints.","title":"The fundamental theorem of calculus for line integrals, for a `C\u00b9` form `\u03c8` of degree zero"},{"id":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/tauceti-integraltopform-mpullback-icc-eq-pathint","kind":"node","lean":"TauCeti.integralTopForm_mpullback_Icc_eq_pathIntegral","parent":2463,"result":true,"route":"../roadmap/tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/index.html#tauceti-integraltopform-mpullback-icc-eq-pathint","scope":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01","status":"fully_proved","summary":"For `a < b`, the line integral of a continuous real `1`-form along a `C\u00b9` curve is the canonical oriented integral over `[a, b]` of the pullback of the form along the curve.","title":"TauCeti.integralTopForm_mpullback_Icc_eq_pathIntegral"},{"id":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/the-compactly-supported-poincare-lemma-on-the-li","kind":"node","lean":"TauCeti.mem_exactCompactlySupportedForms_one_real, TauCeti.mem_exactCompactlySupportedForms_iff_integralTopForm_eq_zero_normedSpace, TauCeti.mem_exactCompactlySupportedForms_of_integralTopForm_eq_zero_normedSpace","parent":2463,"result":true,"route":"../roadmap/tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/index.html#the-compactly-supported-poincare-lemma-on-the-li","scope":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01","status":"fully_proved","summary":"A compactly supported `1`-form on `\u211d` (automatically closed, as every `1`-form on the line is) whose integral against Lebesgue measure vanishes is the exterior derivative of a compactly supported function.","title":"The compactly supported Poincar\u00e9 lemma on the line"},{"id":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/the-fundamental-theorem-of-calculus-for-path-cla","kind":"node","lean":"TauCeti.pathClassIntegral_mextDeriv_ofFunction, TauCeti.pathClassIntegral_mextDeriv","parent":2463,"result":true,"route":"../roadmap/tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/index.html#the-fundamental-theorem-of-calculus-for-path-cla","scope":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01","status":"fully_proved","summary":"The integral of the differential of a `C\u00b2` function along a path class is the difference of the values of the function at the endpoints.","title":"The fundamental theorem of calculus for path classes"},{"id":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/stokes-formula-for-a-homotopy-of-maps-of-a-recta","kind":"node","lean":"TauCeti.rectangleIntegral_sub_rectangleIntegral_eq_sides, TauCeti.rectangleIntegral_mextDeriv","parent":2463,"result":true,"route":"../roadmap/tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01/index.html#stokes-formula-for-a-homotopy-of-maps-of-a-recta","scope":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-01","status":"fully_proved","summary":"Let `\u03c6` be a closed `C\u00b9` two-form and `\u03a6 : (\u211d \u00d7 \u211d) \u00d7 \u211d \u2192 M` a `C\u00b2` homotopy from `\u03a6(\u00b7, 0)` to `\u03a6(\u00b7, 1)`. The integral of `\u03a6^* \u03c6` over the boundary of the cube `[a, b] \u00d7 [0, 1]` vanishes: the change of the integral over the rectangle `[a, b]` from `\u03a6(\u00b7, 0)` to `\u03a6(\u00b7, 1)` is the signed sum of the integrals over the side faces `(t, c) \u21a6 \u03a6((a.1, t), c)`, `(t, c) \u21a6 \u03a6((b.1, t), c)`, `(s, c) \u21a6 \u03a6((s, a.2), c)` and\u2026","title":"Stokes' formula for a homotopy of maps of a rectangle"},{"data_url":"scopes/tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-02.json","id":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-02","kind":"scope","navigation":"transparent","parent":2462,"result":false,"route":"../roadmap/tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-02/index.html","scope":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-02","status":"fully_proved","title":"Stokes' theorem on singular\u2026"},{"id":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-02/stokes-theorem-on-singular-1-chains","kind":"node","lean":"TauCeti.chainsDesc_pathClassIntegral_mextDeriv","parent":2484,"result":true,"route":"../roadmap/tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-02/index.html#stokes-theorem-on-singular-1-chains","scope":"tauceti/geometry/part-01/differential-geometry-forms-de-rham-cohomology-f/part-02","status":"fully_proved","summary":"The integral of `d\u03c8` along a singular `1`-chain is the evaluation of the `0`-form `\u03c8` on its boundary.","title":"Stokes' theorem on singular `1`-chains"},{"data_url":"scopes/tauceti/geometry/part-01/geodesics-the-exponential-map-and-the-hopf-rinow.json","id":"tauceti/geometry/part-01/geodesics-the-exponential-map-and-the-hopf-rinow","kind":"scope","parent":2425,"result":false,"route":"../roadmap/tauceti/geometry/part-01/geodesics-the-exponential-map-and-the-hopf-rinow/index.html","scope":"tauceti/geometry/part-01/geodesics-the-exponential-map-and-the-hopf-rinow","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Geodesics, the exponential map, and the Hopf\u2013Rinow theorem"},{"id":"tauceti/geometry/part-01/geodesics-the-exponential-map-and-the-hopf-rinow/the-hopf-rinow-theorem","kind":"node","lean":"TauCeti.Manifold.tfae_expDomain_eq_univ","parent":2486,"result":true,"route":"../roadmap/tauceti/geometry/part-01/geodesics-the-exponential-map-and-the-hopf-rinow/index.html#the-hopf-rinow-theorem","scope":"tauceti/geometry/part-01/geodesics-the-exponential-map-and-the-hopf-rinow","status":"fully_proved","summary":"For a point `p` of a Riemannian manifold `M` whose distance is the Riemannian distance, the following are equivalent: the exponential map at `p` is defined on all of `T_p M`; `M` is proper; `M` is complete; every maximal geodesic of `M` is defined for all time; and `M` has an increasing exhaustion by compact sets `K n` such that `dist p (q n) \u2192 \u221e` whenever `q n \u2209 K n` for every `n`.","title":"The Hopf\u2013Rinow theorem"},{"id":"tauceti/geometry/part-01/geodesics-the-exponential-map-and-the-hopf-rinow/a-compact-riemannian-manifold-is-geodesically-co","kind":"node","lean":"TauCeti.Manifold.isGeodesicallyCompleteAt_of_compactSpace","parent":2486,"result":true,"route":"../roadmap/tauceti/geometry/part-01/geodesics-the-exponential-map-and-the-hopf-rinow/index.html#a-compact-riemannian-manifold-is-geodesically-co","scope":"tauceti/geometry/part-01/geodesics-the-exponential-map-and-the-hopf-rinow","status":"fully_proved","summary":"every maximal geodesic leaving a point `p` is defined for all time. This is do Carmo's corollary to the Hopf\u2013Rinow theorem.","title":"A compact Riemannian manifold is geodesically complete"},{"id":"tauceti/geometry/part-01/geodesics-the-exponential-map-and-the-hopf-rinow/the-gauss-lemma","kind":"node","lean":"TauCeti.Manifold.inner_mfderiv_riemannianExp_radial, TauCeti.Manifold.inner_mfderiv_riemannianExp_model_space","parent":2486,"result":true,"route":"../roadmap/tauceti/geometry/part-01/geodesics-the-exponential-map-and-the-hopf-rinow/index.html#the-gauss-lemma","scope":"tauceti/geometry/part-01/geodesics-the-exponential-map-and-the-hopf-rinow","status":"fully_proved","summary":"The differential of the Riemannian exponential map preserves the inner product with the radial direction at every vector in its natural domain.","title":"The Gauss lemma"},{"id":"tauceti/geometry/part-01/geodesics-the-exponential-map-and-the-hopf-rinow/the-variational-characterization-of-geodesics","kind":"node","lean":"TauCeti.Manifold.isEnergyCritical_iff_isGeodesicCurveOn","parent":2486,"result":true,"route":"../roadmap/tauceti/geometry/part-01/geodesics-the-exponential-map-and-the-hopf-rinow/index.html#the-variational-characterization-of-geodesics","scope":"tauceti/geometry/part-01/geodesics-the-exponential-map-and-the-hopf-rinow","status":"fully_proved","summary":"On a boundaryless `C^n` manifold with `2 \u2264 n \u2264 \u221e`, a curve which is `C^n` near every point of `[a, b]` is a critical point of the energy between `a` and `b` among `C^n` variations with fixed endpoints exactly when it is a geodesic on the open interval between `a` and `b`. For `n = \u221e`: a smooth curve is critical among smooth variations exactly when it is a geodesic.","title":"The variational characterization of geodesics"},{"id":"tauceti/geometry/part-01/geodesics-the-exponential-map-and-the-hopf-rinow/the-first-variation-of-the-levi-civita-connectio","kind":"node","lean":"TauCeti.Manifold.two_mul_apply_deriv_eq_derivBilinearTensor_of_isLeviCivitaConnection","parent":2486,"result":true,"route":"../roadmap/tauceti/geometry/part-01/geodesics-the-exponential-map-and-the-hopf-rinow/index.html#the-first-variation-of-the-levi-civita-connectio","scope":"tauceti/geometry/part-01/geodesics-the-exponential-map-and-the-hopf-rinow","status":"fully_proved","summary":"Let `g_s` be a family of smooth nondegenerate metrics, jointly `C\u00b2` in `(s, y)` at `(t, x)`, with parameter derivative `h = \u2202\u209b g_s` at `s = t`, and let `\u2207\u02e2` be a Levi-Civita connection of `g_s` for every `s`. Then for every vector field `Y` which is differentiable at `x` and all `v, w \u2208 T_x M`,","title":"The first variation of the Levi-Civita connection"},{"id":"tauceti/geometry/part-01/geodesics-the-exponential-map-and-the-hopf-rinow/tauceti-manifold-contmdiffnondegeneratemetric-le","kind":"node","lean":"TauCeti.Manifold.ContMDiffNondegenerateMetric.leviCivitaConnection","parent":2486,"result":true,"route":"../roadmap/tauceti/geometry/part-01/geodesics-the-exponential-map-and-the-hopf-rinow/index.html#tauceti-manifold-contmdiffnondegeneratemetric-le","scope":"tauceti/geometry/part-01/geodesics-the-exponential-map-and-the-hopf-rinow","status":"fully_proved","summary":"The Levi-Civita connection of a smooth symmetric nondegenerate metric. It is chosen to vanish on vector fields that are not differentiable at the evaluation point.","title":"TauCeti.Manifold.ContMDiffNondegenerateMetric.leviCivitaConnection"},{"id":"tauceti/geometry/part-01/geodesics-the-exponential-map-and-the-hopf-rinow/tauceti-manifold-riemannianexp","kind":"node","lean":"TauCeti.Manifold.riemannianExp","parent":2486,"result":true,"route":"../roadmap/tauceti/geometry/part-01/geodesics-the-exponential-map-and-the-hopf-rinow/index.html#tauceti-manifold-riemannianexp","scope":"tauceti/geometry/part-01/geodesics-the-exponential-map-and-the-hopf-rinow","status":"fully_proved","summary":"The Riemannian exponential map at `p`: the value at time `1` of the maximal geodesic from `p` with initial velocity `v`. Outside `expDomain I M p` it takes the junk value `p`.","title":"TauCeti.Manifold.riemannianExp"},{"id":"tauceti/geometry/part-01/geodesics-the-exponential-map-and-the-hopf-rinow/tauceti-manifold-contmdiffnondegeneratemetricfam","kind":"node","lean":"TauCeti.Manifold.ContMDiffNondegenerateMetricFamily","parent":2486,"result":true,"route":"../roadmap/tauceti/geometry/part-01/geodesics-the-exponential-map-and-the-hopf-rinow/index.html#tauceti-manifold-contmdiffnondegeneratemetricfam","scope":"tauceti/geometry/part-01/geodesics-the-exponential-map-and-the-hopf-rinow","status":"fully_proved","summary":"A smooth family, parametrized by `P`, of symmetric nondegenerate real bilinear forms on the tangent bundle of `M`.","title":"TauCeti.Manifold.ContMDiffNondegenerateMetricFamily"},{"data_url":"scopes/tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro.json","id":"tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro","kind":"scope","parent":2425,"result":false,"route":"../roadmap/tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro/index.html","scope":"tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Geometric topology and the solved Kirby-list problems"},{"id":"tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro/preabstractsimplicialcomplex-iscombinatorialmani","kind":"node","lean":"PreAbstractSimplicialComplex.IsCombinatorialManifold.exists_chartedSpace_hasGroupoid_PLGroupoid","parent":2495,"result":true,"route":"../roadmap/tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro/index.html#preabstractsimplicialcomplex-iscombinatorialmani","scope":"tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro","status":"fully_proved","summary":"The weak realization of a positive-dimensional combinatorial manifold is a PL manifold, possibly with boundary: it has an atlas modelled on the half-space in `(Fin n \u2192 \u211d) \u00d7 \u211d` whose chart transitions lie in the PL groupoid. No finiteness or countability of the complex is assumed.","title":"PreAbstractSimplicialComplex.IsCombinatorialManifold.exists_chartedSpace_hasGroupoid_PLGroupoid"},{"id":"tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro/the-homfly-polynomial-is-a-link-invariant-on-bra","kind":"node","lean":"TauCeti.MarkovEquiv.homfly_eq","parent":2495,"result":true,"route":"../roadmap/tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro/index.html#the-homfly-polynomial-is-a-link-invariant-on-bra","scope":"tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro","status":"fully_proved","summary":"Markov-equivalent braids have the same HOMFLY polynomial.","title":"The HOMFLY polynomial is a link invariant on braid closures"},{"id":"tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro/diffeomorph-istopologicalgroup","kind":"node","lean":"Diffeomorph.isTopologicalGroup","parent":2495,"result":true,"route":"../roadmap/tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro/index.html#diffeomorph-istopologicalgroup","scope":"tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro","status":"fully_proved","summary":"Self-diffeomorphisms of a compact Hausdorff manifold with a locally compact model space form a topological group for the weak Whitney topology.","title":"Diffeomorph.isTopologicalGroup"},{"id":"tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro/tauceti-nil-semidirectproductisommulequiv","kind":"node","lean":"TauCeti.Nil.semidirectProductIsomMulEquiv","parent":2495,"result":true,"route":"../roadmap/tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro/index.html#tauceti-nil-semidirectproductisommulequiv","scope":"tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro","status":"fully_proved","summary":"The full Riemannian isometry group of Nil is `Nil \u22ca O(2)`, with its canonical translation and orthogonal actions. This is an isomorphism of groups.","title":"TauCeti.Nil.semidirectProductIsomMulEquiv"},{"id":"tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro/tauceti-upperhalfspace-hyperbolicmetric","kind":"node","lean":"TauCeti.UpperHalfSpace.hyperbolicMetric","parent":2495,"result":true,"route":"../roadmap/tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro/index.html#tauceti-upperhalfspace-hyperbolicmetric","scope":"tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro","status":"fully_proved","summary":"The standard complete metric of constant curvature `-1` on upper half-space.","title":"TauCeti.UpperHalfSpace.hyperbolicMetric"},{"id":"tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro/tauceti-knotexterior","kind":"node","lean":"TauCeti.knotExterior","parent":2495,"result":true,"route":"../roadmap/tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro/index.html#tauceti-knotexterior","scope":"tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro","status":"fully_proved","summary":"The closed exterior of a solid-torus neighbourhood: the complement of its open solid-torus image.","title":"TauCeti.knotExterior"},{"id":"tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro/tauceti-issolidtorusneighborhood-framingnumber","kind":"node","lean":"TauCeti.IsSolidTorusNeighborhood.framingNumber","parent":2495,"result":true,"route":"../roadmap/tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro/index.html#tauceti-issolidtorusneighborhood-framingnumber","scope":"tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro","status":"fully_proved","summary":"The framing number of the solid torus neighbourhood `\u03a6`: the longitude of `\u03a6` is homologous in the knot complement to `framingNumber` times the meridian. Equivalently, the longitude of `\u03a6` is `\u03bb\u2080 + n \u00b7 \u03bc` for the preferred longitude `\u03bb\u2080` (`TauCeti.IsSolidTorusNeighborhood.preferredFramedBoundaryTorus_basis_one`). For a knot in `S\u00b3` it is the linking number of the longitude of `\u03a6` with the knot.","title":"TauCeti.IsSolidTorusNeighborhood.framingNumber"},{"id":"tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro/tauceti-foliation-eulerclass","kind":"node","lean":"TauCeti.Foliation.eulerClass","parent":2495,"result":true,"route":"../roadmap/tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro/index.html#tauceti-foliation-eulerclass","scope":"tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro","status":"fully_proved","summary":"The Euler class `H\u1d4f(TF, TF\u2080) \u27f6 H\u1d4f(M)` of the tangent field of a foliation of rank `k`: `TauCeti.eulerClass`, restriction to the total space followed by pullback along the zero section, read in degree `k`.","title":"TauCeti.Foliation.eulerClass"},{"id":"tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro/thurstons-euler-class-inequality","kind":"node","lean":"TauCeti.ThurstonEulerClassInequality","parent":2495,"result":true,"route":"../roadmap/tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro/index.html#thurstons-euler-class-inequality","scope":"tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro","status":"fully_proved","summary":"on a closed orientable hyperbolic `3`-manifold `M`, that is a compact connected orientable smooth `3`-manifold without boundary admitting a complete Riemannian metric of constant curvature `-1`, every Euler class of a smooth taut codimension-one foliation satisfies the Euler class inequality.","title":"Thurston's Euler class inequality"},{"id":"tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro/yazdis-theorem-refutes-the-euler-class-one-conje","kind":"node","lean":"TauCeti.YazdiEulerClassOneTheorem.not_eulerClassOneConjecture","parent":2495,"result":true,"route":"../roadmap/tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro/index.html#yazdis-theorem-refutes-the-euler-class-one-conje","scope":"tauceti/geometry/part-01/geometric-topology-and-the-solved-kirby-list-pro","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.YazdiEulerClassOneTheorem.not_eulerClassOneConjecture`, assuming `TauCeti.YazdiEulerClassOneTheorem`, a proposition Tau Ceti defines.","title":"Yazdi's theorem refutes the Euler class one conjecture"},{"data_url":"scopes/tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a.json","id":"tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a","kind":"scope","parent":2425,"result":false,"route":"../roadmap/tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a/index.html","scope":"tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Hamiltonian Floer operations: geometric product and multiplicative PSS"},{"id":"tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a/the-energy-identity-on-a-rectangle","kind":"node","lean":"TauCeti.SymplecticForm.integral_floerEnergyDensity","parent":2506,"result":true,"route":"../roadmap/tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a/index.html#the-energy-identity-on-a-rectangle","scope":"tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a","status":"fully_proved","summary":"For a map `u` and perturbation data that are `C\u00b9` over an open neighbourhood of a rectangle `[a\u2081, b\u2081] \u00d7 [a\u2082, b\u2082]`, the integral of the Floer energy density over the rectangle is the symplectic area, plus the integral of the pulled-back one-form `K(u) = K_s(u) ds + K_t(u) dt` over the positively oriented boundary, minus the integral of the curvature `R_K(u)`.","title":"The energy identity on a rectangle"},{"id":"tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a/tauceti-floer-capclass-mk-eq-mk-iff-action-eq-an","kind":"node","lean":"TauCeti.Floer.CapClass.mk_eq_mk_iff_action_eq_and_index_eq","parent":2506,"result":true,"route":"../roadmap/tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a/index.html#tauceti-floer-capclass-mk-eq-mk-iff-action-eq-an","scope":"tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a","status":"fully_proved","summary":"Two raw caps have the same class exactly when they have the same action and the same grading, for functions `a` and `m` on raw caps obeying the recapping laws for area and twice the first Chern number.","title":"TauCeti.Floer.CapClass.mk_eq_mk_iff_action_eq_and_index_eq"},{"id":"tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a/tauceti-floer-add-sub-add-two-mul-chern-productc","kind":"node","lean":"TauCeti.Floer.add_sub_add_two_mul_chern_productClass_eq","parent":2506,"result":true,"route":"../roadmap/tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a/index.html#tauceti-floer-add-sub-add-two-mul-chern-productc","scope":"tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a","status":"fully_proved","summary":"Cap independence of the index `\u03bc\u2081(D\u2081) + \u03bc\u2082(D\u2082) \u2212 \u03bc\u2080(D\u2080) + 2 c\u2081(A)` of a product coefficient, for gradings `m\u1d62` obeying the recapping law for twice the first Chern number. The Fredholm index of a pair of pants in class `A` is this quantity minus half the dimension of the target.","title":"TauCeti.Floer.add_sub_add_two_mul_chern_productClass_eq"},{"id":"tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a/tauceti-floer-novikovring-forall-finite-lt-iff-b","kind":"node","lean":"TauCeti.Floer.NovikovRing.forall_finite_lt_iff_bddAbove_degree","parent":2506,"result":true,"route":"../roadmap/tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a/index.html#tauceti-floer-novikovring-forall-finite-lt-iff-b","scope":"tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a","status":"fully_proved","summary":"For a positively monotone pair, a coefficient function on the period group satisfies the Novikov condition of the Floer Novikov ring exactly when the degrees of its support are bounded above.","title":"TauCeti.Floer.NovikovRing.forall_finite_lt_iff_bddAbove_degree"},{"id":"tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a/tauceti-floer-periodgroupcomparison-single-eq-if","kind":"node","lean":"TauCeti.Floer.periodGroupComparison_single_eq_iff","parent":2506,"result":true,"route":"../roadmap/tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a/index.html#tauceti-floer-periodgroupcomparison-single-eq-if","scope":"tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a","status":"fully_proved","summary":"Monomial descent identifies precisely the joint periods, retaining equal-area classes of different Chern number. The coefficient is assumed nonzero.","title":"TauCeti.Floer.periodGroupComparison_single_eq_iff"},{"id":"tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a/tauceti-floer-novikovring","kind":"node","lean":"TauCeti.Floer.NovikovRing","parent":2506,"result":true,"route":"../roadmap/tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a/index.html#tauceti-floer-novikovring","scope":"tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a","status":"fully_proved","summary":"The Floer Novikov ring over `R`: Novikov series `\u2211 a_\u03b3 e^\u03b3` over the period group, with valuation `\u03bd(\u03b3) = -\u03c9(\u03b3)`, so that for every bound `C` only finitely many classes with nonzero coefficient have area greater than `-C`. It is graded by `PeriodGroup.degree`.","title":"TauCeti.Floer.NovikovRing"},{"id":"tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a/tauceti-floer-periodgroup","kind":"node","lean":"TauCeti.Floer.PeriodGroup","parent":2506,"result":true,"route":"../roadmap/tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a/index.html#tauceti-floer-periodgroup","scope":"tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a","status":"fully_proved","summary":"The period group `\u0393 = S \u29f8 (ker \u03c9 \u2293 ker c)` of a group of spherical classes `S` with symplectic area `\u03c9` and first Chern number `c`, written as the quotient by the kernel of `s \u21a6 (\u03c9 s, c s)`.","title":"TauCeti.Floer.PeriodGroup"},{"id":"tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a/tauceti-floer-coefficientcomparison","kind":"node","lean":"TauCeti.Floer.coefficientComparison","parent":2506,"result":true,"route":"../roadmap/tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a/index.html#tauceti-floer-coefficientcomparison","scope":"tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a","status":"fully_proved","summary":"The completed coefficient comparison `q^A \u21a6 e^(-q A)`, with source period `\u03c9` and target period `-\u03a9`. Area preservation makes it a unital ring homomorphism.","title":"TauCeti.Floer.coefficientComparison"},{"id":"tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a/tauceti-novikovmodule-kernelproduct","kind":"node","lean":"TauCeti.NovikovModule.kernelProduct","parent":2506,"result":true,"route":"../roadmap/tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a/index.html#tauceti-novikovmodule-kernelproduct","scope":"tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a","status":"fully_proved","summary":"The bilinear product of completed combinations given by a kernel `n` with potential bound `C` and completed outputs: the coefficient of `p\u2080` in the product of `a` and `b` is the finite sum `\u2211 a(p\u2081) b(p\u2082) n(p\u2081, p\u2082, p\u2080)` over all pairs of generators.","title":"TauCeti.NovikovModule.kernelProduct"},{"id":"tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a/tauceti-novikovmodule-kernelmatrix-mem-periodfil","kind":"node","lean":"TauCeti.NovikovModule.kernelMatrix_mem_periodFiltration","parent":2506,"result":true,"route":"../roadmap/tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a/index.html#tauceti-novikovmodule-kernelmatrix-mem-periodfil","scope":"tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a","status":"fully_proved","summary":"Under a potential bound with constant `C`, the `(i, j, k)` entry of the kernel matrix lies in the period filtration at level `F\u2080(r\u2080 k) - F\u2081(r\u2081 i) - F\u2082(r\u2082 j) - C`.","title":"TauCeti.NovikovModule.kernelMatrix_mem_periodFiltration"},{"id":"tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a/tauceti-novikovmodule-novikovkernelproduct","kind":"node","lean":"TauCeti.NovikovModule.novikovKernelProduct","parent":2506,"result":true,"route":"../roadmap/tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a/index.html#tauceti-novikovmodule-novikovkernelproduct","scope":"tauceti/geometry/part-01/hamiltonian-floer-operations-geometric-product-a","status":"fully_proved","summary":"The product of an invariant kernel, as a bilinear map over the Novikov ring, for the Novikov module structures of `TauCeti.NovikovModule.module`.","title":"TauCeti.NovikovModule.novikovKernelProduct"},{"data_url":"scopes/tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized.json","id":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized","kind":"scope","parent":2425,"result":false,"route":"../roadmap/tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/index.html","scope":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Hodge structures (pure, mixed, and polarized)"},{"id":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/categorical-semisimplicity-of-polarizable-ration","kind":"node","lean":"TauCeti.Hodge.PolarizableHodgeStructureCat.exists_iso_biproduct_simple","parent":2518,"result":true,"route":"../roadmap/tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/index.html#categorical-semisimplicity-of-polarizable-ration","scope":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized","status":"fully_proved","summary":"Every object is isomorphic to a finite biproduct of simple objects induced by rational Hodge substructures.","title":"Categorical semisimplicity of polarizable rational Hodge structures"},{"id":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/semisimplicity-of-polarizable-rational-hodge-str","kind":"node","lean":"TauCeti.Hodge.exists_isCompl_of_isPolarizable","parent":2518,"result":true,"route":"../roadmap/tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/index.html#semisimplicity-of-polarizable-rational-hodge-str","scope":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized","status":"fully_proved","summary":"Every rational Hodge substructure of a polarizable pure Hodge structure has a complement, both over `\u211a` and after complexification, orthogonal to it for some polarization.","title":"Semisimplicity of polarizable rational Hodge structures"},{"id":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/the-hodge-decomposition","kind":"node","lean":"TauCeti.Hodge.HodgeStructureOn.isInternal_piece, TauCeti.Hodge.MixedHodgeStructure.isInternal_complexGradedHodgeStructure_piece","parent":2518,"result":true,"route":"../roadmap/tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/index.html#the-hodge-decomposition","scope":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized","status":"fully_proved","summary":"The Hodge components `H^{p,n-p}` form an internal direct sum of the ambient complex vector space.","title":"The Hodge decomposition"},{"id":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/the-hodge-numbers-partition-the-dimension","kind":"node","lean":"TauCeti.Hodge.HodgeStructureOn.finsum_hodgeNumber_eq_finrank","parent":2518,"result":true,"route":"../roadmap/tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/index.html#the-hodge-numbers-partition-the-dimension","scope":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized","status":"fully_proved","summary":"the Hodge numbers of a weight-`n` Hodge structure on a finite-dimensional complex vector space sum to its dimension.","title":"The Hodge numbers partition the dimension"},{"id":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/delignes-theorem-tauceti-hodge-mixedhodgestructu","kind":"node","lean":"TauCeti.Hodge.MixedHodgeStructure.isInternal_deligneSplittingFamily","parent":2518,"result":true,"route":"../roadmap/tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/index.html#delignes-theorem-tauceti-hodge-mixedhodgestructu","scope":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized","status":"fully_proved","summary":"The pieces `I^{p,q}` of Deligne's bigrading form an internal direct sum of the complex vector space underlying a mixed Hodge structure.","title":"Deligne's theorem (`TauCeti.Hodge.MixedHodgeStructure.isInternal_deligneSplittingFamily`)"},{"id":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/strictness-of-a-morphism-of-mixed-hodge-structur","kind":"node","lean":"TauCeti.Hodge.MixedHodgeStructure.Hom.range_inf_F_eq_map_F","parent":2518,"result":true,"route":"../roadmap/tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/index.html#strictness-of-a-morphism-of-mixed-hodge-structur","scope":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized","status":"fully_proved","summary":"A complex vector lying both in the image of `f` and in the target's Hodge step `F^p` is the image of a vector of the source's `F^p`.","title":"Strictness of a morphism of mixed Hodge structures for the Hodge filtration"},{"id":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/tauceti-hodge-mixedhodgestructurecat-abelian","kind":"node","lean":"TauCeti.Hodge.MixedHodgeStructureCat.abelian","parent":2518,"result":true,"route":"../roadmap/tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/index.html#tauceti-hodge-mixedhodgestructurecat-abelian","scope":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized","status":"fully_proved","summary":"Mixed Hodge structures, with rational Hodge morphisms, form an abelian category.","title":"TauCeti.Hodge.MixedHodgeStructureCat.abelian"},{"id":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/delignes-fine-conjugation-relation","kind":"node","lean":"TauCeti.Hodge.MixedHodgeStructure.map_latticeConj_deligneSplitting_sup_below","parent":2518,"result":true,"route":"../roadmap/tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/index.html#delignes-fine-conjugation-relation","scope":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized","status":"fully_proved","summary":"Conjugation exchanges `I^{p,q}` and `I^{q,p}` modulo the sum of the components `I^{r,s}` with `r < q` and `s < p`.","title":"Deligne's fine conjugation relation"},{"id":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/when-v-is-flat-over-z-the-riemann-forms-for-j-ar","kind":"node","lean":"TauCeti.AlmostComplexStructure.isRiemannForm_iff_isPolarization","parent":2518,"result":true,"route":"../roadmap/tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/index.html#when-v-is-flat-over-z-the-riemann-forms-for-j-ar","scope":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized","status":"fully_proved","summary":"Flatness is used only in the forward direction, for nondegeneracy.","title":"When `V` is flat over `\u2124`, the Riemann forms for `J` are exactly the forms polarizing its weight-one Hodge structure"},{"id":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/every-form-polarizing-the-weight-one-hodge-struc","kind":"node","lean":"TauCeti.AlmostComplexStructure.isRiemannForm_of_isPolarization","parent":2518,"result":true,"route":"../roadmap/tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/index.html#every-form-polarizing-the-weight-one-hodge-struc","scope":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized","status":"fully_proved","summary":"`J`-invariance of `E_\u211d` says that the Weil operator is an isometry of the polarization, and positivity of `E_\u211d (J x) x` is positivity of the Hodge form on real vectors.","title":"Every form polarizing the weight-one Hodge structure of `J` is a Riemann form for `J`, with no flatness assumption"},{"id":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/self-duality-is-invertible-up-to-a-nonzero-integ","kind":"node","lean":"TauCeti.Hodge.Polarization.exists_comp_toDualHom_eq_nsmul_id","parent":2518,"result":true,"route":"../roadmap/tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/index.html#self-duality-is-invertible-up-to-a-nonzero-integ","scope":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized","status":"fully_proved","summary":"For a polarization of `V`, there is a Hodge morphism `g : V^*(-n) \u2192 V` such that composing it with self-duality, in either order, is multiplication by a nonzero natural number `c`. Over `\u211a`, `c\u207b\u00b9 \u2022 g` is the inverse of self-duality.","title":"Self-duality is invertible up to a nonzero integer"},{"id":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/the-dual-of-a-polarizable-pure-hodge-structure-i","kind":"node","lean":"TauCeti.Hodge.IsPolarizable.dual","parent":2518,"result":true,"route":"../roadmap/tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/index.html#the-dual-of-a-polarizable-pure-hodge-structure-i","scope":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.Hodge.IsPolarizable.dual`.","title":"The dual of a polarizable pure Hodge structure is polarizable"},{"id":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/polarizable-pure-hodge-structures-are-closed-und","kind":"node","lean":"TauCeti.Hodge.IsPolarizable.internalHom","parent":2518,"result":true,"route":"../roadmap/tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/index.html#polarizable-pure-hodge-structures-are-closed-und","scope":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized","status":"fully_proved","summary":"The internal hom of weights `n\u2081` and `n\u2082` is polarizable of weight `n\u2082 - n\u2081`.","title":"Polarizable pure Hodge structures are closed under internal homs"},{"id":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/tauceti-hodge-hodgestructureon","kind":"node","lean":"TauCeti.Hodge.HodgeStructureOn","parent":2518,"result":true,"route":"../roadmap/tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/index.html#tauceti-hodge-hodgestructureon","scope":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized","status":"fully_proved","summary":"A pure Hodge structure of weight `n` on a complex vector space with conjugation `\u03c9`.","title":"TauCeti.Hodge.HodgeStructureOn"},{"id":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/tauceti-hodge-ispolarization","kind":"node","lean":"TauCeti.Hodge.IsPolarization","parent":2518,"result":true,"route":"../roadmap/tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/index.html#tauceti-hodge-ispolarization","scope":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized","status":"fully_proved","summary":"The Hodge\u2013Riemann bilinear relations for a fixed integral bilinear form `Q` on the lattice of a weight-`n` Hodge structure.","title":"TauCeti.Hodge.IsPolarization"},{"id":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/tauceti-hodge-polarization-q","kind":"node","lean":"TauCeti.Hodge.Polarization.Q","parent":2518,"result":true,"route":"../roadmap/tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/index.html#tauceti-hodge-polarization-q","scope":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized","status":"fully_proved","summary":"The complex bilinear form of a polarization, derived by complexifying the integral form.","title":"TauCeti.Hodge.Polarization.Q"},{"id":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/isbasechange-bilinform","kind":"node","lean":"IsBaseChange.bilinForm","parent":2518,"result":true,"route":"../roadmap/tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/index.html#isbasechange-bilinform","scope":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized","status":"fully_proved","summary":"Extend a bilinear form to an abstract base change, transporting Mathlib's tensor-product base change along the canonical equivalence.","title":"IsBaseChange.bilinForm"},{"id":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/tauceti-hodge-polarizablehodgestructurecat","kind":"node","lean":"TauCeti.Hodge.PolarizableHodgeStructureCat","parent":2518,"result":true,"route":"../roadmap/tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized/index.html#tauceti-hodge-polarizablehodgestructurecat","scope":"tauceti/geometry/part-01/hodge-structures-pure-mixed-and-polarized","status":"fully_proved","summary":"The category of finite-dimensional polarizable rational Hodge structures of weight `n`, with integral, rational, and complex models in `Type u`.","title":"TauCeti.Hodge.PolarizableHodgeStructureCat"},{"data_url":"scopes/tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott.json","id":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott","kind":"scope","parent":2425,"result":false,"route":"../roadmap/tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/index.html","scope":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Morse functions, sublevel topology and Morse\u2013Bott attachments"},{"id":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/the-lusternik-schnirelmann-theorem","kind":"node","lean":"TauCeti.lsCategoryCount_univ_le_totalCriticalCount","parent":2537,"result":true,"route":"../roadmap/tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/index.html#the-lusternik-schnirelmann-theorem","scope":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott","status":"fully_proved","summary":"On a compact boundaryless manifold `M`, the Lusternik\u2013Schnirelmann count of `M` is at most the number of critical points of any smooth function on `M`. The critical points may be degenerate; if there are infinitely many, their count is `\u22a4`.","title":"The Lusternik\u2013Schnirelmann theorem"},{"id":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/deformation-of-sublevels-across-a-band-without-c","kind":"node","lean":"TauCeti.exists_sublevelDeformation","parent":2537,"result":true,"route":"../roadmap/tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/index.html#deformation-of-sublevels-across-a-band-without-c","scope":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott","status":"fully_proved","summary":"If `f` is smooth, `a \u2264 b`, and the band `f\u207b\u00b9[a, b]` is compact and contains no critical point of `f`, then the sublevel `{f \u2264 a}` is a strong deformation retract of `{f \u2264 b}`: there is a retraction `r` of `{f \u2264 b}` onto `{f \u2264 a}` and a homotopy from the inclusion of `r` to the identity fixing `{f \u2264 a}` pointwise.","title":"Deformation of sublevels across a band without critical points (Milnor, Theorem 3.1)"},{"id":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/sublevels-on-either-side-of-a-band-without-criti","kind":"node","lean":"TauCeti.exists_diffeomorph_sublevel","parent":2537,"result":true,"route":"../roadmap/tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/index.html#sublevels-on-either-side-of-a-band-without-criti","scope":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott","status":"fully_proved","summary":"If `f` is smooth, `a \u2264 b`, and the band `f\u207b\u00b9[a, b]` is compact and contains no critical point of `f`, then there is a diffeomorphism `\u03a6` of `M` carrying the sublevel `{f \u2264 b}` onto `{f \u2264 a}` and the level `f\u207b\u00b9(b)` onto `f\u207b\u00b9(a)`.","title":"Sublevels on either side of a band without critical points are diffeomorphic (Milnor, Theorem 3.1)"},{"id":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/the-normalized-morse-lemma-on-a-manifold","kind":"node","lean":"TauCeti.IsManifoldNondegenerateCriticalPoint.exists_normalized_morseChart, TauCeti.MorseChart.eq_quadratic_normalizedChart, TauCeti.IsManifoldNondegenerateCriticalPoint.exists_normalized_morseChart_subset","parent":2537,"result":true,"route":"../roadmap/tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/index.html#the-normalized-morse-lemma-on-a-manifold","scope":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott","status":"fully_proved","summary":"At a nondegenerate critical point `x` of a function `f` which is smooth near `x`, on a manifold with a boundaryless model, there is a partial diffeomorphism `\u03c8` onto an open subset of `EuclideanSpace \u211d (Fin m)`, `m = dim M`, with `\u03c8 x = 0` and","title":"The normalized Morse lemma on a manifold"},{"id":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/a-trajectory-converging-to-a-nondegenerate-criti","kind":"node","lean":"TauCeti.exists_riemannianEDist_le_mul_exp_maximalIntegralCurve_neg_mgradient","parent":2537,"result":true,"route":"../roadmap/tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/index.html#a-trajectory-converging-to-a-nondegenerate-criti","scope":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott","status":"fully_proved","summary":"If the maximal integral curve `\u03b3` of `-grad f` through `x` exists for all positive time and converges to a nondegenerate critical point `p`, then the Riemannian distance from `\u03b3 t` to `p` is at most `C * exp (-\u03bc * t)` for some `\u03bc, C > 0` and all large `t`. This is the manifold counterpart, for an arbitrary `C\u00b9` metric, of the vector-space theorem `IsIntegralCurveOn.exists_norm_sub_le_mul_exp_atTop`.","title":"A trajectory converging to a nondegenerate critical point converges exponentially fast in the Riemannian distance"},{"id":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/negative-gradient-trajectories-of-a-proper-morse","kind":"node","lean":"TauCeti.IsMorse.exists_tendsto_maximalIntegralCurve_neg_mgradient","parent":2537,"result":true,"route":"../roadmap/tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/index.html#negative-gradient-trajectories-of-a-proper-morse","scope":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott","status":"fully_proved","summary":"If `f` is a proper Morse function bounded below, the maximal integral curve of `-grad f` through any point converges to a critical point of `f`, for any `C\u00b9` Riemannian metric.","title":"Negative gradient trajectories of a proper Morse function converge"},{"id":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/tauceti-criticalhessian","kind":"node","lean":"TauCeti.criticalHessian","parent":2537,"result":true,"route":"../roadmap/tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/index.html#tauceti-criticalhessian","scope":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott","status":"fully_proved","summary":"The critical Hessian of a real function `f` at a point `x` of a manifold: the second derivative at `extChartAt I x x` of the coordinate expression of `f` in the preferred extended chart, as a bilinear form on the tangent space at `x`.","title":"TauCeti.criticalHessian"},{"id":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/admissible-morse-charts-at-a-critical-level","kind":"node","lean":"TauCeti.IsMorse.exists_admissible_morseCharts","parent":2537,"result":true,"route":"../roadmap/tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/index.html#admissible-morse-charts-at-a-critical-level","scope":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott","status":"fully_proved","summary":"Let `f` be a Morse function on a Hausdorff manifold whose band `f\u207b\u00b9[c - \u03b4, c + \u03b4]` is compact, for some `0 < \u03b4`. Then there are a width `0 < \u03b5 \u2264 \u03b4` and normalized Morse charts `\u03c8 p`, indexed by the critical points `p` on the level `f\u207b\u00b9(c)`, such that","title":"Admissible Morse charts at a critical level"},{"id":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/milnors-handle-retraction","kind":"node","lean":"TauCeti.exists_inter_quadraticSublevelUnionCell_deformation","parent":2537,"result":true,"route":"../roadmap/tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/index.html#milnors-handle-retraction","scope":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott","status":"fully_proved","summary":"If every fibre `{v | (u, v) \u2208 X}` of `X` is star-shaped about the origin, then `X \u2229 quadraticSublevelUnionCell \u03b5` is a strong deformation retract of `X`: there is a retraction of `X` onto it, and a homotopy in `X` from this retraction to the identity which fixes `X \u2229 quadraticSublevelUnionCell \u03b5`. Both are restrictions of `TauCeti.quadraticHandleRetraction \u03b5` and `TauCeti.quadraticHandleDeformation \u03b5`.","title":"Milnor's handle retraction"},{"id":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/fermats-theorem-on-a-manifold","kind":"node","lean":"IsLocalExtr.mem_criticalPoints","parent":2537,"result":true,"route":"../roadmap/tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/index.html#fermats-theorem-on-a-manifold","scope":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott","status":"fully_proved","summary":"A local extremum is a critical point.","title":"Fermat's theorem on a manifold"},{"id":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/the-lusternik-schnirelmann-bound-on-sublevels","kind":"node","lean":"TauCeti.lsCategoryCount_sublevel_le_encard, TauCeti.lsCategoryCount_univ_le_encard_criticalPoints_inter","parent":2537,"result":true,"route":"../roadmap/tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/index.html#the-lusternik-schnirelmann-bound-on-sublevels","scope":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott","status":"fully_proved","summary":"Let `f` be smooth and `t < b`, and suppose that the sublevel `{f \u2264 b}` is compact and contains only finitely many critical points of `f`. Then the Lusternik\u2013Schnirelmann count of `{f \u2264 t}` is at most the number of critical points of `f` in `{f \u2264 t}`. The critical points may be degenerate.","title":"The Lusternik\u2013Schnirelmann bound on sublevels"},{"id":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/the-lusternik-schnirelmann-lower-bound-for-c-m","kind":"node","lean":"TauCeti.lsCategoryCount_univ_le_minCriticalCount","parent":2537,"result":true,"route":"../roadmap/tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/index.html#the-lusternik-schnirelmann-lower-bound-for-c-m","scope":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott","status":"fully_proved","summary":"On a compact boundaryless manifold, every smooth function has at least as many critical points as the Lusternik\u2013Schnirelmann count of `M`, so `c(M)` is bounded below by that count.","title":"The Lusternik\u2013Schnirelmann lower bound for `c(M)`"},{"id":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/the-morse-form-of-ojasiewiczs-gradient-inequalit","kind":"node","lean":"TauCeti.IsManifoldNondegenerateCriticalPoint.exists_mul_abs_sub_le_norm_mgradient_sq","parent":2537,"result":true,"route":"../roadmap/tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/index.html#the-morse-form-of-ojasiewiczs-gradient-inequalit","scope":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott","status":"fully_proved","summary":"Near a nondegenerate critical point `p` of `f`, the absolute energy difference `|f y - f p|` is bounded by a multiple of the squared Riemannian length of the gradient of `f` at `y`, for any continuous Riemannian metric.","title":"The Morse form of \u0141ojasiewicz's gradient inequality on a Riemannian manifold"},{"id":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/the-morse-lemma-on-a-manifold","kind":"node","lean":"TauCeti.IsManifoldNondegenerateCriticalPoint.nonempty_morseChart","parent":2537,"result":true,"route":"../roadmap/tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/index.html#the-morse-lemma-on-a-manifold","scope":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott","status":"fully_proved","summary":"At a nondegenerate critical point `x` of a function `f` that is smooth at every point of a neighbourhood of `x`, on a manifold modelled on a finite-dimensional real normed space `E`, there is a Morse chart: a chart `\u03c8` of the maximal atlas, centred at `x`, and a linear change of coordinates `L : E \u2243 (Fin n \u2192 \u211d)` such that on the source of `\u03c8`","title":"The Morse lemma on a manifold"},{"id":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/the-morse-polynomial-identity-tauceti-ismorse-ex","kind":"node","lean":"TauCeti.IsMorse.exists_sum_range_monomial_eq_add_mul, TauCeti.IsMorse.exists_sum_range_monomial_sublevelPair_eq_add_mul","parent":2537,"result":true,"route":"../roadmap/tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/index.html#the-morse-polynomial-identity-tauceti-ismorse-ex","scope":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott","status":"fully_proved","summary":"For a Morse function `f` on a compact boundaryless manifold `M`, write `b\u1d62` for the dimension of `H\u1d62(M; V)` and `c\u1d62` for the number of critical points of index `i`. Then for every `n \u2265 dim M`, `\u2211\u1d62\u208c\u2080\u207f c\u1d62 dim V \u00b7 t\u2071 = \u2211\u1d62\u208c\u2080\u207f b\u1d62 t\u2071 + (1 + t) Q(t)` for a polynomial `Q` with natural number coefficients. The coefficient of `t\u2071` in `Q` is the gap `(c\u1d62 - c\u1d62\u208b\u2081 + \u22ef \u00b1 c\u2080) dim V - (b\u1d62 - b\u1d62\u208b\u2081 + \u22ef \u00b1 b\u2080)` in the `i`-th strong\u2026","title":"The Morse polynomial identity (`TauCeti.IsMorse.exists_sum_range_monomial_eq_add_mul`)"},{"id":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/reebs-theorem-milnor-morse-theory-theorem-4-1","kind":"node","lean":"TauCeti.IsMorse.nonempty_homeomorph_sphere_of_totalCriticalCount_eq_two","parent":2537,"result":true,"route":"../roadmap/tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/index.html#reebs-theorem-milnor-morse-theory-theorem-4-1","scope":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott","status":"fully_proved","summary":"A compact boundaryless manifold modelled on a real vector space of dimension `n` which carries a Morse function with exactly two critical points is homeomorphic to the sphere `S\u207f`.","title":"Reeb's theorem (Milnor, *Morse Theory*, Theorem 4.1)"},{"id":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/the-strong-morse-inequalities-tauceti-ismorse-su","kind":"node","lean":"TauCeti.IsMorse.sum_range_neg_one_pow_mul_finrank_singularHomology_le, TauCeti.IsMorse.sum_range_neg_one_pow_mul_finrank_singularHomology_sublevelPair_le","parent":2537,"result":true,"route":"../roadmap/tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/index.html#the-strong-morse-inequalities-tauceti-ismorse-su","scope":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott","status":"fully_proved","summary":"For a Morse function `f` on a compact boundaryless manifold `M`, write `b\u1d62` for the dimension of `H\u1d62(M; V)` and `c\u1d62` for the number of critical points of index `i`. Then `b\u2099 - b\u2099\u208b\u2081 + \u22ef \u00b1 b\u2080 \u2264 (c\u2099 - c\u2099\u208b\u2081 + \u22ef \u00b1 c\u2080) dim V`.","title":"The strong Morse inequalities (`TauCeti.IsMorse.sum_range_neg_one_pow_mul_finrank_singularHomology_le`)"},{"id":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/the-weak-morse-inequalities","kind":"node","lean":"TauCeti.IsMorse.finrank_singularHomology_le, TauCeti.IsMorse.exists_isZero_and_finrank_le_singularHomology_sublevelPair, TauCeti.IsMorse.finrank_singularHomology_sublevelPair_le","parent":2537,"result":true,"route":"../roadmap/tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott/index.html#the-weak-morse-inequalities","scope":"tauceti/geometry/part-01/morse-functions-sublevel-topology-and-morse-bott","status":"fully_proved","summary":"For a Morse function `f` on a compact boundaryless manifold `M`, the singular homology `H\u2099(M; V)` has dimension at most `dim V` times the number of critical points of index `n`.","title":"The weak Morse inequalities"},{"data_url":"scopes/tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism.json","id":"tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism","kind":"scope","parent":2425,"result":false,"route":"../roadmap/tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism/index.html","scope":"tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Positive scalar curvature, surgery and bordism"},{"id":"tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism/bundle-contmdiffriemannianmetric-equivalence-con","kind":"node","lean":"Bundle.ContMDiffRiemannianMetric.equivalence_concordant","parent":2556,"result":true,"route":"../roadmap/tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism/index.html#bundle-contmdiffriemannianmetric-equivalence-con","scope":"tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism","status":"fully_proved","summary":"Concordance is an equivalence relation on the metrics of positive scalar curvature on `M`.","title":"Bundle.ContMDiffRiemannianMetric.equivalence_concordant"},{"id":"tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism/bundle-contmdiffriemannianmetric-concordant-iff","kind":"node","lean":"Bundle.ContMDiffRiemannianMetric.concordant_iff_exists_isCollaredConcordance","parent":2556,"result":true,"route":"../roadmap/tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism/index.html#bundle-contmdiffriemannianmetric-concordant-iff","scope":"tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism","status":"fully_proved","summary":"Two metrics are concordant exactly when there is a collared concordance from one to the other.","title":"Bundle.ContMDiffRiemannianMetric.concordant_iff_exists_isCollaredConcordance"},{"id":"tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism/compact-families-of-metrics-of-positive-scalar-c","kind":"node","lean":"TauCeti.Manifold.exists_pos_forall_le_scalarCurvature","parent":2556,"result":true,"route":"../roadmap/tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism/index.html#compact-families-of-metrics-of-positive-scalar-c","scope":"tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism","status":"fully_proved","summary":"On a compact manifold, a family of metrics of positive scalar curvature indexed by a compact space, whose chart metrics have Gram matrices, first and second derivatives in the basis `b` depending continuously on the parameter and the chart point, has scalar curvature bounded below by a common positive constant.","title":"Compact families of metrics of positive scalar curvature have a uniform scalar gap"},{"id":"tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism/the-scalar-curvature-of-a-product-metric","kind":"node","lean":"Bundle.ContMDiffRiemannianMetric.scalarCurvature_prodTangentSpace","parent":2556,"result":true,"route":"../roadmap/tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism/index.html#the-scalar-curvature-of-a-product-metric","scope":"tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism","status":"fully_proved","summary":"The scalar curvature of a product metric `g\u2081 \u00d7 g\u2082` is the sum of the scalar curvatures of the factors: `Scal(x, y) = Scal_{g\u2081}(x) + Scal_{g\u2082}(y)`.","title":"The scalar curvature of a product metric"},{"id":"tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism/shrinking-a-compact-factor-of-positive-scalar-cu","kind":"node","lean":"Bundle.ContMDiffRiemannianMetric.exists_pos_hasPositiveScalarCurvature_prodTangentSpace_rescale","parent":2556,"result":true,"route":"../roadmap/tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism/index.html#shrinking-a-compact-factor-of-positive-scalar-cu","scope":"tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism","status":"fully_proved","summary":"Let `g\u2082` be a metric of positive scalar curvature on a compact manifold `N`, and let `g` be a metric whose scalar curvature is bounded below. Then there is `\u03b5 > 0` such that `g \u00d7 c g\u2082` has positive scalar curvature for every `0 < c < \u03b5`.","title":"Shrinking a compact factor of positive scalar curvature"},{"id":"tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism/scalar-curvature-of-the-round-sphere","kind":"node","lean":"TauCeti.scalarCurvature_sphereRoundMetricOfRadius","parent":2556,"result":true,"route":"../roadmap/tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism/index.html#scalar-curvature-of-the-round-sphere","scope":"tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism","status":"fully_proved","summary":"The round `n`-sphere of radius `r` has scalar curvature `n (n - 1) / r\u00b2`.","title":"Scalar curvature of the round sphere"},{"id":"tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism/bundle-contmdiffriemannianmetric-haspositivescal","kind":"node","lean":"Bundle.ContMDiffRiemannianMetric.HasPositiveScalarCurvature","parent":2556,"result":true,"route":"../roadmap/tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism/index.html#bundle-contmdiffriemannianmetric-haspositivescal","scope":"tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism","status":"fully_proved","summary":"A smooth Riemannian metric has positive scalar curvature if its scalar curvature is positive at every point.","title":"Bundle.ContMDiffRiemannianMetric.HasPositiveScalarCurvature"},{"id":"tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism/tauceti-manifold-scalarcurvatureofjet","kind":"node","lean":"TauCeti.Manifold.scalarCurvatureOfJet","parent":2556,"result":true,"route":"../roadmap/tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism/index.html#tauceti-manifold-scalarcurvatureofjet","scope":"tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism","status":"fully_proved","summary":"The scalar curvature of a coordinate jet `(g, \u2202g, \u2202\u00b2g)`: the contraction `\u03a3\u2c7c\u2096 g^(jk) \u03a3\u1d62 R\u2071\u1d62\u2c7c\u2096` of the curvature coefficients `riemannOfJet` against the inverse of the Gram matrix `g`.","title":"TauCeti.Manifold.scalarCurvatureOfJet"},{"id":"tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism/scalar-curvature-is-continuous-in-the-second-ord","kind":"node","lean":"ContinuousAt.scalarCurvatureOfJet","parent":2556,"result":true,"route":"../roadmap/tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism/index.html#scalar-curvature-is-continuous-in-the-second-ord","scope":"tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism","status":"fully_proved","summary":"The scalar curvature of a coordinate jet depends continuously on the jet at every jet whose Gram matrix is invertible.","title":"Scalar curvature is continuous in the second-order jet of the metric"},{"id":"tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism/bundle-contmdiffriemannianmetric-positivedefinit","kind":"node","lean":"Bundle.ContMDiffRiemannianMetric.positiveDefiniteSymmetricSectionEquiv","parent":2556,"result":true,"route":"../roadmap/tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism/index.html#bundle-contmdiffriemannianmetric-positivedefinit","scope":"tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism","status":"fully_proved","summary":"Native finite-rank `C^n` metrics are exactly the positive-definite `C^n` sections of the bundle of symmetric bilinear forms.","title":"Bundle.ContMDiffRiemannianMetric.positiveDefiniteSymmetricSectionEquiv"},{"id":"tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism/contmdiffvectorbundle-symmetricbilinearmaps","kind":"node","lean":"ContMDiffVectorBundle.symmetricBilinearMaps","parent":2556,"result":true,"route":"../roadmap/tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism/index.html#contmdiffvectorbundle-symmetricbilinearmaps","scope":"tauceti/geometry/part-01/positive-scalar-curvature-surgery-and-bordism","status":"fully_proved","summary":"When two is invertible, the symmetric bilinear maps on the fibres of a `C^n` vector bundle form a `C^n` vector bundle.","title":"ContMDiffVectorBundle.symmetricBilinearMaps"},{"data_url":"scopes/tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c.json","id":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c","kind":"scope","parent":2425,"result":false,"route":"../roadmap/tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/index.html","scope":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Real algebraic geometry: sign determination and cylindrical decomposition"},{"id":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/validity-of-lazards-projection","kind":"node","lean":"TauCeti.exists_isCAD_lazardValuationReindex_eq","parent":2568,"result":true,"route":"../roadmap/tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/index.html#validity-of-lazards-projection","scope":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c","status":"fully_proved","summary":"Every finite family of real polynomials in `n` variables has a cylindrical algebraic decomposition of `\u211d ^ n` into analytic submanifolds on each cell of which every member has constant Lazard valuation, when the coordinates are read in reverse order, from the coordinate added first by the cylindrical structure to the one added last. Every member is also sign-invariant on each cell. No well-orientedness hypothesis\u2026","title":"Validity of Lazard's projection"},{"id":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/correctness-of-mccallums-projection","kind":"node","lean":"TauCeti.exists_isCAD_orderAt_eq_of_isMcCallumWellOriented","parent":2568,"result":true,"route":"../roadmap/tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/index.html#correctness-of-mccallums-projection","scope":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c","status":"fully_proved","summary":"A finite family of real polynomials that is well-oriented in McCallum's sense has a cylindrical algebraic decomposition into analytic submanifolds, on each cell of which every member of the family has constant ambient order.","title":"Correctness of McCallum's projection"},{"id":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/recursive-sign-invariant-mccallum-correctness","kind":"node","lean":"TauCeti.exists_isCAD_signInvariant_of_isMcCallumWellOriented_projection","parent":2568,"result":true,"route":"../roadmap/tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/index.html#recursive-sign-invariant-mccallum-correctness","scope":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c","status":"fully_proved","summary":"If the McCallum projection of a squarefree basis of `F` is well-oriented, there is a sign-invariant CAD for `F` with analytic cells. Its base CAD is order-invariant for that projection. No restriction is imposed on nullification of the top-level basis, including over positive-dimensional base cells.","title":"Recursive sign-invariant McCallum correctness"},{"id":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/existence-of-adapted-cylindrical-algebraic-decom","kind":"node","lean":"TauCeti.exists_isCAD_signInvariant","parent":2568,"result":true,"route":"../roadmap/tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/index.html#existence-of-adapted-cylindrical-algebraic-decom","scope":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c","status":"fully_proved","summary":"For every finite set `F` of real polynomials in `n` variables there is a cylindrical algebraic decomposition of `\u211d ^ n` on each cell of which every member of `F` is sign-invariant.","title":"Existence of adapted cylindrical algebraic decompositions"},{"id":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/the-tarski-seidenberg-theorem","kind":"node","lean":"TauCeti.instHasSemialgebraicProjectionsReal","parent":2568,"result":true,"route":"../roadmap/tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/index.html#the-tarski-seidenberg-theorem","scope":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c","status":"fully_proved","summary":"The projection of a semialgebraic subset of `\u211d ^ (n + 1)` forgetting the coordinate `0` is semialgebraic. So quantifier elimination, the identification of semialgebraic sets with definable sets, and the image and composition laws of semialgebraic functions, which assume `HasSemialgebraicProjections`, hold over `\u211d` unconditionally.","title":"The Tarski\u2013Seidenberg theorem"},{"id":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/quantifier-elimination-given-projection-closure","kind":"node","lean":"TauCeti.exists_isQF_realize_iff","parent":2568,"result":true,"route":"../roadmap/tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/index.html#quantifier-elimination-given-projection-closure","scope":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c","status":"fully_proved","summary":"Every formula of ordered rings whose free variables `Sum.inl r` are assigned the parameters `r \u2208 R` is equivalent, for every assignment `v` of the remaining free variables, to a quantifier-free formula with parameters.","title":"Quantifier elimination, given projection closure"},{"id":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/tarskis-theorem-for-the-real-field","kind":"node","lean":"TauCeti.isSemialgebraic_iff_definable_ring_real, TauCeti.isSemialgebraic_iff_definable_ring","parent":2568,"result":true,"route":"../roadmap/tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/index.html#tarskis-theorem-for-the-real-field","scope":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c","status":"fully_proved","summary":"A subset of `\u03c3 \u2192 \u211d` is semialgebraic if and only if it is definable with parameters in the field `\u211d`, that is, in the language of rings with its usual interpretation.","title":"Tarski's theorem for the real field"},{"id":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/analytic-delineability-from-constant-discriminan","kind":"node","lean":"TauCeti.exists_analytic_delineation_of_orderAt_discr_eq","parent":2568,"result":true,"route":"../roadmap/tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/index.html#analytic-delineability-from-constant-discriminan","scope":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c","status":"fully_proved","summary":"A polynomial with nonzero fibers of constant degree over a preconnected analytic submanifold, whose nonzero formal discriminant has constant ambient order on the base, has a global delineation with intrinsically analytic root functions. Each root function is represented by an ambient function whose values off the base are immaterial. Along each root section, the polynomial, viewed as a polynomial in all `n + 1`\u2026","title":"Analytic delineability from constant discriminant order"},{"id":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/finset-issquarefreebasis","kind":"node","lean":"Finset.IsSquarefreeBasis","parent":2568,"result":true,"route":"../roadmap/tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/index.html#finset-issquarefreebasis","scope":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c","status":"fully_proved","summary":"`B` is a squarefree basis of the finite family `F` of polynomials over `D`: its members are primitive, squarefree, of positive degree and pairwise relatively prime, and every member of `F` is a constant times a product of powers of members of `B`.","title":"Finset.IsSquarefreeBasis"},{"id":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/tauceti-ismccallumwelloriented","kind":"node","lean":"TauCeti.IsMcCallumWellOriented","parent":2568,"result":true,"route":"../roadmap/tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/index.html#tauceti-ismccallumwelloriented","scope":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c","status":"fully_proved","summary":"A finite family `F` of real polynomials in `n` variables is *well-oriented* in McCallum's sense if, when `n` is positive, singling out the distinguished variable `X 0` gives a family with a squarefree basis `B` such that each member of `B` is nullified at only finitely many points of `\u211d ^ (n - 1)`, and the McCallum projection of the family with respect to `B` is again well-oriented. Every family of constants is\u2026","title":"TauCeti.IsMcCallumWellOriented"},{"id":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/mccallum-lifting-fails-over-positive-dimensional","kind":"node","lean":"TauCeti.exists_nullified_mcCallumProjection_orderAt_ne","parent":2568,"result":true,"route":"../roadmap/tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/index.html#mccallum-lifting-fails-over-positive-dimensional","scope":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c","status":"fully_proved","summary":"The hypothesis of `TauCeti.exists_isSemialgebraicStack_orderAt_eq_of_mcCallumProjection` that no basis member is nullified over a cell with more than one point cannot be dropped, even over a connected semialgebraic analytic submanifold on which the McCallum projection is order-invariant.","title":"McCallum lifting fails over positive-dimensional nullifying cells"},{"id":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/sign-invariance-of-the-mccallum-projection-does","kind":"node","lean":"TauCeti.exists_signInvariant_mcCallumProjection_stackCells_not_signInvariant","parent":2568,"result":true,"route":"../roadmap/tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/index.html#sign-invariance-of-the-mccallum-projection-does","scope":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c","status":"fully_proved","summary":"In `Finset.IsSquarefreeBasis.exists_isSemialgebraicStack_signInvariant_of_mcCallumProjection`, the hypothesis that the McCallum projection is order-invariant on the base cannot be weakened to sign-invariance, even for a nonnullified squarefree basis over a connected semialgebraic analytic submanifold. So a lower level of a CAD that is merely sign-invariant for the projection cannot replace an order-invariant one.","title":"Sign-invariance of the McCallum projection does not suffice for lifting"},{"id":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/lazard-lifting-of-a-family","kind":"node","lean":"TauCeti.nonempty_lazardDelineation_of_lazardProjection","parent":2568,"result":true,"route":"../roadmap/tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/index.html#lazard-lifting-of-a-family","scope":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c","status":"fully_proved","summary":"Over a preconnected analytic submanifold on which the Lazard projection of a family with respect to an irreducible basis is valuation-invariant, the family has a Lazard delineation. Its members may be nullified at points of the base.","title":"Lazard lifting of a family"},{"id":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/cad-lifting-by-lazard-delineations-in-a-reordere","kind":"node","lean":"TauCeti.IsCAD.exists_isCAD_lazardValuation_eq_of_lazardDelineation_reindex","parent":2568,"result":true,"route":"../roadmap/tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/index.html#cad-lifting-by-lazard-delineations-in-a-reordere","scope":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c","status":"fully_proved","summary":"Suppose every cell `C` of a cylindrical algebraic decomposition of `\u211d ^ n` is an analytic submanifold, and its reordering `(\u00b7 \u2218 e.symm) \u207b\u00b9' C` by a permutation `e` of the base coordinates carries a Lazard delineation of a fixed polynomial family. Then the stacks of those delineations, pulled back to the original cells, form a cylindrical algebraic decomposition of `\u211d ^ (n + 1)` with the original decomposition as\u2026","title":"CAD lifting by Lazard delineations in a reordered base"},{"id":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/finset-integerprojection","kind":"node","lean":"Finset.integerProjection","parent":2568,"result":true,"route":"../roadmap/tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/index.html#finset-integerprojection","scope":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c","status":"fully_proved","summary":"The Collins projection of a finite family `P` of integer polynomials in the variables `X 0, \u2026, X n`, computed over `\u2124`: single out the distinguished variable `X 0` and take the Collins projection of the resulting family of polynomials in `X 0` over `\u2124[X 1, \u2026, X n]`.","title":"Finset.integerProjection"},{"id":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/the-integer-projection-computes-the-real-project","kind":"node","lean":"Finset.image_map_integerProjection","parent":2568,"result":true,"route":"../roadmap/tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c/index.html#the-integer-projection-computes-the-real-project","scope":"tauceti/geometry/part-01/real-algebraic-geometry-sign-determination-and-c","status":"fully_proved","summary":"Mapping the integer projection of `P` to `\u211d` gives the Collins projection of the real family of `P`.","title":"The integer projection computes the real projection"},{"data_url":"scopes/tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy.json","id":"tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy","kind":"scope","parent":2425,"result":false,"route":"../roadmap/tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy/index.html","scope":"tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Smooth orbifolds with finite isotropy"},{"id":"tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy/the-linear-local-model-at-a-fixed-point","kind":"node","lean":"TauCeti.exists_equivariant_chart_target_eq_sumOfConjugatesInnerSublevel_of_contMDiffAt","parent":2585,"result":true,"route":"../roadmap/tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy/index.html#the-linear-local-model-at-a-fixed-point","scope":"tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy","status":"fully_proved","summary":"Let a finite group `G` act continuously on a finite-dimensional real manifold without boundary, fixing `y`, by maps that are `C\u00b9` at `y`. Inside any neighbourhood `s` of `y` there is a chart `e` sending `y` to `0` whose domain is `G`-invariant and preconnected and whose target is a positive strict sublevel set of a `\u03c3`-invariant inner product, in which `G` acts linearly through a continuous representation `\u03c3`.\u2026","title":"The linear local model at a fixed point"},{"id":"tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy/tauceti-germkernel-eq-ker","kind":"node","lean":"TauCeti.germKernel_eq_ker","parent":2585,"result":true,"route":"../roadmap/tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy/index.html#tauceti-germkernel-eq-ker","scope":"tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy","status":"fully_proved","summary":"For a group with torsion stabilizers acting by `C\u00b9` maps on a preconnected Hausdorff manifold without boundary, the germ kernel at every point is the kernel of the action: an element acting as the identity near one point acts as the identity everywhere.","title":"TauCeti.germKernel_eq_ker"},{"id":"tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy/the-principal-part-of-a-finite-smooth-action","kind":"node","lean":"TauCeti.dense_setOf_stabilizer_eq_ker_of_contMDiff","parent":2585,"result":true,"route":"../roadmap/tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy/index.html#the-principal-part-of-a-finite-smooth-action","scope":"tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy","status":"fully_proved","summary":"For a finite group acting by `C\u00b9` maps on a preconnected Hausdorff manifold without boundary, the points whose stabilizer is the kernel of the action form a dense set. The action need not be faithful: elements acting trivially are retained in every stabilizer, and generically they are the only ones. The set is also open, by `TauCeti.isOpen_setOf_stabilizer_eq_ker`.","title":"The principal part of a finite smooth action"},{"id":"tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy/generic-stabilizer-of-a-finite-linear-action","kind":"node","lean":"TauCeti.dense_setOf_stabilizer_eq_ker","parent":2585,"result":true,"route":"../roadmap/tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy/index.html#generic-stabilizer-of-a-finite-linear-action","scope":"tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy","status":"fully_proved","summary":"For a finite group acting linearly and continuously on a Hausdorff topological module over a ring in which `0` is a limit of units, for instance a finite-dimensional real vector space, the points whose stabilizer is the kernel of the action form a dense set. The action need not be faithful. The set is also open, by `TauCeti.isOpen_setOf_stabilizer_eq_ker`.","title":"Generic stabilizer of a finite linear action"},{"id":"tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy/the-averaging-contraction","kind":"node","lean":"TauCeti.groupCohomology.averagingHomotopy","parent":2585,"result":true,"route":"../roadmap/tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy/index.html#the-averaging-contraction","scope":"tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy","status":"fully_proved","summary":"The identity of the complex of inhomogeneous cochains is homotopic to the averaging projection onto the invariants placed in degree `0`, through the homotopy whose component from degree `n + 1` to degree `n` is `(-1)^(n+1)` times the average over the last argument (`TauCeti.groupCohomology.averagingHomotopy_hom_succ`).","title":"The averaging contraction"},{"id":"tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy/categorytheory-actioncategory-stabilizermulequiv","kind":"node","lean":"CategoryTheory.ActionCategory.stabilizerMulEquivAut","parent":2585,"result":true,"route":"../roadmap/tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy/index.html#categorytheory-actioncategory-stabilizermulequiv","scope":"tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy","status":"fully_proved","summary":"The stabilizer of `x` is the automorphism group of the corresponding object of the action groupoid. Every element of the stabilizer, including one acting trivially on all of `X`, is kept as its own automorphism, and the multiplication matches `g * h = h \u226b g`.","title":"CategoryTheory.ActionCategory.stabilizerMulEquivAut"},{"id":"tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy/categorytheory-actioncategory-isempty-equivalenc","kind":"node","lean":"CategoryTheory.ActionCategory.isEmpty_equivalence_punit","parent":2585,"result":true,"route":"../roadmap/tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy/index.html#categorytheory-actioncategory-isempty-equivalenc","scope":"tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy","status":"fully_proved","summary":"If the group is nontrivial, the point quotient `[*/G]` is not equivalent to a thin category; in particular it is not equivalent to the point with trivial isotropy.","title":"CategoryTheory.ActionCategory.isEmpty_equivalence_punit"},{"id":"tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy/tauceti-mulaction-germkernel","kind":"node","lean":"TauCeti.MulAction.germKernel","parent":2585,"result":true,"route":"../roadmap/tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy/index.html#tauceti-mulaction-germkernel","scope":"tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy","status":"fully_proved","summary":"The germ kernel of the action of `G` at `x`: the subgroup of the elements of `G` that act as the identity on a neighbourhood of `x`.","title":"TauCeti.MulAction.germKernel"},{"id":"tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy/tauceti-diffeology-instdiffeologicalspacequotien","kind":"node","lean":"TauCeti.Diffeology.instDiffeologicalSpaceQuotient","parent":2585,"result":true,"route":"../roadmap/tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy/index.html#tauceti-diffeology-instdiffeologicalspacequotien","scope":"tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy","status":"fully_proved","summary":"The quotient diffeology on a quotient type: the diffeology coinduced by the projection.","title":"TauCeti.Diffeology.instDiffeologicalSpaceQuotient"},{"id":"tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy/tauceti-topcat-normalspace-of-ispushout","kind":"node","lean":"TauCeti.TopCat.normalSpace_of_isPushout","parent":2585,"result":true,"route":"../roadmap/tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy/index.html#tauceti-topcat-normalspace-of-ispushout","scope":"tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy","status":"fully_proved","summary":"Attaching a normal space `B` to a normal space `X` along a closed subspace of `B` gives a normal space: in a pushout square in `TopCat` whose side `m : A \u27f6 B` is a closed embedding, the corner `P` is normal.","title":"TauCeti.TopCat.normalSpace_of_isPushout"},{"id":"tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy/tauceti-topcat-normalspace-of-iscolimit","kind":"node","lean":"TauCeti.TopCat.normalSpace_of_isColimit","parent":2585,"result":true,"route":"../roadmap/tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy/index.html#tauceti-topcat-normalspace-of-iscolimit","scope":"tauceti/geometry/part-01/smooth-orbifolds-with-finite-isotropy","status":"fully_proved","summary":"The colimit of a sequence of normal spaces in `TopCat` whose transition maps are closed embeddings is normal.","title":"TauCeti.TopCat.normalSpace_of_isColimit"},{"data_url":"scopes/tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics.json","id":"tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics","kind":"scope","parent":2425,"result":false,"route":"../roadmap/tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics/index.html","scope":"tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Symplectic geometry and Hamiltonian mechanics"},{"id":"tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics/tauceti-cotangentbundle-issymplectic-symplecticf","kind":"node","lean":"TauCeti.CotangentBundle.isSymplectic_symplecticForm","parent":2597,"result":true,"route":"../roadmap/tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics/index.html#tauceti-cotangentbundle-issymplectic-symplecticf","scope":"tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics","status":"fully_proved","summary":"The canonical symplectic form of the cotangent bundle is closed and nondegenerate.","title":"TauCeti.CotangentBundle.isSymplectic_symplecticForm"},{"id":"tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics/chart-formula-for-the-canonical-symplectic-form","kind":"node","lean":"TauCeti.CotangentBundle.inChartAt_symplecticForm_apply","parent":2597,"result":true,"route":"../roadmap/tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics/index.html#chart-formula-for-the-canonical-symplectic-form","scope":"tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics","status":"fully_proved","summary":"In every preferred chart of the cotangent bundle, the canonical symplectic form is the linear form `TauCeti.strongDualCotangentSymplecticForm`, `\u03c9((\u03b4q, \u03b4p), (\u03b4q', \u03b4p')) = \u03b4p'(\u03b4q) - \u03b4p(\u03b4q')`.","title":"Chart formula for the canonical symplectic form"},{"id":"tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics/the-liouville-form-is-tautological","kind":"node","lean":"TauCeti.CotangentBundle.mpullback_liouvilleForm","parent":2597,"result":true,"route":"../roadmap/tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics/index.html#the-liouville-form-is-tautological","scope":"tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics","status":"fully_proved","summary":"Pulling it back along the smooth section of the cotangent bundle defined by a smooth one-form `\u03b1` gives back `\u03b1`.","title":"The Liouville form is tautological"},{"id":"tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics/cotangent-lifts-are-symplectic","kind":"node","lean":"Diffeomorph.mpullback_cotangent_symplecticForm","parent":2597,"result":true,"route":"../roadmap/tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics/index.html#cotangent-lifts-are-symplectic","scope":"tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics","status":"fully_proved","summary":"The cotangent lift of a smooth diffeomorphism pulls the canonical symplectic form of `T*N` back to the canonical symplectic form of `T*M`.","title":"Cotangent lifts are symplectic"},{"id":"tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics/fibre-translation-by-a-closed-one-form-is-symple","kind":"node","lean":"TauCeti.CotangentBundle.mpullback_vectorBundleTranslateDiffeomorph_symplecticForm_eq_self_iff","parent":2597,"result":true,"route":"../roadmap/tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics/index.html#fibre-translation-by-a-closed-one-form-is-symple","scope":"tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics","status":"fully_proved","summary":"Translating the fibres of the cotangent bundle by a one-form `\u03b1` preserves the canonical symplectic form exactly when `\u03b1` is closed.","title":"Fibre translation by a closed one-form is symplectic"},{"id":"tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics/tauceti-symplecticform-islagrangian-iff-maximal","kind":"node","lean":"TauCeti.SymplecticForm.isLagrangian_iff_maximal_isIsotropic","parent":2597,"result":true,"route":"../roadmap/tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics/index.html#tauceti-symplecticform-islagrangian-iff-maximal","scope":"tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics","status":"fully_proved","summary":"Lagrangian subspaces are exactly the maximal isotropic subspaces, in any dimension.","title":"TauCeti.SymplecticForm.isLagrangian_iff_maximal_isIsotropic"},{"id":"tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics/tauceti-cotangentbundle","kind":"node","lean":"TauCeti.CotangentBundle","parent":2597,"result":true,"route":"../roadmap/tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics/index.html#tauceti-cotangentbundle","scope":"tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics","status":"fully_proved","summary":"The cotangent bundle of a manifold `M` modelled on `I`: the total space of the bundle whose fibre at `x` is the space of continuous linear functionals on `TangentSpace I x`. It is a manifold modelled on `I.prod \ud835\udcd8(\ud835\udd5c, E \u2192L[\ud835\udd5c] \ud835\udd5c)`.","title":"TauCeti.CotangentBundle"},{"id":"tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics/transition-maps-of-the-cotangent-bundle","kind":"node","lean":"TauCeti.CotangentBundle.extChartAt_extChartAt_symm","parent":2597,"result":true,"route":"../roadmap/tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics/index.html#transition-maps-of-the-cotangent-bundle","scope":"tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics","status":"fully_proved","summary":"Changing from the preferred chart at `z\u2080` to the preferred chart at `z\u2081` sends `(q, p)` to `(\u03c6 q, p \u2218 (d\u03c6_q)\u207b\u00b9)`, where `\u03c6` is the change of base charts; `(d\u03c6_q)\u207b\u00b9` is the derivative of the inverse change of charts at `\u03c6 q`.","title":"Transition maps of the cotangent bundle"},{"id":"tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics/bundle-issubmersion-proj","kind":"node","lean":"Bundle.isSubmersion_proj","parent":2597,"result":true,"route":"../roadmap/tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics/index.html#bundle-issubmersion-proj","scope":"tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics","status":"fully_proved","summary":"The projection of a fibre bundle is a submersion.","title":"Bundle.isSubmersion_proj"},{"id":"tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics/tauceti-symplecticform-issymplectic","kind":"node","lean":"TauCeti.SymplecticForm.IsSymplectic","parent":2597,"result":true,"route":"../roadmap/tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics/index.html#tauceti-symplecticform-issymplectic","scope":"tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics","status":"fully_proved","summary":"A submodule is symplectic if it meets its symplectic complement only in zero, equivalently (`isSymplectic_iff_nondegenerate_restrict`) `\u03c9` restricts to it nondegenerately.","title":"TauCeti.SymplecticForm.IsSymplectic"},{"id":"tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics/addcircle-instismanifold","kind":"node","lean":"AddCircle.instIsManifold","parent":2597,"result":true,"route":"../roadmap/tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics/index.html#addcircle-instismanifold","scope":"tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics","status":"fully_proved","summary":"The circle `AddCircle p` is an analytic manifold modelled on `\u211d`.","title":"AddCircle.instIsManifold"},{"id":"tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics/addcircle-islocaldiffeomorph-coe","kind":"node","lean":"AddCircle.isLocalDiffeomorph_coe","parent":2597,"result":true,"route":"../roadmap/tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics/index.html#addcircle-islocaldiffeomorph-coe","scope":"tauceti/geometry/part-01/symplectic-geometry-and-hamiltonian-mechanics","status":"fully_proved","summary":"The quotient map `\u211d \u2192 AddCircle p` is an analytic local diffeomorphism.","title":"AddCircle.isLocalDiffeomorph_coe"},{"data_url":"scopes/tauceti/geometry/part-01/taubes-gromov-theory.json","id":"tauceti/geometry/part-01/taubes-gromov-theory","kind":"scope","parent":2425,"result":false,"route":"../roadmap/tauceti/geometry/part-01/taubes-gromov-theory/index.html","scope":"tauceti/geometry/part-01/taubes-gromov-theory","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Taubes\u2013Gromov theory"},{"id":"tauceti/geometry/part-01/taubes-gromov-theory/the-good-curve-component-inequality","kind":"node","lean":"TauCeti.sum_mul_apply_self_sub_apply_canonical_le","parent":2610,"result":true,"route":"../roadmap/tauceti/geometry/part-01/taubes-gromov-theory/index.html#the-good-curve-component-inequality","scope":"tauceti/geometry/part-01/taubes-gromov-theory","status":"fully_proved","summary":"If `e = \u2211 m i \u2022 C i` is a combination with natural coefficients of classes which meet pairwise nonnegatively, and `e` is at least `-1` on every class `C i` of negative square (as for a good class and exceptional spheres), then the virtual dimensions `d(x) = x \u00b7 x - \u03ba \u00b7 x` satisfy `\u2211 m i d(C i) \u2264 d(e)`.","title":"The good-curve component inequality"},{"id":"tauceti/geometry/part-01/taubes-gromov-theory/tauceti-isgoodcycle-apply-self-eq-zero-of-one-lt","kind":"node","lean":"TauCeti.IsGoodCycle.apply_self_eq_zero_of_one_lt_mult","parent":2610,"result":true,"route":"../roadmap/tauceti/geometry/part-01/taubes-gromov-theory/index.html#tauceti-isgoodcycle-apply-self-eq-zero-of-one-lt","scope":"tauceti/geometry/part-01/taubes-gromov-theory","status":"fully_proved","summary":"An image occurring with multiplicity at least two has square zero.","title":"TauCeti.IsGoodCycle.apply_self_eq_zero_of_one_lt_mult"},{"id":"tauceti/geometry/part-01/taubes-gromov-theory/independent-component-expansion-of-the-ray-serie","kind":"node","lean":"TauCeti.coeff_rayTorusWeightSeries","parent":2610,"result":true,"route":"../roadmap/tauceti/geometry/part-01/taubes-gromov-theory/index.html#independent-component-expansion-of-the-ray-serie","scope":"tauceti/geometry/part-01/taubes-gromov-theory","status":"fully_proved","summary":"its `n`-th coefficient is the sum, over all configurations `l` of total class `n \u2022 A\u2080` (the elements of `s.weightedFinsuppAntidiag m n`), of the products of the local weights `w_(\u03c3 c, i c)(l c)`.","title":"Independent-component expansion of the ray series"},{"id":"tauceti/geometry/part-01/taubes-gromov-theory/the-weitzenbock-identity-at-the-vacuum-on-temper","kind":"node","lean":"TauCeti.Vortex.temperedVacuumAdjoint_temperedVacuumOperator, TauCeti.Vortex.vacuumAdjoint_vacuumOperator","parent":2610,"result":true,"route":"../roadmap/tauceti/geometry/part-01/taubes-gromov-theory/index.html#the-weitzenbock-identity-at-the-vacuum-on-temper","scope":"tauceti/geometry/part-01/taubes-gromov-theory","status":"fully_proved","summary":"`L\u2080* L\u2080 = -\u0394 + 1 / 4`.","title":"The Weitzenb\u00f6ck identity at the vacuum on tempered fields"},{"id":"tauceti/geometry/part-01/taubes-gromov-theory/the-vacuum-operator-is-invertible-on-tempered-fi","kind":"node","lean":"TauCeti.Vortex.temperedVacuumOperatorEquiv","parent":2610,"result":true,"route":"../roadmap/tauceti/geometry/part-01/taubes-gromov-theory/index.html#the-vacuum-operator-is-invertible-on-tempered-fi","scope":"tauceti/geometry/part-01/taubes-gromov-theory","status":"fully_proved","summary":"`L\u2080` as a continuous linear automorphism of the fields of tempered distributions. Its inverse is `L\u2080\u207b\u00b9 = L\u2080* (-\u0394 + 1 / 4)\u207b\u00b9` (`temperedVacuumOperatorEquiv_symm_apply`).","title":"The vacuum operator is invertible on tempered fields"},{"id":"tauceti/geometry/part-01/taubes-gromov-theory/the-vacuum-operator-restricts-to-a-bijection-h-s","kind":"node","lean":"TauCeti.Vortex.memSobolev_temperedVacuumOperator_iff","parent":2610,"result":true,"route":"../roadmap/tauceti/geometry/part-01/taubes-gromov-theory/index.html#the-vacuum-operator-restricts-to-a-bijection-h-s","scope":"tauceti/geometry/part-01/taubes-gromov-theory","status":"fully_proved","summary":"a tempered field `x` lies in `H^s` if and only if `L\u2080 x` lies in `H^{s-1}`. Since `L\u2080` is bijective on tempered fields (`temperedVacuumOperatorEquiv`), it restricts to a bijection from `H^s` onto `H^{s-1}`, for every real `s`.","title":"The vacuum operator restricts to a bijection `H^s \u2192 H^{s-1}`"},{"id":"tauceti/geometry/part-01/taubes-gromov-theory/the-integral-wang-torsion-order","kind":"node","lean":"TauCeti.MappingTorus.card_torsion_homology_one_eq_natAbs_det","parent":2610,"result":true,"route":"../roadmap/tauceti/geometry/part-01/taubes-gromov-theory/index.html#the-integral-wang-torsion-order","scope":"tauceti/geometry/part-01/taubes-gromov-theory","status":"fully_proved","summary":"If `H\u2081(F; \u2124)` is free of finite rank and `det (1 - \u03c6_*) \u2260 0` on it, the torsion subgroup of `H\u2081(T_\u03c6; \u2124)` has order `|det (1 - \u03c6_*)|`.","title":"The integral Wang torsion order"},{"id":"tauceti/geometry/part-01/taubes-gromov-theory/orbit-torus-weights-are-local-zeta-factors","kind":"node","lean":"TauCeti.X_mul_derivative_torusWeightSeries_of_det_eq_one","parent":2610,"result":true,"route":"../roadmap/tauceti/geometry/part-01/taubes-gromov-theory/index.html#orbit-torus-weights-are-local-zeta-factors","scope":"tauceti/geometry/part-01/taubes-gromov-theory","status":"fully_proved","summary":"Let `P` be a `2 \u00d7 2` matrix of determinant one all of whose iterates are nondegenerate, `det (1 - P ^ k) \u2260 0` for `k \u2265 1`. Read off the label `\u03c3 = (0 < det (1 - P))` and `i = 1` if `det (1 + P) < 0`, `i = 0` otherwise. Then the local series `f = f_(\u03c3, i)` satisfies `t f' = f \u2211_{k \u2265 1} sgn det (1 - P ^ k) t ^ k`; over `\u211a` this says `log f = \u2211_{k \u2265 1} sgn det (1 - P ^ k) t ^ k / k`.","title":"Orbit-torus weights are local zeta factors"},{"id":"tauceti/geometry/part-01/taubes-gromov-theory/tauceti-torusweightseries","kind":"node","lean":"TauCeti.torusWeightSeries","parent":2610,"result":true,"route":"../roadmap/tauceti/geometry/part-01/taubes-gromov-theory/index.html#tauceti-torusweightseries","scope":"tauceti/geometry/part-01/taubes-gromov-theory","status":"fully_proved","summary":"The local multiplicity series `f_(\u03c3, i) \u2208 \u2124\u27e6X\u27e7` attached to an embedded square-zero pseudoholomorphic torus with determinant sign `\u03c3` (`true` for `+`, `false` for `-`) and with `i` negative signs among its three nontrivially twisted normal operators. The positive series are `1 / (1 - X)`, `1 + X`, `(1 + X) / (1 + X ^ 2)` and `(1 + X) (1 - X ^ 2) / (1 + X ^ 2)`, and each negative series is the inverse of the\u2026","title":"TauCeti.torusWeightSeries"},{"id":"tauceti/geometry/part-01/taubes-gromov-theory/tempereddistribution-laplacianresolvent","kind":"node","lean":"TemperedDistribution.laplacianResolvent","parent":2610,"result":true,"route":"../roadmap/tauceti/geometry/part-01/taubes-gromov-theory/index.html#tempereddistribution-laplacianresolvent","scope":"tauceti/geometry/part-01/taubes-gromov-theory","status":"fully_proved","summary":"The resolvent of the Laplacian `(c - \u0394)\u207b\u00b9` on tempered distributions: the Fourier multiplier with symbol `((2\u03c0)\u00b2 \u2016\u03be\u2016\u00b2 + c)\u207b\u00b9`. For `c > 0` it is the two-sided inverse of `-\u0394 + c` (`laplacianResolvent_neg_laplacian_add_smul` and `neg_laplacian_laplacianResolvent_add_smul`); for `c \u2264 0` the value is not meaningful.","title":"TemperedDistribution.laplacianResolvent"},{"id":"tauceti/geometry/part-01/taubes-gromov-theory/elliptic-regularity-for-the-shifted-laplacian","kind":"node","lean":"TemperedDistribution.memSobolev_add_two_iff_neg_laplacian_add_smul","parent":2610,"result":true,"route":"../roadmap/tauceti/geometry/part-01/taubes-gromov-theory/index.html#elliptic-regularity-for-the-shifted-laplacian","scope":"tauceti/geometry/part-01/taubes-gromov-theory","status":"fully_proved","summary":"for `c > 0`, a tempered distribution `f` lies in the Sobolev space `H^{s+2}` if and only if `-\u0394 f + c f` lies in `H^s`.","title":"Elliptic regularity for the shifted Laplacian"},{"id":"tauceti/geometry/part-01/taubes-gromov-theory/tauceti-isgoodcycle","kind":"node","lean":"TauCeti.IsGoodCycle","parent":2610,"result":true,"route":"../roadmap/tauceti/geometry/part-01/taubes-gromov-theory/index.html#tauceti-isgoodcycle","scope":"tauceti/geometry/part-01/taubes-gromov-theory","status":"fully_proved","summary":"The numerical properties of a positive cycle `\u2211 m i \u2022 C i` of distinct simple pseudoholomorphic images in a good class `e` of a symplectic four-manifold with canonical class `\u03ba`, passing through generic points indexed by `P`, read in the intersection form `B` on integral homology. The image `C i` has singularity number `\u03b4 i` and contains the points `pts i`. The fields say: the multiplicities are positive; distinct\u2026","title":"TauCeti.IsGoodCycle"},{"data_url":"scopes/tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur.json","id":"tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur","kind":"scope","parent":2425,"result":false,"route":"../roadmap/tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur/index.html","scope":"tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"The Yamabe problem and conformal scalar curvature"},{"id":"tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur/scalar-curvature-of-a-conformal-metric","kind":"node","lean":"TauCeti.Yamabe.scalarCurvature_rescale_eq_rpow_mul_conformalLaplacian","parent":2623,"result":true,"route":"../roadmap/tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur/index.html#scalar-curvature-of-a-conformal-metric","scope":"tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur","status":"fully_proved","summary":"In dimension `n \u2260 2`, let `u` be a positive function such that `\u011d = u ^ (4 / (n - 2)) g` is a smooth metric. Then the scalar curvature of `\u011d` is `Scal_\u011d = u ^ (-(n + 2) / (n - 2)) L_g u`, where `L_g` is the conformal Laplacian of `g`. Such a `u` is automatically smooth.","title":"Scalar curvature of a conformal metric"},{"id":"tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur/the-yamabe-equation","kind":"node","lean":"TauCeti.Yamabe.scalarCurvature_rescale_eq_iff_conformalLaplacian_eq","parent":2623,"result":true,"route":"../roadmap/tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur/index.html#the-yamabe-equation","scope":"tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur","status":"fully_proved","summary":"In dimension `n \u2260 2`, let `u` be a positive function such that `\u011d = u ^ (4 / (n - 2)) g` is a smooth metric. Then `\u011d` has scalar curvature `c` at a point exactly when `u` solves the Yamabe equation `L_g u = c u ^ (p\u2099 - 1)` there, where `p\u2099 = 2 n / (n - 2)`.","title":"The Yamabe equation"},{"id":"tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur/volume-normalized-total-scalar-curvature-of-a-co","kind":"node","lean":"TauCeti.Yamabe.integral_scalarCurvature_rescale_div_riemannianTotalVolume_rpow","parent":2623,"result":true,"route":"../roadmap/tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur/index.html#volume-normalized-total-scalar-curvature-of-a-co","scope":"tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur","status":"fully_proved","summary":"On a compact manifold of dimension `n \u2260 2`, let `u` be a positive function such that `\u011d = u ^ (4 / (n - 2)) g` is a smooth metric, and let `p\u2099 = 2 n / (n - 2)`. Then `(\u222b Scal_\u011d dV_\u011d) / Vol(\u011d) ^ (2 / p\u2099) = (\u222b u L_g u dV_g) / (\u222b u ^ p\u2099 dV_g) ^ (2 / p\u2099)`, where `L_g` is the conformal Laplacian of `g`.","title":"Volume-normalized total scalar curvature of a conformal metric"},{"id":"tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur/weyl-tensor-under-conformal-change","kind":"node","lean":"Bundle.ContMDiffRiemannianMetric.weylTensor_leviCivitaConnection_rescale","parent":2623,"result":true,"route":"../roadmap/tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur/index.html#weyl-tensor-under-conformal-change","scope":"tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur","status":"fully_proved","summary":"Away from dimension `2`, the fully covariant Weyl tensor of a conformally rescaled smooth metric `f g` is `f` times that of `g`: `W^(f g)(u, v, w, z) = f W^g(u, v, w, z)`. Equivalently, the Weyl tensor of type `(1, 3)` is conformally invariant.","title":"Weyl tensor under conformal change"},{"id":"tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur/local-conformal-flatness-forces-the-weyl-tensor","kind":"node","lean":"Bundle.ContMDiffRiemannianMetric.IsLocallyConformallyFlat.weylTensor_eq_zero","parent":2623,"result":true,"route":"../roadmap/tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur/index.html#local-conformal-flatness-forces-the-weyl-tensor","scope":"tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur","status":"fully_proved","summary":"Away from dimension `2`, the Weyl tensor of a locally conformally flat metric vanishes at every point. This is the easy half of the Weyl\u2013Schouten theorem.","title":"Local conformal flatness forces the Weyl tensor to vanish"},{"id":"tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur/the-bubbles-attain-the-yamabe-constant-of-the-sp","kind":"node","lean":"TauCeti.Yamabe.conformalCoefficient_mul_integral_norm_fderiv_bubble_sq_eq_sphereYamabeConstant","parent":2623,"result":true,"route":"../roadmap/tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur/index.html#the-bubbles-attain-the-yamabe-constant-of-the-sp","scope":"tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur","status":"fully_proved","summary":"in dimension `n > 2`, for Lebesgue measure, `a\u2099 \u222b \u2016dU_{t,b}\u2016\u00b2 = Y(S\u207f) (\u222b U_{t,b} ^ p\u2099) ^ (2 / p\u2099)`. This is the equality case of the sharp Euclidean Sobolev inequality.","title":"The bubbles attain the Yamabe constant of the sphere"},{"id":"tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur/the-area-of-the-unit-round-sphere","kind":"node","lean":"TauCeti.roundSphereVolume_eq_two_mul_pi_rpow_div_Gamma","parent":2623,"result":true,"route":"../roadmap/tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur/index.html#the-area-of-the-unit-round-sphere","scope":"tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur","status":"fully_proved","summary":"`|S\u207f| = 2 \u03c0 ^ ((n + 1) / 2) / \u0393((n + 1) / 2)`.","title":"The area of the unit round sphere"},{"id":"tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur/tauceti-yamabe-conformallaplacian","kind":"node","lean":"TauCeti.Yamabe.conformalLaplacian","parent":2623,"result":true,"route":"../roadmap/tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur/index.html#tauceti-yamabe-conformallaplacian","scope":"tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur","status":"fully_proved","summary":"The conformal Laplacian `L_g u = a\u2099 \u0394\u207a_g u + Scal_g u` of a function `u` on a smooth Riemannian manifold of dimension `n`, where `a\u2099 = 4 (n - 1) / (n - 2)`, `\u0394\u207a_g = -\u0394_g` is the nonnegative Laplacian and `Scal_g` is the scalar curvature.","title":"TauCeti.Yamabe.conformalLaplacian"},{"id":"tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur/bundle-contmdiffriemannianmetric-islocallyconfor","kind":"node","lean":"Bundle.ContMDiffRiemannianMetric.IsLocallyConformallyFlat","parent":2623,"result":true,"route":"../roadmap/tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur/index.html#bundle-contmdiffriemannianmetric-islocallyconfor","scope":"tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur","status":"fully_proved","summary":"A smooth Riemannian metric `g` is locally conformally flat if every point `x` has a neighbourhood on which a conformal rescaling `f g`, by a smooth positive function `f`, is flat: the curvature tensor of the Levi-Civita connection of `f g` vanishes at every point near `x`.","title":"Bundle.ContMDiffRiemannianMetric.IsLocallyConformallyFlat"},{"id":"tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur/covariantderivative-cottontensor","kind":"node","lean":"CovariantDerivative.cottonTensor","parent":2623,"result":true,"route":"../roadmap/tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur/index.html#covariantderivative-cottontensor","scope":"tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur","status":"fully_proved","summary":"The Cotton tensor `C(u, v, w) = (\u2207_w P)(u, v) - (\u2207_v P)(u, w)` of a smooth connection, where `P` is the Schouten tensor and `\u2207P` is `CovariantDerivative.derivSchoutenTensor`. The usual formula on vector fields holds for a smooth metric (see `CovariantDerivative.cottonTensor_apply_of_mdifferentiableAt`); otherwise `\u2207P`, and hence the Cotton tensor, may be `0`.","title":"CovariantDerivative.cottonTensor"},{"id":"tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur/weyl-schouten-in-dimension-three-on-a-model-spac","kind":"node","lean":"Bundle.ContMDiffRiemannianMetric.isLocallyConformallyFlat_iff_cottonTensor_eq_zero_model_space, Bundle.ContMDiffRiemannianMetric.isLocallyConformallyFlat_iff_weylTensor_eq_zero_model_space","parent":2623,"result":true,"route":"../roadmap/tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur/index.html#weyl-schouten-in-dimension-three-on-a-model-spac","scope":"tauceti/geometry/part-01/the-yamabe-problem-and-conformal-scalar-curvatur","status":"fully_proved","summary":"An arbitrary smooth Riemannian metric is locally conformally flat if and only if its Cotton tensor vanishes.","title":"Weyl\u2013Schouten in dimension three, on a model space"},{"data_url":"scopes/tauceti/geometry/part-01/convex.json","id":"tauceti/geometry/part-01/convex","kind":"scope","parent":2425,"result":false,"route":"../roadmap/tauceti/geometry/part-01/convex/index.html","scope":"tauceti/geometry/part-01/convex","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Geometry.Convex` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Convex"},{"id":"tauceti/geometry/part-01/convex/gordans-theorem","kind":"node","lean":"TauCeti.exists_forall_dual_pos_iff","parent":2635,"result":true,"route":"../roadmap/tauceti/geometry/part-01/convex/index.html#gordans-theorem","scope":"tauceti/geometry/part-01/convex","status":"fully_proved","summary":"Some linear functional is strictly positive on every vector of a finite family exactly when the only nonnegative linear relation among the vectors is the trivial one.","title":"Gordan's theorem"},{"id":"tauceti/geometry/part-01/convex/hoffmans-error-bound-for-a-polyhedral-cone","kind":"node","lean":"TauCeti.exists_forall_abs_apply_sub_le_of_forall_neg_le","parent":2635,"result":true,"route":"../roadmap/tauceti/geometry/part-01/convex/index.html#hoffmans-error-bound-for-a-polyhedral-cone","scope":"tauceti/geometry/part-01/convex","status":"fully_proved","summary":"Let `a j` be finitely many functionals on a finite-dimensional space. There is a constant `C` such that every `w` with `-c \u2264 a j w` for all `j`, where `0 \u2264 c`, lies within `C * c` of the cone `{w' | \u2200 j, 0 \u2264 a j w'}`, as measured by finitely many prescribed functionals `b k`. The constant `C` is nonnegative.","title":"Hoffman's error bound for a polyhedral cone"},{"id":"tauceti/geometry/part-01/convex/stiemkes-theorem","kind":"node","lean":"Submodule.exists_pos_dotProduct_eq_zero_iff","parent":2635,"result":true,"route":"../roadmap/tauceti/geometry/part-01/convex/index.html#stiemkes-theorem","scope":"tauceti/geometry/part-01/convex","status":"fully_proved","summary":"A subspace `S` of `\u03b9 \u2192 K` contains no nonzero nonnegative vector exactly when some vector with strictly positive coordinates is orthogonal to all of `S`.","title":"Stiemke's theorem"},{"id":"tauceti/geometry/part-01/convex/tuckers-key-lemma","kind":"node","lean":"TauCeti.exists_nonneg_sum_smul_eq_zero_and_coeff_add_dual_pos_at","parent":2635,"result":true,"route":"../roadmap/tauceti/geometry/part-01/convex/index.html#tuckers-key-lemma","scope":"tauceti/geometry/part-01/convex","status":"fully_proved","summary":"For a finite family of vectors `a j` and an index `k`, there are a nonnegative linear relation `x` among the `a j` and a linear functional `y` that is nonnegative on every `a j`, such that `x k + y (a k) > 0`: either `a k` occurs in a nonnegative relation, or some functional nonnegative on the family is strictly positive on `a k`.","title":"Tucker's key lemma"},{"id":"tauceti/geometry/part-01/convex/tuckers-theorem","kind":"node","lean":"TauCeti.exists_nonneg_sum_smul_eq_zero_and_forall_coeff_add_dual_pos, TauCeti.exists_nat_sum_nsmul_eq_zero_and_forall_coeff_add_dual_pos","parent":2635,"result":true,"route":"../roadmap/tauceti/geometry/part-01/convex/index.html#tuckers-theorem","scope":"tauceti/geometry/part-01/convex","status":"fully_proved","summary":"For a finite family of vectors `a j` there are a nonnegative linear relation `x` among the `a j` and a linear functional `y` that is nonnegative on every `a j`, such that `x j + y (a j) > 0` for every `j`: each vector either occurs in the relation or is strictly positive under `y`.","title":"Tucker's theorem"},{"data_url":"scopes/tauceti/geometry/part-01/lie.json","id":"tauceti/geometry/part-01/lie","kind":"scope","parent":2425,"result":false,"route":"../roadmap/tauceti/geometry/part-01/lie/index.html","scope":"tauceti/geometry/part-01/lie","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Geometry.Lie` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Lie"},{"id":"tauceti/geometry/part-01/lie/the-closed-subgroup-theorem-with-its-model-space","kind":"node","lean":"TauCeti.Lie.nonempty_embeddedLieSubgroupData_lieSubalgebraOfSubgroup_of_isClosed","parent":2641,"result":true,"route":"../roadmap/tauceti/geometry/part-01/lie/index.html#the-closed-subgroup-theorem-with-its-model-space","scope":"tauceti/geometry/part-01/lie","status":"fully_proved","summary":"A closed subgroup of a finite-dimensional real Lie group carries embedded-Lie-subgroup data whose model vector space is the Lie algebra `TauCeti.Lie.lieSubalgebraOfSubgroup` of the subgroup.","title":"The closed-subgroup theorem, with its model space"},{"id":"tauceti/geometry/part-01/lie/hunts-lemma","kind":"node","lean":"LeftInvariantDerivation.exists_Ad_lie_eq_zero","parent":2641,"result":true,"route":"../roadmap/tauceti/geometry/part-01/lie/index.html#hunts-lemma","scope":"tauceti/geometry/part-01/lie","status":"fully_proved","summary":"For `X` and `Y` in the Lie algebra of a compact Lie group, some adjoint translate of `X` brackets to zero with `Y`.","title":"Hunt's lemma"},{"id":"tauceti/geometry/part-01/lie/kroneckers-criterion","kind":"node","lean":"TauCeti.closedZpowers_lieExp_eq_top_iff","parent":2641,"result":true,"route":"../roadmap/tauceti/geometry/part-01/lie/index.html#kroneckers-criterion","scope":"tauceti/geometry/part-01/lie","status":"fully_proved","summary":"`lieExp X` is a topological generator of a connected commutative Lie group if and only if `\u039b + \u2124 X` is dense in the Lie algebra, `\u039b` the kernel of the exponential.","title":"Kronecker's criterion"},{"data_url":"scopes/tauceti/geometry/part-01/manifold.json","id":"tauceti/geometry/part-01/manifold","kind":"scope","parent":2425,"result":false,"route":"../roadmap/tauceti/geometry/part-01/manifold/index.html","scope":"tauceti/geometry/part-01/manifold","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Geometry.Manifold` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Manifold"},{"data_url":"scopes/tauceti/geometry/part-01/manifold/part-01.json","id":"tauceti/geometry/part-01/manifold/part-01","kind":"scope","navigation":"transparent","parent":2645,"result":false,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-01/index.html","scope":"tauceti/geometry/part-01/manifold/part-01","status":"fully_proved","title":"The ambient disc theorem in a\u2026 to Green's second identity on a\u2026"},{"id":"tauceti/geometry/part-01/manifold/part-01/the-ambient-disc-theorem-in-a-manifold","kind":"node","lean":"TauCeti.SmoothEmbedding.exists_diffeotopy_eqOn_of_det_chartAt_mul_pos, TauCeti.SmoothEmbedding.exists_diffeotopy_eqOn_of_isOrientationPreservingAt","parent":2646,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-01/index.html#the-ambient-disc-theorem-in-a-manifold","scope":"tauceti/geometry/part-01/manifold/part-01","status":"fully_proved","summary":"Let `f` and `g` be smooth embeddings of `E` into a Hausdorff, \u03c3-compact boundaryless manifold modelled on `E`, with the same centre `f 0 = g 0` and derivatives at `0` of the same orientation, compared in the chart at the centre. On every compact subset `A` of `E`, `g` is `f` followed by the final diffeomorphism of a smooth diffeotopy of `M` fixing every point outside a compact set.","title":"The ambient disc theorem in a manifold"},{"id":"tauceti/geometry/part-01/manifold/part-01/andreotti-frankel-theorem-for-proper-holomorphic","kind":"node","lean":"TauCeti.isZero_singularHomology_of_isProperMap_holomorphic_immersion","parent":2646,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-01/index.html#andreotti-frankel-theorem-for-proper-holomorphic","scope":"tauceti/geometry/part-01/manifold/part-01","status":"fully_proved","summary":"If a complex manifold `M` of complex dimension `n` admits a smooth proper holomorphic immersion into a proper complex inner product space `F`, then `H\u2096(M; R) = 0` for `k > n` and every coefficient object `R` of an abelian category with coproducts and exact sequential colimits.","title":"Andreotti\u2013Frankel theorem for proper holomorphic immersions"},{"id":"tauceti/geometry/part-01/manifold/part-01/the-bochner-formula","kind":"node","lean":"TauCeti.Manifold.laplaceBeltrami_norm_sq_mgradient","parent":2646,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-01/index.html#the-bochner-formula","scope":"tauceti/geometry/part-01/manifold/part-01","status":"fully_proved","summary":"For a smooth function `f` and every orthonormal basis `e\u1d62` of the tangent space at `x`, `\u0394 |grad f|\u00b2 = 2 \u2211\u1d62\u2c7c Hess f(e\u1d62, e\u2c7c)\u00b2 + 2 d(\u0394 f)(grad f) + 2 Ric(grad f, grad f)`, where `\u0394 = div \u2218 grad` is `TauCeti.Manifold.laplaceBeltrami` and `Ric` is the Ricci tensor of the Levi-Civita connection. For the nonnegative Laplacian `\u0394\u207a = -\u0394` this is `\u0394\u207a |df|\u00b2 + 2 |Hess f|\u00b2 + 2 Ric(grad f, grad f) = 2 \u27e8df, d \u0394\u207a f\u27e9`.","title":"The Bochner formula"},{"id":"tauceti/geometry/part-01/manifold/part-01/browns-bicollaring-theorem","kind":"node","lean":"TauCeti.brownBicollaring, TauCeti.IsLocallyBicollaredWithSides.exists_isBicollar, TauCeti.IsLocallyBicollared.exists_isBicollar, TauCeti.IsLocallyBicollared.isBicollared_of_not_isPreconnected_compl_range, TauCeti.IsLocallyFlat.isBicollared_of_not_isPreconnected_compl_range","parent":2646,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-01/index.html#browns-bicollaring-theorem","scope":"tauceti/geometry/part-01/manifold/part-01","status":"fully_proved","summary":"every locally flat embedding of the standard `n`-sphere in the standard `(n + 1)`-sphere is globally bicollared, including `n = 0`.","title":"Brown's bicollaring theorem"},{"id":"tauceti/geometry/part-01/manifold/part-01/browns-collaring-theorem-for-a-compact-domain","kind":"node","lean":"TauCeti.IsLocallyCollared.isCollared","parent":2646,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-01/index.html#browns-collaring-theorem-for-a-compact-domain","scope":"tauceti/geometry/part-01/manifold/part-01","status":"fully_proved","summary":"an injective, locally collared map from a compact space into a Hausdorff space is collared. The local collars are assembled into one by pushing an external collar inward along each of them in turn (Connelly's proof).","title":"Brown's collaring theorem, for a compact domain"},{"id":"tauceti/geometry/part-01/manifold/part-01/cartans-formula-for-the-levi-form","kind":"node","lean":"TauCeti.RoughForm.apply_mlieBracket_eq_neg_mextDeriv","parent":2646,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-01/index.html#cartans-formula-for-the-levi-form","scope":"tauceti/geometry/part-01/manifold/part-01","status":"fully_proved","summary":"For vector fields `V` and `W` tangent to the kernel of a one-form `\u03b1` near `x`, `\u03b1_x ([V, W]_x) = -d\u03b1_x (V_x, W_x)`.","title":"Cartan's formula for the Levi form"},{"id":"tauceti/geometry/part-01/manifold/part-01/cartans-magic-formula","kind":"node","lean":"TauCeti.mlieDerivFormWithin_eq_interior_mextDerivWithin_add, TauCeti.mlieDerivForm_apply_eq_mextDeriv_add, TauCeti.mlieDerivForm_eq_interior_mextDeriv_add, TauCeti.mlieDerivForm_eq_interior_mextDeriv_add_apply","parent":2646,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-01/index.html#cartans-magic-formula","scope":"tauceti/geometry/part-01/manifold/part-01","status":"fully_proved","summary":"Cartan's magic formula within a set, `L_V \u03c6 = \u03b9_V d\u03c6 + d (\u03b9_V \u03c6)`, for a form and a vector field differentiable within the set at the point.","title":"Cartan's magic formula"},{"id":"tauceti/geometry/part-01/manifold/part-01/the-christoffel-formula-of-the-first-kind","kind":"node","lean":"TauCeti.Manifold.two_mul_leviCivitaConnection_model_space","parent":2646,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-01/index.html#the-christoffel-formula-of-the-first-kind","scope":"tauceti/geometry/part-01/manifold/part-01","status":"fully_proved","summary":"If the Riemannian metric of the model space `F` is the field of bilinear forms `g`, differentiable at `x`, then the Levi-Civita derivative of a vector field `V` differentiable at `x` satisfies `2 g(\u2207_u V, w) = 2 g(DV(u), w) + (Dg(u))(V, w) + (Dg(V))(w, u) - (Dg(w))(u, V)`. Tangent vectors are read in the model space through `tangentSpaceCastModel`.","title":"The Christoffel formula of the first kind"},{"id":"tauceti/geometry/part-01/manifold/part-01/closed-normal-disc-theorem-for-a-closed-smooth-s","kind":"node","lean":"TauCeti.SmoothEmbedding.exists_isClosedEmbedding_normalExp_norm_le_subset","parent":2646,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-01/index.html#closed-normal-disc-theorem-for-a-closed-smooth-s","scope":"tauceti/geometry/part-01/manifold/part-01","status":"fully_proved","summary":"The normal exponential embeds closed discs and spheres of a positive smooth radius as closed subsets of any prescribed open neighbourhood of the core. The open discs embed openly. The core may be noncompact, and the ambient Riemannian metric need not be complete.","title":"Closed normal disc theorem for a closed smooth submanifold"},{"id":"tauceti/geometry/part-01/manifold/part-01/the-contracted-ricci-identity-for-the-gradient","kind":"node","lean":"TauCeti.Manifold.sum_inner_secondCovariantDerivative_mgradient_self","parent":2646,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-01/index.html#the-contracted-ricci-identity-for-the-gradient","scope":"tauceti/geometry/part-01/manifold/part-01","status":"fully_proved","summary":"In every orthonormal basis `e\u1d62` of the tangent space at `x`, `\u2211\u1d62 \u27ea\u2207\u00b2(grad f)(e\u1d62, e\u1d62), w\u27eb = d(\u0394 f)(w) + Ric(w, grad f)`, where `\u0394 = div \u2218 grad` is `TauCeti.Manifold.laplaceBeltrami`. Equivalently, the rough Laplacian of `grad f` is `grad \u0394 f + Ric(grad f)`.","title":"The contracted Ricci identity for the gradient"},{"id":"tauceti/geometry/part-01/manifold/part-01/the-disc-theorem-in-a-ball","kind":"node","lean":"TauCeti.SmoothEmbedding.exists_smoothIsotopy_mapsTo_ball","parent":2646,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-01/index.html#the-disc-theorem-in-a-ball","scope":"tauceti/geometry/part-01/manifold/part-01","status":"fully_proved","summary":"Smooth self-embeddings with values in a Euclidean ball, centred at its centre, with the same derivative orientation there, are smoothly isotopic relative to the centre through embeddings with values in the ball.","title":"The disc theorem in a ball"},{"id":"tauceti/geometry/part-01/manifold/part-01/the-disc-theorem-in-a-manifold","kind":"node","lean":"TauCeti.SmoothEmbedding.exists_smoothIsotopy_of_det_chartAt_mul_pos","parent":2646,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-01/index.html#the-disc-theorem-in-a-manifold","scope":"tauceti/geometry/part-01/manifold/part-01","status":"fully_proved","summary":"Two smooth embeddings of `E` into a boundaryless manifold modelled on `E`, with the same centre `f 0 = g 0` and derivatives at `0` of the same orientation, are smoothly isotopic relative to that centre. The orientations are compared in the chart at the centre: the product of the derivative determinants there is positive. This condition does not depend on the chart, since a change of chart multiplies both\u2026","title":"The disc theorem in a manifold"},{"id":"tauceti/geometry/part-01/manifold/part-01/the-divergence-theorem-on-a-manifold-without-bou","kind":"node","lean":"TauCeti.Manifold.integral_divergence_eq_zero","parent":2646,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-01/index.html#the-divergence-theorem-on-a-manifold-without-bou","scope":"tauceti/geometry/part-01/manifold/part-01","status":"fully_proved","summary":"the divergence of a compactly supported `C\u00b9` vector field has integral zero against the Riemannian volume.","title":"The divergence theorem on a manifold without boundary"},{"id":"tauceti/geometry/part-01/manifold/part-01/einsteins-constancy-theorem","kind":"node","lean":"CovariantDerivative.eq_of_ricciTensor_eq_smul_inner, CovariantDerivative.mfderiv_eq_zero_of_ricciTensor_eq_smul_inner","parent":2646,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-01/index.html#einsteins-constancy-theorem","scope":"tauceti/geometry/part-01/manifold/part-01","status":"fully_proved","summary":"Let a metric-compatible torsion-free connection on a preconnected manifold, of dimension other than two, have Ricci tensor `Ric = a g` for a function `a`. Then `a` is constant.","title":"Einstein's constancy theorem"},{"id":"tauceti/geometry/part-01/manifold/part-01/euclidean-disc-theorem","kind":"node","lean":"TauCeti.SmoothEmbedding.exists_smoothIsotopy_of_det_mul_pos, TauCeti.SmoothEmbedding.exists_smoothIsotopy_of_det_pos","parent":2646,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-01/index.html#euclidean-disc-theorem","scope":"tauceti/geometry/part-01/manifold/part-01","status":"fully_proved","summary":"Smooth self-embeddings with a common centre and the same derivative orientation there are smoothly isotopic relative to that centre. The determinant condition permits either common orientation and includes dimension zero.","title":"Euclidean disc theorem"},{"id":"tauceti/geometry/part-01/manifold/part-01/the-frobenius-theorem-in-coordinates","kind":"node","lean":"TauCeti.IsInvolutiveDistribution.exists_eventually_hasFDerivAt, TauCeti.isInvolutiveDistribution_graphDistribution_iff_forall_exists, TauCeti.IsInvolutiveDistribution.exists_eventually_eq_graph_fderiv","parent":2646,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-01/index.html#the-frobenius-theorem-in-coordinates","scope":"tauceti/geometry/part-01/manifold/part-01","status":"fully_proved","summary":"Let `f` be of class `C^(n+1)` near `(x\u2080, y\u2080)`, with an involutive graph distribution. Then the distribution has an integral manifold through `(x\u2080, y\u2080)`: the graph of a function `u`, of class `C^(n+1)` at `x\u2080`, with `u x\u2080 = y\u2080` and `D u x = f (x, u x)` near `x\u2080`, so that the tangent space of the graph at `(x, u x)` is the graph of `f (x, u x)`.","title":"The Frobenius theorem in coordinates"},{"id":"tauceti/geometry/part-01/manifold/part-01/the-gauss-formula","kind":"node","lean":"CovariantDerivative.IsLeviCivitaConnection.inner_apply_eq_inner_mvfderiv","parent":2646,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-01/index.html#the-gauss-formula","scope":"tauceti/geometry/part-01/manifold/part-01","status":"fully_proved","summary":"If `f : M \u2192 F` is a `C\u00b2` map into a real inner product space whose differential preserves inner products, then `\u27ea\u2207_X Y, w\u27eb = \u27eaD_X (df Y), df w\u27eb` for a Levi-Civita connection `\u2207`: it is the tangential part of the ambient derivative of `df Y : y \u21a6 df_y (Y y)`.","title":"The Gauss formula"},{"id":"tauceti/geometry/part-01/manifold/part-01/greens-first-identity-on-a-manifold-without-boun","kind":"node","lean":"TauCeti.Manifold.integral_mul_laplaceBeltrami","parent":2646,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-01/index.html#greens-first-identity-on-a-manifold-without-boun","scope":"tauceti/geometry/part-01/manifold/part-01","status":"fully_proved","summary":"for a compactly supported `C\u00b9` function `f` and a `C\u00b2` function `g`, `\u222b f \u0394g dV = -\u222b \u27eagrad f, grad g\u27eb dV`.","title":"Green's first identity on a manifold without boundary"},{"id":"tauceti/geometry/part-01/manifold/part-01/greens-formula-for-d-d","kind":"node","lean":"TauCeti.SmoothForm.integralTopForm_inner_codifferential_mextDeriv, TauCeti.SmoothForm.integralTopForm_inner_mextDeriv_codifferential, TauCeti.SmoothForm.mextDeriv_wedge_hodgeStar","parent":2646,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-01/index.html#greens-formula-for-d-d","scope":"tauceti/geometry/part-01/manifold/part-01","status":"fully_proved","summary":"on an oriented boundaryless Riemannian manifold, `\u222b_M \u27e8d*d\u03b1, \u03b2\u27e9 vol = \u222b_M \u27e8d\u03b1, d\u03b2\u27e9 vol` for smooth `k`-forms `\u03b1` and `\u03b2`, one of them compactly supported.","title":"Green's formula for `d*d`"},{"id":"tauceti/geometry/part-01/manifold/part-01/greens-second-identity-on-a-manifold-without-bou","kind":"node","lean":"TauCeti.Manifold.integral_mul_laplaceBeltrami_right_eq_left","parent":2646,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-01/index.html#greens-second-identity-on-a-manifold-without-bou","scope":"tauceti/geometry/part-01/manifold/part-01","status":"fully_proved","summary":"for a compactly supported `C\u00b2` function `f` and a `C\u00b2` function `g`, `\u222b f \u0394g dV = \u222b (\u0394f) g dV`.","title":"Green's second identity on a manifold without boundary"},{"data_url":"scopes/tauceti/geometry/part-01/manifold/part-02.json","id":"tauceti/geometry/part-01/manifold/part-02","kind":"scope","navigation":"transparent","parent":2645,"result":false,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-02/index.html","scope":"tauceti/geometry/part-01/manifold/part-02","status":"fully_proved","title":"Hadamard's identity in the\u2026 to Riemann's removable singularity\u2026"},{"id":"tauceti/geometry/part-01/manifold/part-02/hadamards-identity-in-the-algebra-of-smooth-func","kind":"node","lean":"TauCeti.sub_algebraMap_eq_sum_mul_sub_algebraMap","parent":2667,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-02/index.html#hadamards-identity-in-the-algebra-of-smooth-func","scope":"tauceti/geometry/part-01/manifold/part-02","status":"fully_proved","summary":"A pointwise factorization of `g - g x` through the functions `c i`, with coefficients `a i`, is an identity between the bundled smooth maps themselves, the constants appearing as the images of `algebraMap`.","title":"Hadamard's identity in the algebra of smooth functions"},{"id":"tauceti/geometry/part-01/manifold/part-02/hadamards-lemma-in-coordinates","kind":"node","lean":"ContMDiffMap.exists_hadamard_coords, OpenPartialHomeomorph.exists_eq_sum_holomorphicLinearCoordinateGerm_mul, TauCeti.mem_smoothVanishingIdeal_sq_of_mvfderiv_eq_zero","parent":2667,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-02/index.html#hadamards-lemma-in-coordinates","scope":"tauceti/geometry/part-01/manifold/part-02","status":"fully_proved","summary":"Subtracting its value at `x` writes a smooth real function as a combination of the coordinate functions of any finitely-indexed basis, each taken relative to its own value at `x`, with smooth coefficients. At the base point those coefficients are the directional derivatives along the basis.","title":"Hadamard's lemma in coordinates"},{"id":"tauceti/geometry/part-01/manifold/part-02/the-identity-theorem-on-analytic-manifolds","kind":"node","lean":"ContMDiff.eq_of_eventuallyEq, TauCeti.eqOn_of_preconnected_of_eventuallyEq_of_contMDiffAt","parent":2667,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-02/index.html#the-identity-theorem-on-analytic-manifolds","scope":"tauceti/geometry/part-01/manifold/part-02","status":"fully_proved","summary":"On a preconnected `C^\u03c9` manifold modelled without boundary on a locally connected space, two `C^\u03c9` functions with values in a normed space that agree on a neighbourhood of a point are equal.","title":"The identity theorem on analytic manifolds"},{"id":"tauceti/geometry/part-01/manifold/part-02/the-integrated-bochner-identity","kind":"node","lean":"TauCeti.Manifold.integral_sum_hessian_sq_add_ricci","parent":2667,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-02/index.html#the-integrated-bochner-identity","scope":"tauceti/geometry/part-01/manifold/part-02","status":"fully_proved","summary":"For a compactly supported smooth function on a boundaryless manifold, `\u222b (|Hess f|\u00b2 + Ric(grad f, grad f)) = \u222b (\u0394 f)\u00b2`. Both integrands are integrable, and the identity holds in every pointwise orthonormal basis.","title":"The integrated Bochner identity"},{"id":"tauceti/geometry/part-01/manifold/part-02/the-inverse-function-theorem-for-manifolds","kind":"node","lean":"TauCeti.isLocalDiffeomorphAt_of_mfderiv_eq, TauCeti.isLocalDiffeomorph_of_mfderiv_eq, TauCeti.isLocalDiffeomorph_of_deriv_ne_zero","parent":2667,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-02/index.html#the-inverse-function-theorem-for-manifolds","scope":"tauceti/geometry/part-01/manifold/part-02","status":"fully_proved","summary":"If `f` is `C^n` on an open set `s` with `1 \u2264 n`, `x` belongs to `s` and is an interior point, and the differential at `x` is a continuous linear equivalence, then `f` is a `C^n` local diffeomorphism at `x`.","title":"The inverse function theorem for manifolds"},{"id":"tauceti/geometry/part-01/manifold/part-02/jacobis-formula-on-a-manifold","kind":"node","lean":"TauCeti.mvfderiv_det_apply","parent":2667,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-02/index.html#jacobis-formula-on-a-manifold","scope":"tauceti/geometry/part-01/manifold/part-02","status":"fully_proved","summary":"If the entries of a matrix-valued function `A` are differentiable at `x` and `A x` has invertible determinant, then along a tangent vector `v` the derivative of `det A` is `det (A x) * trace ((A x)\u207b\u00b9 * dA (v))`, where `dA (v)` is the matrix of the derivatives of the entries of `A` along `v`.","title":"Jacobi's formula on a manifold"},{"id":"tauceti/geometry/part-01/manifold/part-02/the-joint-holomorphic-implicit-function-theorem","kind":"node","lean":"TauCeti.exists_jointImplicitFunction_of_eventually_mdifferentiableAt","parent":2667,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-02/index.html#the-joint-holomorphic-implicit-function-theorem","scope":"tauceti/geometry/part-01/manifold/part-02","status":"fully_proved","summary":"Suppose `G : M \u00d7 N \u2192 P` is holomorphic near `u`, and the partial derivative in the `N` coordinate of its representative in holomorphic charts `e`, `d`, `c` is invertible at `u`. There is a solution `\u03c8 : M \u00d7 P \u2192 N`, holomorphic near `(u.1, G u)`, taking that point to `u.2`. Near `(u, G u)`, the equation `G (z, v) = w` is equivalent to `\u03c8 (z, w) = v`. In particular solutions sufficiently near `u.2` are unique. Only\u2026","title":"The joint holomorphic implicit function theorem in charts"},{"id":"tauceti/geometry/part-01/manifold/part-02/the-koszul-formula","kind":"node","lean":"CovariantDerivative.IsLeviCivitaConnection.two_inner_eq_koszul, CovariantDerivative.two_inner_leviCivitaConnection_eq_koszul, TauCeti.Manifold.ContMDiffNondegenerateMetric.IsLeviCivitaConnection.two_apply_eq_koszul, TauCeti.Manifold.ContMDiffNondegenerateMetric.IsLocalIsometry.two_apply_mfderiv_eq_koszul","parent":2667,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-02/index.html#the-koszul-formula","scope":"tauceti/geometry/part-01/manifold/part-02","status":"fully_proved","summary":"for a Levi-Civita connection, `2 \u27ea\u2207_X Y, Z\u27eb` is the Koszul expression of the metric. Mathlib's `CovariantDerivative.IsLeviCivitaConnection.apply_eq` is the same identity with the expression written out; this form is the one `TauCeti.Manifold.koszul` controls.","title":"The Koszul formula"},{"id":"tauceti/geometry/part-01/manifold/part-02/the-lichnerowicz-estimate","kind":"node","lean":"TauCeti.Manifold.finrank_mul_le_of_laplaceBeltrami_eq_neg_mul","parent":2667,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-02/index.html#the-lichnerowicz-estimate","scope":"tauceti/geometry/part-01/manifold/part-02","status":"fully_proved","summary":"On a closed manifold of dimension `n \u2265 2` with `Ric \u2265 (n - 1) k g`, every positive eigenvalue with a nonzero smooth eigenfunction is at least `n k`. The eigen-equation is stated for the native `\u0394 = div grad`. The real lower bound `k` need not be positive.","title":"The Lichnerowicz estimate"},{"id":"tauceti/geometry/part-01/manifold/part-02/the-local-frobenius-theorem-tauceti-isinvolutive","kind":"node","lean":"TauCeti.IsInvolutiveDistribution.exists_flatChart, TauCeti.IsDistributionFrameOn.exists_openPartialHomeomorph_map_fderiv_eq_ker_snd","parent":2667,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-02/index.html#the-local-frobenius-theorem-tauceti-isinvolutive","scope":"tauceti/geometry/part-01/manifold/part-02","status":"fully_proved","summary":"An involutive `C^m` distribution `D` of rank `k`, for `1 \u2264 m`, on a `C^n` manifold with `m + 1 \u2264 n`, modelled on a finite-dimensional real space, is constant in a suitable chart around any interior point `x`: there are a chart `e` of the `C^m` maximal atlas whose domain contains `x` and a `k`-dimensional subspace `W` of the model space such that the differential of `I \u2218 e` carries `D y` onto `W` at every point `y`\u2026","title":"The local Frobenius theorem (`TauCeti.IsInvolutiveDistribution.exists_flatChart`)"},{"id":"tauceti/geometry/part-01/manifold/part-02/milnors-cell-attachment-theorem","kind":"node","lean":"TauCeti.IsMorse.exists_sublevel_union_cells_deformation, TauCeti.IsMorse.exists_relCWComplex_sublevel_deformation, TauCeti.IsMorse.exists_morseCellAttachment_deformation","parent":2667,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-02/index.html#milnors-cell-attachment-theorem","scope":"tauceti/geometry/part-01/manifold/part-02","status":"fully_proved","summary":"Let `f` be a Morse function on a Hausdorff manifold whose band `f\u207b\u00b9[c - \u03b4, c + \u03b4]` is compact, for some `0 < \u03b4`. Then there are a width `0 < \u03b5\u2080 \u2264 \u03b4` and normalized Morse charts `\u03c8 p` with pairwise disjoint sources, indexed by the critical points `p` on the level `f\u207b\u00b9(c)`, with `\u03c8 p p = 0`, with the closed ball of radius `\u221a(2\u03b5\u2080)` in the target of `\u03c8 p`, and in which `f = c - \u03a3_{i < \u03bb} y\u1d62\u00b2 + \u03a3_{i \u2265 \u03bb} y\u1d62\u00b2`, where `\u03bb`\u2026","title":"Milnor's cell attachment theorem"},{"id":"tauceti/geometry/part-01/manifold/part-02/milnors-index-bound-on-a-smooth-affine-complete","kind":"node","lean":"TauCeti.manifoldMorseIndex_norm_sq_sub_toLp_affineZeroLocus_le","parent":2667,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-02/index.html#milnors-index-bound-on-a-smooth-affine-complete","scope":"tauceti/geometry/part-01/manifold/part-02","status":"fully_proved","summary":"On the common zero set in `\u2102^\u03c3` of a polynomial family with surjective Jacobian at every common zero, every critical point of the Euclidean squared distance from a point `a` has real Morse index at most the complex dimension `|\u03c3| - |\u03b9|` of the zero set.","title":"Milnor's index bound on a smooth affine complete intersection"},{"id":"tauceti/geometry/part-01/manifold/part-02/noncompact-euclidean-tubular-neighbourhood-theor","kind":"node","lean":"TauCeti.exists_isOpenEmbedding_normalTubeOfRadius","parent":2667,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-02/index.html#noncompact-euclidean-tubular-neighbourhood-theor","scope":"tauceti/geometry/part-01/manifold/part-02","status":"fully_proved","summary":"A `C\u00b2` immersion inducing the source topology admits a positive continuous radius on which its normal addition map is an open embedding.","title":"Noncompact Euclidean tubular neighbourhood theorem"},{"id":"tauceti/geometry/part-01/manifold/part-02/the-oriented-common-centre-disc-theorem","kind":"node","lean":"TauCeti.SmoothEmbedding.exists_smoothIsotopy_of_isOrientationPreservingAt","parent":2667,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-02/index.html#the-oriented-common-centre-disc-theorem","scope":"tauceti/geometry/part-01/manifold/part-02","status":"fully_proved","summary":"Two smooth embeddings that preserve the chosen orientations at the origin and have the same centre are smoothly isotopic relative to that centre. No orientation condition away from the origin is required.","title":"The oriented common-centre disc theorem"},{"id":"tauceti/geometry/part-01/manifold/part-02/the-poincare-lemma-on-manifolds","kind":"node","lean":"TauCeti.exists_mextDeriv_eq_nhds, TauCeti.exactForms_succ_eq_closedForms_normedSpace","parent":2667,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-02/index.html#the-poincare-lemma-on-manifolds","scope":"tauceti/geometry/part-01/manifold/part-02","status":"fully_proved","summary":"A `C^r` form of positive degree, `r \u2265 1`, that is closed near a point `x` is, on a neighbourhood of `x`, the exterior derivative of a `C^r` form defined on the whole manifold. The manifold may have boundary or corners.","title":"The Poincar\u00e9 lemma on manifolds"},{"id":"tauceti/geometry/part-01/manifold/part-02/the-poincare-wirtinger-inequality-on-a-closed-ri","kind":"node","lean":"TauCeti.Manifold.exists_eLpNorm_sub_average_le_mul_eLpNorm_mgradient","parent":2667,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-02/index.html#the-poincare-wirtinger-inequality-on-a-closed-ri","scope":"tauceti/geometry/part-01/manifold/part-02","status":"fully_proved","summary":"Let `1 \u2264 p < \u221e`, on a compact connected manifold without boundary with a continuous Riemannian metric. There is a constant `C` such that `\u2016u - \u2a0d u\u2016_{L\u1d56} \u2264 C \u2016\u2207u\u2016_{L\u1d56}` for every `C\u00b9` function `u`, where `\u2a0d u` is the mean of `u` with respect to the Riemannian volume.","title":"The Poincar\u00e9\u2013Wirtinger inequality on a closed Riemannian manifold"},{"id":"tauceti/geometry/part-01/manifold/part-02/the-regular-fibre-theorem","kind":"node","lean":"TauCeti.exists_regularFiber_smoothEmbedding","parent":2667,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-02/index.html#the-regular-fibre-theorem","scope":"tauceti/geometry/part-01/manifold/part-02","status":"fully_proved","summary":"A regular fibre over an interior value of a `C^m` map, `m \u2265 1`, from a finite-dimensional boundaryless real manifold to a finite-dimensional real manifold admits a `C^m` manifold structure of dimension `dim M - dim N`, with its subspace topology, for which the inclusion is a `C^m` smooth embedding and has tangent image `ker df`. No separation or countability assumptions are needed, and an empty fibre is allowed.","title":"The regular-fibre theorem"},{"id":"tauceti/geometry/part-01/manifold/part-02/the-relative-ambient-disc-theorem","kind":"node","lean":"TauCeti.SmoothEmbedding.exists_diffeotopy_eqOn_of_isOrientationPreservingAt_of_range_subset","parent":2667,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-02/index.html#the-relative-ambient-disc-theorem","scope":"tauceti/geometry/part-01/manifold/part-02","status":"fully_proved","summary":"Let `f` and `g` be smooth embeddings of `E` into a Hausdorff boundaryless manifold `M` modelled on `E`, preserving the orientations `o` of `E` and `oM` of `M` at the origin, with images in a preconnected open set `U`. On every compact subset `A` of `E`, `g` is `f` followed by the final diffeomorphism of a smooth diffeotopy of `M` fixing every point outside a compact subset of `U`. Points outside `U`, for instance\u2026","title":"The relative ambient disc theorem"},{"id":"tauceti/geometry/part-01/manifold/part-02/the-ricci-identity","kind":"node","lean":"CovariantDerivative.secondCovariantDerivative_sub_swap_of_isTorsionFree, CovariantDerivative.secondCovariantDerivative_sub_swap, TauCeti.Manifold.mvfderiv_hessian_sub_swap","parent":2667,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-02/index.html#the-ricci-identity","scope":"tauceti/geometry/part-01/manifold/part-02","status":"fully_proved","summary":"For a torsion-free connection `\u2207\u1d40` on the tangent bundle, the antisymmetric part of the second covariant derivative of a smooth section `\u03c3` is the curvature of `\u2207` applied to `\u03c3`: `\u2207\u00b2\u03c3(u, v) - \u2207\u00b2\u03c3(v, u) = R(u, v) \u03c3`.","title":"The Ricci identity"},{"id":"tauceti/geometry/part-01/manifold/part-02/riemanns-removable-singularity-theorem-on-comple","kind":"node","lean":"TauCeti.mdifferentiableAt_of_continuousAt_of_eventually_nhdsNE","parent":2667,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-02/index.html#riemanns-removable-singularity-theorem-on-comple","scope":"tauceti/geometry/part-01/manifold/part-02","status":"fully_proved","summary":"A map between complex curves that is continuous at `x` and holomorphic at every point of a punctured neighbourhood of `x` is holomorphic at `x`. This is Mathlib's `Complex.differentiableOn_compl_singleton_and_continuousAt_iff` read in charts.","title":"Riemann's removable singularity theorem on complex curves"},{"data_url":"scopes/tauceti/geometry/part-01/manifold/part-03.json","id":"tauceti/geometry/part-01/manifold/part-03","kind":"scope","navigation":"transparent","parent":2645,"result":false,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-03/index.html","scope":"tauceti/geometry/part-01/manifold/part-03","status":"fully_proved","title":"Robbin\u2013Salamon Corrigendum\u2026 to Weierstrass preparation theorem\u2026"},{"id":"tauceti/geometry/part-01/manifold/part-03/robbin-salamon-corrigendum-lemma-b-on-arbitrary","kind":"node","lean":"TauCeti.ComplexStructureFamily.lieDerivEndomorphism_horizontalTerm_eq_zero","parent":2688,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-03/index.html#robbin-salamon-corrigendum-lemma-b-on-arbitrary","scope":"tauceti/geometry/part-01/manifold/part-03","status":"fully_proved","summary":"if `i\u00b2 = -1`, `j(q)\u00b2 = -1`, `\u03b1(q)(i a) + j(q)(\u03b1(q) a) = 0`, `mixedTerm i \u03b1 j q = 0`, and `verticalTerm j q = 0` on a neighborhood of `p`, then the fibre vector field `y \u21a6 horizontalTerm i \u03b1 j (p.1, y) a b` has vanishing endomorphism Lie derivative on `j(p.1, \u00b7)` at `p.2`.","title":"Robbin\u2013Salamon Corrigendum Lemma B (on arbitrary real normed spaces under `C\u00b2` regularity)"},{"id":"tauceti/geometry/part-01/manifold/part-03/sards-theorem-for-smooth-maps","kind":"node","lean":"ContMDiff.isNullSet_image_criticalPoints, ContMDiff.isNullSet_image_criticalPoints_of_finrank_le_add_two, TauCeti.isNullSet_image_of_not_isRegularPoint_of_finrank_le_add_two, TauCeti.isNullSet_image_of_not_isRegularPoint_of_contMDiffAt, TauCeti.isNullSet_image_criticalPoints, TauCeti.isNullSet_image_of_not_isRegularPoint, TauCeti.isNullSet_image_of_finrank_lt_finrank, TauCeti.isMeagre_image_of_not_isRegularPoint","parent":2688,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-03/index.html#sards-theorem-for-smooth-maps","scope":"tauceti/geometry/part-01/manifold/part-03","status":"fully_proved","summary":"The critical values of a smooth map from a boundaryless Lindel\u00f6f manifold with finite-dimensional model to a manifold with finite-dimensional model form a null set. The dimensions may differ.","title":"Sard's theorem for smooth maps"},{"id":"tauceti/geometry/part-01/manifold/part-03/schurs-theorem-covariantderivative-eq-of-forall","kind":"node","lean":"CovariantDerivative.eq_of_forall_hasSectionalCurvatureAt, CovariantDerivative.hasConstantSectionalCurvature_of_forall_hasSectionalCurvatureAt, CovariantDerivative.mfderiv_eq_zero_of_curvatureTensor_eq_smul_inner_sub","parent":2688,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-03/index.html#schurs-theorem-covariantderivative-eq-of-forall","scope":"tauceti/geometry/part-01/manifold/part-03","status":"fully_proved","summary":"Let a metric-compatible torsion-free connection on a preconnected manifold of dimension at least three have, at every point `x`, the same sectional curvature `\u03ba x` on all tangent two-planes at `x`. Then `\u03ba` is constant.","title":"Schur's theorem (`CovariantDerivative.eq_of_forall_hasSectionalCurvatureAt`)"},{"id":"tauceti/geometry/part-01/manifold/part-03/sharp-local-lipschitz-bound-for-the-extended-cha","kind":"node","lean":"TauCeti.Manifold.eventually_enorm_apply_extChartAt_sub_le","parent":2688,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-03/index.html#sharp-local-lipschitz-bound-for-the-extended-cha","scope":"tauceti/geometry/part-01/manifold/part-03","status":"fully_proved","summary":"For `r > 1`, near `x` every continuous linear map `\u2113` from `T_x M` to a real normed space, read in the extended chart at `x`, is `(r \u2016\u2113\u2016)`-Lipschitz at `x` for the Riemannian distance.","title":"Sharp local Lipschitz bound for the extended chart"},{"id":"tauceti/geometry/part-01/manifold/part-03/smooth-euclidean-tubular-neighbourhood-theorem","kind":"node","lean":"TauCeti.exists_diffeomorph_wholeNormalBundle_subset","parent":2688,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-03/index.html#smooth-euclidean-tubular-neighbourhood-theorem","scope":"tauceti/geometry/part-01/manifold/part-03","status":"fully_proved","summary":"The whole orthogonal normal bundle of a smooth immersion inducing the source topology is diffeomorphic to an open neighbourhood of its image, inside any prescribed open neighbourhood. The identification fixes the core. Neither compactness nor closedness of the image is required.","title":"Smooth Euclidean tubular neighbourhood theorem"},{"id":"tauceti/geometry/part-01/manifold/part-03/smooth-riemannian-tubular-neighbourhood-theorem","kind":"node","lean":"TauCeti.SmoothEmbedding.exists_diffeomorph_normalBundle_riemannian_subset","parent":2688,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-03/index.html#smooth-riemannian-tubular-neighbourhood-theorem","scope":"tauceti/geometry/part-01/manifold/part-03","status":"fully_proved","summary":"The entire intrinsic normal bundle of a smooth embedding is diffeomorphic to an open neighbourhood inside any prescribed open neighbourhood of its image. The zero section is sent to the original embedding. Neither compactness nor closedness of the embedded image nor completeness of the ambient metric is required.","title":"Smooth Riemannian tubular neighbourhood theorem"},{"id":"tauceti/geometry/part-01/manifold/part-03/smooth-tubular-neighbourhood-theorem","kind":"node","lean":"TauCeti.SmoothEmbedding.exists_diffeomorph_normalBundle_subset","parent":2688,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-03/index.html#smooth-tubular-neighbourhood-theorem","scope":"tauceti/geometry/part-01/manifold/part-03","status":"fully_proved","summary":"The entire intrinsic normal bundle of a smooth embedding is diffeomorphic to an arbitrarily small open neighbourhood of its image, with the zero section sent to the embedding. No Riemannian metric is part of the input or output, and the image need not be closed or compact.","title":"Smooth tubular neighbourhood theorem"},{"id":"tauceti/geometry/part-01/manifold/part-03/the-sobolev-inequality-near-a-point","kind":"node","lean":"TauCeti.Manifold.exists_mem_nhds_eLpNorm_restrict_le_mul_eLpNorm_mgradient_add, TauCeti.Manifold.exists_eLpNorm_le_mul_eLpNorm_mgradient_two_add, TauCeti.Manifold.exists_eLpNorm_le_mul_eLpNorm_mgradient_add","parent":2688,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-03/index.html#the-sobolev-inequality-near-a-point","scope":"tauceti/geometry/part-01/manifold/part-03","status":"fully_proved","summary":"Let `1 \u2264 p` and let `p'` be the Sobolev conjugate exponent, `1 / p' = 1 / p - 1 / n`, on a manifold without boundary of dimension `n > 0` with a continuous Riemannian metric. Every point `c` has a neighbourhood `V` with a constant `C` such that `\u2016u\u2016_{L^{p'}(V)} \u2264 C (\u2016\u2207u\u2016_{L\u1d56} + \u2016u\u2016_{L\u1d56})` for every `C\u00b9` function `u`.","title":"The Sobolev inequality near a point"},{"id":"tauceti/geometry/part-01/manifold/part-03/the-straightening-theorem-lee-thm-9-22","kind":"node","lean":"TauCeti.exists_chart_mfderiv_eq_const","parent":2688,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-03/index.html#the-straightening-theorem-lee-thm-9-22","scope":"tauceti/geometry/part-01/manifold/part-03","status":"fully_proved","summary":"Let `v` be a vector field on a `C^n` manifold with finite-dimensional boundaryless model, `C^m` on a neighbourhood of `x`, with `1 \u2264 m`, `m + 1 \u2264 n` and `v x \u2260 0`. Then some chart `e` of the `C^m` maximal atlas whose domain contains `x` carries `v` to a constant nonzero vector field: the differential of `I \u2218 e` sends `v y` to the same vector `u \u2260 0` at every point `y` of the domain of `e`.","title":"The straightening theorem [Lee, Thm. 9.22]"},{"id":"tauceti/geometry/part-01/manifold/part-03/the-transverse-fibre-product-theorem","kind":"node","lean":"TauCeti.exists_isManifold_transverseFiberProduct_isSmoothEmbedding","parent":2688,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-03/index.html#the-transverse-fibre-product-theorem","scope":"tauceti/geometry/part-01/manifold/part-03","status":"fully_proved","summary":"The actual equal-value subtype of two transverse `C^n` maps, `n \u2265 1`, is a `C^n` manifold of dimension `dim X + dim Y - dim N` with its subspace topology. Its inclusion is a smooth embedding, and its tangent image consists of pairs whose images under the two differentials agree. Empty fibre products are allowed; no separation or countability hypotheses are needed.","title":"The transverse fibre-product theorem"},{"id":"tauceti/geometry/part-01/manifold/part-03/trudingers-inequality-near-a-point","kind":"node","lean":"TauCeti.Manifold.exists_mem_nhds_setLIntegral_exp_mul_abs_rpow_le, TauCeti.Manifold.exists_lintegral_exp_mul_abs_rpow_le","parent":2688,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-03/index.html#trudingers-inequality-near-a-point","scope":"tauceti/geometry/part-01/manifold/part-03","status":"fully_proved","summary":"Every point `c` of a manifold without boundary, of dimension `n \u2265 2`, with a continuous Riemannian metric, has a neighbourhood `V` with constants `\u03b2 > 0` and `C` such that `\u222b_V exp (\u03b2 |u| ^ (n / (n - 1))) dV \u2264 C` for every `C\u00b9` function `u` with `\u2016\u2207u\u2016_{L\u207f} + \u2016u\u2016_{L\u207f} \u2264 1`.","title":"Trudinger's inequality near a point"},{"id":"tauceti/geometry/part-01/manifold/part-03/tubular-neighbourhood-theorem","kind":"node","lean":"TauCeti.exists_isOpenEmbedding_normalTube, TauCeti.exists_isTubularNeighborhood_normalBundle","parent":2688,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-03/index.html#tubular-neighbourhood-theorem","scope":"tauceti/geometry/part-01/manifold/part-03","status":"fully_proved","summary":"Tubular neighbourhood theorem for a compact `C\u00b2` embedded submanifold of a Euclidean space: for some `\u03b5 > 0`, the map `(x, v) \u21a6 f x + v` is an open embedding of the normal vectors of length less than `\u03b5` into `V`.","title":"Tubular neighbourhood theorem"},{"id":"tauceti/geometry/part-01/manifold/part-03/weierstrass-division-theorem-in-holomorphic-stal","kind":"node","lean":"TauCeti.holomorphicStructureSheaf.existsUnique_weierstrassDivision","parent":2688,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-03/index.html#weierstrass-division-theorem-in-holomorphic-stal","scope":"tauceti/geometry/part-01/manifold/part-03","status":"fully_proved","summary":"Given germs `f` and `g` at `(x\u2080, w\u2080)` such that the vertical power series of `f` has nonzero image in `(\ud835\udcaa_{E, x\u2080} \u29f8 \ud835\udd2a_{x\u2080})\u27e6X\u27e7`, there exist unique `q \u2208 \ud835\udcaa_{E \u00d7 \u2102, (x\u2080, w\u2080)}` and `r \u2208 (\ud835\udcaa_{E, x\u2080})[X]` forming a Weierstrass division of `g` by `f`.","title":"Weierstrass division theorem in holomorphic stalks"},{"id":"tauceti/geometry/part-01/manifold/part-03/weierstrass-preparation-theorem-in-holomorphic-s","kind":"node","lean":"TauCeti.holomorphicStructureSheaf.existsUnique_weierstrassPreparation","parent":2688,"result":true,"route":"../roadmap/tauceti/geometry/part-01/manifold/part-03/index.html#weierstrass-preparation-theorem-in-holomorphic-s","scope":"tauceti/geometry/part-01/manifold/part-03","status":"fully_proved","summary":"Every holomorphic germ `f` at `(x\u2080, w\u2080)` whose vertical power series has nonzero image in `(\ud835\udcaa_{E, x\u2080} \u29f8 \ud835\udd2a_{x\u2080})\u27e6X\u27e7` admits a unique factorization `f = evalVertical x\u2080 w\u2080 P * u` into a distinguished polynomial `P` over `\ud835\udcaa_{E, x\u2080}` and a unit germ `u` at `(x\u2080, w\u2080)`.","title":"Weierstrass preparation theorem in holomorphic stalks"},{"data_url":"scopes/tauceti/geometry/part-01/real-algebraic.json","id":"tauceti/geometry/part-01/real-algebraic","kind":"scope","parent":2425,"result":false,"route":"../roadmap/tauceti/geometry/part-01/real-algebraic/index.html","scope":"tauceti/geometry/part-01/real-algebraic","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Geometry.RealAlgebraic` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Real Algebraic"},{"id":"tauceti/geometry/part-01/real-algebraic/finite-fiber-dimension-theorem","kind":"node","lean":"TauCeti.IsSemialgebraicOn.semialgebraicDim_image_of_finite, TauCeti.IsSemialgebraic.semialgebraicDim_image_comp_of_finite","parent":2703,"result":true,"route":"../roadmap/tauceti/geometry/part-01/real-algebraic/index.html#finite-fiber-dimension-theorem","scope":"tauceti/geometry/part-01/real-algebraic","status":"fully_proved","summary":"A semialgebraic function with finite fibers on its domain preserves dimension. In particular this applies to semialgebraic bijections, without requiring continuity of the function or its inverse.","title":"Finite-fiber dimension theorem"},{"id":"tauceti/geometry/part-01/real-algebraic/the-local-nash-graph-criterion","kind":"node","lean":"TauCeti.isNashAt_iff_exists_isSemialgebraicOn, TauCeti.forall_isNashAt_apply_iff_exists_isSemialgebraicOn","parent":2703,"result":true,"route":"../roadmap/tauceti/geometry/part-01/real-algebraic/index.html#the-local-nash-graph-criterion","scope":"tauceti/geometry/part-01/real-algebraic","status":"fully_proved","summary":"A scalar function is a Nash germ at `x` exactly when it is analytic there and its graph is semialgebraic on some open semialgebraic neighbourhood.","title":"The local Nash graph criterion"},{"id":"tauceti/geometry/part-01/real-algebraic/nash-implicit-function-theorem-for-systems","kind":"node","lean":"HasStrictFDerivAt.isNashAt_implicitFunctionOfProdDomain","parent":2703,"result":true,"route":"../roadmap/tauceti/geometry/part-01/real-algebraic/index.html#nash-implicit-function-theorem-for-systems","scope":"tauceti/geometry/part-01/real-algebraic","status":"fully_proved","summary":"If the coordinates of a finite system are Nash at `u` and its partial derivative in the unknown variables is invertible, every coordinate of Mathlib's chosen implicit function is a Nash germ at the parameter point.","title":"Nash implicit function theorem for systems"},{"id":"tauceti/geometry/part-01/real-algebraic/the-nash-implicit-root-theorem","kind":"node","lean":"TauCeti.IsNashAt.exists_isNashAt_eventually_eq_zero_iff","parent":2703,"result":true,"route":"../roadmap/tauceti/geometry/part-01/real-algebraic/index.html#the-nash-implicit-root-theorem","scope":"tauceti/geometry/part-01/real-algebraic","status":"fully_proved","summary":"A Nash scalar equation with a simple zero admits a Nash root germ in the parameter variables. Near the given point its zero set is exactly the graph of this root. No polynomial equation or global semialgebraic domain is required.","title":"The Nash implicit root theorem"},{"id":"tauceti/geometry/part-01/real-algebraic/the-nash-inverse-function-theorem","kind":"node","lean":"TauCeti.exists_openPartialHomeomorph_of_forall_isNashAt","parent":2703,"result":true,"route":"../roadmap/tauceti/geometry/part-01/real-algebraic/index.html#the-nash-inverse-function-theorem","scope":"tauceti/geometry/part-01/real-algebraic","status":"fully_proved","summary":"A map with Nash coordinate germs and invertible Fr\u00e9chet derivative at `a` restricts, inside any prescribed neighbourhood, to a local homeomorphism with open semialgebraic source and target and semialgebraic graphs in both directions. Every coordinate of the map and its inverse is Nash throughout those domains. Empty coordinate index types are allowed.","title":"The Nash inverse function theorem"},{"id":"tauceti/geometry/part-01/real-algebraic/thoms-lemma-for-delineations","kind":"node","lean":"TauCeti.Delineation.root_notMem_uIcc_of_sign_eval_eq","parent":2703,"result":true,"route":"../roadmap/tauceti/geometry/part-01/real-algebraic/index.html#thoms-lemma-for-delineations","scope":"tauceti/geometry/part-01/real-algebraic","status":"fully_proved","summary":"Suppose that at a point `x` of the base the derivative of every member of the family is zero or a member. If every member has the same sign at two distinct points `t \u2260 t'` of the fiber over `x`, then no root function takes a value in `[t, t']` at `x`.","title":"Thom's lemma for delineations"},{"data_url":"scopes/tauceti/geometry/part-02.json","id":"tauceti/geometry/part-02","kind":"scope","navigation":"transparent","parent":2424,"result":false,"route":"../roadmap/tauceti/geometry/part-02/index.html","scope":"tauceti/geometry/part-02","status":"fully_proved","title":"Symplectic to Other results"},{"data_url":"scopes/tauceti/geometry/part-02/symplectic.json","id":"tauceti/geometry/part-02/symplectic","kind":"scope","parent":2710,"result":false,"route":"../roadmap/tauceti/geometry/part-02/symplectic/index.html","scope":"tauceti/geometry/part-02/symplectic","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Geometry.Symplectic` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Symplectic"},{"id":"tauceti/geometry/part-02/symplectic/cauchy-schwarz-along-a-complex-line","kind":"node","lean":"TauCeti.SymplecticForm.Compatible.sq_associatedBilinForm_add_sq_le","parent":2711,"result":true,"route":"../roadmap/tauceti/geometry/part-02/symplectic/index.html#cauchy-schwarz-along-a-complex-line","scope":"tauceti/geometry/part-02/symplectic","status":"fully_proved","summary":"For a compatible pair, the squares of the components `g(v, w)` and `\u03c9(v, w) = g(J v, w)` of `w` along the orthogonal pair `v`, `J v` add up to at most `g(v, v) g(w, w)`.","title":"Cauchy--Schwarz along a complex line"},{"id":"tauceti/geometry/part-02/symplectic/eulers-identity-for-the-kinetic-energy","kind":"node","lean":"TauCeti.CotangentBundle.mvfderiv_kineticEnergy_vectorBundleEulerField","parent":2711,"result":true,"route":"../roadmap/tauceti/geometry/part-02/symplectic/index.html#eulers-identity-for-the-kinetic-energy","scope":"tauceti/geometry/part-02/symplectic","status":"fully_proved","summary":"The kinetic energy is homogeneous of degree two in the fibres, so the Euler vector field `Z = p \u2202_p` satisfies `dK(Z) = 2K = \u2016p\u2016\u00b2`.","title":"Euler's identity for the kinetic energy"},{"id":"tauceti/geometry/part-02/symplectic/hamiltons-principle-of-stationary-action","kind":"node","lean":"TauCeti.isActionStationary_iff_eulerLagrange","parent":2711,"result":true,"route":"../roadmap/tauceti/geometry/part-02/symplectic/index.html#hamiltons-principle-of-stationary-action","scope":"tauceti/geometry/part-02/symplectic","status":"fully_proved","summary":"For `2 \u2264 n \u2264 \u221e`, let `q` be `C^n` at every point of `[a, b]` and let `L` be jointly `C\u00b2` along `q`. Then `q` is a stationary point of the action of `L` among `C^n` variations with fixed endpoints between `a` and `b` exactly when it solves the Euler\u2013Lagrange equations `d/dt \u2202_v L(t, q, q') = \u2202_q L(t, q, q')` at every time strictly between `a` and `b`.","title":"Hamilton's principle of stationary action"},{"id":"tauceti/geometry/part-02/symplectic/integrated-wirtinger-inequality","kind":"node","lean":"TauCeti.SymplecticForm.lintegral_symplecticForm_le_stdComplexLineEnergy, TauCeti.SmoothTwoForm.Compatible.lintegral_twoForm_le_stdComplexLineEnergy, TauCeti.SymplecticForm.lintegral_symplecticForm_le_stdComplexLineArea","parent":2711,"result":true,"route":"../roadmap/tauceti/geometry/part-02/symplectic/index.html#integrated-wirtinger-inequality","scope":"tauceti/geometry/part-02/symplectic","status":"fully_proved","summary":"For a compatible pair, the integral of the positive symplectic area density is at most the energy.","title":"Integrated Wirtinger inequality"},{"id":"tauceti/geometry/part-02/symplectic/the-jacobi-identity-for-the-poisson-bracket","kind":"node","lean":"TauCeti.SmoothTwoForm.IsNondegenerate.poissonBracket_jacobi","parent":2711,"result":true,"route":"../roadmap/tauceti/geometry/part-02/symplectic/index.html#the-jacobi-identity-for-the-poisson-bracket","scope":"tauceti/geometry/part-02/symplectic","status":"fully_proved","summary":"For a nondegenerate closed form on a finite-dimensional manifold and `C\u00b2` functions `f`, `g`, `k`, `{f, {g, k}} + {g, {k, f}} + {k, {f, g}} = 0`.","title":"The Jacobi identity for the Poisson bracket"},{"id":"tauceti/geometry/part-02/symplectic/keplers-third-law","kind":"node","lean":"TauCeti.kepler_radialPeriod","parent":2711,"result":true,"route":"../roadmap/tauceti/geometry/part-02/symplectic/index.html#keplers-third-law","scope":"tauceti/geometry/part-02/symplectic","status":"fully_proved","summary":"Between two distinct turning points `0 < r\u2081 < r\u2082` of a Kepler orbit with `\u2113 \u2260 0`, the radial period is `2\u03c0 \u221a(m / \u03ba) a^(3/2)` for the semi-major axis `a = (r\u2081 + r\u2082) / 2`.","title":"Kepler's third law"},{"id":"tauceti/geometry/part-02/symplectic/linear-darboux-theorem","kind":"node","lean":"TauCeti.SymplecticForm.exists_isSymplectomorphism","parent":2711,"result":true,"route":"../roadmap/tauceti/geometry/part-02/symplectic/index.html#linear-darboux-theorem","scope":"tauceti/geometry/part-02/symplectic","status":"fully_proved","summary":"Two symplectic vector spaces of the same finite dimension are linearly symplectomorphic.","title":"Linear Darboux theorem"},{"id":"tauceti/geometry/part-02/symplectic/noethers-theorem","kind":"node","lean":"TauCeti.SmoothTwoForm.IsMomentMap.poissonBracket_eq_zero_iff, TauCeti.hasDerivAt_noetherCharge, TauCeti.noetherCharge_eq_of_eulerLagrange","parent":2711,"result":true,"route":"../roadmap/tauceti/geometry/part-02/symplectic/index.html#noethers-theorem","scope":"tauceti/geometry/part-02/symplectic","status":"fully_proved","summary":"The derivative of a function `f` along the generated vector field `Z_M` vanishes exactly when `f` Poisson-commutes with the component `\u03bc_Z` of a moment map.","title":"Noether's theorem"},{"id":"tauceti/geometry/part-02/symplectic/the-relative-darboux-theorem-in-adapted-coordina","kind":"node","lean":"TauCeti.exists_diffeomorph_fix_zeroSection_eventually_compContinuousLinearMap_fderiv_eq","parent":2711,"result":true,"route":"../roadmap/tauceti/geometry/part-02/symplectic/index.html#the-relative-darboux-theorem-in-adapted-coordina","scope":"tauceti/geometry/part-02/symplectic","status":"fully_proved","summary":"Let `\u03c9\u2080`, `\u03c9\u2081` be closed `C^r` two-forms (`r \u2265 1`) on an open subset `W` of `E \u00d7 N` containing `(q\u2080, 0)`, both nondegenerate at `(q\u2080, 0)`, with the same pullback to the zero section `E \u00d7 {0}`. Then some `C^r` diffeomorphism `\u03c8` of `E \u00d7 N` fixes every point `(q, 0)` with `q` near `q\u2080` and pulls `\u03c9\u2081` back to `\u03c9\u2080` near `(q\u2080, 0)`. The pullback to the zero section may be degenerate, of any rank.","title":"The relative Darboux theorem in adapted coordinates"},{"id":"tauceti/geometry/part-02/symplectic/stokes-formula-for-symplectic-area-on-an-annulus","kind":"node","lean":"TauCeti.SymplecticForm.integral_fderiv_apply_annulus, TauCeti.SymplecticForm.integral_fderiv_apply_prod_Icc, TauCeti.SymplecticForm.integral_fderiv_apply_truncation, TauCeti.SymplecticForm.integral_fderiv_apply_punctured_disc","parent":2711,"result":true,"route":"../roadmap/tauceti/geometry/part-02/symplectic/index.html#stokes-formula-for-symplectic-area-on-an-annulus","scope":"tauceti/geometry/part-02/symplectic","status":"fully_proved","summary":"The pullback area of a `C\u00b2` map is the outer boundary action minus the inner boundary action for the primitive `\u03bb\u2093(v) = \u00bd \u03c9(x, v)`. This identity does not require any regularity on the inner disc or a holomorphicity hypothesis.","title":"Stokes' formula for symplectic area on an annulus"},{"id":"tauceti/geometry/part-02/symplectic/vietas-formula-for-the-kepler-sum-of-turning-poi","kind":"node","lean":"TauCeti.kepler_kappa_eq, TauCeti.isotropicOscillator_sq_eq_sq_mul, TauCeti.isotropicOscillator_energy_eq, TauCeti.kepler_sq_eq_neg_two_mul_mul_mul","parent":2711,"result":true,"route":"../roadmap/tauceti/geometry/part-02/symplectic/index.html#vietas-formula-for-the-kepler-sum-of-turning-poi","scope":"tauceti/geometry/part-02/symplectic","status":"fully_proved","summary":"If `r\u2081 \u2260 r\u2082` are nonzero turning points of a Kepler orbit with `m \u2260 0`, then `\u03ba = -E (r\u2081 + r\u2082)`; equivalently, for `E \u2260 0` the semi-major axis `(r\u2081 + r\u2082) / 2` is `\u03ba / (-2 E)`.","title":"Vieta's formula for the Kepler sum of turning points"},{"id":"tauceti/geometry/part-02/symplectic/wirtinger-identity-on-manifolds","kind":"node","lean":"TauCeti.SmoothTwoForm.Invariant.stdComplexLineEnergyDensity_sub_two_mul_twoForm","parent":2711,"result":true,"route":"../roadmap/tauceti/geometry/part-02/symplectic/index.html#wirtinger-identity-on-manifolds","scope":"tauceti/geometry/part-02/symplectic","status":"fully_proved","summary":"For an invariant pair `(form, J)`, the standard energy density minus twice the symplectic area density is the Cauchy--Riemann defect square: \\omega_x(F \\partial_s + J_x F \\partial_t, J_x(F \\partial_s + J_x F \\partial_t)).$$","title":"Wirtinger identity on manifolds"},{"id":"tauceti/geometry/part-02/symplectic/wirtinger-inequality","kind":"node","lean":"TauCeti.SymplecticForm.two_mul_symplecticForm_le_stdComplexLineEnergyDensity, TauCeti.SymplecticForm.Compatible.apply_le_wedgeNorm, TauCeti.SmoothTwoForm.Compatible.two_mul_twoForm_le_stdComplexLineEnergyDensity","parent":2711,"result":true,"route":"../roadmap/tauceti/geometry/part-02/symplectic/index.html#wirtinger-inequality","scope":"tauceti/geometry/part-02/symplectic","status":"fully_proved","summary":"For a compatible pair `(\u03c9, J)` and any real-linear map out of the standard complex line, twice the symplectic area density of the ordered coordinate pair is at most the standard energy density: the local form of `E(u) \u2265 \u222b u^*\u03c9`.","title":"Wirtinger inequality"},{"id":"tauceti/geometry/part-02/symplectic/witts-extension-theorem","kind":"node","lean":"TauCeti.SymplecticForm.exists_isSymplectomorphism_extend_linearEquiv, TauCeti.SymplecticForm.exists_isSymplectomorphism_extend","parent":2711,"result":true,"route":"../roadmap/tauceti/geometry/part-02/symplectic/index.html#witts-extension-theorem","scope":"tauceti/geometry/part-02/symplectic","status":"fully_proved","summary":"Witt's extension theorem, for a linear isomorphism `f : S \u2243\u2097[\u211d] T` between subspaces of two symplectic vector spaces of the same finite dimension: if `f` preserves the restricted forms, it is the restriction of a linear symplectomorphism.","title":"Witt's extension theorem"},{"data_url":"scopes/tauceti/geometry/part-02/other-results.json","id":"tauceti/geometry/part-02/other-results","kind":"scope","parent":2710,"result":false,"route":"../roadmap/tauceti/geometry/part-02/other-results/index.html","scope":"tauceti/geometry/part-02/other-results","status":"fully_proved","summary":"Results named in the rest of Tau Ceti's `TauCeti.Geometry` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Other results"},{"id":"tauceti/geometry/part-02/other-results/cartans-formula-for-a-contact-hamiltonian-vector","kind":"node","lean":"TauCeti.SmoothForm.IsContactHamiltonianVectorField.mlieDerivForm_eq, TauCeti.SmoothForm.IsContactHamiltonianVectorField.mlieDerivForm_apply_eq","parent":2726,"result":true,"route":"../roadmap/tauceti/geometry/part-02/other-results/index.html#cartans-formula-for-a-contact-hamiltonian-vector","scope":"tauceti/geometry/part-02/other-results","status":"fully_proved","summary":"a differentiable contact Hamiltonian vector field `X` of `f` satisfies `L_X \u03b1 = (R_\u03b1 f) \u03b1`.","title":"Cartan's formula for a contact Hamiltonian vector field"},{"id":"tauceti/geometry/part-02/other-results/cerfs-theorem-as-a-count-of-path-components","kind":"node","lean":"TauCeti.cerfTheorem_iff_natCard_zerothHomotopy_eq_two","parent":2726,"result":true,"route":"../roadmap/tauceti/geometry/part-02/other-results/index.html#cerfs-theorem-as-a-count-of-path-components","scope":"tauceti/geometry/part-02/other-results","status":"fully_proved","summary":"Cerf's theorem holds exactly when the diffeomorphism group of `S\u00b3` has two path components.","title":"Cerf's theorem as a count of path components"},{"id":"tauceti/geometry/part-02/other-results/the-density-theorem-for-rectifiable-sets","kind":"node","lean":"TauCeti.CountablyRectifiable.ae_tendsto_euclideanHausdorffMeasure_inter_closedBall_div","parent":2726,"result":true,"route":"../roadmap/tauceti/geometry/part-02/other-results/index.html#the-density-theorem-for-rectifiable-sets","scope":"tauceti/geometry/part-02/other-results","status":"fully_proved","summary":"A countably `k`-rectifiable subset `E` of a finite-dimensional real inner product space, measurable for `\u03bcHE[k]` and with locally finite `\u03bcHE[k]` measure, has `k`-dimensional density one at `\u03bcHE[k]`-almost every point of `E`: `\u03bcHE[k] (E \u2229 closedBall x r) / (\u03c9_k r ^ k) \u2192 1` as `r \u2192 0\u207a`, where `\u03c9_k r ^ k` is the volume of a ball of radius `r` in `EuclideanSpace \u211d (Fin k)`.","title":"The density theorem for rectifiable sets"},{"id":"tauceti/geometry/part-02/other-results/gordans-lemma","kind":"node","lean":"TauCeti.Toric.IsLatticeRational.fg_dualSemigroup","parent":2726,"result":true,"route":"../roadmap/tauceti/geometry/part-02/other-results/index.html#gordans-lemma","scope":"tauceti/geometry/part-02/other-results","status":"fully_proved","summary":"The dual semigroup of a lattice-rational cone in an integral lattice is finitely generated. Salience is not required.","title":"Gordan's lemma"},{"id":"tauceti/geometry/part-02/other-results/the-jacobi-identity-for-the-jacobi-bracket","kind":"node","lean":"TauCeti.SmoothForm.IsContactForm.jacobiBracket_jacobi","parent":2726,"result":true,"route":"../roadmap/tauceti/geometry/part-02/other-results/index.html#the-jacobi-identity-for-the-jacobi-bracket","scope":"tauceti/geometry/part-02/other-results","status":"fully_proved","summary":"For a contact form on a finite-dimensional manifold and `C\u00b2` functions `f`, `g`, `k`, `{f, {g, k}_\u03b1}_\u03b1 + {g, {k, f}_\u03b1}_\u03b1 + {k, {f, g}_\u03b1}_\u03b1 = 0`.","title":"The Jacobi identity for the Jacobi bracket"},{"id":"tauceti/geometry/part-02/other-results/lebesgues-density-theorem-in-a-k-plane","kind":"node","lean":"Isometry.ae_tendsto_euclideanHausdorffMeasure_inter_closedBall_div","parent":2726,"result":true,"route":"../roadmap/tauceti/geometry/part-02/other-results/index.html#lebesgues-density-theorem-in-a-k-plane","scope":"tauceti/geometry/part-02/other-results","status":"fully_proved","summary":"Let `\u03b9` be an isometry from `k`-dimensional Euclidean space into `X`. Every subset `S` of its range has `k`-dimensional density one at `\u03bcHE[k]`-almost every point of `S`. No measurability of `S` is needed.","title":"Lebesgue's density theorem in a `k`-plane"},{"id":"tauceti/geometry/part-02/other-results/the-smale-conjecture-implies-cerfs-theorem","kind":"node","lean":"TauCeti.SmaleConjecture.cerfTheorem","parent":2726,"result":true,"route":"../roadmap/tauceti/geometry/part-02/other-results/index.html#the-smale-conjecture-implies-cerfs-theorem","scope":"tauceti/geometry/part-02/other-results","status":"fully_proved","summary":"A homotopy equivalence is surjective on path components: if `g` is a homotopy inverse of the inclusion `O(4) \u2192 Diff(S\u00b3)`, then every `f` is joined to the image of `g f` by the homotopy from the composite to the identity.","title":"The Smale conjecture implies Cerf's theorem"},{"data_url":"scopes/tauceti/group-theory.json","id":"tauceti/group-theory","kind":"scope","parent":1095,"result":false,"route":"../roadmap/tauceti/group-theory/index.html","scope":"tauceti/group-theory","status":"fully_proved","summary":"Results in Tau Ceti's `TauCeti.GroupTheory` modules, and the roadmaps whose articles are mostly there: by roadmap, then by area.","title":"Group Theory"},{"data_url":"scopes/tauceti/group-theory/amenable-groups.json","id":"tauceti/group-theory/amenable-groups","kind":"scope","parent":2734,"result":false,"route":"../roadmap/tauceti/group-theory/amenable-groups/index.html","scope":"tauceti/group-theory/amenable-groups","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Amenable groups"},{"id":"tauceti/group-theory/amenable-groups/tauceti-amenable-of-hasfoelnersets","kind":"node","lean":"TauCeti.Amenable.of_hasFoelnerSets","parent":2735,"result":true,"route":"../roadmap/tauceti/group-theory/amenable-groups/index.html#tauceti-amenable-of-hasfoelnersets","scope":"tauceti/group-theory/amenable-groups","status":"fully_proved","summary":"A group with F\u00f8lner sets is amenable, for every group topology.","title":"TauCeti.Amenable.of_hasFoelnerSets"},{"id":"tauceti/group-theory/amenable-groups/tauceti-amenable-of-tendsto","kind":"node","lean":"TauCeti.Amenable.of_tendsto","parent":2735,"result":true,"route":"../roadmap/tauceti/group-theory/amenable-groups/index.html#tauceti-amenable-of-tendsto","scope":"tauceti/group-theory/amenable-groups","status":"fully_proved","summary":"Cluster-point criterion: a topological group is amenable when it carries means `ms i` which are approximately left invariant along a nontrivial filter, that is, `ms i (L_g f) - ms i f \u2192 0` for every `g` and `f`.","title":"TauCeti.Amenable.of_tendsto"},{"id":"tauceti/group-theory/amenable-groups/tauceti-amenable-iff-forall-finset-exists","kind":"node","lean":"TauCeti.amenable_iff_forall_finset_exists","parent":2735,"result":true,"route":"../roadmap/tauceti/group-theory/amenable-groups/index.html#tauceti-amenable-iff-forall-finset-exists","scope":"tauceti/group-theory/amenable-groups","status":"fully_proved","summary":"Finite-intersection criterion: a topological group is amenable exactly when every finite family of invariance equations `m (L_g f) = m f` is satisfied by some mean.","title":"TauCeti.amenable_iff_forall_finset_exists"},{"id":"tauceti/group-theory/amenable-groups/tauceti-amenable-of-directed-of-isup-eq-top","kind":"node","lean":"TauCeti.Amenable.of_directed_of_iSup_eq_top","parent":2735,"result":true,"route":"../roadmap/tauceti/group-theory/amenable-groups/index.html#tauceti-amenable-of-directed-of-isup-eq-top","scope":"tauceti/group-theory/amenable-groups","status":"fully_proved","summary":"A topological group which is the union of a nonempty directed family of amenable subgroups is amenable.","title":"TauCeti.Amenable.of_directed_of_iSup_eq_top"},{"id":"tauceti/group-theory/amenable-groups/tauceti-amenable-of-iscolimit","kind":"node","lean":"TauCeti.Amenable.of_isColimit","parent":2735,"result":true,"route":"../roadmap/tauceti/group-theory/amenable-groups/index.html#tauceti-amenable-of-iscolimit","scope":"tauceti/group-theory/amenable-groups","status":"fully_proved","summary":"The apex of a colimit cocone over a filtered diagram of amenable topological groups in `GrpCat` is amenable, provided the legs of the cocone are continuous.","title":"TauCeti.Amenable.of_isColimit"},{"id":"tauceti/group-theory/amenable-groups/reiter-implies-flner","kind":"node","lean":"TauCeti.exists_finset_card_smul_symmDiff_lt_of_sum_tsum_lt","parent":2735,"result":true,"route":"../roadmap/tauceti/group-theory/amenable-groups/index.html#reiter-implies-flner","scope":"tauceti/group-theory/amenable-groups","status":"fully_proved","summary":"If a nonnegative finitely supported function `p` on `X` satisfies `\u2211_{g \u2208 K} \u2211_x |p (g\u207b\u00b9 \u2022 x) - p x| < \u03b5 \u00b7 \u2211_x p x`, then some finite nonempty set `F \u2286 X` satisfies `|g \u2022 F \u2206 F| < \u03b5 |F|` for every `g \u2208 K`.","title":"Reiter implies F\u00f8lner"},{"id":"tauceti/group-theory/amenable-groups/tauceti-amenable-of-finite","kind":"node","lean":"TauCeti.Amenable.of_finite","parent":2735,"result":true,"route":"../roadmap/tauceti/group-theory/amenable-groups/index.html#tauceti-amenable-of-finite","scope":"tauceti/group-theory/amenable-groups","status":"fully_proved","summary":"A finite topological group is amenable, for any group topology.","title":"TauCeti.Amenable.of_finite"},{"id":"tauceti/group-theory/amenable-groups/tauceti-mean-isleftinvariant-iff-eq-uniform","kind":"node","lean":"TauCeti.Mean.isLeftInvariant_iff_eq_uniform","parent":2735,"result":true,"route":"../roadmap/tauceti/group-theory/amenable-groups/index.html#tauceti-mean-isleftinvariant-iff-eq-uniform","scope":"tauceti/group-theory/amenable-groups","status":"fully_proved","summary":"On a finite group, the uniform mean is the unique left-invariant mean.","title":"TauCeti.Mean.isLeftInvariant_iff_eq_uniform"},{"id":"tauceti/group-theory/amenable-groups/tauceti-amenable","kind":"node","lean":"TauCeti.Amenable","parent":2735,"result":true,"route":"../roadmap/tauceti/group-theory/amenable-groups/index.html#tauceti-amenable","scope":"tauceti/group-theory/amenable-groups","status":"fully_proved","summary":"A topological group `G` is amenable when `UCB G` carries a left-invariant mean.","title":"TauCeti.Amenable"},{"id":"tauceti/group-theory/amenable-groups/tauceti-ucb-lefttranslation","kind":"node","lean":"TauCeti.UCB.leftTranslation","parent":2735,"result":true,"route":"../roadmap/tauceti/group-theory/amenable-groups/index.html#tauceti-ucb-lefttranslation","scope":"tauceti/group-theory/amenable-groups","status":"fully_proved","summary":"Left translation `g \u21a6 L_g`, where `(L_g f)(x) = f (g\u207b\u00b9 * x)`, as a group homomorphism from `G` to the linear isometry equivalences of `UCB G`; the case `\ud835\udd5c = \u211d` of `ucbSubalgebra.leftTranslation`.","title":"TauCeti.UCB.leftTranslation"},{"id":"tauceti/group-theory/amenable-groups/tauceti-mean-instcompactspace","kind":"node","lean":"TauCeti.Mean.instCompactSpace","parent":2735,"result":true,"route":"../roadmap/tauceti/group-theory/amenable-groups/index.html#tauceti-mean-instcompactspace","scope":"tauceti/group-theory/amenable-groups","status":"fully_proved","summary":"The space of means is weak-* compact.","title":"TauCeti.Mean.instCompactSpace"},{"id":"tauceti/group-theory/amenable-groups/equidecompositions-as-partitions","kind":"node","lean":"Equidecomp.exists_iff_exists_finpartition","parent":2735,"result":true,"route":"../roadmap/tauceti/group-theory/amenable-groups/index.html#equidecompositions-as-partitions","scope":"tauceti/group-theory/amenable-groups","status":"fully_proved","summary":"Two sets `A` and `B` are related by an equidecomposition exactly when there are a finite partition of `A` and a group element `\u03b3 s` for each part `s`, such that the translates `\u03b3 s \u2022 s` are pairwise disjoint with union `B`.","title":"Equidecompositions as partitions"},{"id":"tauceti/group-theory/amenable-groups/equidecomp-equipartition","kind":"node","lean":"Equidecomp.Equipartition","parent":2735,"result":true,"route":"../roadmap/tauceti/group-theory/amenable-groups/index.html#equidecomp-equipartition","scope":"tauceti/group-theory/amenable-groups","status":"fully_proved","summary":"An *equipartition* of `X` for an action of `G`: a finite set of pairs `(A\u1d62, g\u1d62)` of a nonempty source piece `A\u1d62 : Set X` and a group element `g\u1d62 : G`, such that the source pieces `A\u1d62` are pairwise disjoint and the target pieces `g\u1d62 \u2022 A\u1d62` are pairwise disjoint. When `G` is a group acting on `X`, moving each `A\u1d62` by `g\u1d62` is then an equidecomposition of the union of the `A\u1d62` with the union of the `g\u1d62 \u2022 A\u1d62` (see\u2026","title":"Equidecomp.Equipartition"},{"id":"tauceti/group-theory/amenable-groups/tauceti-hasfoelnersets-iff","kind":"node","lean":"TauCeti.hasFoelnerSets_iff","parent":2735,"result":true,"route":"../roadmap/tauceti/group-theory/amenable-groups/index.html#tauceti-hasfoelnersets-iff","scope":"tauceti/group-theory/amenable-groups","status":"fully_proved","summary":"A group has F\u00f8lner sets exactly when for every finite `K \u2286 G` and every `\u03b5 > 0` there is a finite nonempty `F \u2286 G` with `|gF \u25b3 F| < \u03b5 \u00b7 |F|` for all `g \u2208 K`.","title":"TauCeti.hasFoelnerSets_iff"},{"id":"tauceti/group-theory/amenable-groups/schroder-bernstein-theorem-for-equidecomposition","kind":"node","lean":"TauCeti.EquidecomposableLE.antisymm","parent":2735,"result":true,"route":"../roadmap/tauceti/group-theory/amenable-groups/index.html#schroder-bernstein-theorem-for-equidecomposition","scope":"tauceti/group-theory/amenable-groups","status":"fully_proved","summary":"If each of `A` and `B` is equidecomposable with a subset of the other, then `A` and `B` are equidecomposable.","title":"Schr\u00f6der\u2013Bernstein theorem for equidecompositions"},{"data_url":"scopes/tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou.json","id":"tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou","kind":"scope","parent":2734,"result":false,"route":"../roadmap/tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou/index.html","scope":"tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Free groups, equations over groups, and limit groups"},{"id":"tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou/conjugacy-criterion-in-free-groups","kind":"node","lean":"FreeGroup.isConj_iff_isRotated_reduceCyclically","parent":2751,"result":true,"route":"../roadmap/tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou/index.html#conjugacy-criterion-in-free-groups","scope":"tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou","status":"fully_proved","summary":"Two elements of a free group are conjugate exactly when the cyclic reductions of their reduced words are cyclic rotations of each other.","title":"Conjugacy criterion in free groups"},{"id":"tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou/free-groups-are-csa","kind":"node","lean":"IsFreeGroup.isCSA","parent":2751,"result":true,"route":"../roadmap/tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou/index.html#free-groups-are-csa","scope":"tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou","status":"fully_proved","summary":"every maximal abelian subgroup of a free group is malnormal.","title":"Free groups are CSA"},{"id":"tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou/isfreegroup-maximal-ismulcommutative-iff","kind":"node","lean":"IsFreeGroup.maximal_isMulCommutative_iff","parent":2751,"result":true,"route":"../roadmap/tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou/index.html#isfreegroup-maximal-ismulcommutative-iff","scope":"tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou","status":"fully_proved","summary":"The maximal abelian subgroups of a nontrivial free group are exactly the cyclic subgroups generated by elements that are not proper powers.","title":"IsFreeGroup.maximal_isMulCommutative_iff"},{"id":"tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou/nonabelian-free-groups-are-domains-baumslag-myas","kind":"node","lean":"IsFreeGroup.isGDomain_id","parent":2751,"result":true,"route":"../roadmap/tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou/index.html#nonabelian-free-groups-are-domains-baumslag-myas","scope":"tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou","status":"fully_proved","summary":"Formalized in Tau Ceti as `IsFreeGroup.isGDomain_id`.","title":"Nonabelian free groups are domains (Baumslag\u2013Myasnikov\u2013Remeslennikov, Corollary 1)"},{"id":"tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou/csa-groups-are-domains-baumslag-myasnikov-remesl","kind":"node","lean":"TauCeti.GroupAlgGeom.IsGDomain.of_isCSA","parent":2751,"result":true,"route":"../roadmap/tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou/index.html#csa-groups-are-domains-baumslag-myasnikov-remesl","scope":"tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou","status":"fully_proved","summary":"A CSA group in which the images of two coefficients do not commute is a domain over these coefficients.","title":"CSA groups are domains (Baumslag\u2013Myasnikov\u2013Remeslennikov, Proposition 1)"},{"id":"tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou/sanovs-theorem","kind":"node","lean":"TauCeti.injective_freeGroup_lift_transvection_two, TauCeti.injective_freeGroup_lift_transvection_of_two_le_abs","parent":2751,"result":true,"route":"../roadmap/tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou/index.html#sanovs-theorem","scope":"tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou","status":"fully_proved","summary":"The matrices `[[1, 2], [0, 1]]` and `[[1, 0], [2, 1]]` freely generate a free subgroup of rank two of `SL(2, \u2124)`. This is the case `R = \u2124`, `c = d = 2` of `injective_freeGroup_lift_transvection_of_two_le_abs`.","title":"Sanov's theorem"},{"id":"tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou/free-groups-of-countable-rank-are-linear-over-z","kind":"node","lean":"TauCeti.exists_injective_freeGroup_SL","parent":2751,"result":true,"route":"../roadmap/tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou/index.html#free-groups-of-countable-rank-are-linear-over-z","scope":"tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou","status":"fully_proved","summary":"Every free group on a countable basis embeds in `SL(2, \u2124)`.","title":"Free groups of countable rank are linear over `\u2124`"},{"id":"tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou/linear-groups-are-equationally-noetherian","kind":"node","lean":"TauCeti.GroupAlgGeom.isEquationallyNoetherian_generalLinearGroup","parent":2751,"result":true,"route":"../roadmap/tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou/index.html#linear-groups-are-equationally-noetherian","scope":"tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou","status":"fully_proved","summary":"Over a commutative Noetherian ring `R`, the general linear group `GL n R` is equationally Noetherian over any coefficients: every system of equations in finitely many variables has a finite subsystem with the same solutions in `GL n R`.","title":"Linear groups are equationally Noetherian"},{"id":"tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou/tauceti-groupalggeom-coordgroup","kind":"node","lean":"TauCeti.GroupAlgGeom.CoordGroup","parent":2751,"result":true,"route":"../roadmap/tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou/index.html#tauceti-groupalggeom-coordgroup","scope":"tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou","status":"fully_proved","summary":"The coordinate group `G_{R(S)} = G[X] \u29f8 Rad_H(S)` of a system `S` over `H`.","title":"TauCeti.GroupAlgGeom.CoordGroup"},{"id":"tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou/tauceti-groupalggeom-zariskitopology","kind":"node","lean":"TauCeti.GroupAlgGeom.zariskiTopology","parent":2751,"result":true,"route":"../roadmap/tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou/index.html#tauceti-groupalggeom-zariskitopology","scope":"tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou","status":"fully_proved","summary":"The Zariski topology generated by the complements of algebraic sets over `G`.","title":"TauCeti.GroupAlgGeom.zariskiTopology"},{"id":"tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou/group-isprimitive","kind":"node","lean":"Group.IsPrimitive","parent":2751,"result":true,"route":"../roadmap/tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou/index.html#group-isprimitive","scope":"tauceti/group-theory/free-groups-equations-over-groups-and-limit-grou","status":"fully_proved","summary":"An element of a group is primitive when it belongs to some free basis of the group.","title":"Group.IsPrimitive"},{"data_url":"scopes/tauceti/group-theory/stating-the-classification-of-finite-simple-grou.json","id":"tauceti/group-theory/stating-the-classification-of-finite-simple-grou","kind":"scope","parent":2734,"result":false,"route":"../roadmap/tauceti/group-theory/stating-the-classification-of-finite-simple-grou/index.html","scope":"tauceti/group-theory/stating-the-classification-of-finite-simple-grou","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Stating the classification of finite simple groups"},{"id":"tauceti/group-theory/stating-the-classification-of-finite-simple-grou/classification-of-finite-simple-groups-statement","kind":"node","lean":"TauCeti.ClassificationStatement","parent":2763,"result":true,"route":"../roadmap/tauceti/group-theory/stating-the-classification-of-finite-simple-grou/index.html#classification-of-finite-simple-groups-statement","scope":"tauceti/group-theory/stating-the-classification-of-finite-simple-grou","status":"fully_proved","summary":"Every finite simple group is isomorphic to one of the explicitly constructed groups on the classification list.","title":"Classification of finite simple groups, statement only"},{"id":"tauceti/group-theory/stating-the-classification-of-finite-simple-grou/tauceti-validlietypeindex-group","kind":"node","lean":"TauCeti.ValidLieTypeIndex.Group","parent":2763,"result":true,"route":"../roadmap/tauceti/group-theory/stating-the-classification-of-finite-simple-grou/index.html#tauceti-validlietypeindex-group","scope":"tauceti/group-theory/stating-the-classification-of-finite-simple-grou","status":"fully_proved","summary":"The Lie-type candidate is the derived subgroup of the fixed points modulo its own centre. No finiteness or simplicity instance is assumed or supplied.","title":"TauCeti.ValidLieTypeIndex.Group"},{"id":"tauceti/group-theory/stating-the-classification-of-finite-simple-grou/the-steinberg-endomorphism-is-the-graph-automorp","kind":"node","lean":"TauCeti.GraphTwistedIndex.steinberg_eq_graphAut_comp_frobenius","parent":2763,"result":true,"route":"../roadmap/tauceti/group-theory/stating-the-classification-of-finite-simple-grou/index.html#the-steinberg-endomorphism-is-the-graph-automorp","scope":"tauceti/group-theory/stating-the-classification-of-finite-simple-grou","status":"fully_proved","summary":"The Steinberg endomorphism is the graph automorphism composed with the Frobenius, uniformly in the thirteen ordinary and graph-twisted families. On the nine untwisted ones the graph factor is trivial and the composite is the Frobenius itself, which is `TauCeti.GraphTwistedIndex.steinberg_eq_frobenius`.","title":"The Steinberg endomorphism is the graph automorphism composed with the Frobenius"},{"id":"tauceti/group-theory/stating-the-classification-of-finite-simple-grou/tauceti-f4shortroot-primefield-specialisogeny-sp","kind":"node","lean":"TauCeti.F4ShortRoot.PrimeField.specialIsogeny_specialIsogeny","parent":2763,"result":true,"route":"../roadmap/tauceti/group-theory/stating-the-classification-of-finite-simple-grou/index.html#tauceti-f4shortroot-primefield-specialisogeny-sp","scope":"tauceti/group-theory/stating-the-classification-of-finite-simple-grou","status":"fully_proved","summary":"Squaring the special endomorphism gives the prime-field Frobenius on all carrier points.","title":"TauCeti.F4ShortRoot.PrimeField.specialIsogeny_specialIsogeny"},{"id":"tauceti/group-theory/stating-the-classification-of-finite-simple-grou/tauceti-sporadicname-presentation","kind":"node","lean":"TauCeti.SporadicName.presentation","parent":2763,"result":true,"route":"../roadmap/tauceti/group-theory/stating-the-classification-of-finite-simple-grou/index.html#tauceti-sporadicname-presentation","scope":"tauceti/group-theory/stating-the-classification-of-finite-simple-grou","status":"fully_proved","summary":"The explicit, cited finite presentation attached to each sporadic-group name.","title":"TauCeti.SporadicName.presentation"},{"id":"tauceti/group-theory/stating-the-classification-of-finite-simple-grou/the-explicit-type-a-carrier-is-equivalent-to-the","kind":"node","lean":"TauCeti.TypeALieIndex.carrierEquivPinned","parent":2763,"result":true,"route":"../roadmap/tauceti/group-theory/stating-the-classification-of-finite-simple-grou/index.html#the-explicit-type-a-carrier-is-equivalent-to-the","scope":"tauceti/group-theory/stating-the-classification-of-finite-simple-grou","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.TypeALieIndex.carrierEquivPinned`.","title":"The explicit type-A carrier is equivalent to the points of the pinned `SL_{r+1}/\u2124` group scheme"},{"id":"tauceti/group-theory/stating-the-classification-of-finite-simple-grou/the-ambient-group-of-a-valid-lie-type-index","kind":"node","lean":"TauCeti.ValidLieTypeIndex.AmbientGroup","parent":2763,"result":true,"route":"../roadmap/tauceti/group-theory/stating-the-classification-of-finite-simple-grou/index.html#the-ambient-group-of-a-valid-lie-type-index","scope":"tauceti/group-theory/stating-the-classification-of-finite-simple-grou","status":"fully_proved","summary":"the group of algebraic-closure-valued points of the explicit carrier assigned to its family. It is generally infinite, and it is not identified with the points of the pinned simply connected group scheme of the diagram. On the thirteen ordinary and graph-twisted constructors it is `TauCeti.GraphTwistedIndex.AmbientGroup`; the Suzuki family runs on the rank-two symplectic carrier, the Ree family of type `G\u2082` on the\u2026","title":"The ambient group of a valid Lie-type index"},{"id":"tauceti/group-theory/stating-the-classification-of-finite-simple-grou/tauceti-fixedpointcandidate-congroffiniteker","kind":"node","lean":"TauCeti.FixedPointCandidate.congrOfFiniteKer","parent":2763,"result":true,"route":"../roadmap/tauceti/group-theory/stating-the-classification-of-finite-simple-grou/index.html#tauceti-fixedpointcandidate-congroffiniteker","scope":"tauceti/group-theory/stating-the-classification-of-finite-simple-grou","status":"fully_proved","summary":"Transport the candidate through an equivariant surjection with finite kernel whose only fixed point is the identity.","title":"TauCeti.FixedPointCandidate.congrOfFiniteKer"},{"id":"tauceti/group-theory/stating-the-classification-of-finite-simple-grou/two-morphisms-out-of-the-short-root-type-f4-carr","kind":"node","lean":"TauCeti.F4ShortRoot.groupScheme_hom_ext_of_rootSubgroup","parent":2763,"result":true,"route":"../roadmap/tauceti/group-theory/stating-the-classification-of-finite-simple-grou/index.html#two-morphisms-out-of-the-short-root-type-f4-carr","scope":"tauceti/group-theory/stating-the-classification-of-finite-simple-grou","status":"fully_proved","summary":"This drops the weight-torus hypothesis of `TauCeti.F4ShortRoot.groupScheme_hom_ext`, which root generation of the carrier makes redundant.","title":"Two morphisms out of the short-root type-`F\u2084` carrier agree as soon as they agree on its eight numbered root subgroups"},{"id":"tauceti/group-theory/stating-the-classification-of-finite-simple-grou/the-compiled-word-denotes-the-direct-interpretat","kind":"node","lean":"TauCeti.Relator.toWord_toFreeGroup","parent":2763,"result":true,"route":"../roadmap/tauceti/group-theory/stating-the-classification-of-finite-simple-grou/index.html#the-compiled-word-denotes-the-direct-interpretat","scope":"tauceti/group-theory/stating-the-classification-of-finite-simple-grou","status":"fully_proved","summary":"Consequently, a reviewer may check the structured `Relator` against a published presentation while the eventual `PresentedGroup` safely uses `FreeGroup.mk r.toWord` as its defining relation.","title":"The compiled word denotes the direct interpretation of the relator expression"},{"id":"tauceti/group-theory/stating-the-classification-of-finite-simple-grou/tauceti-grouppresentation-homequiv","kind":"node","lean":"TauCeti.GroupPresentation.homEquiv","parent":2763,"result":true,"route":"../roadmap/tauceti/group-theory/stating-the-classification-of-finite-simple-grou/index.html#tauceti-grouppresentation-homequiv","scope":"tauceti/group-theory/stating-the-classification-of-finite-simple-grou","status":"fully_proved","summary":"Homomorphisms from a finite presentation correspond to generator assignments satisfying its transcribed relators. No generation or injectivity condition is imposed on the assignment.","title":"TauCeti.GroupPresentation.homEquiv"},{"data_url":"scopes/tauceti/group-theory/coxeter.json","id":"tauceti/group-theory/coxeter","kind":"scope","parent":2734,"result":false,"route":"../roadmap/tauceti/group-theory/coxeter/index.html","scope":"tauceti/group-theory/coxeter","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.GroupTheory.Coxeter` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Coxeter"},{"id":"tauceti/group-theory/coxeter/deodhars-lemma","kind":"node","lean":"CoxeterSystem.IsMinimalCosetRep.simple_mul_or_exists_eq_mul_simple","parent":2775,"result":true,"route":"../roadmap/tauceti/group-theory/coxeter/index.html#deodhars-lemma","scope":"tauceti/group-theory/coxeter","status":"fully_proved","summary":"Let `u` be a minimal coset representative for `cs.parabolic J` and let `s i` be a simple reflection. Then either `s i * u` is again a minimal coset representative, or `s i * u = u * s j` for some `j \u2208 J`, so that `s i * u` lies in the coset of `u`. The second case can only occur when `s i * u` is longer than `u`.","title":"Deodhar's lemma"},{"id":"tauceti/group-theory/coxeter/matsumotos-theorem-relational-form","kind":"node","lean":"TauCeti.braidEquivalent_of_isReduced_wordProd_eq, TauCeti.matsumoto","parent":2775,"result":true,"route":"../roadmap/tauceti/group-theory/coxeter/index.html#matsumotos-theorem-relational-form","scope":"tauceti/group-theory/coxeter","status":"fully_proved","summary":"Two reduced words representing the same element of a Coxeter group are connected by braid moves.","title":"Matsumoto's theorem, relational form"},{"id":"tauceti/group-theory/coxeter/tits-lemma","kind":"node","lean":"CoxeterSystem.exists_wordProd_alternatingWord_mul_of_isLeftDescent_pair","parent":2775,"result":true,"route":"../roadmap/tauceti/group-theory/coxeter/index.html#tits-lemma","scope":"tauceti/group-theory/coxeter","status":"fully_proved","summary":"If `i \u2260 i'` are both left descents of `w`, then `w` splits as a rank-two parabolic factor times a remainder, the lengths adding, and the parabolic factor is spelled by *both* alternating words of length `orderOf (s i * s i')`.","title":"Tits' lemma"},{"data_url":"scopes/tauceti/group-theory/group-action.json","id":"tauceti/group-theory/group-action","kind":"scope","parent":2734,"result":false,"route":"../roadmap/tauceti/group-theory/group-action/index.html","scope":"tauceti/group-theory/group-action","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.GroupTheory.GroupAction` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Group Action"},{"id":"tauceti/group-theory/group-action/burnsides-lemma-on-a-product","kind":"node","lean":"TauCeti.sum_card_fixedBy_mul_card_fixedBy_eq_card_orbits_mul_card_group","parent":2779,"result":true,"route":"../roadmap/tauceti/group-theory/group-action/index.html#burnsides-lemma-on-a-product","scope":"tauceti/group-theory/group-action","status":"fully_proved","summary":"For a finite group `G` acting on two finite sets `X` and `Y`, the sum over `g : G` of the product of the two fixed-point counts is the number of orbits of `G` on `X \u00d7 Y`, times the order of `G`.","title":"Burnside's lemma on a product"},{"id":"tauceti/group-theory/group-action/iwasawas-criterion-for-a-quotient","kind":"node","lean":"MulAction.IwasawaStructure.isSimpleGroup_quotient, MulAction.IwasawaStructure.isSimpleGroup_quotient_center, MulAction.IwasawaStructure.isSimpleGroup_derivedCentralQuotient","parent":2779,"result":true,"route":"../roadmap/tauceti/group-theory/group-action/index.html#iwasawas-criterion-for-a-quotient","scope":"tauceti/group-theory/group-action","status":"fully_proved","summary":"Let a perfect group `G` have an Iwasawa structure on a quasiprimitive action. If a proper normal subgroup `N` contains every element acting trivially, then `G \u29f8 N` is simple.","title":"Iwasawa's criterion for a quotient"},{"id":"tauceti/group-theory/group-action/iwasawas-normal-subgroup-theorem-for-a-perfect-g","kind":"node","lean":"MulAction.IwasawaStructure.fixedPoints_eq_univ_or_eq_top","parent":2779,"result":true,"route":"../roadmap/tauceti/group-theory/group-action/index.html#iwasawas-normal-subgroup-theorem-for-a-perfect-g","scope":"tauceti/group-theory/group-action","status":"fully_proved","summary":"If a perfect group has an Iwasawa structure on a quasiprimitive action, each of its normal subgroups either fixes every point or is the whole group.","title":"Iwasawa's normal-subgroup theorem for a perfect group"},{"data_url":"scopes/tauceti/group-theory/permutations.json","id":"tauceti/group-theory/permutations","kind":"scope","parent":2734,"result":false,"route":"../roadmap/tauceti/group-theory/permutations/index.html","scope":"tauceti/group-theory/permutations","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.GroupTheory.Perm` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Permutations"},{"id":"tauceti/group-theory/permutations/the-euler-bound-for-a-pair-of-permutations","kind":"node","lean":"Equiv.Perm.orbitCount_add_orbitCount_add_orbitCount_mul_le","parent":2783,"result":true,"route":"../roadmap/tauceti/group-theory/permutations/index.html#the-euler-bound-for-a-pair-of-permutations","scope":"tauceti/group-theory/permutations","status":"fully_proved","summary":"For permutations `\u03c3` and `\u03c4` of a finite type, the orbits of `\u03c3`, of `\u03c4` and of `\u03c4 * \u03c3` together number at most the number of points plus twice the number of orbits of the group generated by `\u03c3` and `\u03c4`. For the rotation and edge involution of a ribbon graph these count vertices, edges and faces, and this is the bound `\u03c7 \u2264 2` on each connected component; equality holds exactly when every component is a sphere.","title":"The Euler bound for a pair of permutations"},{"id":"tauceti/group-theory/permutations/hurwitzs-transposition-bound","kind":"node","lean":"TauCeti.card_add_orbitCount_le_length_add_two, TauCeti.card_add_orbitCount_add_two_le_length_add_two_mul_card_orbits, TauCeti.card_add_orbitCount_le_length_add_two_mul_card_orbits","parent":2783,"result":true,"route":"../roadmap/tauceti/group-theory/permutations/index.html#hurwitzs-transposition-bound","scope":"tauceti/group-theory/permutations","status":"fully_proved","summary":"If a list of transpositions generates a group acting transitively on a finite type, its length is at least the number of points plus the number of cycles of its product, minus two. Equivalently, `Nat.card \u03b1 + orbitCount L.prod \u2264 L.length + 2`.","title":"Hurwitz's transposition bound"},{"id":"tauceti/group-theory/permutations/jordans-theorem-fails-in-degree-p-1","kind":"node","lean":"TauCeti.not_forall_alternatingGroup_le_of_isPreprimitive_of_isCycle_mem_of_card_support_add_one_eq","parent":2783,"result":true,"route":"../roadmap/tauceti/group-theory/permutations/index.html#jordans-theorem-fails-in-degree-p-1","scope":"tauceti/group-theory/permutations","status":"fully_proved","summary":"A primitive permutation group of degree `n` containing a cycle of prime length `p` need not contain the alternating group when `p + 1 = n`: the affine group `AGL(1, 8)` of the field with eight elements is primitive on eight points and contains a `7`-cycle. This shows that the bound `p + 3 \u2264 n` of `TauCeti.alternatingGroup_le_of_isPreprimitive_of_isCycle_mem` cannot be weakened to `p + 1 \u2264 n`.","title":"Jordan's theorem fails in degree `p + 1`"},{"id":"tauceti/group-theory/permutations/jordans-theorem-fails-in-degree-p","kind":"node","lean":"TauCeti.not_forall_alternatingGroup_le_of_isPreprimitive_of_isCycle_mem_of_card_support_eq","parent":2783,"result":true,"route":"../roadmap/tauceti/group-theory/permutations/index.html#jordans-theorem-fails-in-degree-p","scope":"tauceti/group-theory/permutations","status":"fully_proved","summary":"A primitive permutation group of degree `n` containing a cycle of prime length `p` need not contain the alternating group when `p = n`: the affine group `AGL(1, 5)` is primitive on five points and contains a `5`-cycle. This shows that the bound `p + 3 \u2264 n` of `TauCeti.alternatingGroup_le_of_isPreprimitive_of_isCycle_mem` cannot be weakened to `p \u2264 n`.","title":"Jordan's theorem fails in degree `p`"},{"id":"tauceti/group-theory/permutations/jordans-theorem-for-a-cycle-of-prime-length-wiel","kind":"node","lean":"TauCeti.alternatingGroup_le_of_isPreprimitive_of_isCycle_mem","parent":2783,"result":true,"route":"../roadmap/tauceti/group-theory/permutations/index.html#jordans-theorem-for-a-cycle-of-prime-length-wiel","scope":"tauceti/group-theory/permutations","status":"fully_proved","summary":"A primitive permutation group of degree at least `p + 3` that contains a cycle of prime length `p` contains the alternating group.","title":"Jordan's theorem for a cycle of prime length (Wielandt, Theorem 13.9)"},{"data_url":"scopes/tauceti/group-theory/specific-groups.json","id":"tauceti/group-theory/specific-groups","kind":"scope","parent":2734,"result":false,"route":"../roadmap/tauceti/group-theory/specific-groups/index.html","scope":"tauceti/group-theory/specific-groups","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.GroupTheory.SpecificGroups` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Specific Groups"},{"id":"tauceti/group-theory/specific-groups/eulers-product-formula-for-the-elements-of-order","kind":"node","lean":"IsCyclic.card_filter_dvd_orderOf_eq_mul_prod_primeFactors, IsAddCyclic.card_filter_dvd_addOrderOf_eq_add_sum_primeFactors","parent":2789,"result":true,"route":"../roadmap/tauceti/group-theory/specific-groups/index.html#eulers-product-formula-for-the-elements-of-order","scope":"tauceti/group-theory/specific-groups","status":"fully_proved","summary":"For `f` dividing the order `h` of a cyclic group, the elements of order divisible by `f` are a proportion `\u220f p \u2223 f, (1 - p ^ -(v_p h - v_p f + 1))` of the group, the product running over the primes of `f`.","title":"Euler's product formula for the elements of order divisible by `f`"},{"id":"tauceti/group-theory/specific-groups/the-finite-field-generation-theorem","kind":"node","lean":"TauCeti.SuzukiLieIndex.finiteFixedSubgroup_le_suzukiGroup","parent":2789,"result":true,"route":"../roadmap/tauceti/group-theory/specific-groups/index.html#the-finite-field-generation-theorem","scope":"tauceti/group-theory/specific-groups","status":"fully_proved","summary":"every Steinberg fixed point over the generator field is a product of the standard generators. A fixed point fixing `\u221e` is a torus element times a unipotent generator, and any other is carried to one by `w u(a, b)` for the unipotent generator moving its last column onto the line through `e\u2080`.","title":"The finite-field generation theorem"},{"id":"tauceti/group-theory/specific-groups/the-rank-one-bruhat-decomposition-sz-q-b-b-w-b-f","kind":"node","lean":"TauCeti.Suzuki.mem_borelSubgroupOf_or_mem_doubleCoset","parent":2789,"result":true,"route":"../roadmap/tauceti/group-theory/specific-groups/index.html#the-rank-one-bruhat-decomposition-sz-q-b-b-w-b-f","scope":"tauceti/group-theory/specific-groups","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.Suzuki.mem_borelSubgroupOf_or_mem_doubleCoset`.","title":"The rank-one Bruhat decomposition `Sz(q) = B \u222a B w B`, for `m \u2260 0`"},{"data_url":"scopes/tauceti/group-theory/tits-system.json","id":"tauceti/group-theory/tits-system","kind":"scope","parent":2734,"result":false,"route":"../roadmap/tauceti/group-theory/tits-system/index.html","scope":"tauceti/group-theory/tits-system","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.GroupTheory.TitsSystem` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Tits System"},{"id":"tauceti/group-theory/tits-system/the-bruhat-decomposition-of-a-standard-parabolic","kind":"node","lean":"TauCeti.TitsSystem.mem_standardParabolic_iff","parent":2793,"result":true,"route":"../roadmap/tauceti/group-theory/tits-system/index.html#the-bruhat-decomposition-of-a-standard-parabolic","scope":"tauceti/group-theory/tits-system","status":"fully_proved","summary":"an element lies in `P_J` exactly when its Bruhat cell is indexed by the Weyl subgroup generated by `J`.","title":"The Bruhat decomposition of a standard parabolic"},{"id":"tauceti/group-theory/tits-system/the-key-step-of-titss-theorem-on-normal-subgroup","kind":"node","lean":"TauCeti.TitsSystem.simple_mul_comm_of_bruhatCell_subset_sup","parent":2793,"result":true,"route":"../roadmap/tauceti/group-theory/tits-system/index.html#the-key-step-of-titss-theorem-on-normal-subgroup","scope":"tauceti/group-theory/tits-system","status":"fully_proved","summary":"If `H` is normal, a simple reflection whose Bruhat cell lies in `H \u2294 B` commutes with every simple reflection whose cell does not.","title":"The key step of Tits's theorem on normal subgroups"},{"id":"tauceti/group-theory/tits-system/titss-simplicity-criterion-for-the-normal-closur","kind":"node","lean":"TauCeti.TitsSystem.isSimpleGroup_quotient_normalCore_subgroupOf","parent":2793,"result":true,"route":"../roadmap/tauceti/group-theory/tits-system/index.html#titss-simplicity-criterion-for-the-normal-closur","scope":"tauceti/group-theory/tits-system","status":"fully_proved","summary":"Under the hypotheses of `TauCeti.TitsSystem.le_normalCore_subgroupOf_or_eq_top_of_normal`, the quotient `H\u207a / (Z \u2229 H\u207a)` of the normal closure `H\u207a` of `U` by its intersection with `Z = \u22c2 g B g\u207b\u00b9` is simple whenever it is nontrivial.","title":"Tits's simplicity criterion for the normal closure of `U`"},{"id":"tauceti/group-theory/tits-system/titss-simplicity-theorem","kind":"node","lean":"TauCeti.TitsSystem.isSimpleGroup_quotient_normalCore, TauCeti.TitsSystem.le_normalCore_or_eq_top_of_normal","parent":2793,"result":true,"route":"../roadmap/tauceti/group-theory/tits-system/index.html#titss-simplicity-theorem","scope":"tauceti/group-theory/tits-system","status":"fully_proved","summary":"Suppose there is at least one simple reflection, the Weyl group is irreducible, `G` is perfect, and `U` is a solvable subgroup normalized by `B` whose conjugates generate `G`. Then the quotient of `G` by the normal core `\u22c2 g B g\u207b\u00b9` of `B` is simple.","title":"Tits's simplicity theorem"},{"id":"tauceti/group-theory/tits-system/titss-theorem-on-normal-subgroups","kind":"node","lean":"TauCeti.TitsSystem.exists_standardParabolic_eq_sup_of_normal, TauCeti.TitsSystem.le_normalCore_subgroupOf_or_eq_top_of_normal","parent":2793,"result":true,"route":"../roadmap/tauceti/group-theory/tits-system/index.html#titss-theorem-on-normal-subgroups","scope":"tauceti/group-theory/tits-system","status":"fully_proved","summary":"For a normal subgroup `H`, the subgroup `H \u2294 B` is a standard parabolic `P_J` for a set `J` of simple reflections that commutes elementwise with its complement.","title":"Tits's theorem on normal subgroups"},{"data_url":"scopes/tauceti/group-theory/other-results.json","id":"tauceti/group-theory/other-results","kind":"scope","parent":2734,"result":false,"route":"../roadmap/tauceti/group-theory/other-results/index.html","scope":"tauceti/group-theory/other-results","status":"fully_proved","summary":"Results named in the rest of Tau Ceti's `TauCeti.GroupTheory` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Other results"},{"id":"tauceti/group-theory/other-results/artins-identity-for-fixed-points","kind":"node","lean":"TauCeti.sum_artinCoeff_mul_card_fixedBy","parent":2799,"result":true,"route":"../roadmap/tauceti/group-theory/other-results/index.html#artins-identity-for-fixed-points","scope":"tauceti/group-theory/other-results","status":"fully_proved","summary":"weighting the permutation character of `G \u29f8 C` by `C.artinCoeff * |C|` and summing over all subgroups gives `|G|`, at every `g : G`.","title":"Artin's identity for fixed points"},{"id":"tauceti/group-theory/other-results/eulers-theorem-on-an-n-th-root-of-unity","kind":"node","lean":"TauCeti.pow_pow_totient_eq_self","parent":2799,"result":true,"route":"../roadmap/tauceti/group-theory/other-results/index.html#eulers-theorem-on-an-n-th-root-of-unity","scope":"tauceti/group-theory/other-results","status":"fully_proved","summary":"if `x ^ n = 1` in a monoid and `q` is prime to `n`, then `x ^ q ^ \u03c6(n) = x`, since `q ^ \u03c6(n) \u2261 1 [MOD n]`.","title":"Euler's theorem on an `n`-th root of unity"},{"id":"tauceti/group-theory/other-results/furtwanglers-theorem-the-group-theoretic-princip","kind":"node","lean":"MonoidHom.transfer_commutator_eq_one","parent":2799,"result":true,"route":"../roadmap/tauceti/group-theory/other-results/index.html#furtwanglers-theorem-the-group-theoretic-princip","scope":"tauceti/group-theory/other-results","status":"fully_proved","summary":"if the commutator subgroup `G'` of `G` is finitely generated and of finite index (for instance if `G` is finite), the transfer `G \u2192 A` of every homomorphism `\u03d5 : G' \u2192* A` to a commutative group is trivial. For `\u03d5` the abelianization map, this says that the transfer `G \u2192 G' \u29f8 G''` is trivial.","title":"Furtw\u00e4ngler's theorem, the group-theoretic principal ideal theorem"},{"id":"tauceti/group-theory/other-results/gaschutzs-counting-lemma","kind":"node","lean":"MonoidHom.card_comp_eq_and_closure_range_eq_congr","parent":2799,"result":true,"route":"../roadmap/tauceti/group-theory/other-results/index.html#gaschutzs-counting-lemma","scope":"tauceti/group-theory/other-results","status":"fully_proved","summary":"For a subgroup `K` of a finite group `G`, the number of `\u03b9`-tuples that generate `K` and lift a given generating tuple of `H` along `f` does not depend on that generating tuple.","title":"Gasch\u00fctz's counting lemma"},{"id":"tauceti/group-theory/other-results/gaschutzs-lemma","kind":"node","lean":"MonoidHom.exists_comp_eq_and_closure_range_eq_top_of_surjective","parent":2799,"result":true,"route":"../roadmap/tauceti/group-theory/other-results/index.html#gaschutzs-lemma","scope":"tauceti/group-theory/other-results","status":"fully_proved","summary":"Let `f : G \u2192* H` be a surjective homomorphism from a finite group, and let `G` be generated by an `\u03b9`-tuple. Then every generating `\u03b9`-tuple of `H` lifts along `f` to a generating `\u03b9`-tuple of `G`.","title":"Gasch\u00fctz's lemma"},{"id":"tauceti/group-theory/other-results/gruns-lemma-for-the-recipe","kind":"node","lean":"TauCeti.DerivedCentralQuotient.center_eq_bot","parent":2799,"result":true,"route":"../roadmap/tauceti/group-theory/other-results/index.html#gruns-lemma-for-the-recipe","scope":"tauceti/group-theory/other-results","status":"fully_proved","summary":"when the derived subgroup is perfect, the derived central quotient has trivial centre.","title":"Gr\u00fcn's lemma for the recipe"},{"id":"tauceti/group-theory/other-results/howsons-theorem","kind":"node","lean":"Subgroup.FG.inf_of_isFreeGroup","parent":2799,"result":true,"route":"../roadmap/tauceti/group-theory/other-results/index.html#howsons-theorem","scope":"tauceti/group-theory/other-results","status":"fully_proved","summary":"in a free group, the intersection of two finitely generated subgroups is finitely generated.","title":"Howson's theorem"},{"id":"tauceti/group-theory/other-results/malcevs-theorem","kind":"node","lean":"Group.ResiduallyFinite.injective_of_surjective","parent":2799,"result":true,"route":"../roadmap/tauceti/group-theory/other-results/index.html#malcevs-theorem","scope":"tauceti/group-theory/other-results","status":"fully_proved","summary":"A surjective endomorphism of a finitely generated residually finite group is injective: such a group is Hopfian.","title":"Malcev's theorem"},{"id":"tauceti/group-theory/other-results/schreiers-index-formula","kind":"node","lean":"Subgroup.exists_finset_card_le_one_add_index_mul, Subgroup.rank_le_one_add_index_mul_rank_sub_one","parent":2799,"result":true,"route":"../roadmap/tauceti/group-theory/other-results/index.html#schreiers-index-formula","scope":"tauceti/group-theory/other-results","status":"fully_proved","summary":"If `H` has finite index `n` in a group generated by a finite set `S` of `d` elements, then `H` is generated by at most `1 + n * (d - 1)` elements. This sharpens Mathlib's `Subgroup.exists_finset_card_le_mul`, which gives `n * d`.","title":"Schreier's index formula"},{"id":"tauceti/group-theory/other-results/tarskis-alternative","kind":"node","lean":"TauCeti.amenable_iff_not_paradoxical_univ, TauCeti.hasFoelnerSets_iff_not_paradoxical_univ","parent":2799,"result":true,"route":"../roadmap/tauceti/group-theory/other-results/index.html#tarskis-alternative","scope":"tauceti/group-theory/other-results","status":"fully_proved","summary":"a discrete group is amenable exactly when it is not paradoxical for its left regular action.","title":"Tarski's alternative"},{"id":"tauceti/group-theory/other-results/the-third-isomorphism-theorem-for-coset-spaces","kind":"node","lean":"QuotientGroup.quotientQuotientEquivQuotientSup, QuotientAddGroup.quotientQuotientEquivQuotientSup","parent":2799,"result":true,"route":"../roadmap/tauceti/group-theory/other-results/index.html#the-third-isomorphism-theorem-for-coset-spaces","scope":"tauceti/group-theory/other-results","status":"fully_proved","summary":"For a normal `N` and an *arbitrary* subgroup `H`, the cosets of the image of `H` in `G \u29f8 N` are the cosets of `H \u2294 N` in `G`, both directions sending the class of `g` to the class of `g`.","title":"The third isomorphism theorem for coset spaces"},{"data_url":"scopes/tauceti/knot-theory.json","id":"tauceti/knot-theory","kind":"scope","parent":1095,"result":false,"route":"../roadmap/tauceti/knot-theory/index.html","scope":"tauceti/knot-theory","status":"fully_proved","summary":"Results in Tau Ceti's `TauCeti.KnotTheory` modules, and the roadmaps whose articles are mostly there: by roadmap, then by area.","title":"Knot Theory"},{"data_url":"scopes/tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology.json","id":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology","kind":"scope","parent":2811,"result":false,"route":"../roadmap/tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/index.html","scope":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Combinatorial Heegaard Floer and grid homology"},{"id":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/the-commutation-map-is-a-chain-map","kind":"node","lean":"TauCeti.GridDiagram.commutationMap_comp_unblockedDifferential","parent":2812,"result":true,"route":"../roadmap/tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/index.html#the-commutation-map-is-a-chain-map","scope":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology","status":"fully_proved","summary":"Over a commutative semiring of characteristic two, counting empty pentagons turning on either side of the replaced grid line intertwines the unblocked differentials of a grid diagram and of the diagram with the two columns of the turn exchanged. This holds at every column turn: the two marking segments need not be non-interleaving.","title":"The commutation map is a chain map"},{"id":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/tauceti-griddiagram-unblockeddifferential-commut","kind":"node","lean":"TauCeti.GridDiagram.unblockedDifferential_commutationHomotopy_add_eq","parent":2812,"result":true,"route":"../roadmap/tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/index.html#tauceti-griddiagram-unblockeddifferential-commut","scope":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology","status":"fully_proved","summary":"The homotopy equation `\u2202 \u2218 H + H \u2218 \u2202 = 1 + \u03a8 \u2218 \u03a6` of a column commutation, over any commutative semiring of characteristic two: the hexagon-counting map `H` is a chain homotopy between the identity of `GC\u207b(G)` and the round trip through the commuted diagram.","title":"TauCeti.GridDiagram.unblockedDifferential_commutationHomotopy_add_eq"},{"id":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/stabilization-invariance-of-gh","kind":"node","lean":"TauCeti.GridDiagram.stabilizeXHomologyMap_bijective","parent":2812,"result":true,"route":"../roadmap/tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/index.html#stabilization-invariance-of-gh","scope":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology","status":"fully_proved","summary":"The map `GH\u207b(G') \u2192 GH\u207b(G)` induced by the stabilization chain map is bijective.","title":"Stabilization invariance of `GH\u207b`"},{"id":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/stabilization-multiplies-the-poincare-series-by","kind":"node","lean":"TauCeti.OddComponentGridDiagram.bigradedFullyBlockedHomologyPoincareSeries_stabilizeX","parent":2812,"result":true,"route":"../roadmap/tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/index.html#stabilization-multiplies-the-poincare-series-by","scope":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology","status":"fully_proved","summary":"The Poincar\u00e9 series of the fully blocked homology of the stabilization splitting an `X`-marking is that of the original diagram times the series `W` of the stabilization factor.","title":"Stabilization multiplies the Poincar\u00e9 series by `W`"},{"id":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/of-a-torus-knot-grid","kind":"node","lean":"TauCeti.GridDiagram.two_mul_tau_torusLink","parent":2812,"result":true,"route":"../roadmap/tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/index.html#of-a-torus-knot-grid","scope":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology","status":"fully_proved","summary":"on the standard grid of the `(p + 1, q + 1)` torus knot, `2 \u03c4 = -p q`.","title":"`\u03c4` of a torus knot grid"},{"id":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/the-structure-theorem-for-finitely-generated-gra","kind":"node","lean":"TauCeti.InternalGrading.exists_isInternal_span_singleton","parent":2812,"result":true,"route":"../roadmap/tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/index.html#the-structure-theorem-for-finitely-generated-gra","scope":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology","status":"fully_proved","summary":"Let `X` lower degree by `d \u2260 0` in a finitely generated graded `k[X]`-module `M` over a field. Then `M` is the internal direct sum of cyclic submodules `k[X] \u2219 t i` generated by homogeneous elements `t i`, indexed by the disjoint union of two finite types: for `i` in the first, `t i` has zero annihilator, so `k[X] \u2219 t i` is a free `X`-tower; for `j` in the second, `t j` has annihilator `(X ^ m j)` with `m j \u2260 0`,\u2026","title":"The structure theorem for finitely generated graded `k[X]`-modules"},{"id":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/the-graded-euler-characteristic-of-fully-blocked","kind":"node","lean":"TauCeti.OddComponentGridDiagram.gradedHomologyEulerChar_eq_smul_T_mul_det_weightMatrix","parent":2812,"result":true,"route":"../roadmap/tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/index.html#the-graded-euler-characteristic-of-fully-blocked","scope":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.OddComponentGridDiagram.gradedHomologyEulerChar_eq_smul_T_mul_det_weightMatrix`.","title":"The graded Euler characteristic of fully blocked grid homology is the normalized grid determinant"},{"id":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/tauceti-gridlink-unknottingnumber","kind":"node","lean":"TauCeti.GridLink.unknottingNumber","parent":2812,"result":true,"route":"../roadmap/tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/index.html#tauceti-gridlink-unknottingnumber","scope":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology","status":"fully_proved","summary":"The unknotting number of a grid link: the least number of crossing changes turning it into the unknot, or `\u22a4` if no sequence of crossing changes does.","title":"TauCeti.GridLink.unknottingNumber"},{"id":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/a-chain-map-of-bidegree-2-1-across-a-crossing-ch","kind":"node","lean":"TauCeti.GridDiagram.commutationBidegree_eq_of_notMem_columnArc","parent":2812,"result":true,"route":"../roadmap/tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/index.html#a-chain-map-of-bidegree-2-1-across-a-crossing-ch","scope":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology","status":"fully_proved","summary":"If the segments of the two columns of the column turn `C` interleave and the `O`-marking of the first column does not lie on the arc of the second, then the commutation map turning in the row of the `X`-marking of the first column has bidegree `(-2, -1)`, the bidegree of multiplication by `U`.","title":"A chain map of bidegree `(-2, -1)` across a crossing change"},{"id":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/the-invariant-of-a-knot-grid-diagram","kind":"node","lean":"TauCeti.GridDiagram.IsKnot.tau","parent":2812,"result":true,"route":"../roadmap/tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/index.html#the-invariant-of-a-knot-grid-diagram","scope":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology","status":"fully_proved","summary":"minus the largest Alexander degree of a homogeneous element of the unblocked grid homology `GH\u207b(G)` that is not `K[U]`-torsion.","title":"The invariant `\u03c4` of a knot grid diagram"},{"id":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/the-commutation-map-on-gh-preserves-the-class-of","kind":"node","lean":"TauCeti.GridDiagram.unblockedHomologyCommutationMap_XMarkingClass","parent":2812,"result":true,"route":"../roadmap/tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/index.html#the-commutation-map-on-gh-preserves-the-class-of","scope":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology","status":"fully_proved","summary":"Under the condition of `commutationMap_single_X`, the induced map on unblocked grid homology sends the class of the `X`-marking state of `G` to that of the commuted diagram.","title":"The commutation map on `GH\u207b` preserves the class of the `X`-marking state"},{"id":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/tauceti-heegaardregionsystem-hatdifferential","kind":"node","lean":"TauCeti.HeegaardRegionSystem.hatDifferential","parent":2812,"result":true,"route":"../roadmap/tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/index.html#tauceti-heegaardregionsystem-hatdifferential","scope":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology","status":"fully_proved","summary":"The hat differential of a Heegaard diagram with region Euler characteristics `\u03c7`: the `ZMod 2`-linear endomorphism of the free module on generators whose coefficient from `x` to `y` is the number, modulo two, of nonnegative domains of Maslov index one from `x` to `y` avoiding the basepoints (`TauCeti.HeegaardRegionSystem.hatCoefficient`).","title":"TauCeti.HeegaardRegionSystem.hatDifferential"},{"id":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/tauceti-heegaardregionsystem-isnice","kind":"node","lean":"TauCeti.HeegaardRegionSystem.IsNice","parent":2812,"result":true,"route":"../roadmap/tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/index.html#tauceti-heegaardregionsystem-isnice","scope":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology","status":"fully_proved","summary":"A pointed Heegaard diagram, with region Euler characteristics `\u03c7`, is *nice* when every region containing no basepoint is a bigon or a square: a disk, of Euler characteristic one, with two or four corners.","title":"TauCeti.HeegaardRegionSystem.IsNice"},{"id":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/euler-poincare-in-one-alexander-degree","kind":"node","lean":"TauCeti.OddComponentGridDiagram.alexanderHomologyEulerChar_eq_alexanderEulerChar","parent":2812,"result":true,"route":"../roadmap/tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/index.html#euler-poincare-in-one-alexander-degree","scope":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology","status":"fully_proved","summary":"The alternating dimension of fully blocked grid homology equals the alternating count of grid states.","title":"Euler--Poincar\u00e9 in one Alexander degree"},{"id":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/the-torsion-part-of-the-structure-theorem","kind":"node","lean":"TauCeti.InternalGrading.exists_iSupIndep_iSup_span_singleton_eq_of_le_torsion","parent":2812,"result":true,"route":"../roadmap/tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology/index.html#the-torsion-part-of-the-structure-theorem","scope":"tauceti/knot-theory/combinatorial-heegaard-floer-and-grid-homology","status":"fully_proved","summary":"Let `X` lower degree by `d \u2260 0` in a finitely generated graded `k[X]`-module over a field. Then every homogeneous `k[X]`-submodule `L` contained in the torsion submodule is the independent sum of finitely many cyclic submodules `k[X] \u2219 t i`, where each `t i` is homogeneous with annihilator `(X ^ m i)` for some `m i \u2260 0`. Thus each summand is isomorphic to `k[X] \u29f8 (X ^ m i)` (`Ideal.quotTorsionOfEquivSpanSingleton`).","title":"The torsion part of the structure theorem"},{"data_url":"scopes/tauceti/knot-theory/khovanov-homology-of-classical-links.json","id":"tauceti/knot-theory/khovanov-homology-of-classical-links","kind":"scope","parent":2811,"result":false,"route":"../roadmap/tauceti/knot-theory/khovanov-homology-of-classical-links/index.html","scope":"tauceti/knot-theory/khovanov-homology-of-classical-links","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Khovanov homology of classical links"},{"id":"tauceti/knot-theory/khovanov-homology-of-classical-links/the-enhanced-state-sum-of-a-nonempty-diagram-is","kind":"node","lean":"TauCeti.OrientedPDCode.enhancedStateSum_eq_mul_eval\u2082_jonesPolynomial","parent":2828,"result":true,"route":"../roadmap/tauceti/knot-theory/khovanov-homology-of-classical-links/index.html#the-enhanced-state-sum-of-a-nonempty-diagram-is","scope":"tauceti/knot-theory/khovanov-homology-of-classical-links","status":"fully_proved","summary":"The hypothesis `0 < componentCount` says that the diagram is nonempty; it excludes the empty diagram, whose enhanced state sum is `1`.","title":"The enhanced state sum of a nonempty diagram is `(q + q\u207b\u00b9)` times its Jones polynomial at `t ^ (1/2) = -q`"},{"id":"tauceti/knot-theory/khovanov-homology-of-classical-links/two-states-differing-exactly-at-a-crossing-do-no","kind":"node","lean":"TauCeti.PDCode.card_circlesAt_add_card_circlesAt_le_three","parent":2828,"result":true,"route":"../roadmap/tauceti/knot-theory/khovanov-homology-of-classical-links/index.html#two-states-differing-exactly-at-a-crossing-do-no","scope":"tauceti/knot-theory/khovanov-homology-of-classical-links","status":"fully_proved","summary":"So along an edge of the cube of resolutions two circles merge into one, one circle splits into two, or one circle through the crossing is replaced by one circle.","title":"Two states differing exactly at a crossing do not both have two circles through it"},{"id":"tauceti/knot-theory/khovanov-homology-of-classical-links/the-boundary-count-criterion-for-noninterleaving","kind":"node","lean":"Function.Involutive.orbitCount_finRotate_mul_add_orbitCount_eq_iff","parent":2828,"result":true,"route":"../roadmap/tauceti/knot-theory/khovanov-homology-of-classical-links/index.html#the-boundary-count-criterion-for-noninterleaving","scope":"tauceti/knot-theory/khovanov-homology-of-classical-links","status":"fully_proved","summary":"A chord diagram on a nonempty boundary circle, with matching an involution `\u03c3` and `m = n - orbitCount \u03c3` bands, has exactly `m + 1` boundary cycles if and only if no two of its bands interleave.","title":"The boundary-count criterion for noninterleaving"},{"id":"tauceti/knot-theory/khovanov-homology-of-classical-links/tauceti-khovanov-isfrobenius","kind":"node","lean":"TauCeti.Khovanov.isFrobenius","parent":2828,"result":true,"route":"../roadmap/tauceti/knot-theory/khovanov-homology-of-classical-links/index.html#tauceti-khovanov-isfrobenius","scope":"tauceti/knot-theory/khovanov-homology-of-classical-links","status":"fully_proved","summary":"The dual numbers with the Khovanov comultiplication satisfy the Frobenius equation.","title":"TauCeti.Khovanov.isFrobenius"},{"id":"tauceti/knot-theory/khovanov-homology-of-classical-links/categorytheory-functor-cubecomplex","kind":"node","lean":"CategoryTheory.Functor.cubeComplex","parent":2828,"result":true,"route":"../roadmap/tauceti/knot-theory/khovanov-homology-of-classical-links/index.html#categorytheory-functor-cubecomplex","scope":"tauceti/knot-theory/khovanov-homology-of-classical-links","status":"fully_proved","summary":"The signed totalization of the commutative cube `F`: in degree `r` it is the biproduct of the vertices `F(S)` of height `#S = r`, with differential the sum of the edge maps `F(S) \u27f6 F(S \u222a {a})`, signed by `(-1)` to the number of elements of `S` below `a`. It is characterised by the summand inclusions `Functor.cubeComplex\u03b9` and projections `Functor.cubeComplex\u03c0`, and by `Functor.\u03b9_cubeComplex_d`.","title":"CategoryTheory.Functor.cubeComplex"},{"id":"tauceti/knot-theory/khovanov-homology-of-classical-links/tauceti-khovanov-circletensor","kind":"node","lean":"TauCeti.Khovanov.CircleTensor","parent":2828,"result":true,"route":"../roadmap/tauceti/knot-theory/khovanov-homology-of-classical-links/index.html#tauceti-khovanov-circletensor","scope":"tauceti/knot-theory/khovanov-homology-of-classical-links","status":"fully_proved","summary":"The circle tensor of a finite set `\u03b9` of circles: the tensor product of one copy of the Khovanov Frobenius algebra `R[\u03b5]` for each circle. It is a commutative `R`-algebra.","title":"TauCeti.Khovanov.CircleTensor"},{"id":"tauceti/knot-theory/khovanov-homology-of-classical-links/tauceti-khovanov-quantumgrading","kind":"node","lean":"TauCeti.Khovanov.quantumGrading","parent":2828,"result":true,"route":"../roadmap/tauceti/knot-theory/khovanov-homology-of-classical-links/index.html#tauceti-khovanov-quantumgrading","scope":"tauceti/knot-theory/khovanov-homology-of-classical-links","status":"fully_proved","summary":"The quantum grading of the Khovanov Frobenius algebra `R[\u03b5]`: the basis vector `1` spans degree `1`, the basis vector `\u03b5` spans degree `-1`, and every other degree is zero. It is the grading of the free module on the basis `1, \u03b5` by `TauCeti.Khovanov.quantumDegree`, transported to `R[\u03b5]`.","title":"TauCeti.Khovanov.quantumGrading"},{"id":"tauceti/knot-theory/khovanov-homology-of-classical-links/tauceti-khovanov-fiberwisesplit","kind":"node","lean":"TauCeti.Khovanov.fiberwiseSplit","parent":2828,"result":true,"route":"../roadmap/tauceti/knot-theory/khovanov-homology-of-classical-links/index.html#tauceti-khovanov-fiberwisesplit","scope":"tauceti/knot-theory/khovanov-homology-of-classical-links","status":"fully_proved","summary":"The split map along `f`, adjoint to multiplication along its fibres. It maps the tensor on `\u03c4` to the tensor on `\u03c3`; a two-element fibre splits that circle.","title":"TauCeti.Khovanov.fiberwiseSplit"},{"id":"tauceti/knot-theory/khovanov-homology-of-classical-links/tauceti-pdcode-resolutionmergemap","kind":"node","lean":"TauCeti.PDCode.resolutionMergeMap","parent":2828,"result":true,"route":"../roadmap/tauceti/knot-theory/khovanov-homology-of-classical-links/index.html#tauceti-pdcode-resolutionmergemap","scope":"tauceti/knot-theory/khovanov-homology-of-classical-links","status":"fully_proved","summary":"Multiplication on the two circles merged at `c`, and identity on every other factor of the actual resolved-circle tensor. The one-circle target condition follows from the source count.","title":"TauCeti.PDCode.resolutionMergeMap"},{"id":"tauceti/knot-theory/khovanov-homology-of-classical-links/tauceti-pdcode-resolutionsplitmap","kind":"node","lean":"TauCeti.PDCode.resolutionSplitMap","parent":2828,"result":true,"route":"../roadmap/tauceti/knot-theory/khovanov-homology-of-classical-links/index.html#tauceti-pdcode-resolutionsplitmap","scope":"tauceti/knot-theory/khovanov-homology-of-classical-links","status":"fully_proved","summary":"Comultiplication on the circle through `c`, splitting it into the two circles of the target through `c`, and identity on every other factor of the actual resolved-circle tensor. The one-circle source condition follows from the target count.","title":"TauCeti.PDCode.resolutionSplitMap"},{"data_url":"scopes/tauceti/knot-theory/other-results.json","id":"tauceti/knot-theory/other-results","kind":"scope","parent":2811,"result":false,"route":"../roadmap/tauceti/knot-theory/other-results/index.html","scope":"tauceti/knot-theory/other-results","status":"fully_proved","summary":"Results named in the rest of Tau Ceti's `TauCeti.KnotTheory` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Other results"},{"id":"tauceti/knot-theory/other-results/eulers-formula-for-a-pd-code","kind":"node","lean":"TauCeti.PDCode.eulerChar_toPermutationTriple","parent":2839,"result":true,"route":"../roadmap/tauceti/knot-theory/other-results/index.html#eulers-formula-for-a-pd-code","scope":"tauceti/knot-theory/other-results","status":"fully_proved","summary":"The underlying graph has `n` vertices and `2 * n` edges, so its Euler characteristic is the number of faces less the number of crossings.","title":"Euler's formula for a PD-code"},{"id":"tauceti/knot-theory/other-results/euler-poincare-for-khovanov-homology","kind":"node","lean":"TauCeti.OrientedPDCode.homologyEulerChar_khovanovComplex, TauCeti.PDCode.homologyEulerChar_khovanovBracket","parent":2839,"result":true,"route":"../roadmap/tauceti/knot-theory/other-results/index.html#euler-poincare-for-khovanov-homology","scope":"tauceti/knot-theory/other-results","status":"fully_proved","summary":"over a Noetherian ring with the rank-nullity property, such as `\u2124` or a field, the homology Euler characteristic of the quantum component `j` of the normalized Khovanov complex is the coefficient of `q\u02b2` in the enhanced state sum, the signed count of enhanced states of quantum degree `j`.","title":"Euler\u2013Poincar\u00e9 for Khovanov homology"},{"id":"tauceti/knot-theory/other-results/khovanovs-theorem","kind":"node","lean":"TauCeti.OrientedPDCode.khovanovEulerChar_eq_mul_eval\u2082_jonesPolynomial","parent":2839,"result":true,"route":"../roadmap/tauceti/knot-theory/other-results/index.html#khovanovs-theorem","scope":"tauceti/knot-theory/other-results","status":"fully_proved","summary":"the graded Euler characteristic of the Khovanov homology of a nonempty planar oriented code is `(q + q\u207b\u00b9)` times its Jones polynomial at `t ^ (1/2) = -q`, over a Noetherian ring with the rank-nullity property, such as `\u2124` or a field. The hypothesis `0 < componentCount` says that the diagram is nonempty; the empty diagram has graded Euler characteristic `1` (`TauCeti.OrientedPDCode.khovanovEulerChar_empty`).","title":"Khovanov's theorem"},{"id":"tauceti/knot-theory/other-results/levines-theorem-gives-a-boundary-that-is-not-s3","kind":"node","lean":"TauCeti.LevineHomologyConcordanceTheorem.exists_isEmpty_diffeomorph_sphere","parent":2839,"result":true,"route":"../roadmap/tauceti/knot-theory/other-results/index.html#levines-theorem-gives-a-boundary-that-is-not-s3","scope":"tauceti/knot-theory/other-results","status":"fully_proved","summary":"Assuming Levine's theorem, some smooth compact contractible `4`-manifold `X` has as boundary a closed `3`-manifold `Y` not diffeomorphic to `S\u00b3`. A diffeomorphism `e : Y \u2243\u2098 S\u00b3` would make the knot `K` of Levine's theorem homology concordant to the knot `e \u2218 K` in `S\u00b3` (`TauCeti.SmoothEmbedding.homologyConcordant_transDiffeomorph`).","title":"Levine's theorem gives a boundary that is not `S\u00b3`"},{"data_url":"scopes/tauceti/linear-algebra.json","id":"tauceti/linear-algebra","kind":"scope","parent":1095,"result":false,"route":"../roadmap/tauceti/linear-algebra/index.html","scope":"tauceti/linear-algebra","status":"fully_proved","summary":"Results in Tau Ceti's `TauCeti.LinearAlgebra` modules, and the roadmaps whose articles are mostly there: by roadmap, then by area.","title":"Linear Algebra"},{"data_url":"scopes/tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions.json","id":"tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions","kind":"scope","parent":2844,"result":false,"route":"../roadmap/tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions/index.html","scope":"tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Algebraic codes and code-lattice constructions"},{"id":"tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions/the-macwilliams-identity","kind":"node","lean":"Submodule.natCard_mul_weightEnumerator_euclideanDual","parent":2845,"result":true,"route":"../roadmap/tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions/index.html#the-macwilliams-identity","scope":"tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions","status":"fully_proved","summary":"for a linear code `C` over a finite field `F` with `q` elements, `#C \u00b7 W_{C\u22a5}(X, Y) = W_C(X + (q - 1) Y, X - Y)` in `\u2124[X, Y]`.","title":"The MacWilliams identity"},{"id":"tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions/tauceti-binarycode-istypeii-eight-dvd-card","kind":"node","lean":"TauCeti.BinaryCode.IsTypeII.eight_dvd_card","parent":2845,"result":true,"route":"../roadmap/tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions/index.html#tauceti-binarycode-istypeii-eight-dvd-card","scope":"tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions","status":"fully_proved","summary":"A Type II binary code has length divisible by eight.","title":"TauCeti.BinaryCode.IsTypeII.eight_dvd_card"},{"id":"tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions/tauceti-binarygolay-weightenumerator-code","kind":"node","lean":"TauCeti.BinaryGolay.weightEnumerator_code","parent":2845,"result":true,"route":"../roadmap/tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions/index.html#tauceti-binarygolay-weightenumerator-code","scope":"tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions","status":"fully_proved","summary":"The homogeneous weight enumerator is `X\u00b2\u2074 + 759 X\u00b9\u2076Y\u2078 + 2576 X\u00b9\u00b2Y\u00b9\u00b2 + 759 X\u2078Y\u00b9\u2076 + Y\u00b2\u2074`.","title":"TauCeti.BinaryGolay.weightEnumerator_code"},{"id":"tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions/gluing-2-z24-along-the-golay-code-gives-an-even","kind":"node","lean":"TauCeti.BinaryGolay.isEven_ofIsotropicSubgroup_codeInZeroLatticeDiscriminantGroup","parent":2845,"result":true,"route":"../roadmap/tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions/index.html#gluing-2-z24-along-the-golay-code-gives-an-even","scope":"tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.BinaryGolay.isEven_ofIsotropicSubgroup_codeInZeroLatticeDiscriminantGroup`.","title":"Gluing `2 \u2124\u00b2\u2074` along the Golay code gives an even lattice, because the glue is quadratic-isotropic"},{"id":"tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions/gluing-2-z24-along-the-golay-code-gives-a-unimod","kind":"node","lean":"TauCeti.BinaryGolay.isUnimodular_ofIsotropicSubgroup_codeInZeroLatticeDiscriminantGroup","parent":2845,"result":true,"route":"../roadmap/tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions/index.html#gluing-2-z24-along-the-golay-code-gives-a-unimod","scope":"tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.BinaryGolay.isUnimodular_ofIsotropicSubgroup_codeInZeroLatticeDiscriminantGroup`.","title":"Gluing `2 \u2124\u00b2\u2074` along the Golay code gives a unimodular lattice, because the glue is Lagrangian"},{"id":"tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions/the-golay-root-lattice-is-a124","kind":"node","lean":"TauCeti.BinaryGolay.rootLatticeIsometry","parent":2845,"result":true,"route":"../roadmap/tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions/index.html#the-golay-root-lattice-is-a124","scope":"tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions","status":"fully_proved","summary":"The zero-code lattice generated by all Golay roots is isometric to the orthogonal sum of twenty-four copies of the existing `A\u2081` lattice. The ambient isometry doubles every coordinate.","title":"The Golay root lattice is `A\u2081\u00b2\u2074`"},{"id":"tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions/over-z-2-r-with-r-1-a-self-orthogonal-code-has-a","kind":"node","lean":"TauCeti.ConstructionA.isEven_and_isUnimodular_integralLattice_iff_isTypeII","parent":2845,"result":true,"route":"../roadmap/tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions/index.html#over-z-2-r-with-r-1-a-self-orthogonal-code-has-a","scope":"tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.ConstructionA.isEven_and_isUnimodular_integralLattice_iff_isTypeII`.","title":"Over `\u2124/2^r` with `r \u2265 1`, a self-orthogonal code has an even unimodular Construction A lattice exactly when it is a Type II code"},{"id":"tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions/the-discriminant-bilinear-module-of-a-constructi","kind":"node","lean":"TauCeti.ConstructionA.discriminantBilinearOrthogonalQuotientIsometry","parent":2845,"result":true,"route":"../roadmap/tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions/index.html#the-discriminant-bilinear-module-of-a-constructi","scope":"tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions","status":"fully_proved","summary":"For a self-orthogonal additive code `C` over `\u2124/m`, the discriminant bilinear module of `P_m(C)` is the orthogonal quotient of `C` in the coordinate alphabet `(\u2124/m)^\u03b9`.","title":"The discriminant bilinear module of a Construction A lattice is `C\u22a5 / C`"},{"id":"tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions/gluing-along-a-code-gives-a-unimodular-lattice-e","kind":"node","lean":"TauCeti.ConstructionA.isUnimodular_ofIsotropicSubgroup_codeInZeroLatticeDiscriminantGroup_iff","parent":2845,"result":true,"route":"../roadmap/tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions/index.html#gluing-along-a-code-gives-a-unimodular-lattice-e","scope":"tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions","status":"fully_proved","summary":"For an even modulus and a quadratically isotropic code `C`, the overlattice of the zero-code lattice `m \u2124^\u03b9` glued along the discriminant subgroup of `C` is unimodular exactly when `C` equals its orthogonal complement in the coordinate alphabet `(\u2124/m)^\u03b9`.","title":"Gluing along a code gives a unimodular lattice exactly when the code is Lagrangian"},{"id":"tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions/set-weightenumerator","kind":"node","lean":"Set.weightEnumerator","parent":2845,"result":true,"route":"../roadmap/tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions/index.html#set-weightenumerator","scope":"tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions","status":"fully_proved","summary":"The homogeneous Hamming weight enumerator `W_C(X, Y) = \u2211_w A_w(C) X^(n-w) Y^w`, where `n` is the length, `A_w(C)` is the weight distribution, and the variables `0, 1` are `X, Y`.","title":"Set.weightEnumerator"},{"id":"tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions/tauceti-constructiona-lattice","kind":"node","lean":"TauCeti.ConstructionA.lattice","parent":2845,"result":true,"route":"../roadmap/tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions/index.html#tauceti-constructiona-lattice","scope":"tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions","status":"fully_proved","summary":"The rational carrier of Construction A: the integer vectors reducing to the additive code `C` modulo `m`.","title":"TauCeti.ConstructionA.lattice"},{"id":"tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions/tauceti-monomialautequiveuclideandual","kind":"node","lean":"TauCeti.monomialAutEquivEuclideanDual","parent":2845,"result":true,"route":"../roadmap/tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions/index.html#tauceti-monomialautequiveuclideandual","scope":"tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions","status":"fully_proved","summary":"The monomial automorphism groups of a linear code and of its Euclidean dual are isomorphic, by taking contragredients.","title":"TauCeti.monomialAutEquivEuclideanDual"},{"id":"tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions/tauceti-ismonomialequivalent-hammingmindist-eq","kind":"node","lean":"TauCeti.IsMonomialEquivalent.hammingMinDist_eq","parent":2845,"result":true,"route":"../roadmap/tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions/index.html#tauceti-ismonomialequivalent-hammingmindist-eq","scope":"tauceti/linear-algebra/algebraic-codes-and-code-lattice-constructions","status":"fully_proved","summary":"Monomially equivalent codes have the same minimum distance.","title":"TauCeti.IsMonomialEquivalent.hammingMinDist_eq"},{"data_url":"scopes/tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome.json","id":"tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome","kind":"scope","parent":2844,"result":false,"route":"../roadmap/tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome/index.html","scope":"tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Clifford bundles, spin structures, and Dirac geometry"},{"id":"tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome/cliffordalgebra-spingroupequivneg","kind":"node","lean":"CliffordAlgebra.spinGroupEquivNeg","parent":2859,"result":true,"route":"../roadmap/tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome/index.html#cliffordalgebra-spingroupequivneg","scope":"tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome","status":"fully_proved","summary":"The Spin groups of `Q` and `-Q` are isomorphic, through Mathlib's identification `evenEquivEvenNeg` of their even Clifford algebras. A product of two vectors `\u03b9 Q a * \u03b9 Q b` goes to `-(\u03b9 (-Q) a * \u03b9 (-Q) b)`, and the vector actions agree.","title":"CliffordAlgebra.spinGroupEquivNeg"},{"id":"tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome/cliffordalgebra-realcliffordspingroupequivneg","kind":"node","lean":"CliffordAlgebra.realCliffordSpinGroupEquivNeg","parent":2859,"result":true,"route":"../roadmap/tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome/index.html#cliffordalgebra-realcliffordspingroupequivneg","scope":"tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome","status":"fully_proved","summary":"The Spin group of signature `(p, q)`, transported into the Clifford algebra of the negated signature `(q, p)`. For `p = n` and `q = 0` this is the compact Spin group `Spin(n)` in the Clifford algebra of the negative-definite coordinate form `realCliffordForm 0 n`.","title":"CliffordAlgebra.realCliffordSpinGroupEquivNeg"},{"id":"tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome/the-degree-scaling-by-2-intertwines-the-native-a","kind":"node","lean":"TauCeti.AlmostComplexStructure.degreeScaling_comp_spinAction_eq_fockAction_comp","parent":2859,"result":true,"route":"../roadmap/tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome/index.html#the-degree-scaling-by-2-intertwines-the-native-a","scope":"tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome","status":"fully_proved","summary":"`T \u2218 spinAction a = fockAction a \u2218 T`, where `T` multiplies `\u22c0^p W` by `\u221a2 ^ p` (`TauCeti.ExteriorAlgebra.degreeScaling_of_mem_exteriorPower`).","title":"The degree scaling by `\u221a2` intertwines the native and the normalized Fock actions"},{"id":"tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome/the-normalized-fock-action-of-a-real-vector","kind":"node","lean":"TauCeti.AlmostComplexStructure.fockAction_\u03b9_one_tmul","parent":2859,"result":true,"route":"../roadmap/tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome/index.html#the-normalized-fock-action-of-a-real-vector","scope":"tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome","status":"fully_proved","summary":"`c(v)` is `\u221a2` times exterior multiplication by `v^(1,0)` plus `\u221a2 / 2` times contraction against `QuadraticMap.polar (Q.baseChange \u2102) \u00b7 v^(0,1)`. In the geometric convention `Q (v) = -g (v, v)` that contraction is `-2 \u03b9(v^(0,1))` for the contraction `\u03b9` against `g_\u2102`, so this is `c(v) = \u221a2 (\u03b5(v^(1,0)) - \u03b9(v^(0,1)))`.","title":"The normalized Fock action of a real vector"},{"id":"tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome/every-module-over-a-end-k-s-is-s-tensored-with-i","kind":"node","lean":"TauCeti.isotypicTensorEquivOfBijective","parent":2859,"result":true,"route":"../roadmap/tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome/index.html#every-module-over-a-end-k-s-is-s-tensored-with-i","scope":"tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome","status":"fully_proved","summary":"Evaluation `s \u2297 T \u21a6 T s` is an isomorphism of `A`-modules `S \u2297[k] (S \u2192\u2097[A] M) \u2243\u2097[A] M`, with `A` acting on the left factor.","title":"Every module over `A \u2245 End_k S` is `S` tensored with its multiplicity space"},{"id":"tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome/the-scalar-part-is-the-normalized-regular-trace","kind":"node","lean":"CliffordAlgebra.trace_mulLeft_eq_two_pow_mul_scalarTrace","parent":2859,"result":true,"route":"../roadmap/tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome/index.html#the-scalar-part-is-the-normalized-regular-trace","scope":"tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome","status":"fully_proved","summary":"over a field in which `2` is invertible, for every quadratic form on a finite-dimensional space, `trace (x * \u00b7) = 2 ^ finrank K V * scalarTrace Q x`. Every such form has an orthogonal basis, so `CliffordAlgebra.trace_mulLeft_eq_finrank_mul_scalarTrace` applies.","title":"The scalar part is the normalized regular trace"},{"id":"tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome/positivity-of-the-trace-pairing","kind":"node","lean":"CliffordAlgebra.scalarTrace_reverse_mul_self_nonneg","parent":2859,"result":true,"route":"../roadmap/tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome/index.html#positivity-of-the-trace-pairing","scope":"tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome","status":"fully_proved","summary":"if `Q` is nonnegative on an orthogonal basis, then `\u03c4 (reverse x * x) \u2265 0` for every `x`.","title":"Positivity of the trace pairing"},{"id":"tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome/faithfulness-of-the-trace-pairing","kind":"node","lean":"CliffordAlgebra.scalarTrace_reverse_mul_self_eq_zero_iff","parent":2859,"result":true,"route":"../roadmap/tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome/index.html#faithfulness-of-the-trace-pairing","scope":"tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome","status":"fully_proved","summary":"if `Q` is positive on an orthogonal basis, then `\u03c4 (reverse x * x) = 0` only for `x = 0`.","title":"Faithfulness of the trace pairing"},{"id":"tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome/the-polarization-of-a-complexified-quadratic-spa","kind":"node","lean":"TauCeti.AlmostComplexStructure.spinPolarizationData","parent":2859,"result":true,"route":"../roadmap/tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome/index.html#the-polarization-of-a-complexified-quadratic-spa","scope":"tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome","status":"fully_proved","summary":"For a nondegenerate quadratic form `Q` on a finite-dimensional real vector space and an almost complex structure `J` preserving `Q`, the `i`- and `-i`-eigenspaces of the complexification of `J` are the two isotropic summands of a polarization of `Q.baseChange \u2102`, with no orthogonal remainder. Its pairing field identifies the `-i`-eigenspace with the dual of the `i`-eigenspace through the polar form.","title":"The polarization of a complexified quadratic space by a complex structure"},{"id":"tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome/the-sesquilinear-complexification-of-a-real-bili","kind":"node","lean":"LinearMap.BilinForm.sesqComplexification","parent":2859,"result":true,"route":"../roadmap/tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome/index.html#the-sesquilinear-complexification-of-a-real-bili","scope":"tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome","status":"fully_proved","summary":"the form `h (x, y) = B_\u2102 (conj x, y)` on `\u2102 \u2297[\u211d] V`, conjugate-linear in the first argument and linear in the second, with `h (z \u2297 v, w \u2297 u) = conj z * w * B (v, u)` (`LinearMap.BilinForm.sesqComplexification_tmul_tmul`). For a Euclidean metric it is the induced positive Hermitian form on the complexification.","title":"The sesquilinear complexification of a real bilinear form `B`"},{"id":"tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome/the-hermitian-form-of-a-j-invariant-symplectic-f","kind":"node","lean":"TauCeti.SymplecticForm.Invariant.complexAssociatedForm_eq_two_mul_sesqComplexification","parent":2859,"result":true,"route":"../roadmap/tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome/index.html#the-hermitian-form-of-a-j-invariant-symplectic-f","scope":"tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome","status":"fully_proved","summary":"with `g = \u03c9 (\u00b7, J \u00b7)` and `h` its sesquilinear complexification, `\u03c9 (v, J w) + i \u03c9 (v, w) = 2 h (v^(1,0), w^(1,0))`.","title":"The Hermitian form of a `J`-invariant symplectic form is twice the pullback of the induced Hermitian form along `v \u21a6 v^(1,0)`"},{"id":"tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome/cliffordalgebra-conjstar","kind":"node","lean":"CliffordAlgebra.conjStar","parent":2859,"result":true,"route":"../roadmap/tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome/index.html#cliffordalgebra-conjstar","scope":"tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome","status":"fully_proved","summary":"The conjugate-linear Clifford involution of the complexified Clifford algebra: Clifford conjugation `star = reverse \u2218 involute` followed by complex conjugation of the coefficients. It is an involutive anti-automorphism (`CliffordAlgebra.conjStar_mul`, `CliffordAlgebra.conjStar_conjStar`) negating the real vectors.","title":"CliffordAlgebra.conjStar"},{"id":"tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome/cliffordalgebra-scalartrace","kind":"node","lean":"CliffordAlgebra.scalarTrace","parent":2859,"result":true,"route":"../roadmap/tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome/index.html#cliffordalgebra-scalartrace","scope":"tauceti/linear-algebra/clifford-bundles-spin-structures-and-dirac-geome","status":"fully_proved","summary":"The scalar part of an element of a Clifford algebra: its degree-zero component under the module isomorphism `CliffordAlgebra.equivExterior` with the exterior algebra.","title":"CliffordAlgebra.scalarTrace"},{"data_url":"scopes/tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat.json","id":"tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat","kind":"scope","parent":2844,"result":false,"route":"../roadmap/tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat/index.html","scope":"tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Integral lattices, discriminant forms, and overlattices"},{"id":"tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat/the-e8-root-lattice-is-isometric-to-the-spinor-g","kind":"node","lean":"TauCeti.IntegralLattice.typeE\u2088IsometryD8Plus","parent":2873,"result":true,"route":"../roadmap/tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat/index.html#the-e8-root-lattice-is-isometric-to-the-spinor-g","scope":"tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat","status":"fully_proved","summary":"The underlying rational equivalence sends the `i`-th simple root of `E\u2088` to the `i`-th vector of Bourbaki's plate VII in Conway--Sloane coordinates.","title":"The `E\u2088` root lattice is isometric to the spinor glue lattice `D\u2088\u207a`"},{"id":"tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat/even-overlattices-correspond-to-quadratic-isotro","kind":"node","lean":"TauCeti.IntegralLattice.evenIntermediateCarrierOrderIsoIsotropicSubgroup","parent":2873,"result":true,"route":"../roadmap/tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat/index.html#even-overlattices-correspond-to-quadratic-isotro","scope":"tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat","status":"fully_proved","summary":"For an even lattice, the intermediate-carrier order isomorphism restricts to the even carriers on one side and the quadratic-isotropic subgroups of the discriminant group on the other.","title":"Even overlattices correspond to quadratic-isotropic subgroups"},{"id":"tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat/the-standard-model-is-isometric-to-the-discrimin","kind":"node","lean":"TauCeti.IntegralLattice.checkerboardDiscriminantQuadraticIsometry","parent":2873,"result":true,"route":"../roadmap/tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat/index.html#the-standard-model-is-isometric-to-the-discrimin","scope":"tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat","status":"fully_proved","summary":"The standard model is isometric to the discriminant quadratic module of an even-rank checkerboard lattice, by the identification carrying `(1, 0)` to the vector class and `(0, 1)` to the spinor class.","title":"The standard model is isometric to the discriminant quadratic module of an even-rank checkerboard lattice"},{"id":"tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat/positive-definite-unimodular-lattices-of-rank-at","kind":"node","lean":"TauCeti.IntegralLattice.IsPosDef.nonempty_isometry_of_isUnimodular_of_finrank_le_nine","parent":2873,"result":true,"route":"../roadmap/tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat/index.html#positive-definite-unimodular-lattices-of-rank-at","scope":"tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat","status":"fully_proved","summary":"Such a lattice is the standard lattice `I\u2099`, the root lattice `E\u2088`, or `I\u2081 \u2295 E\u2088`.","title":"Positive definite unimodular lattices of rank at most nine"},{"id":"tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat/e8-is-the-unique-even-unimodular-positive-defini","kind":"node","lean":"TauCeti.IntegralLattice.IsPosDef.nonempty_isometry_typeE\u2088RootLattice_of_isEven","parent":2873,"result":true,"route":"../roadmap/tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat/index.html#e8-is-the-unique-even-unimodular-positive-defini","scope":"tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.IntegralLattice.IsPosDef.nonempty_isometry_typeE\u2088RootLattice_of_isEven`.","title":"`E\u2088` is the unique even unimodular positive definite lattice of rank eight"},{"id":"tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat/the-reflections-in-the-240-roots-generate-the-is","kind":"node","lean":"TauCeti.IntegralLattice.weylGroup_typeE\u2088RootLattice_eq_top","parent":2873,"result":true,"route":"../roadmap/tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat/index.html#the-reflections-in-the-240-roots-generate-the-is","scope":"tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat","status":"fully_proved","summary":"its Weyl group is the whole of `O(E\u2088)`.","title":"The reflections in the 240 roots generate the isometry group of the E\u2088 lattice"},{"id":"tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat/tauceti-integrallattice","kind":"node","lean":"TauCeti.IntegralLattice","parent":2873,"result":true,"route":"../roadmap/tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat/index.html#tauceti-integrallattice","scope":"tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat","status":"fully_proved","summary":"An integral symmetric lattice in a rational vector space.","title":"TauCeti.IntegralLattice"},{"id":"tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat/tauceti-integrallattice-discriminantquadraticmod","kind":"node","lean":"TauCeti.IntegralLattice.discriminantQuadraticModule","parent":2873,"result":true,"route":"../roadmap/tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat/index.html#tauceti-integrallattice-discriminantquadraticmod","scope":"tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat","status":"fully_proved","summary":"The discriminant group of an even nondegenerate integral lattice, equipped with its canonical half-norm quadratic map and discriminant polar pairing.","title":"TauCeti.IntegralLattice.discriminantQuadraticModule"},{"id":"tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat/a-positive-definite-lattice-is-i-k-plus-a-lattic","kind":"node","lean":"TauCeti.IntegralLattice.IsPosDef.unitNormDecomposition","parent":2873,"result":true,"route":"../roadmap/tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat/index.html#a-positive-definite-lattice-is-i-k-plus-a-lattic","scope":"tauceti/linear-algebra/integral-lattices-discriminant-forms-and-overlat","status":"fully_proved","summary":"The standard lattice `I_k` is carried onto the unit-norm sublattice, with `k` its rank, and the second summand is the complement `unitNormComplement L`.","title":"A positive definite lattice is `I_k` plus a lattice without vectors of norm one"},{"data_url":"scopes/tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi.json","id":"tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi","kind":"scope","parent":2844,"result":false,"route":"../roadmap/tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi/index.html","scope":"tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Niemeier glue data and explicit Leech constructions"},{"id":"tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi/the-leech-lattice","kind":"node","lean":"TauCeti.leechLattice","parent":2883,"result":true,"route":"../roadmap/tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi/index.html#the-leech-lattice","scope":"tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi","status":"fully_proved","summary":"The Leech lattice, glued from the Golay Construction B lattice along the order-two subgroup generated by the odd shift `(-3/2, 1/2, \u2026, 1/2)`. Its carrier is the union of the Construction B lattice and its translate by the shift, by `mem_leechLattice_carrier_iff`.","title":"The Leech lattice"},{"id":"tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi/the-leech-lattice-is-unimodular-being-glued-alon","kind":"node","lean":"TauCeti.isUnimodular_leechLattice","parent":2883,"result":true,"route":"../roadmap/tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi/index.html#the-leech-lattice-is-unimodular-being-glued-alon","scope":"tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.isUnimodular_leechLattice`.","title":"The Leech lattice is unimodular, being glued along a Lagrangian subgroup"},{"id":"tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi/the-leech-lattice-is-even","kind":"node","lean":"TauCeti.isEven_leechLattice","parent":2883,"result":true,"route":"../roadmap/tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi/index.html#the-leech-lattice-is-even","scope":"tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.isEven_leechLattice`.","title":"The Leech lattice is even"},{"id":"tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi/the-carrier-decomposition-of-the-leech-lattice","kind":"node","lean":"TauCeti.mem_leechLattice_carrier_iff","parent":2883,"result":true,"route":"../roadmap/tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi/index.html#the-carrier-decomposition-of-the-leech-lattice","scope":"tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi","status":"fully_proved","summary":"a vector lies in the Leech lattice exactly when it or its difference with the odd shift lies in the Golay Construction B lattice.","title":"The carrier decomposition of the Leech lattice"},{"id":"tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi/the-norm-bound-for-the-leech-lattice-in-congruen","kind":"node","lean":"TauCeti.BinaryGolay.thirtyTwo_le_sum_sq","parent":2883,"result":true,"route":"../roadmap/tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi/index.html#the-norm-bound-for-the-leech-lattice-in-congruen","scope":"tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi","status":"fully_proved","summary":"Let `m \u2208 {0, 1}` and let `b` be an integer vector reducing to a Golay codeword modulo two, and put `a = 2b + m \ud835\udfd9`. If `\u2211 a\u1d62 \u2261 4m (mod 8)` and `a \u2260 0`, then `\u2211 a\u1d62\u00b2 \u2265 32`. These are the congruence conditions that describe the Leech lattice in the coordinates `x = a / 2`, in which the halved dot product of `x` with itself is `\u2211 a\u1d62\u00b2 / 8`; so the bound says that such an `x` has norm at least `4`.","title":"The norm bound for the Leech lattice in congruence coordinates"},{"id":"tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi/the-minimum-norm-of-the-discriminant-class-label","kind":"node","lean":"TauCeti.IntegralLattice.isLeast_form_self_typeADiscriminantGroupEquiv","parent":2883,"result":true,"route":"../roadmap/tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi/index.html#the-minimum-norm-of-the-discriminant-class-label","scope":"tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi","status":"fully_proved","summary":"The class `typeADiscriminantGroupEquiv n a = a.val \u2022 \u03c9\u2081` of `A\u2099` has minimum norm `a.val (n + 1 - a.val) / (n + 1)`, the shortest norm of the Conway\u2013Sloane glue label `[a]` of an `A\u2099` component.","title":"The minimum norm of the discriminant class labelled `a : ZMod (n + 1)`"},{"id":"tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi/tauceti-integrallattice-isleast-form-self-orthog","kind":"node","lean":"TauCeti.IntegralLattice.isLeast_form_self_orthogonalSum","parent":2883,"result":true,"route":"../roadmap/tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi/index.html#tauceti-integrallattice-isleast-form-self-orthog","scope":"tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi","status":"fully_proved","summary":"Attained least norms in two discriminant cosets add to the attained least norm of the corresponding coset of the orthogonal sum.","title":"TauCeti.IntegralLattice.isLeast_form_self_orthogonalSum"},{"id":"tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi/the-positive-root-lattice-of-a-simply-laced-dynk","kind":"node","lean":"TauCeti.DynkinType.rootLattice","parent":2883,"result":true,"route":"../roadmap/tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi/index.html#the-positive-root-lattice-of-a-simply-laced-dynk","scope":"tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi","status":"fully_proved","summary":"the integral lattice on `Fin t.rank \u2192 \u211a` whose Gram matrix in the standard basis is the Cartan matrix `t.cartanMatrix`. The standard coordinate vectors are the simple roots, in Bourbaki numbering.","title":"The positive root lattice of a simply-laced Dynkin type `t`"},{"id":"tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi/the-root-lattice-of-type-dn-is-the-checkerboard","kind":"node","lean":"TauCeti.DynkinType.rootLatticeIsometryCheckerboard","parent":2883,"result":true,"route":"../roadmap/tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi/index.html#the-root-lattice-of-type-dn-is-the-checkerboard","scope":"tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi","status":"fully_proved","summary":"For `4 \u2264 n`, the isometry sends the `i`-th simple root to the `i`-th Bourbaki simple root of the Conway--Sloane coordinate model `{x \u2208 \u2124\u207f | \u2211 x\u1d62 even}`; both bases have Gram matrix `CartanMatrix.D n`.","title":"The root lattice of type `D\u2099` is the checkerboard lattice"},{"id":"tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi/construction-b","kind":"node","lean":"TauCeti.ConstructionB.integralLattice","parent":2883,"result":true,"route":"../roadmap/tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi/index.html#construction-b","scope":"tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi","status":"fully_proved","summary":"Construction B of a self-orthogonal binary linear code, as an integral lattice: the Construction A vectors whose coordinate sum is divisible by four, with the halved dot product.","title":"Construction B"},{"id":"tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi/construction-b-vectors-have-norm-at-least-four","kind":"node","lean":"TauCeti.ConstructionB.four_le_form_self","parent":2883,"result":true,"route":"../roadmap/tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi/index.html#construction-b-vectors-have-norm-at-least-four","scope":"tauceti/linear-algebra/niemeier-glue-data-and-explicit-leech-constructi","status":"fully_proved","summary":"If every nonzero codeword of `C` has weight at least eight, every nonzero vector of the Construction B carrier has Construction A norm, the dot product halved, at least `4`.","title":"Construction B vectors have norm at least four"},{"data_url":"scopes/tauceti/linear-algebra/orthogonal-and-spin-groups.json","id":"tauceti/linear-algebra/orthogonal-and-spin-groups","kind":"scope","parent":2844,"result":false,"route":"../roadmap/tauceti/linear-algebra/orthogonal-and-spin-groups/index.html","scope":"tauceti/linear-algebra/orthogonal-and-spin-groups","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Orthogonal and spin groups"},{"id":"tauceti/linear-algebra/orthogonal-and-spin-groups/cliffordalgebra-range-spintospecialorthogonal-eq","kind":"node","lean":"CliffordAlgebra.range_spinToSpecialOrthogonal_eq_ker_spinorNorm","parent":2895,"result":true,"route":"../roadmap/tauceti/linear-algebra/orthogonal-and-spin-groups/index.html#cliffordalgebra-range-spintospecialorthogonal-eq","scope":"tauceti/linear-algebra/orthogonal-and-spin-groups","status":"fully_proved","summary":"The image of the Spin action on `SO(Q)` is exactly the kernel of the spinor norm.","title":"CliffordAlgebra.range_spinToSpecialOrthogonal_eq_ker_spinorNorm"},{"id":"tauceti/linear-algebra/orthogonal-and-spin-groups/spin-is-the-even-unitary-group-in-dimensions-one","kind":"node","lean":"CliffordAlgebra.evenUnitaryGroup_le_lipschitzGroup","parent":2895,"result":true,"route":"../roadmap/tauceti/linear-algebra/orthogonal-and-spin-groups/index.html#spin-is-the-even-unitary-group-in-dimensions-one","scope":"tauceti/linear-algebra/orthogonal-and-spin-groups","status":"fully_proved","summary":"For a nondegenerate quadratic space of dimension between one and five, the even unitary carrier lies in the Lipschitz group. Dimension zero is excluded, and the exclusion is not decoration: there the Lipschitz group is trivial while the even unitary carrier is `\u03bc\u2082`. The bound five is sharp, by `CliffordAlgebra.not_evenUnitaryGroup_le_lipschitzGroup_of_six_le_finrank`.","title":"Spin is the even unitary group in dimensions one to five"},{"id":"tauceti/linear-algebra/orthogonal-and-spin-groups/the-spin-group-of-three-hyperbolic-planes-is-sl4","kind":"node","lean":"TauCeti.hyperbolicSixSpinEquivSpecialLinear","parent":2895,"result":true,"route":"../roadmap/tauceti/linear-algebra/orthogonal-and-spin-groups/index.html#the-spin-group-of-three-hyperbolic-planes-is-sl4","scope":"tauceti/linear-algebra/orthogonal-and-spin-groups","status":"fully_proved","summary":"This is `TauCeti.spinGroupEquivSpecialLinearGroupOfFinrankEqSix` for the canonical split polarization, so a Spin element goes to the matrix of its even half-spin action in `TauCeti.spinSixHalfSpinStandardBasis`. By `TauCeti.coe_hyperbolicSixSpinEquivSpecialLinear_apply` it is the restriction of `TauCeti.hyperbolicSixEvenUnitaryEquivGeneralLinear` to Spin, so `Spin(Q)` is the determinant-one subgroup of the\u2026","title":"The Spin group of three hyperbolic planes is `SL\u2084(K)`"},{"id":"tauceti/linear-algebra/orthogonal-and-spin-groups/split-spin5-is-sp4","kind":"node","lean":"CliffordAlgebra.exists_spinGroupEquivSymplecticGroup_of_finrank_eq_five","parent":2895,"result":true,"route":"../roadmap/tauceti/linear-algebra/orthogonal-and-spin-groups/index.html#split-spin5-is-sp4","scope":"tauceti/linear-algebra/orthogonal-and-spin-groups","status":"fully_proved","summary":"For a nondegenerate quadratic form on a five-dimensional space whose even Clifford algebra splits, `C\u2080 \u2243 M\u2084(K)`, the group `Spin(Q)` is isomorphic to the symplectic group `Sp\u2084(K)`.","title":"Split `Spin\u2085` is `Sp\u2084`"},{"id":"tauceti/linear-algebra/orthogonal-and-spin-groups/spin5-is-a-twisted-form-of-sp4","kind":"node","lean":"CliffordAlgebra.exists_injective_spinGroup_symplecticGroup_of_finrank_eq_five","parent":2895,"result":true,"route":"../roadmap/tauceti/linear-algebra/orthogonal-and-spin-groups/index.html#spin5-is-a-twisted-form-of-sp4","scope":"tauceti/linear-algebra/orthogonal-and-spin-groups","status":"fully_proved","summary":"For a nondegenerate quadratic form on a five-dimensional space and any separably closed extension `L / K`, there are a splitting `e : C\u2080 \u2297 L \u2243 M\u2084(L)` carrying reversal to the symplectic adjoint and an injective homomorphism `f : Spin(Q) \u2192 Sp\u2084(L)` with `e\u207b\u00b9 (f s)` the scalar extension of `s`. Its image consists of exactly those symplectic matrices `g` for which `e\u207b\u00b9 g` comes from `C\u2080`, so `Spin(Q)` is the group\u2026","title":"`Spin\u2085` is a twisted form of `Sp\u2084`"},{"id":"tauceti/linear-algebra/orthogonal-and-spin-groups/tauceti-quadraticmap-iscompact-orthogonalgroup-i","kind":"node","lean":"TauCeti.QuadraticMap.isCompact_orthogonalGroup_iff","parent":2895,"result":true,"route":"../roadmap/tauceti/linear-algebra/orthogonal-and-spin-groups/index.html#tauceti-quadraticmap-iscompact-orthogonalgroup-i","scope":"tauceti/linear-algebra/orthogonal-and-spin-groups","status":"fully_proved","summary":"The orthogonal group of a quadratic form on a finite-dimensional space over a locally compact nontrivially normed field is compact exactly when the form is anisotropic.","title":"TauCeti.QuadraticMap.isCompact_orthogonalGroup_iff"},{"id":"tauceti/linear-algebra/orthogonal-and-spin-groups/tauceti-quadraticmap-orthogonalcompactopens-disc","kind":"node","lean":"TauCeti.QuadraticMap.OrthogonalCompactOpens.discreteTopology_range_fullAdelicSpinDiagonal","parent":2895,"result":true,"route":"../roadmap/tauceti/linear-algebra/orthogonal-and-spin-groups/index.html#tauceti-quadraticmap-orthogonalcompactopens-disc","scope":"tauceti/linear-algebra/orthogonal-and-spin-groups","status":"fully_proved","summary":"Rational Spin points of a nondegenerate rational quadratic space form a discrete subgroup of the full adelic Spin group for every compatible compact-open reference family.","title":"TauCeti.QuadraticMap.OrthogonalCompactOpens.discreteTopology_range_fullAdelicSpinDiagonal"},{"id":"tauceti/linear-algebra/orthogonal-and-spin-groups/tauceti-quadraticmap-orthogonalcompactopens-not","kind":"node","lean":"TauCeti.QuadraticMap.OrthogonalCompactOpens.not_discreteTopology_range_finiteAdelicSpecialOrthogonalDiagonal","parent":2895,"result":true,"route":"../roadmap/tauceti/linear-algebra/orthogonal-and-spin-groups/index.html#tauceti-quadraticmap-orthogonalcompactopens-not","scope":"tauceti/linear-algebra/orthogonal-and-spin-groups","status":"fully_proved","summary":"For a nondegenerate isotropic rational quadratic space of dimension at least three, the rational diagonal is not discrete in finite adelic `SO`. In dimension two the transvection parameter quotient is zero, so the dimension hypothesis is essential to this argument.","title":"TauCeti.QuadraticMap.OrthogonalCompactOpens.not_discreteTopology_range_finiteAdelicSpecialOrthogonalDiagonal"},{"id":"tauceti/linear-algebra/orthogonal-and-spin-groups/tauceti-quadraticmap-orthogonalcompactopens-full","kind":"node","lean":"TauCeti.QuadraticMap.OrthogonalCompactOpens.fullAdelicSpin","parent":2895,"result":true,"route":"../roadmap/tauceti/linear-algebra/orthogonal-and-spin-groups/index.html#tauceti-quadraticmap-orthogonalcompactopens-full","scope":"tauceti/linear-algebra/orthogonal-and-spin-groups","status":"fully_proved","summary":"The full adelic Spin group `Spin(V)(\ud835\udd38)`, with its unrestricted real factor.","title":"TauCeti.QuadraticMap.OrthogonalCompactOpens.fullAdelicSpin"},{"id":"tauceti/linear-algebra/orthogonal-and-spin-groups/tauceti-quadraticmap-orthogonalcompactopens-away","kind":"node","lean":"TauCeti.QuadraticMap.OrthogonalCompactOpens.awayAdelicSpin","parent":2895,"result":true,"route":"../roadmap/tauceti/linear-algebra/orthogonal-and-spin-groups/index.html#tauceti-quadraticmap-orthogonalcompactopens-away","scope":"tauceti/linear-algebra/orthogonal-and-spin-groups","status":"fully_proved","summary":"The Spin group `Spin(V)(\ud835\udd38^S)` of the adeles away from `S`: the restricted product over the primes outside `S` of the local Spin groups relative to the reference subgroups `U.spin p`.","title":"TauCeti.QuadraticMap.OrthogonalCompactOpens.awayAdelicSpin"},{"id":"tauceti/linear-algebra/orthogonal-and-spin-groups/tauceti-quadraticmap-orthogonalcompactopens-fini","kind":"node","lean":"TauCeti.QuadraticMap.OrthogonalCompactOpens.finiteAdelicSpinDoubleCoset","parent":2895,"result":true,"route":"../roadmap/tauceti/linear-algebra/orthogonal-and-spin-groups/index.html#tauceti-quadraticmap-orthogonalcompactopens-fini","scope":"tauceti/linear-algebra/orthogonal-and-spin-groups","status":"fully_proved","summary":"The finite adelic Spin double-coset set `Spin(V)(\u211a) \\ Spin(V)(\ud835\udd38_f) / \u220f_p U_p^{Spin}`.","title":"TauCeti.QuadraticMap.OrthogonalCompactOpens.finiteAdelicSpinDoubleCoset"},{"data_url":"scopes/tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t.json","id":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t","kind":"scope","parent":2844,"result":false,"route":"../roadmap/tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/index.html","scope":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Schubert geometry, Demazure sections and flag K-theory"},{"id":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/tauceti-plpath-lower-eq-some-iff","kind":"node","lean":"TauCeti.PLPath.lower_eq_some_iff","parent":2907,"result":true,"route":"../roadmap/tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/index.html#tauceti-plpath-lower-eq-some-iff","scope":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t","status":"fully_proved","summary":"Littelmann's root operators are partial inverses: lowering `p` gives `q` exactly when raising `q` gives `p`.","title":"TauCeti.PLPath.lower_eq_some_iff"},{"id":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/the-stabilizer-of-a-dominant-weight-is-a-standar","kind":"node","lean":"RootPairing.parabolic_wallIndices_eq_stabilizer","parent":2907,"result":true,"route":"../roadmap/tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/index.html#the-stabilizer-of-a-dominant-weight-is-a-standar","scope":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t","status":"fully_proved","summary":"the one generated by the simple reflections in the walls through it.","title":"The stabilizer of a dominant weight is a standard parabolic subgroup"},{"id":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/a-cover-of-w-w-has-a-unique-positive-root-label","kind":"node","lean":"TauCeti.existsUnique_isPos_reflection_smul_eq_of_minimalCosetRepCovBy","parent":2907,"result":true,"route":"../roadmap/tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/index.html#a-cover-of-w-w-has-a-unique-positive-root-label","scope":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t","status":"fully_proved","summary":"If `y` is covered by `x` in the Bruhat order of the minimal coset representatives of the stabilizer of the dominant weight `\u03bb`, there is exactly one positive root `\u03b2` with `y \u03bb = s_\u03b2 (x \u03bb)`.","title":"A cover of `W / W_\u03bb` has a unique positive root label"},{"id":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/the-bond-of-a-cover-of-w-w-is-a-positive-integer","kind":"node","lean":"TauCeti.exists_nat_reflectionBond_eq_of_minimalCosetRepCovBy","parent":2907,"result":true,"route":"../roadmap/tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/index.html#the-bond-of-a-cover-of-w-w-is-a-positive-integer","scope":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t","status":"fully_proved","summary":"The bond of a cover of `W / W_\u03bb` is a positive integer when the dominant weight `\u03bb` is integral, that is when every simple coroot takes an integer value on `\u03bb`.","title":"The bond of a cover of `W / W_\u03bb` is a positive integer"},{"id":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/the-localization-identity-of-the-demazure-operat","kind":"node","lean":"RootPairing.one_sub_single_neg_root_mul_demazureOperator","parent":2907,"result":true,"route":"../roadmap/tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/index.html#the-localization-identity-of-the-demazure-operat","scope":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t","status":"fully_proved","summary":"`(1 - e^{-\u03b1\u1d62}) D\u1d62(f) = f - e^{-\u03b1\u1d62} s\u1d62(f)`, where `s\u1d62` reindexes `\u2124[M]` along the reflection of `M`. Equivalently, `D\u1d62(e^x) = (e^x - e^{-\u03b1\u1d62} e^{s\u1d62 x}) / (1 - e^{-\u03b1\u1d62})`.","title":"The localization identity of the Demazure operator"},{"id":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/the-demazure-operator-is-idempotent-di2-di","kind":"node","lean":"RootPairing.demazureOperator_demazureOperator","parent":2907,"result":true,"route":"../roadmap/tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/index.html#the-demazure-operator-is-idempotent-di2-di","scope":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t","status":"fully_proved","summary":"Formalized in Tau Ceti as `RootPairing.demazureOperator_demazureOperator`.","title":"The Demazure operator is idempotent, `D\u1d62\u00b2 = D\u1d62`"},{"id":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/tauceti-normalcrystal-tensor-eq-max","kind":"node","lean":"TauCeti.NormalCrystal.tensor_\u03b5_eq_max","parent":2907,"result":true,"route":"../roadmap/tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/index.html#tauceti-normalcrystal-tensor-eq-max","scope":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t","status":"fully_proved","summary":"Kashiwara's form of the raising string length of a tensor product: `\u03b5 i (x\u2081 \u2297 x\u2082) = max (\u03b5 i x\u2081) (\u03b5 i x\u2082 - \u27e8wt x\u2081, \u03b1_i^\u2228\u27e9)`.","title":"TauCeti.NormalCrystal.tensor_\u03b5_eq_max"},{"id":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/tauceti-normalcrystal","kind":"node","lean":"TauCeti.NormalCrystal","parent":2907,"result":true,"route":"../roadmap/tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/index.html#tauceti-normalcrystal","scope":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t","status":"fully_proved","summary":"A normal crystal over the root datum `P` with base `b`: a weight map, partial raising and lowering operators `e i`, `f i` indexed by the simple roots, and natural string lengths `\u03b5 i`, `\u03c6 i`, satisfying Kashiwara's crystal axioms together with normality (`\u03b5 i x` and `\u03c6 i x` are the lengths of the `e i`- and `f i`-strings through `x`). The value `none` of `e i` or `f i` stands for Kashiwara's `0`.","title":"TauCeti.NormalCrystal"},{"id":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/tauceti-normalcrystal-tensor","kind":"node","lean":"TauCeti.NormalCrystal.tensor","parent":2907,"result":true,"route":"../roadmap/tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/index.html#tauceti-normalcrystal-tensor","scope":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t","status":"fully_proved","summary":"The tensor product `B\u2081 \u2297 B\u2082` of two normal crystals, on the product type, with Kashiwara's tie convention: on a tie `\u03c6 i x\u2081 = \u03b5 i x\u2082` the raising operator acts on the left factor and the lowering operator on the right factor.","title":"TauCeti.NormalCrystal.tensor"},{"id":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/tauceti-plpath-lower","kind":"node","lean":"TauCeti.PLPath.lower","parent":2907,"result":true,"route":"../roadmap/tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/index.html#tauceti-plpath-lower","scope":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t","status":"fully_proved","summary":"Littelmann's lowering operator `f_\u03b1` on paths, for the root `\u03b1 = x` and the coroot pairing `f = \u27e8-, \u03b1^\u2228\u27e9`. Let `h` be the minimal height, `t\u2080` the last time at which the height equals `h` and `t\u2081` the first time from which on the height is at least `h + 1`. Between `t\u2080` and `t\u2081`, the pieces along which the height increases and stays at most every later height are reflected by `Module.preReflection x f`, the\u2026","title":"TauCeti.PLPath.lower"},{"id":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/tauceti-plpath-raise","kind":"node","lean":"TauCeti.PLPath.raise","parent":2907,"result":true,"route":"../roadmap/tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/index.html#tauceti-plpath-raise","scope":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t","status":"fully_proved","summary":"Littelmann's raising operator `e_\u03b1` on paths, the conjugate of `TauCeti.PLPath.lower` by the reversal of paths. Let `h` be the minimal height, `t\u2081` the first time at which the height equals `h` and `t\u2080` the last time up to which the height is at least `h + 1`. Between `t\u2080` and `t\u2081`, the pieces along which the height decreases and stays at most every earlier height are reflected by `Module.preReflection x f`, the\u2026","title":"TauCeti.PLPath.raise"},{"id":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/rootpairing-reflectionbond","kind":"node","lean":"RootPairing.reflectionBond","parent":2907,"result":true,"route":"../roadmap/tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/index.html#rootpairing-reflectionbond","scope":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t","status":"fully_proved","summary":"The bond of two weights related by a reflection in a positive root: if `y = s_\u03b2 x` for a positive root `\u03b2`, it is `\u27e8y, \u03b2^\u2228\u27e9 = -\u27e8x, \u03b2^\u2228\u27e9`. For `y \u2260 x` and a reduced root pairing the positive root `\u03b2` is unique (`TauCeti.eq_of_isPos_of_reflection_eq`), so the bond is well defined (`RootPairing.reflectionBond_eq_coroot'`). If no reflection in a positive root sends `x` to `y`, the bond is `0`. For a cover `y \u22d6 x` of\u2026","title":"RootPairing.reflectionBond"},{"id":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/coxetersystem-minimalcosetrepcovby","kind":"node","lean":"CoxeterSystem.MinimalCosetRepCovBy","parent":2907,"result":true,"route":"../roadmap/tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/index.html#coxetersystem-minimalcosetrepcovby","scope":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t","status":"fully_proved","summary":"`y` is covered by `x` in the Bruhat order of the parabolic quotient `W / W_J`, modelled on the minimal coset representatives: both are minimal coset representatives, `y < x`, and no minimal coset representative lies strictly between them.","title":"CoxeterSystem.MinimalCosetRepCovBy"},{"id":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/demazures-formula-on-a-basis-element","kind":"node","lean":"RootPairing.weylDenominator_mul_prod_map_demazureOperator_single, RootPairing.weylDenominator_mul_prod_map_demazureOperator","parent":2907,"result":true,"route":"../roadmap/tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/index.html#demazures-formula-on-a-basis-element","scope":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t","status":"fully_proved","summary":"`D_{w\u2080}(e^\u03bb)` is Weyl's quotient. For a reduced word `\u03c9` of the longest element `w\u2080` and a weight `\u03bb`,","title":"Demazure's formula on a basis element"},{"id":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/the-weyl-dimension-formula-tauceti-sum-coeff-mul","kind":"node","lean":"TauCeti.sum_coeff_mul_prod_coroot'_weylVector_eq, TauCeti.sum_coeff_mul_prod_form_weylVector_root_eq","parent":2907,"result":true,"route":"../roadmap/tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t/index.html#the-weyl-dimension-formula-tauceti-sum-coeff-mul","scope":"tauceti/linear-algebra/schubert-geometry-demazure-sections-and-flag-k-t","status":"fully_proved","summary":"If `f \u00b7 \u0394 = N(\u03bb)` and `g \u00b7 \u0394 = N(0)` in `\u2124[M]`, then the sums of the coefficients of `f` and `g` satisfy","title":"The Weyl dimension formula (`TauCeti.sum_coeff_mul_prod_coroot'_weylVector_eq`)"},{"data_url":"scopes/tauceti/linear-algebra/finite-bilinear-module.json","id":"tauceti/linear-algebra/finite-bilinear-module","kind":"scope","parent":2844,"result":false,"route":"../roadmap/tauceti/linear-algebra/finite-bilinear-module/index.html","scope":"tauceti/linear-algebra/finite-bilinear-module","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.LinearAlgebra.FiniteBilinearModule` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Finite Bilinear Module"},{"id":"tauceti/linear-algebra/finite-bilinear-module/nikulins-comparison-theorem","kind":"node","lean":"TauCeti.FiniteQuadraticModule.nonempty_isometry_iff_bilinear_isometry_and_gaussSign_eq","parent":2923,"result":true,"route":"../roadmap/tauceti/linear-algebra/finite-bilinear-module/index.html#nikulins-comparison-theorem","scope":"tauceti/linear-algebra/finite-bilinear-module","status":"fully_proved","summary":"Two nondegenerate finite quadratic modules are isometric exactly when their polar bilinear modules are isometric and their Gauss-sum invariants agree.","title":"Nikulin's comparison theorem"},{"id":"tauceti/linear-algebra/finite-bilinear-module/nikulins-decomposition-of-nondegenerate-quadrati","kind":"node","lean":"TauCeti.FiniteQuadraticModule.exists_isometry_pi_oddCyclic","parent":2923,"result":true,"route":"../roadmap/tauceti/linear-algebra/finite-bilinear-module/index.html#nikulins-decomposition-of-nondegenerate-quadrati","scope":"tauceti/linear-algebra/finite-bilinear-module","status":"fully_proved","summary":"A nondegenerate finite quadratic module of odd order is isometric to an orthogonal sum of odd cyclic modules `q_\u03b8^{(p)}(p^k)`, each of odd prime-power order `m = p^k` with coefficient `\u03b8` prime to `m`.","title":"Nikulin's decomposition of nondegenerate quadratic modules of odd order"},{"id":"tauceti/linear-algebra/finite-bilinear-module/nikulins-generator-decomposition","kind":"node","lean":"TauCeti.FiniteQuadraticModule.exists_isometry_prod_pi_generators","parent":2923,"result":true,"route":"../roadmap/tauceti/linear-algebra/finite-bilinear-module/index.html#nikulins-generator-decomposition","scope":"tauceti/linear-algebra/finite-bilinear-module","status":"fully_proved","summary":"A nondegenerate finite quadratic module is isometric to an orthogonal sum of Nikulin's generators: odd cyclic modules `q_\u03b8^{(p)}(p^k)` of odd prime-power order `m = p^k` with `\u03b8` prime to `m`, dyadic cyclic modules `q_\u03b7^{(2)}(2^{k+1})` with `\u03b7` odd, and rank-two modules `u^{(2)}(2^{k+1})` and `v^{(2)}(2^{k+1})`.","title":"Nikulin's generator decomposition"},{"data_url":"scopes/tauceti/linear-algebra/integral-lattice.json","id":"tauceti/linear-algebra/integral-lattice","kind":"scope","parent":2844,"result":false,"route":"../roadmap/tauceti/linear-algebra/integral-lattice/index.html","scope":"tauceti/linear-algebra/integral-lattice","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.LinearAlgebra.IntegralLattice` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Integral Lattice"},{"id":"tauceti/linear-algebra/integral-lattice/milgrams-theorem","kind":"node","lean":"TauCeti.IntegralLattice.gaussSign_discriminantQuadraticModule_eq_sigPos_sub_sigNeg, TauCeti.IntegralLattice.gaussSign_discriminantQuadraticModule_rankOne","parent":2927,"result":true,"route":"../roadmap/tauceti/linear-algebra/integral-lattice/index.html#milgrams-theorem","scope":"tauceti/linear-algebra/integral-lattice","status":"fully_proved","summary":"The Gauss-sum invariant of the discriminant form of an even nondegenerate lattice is its signature `t\u208a - t\u208b` modulo `8`.","title":"Milgram's theorem"},{"id":"tauceti/linear-algebra/integral-lattice/minkowskis-bound-for-the-minimum","kind":"node","lean":"TauCeti.IntegralLattice.IsPosDef.minimum_le_mul_determinant_rpow, TauCeti.IntegralLattice.IsPosDef.minimum_pow_mul_pi_pow_le","parent":2927,"result":true,"route":"../roadmap/tauceti/linear-algebra/integral-lattice/index.html#minkowskis-bound-for-the-minimum","scope":"tauceti/linear-algebra/integral-lattice","status":"fully_proved","summary":"A positive definite integral lattice of rank `n` satisfies `min L \u2264 (4 / \u03c0) \u00b7 \u0393(n/2 + 1) ^ (2/n) \u00b7 (det L) ^ (1/n)`.","title":"Minkowski's bound for the minimum"},{"id":"tauceti/linear-algebra/integral-lattice/minkowskis-convex-body-theorem-for-a-realized-la","kind":"node","lean":"TauCeti.IntegralLattice.exists_ne_zero_mem_of_two_pow_mul_sqrt_determinant_lt","parent":2927,"result":true,"route":"../roadmap/tauceti/linear-algebra/integral-lattice/index.html#minkowskis-convex-body-theorem-for-a-realized-la","scope":"tauceti/linear-algebra/integral-lattice","status":"fully_proved","summary":"Let `\u03c6` realize a nondegenerate integral lattice `L` in a real inner product space `E` of dimension `n`. A convex set `K \u2286 E`, symmetric about the origin, of volume greater than `2\u207f \u00b7 \u221a(det L)` contains `\u03c6 y` for some nonzero `y \u2208 L`.","title":"Minkowski's convex body theorem for a realized lattice"},{"id":"tauceti/linear-algebra/integral-lattice/minkowskis-second-theorem","kind":"node","lean":"TauCeti.IntegralLattice.IsPosDef.prod_successiveMinimum_mul_pi_pow_le","parent":2927,"result":true,"route":"../roadmap/tauceti/linear-algebra/integral-lattice/index.html#minkowskis-second-theorem","scope":"tauceti/linear-algebra/integral-lattice","status":"fully_proved","summary":"The successive minima `\u03bb\u2080, \u2026, \u03bb_{n-1}` of a positive definite integral lattice of rank `n` satisfy `(\u220f\u1d62 \u03bb\u1d62) \u00b7 \u03c0\u207f \u2264 4\u207f \u00b7 \u0393(n/2 + 1)\u00b2 \u00b7 det L`, that is, `\u220f\u1d62 \u03bb\u1d62 \u2264 c\u2099\u207f \u00b7 det L` with `c\u2099 = (4 / \u03c0) \u00b7 \u0393(n/2 + 1) ^ (2/n)` the constant of Minkowski's bound for the minimum.","title":"Minkowski's second theorem"},{"data_url":"scopes/tauceti/linear-algebra/matrix.json","id":"tauceti/linear-algebra/matrix","kind":"scope","parent":2844,"result":false,"route":"../roadmap/tauceti/linear-algebra/matrix/index.html","scope":"tauceti/linear-algebra/matrix","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.LinearAlgebra.Matrix` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Matrix"},{"id":"tauceti/linear-algebra/matrix/the-bruhat-decomposition-counted","kind":"node","lean":"TauCeti.GL2Borel.card_doubleCosetQuotient_eq_two, TauCeti.GL2Borel.union_doubleCoset_weyl_eq_univ, TauCeti.GL2Borel.disjoint_doubleCoset_weyl","parent":2932,"result":true,"route":"../roadmap/tauceti/linear-algebra/matrix/index.html#the-bruhat-decomposition-counted","scope":"tauceti/linear-algebra/matrix","status":"fully_proved","summary":"the Borel subgroup of `GL\u2082` over a field has exactly two double cosets, indexed by the Weyl group `S\u2082` of `GL\u2082`. This is the input to the Mackey irreducibility criterion for the principal series.","title":"The Bruhat decomposition, counted"},{"id":"tauceti/linear-algebra/matrix/cauchy-binet-for-2-2-minors","kind":"node","lean":"Matrix.pairMinor_mul, Matrix.pairMinor_mul_fin_four","parent":2932,"result":true,"route":"../roadmap/tauceti/linear-algebra/matrix/index.html#cauchy-binet-for-2-2-minors","scope":"tauceti/linear-algebra/matrix","status":"fully_proved","summary":"A minor of a product is the sum, over the increasing pairs of the middle index type, of the products of the corresponding minors of the two factors.","title":"Cauchy--Binet for `2 \u00d7 2` minors"},{"id":"tauceti/linear-algebra/matrix/the-cauchy-binet-identity-for-the-cross-product","kind":"node","lean":"Matrix.mulVec_cross_mulVec","parent":2932,"result":true,"route":"../roadmap/tauceti/linear-algebra/matrix/index.html#the-cauchy-binet-identity-for-the-cross-product","scope":"tauceti/linear-algebra/matrix","status":"fully_proved","summary":"A matrix `M` sends the cross product of `u` and `w` to the cross product of their images up to the transposed adjugate of `M`: `(M u) \u2a2f\u2083 (M w) = (adj M)\u1d40 (u \u2a2f\u2083 w)`.","title":"The Cauchy--Binet identity for the cross product"},{"id":"tauceti/linear-algebra/matrix/cayley-hamilton-in-size-two","kind":"node","lean":"Matrix.sq_eq_trace_smul_sub_det_smul_one_fin_two, Matrix.Represents.aeval_charpoly_eq_zero, Matrix.pow_add_two_fin_two","parent":2932,"result":true,"route":"../roadmap/tauceti/linear-algebra/matrix/index.html#cayley-hamilton-in-size-two","scope":"tauceti/linear-algebra/matrix","status":"fully_proved","summary":"a `2 \u00d7 2` matrix `A` satisfies `A ^ 2 = (trace A) \u2022 A - (det A) \u2022 1`.","title":"Cayley\u2013Hamilton in size two"},{"id":"tauceti/linear-algebra/matrix/collatz-wielandt-bound","kind":"node","lean":"Matrix.le_of_smul_le_mulVec, Matrix.ofReal_le_spectralRadius_of_smul_le_mulVec","parent":2932,"result":true,"route":"../roadmap/tauceti/linear-algebra/matrix/index.html#collatz-wielandt-bound","scope":"tauceti/linear-algebra/matrix","status":"fully_proved","summary":"Let `A` be entrywise nonnegative with `A v \u2264 r v` for a vector `v` all of whose entries are strictly positive. If `t y \u2264 A y` entrywise for some nonzero nonnegative vector `y`, then `t \u2264 r`.","title":"Collatz\u2013Wielandt bound"},{"id":"tauceti/linear-algebra/matrix/gerstenhabers-theorem","kind":"node","lean":"Matrix.mem_closure_orbit_iff_forall_rank_pow_le, Matrix.mem_zeroLocus_vanishingIdeal_orbit_iff_forall_rank_pow_le","parent":2932,"result":true,"route":"../roadmap/tauceti/linear-algebra/matrix/index.html#gerstenhabers-theorem","scope":"tauceti/linear-algebra/matrix","status":"fully_proved","summary":"Over a field with continuous ring operations and closed points in which `0` is not isolated, such as `\u2102`, a matrix `Y` lies in the closure of the conjugacy class of a nilpotent matrix `X` exactly when `rank (Y ^ k) \u2264 rank (X ^ k)` for every `k`.","title":"Gerstenhaber's theorem"},{"id":"tauceti/linear-algebra/matrix/hadamards-bound-l1-row-form","kind":"node","lean":"Matrix.abv_det_le_prod_sum_abv, Matrix.abv_det_le_prod_sum_abv', Matrix.abs_det_le_prod_sum_abs', Matrix.abs_det_le_prod_sum_abs","parent":2932,"result":true,"route":"../roadmap/tauceti/linear-algebra/matrix/index.html#hadamards-bound-l1-row-form","scope":"tauceti/linear-algebra/matrix","status":"fully_proved","summary":"The absolute value of a determinant is at most the product of the `\u2113\u00b9` norms of the rows.","title":"Hadamard's bound, `\u2113\u00b9` row form"},{"id":"tauceti/linear-algebra/matrix/iwasawas-theorem-for-the-projective-symplectic-g","kind":"node","lean":"TauCeti.GLSymplecticFin.isSimpleGroup_quotient_center","parent":2932,"result":true,"route":"../roadmap/tauceti/linear-algebra/matrix/index.html#iwasawas-theorem-for-the-projective-symplectic-g","scope":"tauceti/linear-algebra/matrix","status":"fully_proved","summary":"In positive rank, the quotient `PSp_{2m}(K)` of the symplectic group by its centre is simple as soon as `Sp_{2m}(K)` is perfect.","title":"Iwasawa's theorem for the projective symplectic group"},{"id":"tauceti/linear-algebra/matrix/jacobis-complementary-minor-theorem","kind":"node","lean":"Matrix.det_mul_det_submatrix_adjugate","parent":2932,"result":true,"route":"../roadmap/tauceti/linear-algebra/matrix/index.html#jacobis-complementary-minor-theorem","scope":"tauceti/linear-algebra/matrix","status":"fully_proved","summary":"For `#I = #J = k`, the product of the determinant of `A` and the complementary minor of the adjugate, with rows `J\u1d9c` and columns `I\u1d9c`, is `(-1) ^ (\u2211 I + \u2211 J)` times `(det A) ^ (n - k)` times the minor `A[I, J]`.","title":"Jacobi's complementary minor theorem"},{"id":"tauceti/linear-algebra/matrix/jacobis-complementary-minor-identity","kind":"node","lean":"Matrix.det_submatrix_comp_inr_eq_sign_mul_det_mul_det_submatrix_comp_inl, Matrix.det_submatrix_inr_eq_sign_mul_det_mul_det_submatrix_inl","parent":2932,"result":true,"route":"../roadmap/tauceti/linear-algebra/matrix/index.html#jacobis-complementary-minor-identity","scope":"tauceti/linear-algebra/matrix","status":"fully_proved","summary":"Jacobi's complementary-minor identity for an arbitrary index type `n`, split into two groups of indices by an equivalence `e : m \u2295 p \u2243 n`.","title":"Jacobi's complementary-minor identity"},{"id":"tauceti/linear-algebra/matrix/jacobis-formula-by-rows","kind":"node","lean":"Matrix.sum_det_updateRow_eq_trace_adjugate_mul, Matrix.sum_det_updateCol_eq_trace_adjugate_mul","parent":2932,"result":true,"route":"../roadmap/tauceti/linear-algebra/matrix/index.html#jacobis-formula-by-rows","scope":"tauceti/linear-algebra/matrix","status":"fully_proved","summary":"Replacing the `i`-th row of `A` by the `i`-th row of `B` and summing the determinants over `i` gives `trace (adj A * B)`, the derivative of `det` at `A` in the direction `B`.","title":"Jacobi's formula, by rows"},{"id":"tauceti/linear-algebra/matrix/perron-frobenius-theorem-for-nonnegative-matrice","kind":"node","lean":"Matrix.exists_mulVec_eq_smul_of_nonneg, Matrix.IsIrreducible.rootMultiplicity_charpoly_eq_one, Matrix.IsPrimitive.eq_of_norm_eq_of_mulVec_eq_smul, Matrix.IsPrimitive.norm_lt_of_mem_spectrum, Matrix.IsPrimitive.norm_lt_of_mulVec_eq_smul, Matrix.eq_of_norm_eq_of_mulVec_eq_smul, Matrix.spectralRadius_eq_of_mulVec_eq_smul, Matrix.IsIrreducible.exists_mulVec_eq_smul, Matrix.exists_mulVec_eq_smul_of_pos","parent":2932,"result":true,"route":"../roadmap/tauceti/linear-algebra/matrix/index.html#perron-frobenius-theorem-for-nonnegative-matrice","scope":"tauceti/linear-algebra/matrix","status":"fully_proved","summary":"an entrywise nonnegative real square matrix has its spectral radius `r`, viewed as a complex matrix, as an eigenvalue, with an eigenvector whose entries are nonnegative and not all zero.","title":"Perron\u2013Frobenius theorem for nonnegative matrices"},{"id":"tauceti/linear-algebra/matrix/sherman-morrison-formula-over-a-commutative-ring","kind":"node","lean":"Matrix.inv_add_vecMulVec","parent":2932,"result":true,"route":"../roadmap/tauceti/linear-algebra/matrix/index.html#sherman-morrison-formula-over-a-commutative-ring","scope":"tauceti/linear-algebra/matrix","status":"fully_proved","summary":"The rank-one correction is the outer product of the updated column and row, divided by the scalar denominator.","title":"Sherman\u2013Morrison formula over a commutative ring"},{"id":"tauceti/linear-algebra/matrix/skolem-noether-for-matrix-algebras-over-a-local","kind":"node","lean":"Matrix.GeneralLinearGroup.innerAut_surjective","parent":2932,"result":true,"route":"../roadmap/tauceti/linear-algebra/matrix/index.html#skolem-noether-for-matrix-algebras-over-a-local","scope":"tauceti/linear-algebra/matrix","status":"fully_proved","summary":"every algebra automorphism of a matrix algebra over a commutative local semiring, in particular over a field, is inner.","title":"Skolem\u2013Noether for matrix algebras over a local semiring"},{"id":"tauceti/linear-algebra/matrix/sylvesters-rank-inequality","kind":"node","lean":"Matrix.rank_add_rank_le_rank_mul_add_card","parent":2932,"result":true,"route":"../roadmap/tauceti/linear-algebra/matrix/index.html#sylvesters-rank-inequality","scope":"tauceti/linear-algebra/matrix","status":"fully_proved","summary":"for an `m \u00d7 n` matrix `A` and an `n \u00d7 o` matrix `B` over a field, `rank A + rank B \u2264 rank (A * B) + n`. Equivalently, multiplying by `A` lowers the rank of `B` by at most the nullity of `A`.","title":"Sylvester's rank inequality"},{"id":"tauceti/linear-algebra/matrix/ultrametric-hadamard-bound-row-form","kind":"node","lean":"Matrix.abv_det_le_prod_of_le, Matrix.abv_det_le_prod_of_le'","parent":2932,"result":true,"route":"../roadmap/tauceti/linear-algebra/matrix/index.html#ultrametric-hadamard-bound-row-form","scope":"tauceti/linear-algebra/matrix","status":"fully_proved","summary":"For a nonarchimedean absolute value `abv`, if row `i` of `A` is bounded entrywise in `abv` by `b i`, then `abv A.det \u2264 \u220f i, b i`.","title":"Ultrametric Hadamard bound, row form"},{"id":"tauceti/linear-algebra/matrix/whiteheads-lemma","kind":"node","lean":"TauCeti.fromBlocksGL_inv_mem_elementarySubgroup","parent":2932,"result":true,"route":"../roadmap/tauceti/linear-algebra/matrix/index.html#whiteheads-lemma","scope":"tauceti/linear-algebra/matrix","status":"fully_proved","summary":"For an invertible matrix `u`, the block diagonal matrix `u \u2295 u\u207b\u00b9` is elementary: it is the product of six block unipotent matrices, `u \u2295 u\u207b\u00b9 = U(u) L(-u\u207b\u00b9) U(u) U(-1) L(1) U(-1)`.","title":"Whitehead's lemma"},{"data_url":"scopes/tauceti/linear-algebra/quadratic-form.json","id":"tauceti/linear-algebra/quadratic-form","kind":"scope","parent":2844,"result":false,"route":"../roadmap/tauceti/linear-algebra/quadratic-form/index.html","scope":"tauceti/linear-algebra/quadratic-form","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.LinearAlgebra.QuadraticForm` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Quadratic Form"},{"id":"tauceti/linear-algebra/quadratic-form/eulers-four-square-identity-for-a-quadratic-map","kind":"node","lean":"QuadraticMap.euler_four_squares","parent":2950,"result":true,"route":"../roadmap/tauceti/linear-algebra/quadratic-form/index.html#eulers-four-square-identity-for-a-quadratic-map","scope":"tauceti/linear-algebra/quadratic-form","status":"fully_proved","summary":"Multiplying `(x, y, z, w)` by the quaternion `a + bi + cj + dk` multiplies `Q x + Q y + Q z + Q w` by `a\u00b2 + b\u00b2 + c\u00b2 + d\u00b2`. For `Q = x \u21a6 x\u00b2` on a commutative ring this is `euler_four_squares`.","title":"Euler's four-square identity for a quadratic map"},{"id":"tauceti/linear-algebra/quadratic-form/frobenius-reciprocity-on-regular-form-classes","kind":"node","lean":"TauCeti.RegularFormClass.scharlauTransfer_mul_baseChange","parent":2950,"result":true,"route":"../roadmap/tauceti/linear-algebra/quadratic-form/index.html#frobenius-reciprocity-on-regular-form-classes","scope":"tauceti/linear-algebra/quadratic-form","status":"fully_proved","summary":"Transfer of a class over `L` multiplied by a scalar-extended class from `K` is the transfer multiplied by the original base class.","title":"Frobenius reciprocity on regular-form classes"},{"id":"tauceti/linear-algebra/quadratic-form/serres-connectivity-theorem-for-orthogonal-bases","kind":"node","lean":"Module.Basis.reflTransGen_isOrtho_isContiguous","parent":2950,"result":true,"route":"../roadmap/tauceti/linear-algebra/quadratic-form/index.html#serres-connectivity-theorem-for-orthogonal-bases","scope":"tauceti/linear-algebra/quadratic-form","status":"fully_proved","summary":"Orthogonal bases of a regular quadratic space in dimension at least three are joined by a finite chain whose every step consists of two orthogonal bases with a vector in common.","title":"Serre's connectivity theorem for orthogonal bases"},{"id":"tauceti/linear-algebra/quadratic-form/sylvesters-law-of-inertia-for-regular-forms","kind":"node","lean":"QuadraticForm.equivalent_iff_sigPos_eq_and_sigNeg_eq, QuadraticForm.equivalent_iff_finrank_eq_and_sigPos_eq_and_sigNeg_eq, TauCeti.RegularFormClass.eq_iff_rank_eq_and_signature_eq","parent":2950,"result":true,"route":"../roadmap/tauceti/linear-algebra/quadratic-form/index.html#sylvesters-law-of-inertia-for-regular-forms","scope":"tauceti/linear-algebra/quadratic-form","status":"fully_proved","summary":"two nondegenerate real quadratic forms are isometric exactly when their signatures agree.","title":"Sylvester's law of inertia for regular forms"},{"id":"tauceti/linear-algebra/quadratic-form/witt-decomposition","kind":"node","lean":"TauCeti.RegularFormClass.wittDecomposition, QuadraticForm.exists_equivalent_hyperbolicPresentation_prod, TauCeti.eq_of_nsmul_hyperbolicClass_add_eq, TauCeti.exists_nsmul_hyperbolicClass_add","parent":2950,"result":true,"route":"../roadmap/tauceti/linear-algebra/quadratic-form/index.html#witt-decomposition","scope":"tauceti/linear-algebra/quadratic-form","status":"fully_proved","summary":"a class is its Witt index worth of hyperbolic planes plus its anisotropic part.","title":"Witt decomposition"},{"id":"tauceti/linear-algebra/quadratic-form/witts-chain-theorem-in-its-nontrivial-range","kind":"node","lean":"TauCeti.diagonalChain_iff_equivalent","parent":2950,"result":true,"route":"../roadmap/tauceti/linear-algebra/quadratic-form/index.html#witts-chain-theorem-in-its-nontrivial-range","scope":"tauceti/linear-algebra/quadratic-form","status":"fully_proved","summary":"two diagonal forms of rank at least two are isometric if and only if their coefficient families are connected by a diagonal chain.","title":"Witt's chain theorem in its nontrivial range"},{"id":"tauceti/linear-algebra/quadratic-form/witts-extension-theorem-lam-i-4-9","kind":"node","lean":"QuadraticMap.IsometryEquiv.exists_extension","parent":2950,"result":true,"route":"../roadmap/tauceti/linear-algebra/quadratic-form/index.html#witts-extension-theorem-lam-i-4-9","scope":"tauceti/linear-algebra/quadratic-form","status":"fully_proved","summary":"An isometry between regular subspaces of a finite-dimensional quadratic space extends to an isometry of the whole space. No regularity hypothesis on the ambient form is needed.","title":"Witt's extension theorem (Lam I.4.9)"},{"data_url":"scopes/tauceti/linear-algebra/root-system.json","id":"tauceti/linear-algebra/root-system","kind":"scope","parent":2844,"result":false,"route":"../roadmap/tauceti/linear-algebra/root-system/index.html","scope":"tauceti/linear-algebra/root-system","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.LinearAlgebra.RootSystem` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Root System"},{"id":"tauceti/linear-algebra/root-system/the-cartan-killing-classification","kind":"node","lean":"RootPairing.existsUnique_nonempty_equiv_rationalRootSystem, TauCeti.existsUnique_dynkinType","parent":2958,"result":true,"route":"../roadmap/tauceti/linear-algebra/root-system/index.html#the-cartan-killing-classification","scope":"tauceti/linear-algebra/root-system","status":"fully_proved","summary":"Every irreducible reduced crystallographic finite root system over `\u211a` is isomorphic to the model root system `TauCeti.DynkinType.rationalRootSystem t` of a unique valid Dynkin type `t`.","title":"The Cartan-Killing classification"},{"id":"tauceti/linear-algebra/root-system/chevalleys-theorem-on-structure-constants","kind":"node","lean":"TauCeti.DynkinType.structureConstant_eq_natCast_or_eq_neg_natCast","parent":2958,"result":true,"route":"../roadmap/tauceti/linear-algebra/root-system/index.html#chevalleys-theorem-on-structure-constants","scope":"tauceti/linear-algebra/root-system","status":"fully_proved","summary":"If `\u03b1 + \u03b2` is a root, then `N_{\u03b1,\u03b2} = \u00b1(r + 1)`, where `r = chainBotCoeff \u03b1 \u03b2` is the largest integer with `\u03b2 - r \u03b1` a root of the pinned datum.","title":"Chevalley's theorem on structure constants"},{"id":"tauceti/linear-algebra/root-system/kostants-multiplicity-formula-stated-before-any","kind":"node","lean":"RootPairing.coeff_eq_kostantMultiplicity_of_mul_weylDenominator_eq_weylNumerator","parent":2958,"result":true,"route":"../roadmap/tauceti/linear-algebra/root-system/index.html#kostants-multiplicity-formula-stated-before-any","scope":"tauceti/linear-algebra/root-system","status":"fully_proved","summary":"an element of `\u2124[M]` whose product with the Weyl denominator is the Weyl numerator of `\u03bb` has `\u03bc`-th coefficient `\u2211_{w \u2208 W} sgn(w) P(w \u2b1d \u03bb - \u03bc)`.","title":"Kostant's multiplicity formula, stated before any module is named"},{"id":"tauceti/linear-algebra/root-system/tits-theorem-the-braid-relations-are-the-only-re","kind":"node","lean":"TauCeti.weylCoxeterHom_injective","parent":2958,"result":true,"route":"../roadmap/tauceti/linear-algebra/root-system/index.html#tits-theorem-the-braid-relations-are-the-only-re","scope":"tauceti/linear-algebra/root-system","status":"fully_proved","summary":"The canonical homomorphism from the Coxeter group of a base to the Weyl group is injective.","title":"Tits' theorem: the braid relations are the only relations among the simple reflections"},{"id":"tauceti/linear-algebra/root-system/the-weyl-denominator-identity","kind":"node","lean":"TauCeti.weylDenominator_eq_weylNumerator_zero, RootPairing.dotAlternatization_one","parent":2958,"result":true,"route":"../roadmap/tauceti/linear-algebra/root-system/index.html#the-weyl-denominator-identity","scope":"tauceti/linear-algebra/root-system","status":"fully_proved","summary":"the Weyl denominator is the Weyl numerator of the weight `0`,","title":"The Weyl denominator identity"},{"data_url":"scopes/tauceti/linear-algebra/other-results.json","id":"tauceti/linear-algebra/other-results","kind":"scope","parent":2844,"result":false,"route":"../roadmap/tauceti/linear-algebra/other-results/index.html","scope":"tauceti/linear-algebra/other-results","status":"fully_proved","summary":"Results named in the rest of Tau Ceti's `TauCeti.LinearAlgebra` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Other results"},{"id":"tauceti/linear-algebra/other-results/the-cauchy-binet-formula","kind":"node","lean":"TauCeti.det_mul_transpose_eq_sum_plucker_mul, Matrix.det_mul_transpose_eq_sum_plucker_sq, TauCeti.det_mul_conjTranspose_eq_sum_plucker_mul_star","parent":2964,"result":true,"route":"../roadmap/tauceti/linear-algebra/other-results/index.html#the-cauchy-binet-formula","scope":"tauceti/linear-algebra/other-results","status":"fully_proved","summary":"The determinant of the `p \u00d7 p` matrix of pairwise products `\u2211\u2096 x i k * y j k` is the sum over the `p`-element subsets `s` of `\u03b9` of the products of the Pl\u00fccker coordinates of `x` and `y` at `s`. For `p > #\u03b9` both sides vanish.","title":"The Cauchy\u2013Binet formula"},{"id":"tauceti/linear-algebra/other-results/cayley-hamilton-on-a-finite-projective-module","kind":"node","lean":"LinearMap.exists_aeval_reflect_projectiveCharpolyRev_eq_zero","parent":2964,"result":true,"route":"../roadmap/tauceti/linear-algebra/other-results/index.html#cayley-hamilton-on-a-finite-projective-module","scope":"tauceti/linear-algebra/other-results","status":"fully_proved","summary":"For some `n` bounding the degree of the reversed characteristic polynomial `Q` of `f`, the reflection `X ^ n \u00b7 Q(1/X)` of `Q` kills `f`; it is monic (`monic_reflect_projectiveCharpolyRev`).","title":"Cayley\u2013Hamilton on a finite projective module"},{"id":"tauceti/linear-algebra/other-results/the-choie-zagier-criterion-popa-zagier-lemma-1","kind":"node","lean":"TauCeti.End.one_sub_apply_mem_range_one_sub_mul_iff, TauCeti.End.one_sub_apply_mem_range_one_sub_mul_of_mem_sup","parent":2964,"result":true,"route":"../roadmap/tauceti/linear-algebra/other-results/index.html#the-choie-zagier-criterion-popa-zagier-lemma-1","scope":"tauceti/linear-algebra/other-results","status":"fully_proved","summary":"for `\u03c3 ^ 2 = 1` and `\u03c5 ^ 3 = 1` with `2` and `3` invertible and `ker (1 + \u03c3)`, `ker (1 + \u03c5 + \u03c5\u00b2)` disjoint, a vector `x` lies in `range (1 + \u03c3) \u2294 range (1 + \u03c5 + \u03c5\u00b2)` exactly when `(1 - \u03c3) x \u2208 range (1 - \u03c5 \u03c3)`.","title":"The Choie\u2013Zagier criterion (Popa\u2013Zagier, Lemma 1)"},{"id":"tauceti/linear-algebra/other-results/dedekinds-bound","kind":"node","lean":"TauCeti.natCard_monoidHom_le","parent":2964,"result":true,"route":"../roadmap/tauceti/linear-algebra/other-results/index.html#dedekinds-bound","scope":"tauceti/linear-algebra/other-results","status":"fully_proved","summary":"A finite monoid `G` has at most `#G` homomorphisms into a domain `M`, since they are linearly independent in the `M`-module `G \u2192 M` of rank `#G`.","title":"Dedekind's bound"},{"id":"tauceti/linear-algebra/other-results/eichlers-theorem","kind":"node","lean":"LinearMap.BilinForm.eq_orthComponents_iff","parent":2964,"result":true,"route":"../roadmap/tauceti/linear-algebra/other-results/index.html#eichlers-theorem","scope":"tauceti/linear-algebra/other-results","status":"fully_proved","summary":"For a reflexive positive definite integral form, a set of submodules consists of pairwise orthogonal orthogonally indecomposable submodules spanning `M` exactly when it is the set of orthogonal components. In particular such a decomposition exists and is unique.","title":"Eichler's theorem"},{"id":"tauceti/linear-algebra/other-results/eulers-identity-twisted-by-an-endomorphism","kind":"node","lean":"AlgHom.apply_derivationOfEnd_eq_mul_of_mem_homogeneousSubmodule","parent":2964,"result":true,"route":"../roadmap/tauceti/linear-algebra/other-results/index.html#eulers-identity-twisted-by-an-endomorphism","scope":"tauceti/linear-algebra/other-results","status":"fully_proved","summary":"If an algebra map `\u03c6` turns `f` into multiplication by `c` on generators, `\u03c6 (\u03b9 (f x)) = c * \u03c6 (\u03b9 x)`, then on a homogeneous element of degree `n` it turns the derivation induced by `f` into multiplication by `n * c`. For `f = 1` and `\u03c6` the identity this is Euler's identity `D p = n \u2022 p` for the Euler derivation `D`.","title":"Euler's identity, twisted by an endomorphism"},{"id":"tauceti/linear-algebra/other-results/exactness-in-popa-zagiers-3-theorem-2","kind":"node","lean":"TauCeti.End.mem_sup_range_one_add_mul_one_sub","parent":2964,"result":true,"route":"../roadmap/tauceti/linear-algebra/other-results/index.html#exactness-in-popa-zagiers-3-theorem-2","scope":"tauceti/linear-algebra/other-results","status":"fully_proved","summary":"for `\u03c3 ^ 2 = \u03c3' ^ 2 = 1` and `\u03c5 ^ 3 = \u03c5' ^ 3 = 1`, with `\u03c3` and `\u03c5` commuting with `\u03c3'` and `\u03c5'`, `2` and `3` invertible and `ker (1 + \u03c3')`, `ker (1 + \u03c5' + \u03c5'\u00b2)` disjoint: if `\u03be \u2208 range (1 - \u03c3') \u2294 range (1 - \u03c5')`, `(1 + \u03c3') \u03be \u2208 range (1 + \u03c5 + \u03c5\u00b2)` and `(1 + \u03c5' + \u03c5'\u00b2) \u03be \u2208 range (1 + \u03c3)`, then `\u03be \u2208 range ((1 + \u03c3) (1 - \u03c3')) \u2294 range ((1 + \u03c5 + \u03c5\u00b2) (1 - \u03c5'))`.","title":"Exactness in Popa\u2013Zagier's \u00a73 Theorem 2"},{"id":"tauceti/linear-algebra/other-results/the-graded-jacobi-identity-for-the-gerstenhaber","kind":"node","lean":"TauCeti.ReducedTensorWords.leibniz_gerstenhaberBracket","parent":2964,"result":true,"route":"../roadmap/tauceti/linear-algebra/other-results/index.html#the-graded-jacobi-identity-for-the-gerstenhaber","scope":"tauceti/linear-algebra/other-results","status":"fully_proved","summary":"bracketing with `F` of degree `p` is a graded derivation of the bracket of `g` of degree `q` and `h` of degree `r`, `[F, [g, h]] = [[F, g], h] + (-1)^(p * q) \u2022 [g, [F, h]]`.","title":"The graded Jacobi identity for the Gerstenhaber bracket, in Leibniz form"},{"id":"tauceti/linear-algebra/other-results/hermites-inequality","kind":"node","lean":"LinearMap.BilinForm.exists_ne_zero_three_pow_mul_pow_le_four_pow_mul_det","parent":2964,"result":true,"route":"../roadmap/tauceti/linear-algebra/other-results/index.html#hermites-inequality","scope":"tauceti/linear-algebra/other-results","status":"fully_proved","summary":"A symmetric positive definite integral bilinear form `B` on a free `\u2124`-module of rank `n \u2265 1` takes on some nonzero vector `x` a value with `3 ^ (n choose 2) \u00b7 B(x, x) ^ n \u2264 4 ^ (n choose 2) \u00b7 det B`, where `det B` is the Gram determinant of `B` in any basis. Equivalently `B(x, x) \u2264 (4 / 3) ^ ((n - 1) / 2) \u00b7 (det B) ^ (1 / n)`.","title":"Hermite's inequality"},{"id":"tauceti/linear-algebra/other-results/iwasawas-theorem-for-psl","kind":"node","lean":"Matrix.ProjectiveSpecialLinearGroup.isSimpleGroup_of_commutator_eq_top","parent":2964,"result":true,"route":"../roadmap/tauceti/linear-algebra/other-results/index.html#iwasawas-theorem-for-psl","scope":"tauceti/linear-algebra/other-results","status":"fully_proved","summary":"On at least two coordinates, the projective special linear group is simple as soon as the special linear group is perfect.","title":"Iwasawa's theorem for `PSL`"},{"id":"tauceti/linear-algebra/other-results/jacobis-formula-for-a-determinant-in-row-form","kind":"node","lean":"Matrix.sum_det_updateRow_mul_row","parent":2964,"result":true,"route":"../roadmap/tauceti/linear-algebra/other-results/index.html#jacobis-formula-for-a-determinant-in-row-form","scope":"tauceti/linear-algebra/other-results","status":"fully_proved","summary":"Rescale one row of a matrix entry by entry along a fixed vector of factors, and sum the resulting determinants over the rows: the answer is the determinant multiplied by the total of the factors.","title":"Jacobi's formula for a determinant, in row form"},{"id":"tauceti/linear-algebra/other-results/the-jacobian-criterion","kind":"node","lean":"SymmetricAlgebra.jacobian_ne_zero","parent":2964,"result":true,"route":"../roadmap/tauceti/linear-algebra/other-results/index.html#the-jacobian-criterion","scope":"tauceti/linear-algebra/other-results","status":"fully_proved","summary":"Over a domain `R` of characteristic zero, if every generator `\u03b9 (b j)` is algebraic over the subalgebra generated by the family `f`, then the Jacobian of `f` is nonzero. This applies to a system of basic invariants of a finite group, over which the polynomial algebra is integral.","title":"The Jacobian criterion"},{"id":"tauceti/linear-algebra/other-results/jacobsons-lemma-on-commutators","kind":"node","lean":"LinearMap.isNilpotent_lie_of_lie_lie_eq_zero","parent":2964,"result":true,"route":"../roadmap/tauceti/linear-algebra/other-results/index.html#jacobsons-lemma-on-commutators","scope":"tauceti/linear-algebra/other-results","status":"fully_proved","summary":"In characteristic zero, if `A` commutes with `\u2045A, B\u2046`, then `\u2045A, B\u2046` is nilpotent: each of its positive powers is a commutator with `A`, hence traceless.","title":"Jacobson's lemma on commutators"},{"id":"tauceti/linear-algebra/other-results/kolchins-common-fixed-vector-theorem","kind":"node","lean":"Representation.exists_common_fixed_vector_of_isUnipotent","parent":2964,"result":true,"route":"../roadmap/tauceti/linear-algebra/other-results/index.html#kolchins-common-fixed-vector-theorem","scope":"tauceti/linear-algebra/other-results","status":"fully_proved","summary":"If every element of a monoid acts unipotently on a nonzero finite-dimensional vector space over a field, then the monoid fixes a nonzero vector.","title":"Kolchin's common fixed vector theorem"},{"id":"tauceti/linear-algebra/other-results/laplaces-identity-bombieri-gubler-7-16","kind":"node","lean":"TauCeti.wedgeForm_plucker","parent":2964,"result":true,"route":"../roadmap/tauceti/linear-algebra/other-results/index.html#laplaces-identity-bombieri-gubler-7-16","scope":"tauceti/linear-algebra/other-results","status":"fully_proved","summary":"the wedge of `p` forms takes the value `det (l a (y b))` on the Pl\u00fccker vector of `p` vectors `y`.","title":"Laplace's identity (Bombieri\u2013Gubler (7.16))"},{"data_url":"scopes/tauceti/measure-theory.json","id":"tauceti/measure-theory","kind":"scope","parent":1095,"result":false,"route":"../roadmap/tauceti/measure-theory/index.html","scope":"tauceti/measure-theory","status":"fully_proved","summary":"Results in Tau Ceti's `TauCeti.MeasureTheory` modules, and the roadmaps whose articles are mostly there: by roadmap, then by area.","title":"Measure Theory"},{"data_url":"scopes/tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri.json","id":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri","kind":"scope","parent":2980,"result":false,"route":"../roadmap/tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/index.html","scope":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Euclidean isoperimetry, sharp inequalities, and rigidity"},{"id":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/tauceti-hasprekopaleindler-volume","kind":"node","lean":"TauCeti.hasPrekopaLeindler_volume","parent":2981,"result":true,"route":"../roadmap/tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/index.html#tauceti-hasprekopaleindler-volume","scope":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri","status":"fully_proved","summary":"The Pr\u00e9kopa\u2013Leindler inequality on a finite-dimensional real inner product space, such as `EuclideanSpace \u211d (Fin n)`: its volume has the Pr\u00e9kopa\u2013Leindler property at every weight `0 \u2264 \u03b8 \u2264 1`.","title":"TauCeti.hasPrekopaLeindler_volume"},{"id":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/brunn-minkowski-inequality","kind":"node","lean":"TauCeti.volume_rpow_add_volume_rpow_le","parent":2981,"result":true,"route":"../roadmap/tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/index.html#brunn-minkowski-inequality","scope":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri","status":"fully_proved","summary":"Brunn\u2013Minkowski inequality for nonempty Lebesgue-measurable sets in a positive-dimensional real inner product space. The sum has its outer volume and need not be measurable or have finite volume.","title":"Brunn\u2013Minkowski inequality"},{"id":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/equality-cases-of-multiplicative-brunn-minkowski","kind":"node","lean":"Convex.volume_smul_add_smul_eq_iff","parent":2981,"result":true,"route":"../roadmap/tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/index.html#equality-cases-of-multiplicative-brunn-minkowski","scope":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri","status":"fully_proved","summary":"Let `K` and `L` be compact convex sets of positive volume and `0 < \u03b8 < 1`. Then `volume ((1 - \u03b8) \u2022 K + \u03b8 \u2022 L) = volume K ^ (1 - \u03b8) * volume L ^ \u03b8` exactly when `L` is a translate of `K`.","title":"Equality cases of multiplicative Brunn\u2013Minkowski for convex bodies"},{"id":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/equality-in-the-area-energy-inequality","kind":"node","lean":"TauCeti.mul_integral_norm_sq_eq_four_pi_mul_abs_signedArea_iff","parent":2981,"result":true,"route":"../roadmap/tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/index.html#equality-in-the-area-energy-inequality","scope":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri","status":"fully_proved","summary":"For an absolutely continuous closed plane curve with square-integrable derivative, `(b - a) \u222b\u2090\u1d47 \u2016\u03b3'\u2016\u00b2 = 4\u03c0 |signedArea \u03b3 a b|` exactly when `\u03b3` is constant or traverses a circle once at constant angular speed, in either orientation.","title":"Equality in the area\u2013energy inequality"},{"id":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/ball-rigidity-from-the-sharp-first-variation-of","kind":"node","lean":"ConvexBody.hasDerivWithinAt_volume_real_add_smul_closedBall_rpow_iff_exists_eq_closedBall","parent":2981,"result":true,"route":"../roadmap/tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/index.html#ball-rigidity-from-the-sharp-first-variation-of","scope":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri","status":"fully_proved","summary":"A convex body `K` of positive volume has parallel-volume root `t \u21a6 volume (K + t \u2022 closedBall 0 1) ^ (1 / n)` with right derivative `volume (closedBall 0 1) ^ (1 / n)` at zero exactly when `K` is a closed ball of positive radius.","title":"Ball rigidity from the sharp first variation of parallel volume"},{"id":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/tauceti-steinersymmetrization","kind":"node","lean":"TauCeti.steinerSymmetrization","parent":2981,"result":true,"route":"../roadmap/tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/index.html#tauceti-steinersymmetrization","scope":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri","status":"fully_proved","summary":"Steiner symmetrization along the unit direction `e`, centered on its orthogonal hyperplane. The section of length `\u2113` is the open centered interval of radius `\u2113 / 2`; this includes empty intervals for `\u2113 = 0` and the whole line for `\u2113 = \u221e`.","title":"TauCeti.steinerSymmetrization"},{"id":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/tauceti-schwarzrearrangement","kind":"node","lean":"TauCeti.schwarzRearrangement","parent":2981,"result":true,"route":"../roadmap/tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/index.html#tauceti-schwarzrearrangement","scope":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri","status":"fully_proved","summary":"The Schwarz rearrangement of `f : E \u2192 \u211d\u22650\u221e` with respect to `\u03bc`: the radially nonincreasing function whose superlevel set `{t < f*}` is `{x | \u03bc (ball 0 \u2016x\u2016) < \u03bc {t < f}}`. For an additive Haar measure on a nontrivial finite-dimensional real normed space, this is the centered open ball of the same measure when `\u03bc {t < f}` is finite, and the whole space when it is infinite.","title":"TauCeti.schwarzRearrangement"},{"id":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/de-la-vallee-poussins-lemma","kind":"node","lean":"MeasureTheory.Integrable.exists_convexOn_superlinear_integrable","parent":2981,"result":true,"route":"../roadmap/tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/index.html#de-la-vallee-poussins-lemma","scope":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri","status":"fully_proved","summary":"Every integrable function `f` remains integrable after composing its norm with some convex, monotone function `\u03a6` with `\u03a6 0 = 0` that grows superlinearly, `\u03a6 t / t \u2192 \u221e`.","title":"De la Vall\u00e9e Poussin's lemma"},{"id":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/the-isoperimetric-inequality-on-the-line","kind":"node","lean":"TauCeti.Real.two_le_perimeter","parent":2981,"result":true,"route":"../roadmap/tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/index.html#the-isoperimetric-inequality-on-the-line","scope":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri","status":"fully_proved","summary":"A null-measurable set of positive finite measure in `\u211d` has perimeter at least two.","title":"The isoperimetric inequality on the line"},{"id":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/brunns-concavity-principle","kind":"node","lean":"Convex.concaveOn_volume_real_preimage_prodMk_rpow, Convex.concaveOn_volume_real_hyperplaneSection_rpow, Convex.mul_volume_hyperplaneSection_rpow_add_mul_volume_hyperplaneSection_rpow_le, Convex.mul_volume_preimage_prodMk_rpow_add_mul_volume_preimage_prodMk_rpow_le","parent":2981,"result":true,"route":"../roadmap/tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/index.html#brunns-concavity-principle","scope":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri","status":"fully_proved","summary":"if every fiber of a convex set has finite volume, the dimension root of the fiber volume is concave on the projection of the set.","title":"Brunn's concavity principle"},{"id":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/brunns-principle-in-log-concave-form","kind":"node","lean":"Convex.logConcave_measure_preimage_prodMk","parent":2981,"result":true,"route":"../roadmap/tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/index.html#brunns-principle-in-log-concave-form","scope":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri","status":"fully_proved","summary":"For a measurable convex set `s \u2286 E \u00d7 F` and a measure `\u03bd` with the Pr\u00e9kopa\u2013Leindler property at every weight, the measure of the slices, `x \u21a6 \u03bd (Prod.mk x \u207b\u00b9' s)`, is log-concave.","title":"Brunn's principle, in log-concave form"},{"id":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/brunn-minkowski-for-a-shell","kind":"node","lean":"TauCeti.finrank_mul_rpow_pow_mul_le_volume_real_sub","parent":2981,"result":true,"route":"../roadmap/tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/index.html#brunn-minkowski-for-a-shell","scope":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri","status":"fully_proved","summary":"If the `r`-neighbourhood `G + ball 0 r` of a nonempty null-measurable set `G` lies in a set `F` of finite volume, then in dimension `n \u2265 1` `volume F - volume G \u2265 n (volume G ^ (1 / n)) ^ (n - 1) r \u03c9_n ^ (1 / n)`.","title":"Brunn\u2013Minkowski for a shell"},{"id":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/de-la-vallee-poussins-criterion","kind":"node","lean":"TauCeti.unifIntegrable_of_lintegral_ofReal_le, TauCeti.exists_setLIntegral_enorm_le_mul_add","parent":2981,"result":true,"route":"../roadmap/tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/index.html#de-la-vallee-poussins-criterion","scope":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri","status":"fully_proved","summary":"A family of a.e. strongly measurable functions with `\u222b \u03a6(\u2016f i\u2016) \u2264 C < \u221e` for a superlinear `\u03a6` is uniformly integrable.","title":"De la Vall\u00e9e Poussin's criterion"},{"id":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/hardy-littlewood-rearrangement-inequality","kind":"node","lean":"TauCeti.lintegral_mul_le_lintegral_schwarzRearrangement_mul","parent":2981,"result":true,"route":"../roadmap/tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/index.html#hardy-littlewood-rearrangement-inequality","scope":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri","status":"fully_proved","summary":"For two nonnegative a.e.-measurable functions on a finite-dimensional real normed space, Schwarz rearrangement can only increase the integral of their product. The statement includes infinite integrals and dimension zero.","title":"Hardy--Littlewood rearrangement inequality"},{"id":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/prekopa-leindler-inequality-for-the-minimum-on-r","kind":"node","lean":"TauCeti.Real.mul_lintegral_add_mul_lintegral_le","parent":2981,"result":true,"route":"../roadmap/tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri/index.html#prekopa-leindler-inequality-for-the-minimum-on-r","scope":"tauceti/measure-theory/euclidean-isoperimetry-sharp-inequalities-and-ri","status":"fully_proved","summary":"If `f` and `g` have the same supremum and `h (c * x + d * y)` dominates `min (f x) (g y)` for all `x` and `y`, then `c \u222b f + d \u222b g \u2264 \u222b h`. The coefficients may vanish, and the common supremum and the integrals may be infinite.","title":"Pr\u00e9kopa\u2013Leindler inequality for the minimum on `\u211d`"},{"data_url":"scopes/tauceti/measure-theory/optimal-transport-and-wasserstein-geometry.json","id":"tauceti/measure-theory/optimal-transport-and-wasserstein-geometry","kind":"scope","parent":2980,"result":false,"route":"../roadmap/tauceti/measure-theory/optimal-transport-and-wasserstein-geometry/index.html","scope":"tauceti/measure-theory/optimal-transport-and-wasserstein-geometry","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Optimal transport and Wasserstein geometry"},{"id":"tauceti/measure-theory/optimal-transport-and-wasserstein-geometry/breniers-theorem","kind":"node","lean":"TauCeti.exists_isKantorovichOptimalTransportMap_gradient, TauCeti.IsOptimalCoupling.exists_eq_graphPlan_of_absolutelyContinuous, TauCeti.exists_isKantorovichOptimalTransportMap_gradient_and_gradient_fenchelConjugate","parent":2997,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport-and-wasserstein-geometry/index.html#breniers-theorem","scope":"tauceti/measure-theory/optimal-transport-and-wasserstein-geometry","status":"fully_proved","summary":"Let `\u03bc` and `\u03bd` be probability measures on a finite-dimensional real inner product space `E`, with `\u03bc` absolutely continuous with respect to an additive Haar measure, and suppose the optimal cost for `\u2016x - y\u2016 ^ 2 / 2` is finite, as it is for laws with finite second moment (`TauCeti.transportCost_norm_sub_sq_div_two_ne_top`). Then there is a convex, lower semicontinuous `u : E \u2192 EReal` that never takes the value\u2026","title":"Brenier's theorem"},{"id":"tauceti/measure-theory/optimal-transport-and-wasserstein-geometry/the-benamou-brenier-formula","kind":"node","lean":"TauCeti.isLeast_lintegral_isContinuityEquation, TauCeti.wassersteinEDist_rpow_eq_sInf_lintegral_isContinuityEquation","parent":2997,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport-and-wasserstein-geometry/index.html#the-benamou-brenier-formula","scope":"tauceti/measure-theory/optimal-transport-and-wasserstein-geometry","status":"fully_proved","summary":"For `1 \u2264 p < \u221e` and probability measures `\u03bc\u2080`, `\u03bc\u2081` on a finite-dimensional real normed space at finite Wasserstein distance, `W_p (\u03bc\u2080, \u03bc\u2081) ^ p` is the least `p`-action `\u222b\u2080\u00b9 \u222b \u2016v\u209c\u2016 ^ p d\u03bd\u209c dt` of a curve `t \u21a6 \u03bd\u209c` of finite measures, narrowly continuous on `[0, 1]` and joining `\u03bc\u2080` to `\u03bc\u2081`, solving the continuity equation with velocity `v` on `(0, 1)`. The minimum is attained by the displacement interpolation of an\u2026","title":"The Benamou\u2013Brenier formula"},{"id":"tauceti/measure-theory/optimal-transport-and-wasserstein-geometry/lisinis-superposition-principle-in-a-complete-se","kind":"node","lean":"TauCeti.exists_map_IccExtend_eq_lintegral_curveAction_eq","parent":2997,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport-and-wasserstein-geometry/index.html#lisinis-superposition-principle-in-a-complete-se","scope":"tauceti/measure-theory/optimal-transport-and-wasserstein-geometry","status":"fully_proved","summary":"Let `1 < p < \u221e` and let `\u03bc` be a curve in `P_p(X)`, absolutely continuous on `[a, b]` with finite `p`-action, where `a < b`. There is a probability measure `\u03b7` on `C([a, b], X)`, concentrated on absolutely continuous paths, whose law at each time `t \u2208 [a, b]` is `\u03bc t`, whose expected `p`-action is the `p`-action of `\u03bc`, and whose averaged speed is the metric speed of `\u03bc`: `|\u03bc'|(t) ^ p = \u222b |\u03b3'|(t) ^ p d\u03b7` at almost\u2026","title":"Lisini's superposition principle in a complete separable metric space"},{"id":"tauceti/measure-theory/optimal-transport-and-wasserstein-geometry/uniqueness-of-wasserstein-geodesics","kind":"node","lean":"TauCeti.WassersteinSpace.eqOn_of_isGeodesicSegment_of_ae_subsingleton","parent":2997,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport-and-wasserstein-geometry/index.html#uniqueness-of-wasserstein-geodesics","scope":"tauceti/measure-theory/optimal-transport-and-wasserstein-geometry","status":"fully_proved","summary":"Let `X` be a complete separable metric space and `1 < p < \u221e`. Suppose that every optimal coupling of `\u03bc` and `\u03bd` for the cost `d ^ p` is `\u03c0`, and that `\u03c0`-almost every pair of points of `X` is joined by at most one geodesic path. Then any two geodesic segments of `P_p (X)` from `\u03bc` to `\u03bd` agree on `[0, 1]`.","title":"Uniqueness of Wasserstein geodesics"},{"id":"tauceti/measure-theory/optimal-transport-and-wasserstein-geometry/mccanns-displacement-convexity-of-internal-energ","kind":"node","lean":"TauCeti.internalEnergy_map_smul_add_smul_gradient_le","parent":2997,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport-and-wasserstein-geometry/index.html#mccanns-displacement-convexity-of-internal-energ","scope":"tauceti/measure-theory/optimal-transport-and-wasserstein-geometry","status":"fully_proved","summary":"Let `E` be a real inner product space of finite dimension `n \u2265 1` with an additive Haar measure `\u03c1`, and let `U : \u211d \u2192 \u211d` satisfy `U 0 = 0` and McCann's condition: `s \u21a6 s\u207f U(s\u207b\u207f)` is convex and nonincreasing on `(0, \u221e)`. Let `f : E \u2192 EReal` be convex (convex real epigraph, never `\u22a5`), with real representative `F`, and let `\u03bc` be a \u03c3-finite measure, absolutely continuous with respect to `\u03c1` and carried by the\u2026","title":"McCann's displacement convexity of internal energies"},{"id":"tauceti/measure-theory/optimal-transport-and-wasserstein-geometry/tauceti-convexsection","kind":"node","lean":"TauCeti.convexSection","parent":2997,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport-and-wasserstein-geometry/index.html#tauceti-convexsection","scope":"tauceti/measure-theory/optimal-transport-and-wasserstein-geometry","status":"fully_proved","summary":"The section of `u` on `\u03a9` at `x`, with slope `\u2113` and strict height `h`.","title":"TauCeti.convexSection"},{"id":"tauceti/measure-theory/optimal-transport-and-wasserstein-geometry/tauceti-gromovwassersteinspace","kind":"node","lean":"TauCeti.GromovWassersteinSpace","parent":2997,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport-and-wasserstein-geometry/index.html#tauceti-gromovwassersteinspace","scope":"tauceti/measure-theory/optimal-transport-and-wasserstein-geometry","status":"fully_proved","summary":"Finite-size standard Borel measured kernels modulo weak isomorphism, with the full Gromov\u2013Wasserstein metric. The carrier universe is fixed, but is independent of the target.","title":"TauCeti.GromovWassersteinSpace"},{"id":"tauceti/measure-theory/optimal-transport-and-wasserstein-geometry/tauceti-metricmeasurespace-measurecontraction","kind":"node","lean":"TauCeti.MetricMeasureSpace.MeasureContraction","parent":2997,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport-and-wasserstein-geometry/index.html#tauceti-metricmeasurespace-measurecontraction","scope":"tauceti/measure-theory/optimal-transport-and-wasserstein-geometry","status":"fully_proved","summary":"The measure-contraction property `MCP(K, N)` of Ohta and Sturm, in the form of Cavalletti\u2013Mondino: for every point `o` of the support of the reference measure `m` and every measurable `A` with `0 < m(A) < \u221e`, some dynamic plan `\u03b7` concentrated on geodesics has law `m[|A] = m(A)\u207b\u00b9 m|_A` at time `0` and law `\u03b4_o` at time `1`, and for every time `t` the law at time `t` of `\u03b7` weighted by `\u03c4_{K,N}^{(1-t)}(d(\u03b3\u2080, \u03b3\u2081))^N`\u2026","title":"TauCeti.MetricMeasureSpace.MeasureContraction"},{"id":"tauceti/measure-theory/optimal-transport-and-wasserstein-geometry/the-benamou-brenier-lower-bound","kind":"node","lean":"TauCeti.IsContinuityEquation.wassersteinEDist_rpow_le","parent":2997,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport-and-wasserstein-geometry/index.html#the-benamou-brenier-lower-bound","scope":"tauceti/measure-theory/optimal-transport-and-wasserstein-geometry","status":"fully_proved","summary":"Let `t \u21a6 \u03bd\u209c` be a curve of finite measures, narrowly continuous on `[a, b]` with `a \u2264 b`, solving the continuity equation with velocity `v` on `(a, b)`, whose initial law `\u03bd_a` is a probability measure (by conservation of mass, so is `\u03bd_b`). Then for `1 \u2264 p < \u221e` the `p`-Wasserstein distance between the endpoint laws is controlled by the `p`-action of `v`:\u2026","title":"The Benamou\u2013Brenier lower bound"},{"id":"tauceti/measure-theory/optimal-transport-and-wasserstein-geometry/the-bonnet-myers-diameter-bound-under-mcp-k-n","kind":"node","lean":"TauCeti.MetricMeasureSpace.MeasureContraction.dist_le, TauCeti.MetricMeasureSpace.DistortedCurvatureDimension.diam_univ_le, TauCeti.MetricMeasureSpace.CurvatureDimension.diam_support_le, TauCeti.MetricMeasureSpace.MeasureContraction.diam_support_le","parent":2997,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport-and-wasserstein-geometry/index.html#the-bonnet-myers-diameter-bound-under-mcp-k-n","scope":"tauceti/measure-theory/optimal-transport-and-wasserstein-geometry","status":"fully_proved","summary":"If `M` satisfies `MCP(K, N)` with `K > 0` and `N > 0`, any two points of the support of the reference measure are at distance at most `\u03c0 \u221a((N - 1) / K)`.","title":"The Bonnet\u2013Myers diameter bound under `MCP(K, N)`"},{"id":"tauceti/measure-theory/optimal-transport-and-wasserstein-geometry/the-knott-smith-optimality-criterion","kind":"node","lean":"TauCeti.IsCoupling.isOptimalCoupling_iff_exists_ae_mem_subdifferential","parent":2997,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport-and-wasserstein-geometry/index.html#the-knott-smith-optimality-criterion","scope":"tauceti/measure-theory/optimal-transport-and-wasserstein-geometry","status":"fully_proved","summary":"On a separable real inner product space, a coupling `\u03c0` of finite measures `\u03bc` and `\u03bd` with finite second moments is an optimal plan for the cost `\u2016x - y\u2016 ^ 2 / 2` exactly when there is a proper (never `\u22a5`, and finite somewhere) convex, lower semicontinuous `u : E \u2192 EReal` such that `y \u2208 \u2202u(x)` for `\u03c0`-almost every `(x, y)`.","title":"The Knott\u2013Smith optimality criterion"},{"data_url":"scopes/tauceti/measure-theory/successive-minima-and-minkowskis-theorems.json","id":"tauceti/measure-theory/successive-minima-and-minkowskis-theorems","kind":"scope","parent":2980,"result":false,"route":"../roadmap/tauceti/measure-theory/successive-minima-and-minkowskis-theorems/index.html","scope":"tauceti/measure-theory/successive-minima-and-minkowskis-theorems","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Successive minima and Minkowski's theorems"},{"id":"tauceti/measure-theory/successive-minima-and-minkowskis-theorems/orthonormalbasis-volume-weighted-crosspolytope","kind":"node","lean":"OrthonormalBasis.volume_weighted_crosspolytope","parent":3009,"result":true,"route":"../roadmap/tauceti/measure-theory/successive-minima-and-minkowskis-theorems/index.html#orthonormalbasis-volume-weighted-crosspolytope","scope":"tauceti/measure-theory/successive-minima-and-minkowskis-theorems","status":"fully_proved","summary":"The canonical Euclidean volume of a weighted cross-polytope in an arbitrary real basis. The index type may be empty, in which case the volume and the empty products are one.","title":"OrthonormalBasis.volume_weighted_crosspolytope"},{"data_url":"scopes/tauceti/measure-theory/constructions.json","id":"tauceti/measure-theory/constructions","kind":"scope","parent":2980,"result":false,"route":"../roadmap/tauceti/measure-theory/constructions/index.html","scope":"tauceti/measure-theory/constructions","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.MeasureTheory.Constructions` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Constructions"},{"id":"tauceti/measure-theory/constructions/the-borel-graph-theorem","kind":"node","lean":"TauCeti.measurableSet_graph_iff_measurable, TauCeti.measurableSet_graphOn_iff, TauCeti.analyticSet_graph_iff_measurable","parent":3011,"result":true,"route":"../roadmap/tauceti/measure-theory/constructions/index.html#the-borel-graph-theorem","scope":"tauceti/measure-theory/constructions","status":"fully_proved","summary":"A map between standard Borel spaces is measurable if and only if its graph is measurable.","title":"The Borel graph theorem"},{"id":"tauceti/measure-theory/constructions/lusin-souslin-for-vertical-sections","kind":"node","lean":"MeasurableSet.measurableEmbedding_domRestrict_fst","parent":3011,"result":true,"route":"../roadmap/tauceti/measure-theory/constructions/index.html#lusin-souslin-for-vertical-sections","scope":"tauceti/measure-theory/constructions","status":"fully_proved","summary":"If a measurable subset `G` of a product of standard Borel spaces has at most one point above each `x`, its projection `G \u2192 X` is a measurable embedding.","title":"Lusin\u2013Souslin for vertical sections"},{"id":"tauceti/measure-theory/constructions/the-perfect-set-theorem-for-borel-sets-alexandro","kind":"node","lean":"MeasurableSet.exists_nat_bool_injection_of_not_countable","parent":3011,"result":true,"route":"../roadmap/tauceti/measure-theory/constructions/index.html#the-perfect-set-theorem-for-borel-sets-alexandro","scope":"tauceti/measure-theory/constructions","status":"fully_proved","summary":"An uncountable Borel subset of a Polish space contains the range of a continuous injection of Cantor space.","title":"The perfect set theorem for Borel sets (Alexandrov\u2013Hausdorff)"},{"id":"tauceti/measure-theory/constructions/radial-fundamental-theorem-of-calculus","kind":"node","lean":"TauCeti.integral_norm_rpow_neg_finrank_mul_fderiv_apply_self","parent":3011,"result":true,"route":"../roadmap/tauceti/measure-theory/constructions/index.html#radial-fundamental-theorem-of-calculus","scope":"tauceti/measure-theory/constructions","status":"fully_proved","summary":"For a `C\u00b9` function `f` with compact support on a nontrivial finite-dimensional real normed space of dimension `d`, `\u222b \u2016x\u2016 ^ (-d) * f' x x \u2202\u03bc = -(d * \u03bc (ball 0 1)) * f 0`.","title":"Radial fundamental theorem of calculus"},{"id":"tauceti/measure-theory/constructions/souslins-theorem-for-graphs","kind":"node","lean":"MeasureTheory.AnalyticSet.measurable_of_graph","parent":3011,"result":true,"route":"../roadmap/tauceti/measure-theory/constructions/index.html#souslins-theorem-for-graphs","scope":"tauceti/measure-theory/constructions","status":"fully_proved","summary":"A map from a Hausdorff space with its Borel sets measurable to a Polish space, whose graph is analytic, is measurable.","title":"Souslin's theorem for graphs"},{"data_url":"scopes/tauceti/measure-theory/function.json","id":"tauceti/measure-theory/function","kind":"scope","parent":2980,"result":false,"route":"../roadmap/tauceti/measure-theory/function/index.html","scope":"tauceti/measure-theory/function","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.MeasureTheory.Function` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Function"},{"id":"tauceti/measure-theory/function/campanatos-theorem","kind":"node","lean":"TauCeti.MeasureTheory.holderWith_preciseRepresentative_of_setAverage_norm_sub_le","parent":3017,"result":true,"route":"../roadmap/tauceti/measure-theory/function/index.html#campanatos-theorem","scope":"tauceti/measure-theory/function","status":"fully_proved","summary":"Let `\u03b1 > 0`, and let `f` be locally integrable, with mean oscillation on every closed ball of radius `r > 0` at most `M r^\u03b1`. Then the precise representative of `f` is `\u03b1`-H\u00f6lder continuous, with constant `2 \u00b7 2^\u03b1 \u00b7 2\u207f / (1 - 2^(-\u03b1)) \u00b7 M`, where `n` is the dimension. It agrees with `f` almost everywhere (`TauCeti.MeasureTheory.ae_eq_preciseRepresentative`).","title":"Campanato's theorem"},{"id":"tauceti/measure-theory/function/discrete-holder-estimate-for-a-disjoint-family-o","kind":"node","lean":"TauCeti.sum_edist_ceil_div_rpow_le","parent":3017,"result":true,"route":"../roadmap/tauceti/measure-theory/function/index.html#discrete-holder-estimate-for-a-disjoint-family-o","scope":"tauceti/measure-theory/function","status":"fully_proved","summary":"the `p`-th power of the sum of the distances between the piecewise constant interpolant at times `s\u1d62` and `t\u1d62` is at most `(|\u22c3 (s\u1d62, t\u1d62]| + k \u03c4) ^ (p - 1)` times the discrete `p`-action up to time `T`, where `k` is the number of intervals. As `\u03c4 \u2192 0` the factor tends to the one in the H\u00f6lder estimate `AbsolutelyContinuousOnInterval.edist_rpow_le_mul_curveAction` for curves.","title":"Discrete H\u00f6lder estimate for a disjoint family of intervals `(s\u1d62, t\u1d62]` in `[0, T]`"},{"id":"tauceti/measure-theory/function/dunford-pettis-theorem","kind":"node","lean":"MeasureTheory.Lp.isCompact_closure_toWeakSpace_iff_isBounded_and_unifIntegrable, MeasureTheory.Lp.isCompact_closure_toWeakSpace_iff_isBounded_and_forall_setIntegral_le, MeasureTheory.Lp.isCompact_closure_toWeakSpace_iff_uniformIntegrable, MeasureTheory.Lp.isCompact_closure_toWeakSpace_of_uniformIntegrable, MeasureTheory.Lp.uniformIntegrable_of_isCompact_closure_toWeakSpace","parent":3017,"result":true,"route":"../roadmap/tauceti/measure-theory/function/index.html#dunford-pettis-theorem","scope":"tauceti/measure-theory/function","status":"fully_proved","summary":"For a finite measure, a subset of scalar `L\u00b9` has weakly compact closure exactly when it is norm bounded and uniformly integrable.","title":"Dunford\u2013Pettis theorem"},{"id":"tauceti/measure-theory/function/eilenbergs-inequality-integral-form","kind":"node","lean":"LipschitzOnWith.lintegral_hausdorffMeasure_inter_preimage_le, LipschitzOnWith.exists_measurable_hausdorffMeasure_inter_preimage_le","parent":3017,"result":true,"route":"../roadmap/tauceti/measure-theory/function/index.html#eilenbergs-inequality-integral-form","scope":"tauceti/measure-theory/function","status":"fully_proved","summary":"for `f` `K`-Lipschitz on `A` with values in an `m`-dimensional real normed space with additive Haar measure `\u03bc`, and `t` real, `\u222b\u207b y, \u03bcH[t] (A \u2229 f \u207b\u00b9' {y}) \u2202\u03bc \u2264 \u03bc (ball 0 1) * K ^ m * \u03bcH[m + t] A`.","title":"Eilenberg's inequality, integral form"},{"id":"tauceti/measure-theory/function/fatous-lemma-for-the-lp-seminorm-of-r0-valued-fu","kind":"node","lean":"TauCeti.eLpNorm_ennreal_le_of_ae_tendsto, TauCeti.eLpNorm_le_of_ae_tendsto_ennreal","parent":3017,"result":true,"route":"../roadmap/tauceti/measure-theory/function/index.html#fatous-lemma-for-the-lp-seminorm-of-r0-valued-fu","scope":"tauceti/measure-theory/function","status":"fully_proved","summary":"Fatou's lemma for the `L\u1d56` seminorm of `\u211d\u22650\u221e`-valued functions, at every exponent `p : \u211d\u22650\u221e` (including `0` and `\u221e`) and along any countably generated `NeBot` filter. Almost everywhere measurability of the limit is deduced from the pointwise convergence.","title":"Fatou's lemma for the `L\u1d56` seminorm of `\u211d\u22650\u221e`-valued functions"},{"id":"tauceti/measure-theory/function/the-frechet-kolmogorov-compactness-criterion","kind":"node","lean":"TauCeti.totallyBounded_of_comp_add_sub_of_unifTight","parent":3017,"result":true,"route":"../roadmap/tauceti/measure-theory/function/index.html#the-frechet-kolmogorov-compactness-criterion","scope":"tauceti/measure-theory/function","status":"fully_proved","summary":"For `1 \u2264 p < \u221e`, a family `S` of `L\u1d56` functions is totally bounded as soon as it is bounded in `L\u1d56`, its translation increments are uniformly small in `L\u1d56`, and it is uniformly tight in the sense of `MeasureTheory.UnifTight`.","title":"The Fr\u00e9chet--Kolmogorov compactness criterion"},{"id":"tauceti/measure-theory/function/the-frechet-kolmogorov-criterion-for-a-family-va","kind":"node","lean":"TauCeti.totallyBounded_of_comp_add_sub_of_isBounded_of_ae_eq_zero_compl","parent":3017,"result":true,"route":"../roadmap/tauceti/measure-theory/function/index.html#the-frechet-kolmogorov-criterion-for-a-family-va","scope":"tauceti/measure-theory/function","status":"fully_proved","summary":"The Fr\u00e9chet--Kolmogorov criterion for a family vanishing almost everywhere off a fixed bounded set, the form that Rellich--Kondrachov for `W^{1,p}_0(\u03a9)` consumes. For such functions the tightness hypothesis is automatic, so an `L\u1d56`-bounded family whose translation increments are uniformly small in `L\u1d56` is totally bounded.","title":"The Fr\u00e9chet--Kolmogorov criterion for a family vanishing almost everywhere off a fixed bounded set"},{"id":"tauceti/measure-theory/function/fundamental-theorem-for-absolutely-continuous-cu","kind":"node","lean":"AbsolutelyContinuousOnInterval.lintegral_metricDerivative_lt_top, AbsolutelyContinuousOnInterval.ae_tendsto_metricDerivative, AbsolutelyContinuousOnInterval.edist_le_lintegral_metricDerivative","parent":3017,"result":true,"route":"../roadmap/tauceti/measure-theory/function/index.html#fundamental-theorem-for-absolutely-continuous-cu","scope":"tauceti/measure-theory/function","status":"fully_proved","summary":"the metric derivative of an absolutely continuous curve on `[a, b]` has finite integral over `[a, b]`.","title":"Fundamental theorem for absolutely continuous curves, integrability"},{"id":"tauceti/measure-theory/function/holder-estimate-along-a-curve","kind":"node","lean":"AbsolutelyContinuousOnInterval.edist_rpow_le_mul_curveAction","parent":3017,"result":true,"route":"../roadmap/tauceti/measure-theory/function/index.html#holder-estimate-along-a-curve","scope":"tauceti/measure-theory/function","status":"fully_proved","summary":"if `\u03b3` is absolutely continuous on `[a, b]` and `1 \u2264 p`, then `edist (\u03b3 s) (\u03b3 u) ^ p \u2264 edist s u ^ (p - 1) * curveAction p \u03b3 s u` for all `s, u \u2208 [a, b]`. In particular a curve of finite `p`-action with `1 < p` is `(1 - 1 / p)`-H\u00f6lder continuous.","title":"H\u00f6lder estimate along a curve"},{"id":"tauceti/measure-theory/function/holders-inequality-for-the-integral-of-a-product","kind":"node","lean":"TauCeti.enorm_integral_mul_le, TauCeti.enorm_integral_inner_le, TauCeti.rpow_lintegral_le_measure_univ_rpow_mul","parent":3017,"result":true,"route":"../roadmap/tauceti/measure-theory/function/index.html#holders-inequality-for-the-integral-of-a-product","scope":"tauceti/measure-theory/function","status":"fully_proved","summary":"`\u2016\u222b u v\u2016 \u2264 \u2016u\u2016_p \u2016v\u2016_q` for H\u00f6lder conjugate exponents `p` and `q`.","title":"H\u00f6lder's inequality for the integral of a product"},{"id":"tauceti/measure-theory/function/jensens-inequality-for-the-conditional-expectati","kind":"node","lean":"TauCeti.MeasureTheory.lintegral_enorm_condExp_rpow_le","parent":3017,"result":true,"route":"../roadmap/tauceti/measure-theory/function/index.html#jensens-inequality-for-the-conditional-expectati","scope":"tauceti/measure-theory/function","status":"fully_proved","summary":"for `1 \u2264 p`, the `p`-th power of the norm of `\u03bc[f | m]` integrates to at most that of `f`. Unlike Mathlib's `integral_norm_condExp_rpow_le`, this needs no integrability of `\u2016f\u2016 ^ p`.","title":"Jensen's inequality for the conditional expectation, as an inequality of `lintegral`s"},{"id":"tauceti/measure-theory/function/the-local-campanato-estimate","kind":"node","lean":"TauCeti.MeasureTheory.dist_preciseRepresentative_le_of_setAverage_norm_sub_le","parent":3017,"result":true,"route":"../roadmap/tauceti/measure-theory/function/index.html#the-local-campanato-estimate","scope":"tauceti/measure-theory/function","status":"fully_proved","summary":"Let `\u03b1 > 0` and `d = dist x y`. If `f` has mean oscillation at most `M r^\u03b1` on `closedBall x r` for `0 < r \u2264 2 d` and on `closedBall y r` for `0 < r \u2264 d`, and is integrable on `closedBall x (2 d)`, then the precise representatives of `f` at `x` and `y` differ by at most `2 \u00b7 2^\u03b1 \u00b7 2\u207f / (1 - 2^(-\u03b1)) \u00b7 M d^\u03b1`, where `n` is the dimension.","title":"The local Campanato estimate"},{"id":"tauceti/measure-theory/function/lusins-theorem","kind":"node","lean":"Measurable.exists_isClosed_measure_sdiff_lt_continuousOn, AEMeasurable.exists_isClosed_measure_sdiff_lt_continuousOn, MeasureTheory.AEStronglyMeasurable.exists_isClosed_measure_sdiff_lt_continuousOn, MeasureTheory.StronglyMeasurable.exists_isClosed_measure_sdiff_lt_continuousOn","parent":3017,"result":true,"route":"../roadmap/tauceti/measure-theory/function/index.html#lusins-theorem","scope":"tauceti/measure-theory/function","status":"fully_proved","summary":"A measurable map `f` from a weakly regular measure space into a second-countable topological space is continuous on a closed set `F \u2286 s` with `\u03bc (s \\ F) < \u03b5`, for every measurable set `s` of finite measure and every `\u03b5 > 0`.","title":"Lusin's theorem"},{"id":"tauceti/measure-theory/function/pettis-theorem-in-a-separable-inner-product-spac","kind":"node","lean":"TauCeti.aestronglyMeasurable_iff_forall_inner, TauCeti.aestronglyMeasurable_strongDual_iff_forall_apply","parent":3017,"result":true,"route":"../roadmap/tauceti/measure-theory/function/index.html#pettis-theorem-in-a-separable-inner-product-spac","scope":"tauceti/measure-theory/function","status":"fully_proved","summary":"A function into a separable inner product space is almost everywhere strongly measurable exactly when it is weakly measurable, that is, when each of its coordinates `t \u21a6 \u27eav, h t\u27eb` is almost everywhere strongly measurable.","title":"Pettis' theorem in a separable inner product space"},{"id":"tauceti/measure-theory/function/sards-lemma-for-lipschitz-maps","kind":"node","lean":"LipschitzWith.euclideanHausdorffMeasure_image_setOf_normDet_fderiv_eq_zero","parent":3017,"result":true,"route":"../roadmap/tauceti/measure-theory/function/index.html#sards-lemma-for-lipschitz-maps","scope":"tauceti/measure-theory/function","status":"fully_proved","summary":"The image of the set where the `dim U`-dimensional Jacobian of a Lipschitz map `f : U \u2192 V` vanishes is null for the `dim U`-dimensional Euclidean Hausdorff measure.","title":"Sard's lemma for Lipschitz maps"},{"id":"tauceti/measure-theory/function/vitalis-convergence-theorem-for-maps-into-a-pseu","kind":"node","lean":"TauCeti.tendsto_eLpNorm_dist_iff_tendstoInMeasure_and_unifIntegrable, TauCeti.tendsto_eLpNorm_dist_iff_tendstoInMeasure_and_uniformIntegrable","parent":3017,"result":true,"route":"../roadmap/tauceti/measure-theory/function/index.html#vitalis-convergence-theorem-for-maps-into-a-pseu","scope":"tauceti/measure-theory/function","status":"fully_proved","summary":"Let `1 \u2264 p < \u221e`, let `\u03bc` be a finite measure, and let the almost-everywhere measurable maps `T n, T\u2080 : X \u2192 Y` have finite `p`-th moment about a point `y\u2080`. Then `dist (T n x) (T\u2080 x)` tends to `0` in `L\u1d56(\u03bc)` if and only if `T n` converges to `T\u2080` in `\u03bc`-measure and the family `dist (T n \u00b7) y\u2080` is uniformly integrable in `L\u1d56(\u03bc)`.","title":"Vitali's convergence theorem for maps into a pseudometric space"},{"id":"tauceti/measure-theory/function/youngs-inequality-for-an-l1-kernel","kind":"node","lean":"MeasureTheory.Lp.norm_convolutionL_apply_le, TauCeti.norm_convolutionLp_le","parent":3017,"result":true,"route":"../roadmap/tauceti/measure-theory/function/index.html#youngs-inequality-for-an-l1-kernel","scope":"tauceti/measure-theory/function","status":"fully_proved","summary":"`\u2016h \u22c6 f\u2016_p \u2264 \u2016h\u2016\u2081 \u2016f\u2016_p`.","title":"Young's inequality for an `L\u00b9` kernel"},{"data_url":"scopes/tauceti/measure-theory/integral.json","id":"tauceti/measure-theory/integral","kind":"scope","parent":2980,"result":false,"route":"../roadmap/tauceti/measure-theory/integral/index.html","scope":"tauceti/measure-theory/integral","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.MeasureTheory.Integral` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Integral"},{"data_url":"scopes/tauceti/measure-theory/integral/part-01.json","id":"tauceti/measure-theory/integral/part-01","kind":"scope","navigation":"transparent","parent":3035,"result":false,"route":"../roadmap/tauceti/measure-theory/integral/part-01/index.html","scope":"tauceti/measure-theory/integral/part-01","status":"fully_proved","title":"The Calder\u00f3n\u2013Zygmund theorem to The Riesz\u2013Thorin interpolation\u2026"},{"id":"tauceti/measure-theory/integral/part-01/the-calderon-zygmund-theorem","kind":"node","lean":"ContinuousLinearMap.eLpNorm_le_of_hormander, ContinuousLinearMap.eLpNorm_le_of_hormander_adjoint, ContinuousLinearMap.eLpNorm_le_of_setLIntegral_compl_closedBall_adjoint_le, ContinuousLinearMap.eLpNorm_le_of_setLIntegral_compl_closedBall_le, ContinuousLinearMap.mul_volume_lt_enorm_le_of_hormander, ContinuousLinearMap.mul_volume_lt_enorm_le_of_norm_sub_le_mul_rpow, ContinuousLinearMap.mul_volume_lt_enorm_le_of_setLIntegral_compl_closedBall_le, TauCeti.exists_eLpNorm_le_of_setLIntegral_compl_closedBall_le, ContinuousLinearMap.exists_eLpNorm_le_of_norm_sub_le_mul_rpow, ContinuousLinearMap.exists_mul_volume_lt_enorm_le_of_norm_sub_le_mul_rpow, TauCeti.exists_eLpNorm_le_of_hormander, TauCeti.exists_mul_volume_lt_enorm_le_of_hormander","parent":3036,"result":true,"route":"../roadmap/tauceti/measure-theory/integral/part-01/index.html#the-calderon-zygmund-theorem","scope":"tauceti/measure-theory/integral/part-01","status":"fully_proved","summary":"The Calder\u00f3n\u2013Zygmund theorem for an operator given by a kernel, strong type `(p, p)` for `1 < p < 2`. Let `T` be a bounded linear operator on `L\u00b2(\u211d\u207f)` such that `T b x = \u222b K x y (b y) dy` for almost every `x` off any closed ball outside which `b` vanishes, where the kernel `K` satisfies H\u00f6rmander's condition `\u222b_{dist x y' > 2 dist y y'} \u2016K x y - K x y'\u2016 dx \u2264 B` for all `y`, `y'`. Then for every `1 < p < 2` and\u2026","title":"The Calder\u00f3n\u2013Zygmund theorem"},{"id":"tauceti/measure-theory/integral/part-01/cauchy-schwarz-on-a-circle","kind":"node","lean":"TauCeti.circleLIntegral_sq_le","parent":3036,"result":true,"route":"../roadmap/tauceti/measure-theory/integral/part-01/index.html#cauchy-schwarz-on-a-circle","scope":"tauceti/measure-theory/integral/part-01","status":"fully_proved","summary":"The square of the circle integral of `g` is at most `2 \u03c0 \u03c1` \u2014 for `\u03c1 > 0` the circumference of the circle \u2014 times the circle integral of `g ^ 2`.","title":"Cauchy\u2013Schwarz on a circle"},{"id":"tauceti/measure-theory/integral/part-01/the-divergence-theorem-on-a-half-space-tauceti-s","kind":"node","lean":"TauCeti.setIntegral_setOf_inner_lt_trace_fderiv, TauCeti.setIntegral_setOf_inner_lt_fderiv_apply","parent":3036,"result":true,"route":"../roadmap/tauceti/measure-theory/integral/part-01/index.html#the-divergence-theorem-on-a-half-space-tauceti-s","scope":"tauceti/measure-theory/integral/part-01","status":"fully_proved","summary":"For a unit vector `\u03bd` and a compactly supported `C\u00b9` vector field `\u03c6`, the integral of the divergence `tr (D\u03c6)` over the half-space `{x | \u27eax, \u03bd\u27eb < c}` is the flux `\u222b \u27ea\u03c6, \u03bd\u27eb` of `\u03c6` through the boundary hyperplane `{x | \u27eax, \u03bd\u27eb = c}` against `(n - 1)`-dimensional Hausdorff measure.","title":"The divergence theorem on a half-space (`TauCeti.setIntegral_setOf_inner_lt_trace_fderiv`)"},{"id":"tauceti/measure-theory/integral/part-01/the-divergence-theorem-on-an-affine-parallelepip","kind":"node","lean":"ContinuousLinearEquiv.integral_divergence_affineFluxPullback","parent":3036,"result":true,"route":"../roadmap/tauceti/measure-theory/integral/part-01/index.html#the-divergence-theorem-on-an-affine-parallelepip","scope":"tauceti/measure-theory/integral/part-01","status":"fully_proved","summary":"Let `A u = p + L u` and let `F` be a flux field continuous on the parallelepiped `A '' [0, 1]\u207f\u207a\u00b9`, differentiable with derivatives `F' i` at `A u` for every interior point `u` of the unit cube off a countable set, and with `u \u21a6 div F (A u)` integrable on the unit cube. Then `det L` times the integral of `div F \u2218 A` over the unit cube is the boundary sum of the cofactor pullback `H = L.affineFluxPullback p F`: the\u2026","title":"The divergence theorem on an affine parallelepiped"},{"id":"tauceti/measure-theory/integral/part-01/the-dyadic-john-nirenberg-inequality-for-any-add","kind":"node","lean":"TauCeti.measure_lt_enorm_sub_setAverage_le_inv_two_pow","parent":3036,"result":true,"route":"../roadmap/tauceti/measure-theory/integral/part-01/index.html#the-dyadic-john-nirenberg-inequality-for-any-add","scope":"tauceti/measure-theory/integral/part-01","status":"fully_proved","summary":"the superlevel set at `2\u207f\u207a\u00b9 N M` has measure at most `2\u207b\u1d3a` times the cube measure.","title":"The dyadic John\u2013Nirenberg inequality for any additive Haar measure"},{"id":"tauceti/measure-theory/integral/part-01/the-fefferman-stein-inequality","kind":"node","lean":"TauCeti.eLpNorm_le_eLpNorm_sharpMaximalFunction, TauCeti.lintegral_enorm_rpow_le_lintegral_sharpMaximalFunction_rpow","parent":3036,"result":true,"route":"../roadmap/tauceti/measure-theory/integral/part-01/index.html#the-fefferman-stein-inequality","scope":"tauceti/measure-theory/integral/part-01","status":"fully_proved","summary":"For `1 < p < \u221e` and integrable `f` on `\u211d\u207f = \u03b9 \u2192 \u211d`, `n \u2265 1`, the `L\u1d56` norm of `f` is controlled by that of its sharp maximal function `f^#`, with a constant depending only on `n` and `p`:","title":"The Fefferman\u2013Stein inequality"},{"id":"tauceti/measure-theory/integral/part-01/the-full-strong-type-calderon-zygmund-theorem","kind":"node","lean":"ContinuousLinearMap.exists_strongTypeExtension_of_setLIntegral_compl_closedBall_le","parent":3036,"result":true,"route":"../roadmap/tauceti/measure-theory/integral/part-01/index.html#the-full-strong-type-calderon-zygmund-theorem","scope":"tauceti/measure-theory/integral/part-01","status":"fully_proved","summary":"An `L\u00b2`-bounded operator for which both it and its adjoint satisfy the cancellation estimate with finite constants extends to a bounded operator on `L\u1d56(\u211d\u207f; H) \u2192 L\u1d56(\u211d\u207f; K)` for every `1 < p < \u221e`. On `L\u00b2 \u2229 L\u1d56`, the extended map agrees almost everywhere with the given operator.","title":"The full strong-type Calder\u00f3n\u2013Zygmund theorem"},{"id":"tauceti/measure-theory/integral/part-01/the-fundamental-theorem-of-calculus-along-a-coor","kind":"node","lean":"ContDiffOn.integral_fderivWithin_fullSimplex","parent":3036,"result":true,"route":"../roadmap/tauceti/measure-theory/integral/part-01/index.html#the-fundamental-theorem-of-calculus-along-a-coor","scope":"tauceti/measure-theory/integral/part-01","status":"fully_proved","summary":"For a function `g` that is `C\u00b9` on `\u0394\u1d56\u207a\u00b9`, the integral over `\u0394\u1d56\u207a\u00b9` of its partial derivative in the direction `e_j` is the integral over `\u0394\u1d56` of the difference of its values at the points of the faces `\u2211 x\u1d62 = 1` and `x_j = 0` whose other coordinates are given by `y \u2208 \u0394\u1d56`.","title":"The fundamental theorem of calculus along a coordinate of the standard simplex"},{"id":"tauceti/measure-theory/integral/part-01/the-fundamental-theorem-of-calculus-in-the-first","kind":"node","lean":"TauCeti.setIntegral_rect_eq_setIntegral_sub_of_hasDerivAt_fst","parent":3036,"result":true,"route":"../roadmap/tauceti/measure-theory/integral/part-01/index.html#the-fundamental-theorem-of-calculus-in-the-first","scope":"tauceti/measure-theory/integral/part-01","status":"fully_proved","summary":"The integral of a continuous partial derivative `\u03c6 = \u2202\u2081 \u03a6` over the rectangle `[a\u2081, b\u2081] \u00d7 [a\u2082, b\u2082]` is the integral over `[a\u2082, b\u2082]` of the difference of the values of `\u03a6` on the two vertical sides.","title":"The fundamental theorem of calculus in the first coordinate"},{"id":"tauceti/measure-theory/integral/part-01/the-fundamental-theorem-of-calculus-in-the-secon","kind":"node","lean":"TauCeti.setIntegral_rect_eq_setIntegral_sub_of_hasDerivAt_snd","parent":3036,"result":true,"route":"../roadmap/tauceti/measure-theory/integral/part-01/index.html#the-fundamental-theorem-of-calculus-in-the-secon","scope":"tauceti/measure-theory/integral/part-01","status":"fully_proved","summary":"The integral of a continuous partial derivative `\u03c6 = \u2202\u2082 \u03a6` over the rectangle `[a\u2081, b\u2081] \u00d7 [a\u2082, b\u2082]` is the integral over `[a\u2081, b\u2081]` of the difference of the values of `\u03a6` on the two horizontal sides.","title":"The fundamental theorem of calculus in the second coordinate"},{"id":"tauceti/measure-theory/integral/part-01/greens-formula-on-a-closed-annulus","kind":"node","lean":"ContinuousLinearMap.integral_bilinear_fderiv_sub_annulus, ContinuousLinearMap.integral_bilinear_fderiv_sub_prod_Icc","parent":3036,"result":true,"route":"../roadmap/tauceti/measure-theory/integral/part-01/index.html#greens-formula-on-a-closed-annulus","scope":"tauceti/measure-theory/integral/part-01","status":"fully_proved","summary":"For a `C\u00b2` map near the annulus of radii `0 \u2264 a \u2264 b`, the integral of the alternating part of a bilinear pairing of its derivatives equals the outer circle integral of `B u du` minus the inner circle integral. Both circles are parametrized counterclockwise; their difference gives the boundary orientation.","title":"Green's formula on a closed annulus"},{"id":"tauceti/measure-theory/integral/part-01/the-hardy-littlewood-maximal-inequality","kind":"node","lean":"TauCeti.eLpNorm_maximalFunction_le, TauCeti.lintegral_rpow_maximalFunction_le, TauCeti.measure_lt_maximalFunction_le, TauCeti.mul_measure_lt_maximalFunction_le","parent":3036,"result":true,"route":"../roadmap/tauceti/measure-theory/integral/part-01/index.html#the-hardy-littlewood-maximal-inequality","scope":"tauceti/measure-theory/integral/part-01","status":"fully_proved","summary":"The Hardy\u2013Littlewood maximal inequality, the strong-type `(p, p)` bound for `1 < p < \u221e`, in terms of representative-level `L\u1d56` seminorms.","title":"The Hardy\u2013Littlewood maximal inequality"},{"id":"tauceti/measure-theory/integral/part-01/holders-bound-on-an-average","kind":"node","lean":"TauCeti.enorm_setAverage_le, TauCeti.enorm_setAverage_rpow_le","parent":3036,"result":true,"route":"../roadmap/tauceti/measure-theory/integral/part-01/index.html#holders-bound-on-an-average","scope":"tauceti/measure-theory/integral/part-01","status":"fully_proved","summary":"the average of `f` over a set `s` is at most `\u03bc s ^ (-1/p)` times the `L\u1d56` seminorm of `f` on `s`. At `p = 1` this is the bound by the average of `\u2016f\u2016`, and the volume factor sharpens as `p` grows because a larger exponent controls the mass of `s` more efficiently. No measurability or integrability hypothesis is needed, since a function not integrable on `s` has average zero.","title":"H\u00f6lder's bound on an average"},{"id":"tauceti/measure-theory/integral/part-01/holders-inequality-for-youngs-exponents","kind":"node","lean":"TauCeti.lintegral_mul_le_of_inv_add_inv_eq, TauCeti.MeasureTheory.abs_integral_mul_le_eLpNorm_mul_eLpNorm_mul_measureReal_rpow","parent":3036,"result":true,"route":"../roadmap/tauceti/measure-theory/integral/part-01/index.html#holders-inequality-for-youngs-exponents","scope":"tauceti/measure-theory/integral/part-01","status":"fully_proved","summary":"Let `1 \u2264 p \u2264 q` and let `r` be the exponent with `1/p + 1/r = 1 + 1/q`. Then `\u222b\u207b k g \u2264 (\u222b\u207b k ^ r g ^ p) ^ (1/q) (\u222b\u207b k ^ r) ^ (1 - 1/p) (\u222b\u207b g ^ p) ^ (1/p - 1/q)`.","title":"H\u00f6lder's inequality for Young's exponents"},{"id":"tauceti/measure-theory/integral/part-01/jensens-inequality-for-the-square-of-a-mean","kind":"node","lean":"TauCeti.sq_setAverage_norm_le_setAverage_norm_sq","parent":3036,"result":true,"route":"../roadmap/tauceti/measure-theory/integral/part-01/index.html#jensens-inequality-for-the-square-of-a-mean","scope":"tauceti/measure-theory/integral/part-01","status":"fully_proved","summary":"On a set `s` of finite nonzero measure, the square of the mean of `\u2016g\u2016` is at most the mean of `\u2016g\u2016\u00b2`.","title":"Jensen's inequality for the square of a mean"},{"id":"tauceti/measure-theory/integral/part-01/the-john-nirenberg-inequality-on-euclidean-balls","kind":"node","lean":"TauCeti.exists_setLIntegral_exp_mul_norm_sub_setAverage_ball_le, TauCeti.measure_lt_norm_sub_setAverage_closedBall_le, TauCeti.measure_lt_norm_sub_setAverage_le","parent":3036,"result":true,"route":"../roadmap/tauceti/measure-theory/integral/part-01/index.html#the-john-nirenberg-inequality-on-euclidean-balls","scope":"tauceti/measure-theory/integral/part-01","status":"fully_proved","summary":"There are constants `A > 0` and `C`, depending only on the dimension `n`, such that the following holds for every additive Haar measure `\u03bc` on `\u211d\u207f = EuclideanSpace \u211d \u03b9`. Let `f : \u211d\u207f \u2192 E` be integrable on the ball `B(x\u2080, R)`, with mean oscillation `\u2a0d_B \u2016f - f_B\u2016 \u2264 M` on every ball `B \u2286 B(x\u2080, R)`. If `\u03c3 M \u2264 A`, then on the concentric ball `B' = B(x\u2080, R / (2\u221an))`,","title":"The John\u2013Nirenberg inequality on Euclidean balls"},{"id":"tauceti/measure-theory/integral/part-01/minkowskis-integral-inequality","kind":"node","lean":"TauCeti.lintegral_Lp_lintegral_le","parent":3036,"result":true,"route":"../roadmap/tauceti/measure-theory/integral/part-01/index.html#minkowskis-integral-inequality","scope":"tauceti/measure-theory/integral/part-01","status":"fully_proved","summary":"For `1 \u2264 p` and a jointly a.e. measurable `f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e`, the `L\u1d56(\u03bc)` seminorm of `x \u21a6 \u222b\u207b y, f x y \u2202\u03bd` is at most the integral over `\u03bd` of the `L\u1d56(\u03bc)` seminorms of the functions `x \u21a6 f x y`: `(\u222b\u207b x, (\u222b\u207b y, f x y \u2202\u03bd) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u222b\u207b y, (\u222b\u207b x, f x y ^ p \u2202\u03bc) ^ (1 / p) \u2202\u03bd`.","title":"Minkowski's integral inequality"},{"id":"tauceti/measure-theory/integral/part-01/mosers-crossover-estimate-tauceti-exists-setlint","kind":"node","lean":"TauCeti.exists_setLIntegral_exp_mul_mul_setLIntegral_exp_neg_mul_ball_le","parent":3036,"result":true,"route":"../roadmap/tauceti/measure-theory/integral/part-01/index.html#mosers-crossover-estimate-tauceti-exists-setlint","scope":"tauceti/measure-theory/integral/part-01","status":"fully_proved","summary":"There are constants `A > 0` and `C`, depending only on the dimension `n`, such that the following holds for every additive Haar measure `\u03bc` on `\u211d\u207f = EuclideanSpace \u211d \u03b9`. Let `f : \u211d\u207f \u2192 \u211d` be integrable on the ball `B(x\u2080, R)`, with mean oscillation `\u2a0d_B |f - f_B| \u2264 M` on every ball `B \u2286 B(x\u2080, R)`. If `|\u03c3| M \u2264 A`, then on the concentric ball `B' = B(x\u2080, R / (2\u221an))`,","title":"Moser's crossover estimate (`TauCeti.exists_setLIntegral_exp_mul_mul_setLIntegral_exp_neg_mul_ball_le`)"},{"id":"tauceti/measure-theory/integral/part-01/riesz-markov-kakutani-on-a-compact-hausdorff-spa","kind":"node","lean":"LinearMap.exists_regular_integral_eq","parent":3036,"result":true,"route":"../roadmap/tauceti/measure-theory/integral/part-01/index.html#riesz-markov-kakutani-on-a-compact-hausdorff-spa","scope":"tauceti/measure-theory/integral/part-01","status":"fully_proved","summary":"a linear functional on `C(X, \u211d)` that is nonnegative on nonnegative functions is integration against a regular measure.","title":"Riesz\u2013Markov\u2013Kakutani on a compact Hausdorff space"},{"id":"tauceti/measure-theory/integral/part-01/the-riesz-thorin-interpolation-theorem","kind":"node","lean":"TauCeti.eLpNorm_le_rpow_mul_rpow_mul_eLpNorm, TauCeti.eLpNorm_le_rpow_mul_rpow_mul_eLpNorm_of_ne_top","parent":3036,"result":true,"route":"../roadmap/tauceti/measure-theory/integral/part-01/index.html#the-riesz-thorin-interpolation-theorem","scope":"tauceti/measure-theory/integral/part-01","status":"fully_proved","summary":"Let `T` be a complex-linear map from simple functions on `(\u03b1, \u03bc)` to almost-everywhere classes on a \u03c3-finite `(\u03b2, \u03bd)` with `\u2016T f\u2016_{q\u2080} \u2264 M\u2080 \u2016f\u2016_{p\u2080}` and `\u2016T f\u2016_{q\u2081} \u2264 M\u2081 \u2016f\u2016_{p\u2081}` for all simple `f`, where `p\u2080, p\u2081 \u2208 [0, \u221e]` and `1 \u2264 q\u2080, q\u2081 \u2264 \u221e`. For `0 < \u03b8 < 1` and the intermediate exponents `1 / p = (1 - \u03b8) / p\u2080 + \u03b8 / p\u2081`, `1 / q = (1 - \u03b8) / q\u2080 + \u03b8 / q\u2081`,","title":"The Riesz\u2013Thorin interpolation theorem"},{"data_url":"scopes/tauceti/measure-theory/integral/part-02.json","id":"tauceti/measure-theory/integral/part-02","kind":"scope","navigation":"transparent","parent":3035,"result":false,"route":"../roadmap/tauceti/measure-theory/integral/part-02/index.html","scope":"tauceti/measure-theory/integral/part-02","status":"fully_proved","title":"The Vitali covering estimate to Young's inequality for integral\u2026"},{"id":"tauceti/measure-theory/integral/part-02/the-vitali-covering-estimate","kind":"node","lean":"TauCeti.mul_measure_le_of_forall_exists_ball","parent":3057,"result":true,"route":"../roadmap/tauceti/measure-theory/integral/part-02/index.html#the-vitali-covering-estimate","scope":"tauceti/measure-theory/integral/part-02","status":"fully_proved","summary":"If every point of `S` is the centre of a ball of radius at most `R` on which `\u222b\u207b g` is at least `t` times the measure of the ball, then `t * \u03bc S \u2264 4 ^ n * \u222b\u207b g`, with `n` the dimension of the space.","title":"The Vitali covering estimate"},{"id":"tauceti/measure-theory/integral/part-02/wolffs-lemma-tauceti-exists-circlelintegral-sq-l","kind":"node","lean":"TauCeti.exists_circleLIntegral_sq_lt, TauCeti.exists_circleLIntegral_lt_of_lintegral_sq_ne_top","parent":3057,"result":true,"route":"../roadmap/tauceti/measure-theory/integral/part-02/index.html#wolffs-lemma-tauceti-exists-circlelintegral-sq-l","scope":"tauceti/measure-theory/integral/part-02","status":"fully_proved","summary":"If `2 \u03c0` times the plane integral of `g ^ 2` is smaller than `c * log (R / r)`, then some circle of radius `\u03c1` strictly between `r` and `R` has `circleLIntegral g \u03b6 \u03c1 ^ 2 < c`.","title":"Wolff's lemma (`TauCeti.exists_circleLIntegral_sq_lt`)"},{"id":"tauceti/measure-theory/integral/part-02/youngs-inequality-for-integral-kernels","kind":"node","lean":"TauCeti.lintegral_rpow_lintegral_mul_le_of_inv_add_inv_eq","parent":3057,"result":true,"route":"../roadmap/tauceti/measure-theory/integral/part-02/index.html#youngs-inequality-for-integral-kernels","scope":"tauceti/measure-theory/integral/part-02","status":"fully_proved","summary":"Let `1 \u2264 p \u2264 q` and let `r` be the exponent with `1/p + 1/r = 1 + 1/q`. If the `r`-th powers of the kernel `k` have row integrals `\u222b\u207b y, k x y ^ r \u2202\u03bd` at most `A` for almost every `x` and column integrals `\u222b\u207b x, k x y ^ r \u2202\u03bc` at most `B` for almost every `y`, then the integral operator with kernel `k` satisfies `\u222b\u207b x, (\u222b\u207b y, k x y * g y \u2202\u03bd) ^ q \u2202\u03bc \u2264 A ^ (q (1 - 1/p)) * B * (\u222b\u207b y, g y ^ p \u2202\u03bd) ^ (q / p)`.","title":"Young's inequality for integral kernels"},{"data_url":"scopes/tauceti/measure-theory/measurable-space.json","id":"tauceti/measure-theory/measurable-space","kind":"scope","parent":2980,"result":false,"route":"../roadmap/tauceti/measure-theory/measurable-space/index.html","scope":"tauceti/measure-theory/measurable-space","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.MeasureTheory.MeasurableSpace` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Measurable Space"},{"id":"tauceti/measure-theory/measurable-space/the-jankov-von-neumann-uniformization-theorem","kind":"node","lean":"MeasureTheory.AnalyticSet.exists_measurable_uniformization","parent":3061,"result":true,"route":"../roadmap/tauceti/measure-theory/measurable-space/index.html#the-jankov-von-neumann-uniformization-theorem","scope":"tauceti/measure-theory/measurable-space","status":"fully_proved","summary":"An analytic relation `R \u2286 X \u00d7 Y`, with `X` a T\u2081 space, has a uniformization `f : X \u2192 Y` which is measurable for the \u03c3-algebra generated by the analytic subsets of `X`: `(x, f x) \u2208 R` for every `x` in the projection of `R`.","title":"The Jankov\u2013von Neumann uniformization theorem"},{"id":"tauceti/measure-theory/measurable-space/the-kuratowski-ryll-nardzewski-measurable-select","kind":"node","lean":"TauCeti.exists_measurable_forall_mem","parent":3061,"result":true,"route":"../roadmap/tauceti/measure-theory/measurable-space/index.html#the-kuratowski-ryll-nardzewski-measurable-select","scope":"tauceti/measure-theory/measurable-space","status":"fully_proved","summary":"A map `M` from a measurable space to the nonempty closed subsets of a Polish space, which is weakly measurable in the sense that `{z | M z \u2229 U \u2260 \u2205}` is measurable for every open `U`, has a measurable selection.","title":"The Kuratowski\u2013Ryll-Nardzewski measurable selection theorem"},{"id":"tauceti/measure-theory/measurable-space/lusins-universal-measurability-theorem","kind":"node","lean":"TauCeti.MeasureTheory.AnalyticSet.nullMeasurableSet","parent":3061,"result":true,"route":"../roadmap/tauceti/measure-theory/measurable-space/index.html#lusins-universal-measurability-theorem","scope":"tauceti/measure-theory/measurable-space","status":"fully_proved","summary":"An analytic subset of a metrizable space is null-measurable with respect to every s-finite measure on a measurable space containing the Borel sets, equivalently measurable in the completion of that measure.","title":"Lusin's universal-measurability theorem"},{"data_url":"scopes/tauceti/measure-theory/measure.json","id":"tauceti/measure-theory/measure","kind":"scope","parent":2980,"result":false,"route":"../roadmap/tauceti/measure-theory/measure/index.html","scope":"tauceti/measure-theory/measure","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.MeasureTheory.Measure` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Measure"},{"id":"tauceti/measure-theory/measure/helly-bray-theorem","kind":"node","lean":"MeasureTheory.ProbabilityMeasure.tendsto_iff_forall_continuousAt_tendsto_cdf","parent":3065,"result":true,"route":"../roadmap/tauceti/measure-theory/measure/index.html#helly-bray-theorem","scope":"tauceti/measure-theory/measure","status":"fully_proved","summary":"Along a countably generated filter, probability measures on `\u211d` converge weakly if and only if their cumulative distribution functions converge pointwise at every continuity point of the limiting cumulative distribution function.","title":"Helly\u2013Bray theorem"},{"id":"tauceti/measure-theory/measure/holder-markov-estimate","kind":"node","lean":"TauCeti.measure_continuity_modulus_le_lintegral_curveAction","parent":3065,"result":true,"route":"../roadmap/tauceti/measure-theory/measure/index.html#holder-markov-estimate","scope":"tauceti/measure-theory/measure","status":"fully_proved","summary":"Let `P` be a measure on `C([a, b], X)` concentrated on curves absolutely continuous on `[a, b]`, and `1 \u2264 p`. For `\u03b7 > 0` and every `\u03b4`, the curves with two times closer than `\u03b4` at which their values are at least `\u03b7` apart have `P`-measure at most `\u03b4 ^ (p - 1) * (\u222b A_p dP) / \u03b7 ^ p`, where `A_p` is the `p`-action over `[a, b]`.","title":"H\u00f6lder\u2013Markov estimate"},{"id":"tauceti/measure-theory/measure/lebesgues-differentiation-theorem-along-dyadic-c","kind":"node","lean":"TauCeti.ae_tendsto_setLAverage_dyadicCube","parent":3065,"result":true,"route":"../roadmap/tauceti/measure-theory/measure/index.html#lebesgues-differentiation-theorem-along-dyadic-c","scope":"tauceti/measure-theory/measure","status":"fully_proved","summary":"for an integrable `g`, at almost every `x` the averages of `g` over the dyadic cubes containing `x` converge to `g x` as the level tends to `-\u221e`.","title":"Lebesgue's differentiation theorem along dyadic cubes"},{"id":"tauceti/measure-theory/measure/levys-continuity-theorem-including-existence","kind":"node","lean":"TauCeti.exists_probabilityMeasure_of_tendsto_charFun","parent":3065,"result":true,"route":"../roadmap/tauceti/measure-theory/measure/index.html#levys-continuity-theorem-including-existence","scope":"tauceti/measure-theory/measure","status":"fully_proved","summary":"If the characteristic functions of a sequence of probability measures converge pointwise to a function continuous at zero, there is a unique probability measure with that characteristic function, and the sequence converges weakly to it. Continuity is required only at zero; normalization of the limit follows from the probability measures in the sequence.","title":"L\u00e9vy's continuity theorem, including existence"},{"id":"tauceti/measure-theory/measure/prokhorovs-theorem-for-finite-measures","kind":"node","lean":"TauCeti.isCompact_closure_of_isTightMeasureSet_of_mass_le, TauCeti.exists_tendsto_finiteMeasure_of_eventually_measure_compl_eq_zero","parent":3065,"result":true,"route":"../roadmap/tauceti/measure-theory/measure/index.html#prokhorovs-theorem-for-finite-measures","scope":"tauceti/measure-theory/measure","status":"fully_proved","summary":"A tight set of finite measures with uniformly bounded total mass has compact closure in the topology of weak convergence.","title":"Prokhorov's theorem for finite measures"},{"id":"tauceti/measure-theory/measure/sierpinskis-theorem","kind":"node","lean":"MeasureTheory.Measure.exists_subset_measure_eq_of_nullSingleton","parent":3065,"result":true,"route":"../roadmap/tauceti/measure-theory/measure/index.html#sierpinskis-theorem","scope":"tauceti/measure-theory/measure","status":"fully_proved","summary":"An atomless measure on a standard Borel space takes every value between `0` and `\u03bc s` on measurable subsets of a measurable set `s` of finite measure.","title":"Sierpi\u0144ski's theorem"},{"data_url":"scopes/tauceti/measure-theory/optimal-transport.json","id":"tauceti/measure-theory/optimal-transport","kind":"scope","parent":2980,"result":false,"route":"../roadmap/tauceti/measure-theory/optimal-transport/index.html","scope":"tauceti/measure-theory/optimal-transport","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.MeasureTheory.OptimalTransport` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Optimal Transport"},{"data_url":"scopes/tauceti/measure-theory/optimal-transport/part-01.json","id":"tauceti/measure-theory/optimal-transport/part-01","kind":"scope","navigation":"transparent","parent":3072,"result":false,"route":"../roadmap/tauceti/measure-theory/optimal-transport/part-01/index.html","scope":"tauceti/measure-theory/optimal-transport/part-01","status":"fully_proved","title":"Approximate Kantorovich duality\u2026 to Strong Kantorovich duality for\u2026"},{"id":"tauceti/measure-theory/optimal-transport/part-01/approximate-kantorovich-duality-on-compact-space","kind":"node","lean":"TauCeti.exists_continuous_forall_add_le_transportCost_le","parent":3073,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport/part-01/index.html#approximate-kantorovich-duality-on-compact-space","scope":"tauceti/measure-theory/optimal-transport/part-01","status":"fully_proved","summary":"For probability measures on compact pseudometric spaces, a continuous nonnegative cost, and every `\u03b5 > 0`, there is a pair of *continuous* potentials satisfying the Kantorovich constraint whose value is within `\u03b5` of the primal transport cost. Together with weak duality this closes the duality gap.","title":"Approximate Kantorovich duality on compact spaces"},{"id":"tauceti/measure-theory/optimal-transport/part-01/the-benamou-brenier-upper-bound","kind":"node","lean":"TauCeti.exists_isContinuityEquation_map_smul_fst_add_smul_snd_of_isOptimalCoupling_le","parent":3073,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport/part-01/index.html#the-benamou-brenier-upper-bound","scope":"tauceti/measure-theory/optimal-transport/part-01","status":"fully_proved","summary":"Let `1 \u2264 p < \u221e`, and let `\u03c0` be an optimal coupling of a finite measure `\u03bc` and a measure `\u03bd` on `E` for the cost `d(x, y) ^ p`, at finite Wasserstein distance `W_p (\u03bc, \u03bd)`. Then the displacement interpolation, the laws `\u03bc\u209c` of `(1 - t) \u2022 x + t \u2022 y` under `\u03c0`, solves the continuity equation on `(0, 1)` with a velocity field of `p`-action at most `W_p (\u03bc, \u03bd) ^ p`: `\u222b\u2080\u00b9 \u222b \u2016v\u209c\u2016 ^ p d\u03bc\u209c dt \u2264 W_p (\u03bc, \u03bd) ^ p`.","title":"The Benamou\u2013Brenier upper bound"},{"id":"tauceti/measure-theory/optimal-transport/part-01/bounded-lipschitz-duality-in-supremum-form","kind":"node","lean":"TauCeti.transportCost_truncated_dist_eq_iSup","parent":3073,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport/part-01/index.html#bounded-lipschitz-duality-in-supremum-form","scope":"tauceti/measure-theory/optimal-transport/part-01","status":"fully_proved","summary":"A uniform bound `R` on the test functions corresponds to truncation of the ground distance at `2 * R`, for finite measures of equal mass.","title":"Bounded-Lipschitz duality in supremum form"},{"id":"tauceti/measure-theory/optimal-transport/part-01/bounded-lipschitz-kantorovich-rubinstein-duality","kind":"node","lean":"TauCeti.isGreatest_ofReal_integral_sub_integral_boundedLipschitz","parent":3073,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport/part-01/index.html#bounded-lipschitz-kantorovich-rubinstein-duality","scope":"tauceti/measure-theory/optimal-transport/part-01","status":"fully_proved","summary":"The cost capped at `2 * R` is the greatest difference of integrals of a `1`-Lipschitz function bounded by `R`, for finite measures of equal mass. Both constraints are separate; this is the maximum-norm convention, not their sum.","title":"Bounded-Lipschitz Kantorovich\u2013Rubinstein duality with attainment"},{"id":"tauceti/measure-theory/optimal-transport/part-01/cauchy-schwarz-for-the-pairing","kind":"node","lean":"TauCeti.IsOptimalCoupling.abs_integral_inner_le","parent":3073,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport/part-01/index.html#cauchy-schwarz-for-the-pairing","scope":"tauceti/measure-theory/optimal-transport/part-01","status":"fully_proved","summary":"For an optimal coupling `\u03b3` of two laws `\u03bc` and `\u03bd` of `P\u2082(E)` for the quadratic cost and `\u03be \u2208 L\u00b2(\u03bc; E)`, `|\u222b \u27ea\u03be x, y - x\u27eb d\u03b3| \u2264 \u2016\u03be\u2016 W\u2082(\u03bc, \u03bd)`.","title":"Cauchy\u2013Schwarz for the pairing"},{"id":"tauceti/measure-theory/optimal-transport/part-01/finite-multi-marginal-kantorovich-duality","kind":"node","lean":"TauCeti.exists_cost_eq_finiteMultiDualValue","parent":3073,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport/part-01/index.html#finite-multi-marginal-kantorovich-duality","scope":"tauceti/measure-theory/optimal-transport/part-01","status":"fully_proved","summary":"For a nonempty finite family of finite spaces, the primal and dual problems have the same value and both attain it.","title":"Finite multi-marginal Kantorovich duality"},{"id":"tauceti/measure-theory/optimal-transport/part-01/the-frechet-hoeffding-bound-on-a-quadrant","kind":"node","lean":"TauCeti.ofReal_cdf_sub_le_measure","parent":3073,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport/part-01/index.html#the-frechet-hoeffding-bound-on-a-quadrant","scope":"tauceti/measure-theory/optimal-transport/part-01","status":"fully_proved","summary":"Every transport plan of two real laws gives the quadrant `Iic a \u00d7\u02e2 Ioi b` mass at least `cdf \u03bc a - cdf \u03bd b`: the pairs with first coordinate at most `a` carry mass `cdf \u03bc a`, and at most `cdf \u03bd b` of it has second coordinate at most `b`.","title":"The Fr\u00e9chet\u2013Hoeffding bound on a quadrant"},{"id":"tauceti/measure-theory/optimal-transport/part-01/the-generalised-bonnet-myers-theorem-under-cd-k","kind":"node","lean":"TauCeti.MetricMeasureSpace.CurvatureDimension.dist_le","parent":3073,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport/part-01/index.html#the-generalised-bonnet-myers-theorem-under-cd-k","scope":"tauceti/measure-theory/optimal-transport/part-01","status":"fully_proved","summary":"If `M` satisfies `CD(K, N)` with `K > 0` and `N \u2265 1`, any two points of the support of the reference measure are at distance at most `\u03c0 \u221a((N - 1) / K)`.","title":"The generalised Bonnet\u2013Myers theorem under `CD(K, N)`"},{"id":"tauceti/measure-theory/optimal-transport/part-01/the-gibbs-identity","kind":"node","lean":"TauCeti.lintegral_add_mul_klDiv_eq_mul_klDiv_tilted","parent":3073,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport/part-01/index.html#the-gibbs-identity","scope":"tauceti/measure-theory/optimal-transport/part-01","status":"fully_proved","summary":"For probability measures `\u03bc` and `\u03bd`, a cost `c` finite `\u03bc.prod \u03bd`-almost everywhere, and a positive temperature `\u03b5`, every probability measure `\u03c0` on `X \u00d7 Y` satisfies `\u222b c d\u03c0 + \u03b5 * klDiv \u03c0 (\u03bc \u2297 \u03bd) = \u03b5 * klDiv \u03c0 R - \u03b5 * log Z`, where `Z = \u222b e^{-c/\u03b5} d(\u03bc \u2297 \u03bd)` and `R = Z\u207b\u00b9 e^{-c/\u03b5} (\u03bc \u2297 \u03bd)` is the Gibbs measure, written as the tilted measure `(\u03bc.prod \u03bd).tilted fun z \u21a6 -((c z).toReal / \u03b5)`. Both sides may be `\u221e`.","title":"The Gibbs identity"},{"id":"tauceti/measure-theory/optimal-transport/part-01/kantorovich-rubinstein-duality","kind":"node","lean":"TauCeti.wassersteinEDist_one_eq_iSup, TauCeti.wassersteinEDist_one_eq_iSup_of_compactSpace, TauCeti.wassersteinEDist_one_eq_iSup_apply_eq_zero","parent":3073,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport/part-01/index.html#kantorovich-rubinstein-duality","scope":"tauceti/measure-theory/optimal-transport/part-01","status":"fully_proved","summary":"For probability measures with finite first moments on a second-countable pseudometric space whose measurable structure is standard Borel and contains the open sets \u2014 in particular on a Polish metric space with its Borel \u03c3-algebra \u2014 the `1`-Wasserstein distance is the supremum of the differences of expectations of the `1`-Lipschitz real functions.","title":"Kantorovich\u2013Rubinstein duality"},{"id":"tauceti/measure-theory/optimal-transport/part-01/kantorovich-rubinstein-weak-duality","kind":"node","lean":"TauCeti.ofReal_integral_sub_integral_le_wassersteinEDist_one","parent":3073,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport/part-01/index.html#kantorovich-rubinstein-weak-duality","scope":"tauceti/measure-theory/optimal-transport/part-01","status":"fully_proved","summary":"The difference of the expectations of an integrable `1`-Lipschitz real function under two measures is at most their `1`-Wasserstein distance. No measurability of the ground distance and no finiteness of the measures is needed.","title":"Kantorovich\u2013Rubinstein weak duality"},{"id":"tauceti/measure-theory/optimal-transport/part-01/markovs-inequality-for-the-wasserstein-distance","kind":"node","lean":"TauCeti.ofReal_mul_measure_compl_closedBall_le_wassersteinEDist","parent":3073,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport/part-01/index.html#markovs-inequality-for-the-wasserstein-distance","scope":"tauceti/measure-theory/optimal-transport/part-01","status":"fully_proved","summary":"For a probability measure `\u03bc` and a radius `r`, the mass of `\u03bc` outside the closed ball of radius `r` about `x`, multiplied by `r`, is at most `W_p (\u03b4_x, \u03bc)`, for every exponent `1 \u2264 p`.","title":"Markov's inequality for the Wasserstein distance from a Dirac law"},{"id":"tauceti/measure-theory/optimal-transport/part-01/the-one-dimensional-kantorovich-formula","kind":"node","lean":"TauCeti.wassersteinEDist_one_eq_lintegral_enorm_cdf_sub","parent":3073,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport/part-01/index.html#the-one-dimensional-kantorovich-formula","scope":"tauceti/measure-theory/optimal-transport/part-01","status":"fully_proved","summary":"The Wasserstein distance at exponent one of two probability laws on `\u211d` is the area between their cumulative distribution functions.","title":"The one-dimensional Kantorovich formula"},{"id":"tauceti/measure-theory/optimal-transport/part-01/polish-kantorovich-duality-for-costs-bounded-bel","kind":"node","lean":"TauCeti.isLUB_kantorovichDualValue_of_lowerSemicontinuous_residual_bddBelow","parent":3073,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport/part-01/index.html#polish-kantorovich-duality-for-costs-bounded-bel","scope":"tauceti/measure-theory/optimal-transport/part-01","status":"fully_proved","summary":"If the nonnegative residual of an integrable split lower bound is lower semicontinuous, the signed primal value is the least upper bound of the values of all integrable feasible real potentials. The value may be negative or `\u221e`; no integrable split upper envelope or dual attainment is assumed.","title":"Polish Kantorovich duality for costs bounded below"},{"id":"tauceti/measure-theory/optimal-transport/part-01/pratellis-theorem","kind":"node","lean":"TauCeti.mongeCost_eq_transportCost_of_continuous","parent":3073,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport/part-01/index.html#pratellis-theorem","scope":"tauceti/measure-theory/optimal-transport/part-01","status":"fully_proved","summary":"For an atomless finite source measure, a nonempty target type and a continuous cost `c : X \u00d7 Y \u2192 [0, \u221e]` on second-countable standard Borel spaces, for instance Polish spaces, the Monge and Kantorovich values agree, whether finite or not. The cost may be unbounded and take the value `\u221e`, and the Monge infimum need not be attained.","title":"Pratelli's theorem"},{"id":"tauceti/measure-theory/optimal-transport/part-01/prokhorovs-theorem-for-multi-couplings","kind":"node","lean":"TauCeti.isCompact_setOfPred_isMultiCoupling","parent":3073,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport/part-01/index.html#prokhorovs-theorem-for-multi-couplings","scope":"tauceti/measure-theory/optimal-transport/part-01","status":"fully_proved","summary":"The multi-couplings of finitely many tight probability measures form a weakly compact set: it is tight, hence relatively compact, and closed.","title":"Prokhorov's theorem for multi-couplings"},{"id":"tauceti/measure-theory/optimal-transport/part-01/the-pythagorean-identity-of-the-schrodinger-prob","kind":"node","lean":"TauCeti.IsCoupling.klDiv_eq_klDiv_add_klDiv_of_eq_withDensity","parent":3073,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport/part-01/index.html#the-pythagorean-identity-of-the-schrodinger-prob","scope":"tauceti/measure-theory/optimal-transport/part-01","status":"fully_proved","summary":"Let `\u03c0` be a coupling of `\u03bc` and `\u03bd` whose density against the finite reference `R` is `exp (\u03c6(x) + \u03c8(y))`, for measurable potentials `\u03c6 \u2208 L\u00b9(\u03bc)` and `\u03c8 \u2208 L\u00b9(\u03bd)`. Then every coupling `\u03b3` of `\u03bc` and `\u03bd` satisfies `klDiv \u03b3 R = klDiv \u03b3 \u03c0 + klDiv \u03c0 R`.","title":"The Pythagorean identity of the Schr\u00f6dinger problem"},{"id":"tauceti/measure-theory/optimal-transport/part-01/rockafellars-theorem","kind":"node","lean":"TauCeti.IsCyclicallyMonotone.exists_fenchelConjugate_subset_subdifferential, TauCeti.IsCyclicallyMonotone.exists_fenchelConjugate_inner\u2097_subset_subdifferential","parent":3073,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport/part-01/index.html#rockafellars-theorem","scope":"tauceti/measure-theory/optimal-transport/part-01","status":"fully_proved","summary":"A cyclically monotone set lies in the graph of the subdifferential of a Legendre\u2013Fenchel conjugate `g\u22c6`, hence of a convex function that is lower semicontinuous for every topology compatible with the pairing, and is never `\u22a5` as soon as the set is nonempty (`TauCeti.apply_ne_bot_of_mem_subdifferential`).","title":"Rockafellar's theorem"},{"id":"tauceti/measure-theory/optimal-transport/part-01/the-schachermayer-teichmann-theorem","kind":"node","lean":"TauCeti.IsCoupling.isOptimalCoupling_iff_exists_isCyclicallyMonotone","parent":3073,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport/part-01/index.html#the-schachermayer-teichmann-theorem","scope":"tauceti/measure-theory/optimal-transport/part-01","status":"fully_proved","summary":"For a finite-valued lower semicontinuous cost on a product of Polish spaces, a coupling of finite cost between two probability measures is optimal if and only if it is concentrated on a measurable `c`-cyclically monotone set.","title":"The Schachermayer--Teichmann theorem"},{"id":"tauceti/measure-theory/optimal-transport/part-01/strong-kantorovich-duality-for-probability-measu","kind":"node","lean":"TauCeti.toReal_transportCost_eq_sSup, TauCeti.isLUB_kantorovichDualValue_integrable, TauCeti.isLUB_ofReal_kantorovichDualValue_continuous_of_lowerSemicontinuous, TauCeti.isLUB_ofReal_kantorovichDualValue_integrable_of_lowerSemicontinuous, TauCeti.transportCost_eq_ofReal_sSup, TauCeti.isLUB_kantorovichDualValue_continuous, TauCeti.isGreatest_kantorovichDualValue_integrable","parent":3073,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport/part-01/index.html#strong-kantorovich-duality-for-probability-measu","scope":"tauceti/measure-theory/optimal-transport/part-01","status":"fully_proved","summary":"Strong Kantorovich duality for probability measures, read as an equality between the primal value and the supremum of the dual values over the continuous potentials.","title":"Strong Kantorovich duality for probability measures"},{"data_url":"scopes/tauceti/measure-theory/optimal-transport/part-02.json","id":"tauceti/measure-theory/optimal-transport/part-02","kind":"scope","navigation":"transparent","parent":3072,"result":false,"route":"../roadmap/tauceti/measure-theory/optimal-transport/part-02/index.html","scope":"tauceti/measure-theory/optimal-transport/part-02","status":"fully_proved","title":"The theorem of Rockafellar and\u2026 to The twist theorem at a point"},{"id":"tauceti/measure-theory/optimal-transport/part-02/the-theorem-of-rockafellar-and-ruschendorf","kind":"node","lean":"TauCeti.IsCyclicallyMonotone.exists_isCConcave_subset_cSuperdifferential","parent":3094,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport/part-02/index.html#the-theorem-of-rockafellar-and-ruschendorf","scope":"tauceti/measure-theory/optimal-transport/part-02","status":"fully_proved","summary":"Every `c`-cyclically monotone set is contained in the `c`-superdifferential of a `c`-concave potential: there is a `\u03c6` with `\u03c6 x + \u03c6\u1d9c y = c (x, y)` at every point `(x, y)` of the set, where `\u03c6\u1d9c` is the infimal `c`-transform of `\u03c6`.","title":"The theorem of Rockafellar and R\u00fcschendorf"},{"id":"tauceti/measure-theory/optimal-transport/part-02/the-twist-theorem","kind":"node","lean":"TauCeti.ae_subsingleton_setOf_mem_contactSet","parent":3094,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport/part-02/index.html#the-twist-theorem","scope":"tauceti/measure-theory/optimal-transport/part-02","status":"fully_proved","summary":"If the cost is differentiable in its first variable at contact partners and `y \u21a6 D\u2093 c (x, y)` is injective among differentiable sections at almost every `x`, and a dual feasible pair has a first potential finite near, and differentiable at, `\u03bc`-almost every point, then the contact fibres `{y | (x, y) \u2208 contactSet c \u03c6 \u03c8}` are subsingletons for `\u03bc`-almost every `x`. Combined with\u2026","title":"The twist theorem"},{"id":"tauceti/measure-theory/optimal-transport/part-02/the-twist-theorem-at-a-point","kind":"node","lean":"TauCeti.subsingleton_setOf_mem_contactSet","parent":3094,"result":true,"route":"../roadmap/tauceti/measure-theory/optimal-transport/part-02/index.html#the-twist-theorem-at-a-point","scope":"tauceti/measure-theory/optimal-transport/part-02","status":"fully_proved","summary":"Suppose the cost is twisted at `x`: the map `y \u21a6 D\u2093 c (x, y)` is injective on the set of `y` for which the section `c (\u00b7, y)` is differentiable at `x`. If the potential is differentiable at `x`, finite near `x`, and satisfies the dual constraint near `x`, and the cost sections of the contact partners of `x` are differentiable at `x`, then `x` has at most one contact partner.","title":"The twist theorem at a point"},{"data_url":"scopes/tauceti/measure-theory/other-results.json","id":"tauceti/measure-theory/other-results","kind":"scope","parent":2980,"result":false,"route":"../roadmap/tauceti/measure-theory/other-results/index.html","scope":"tauceti/measure-theory/other-results","status":"fully_proved","summary":"Results named in the rest of Tau Ceti's `TauCeti.MeasureTheory` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Other results"},{"id":"tauceti/measure-theory/other-results/lebesgues-density-theorem","kind":"node","lean":"TauCeti.ae_hasDensityAt_indicator, MeasureTheory.NullMeasurableSet.ae_tendsto_measure_inter_div, TauCeti.essentialExterior_ae_eq, TauCeti.essentialInterior_ae_eq, TauCeti.measure_essentialBoundary","parent":3098,"result":true,"route":"../roadmap/tauceti/measure-theory/other-results/index.html#lebesgues-density-theorem","scope":"tauceti/measure-theory/other-results","status":"fully_proved","summary":"At almost every point a null-measurable set has density one or zero, according to whether the point lies in the set.","title":"Lebesgue's density theorem"},{"id":"tauceti/measure-theory/other-results/lebesgues-differentiation-theorem","kind":"node","lean":"Besicovitch.ae_tendsto_average_closedBall, IsUnifLocDoublingMeasure.ae_tendsto_average_closedBall_norm_sub","parent":3098,"result":true,"route":"../roadmap/tauceti/measure-theory/other-results/index.html#lebesgues-differentiation-theorem","scope":"tauceti/measure-theory/other-results","status":"fully_proved","summary":"Lebesgue's differentiation theorem for a locally finite measure in a space with the Besicovitch covering property: for a locally integrable function `f`, at almost every point `x` the averages of `f` over the closed balls around `x` tend to `f x`.","title":"Lebesgue's differentiation theorem"},{"id":"tauceti/measure-theory/other-results/minkowskis-convex-body-theorem-for-the-integer-l","kind":"node","lean":"MeasureTheory.exists_ne_zero_int_vec_mem_of_two_pow_le_volume","parent":3098,"result":true,"route":"../roadmap/tauceti/measure-theory/other-results/index.html#minkowskis-convex-body-theorem-for-the-integer-l","scope":"tauceti/measure-theory/other-results","status":"fully_proved","summary":"A compact convex set `s \u2286 \u03b7 \u2192 \u211d`, symmetric about the origin, of volume at least `2 ^ |\u03b7|` contains a nonzero integer vector.","title":"Minkowski's convex body theorem for the integer lattice"},{"id":"tauceti/measure-theory/other-results/scalar-radon-nikodym-theorem","kind":"node","lean":"MeasureTheory.VectorMeasure.withDensity\u1d65_scalarRNDeriv","parent":3098,"result":true,"route":"../roadmap/tauceti/measure-theory/other-results/index.html#scalar-radon-nikodym-theorem","scope":"tauceti/measure-theory/other-results","status":"fully_proved","summary":"If the variation of a scalar measure is absolutely continuous with respect to a sigma-finite measure, its scalar density reconstructs it.","title":"Scalar Radon\u2013Nikodym theorem"},{"data_url":"scopes/tauceti/model-theory.json","id":"tauceti/model-theory","kind":"scope","parent":1095,"result":false,"route":"../roadmap/tauceti/model-theory/index.html","scope":"tauceti/model-theory","status":"fully_proved","summary":"Results in Tau Ceti's `TauCeti.ModelTheory` modules, and the roadmaps whose articles are mostly there: by roadmap, then by area.","title":"Model Theory"},{"data_url":"scopes/tauceti/model-theory/infinitary-model-existence-omitting-types-and-in.json","id":"tauceti/model-theory/infinitary-model-existence-omitting-types-and-in","kind":"scope","parent":3103,"result":false,"route":"../roadmap/tauceti/model-theory/infinitary-model-existence-omitting-types-and-in/index.html","scope":"tauceti/model-theory/infinitary-model-existence-omitting-types-and-in","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Infinitary model existence, omitting types and interpolation"},{"id":"tauceti/model-theory/infinitary-model-existence-omitting-types-and-in/firstorder-language-theory","kind":"node","lean":"FirstOrder.Language.Theory\u03c9","parent":3104,"result":true,"route":"../roadmap/tauceti/model-theory/infinitary-model-existence-omitting-types-and-in/index.html#firstorder-language-theory","scope":"tauceti/model-theory/infinitary-model-existence-omitting-types-and-in","status":"fully_proved","summary":"An `L_{\u03c9\u2081\u03c9}` theory: a set of `L_{\u03c9\u2081\u03c9}` sentences.","title":"FirstOrder.Language.Theory\u03c9"},{"id":"tauceti/model-theory/infinitary-model-existence-omitting-types-and-in/firstorder-language-theory-model","kind":"node","lean":"FirstOrder.Language.Theory\u03c9.Model","parent":3104,"result":true,"route":"../roadmap/tauceti/model-theory/infinitary-model-existence-omitting-types-and-in/index.html#firstorder-language-theory-model","scope":"tauceti/model-theory/infinitary-model-existence-omitting-types-and-in","status":"fully_proved","summary":"A structure is a model of an `L_{\u03c9\u2081\u03c9}` theory when it satisfies every sentence of the theory.","title":"FirstOrder.Language.Theory\u03c9.Model"},{"id":"tauceti/model-theory/infinitary-model-existence-omitting-types-and-in/firstorder-language-stronghomclass-theory-model","kind":"node","lean":"FirstOrder.Language.StrongHomClass.theory\u03c9_model_iff","parent":3104,"result":true,"route":"../roadmap/tauceti/model-theory/infinitary-model-existence-omitting-types-and-in/index.html#firstorder-language-stronghomclass-theory-model","scope":"tauceti/model-theory/infinitary-model-existence-omitting-types-and-in","status":"fully_proved","summary":"Isomorphic structures are models of the same `L_{\u03c9\u2081\u03c9}` theories.","title":"FirstOrder.Language.StrongHomClass.theory\u03c9_model_iff"},{"id":"tauceti/model-theory/infinitary-model-existence-omitting-types-and-in/firstorder-language-theory-conjunction","kind":"node","lean":"FirstOrder.Language.Theory\u03c9.conjunction","parent":3104,"result":true,"route":"../roadmap/tauceti/model-theory/infinitary-model-existence-omitting-types-and-in/index.html#firstorder-language-theory-conjunction","scope":"tauceti/model-theory/infinitary-model-existence-omitting-types-and-in","status":"fully_proved","summary":"The conjunction of a countable theory: its sentences, enumerated into the carrier `\u2115` along `IndexCoding.ofEncodableWith`, joined by one countable conjunction. Indices that code no sentence carry `\u22a4`.","title":"FirstOrder.Language.Theory\u03c9.conjunction"},{"id":"tauceti/model-theory/infinitary-model-existence-omitting-types-and-in/a-countable-theory-is-one-sentence","kind":"node","lean":"FirstOrder.Language.Theory\u03c9.realize_conjunction_iff","parent":3104,"result":true,"route":"../roadmap/tauceti/model-theory/infinitary-model-existence-omitting-types-and-in/index.html#a-countable-theory-is-one-sentence","scope":"tauceti/model-theory/infinitary-model-existence-omitting-types-and-in","status":"fully_proved","summary":"A structure satisfies the conjunction of a countable theory exactly when it is a model of the theory.","title":"A countable theory is one sentence"},{"data_url":"scopes/tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy.json","id":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy","kind":"scope","parent":3103,"result":false,"route":"../roadmap/tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/index.html","scope":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Infinitary syntax, back-and-forth, and Scott analysis"},{"id":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/scotts-isomorphism-theorem","kind":"node","lean":"TauCeti.scott_isomorphism","parent":3110,"result":true,"route":"../roadmap/tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/index.html#scotts-isomorphism-theorem","scope":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy","status":"fully_proved","summary":"In a countable language, every countable structure `M` has an `L\u03c9\u2081\u03c9` sentence that `M` satisfies and that holds in a countable structure, in any universe, exactly when that structure is isomorphic to `M`.","title":"Scott's isomorphism theorem"},{"id":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/tauceti-realize-scottsentenceat-iff","kind":"node","lean":"TauCeti.realize_scottSentenceAt_iff","parent":3110,"result":true,"route":"../roadmap/tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/index.html#tauceti-realize-scottsentenceat-iff","scope":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy","status":"fully_proved","summary":"At a countable internally stabilizing level, the Scott sentence of a countable structure `M` holds in a countable structure exactly when that structure is isomorphic to `M`.","title":"TauCeti.realize_scottSentenceAt_iff"},{"id":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/karps-theorem-at-a-common-carrier","kind":"node","lean":"TauCeti.karp_theorem_at","parent":3110,"result":true,"route":"../roadmap/tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/index.html#karps-theorem-at-a-common-carrier","scope":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy","status":"fully_proved","summary":"Structures in a relational language that are both coded into `\u03ba` are potentially isomorphic exactly when they satisfy the same infinitary sentences with carrier `\u03ba`.","title":"Karp's theorem at a common carrier"},{"id":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/karps-theorem","kind":"node","lean":"TauCeti.karp_theorem, TauCeti.InfEquivAt.potentialIso, TauCeti.PotentialIso.infEquivAt","parent":3110,"result":true,"route":"../roadmap/tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/index.html#karps-theorem","scope":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy","status":"fully_proved","summary":"Structures in a relational language are potentially isomorphic exactly when they satisfy the same infinitary sentences. The converse uses the carrier `M \u2295 N`.","title":"Karp's theorem"},{"id":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/karps-theorem-for-countable-structures","kind":"node","lean":"TauCeti.countable_infEquivAt_nat_iff_iso","parent":3110,"result":true,"route":"../roadmap/tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/index.html#karps-theorem-for-countable-structures","scope":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy","status":"fully_proved","summary":"Countable structures in a relational language are isomorphic exactly when they satisfy the same `L_{\u03c9\u2081\u03c9}` sentences, the infinitary sentences whose conjunctions and disjunctions are indexed by `\u2115`.","title":"Karp's theorem for countable structures"},{"id":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/tauceti-realize-scottformula-iff-bfequiv","kind":"node","lean":"TauCeti.realize_scottFormula_iff_bfEquiv","parent":3110,"result":true,"route":"../roadmap/tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/index.html#tauceti-realize-scottformula-iff-bfequiv","scope":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy","status":"fully_proved","summary":"A tuple in any structure satisfies the canonical Scott formula exactly when it is back-and-forth equivalent to the original tuple at that countable ordinal.","title":"TauCeti.realize_scottFormula_iff_bfEquiv"},{"id":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/tauceti-internalscottrank-lt-omega-one","kind":"node","lean":"TauCeti.internalScottRank_lt_omega_one","parent":3110,"result":true,"route":"../roadmap/tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/index.html#tauceti-internalscottrank-lt-omega-one","scope":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy","status":"fully_proved","summary":"Every countable structure has internal Scott rank below the first uncountable ordinal. The relational language need not be countable.","title":"TauCeti.internalScottRank_lt_omega_one"},{"id":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/tauceti-selfstabilizationordinal-eq-isup-orbitra","kind":"node","lean":"TauCeti.selfStabilizationOrdinal_eq_iSup_orbitRank","parent":3110,"result":true,"route":"../roadmap/tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/index.html#tauceti-selfstabilizationordinal-eq-isup-orbitra","scope":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy","status":"fully_proved","summary":"The least simultaneous stabilization level is the supremum of the orbit ranks of all finite tuples. This holds for arbitrary structures.","title":"TauCeti.selfStabilizationOrdinal_eq_iSup_orbitRank"},{"id":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/tauceti-elementarilyequivalent-int-sumlex-int","kind":"node","lean":"TauCeti.elementarilyEquivalent_int_sumLex_int","parent":3110,"result":true,"route":"../roadmap/tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/index.html#tauceti-elementarilyequivalent-int-sumlex-int","scope":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy","status":"fully_proved","summary":"`\u2124` and `\u2124 \u2295\u2097 \u2124` satisfy the same first-order sentences in the language of orders.","title":"TauCeti.elementarilyEquivalent_int_sumLex_int"},{"id":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/elementary-equivalence-is-strictly-weaker-than-i","kind":"node","lean":"TauCeti.not_infEquivAt_nat_int_sumLex_int","parent":3110,"result":true,"route":"../roadmap/tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/index.html#elementary-equivalence-is-strictly-weaker-than-i","scope":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy","status":"fully_proved","summary":"The countable orders `\u2124` and `\u2124 \u2295\u2097 \u2124` are elementarily equivalent (`elementarilyEquivalent_int_sumLex_int`), but some `L_{\u03c9\u2081\u03c9}` sentence distinguishes them. A fortiori they are not `L_{\u221e\u03c9}`-equivalent, since `TauCeti.InfEquivW.infEquivAt` turns `L_{\u221e\u03c9}`-equivalence into agreement at `\u2115`.","title":"Elementary equivalence is strictly weaker than infinitary equivalence"},{"id":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/tauceti-exists-dlo-aleph-one-potentialiso-isempt","kind":"node","lean":"TauCeti.exists_dlo_aleph_one_potentialIso_isEmpty_equiv","parent":3110,"result":true,"route":"../roadmap/tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/index.html#tauceti-exists-dlo-aleph-one-potentialiso-isempt","scope":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy","status":"fully_proved","summary":"Potential isomorphism, and hence `L\u221e\u03c9`-equivalence, does not imply isomorphism for uncountable structures of the same cardinality: in every universe there are two dense linear orders without endpoints of cardinality `\u2135\u2081` that are potentially isomorphic, and so `L\u221e\u03c9`-equivalent (`TauCeti.InfEquivW`), but not isomorphic. The witnesses are `\u03c9\u2081 \u00d7\u2097 \u211a` and its order dual.","title":"TauCeti.exists_dlo_aleph_one_potentialIso_isEmpty_equiv"},{"id":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/firstorder-language-boundedformulainf","kind":"node","lean":"FirstOrder.Language.BoundedFormulaInf","parent":3110,"result":true,"route":"../roadmap/tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/index.html#firstorder-language-boundedformulainf","scope":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy","status":"fully_proved","summary":"An infinitary bounded formula of `L_{\u221e\u03c9}`, with infinitary conjunctions and disjunctions branching over the fixed carrier `\u03b9`, free variables indexed by `\u03b1`, and `n` additional bound variables available.","title":"FirstOrder.Language.BoundedFormulaInf"},{"id":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/transport-a-formula-along-a-coding-of-its-carrie","kind":"node","lean":"FirstOrder.Language.BoundedFormulaInf.reindex","parent":3110,"result":true,"route":"../roadmap/tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/index.html#transport-a-formula-along-a-coding-of-its-carrie","scope":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy","status":"fully_proved","summary":"each `\u03b9`-indexed conjunction or disjunction becomes the corresponding coded connective at carrier `\u03ba`.","title":"Transport a formula along a coding of its carrier"},{"id":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/firstorder-language-boundedformulainf-freevarsup","kind":"node","lean":"FirstOrder.Language.BoundedFormulaInf.freeVarSupport","parent":3110,"result":true,"route":"../roadmap/tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/index.html#firstorder-language-boundedformulainf-freevarsup","scope":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy","status":"fully_proved","summary":"The free-variable support of an infinitary formula: the set of free variables that occur in one of its atoms. Infinitary connectives take the union over all their branches, so the support may be infinite.","title":"FirstOrder.Language.BoundedFormulaInf.freeVarSupport"},{"id":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/tauceti-bfequiv","kind":"node","lean":"TauCeti.BFEquiv","parent":3110,"result":true,"route":"../roadmap/tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/index.html#tauceti-bfequiv","scope":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy","status":"fully_proved","summary":"Ordinal back-and-forth equivalence of tuples. At successors the previous level is retained, in addition to the forth and back requirements. The language is relational.","title":"TauCeti.BFEquiv"},{"id":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/tauceti-potentialiso","kind":"node","lean":"TauCeti.PotentialIso","parent":3110,"result":true,"route":"../roadmap/tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/index.html#tauceti-potentialiso","scope":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy","status":"fully_proved","summary":"Two structures are potentially isomorphic if a nonempty set of finitely generated partial equivalences admits extensions containing every element on either side.","title":"TauCeti.PotentialIso"},{"id":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/tauceti-potentialiso-trans","kind":"node","lean":"TauCeti.PotentialIso.trans","parent":3110,"result":true,"route":"../roadmap/tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/index.html#tauceti-potentialiso-trans","scope":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy","status":"fully_proved","summary":"Potential isomorphism is transitive.","title":"TauCeti.PotentialIso.trans"},{"id":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/back-and-forth-equivalence-at-level-gives-agreem","kind":"node","lean":"TauCeti.BFEquiv.realize_iff_of_qrank_le","parent":3110,"result":true,"route":"../roadmap/tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/index.html#back-and-forth-equivalence-at-level-gives-agreem","scope":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy","status":"fully_proved","summary":"Tuples that are back-and-forth equivalent at level `\u03b1` satisfy the same infinitary formulas of quantifier rank at most `\u03b1`, at every carrier. The rank and the level are compared after lifting them to a common universe.","title":"Back-and-forth equivalence at level `\u03b1` gives agreement up to quantifier rank `\u03b1`"},{"id":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/tauceti-exists-complete-self-stabilization","kind":"node","lean":"TauCeti.exists_complete_self_stabilization","parent":3110,"result":true,"route":"../roadmap/tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy/index.html#tauceti-exists-complete-self-stabilization","scope":"tauceti/model-theory/infinitary-syntax-back-and-forth-and-scott-analy","status":"fully_proved","summary":"A countable structure stabilizes below `\u03c9\u2081`. Only its tuples are counted; the language may be uncountable.","title":"TauCeti.exists_complete_self_stabilization"},{"data_url":"scopes/tauceti/model-theory/counting.json","id":"tauceti/model-theory/counting","kind":"scope","parent":3103,"result":false,"route":"../roadmap/tauceti/model-theory/counting/index.html","scope":"tauceti/model-theory/counting","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.ModelTheory.Counting` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Counting"},{"id":"tauceti/model-theory/counting/morleys-cardinal-dichotomy-on-the-tier-n","kind":"node","lean":"TauCeti.mk_quotient_isoSetoidOn_nat_le_aleph_one_or_eq_continuum, TauCeti.mk_isoClasses_le_aleph_one_or_eq_continuum","parent":3130,"result":true,"route":"../roadmap/tauceti/model-theory/counting/index.html#morleys-cardinal-dichotomy-on-the-tier-n","scope":"tauceti/model-theory/counting","status":"fully_proved","summary":"In a relational language with countably many relation symbols of each arity, the models of `\u03c6` on `\u2115` have either at most `\u2135\u2081` or `\ud835\udd20` isomorphism classes.","title":"Morley's cardinal dichotomy on the tier `\u2115`"},{"id":"tauceti/model-theory/counting/morleys-dichotomy-on-the-tier-n","kind":"node","lean":"TauCeti.mk_quotient_isoSetoidOn_nat_le_aleph_one_or_hasPerfectAntichainOn, TauCeti.mk_isoClasses_le_aleph_one_or_hasPerfectAntichainOn","parent":3130,"result":true,"route":"../roadmap/tauceti/model-theory/counting/index.html#morleys-dichotomy-on-the-tier-n","scope":"tauceti/model-theory/counting","status":"fully_proved","summary":"In a relational language with countably many relation symbols of each arity, the models on `\u2115` of an infinitary sentence with countable branching have at most `\u2135\u2081` isomorphism classes, or contain a nonempty perfect set of pairwise non-isomorphic models.","title":"Morley's dichotomy on the tier `\u2115`"},{"id":"tauceti/model-theory/counting/morleys-theorem-cardinal-form","kind":"node","lean":"TauCeti.morley_countableModelCount, TauCeti.morley_countableModelCount_le_aleph_one_or_hasPerfectlyMany","parent":3130,"result":true,"route":"../roadmap/tauceti/model-theory/counting/index.html#morleys-theorem-cardinal-form","scope":"tauceti/model-theory/counting","status":"fully_proved","summary":"In a relational language with countably many relation symbols of each arity, an infinitary sentence with countable branching has either at most `\u2135\u2081` or `\ud835\udd20` countable models up to isomorphism. Neither the continuum hypothesis nor its negation is assumed.","title":"Morley's theorem (cardinal form)"},{"data_url":"scopes/tauceti/model-theory/model-existence.json","id":"tauceti/model-theory/model-existence","kind":"scope","parent":3103,"result":false,"route":"../roadmap/tauceti/model-theory/model-existence/index.html","scope":"tauceti/model-theory/model-existence","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.ModelTheory.ModelExistence` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Model Existence"},{"id":"tauceti/model-theory/model-existence/the-fragment-tarski-vaught-criterion","kind":"node","lean":"FirstOrder.Language.aElementary_of_tarskiVaught","parent":3134,"result":true,"route":"../roadmap/tauceti/model-theory/model-existence/index.html#the-fragment-tarski-vaught-criterion","scope":"tauceti/model-theory/model-existence","status":"fully_proved","summary":"If, for every universal member `\u2200 x, \u03c6` of `A` and every tuple `a` from `N`, whenever `\u03c6` has a counterexample at `f \u2218 a` in `M` it has one in the image of `f`, then `f` is `A`-elementary.","title":"The fragment Tarski\u2013Vaught criterion"},{"id":"tauceti/model-theory/model-existence/the-model-existence-theorem-with-omitted-types","kind":"node","lean":"FirstOrder.Language.ConsistencyProperty.model_existence_omitting","parent":3134,"result":true,"route":"../roadmap/tauceti/model-theory/model-existence/index.html#the-model-existence-theorem-with-omitted-types","scope":"tauceti/model-theory/model-existence","status":"fully_proved","summary":"Over a language with countably many function and relation symbols, let `p i` be countably many countable partial types such that every member `t` of the consistency property `C` and every tuple `c` of Henkin constants admit some `\u03c8 \u2208 p i` for which `t` together with the refutation `\u00ac\u03c8[c]` is again a member. Then every countable member `s` of `C` has a countable nonempty model, in any universe `w`, in which every\u2026","title":"The model existence theorem with omitted types"},{"id":"tauceti/model-theory/model-existence/the-omitting-types-theorem-for-a-countable-forma","kind":"node","lean":"FirstOrder.Language.Theory\u03c9.exists_countable_model_omitting","parent":3134,"result":true,"route":"../roadmap/tauceti/model-theory/model-existence/index.html#the-omitting-types-theorem-for-a-countable-forma","scope":"tauceti/model-theory/model-existence","status":"fully_proved","summary":"A satisfiable theory of sentences of the fragment has a countable nonempty model, in any universe, omitting any countable family of partial types in the fragment, none isolated over the theory in the fragment. Countability of the language follows from countability and formation closure of the fragment.","title":"The omitting types theorem for a countable formation-closed fragment"},{"data_url":"scopes/tauceti/number-theory.json","id":"tauceti/number-theory","kind":"scope","parent":1095,"result":false,"route":"../roadmap/tauceti/number-theory/index.html","scope":"tauceti/number-theory","status":"fully_proved","summary":"Results in Tau Ceti's `TauCeti.NumberTheory` modules, and the roadmaps whose articles are mostly there: by roadmap, then by area.","title":"Number Theory"},{"data_url":"scopes/tauceti/number-theory/part-01.json","id":"tauceti/number-theory/part-01","kind":"scope","navigation":"transparent","parent":3138,"result":false,"route":"../roadmap/tauceti/number-theory/part-01/index.html","scope":"tauceti/number-theory/part-01","status":"fully_proved","title":"Arithmetic Dirichlet series and\u2026 to Theta series of lattices"},{"data_url":"scopes/tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method.json","id":"tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method","kind":"scope","parent":3139,"result":false,"route":"../roadmap/tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method/index.html","scope":"tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Arithmetic Dirichlet series and Tauberian methods"},{"id":"tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method/the-prime-ideal-theorem","kind":"node","lean":"TauCeti.primeIdealTheorem, TauCeti.primeCount_univ_isEquivalent_div_log","parent":3140,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method/index.html#the-prime-ideal-theorem","scope":"tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method","status":"fully_proved","summary":"For a number field `K`, Chebyshev's functions satisfy `\u03c8_K(x) ~ x` and `\u03d1_K(x) ~ x`, and the number `\u03c0_K(x)` of prime ideals of norm at most `x` satisfies `\u03c0_K(x) ~ Li(x)`.","title":"The prime ideal theorem"},{"id":"tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method/the-wiener-ikehara-theorem","kind":"node","lean":"TauCeti.LSeries.wienerIkehara, TauCeti.LSeries.wienerIkehara_nat, TauCeti.LSeries.wienerIkehara_rpow, TauCeti.LSeries.wienerIkehara_of_forall_monotone_dual, TauCeti.LSeries.wienerIkehara_of_eventually_nonneg, TauCeti.LSeries.wienerIkehara_zero","parent":3140,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method/index.html#the-wiener-ikehara-theorem","scope":"tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method","status":"fully_proved","summary":"Let `a n \u2265 0` have Dirichlet series with sum `F s` on `Re s > 1`, and let `G` be continuous on `Re s \u2265 1` with `G s = F s - \u03ba / (s - 1)` on `Re s > 1`. Then `x\u207b\u00b9 \u2211_{1 \u2264 n \u2264 x} a n \u2192 \u03ba` as `x \u2192 \u221e`.","title":"The Wiener--Ikehara theorem"},{"id":"tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method/prime-number-theorem-transfer-from-exact-boundar","kind":"node","lean":"TauCeti.primeNumberTheoremTransfer","parent":3140,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method/index.html#prime-number-theorem-transfer-from-exact-boundar","scope":"tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method","status":"fully_proved","summary":"The three conclusions are, respectively, the von Mangoldt-weighted prime-power asymptotic, the logarithmically weighted prime asymptotic after removing higher powers, and the unweighted prime-counting asymptotic after Abel summation. The error-term formulation includes residue zero.","title":"Prime-number-theorem transfer from exact boundary data"},{"id":"tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method/the-dedekind-zeta-function-has-no-zeros-on-the-l","kind":"node","lean":"TauCeti.ne_zero_of_eqOn_dedekindZeta","parent":3140,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method/index.html#the-dedekind-zeta-function-has-no-zeros-on-the-l","scope":"tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method","status":"fully_proved","summary":"If `f` agrees with `\u03b6_K` on `Re s > 1` and is complex differentiable at a point `s \u2260 1` with `Re s = 1`, then `f s \u2260 0`. By `exists_differentiableOn_eq_dedekindZeta_sub`, such an `f` exists: the meromorphic continuation of `\u03b6_K`.","title":"The Dedekind zeta function has no zeros on the line `Re s = 1` away from its pole"},{"id":"tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method/nonvanishing-when-the-square-also-cancels","kind":"node","lean":"TauCeti.continuedLFunctionOfWeight_ne_zero_of_hasCancellation_sq","parent":3140,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method/index.html#nonvanishing-when-the-square-also-cancels","scope":"tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method","status":"fully_proved","summary":"If a unitary weight `\u03c7` and its pointwise square `\u03c7 ^ 2` both have cancellation, the continued L-function of `\u03c7` has no zeros on the closed half-plane `Re s \u2265 1`.","title":"Nonvanishing when the square also cancels"},{"id":"tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method/the-analytic-euler-product","kind":"node","lean":"TauCeti.EulerProductData.hasProd_eulerFactor","parent":3140,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method/index.html#the-analytic-euler-product","scope":"tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method","status":"fully_proved","summary":"If the ideal-indexed Dirichlet series of `D` converges absolutely at `s`, then its local Euler factors have an unrestricted infinite product over the height-one primes, and that product is the `LSeries` of the norm coefficients of `D`.","title":"The analytic Euler product"},{"id":"tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method/tauceti-eulerproductdata-hasprod-eulerfactor-con","kind":"node","lean":"TauCeti.EulerProductData.hasProd_eulerFactor_convolution","parent":3140,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method/index.html#tauceti-eulerproductdata-hasprod-eulerfactor-con","scope":"tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method","status":"fully_proved","summary":"The Euler factors of a convolution have an unrestricted product equal to the product of the original norm-regrouped Dirichlet series, whenever both ideal-indexed series converge absolutely.","title":"TauCeti.EulerProductData.hasProd_eulerFactor_convolution"},{"id":"tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method/boundary-data-for-the-prime-dirichlet-series-of","kind":"node","lean":"TauCeti.PrimeBoundaryRemainder","parent":3140,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method/index.html#boundary-data-for-the-prime-dirichlet-series-of","scope":"tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method","status":"fully_proved","summary":"The function `series` is the sum of the Dirichlet series of `primeVonMangoldtCoeff K S` on `Re s > 1`. The function `remainder` is continuous on `Re s \u2265 1` and agrees on the open half-plane with `series s - \u03b4 / (s - 1)`.","title":"Boundary data for the prime Dirichlet series of `S`"},{"id":"tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method/boundary-data-for-all-primes-of-a-number-field","kind":"node","lean":"TauCeti.LFunctions.primeIdealVonMangoldtBoundary","parent":3140,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method/index.html#boundary-data-for-all-primes-of-a-number-field","scope":"tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method","status":"fully_proved","summary":"The von Mangoldt series of all primes of `K` sums to `-\u03b6_K'(s)/\u03b6_K(s)` on `Re s > 1`, and `-\u03b6_K'(s)/\u03b6_K(s) - 1/(s - 1)` extends continuously to `Re s \u2265 1` (`TauCeti.exists_continuousOn_eq_neg_deriv_dedekindZeta_div_sub`).","title":"Boundary data for all primes of a number field"},{"id":"tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method/the-continued-l-function-of-a-unitary-weight-def","kind":"node","lean":"TauCeti.continuedLFunctionOfWeight","parent":3140,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method/index.html#the-continued-l-function-of-a-unitary-weight-def","scope":"tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method","status":"fully_proved","summary":"`s * \u222b_{1}^{\u221e} (\u2211_{N(I) \u2264 t} \u03c7(I)) t ^ (-(s + 1)) dt`.","title":"The continued L-function of a unitary weight, defined by partial summation"},{"id":"tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method/tauceti-eulerproductdata-convolution","kind":"node","lean":"TauCeti.EulerProductData.convolution","parent":3140,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method/index.html#tauceti-eulerproductdata-convolution","scope":"tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method","status":"fully_proved","summary":"Convolve two ideal coefficient systems. This operation multiplies local series and is distinct from the pointwise multiplication instance on `EulerProductData`.","title":"TauCeti.EulerProductData.convolution"},{"id":"tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method/tauceti-multiplicativeidealweight-inststarmul","kind":"node","lean":"TauCeti.MultiplicativeIdealWeight.instStarMul","parent":3140,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method/index.html#tauceti-multiplicativeidealweight-inststarmul","scope":"tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method","status":"fully_proved","summary":"Conjugation is the `star` of the weights, as for `MulChar`, `TauCeti.IdealArithmeticFunction` and `TauCeti.EulerProductData`: an involution which is multiplicative, the product being commutative.","title":"TauCeti.MultiplicativeIdealWeight.instStarMul"},{"id":"tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method/the-conditional-prime-ideal-theorem","kind":"node","lean":"TauCeti.primeIdealTheorem_of_boundary","parent":3140,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method/index.html#the-conditional-prime-ideal-theorem","scope":"tauceti/number-theory/part-01/arithmetic-dirichlet-series-and-tauberian-method","status":"fully_proved","summary":"Boundary data for all prime ideals with residue one give the standard asymptotic equivalences for `\u03c8`, `\u03d1`, and `\u03c0`.","title":"The conditional prime ideal theorem"},{"data_url":"scopes/tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c.json","id":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c","kind":"scope","parent":3139,"result":false,"route":"../roadmap/tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/index.html","scope":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Bounded Laplace and ordered Dirichlet boundary convergence"},{"id":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/newmans-tauberian-theorem","kind":"node","lean":"TauCeti.LSeries.newmanLaplace_tendsto_integral, TauCeti.LSeries.newmanLaplace_tendsto_integral_of_analyticAt","parent":3154,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/index.html#newmans-tauberian-theorem","scope":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c","status":"fully_proved","summary":"Let `f : \u211d \u2192 \u2102` be measurable and bounded on `[0, \u221e)`, and let `G` be holomorphic on an open set `\u03a9` containing the closed half-plane `0 \u2264 re z`, agreeing with the Laplace transform `\u222b t in Ici 0, f t * exp (-z t)` on the open half-plane `0 < re z`. Then `\u222b t in 0..T, f t` converges to `G 0` as `T \u2192 \u221e`.","title":"Newman's Tauberian theorem"},{"id":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/newmans-estimate-at-a-fixed-radius","kind":"node","lean":"TauCeti.LSeries.eventually_norm_sub_intervalIntegral_le","parent":3154,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/index.html#newmans-estimate-at-a-fixed-radius","scope":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c","status":"fully_proved","summary":"Let `f` be measurable and bounded by `M` on `[0, \u221e)`, and let `G` be holomorphic on an open set `\u03a9` containing the closed half-plane `0 \u2264 re z` and equal to the Laplace transform of `f` on `0 < re z`. For every `R > 0` and `\u03b5 > 0`, eventually in `T`, `\u2016G 0 - \u222b t in 0..T, f t\u2016 \u2264 2 M / R + \u03b5`.","title":"Newman's estimate at a fixed radius"},{"id":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/newmans-contour-identity","kind":"node","lean":"TauCeti.LSeries.integral_right_semicircle_add_integral_newmanDetour_mul_kernel","parent":3154,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/index.html#newmans-contour-identity","scope":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c","status":"fully_proved","summary":"Let `G` be holomorphic on an open convex set `U` containing the right semicircle of radius `R > 0` and the detour `newmanDetour R \u03b4` with `\u03b4 > 0`, and let `G_T z = \u222b t in 0..T, f t * exp (-z t)`. Against Newman's kernel `k_T z = exp (T z) (1 + z\u00b2 / R\u00b2) / z`, the right semicircle integral of `(G - G_T) k_T` (counterclockwise from `-R * I` to `R * I`) plus the detour integral of `G k_T` (from `R * I` back to\u2026","title":"Newman's contour identity"},{"id":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/holomorphic-functions-have-primitives-on-a-star","kind":"node","lean":"TauCeti.Contour.isExactOn_of_starConvex","parent":3154,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/index.html#holomorphic-functions-have-primitives-on-a-star","scope":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c","status":"fully_proved","summary":"If `U` is open and star-shaped about some point, then every function holomorphic on `U` has a primitive on `U`.","title":"Holomorphic functions have primitives on a star-shaped set"},{"id":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/holomorphy-of-the-step-correction","kind":"node","lean":"TauCeti.LSeries.differentiableOn_stepCorrection","parent":3154,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/index.html#holomorphy-of-the-step-correction","scope":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c","status":"fully_proved","summary":"For coefficients bounded from `n = 1` on, the step correction is holomorphic on the half-plane `-1 < re z`: near `z\u2080`, the terms are bounded uniformly by a multiple of the summable `n ^ (-1 - \u03b5)` for `\u03b5 = (re z\u2080 + 1) / 2`.","title":"Holomorphy of the step correction"},{"id":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/the-step-correction-at-0","kind":"node","lean":"TauCeti.LSeries.stepCorrection_zero","parent":3154,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/index.html#the-step-correction-at-0","scope":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c","status":"fully_proved","summary":"`stepCorrection a 0 = \u2211' n, a n (log (1 + 1 / n) - 1 / n)`.","title":"The step correction at `0`"},{"id":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/newmans-left-detour","kind":"node","lean":"TauCeti.LSeries.newmanDetour","parent":3154,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/index.html#newmans-left-detour","scope":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c","status":"fully_proved","summary":"The parabolic arc `s \u21a6 -\u03b4 (1 - s\u00b2) - R s I`, which on `[-1, 1]` runs from `R * I` to `-R * I`; for `R, \u03b4 \u2265 0` it stays in the strip `-\u03b4 \u2264 re z \u2264 0`, `|im z| \u2264 R`.","title":"Newman's left detour"},{"id":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/the-detour-inside-the-continuation-domain","kind":"node","lean":"TauCeti.LSeries.exists_convex_newmanDetour_subset","parent":3154,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/index.html#the-detour-inside-the-continuation-domain","scope":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c","status":"fully_proved","summary":"If `\u03a9` is open and contains the closed half-plane `0 \u2264 re z`, then for some `\u03b4 > 0` the right semicircle of radius `R` and the detour `newmanDetour R \u03b4` lie in a common open convex subset of `\u03a9`.","title":"The detour inside the continuation domain"},{"id":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/the-detour-contribution-in-newmans-argument-vani","kind":"node","lean":"TauCeti.LSeries.tendsto_integral_newmanDetour_mul_kernel","parent":3154,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/index.html#the-detour-contribution-in-newmans-argument-vani","scope":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c","status":"fully_proved","summary":"If `G` is continuous on the detour `newmanDetour R \u03b4` with `R, \u03b4 > 0`, then the detour integral of `G` against Newman's kernel `exp (T z) (1 + z\u00b2 / R\u00b2) / z` tends to `0` as `T \u2192 \u221e`.","title":"The detour contribution in Newman's argument vanishes in the limit"},{"id":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/truncated-laplace-transforms-are-entire","kind":"node","lean":"TauCeti.differentiable_intervalIntegral_mul_cexp_neg_mul","parent":3154,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/index.html#truncated-laplace-transforms-are-entire","scope":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c","status":"fully_proved","summary":"If `f` is interval integrable on `[a, b]`, then `z \u21a6 \u222b t in a..b, f t * exp (-z t)` is complex differentiable everywhere.","title":"Truncated Laplace transforms are entire"},{"id":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/holomorphy-of-the-laplace-transform","kind":"node","lean":"TauCeti.differentiableOn_integral_mul_cexp_neg_mul","parent":3154,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/index.html#holomorphy-of-the-laplace-transform","scope":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c","status":"fully_proved","summary":"If `f` is bounded on `[0, \u221e)`, its Laplace transform `z \u21a6 \u222b t in Ici 0, f t * exp (-z t)` is complex differentiable on the half-plane `0 < re z`.","title":"Holomorphy of the Laplace transform"},{"id":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/substitution-x-exp-t","kind":"node","lean":"TauCeti.intervalIntegral_ofReal_cpow_neg_sub_one","parent":3154,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/index.html#substitution-x-exp-t","scope":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c","status":"fully_proved","summary":"For `0 < u` and `0 < v`, the integral of `x ^ (-z - 1)` over `[u, v]` is the truncated Laplace transform `\u222b t in log u..log v, exp (-z t)` of the constant function `1`.","title":"Substitution `x = exp t`"},{"id":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/abelian-theorem-for-dirichlet-series-at-s-1","kind":"node","lean":"TauCeti.LSeries.tendsto_LSeries_nhdsGT_one","parent":3154,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/index.html#abelian-theorem-for-dirichlet-series-at-s-1","scope":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c","status":"fully_proved","summary":"If the ordered partial sums `\u2211 n \u2208 Icc 1 N, a n / n` tend to `L` and the Dirichlet series of `a` converges absolutely at every real `\u03c3 > 1`, then `LSeries a \u03c3 \u2192 L` as `\u03c3` decreases to `1`.","title":"Abelian theorem for Dirichlet series at `s = 1`"},{"id":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/cauchys-formula-for-newmans-contour","kind":"node","lean":"TauCeti.LSeries.integral_right_semicircle_add_integral_newmanDetour, TauCeti.LSeries.circleIntegral_intervalIntegral_mul_kernel","parent":3154,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/index.html#cauchys-formula-for-newmans-contour","scope":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c","status":"fully_proved","summary":"Let `g` be holomorphic on an open convex set `U` containing the right semicircle of radius `R > 0` and the detour `newmanDetour R \u03b4` with `\u03b4 > 0`. Then the integral of `g z / z` over the right semicircle, counterclockwise from `-R * I` to `R * I`, plus its integral over the detour, back from `R * I` to `-R * I` through the left half-plane, is `2\u03c0i g 0`. Both integrals include the velocity of their parametrization.","title":"Cauchy's formula for Newman's contour"},{"id":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/the-converse-of-cauchys-theorem","kind":"node","lean":"TauCeti.Contour.isExactOn_of_forall_closed_integral_eq_zero","parent":3154,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c/index.html#the-converse-of-cauchys-theorem","scope":"tauceti/number-theory/part-01/bounded-laplace-and-ordered-dirichlet-boundary-c","status":"fully_proved","summary":"A continuous function on an open set `U` whose contour integral vanishes along every closed piecewise-`C\u00b9` curve in `U` has a primitive on `U`.","title":"The converse of Cauchy's theorem"},{"data_url":"scopes/tauceti/number-theory/part-01/the-chebotarev-density-theorem.json","id":"tauceti/number-theory/part-01/the-chebotarev-density-theorem","kind":"scope","parent":3139,"result":false,"route":"../roadmap/tauceti/number-theory/part-01/the-chebotarev-density-theorem/index.html","scope":"tauceti/number-theory/part-01/the-chebotarev-density-theorem","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"The Chebotarev density theorem"},{"id":"tauceti/number-theory/part-01/the-chebotarev-density-theorem/the-chebotarev-density-theorem","kind":"node","lean":"NumberField.Chebotarev.hasDirichletDensity_frobeniusPrimeSet, NumberField.Chebotarev.frobeniusPsi_asymptotic, NumberField.Chebotarev.tendsto_frobeniusPsi","parent":3170,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/the-chebotarev-density-theorem/index.html#the-chebotarev-density-theorem","scope":"tauceti/number-theory/part-01/the-chebotarev-density-theorem","status":"fully_proved","summary":"For a finite Galois extension `L / K` of number fields and a conjugacy class `C` of `Gal(L/K)`, the primes of `\ud835\udcde K` whose Frobenius in `L` lies in `C` have Dirichlet density `#C / #Gal(L/K)`.","title":"The Chebotarev density theorem"},{"id":"tauceti/number-theory/part-01/the-chebotarev-density-theorem/numberfield-chebotarev-hasnaturaldensity-frobeni","kind":"node","lean":"NumberField.Chebotarev.hasNaturalDensity_frobeniusPrimeSet","parent":3170,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/the-chebotarev-density-theorem/index.html#numberfield-chebotarev-hasnaturaldensity-frobeni","scope":"tauceti/number-theory/part-01/the-chebotarev-density-theorem","status":"fully_proved","summary":"Natural-density Chebotarev: among the primes of `K`, the primes with arithmetic Frobenius class `C` have density `#C / #Gal(L/K)`.","title":"NumberField.Chebotarev.hasNaturalDensity_frobeniusPrimeSet"},{"id":"tauceti/number-theory/part-01/the-chebotarev-density-theorem/weighted-chebotarev-for-abelian-extensions","kind":"node","lean":"NumberField.Chebotarev.frobeniusPsi_asymptotic_of_mul_comm","parent":3170,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/the-chebotarev-density-theorem/index.html#weighted-chebotarev-for-abelian-extensions","scope":"tauceti/number-theory/part-01/the-chebotarev-density-theorem","status":"fully_proved","summary":"If `Gal(L/K)` is abelian, then for every `\u03c3 \u2208 Gal(L/K)` the Frobenius `\u03c8` function of `\u03c3` satisfies `\u03c8_\u03c3(x) = x / #Gal(L/K) + o(x)`.","title":"Weighted Chebotarev for abelian extensions"},{"id":"tauceti/number-theory/part-01/the-chebotarev-density-theorem/non-galois-chebotarev-for-splitting-types","kind":"node","lean":"NumberField.Chebotarev.hasDirichletDensity_setOf_map_inertiaDeg_eq","parent":3170,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/the-chebotarev-density-theorem/index.html#non-galois-chebotarev-for-splitting-types","scope":"tauceti/number-theory/part-01/the-chebotarev-density-theorem","status":"fully_proved","summary":"The primes with residue-degree multiset `\u03bc` in `E` have density the total proportion of elements whose action on `Gal(L/K) \u29f8 Gal(L/E)` has full cycle type `\u03bc`, expressed as a sum of conjugacy-class proportions. No normality of `E/K` or realizability hypothesis on `\u03bc` is needed.","title":"Non-Galois Chebotarev for splitting types"},{"id":"tauceti/number-theory/part-01/the-chebotarev-density-theorem/dirichlets-theorem-on-primes-in-arithmetic-progr","kind":"node","lean":"NumberField.Chebotarev.hasDirichletDensity_primesCongruent","parent":3170,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/the-chebotarev-density-theorem/index.html#dirichlets-theorem-on-primes-in-arithmetic-progr","scope":"tauceti/number-theory/part-01/the-chebotarev-density-theorem","status":"fully_proved","summary":"For `m \u2260 0` and `a` a unit modulo `m`, the primes of `\ud835\udcde \u211a` whose absolute norm is congruent to `a` modulo `m` have Dirichlet density `1 / \u03c6(m)`.","title":"Dirichlet's theorem on primes in arithmetic progressions, with Dirichlet density"},{"id":"tauceti/number-theory/part-01/the-chebotarev-density-theorem/the-primes-of-k-with-artin-class-c-in-l","kind":"node","lean":"NumberField.Chebotarev.frobeniusPrimeSet","parent":3170,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/the-chebotarev-density-theorem/index.html#the-primes-of-k-with-artin-class-c-in-l","scope":"tauceti/number-theory/part-01/the-chebotarev-density-theorem","status":"fully_proved","summary":"A height-one prime `\ud835\udd2d` of `\ud835\udcde K` belongs to `frobeniusPrimeSet K L C` when `\ud835\udd2d` is unramified in `L` \u2014 so that `artinSymbol` is applicable \u2014 and its Artin class is `C`.","title":"The primes of `K` with Artin class `C` in `L`"},{"id":"tauceti/number-theory/part-01/the-chebotarev-density-theorem/the-fixed-field-frobenius-fiber-count","kind":"node","lean":"NumberField.Chebotarev.fixedField_frobenius_fiber_card","parent":3170,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/the-chebotarev-density-theorem/index.html#the-fixed-field-frobenius-fiber-count","scope":"tauceti/number-theory/part-01/the-chebotarev-density-theorem","status":"fully_proved","summary":"Let `sigma` represent the conjugacy class `C`, and let `p` be an unramified prime with Artin class `C`. The number of primes of `L ^ <sigma>` above `p` whose relative Artin class in `L / L ^ <sigma>` is represented by `sigma.toFixedFieldAlgEquiv` is","title":"The fixed-field Frobenius fiber count"},{"id":"tauceti/number-theory/part-01/the-chebotarev-density-theorem/the-crossing-constant","kind":"node","lean":"TauCeti.NumberField.Chebotarev.crossingConstant","parent":3170,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/the-chebotarev-density-theorem/index.html#the-crossing-constant","scope":"tauceti/number-theory/part-01/the-chebotarev-density-theorem","status":"fully_proved","summary":"The number of tagged elements of the auxiliary cyclic group `H` \u2014 those whose order is divisible by `f` \u2014 divided by `#Aut_K(L) * #H`.","title":"The crossing constant"},{"id":"tauceti/number-theory/part-01/the-chebotarev-density-theorem/the-crossing-constant-approaches-1-aut-k-l","kind":"node","lean":"TauCeti.NumberField.Chebotarev.exists_forall_lt_crossingConstant","parent":3170,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/the-chebotarev-density-theorem/index.html#the-crossing-constant-approaches-1-aut-k-l","scope":"tauceti/number-theory/part-01/the-chebotarev-density-theorem","status":"fully_proved","summary":"For every `\u03b4 < 1 / #Aut_K(L)` there is a level `r` such that the crossing constant of every cyclic group whose order is divisible by `f ^ r` exceeds `\u03b4`.","title":"The crossing constant approaches `1 / #Aut_K(L)`"},{"id":"tauceti/number-theory/part-01/the-chebotarev-density-theorem/the-prime-number-theorem-for-arithmetic-progress","kind":"node","lean":"NumberField.Chebotarev.primeCount_primesCongruent_isEquivalent_logIntegral","parent":3170,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/the-chebotarev-density-theorem/index.html#the-prime-number-theorem-for-arithmetic-progress","scope":"tauceti/number-theory/part-01/the-chebotarev-density-theorem","status":"fully_proved","summary":"At every nonzero level `m`, the number of primes of `\ud835\udcaa \u211a` whose norms are congruent to an invertible residue `a` modulo `m` is asymptotic to `Li(x) / \u03c6(m)`. This is the logarithmic-integral form of equidistribution among the invertible residue classes.","title":"The prime number theorem for arithmetic progressions"},{"data_url":"scopes/tauceti/number-theory/part-01/class-field-theory.json","id":"tauceti/number-theory/part-01/class-field-theory","kind":"scope","parent":3139,"result":false,"route":"../roadmap/tauceti/number-theory/part-01/class-field-theory/index.html","scope":"tauceti/number-theory/part-01/class-field-theory","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Class field theory"},{"id":"tauceti/number-theory/part-01/class-field-theory/tates-theorem-for-a-class-formation-in-every-int","kind":"node","lean":"TauCeti.ClassFieldTheory.ClassFormation.tateIso","parent":3181,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/class-field-theory/index.html#tates-theorem-for-a-class-formation-in-every-int","scope":"tauceti/number-theory/part-01/class-field-theory","status":"fully_proved","summary":"the generic theorem applied to the fundamental class, with the three hypotheses discharged by the three individually named consequences of the axioms. This is Artin\u2013Tate's Main Theorem (Chapter XIV \u00a74, Theorem 1).","title":"Tate's theorem for a class formation, in every integer degree"},{"id":"tauceti/number-theory/part-01/class-field-theory/local-reciprocity-for-the-weil-group","kind":"node","lean":"TauCeti.ClassFieldTheory.localWeilArtinEquiv","parent":3181,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/class-field-theory/index.html#local-reciprocity-for-the-weil-group","scope":"tauceti/number-theory/part-01/class-field-theory","status":"fully_proved","summary":"For a nonarchimedean local field `K` of characteristic zero, the local Artin map is an isomorphism of topological groups from `K\u02e3` onto the topological abelianization of the local Weil group (`localWeilArtinEquiv_apply`).","title":"Local reciprocity for the Weil group"},{"id":"tauceti/number-theory/part-01/class-field-theory/finite-global-reciprocity","kind":"node","lean":"TauCeti.ClassFieldTheory.globalArtinEquiv","parent":3181,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/class-field-theory/index.html#finite-global-reciprocity","scope":"tauceti/number-theory/part-01/class-field-theory","status":"fully_proved","summary":"Finite global reciprocity `C_K / N_{L/K}(C_L) \u2243 Gal(L/K)^ab` for a finite Galois extension `L/K` of number fields, embedded in the separable closure of `K` by `\u03b9`: the abstract Artin reciprocity isomorphism `ClassFormation.artinEquiv` of the global class formation on the layer of `L`, read through `globalLayerNormQuotientEquiv K L` and `layerGalEquiv \u03b9`. It does not depend on `\u03b9` (`globalArtinEquiv_eq_of_iota`).","title":"Finite global reciprocity"},{"id":"tauceti/number-theory/part-01/class-field-theory/the-norm-index-theorem","kind":"node","lean":"TauCeti.ClassFieldTheory.card_ideleClassNormQuotient","parent":3181,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/class-field-theory/index.html#the-norm-index-theorem","scope":"tauceti/number-theory/part-01/class-field-theory","status":"fully_proved","summary":"for a finite abelian extension `L/K` of number fields, `[C_K : N_{L/K}(C_L)] = [L : K]`, since finite global reciprocity identifies the norm quotient with `Gal(L/K)^ab`, which is `Gal(L/K)` itself.","title":"The norm index theorem"},{"id":"tauceti/number-theory/part-01/class-field-theory/hilbert-reciprocity","kind":"node","lean":"TauCeti.ClassFieldTheory.hilbertProductFormula","parent":3181,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/class-field-theory/index.html#hilbert-reciprocity","scope":"tauceti/number-theory/part-01/class-field-theory","status":"fully_proved","summary":"For `a, b \u2208 K\u02e3`, the Hilbert invariants of `a` and `b` at all the finite and infinite places of the number field `K` sum to zero.","title":"Hilbert reciprocity"},{"id":"tauceti/number-theory/part-01/class-field-theory/quadratic-reciprocity-is-hilbert-reciprocity-for","kind":"node","lean":"TauCeti.ClassFieldTheory.sum_finiteHilbertInvariantAt_add_sum_infiniteHilbertInvariantAt_eq_zero_iff","parent":3181,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/class-field-theory/index.html#quadratic-reciprocity-is-hilbert-reciprocity-for","scope":"tauceti/number-theory/part-01/class-field-theory","status":"fully_proved","summary":"For distinct odd primes `p` and `q`, the cohomological Hilbert invariants of `p` and `q` sum to zero over the places of `\u211a` exactly when `(p / q) (q / p) = (-1) ^ ((p - 1)/2 \u00b7 (q - 1)/2)`. So Hilbert's reciprocity law `\u2211_v inv_v((p) \u222a (q)) = 0` gives the quadratic reciprocity law, its dyadic factor being the sign.","title":"Quadratic reciprocity is Hilbert reciprocity for two odd primes"},{"id":"tauceti/number-theory/part-01/class-field-theory/the-global-invariant","kind":"node","lean":"TauCeti.ClassFieldTheory.globalInv","parent":3181,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/class-field-theory/index.html#the-global-invariant","scope":"tauceti/number-theory/part-01/class-field-theory","status":"fully_proved","summary":"The global invariant `H\u00b2(Gal(L/F), C_L) \u2192 \u211a/\u2124` of a finite normal layer `L/F` inside `K\u02e2`: the invariant of a class is the sum of the local invariants of a lift to the ideles of its inflation to some refinement of the layer. By `globalInv_eq` it does not depend on the refinement or on the lift, and `globalInv_unique` characterizes it by the equations `globalInv_ideleToClassH2` and `globalInv_infl`.","title":"The global invariant"},{"id":"tauceti/number-theory/part-01/class-field-theory/the-global-invariant-is-injective","kind":"node","lean":"TauCeti.ClassFieldTheory.globalInv_injective","parent":3181,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/class-field-theory/index.html#the-global-invariant-is-injective","scope":"tauceti/number-theory/part-01/class-field-theory","status":"fully_proved","summary":"The global invariant is injective, because the idele-class `H\u00b9` vanishes and the sum of the local invariants vanishes only on the classes that die in the idele classes.","title":"The global invariant is injective"},{"id":"tauceti/number-theory/part-01/class-field-theory/the-global-class-formation","kind":"node","lean":"TauCeti.ClassFieldTheory.globalClassFormation","parent":3181,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/class-field-theory/index.html#the-global-class-formation","scope":"tauceti/number-theory/part-01/class-field-theory","status":"fully_proved","summary":"the idele-class formation of `K` is a class formation, whose invariant on a layer is the global invariant (`globalClassFormation_inv`).","title":"The global class formation"},{"id":"tauceti/number-theory/part-01/class-field-theory/the-local-invariant","kind":"node","lean":"TauCeti.ClassFieldTheory.invMap","parent":3181,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/class-field-theory/index.html#the-local-invariant","scope":"tauceti/number-theory/part-01/class-field-theory","status":"fully_proved","summary":"The local invariant `inv_K : Br K \u2243+ \u211a/\u2124` of a nonarchimedean local field `K`, normalized by arithmetic Frobenius: on the classes inflated from an unramified extension `E/K` inside `K\u02e2` it is the invariant `TauCeti.ClassFieldTheory.unramifiedInv K E` of the layer (`invMap_relBrInfl`).","title":"The local invariant"},{"id":"tauceti/number-theory/part-01/class-field-theory/the-local-class-formation","kind":"node","lean":"TauCeti.ClassFieldTheory.localClassFormation","parent":3181,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/class-field-theory/index.html#the-local-class-formation","scope":"tauceti/number-theory/part-01/class-field-theory","status":"fully_proved","summary":"the formation of the units of a separable closure of a nonarchimedean local field `K` is a class formation, with the invariant `layerInv` of a layer as invariant map.","title":"The local class formation"},{"id":"tauceti/number-theory/part-01/class-field-theory/the-kummer-case-of-global-existence-milne-vii-le","kind":"node","lean":"TauCeti.ClassFieldTheory.exists_abelianLayer_globalNormSubgroup_eq_of_pow_mem","parent":3181,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/class-field-theory/index.html#the-kummer-case-of-global-existence-milne-vii-le","scope":"tauceti/number-theory/part-01/class-field-theory","status":"fully_proved","summary":"Let `K` be a number field containing the `p`-th roots of unity for a prime `p`. Every open subgroup of the idele class group `C_K` containing the `p`-th powers is the norm subgroup of a finite abelian extension of `K`.","title":"The Kummer case of global existence (Milne VII, Lemma 9.3)"},{"id":"tauceti/number-theory/part-01/class-field-theory/the-idelic-consequence-of-furtwanglers-theorem","kind":"node","lean":"TauCeti.ClassFieldTheory.ideleClassExtension_mem_range_ideleClassNormMap_commutator","parent":3181,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/class-field-theory/index.html#the-idelic-consequence-of-furtwanglers-theorem","scope":"tauceti/number-theory/part-01/class-field-theory","status":"fully_proved","summary":"If `Lay` represents `L/F` and `E` is its maximal abelian subextension, every idele class of `F`, extended to `E`, is a norm from `L`. The field `E` is pinned as the fixed field of the preimage of the commutator subgroup of the finite Galois group; no norm or principalization hypothesis is assumed.","title":"The idelic consequence of Furtw\u00e4ngler's theorem"},{"id":"tauceti/number-theory/part-01/class-field-theory/tauceti-classfieldtheory-tatetheorem","kind":"node","lean":"TauCeti.ClassFieldTheory.tateTheorem","parent":3181,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/class-field-theory/index.html#tauceti-classfieldtheory-tatetheorem","scope":"tauceti/number-theory/part-01/class-field-theory","status":"fully_proved","summary":"Tate's theorem (J. Tate, *The higher dimensional cohomology groups of class field theory*, Ann. of Math. 56 (1952); Artin\u2013Tate, Chapter XIV \u00a74), with its hypotheses stated one by one over the finite quotient system `H \u21a6 L.subgroupLayer H`:","title":"TauCeti.ClassFieldTheory.tateTheorem"},{"data_url":"scopes/tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local.json","id":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local","kind":"scope","parent":3139,"result":false,"route":"../roadmap/tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/index.html","scope":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Counting totally ramified extensions of a local field, and Serre's mass formula"},{"id":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/the-orbit-count-serre-1978-remarque-3","kind":"node","lean":"TauCeti.ncard_isomorphic_mul_natCard_algEquiv","parent":3196,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/index.html#the-orbit-count-serre-1978-remarque-3","scope":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local","status":"fully_proved","summary":"for `L \u2208 \u03c3_K(n)`, the isomorphism class of `L` inside `\u03c3_K(n)` has `n / #Aut_K(L)` members, in multiplied form. The extension `L / K` is separable of degree `n`, so it has `n` embeddings into `SeparableClosure K`; their images are the members of the class, and the embeddings with a given image form a torsor under `Aut_K(L)`.","title":"The orbit count ([Serre 1978, Remarque 3\u00b0])"},{"id":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/tauceti-isrepresentativeset-tsum-one-div-natcard","kind":"node","lean":"TauCeti.IsRepresentativeSet.tsum_one_div_natCard_algEquiv_mul_natCard_residueField_pow_wildExponent_eq_one_iff","parent":3196,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/index.html#tauceti-isrepresentativeset-tsum-one-div-natcard","scope":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local","status":"fully_proved","summary":"For `0 < n`, Serre's mass formula over a set of representatives, `\u2211_{M \u2208 R} 1 / (#Aut_K(M) \u00b7 q ^ c(M)) = 1`, is equivalent to the mass formula over `\u03c3_K(n)`, `\u2211_{L \u2208 \u03c3_K(n)} q ^ (-c(L)) = n`.","title":"TauCeti.IsRepresentativeSet.tsum_one_div_natCard_algEquiv_mul_natCard_residueField_pow_wildExponent_eq_one_iff"},{"id":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/the-totally-ramified-quadratic-extensions-of-q-2","kind":"node","lean":"TauCeti.totallyRamifiedOfDegree_two_eq_padicTwo","parent":3196,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/index.html#the-totally-ramified-quadratic-extensions-of-q-2","scope":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local","status":"fully_proved","summary":"If `x\u2081, \u2026, x\u2086` are square roots of `-1, -5, 2, -2, 10, -10` in the separable closure of `\u211a_2`, then `\u03c3_{\u211a_2}(2) = {\u211a_2(x\u2081), \u211a_2(x\u2082), \u211a_2(x\u2083), \u211a_2(x\u2084), \u211a_2(x\u2085), \u211a_2(x\u2086)}`.","title":"The totally ramified quadratic extensions of `\u211a_2`"},{"id":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/serres-mass-formula-for-the-quadratic-extensions","kind":"node","lean":"TauCeti.tsum_one_div_pow_wildExponent_two_padicTwo, TauCeti.Padic.tsum_one_div_pow_wildExponent_two","parent":3196,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/index.html#serres-mass-formula-for-the-quadratic-extensions","scope":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local","status":"fully_proved","summary":"`\u2211_{L \u2208 \u03c3_{\u211a_2}(2)} 2 ^ (-c(L)) = 2 \u00b7 2\u207b\u00b9 + 4 \u00b7 2\u207b\u00b2 = 2`.","title":"Serre's mass formula for the quadratic extensions of `\u211a_2`"},{"id":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/serres-mass-formula-for-quadratic-extensions-in","kind":"node","lean":"TauCeti.tsum_one_div_natCard_residueField_pow_wildExponent_two, TauCeti.tsum_one_div_natCard_pow_wildExponent_laurentSeries","parent":3196,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/index.html#serres-mass-formula-for-quadratic-extensions-in","scope":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local","status":"fully_proved","summary":"`\u2211_{L \u2208 \u03c3_K(2)} q ^ (-c(L)) = 2`, where `q` is the cardinality of the residue field of `K`. Both members of `\u03c3_K(2)` are tamely ramified, with weight `1`.","title":"Serre's mass formula for quadratic extensions in odd residue characteristic"},{"id":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/hensels-lemma-with-an-estimate-over-a-henselian","kind":"node","lean":"Polynomial.exists_isRoot_addVal_eval_eq, Polynomial.existsUnique_isRoot_addVal_lt","parent":3196,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/index.html#hensels-lemma-with-an-estimate-over-a-henselian","scope":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local","status":"fully_proved","summary":"If the order of `F` at `y\u2080` exceeds twice the order of its derivative there, then `F` has a root `z` whose distance to `y\u2080` has order exactly `addVal (F(y\u2080)) - addVal (F'(y\u2080))`. This is the existence part of Mathlib's `hensels_lemma` in `addVal` spelling; the uniqueness of `z` in that ball is `Polynomial.existsUnique_isRoot_addVal_lt`.","title":"Hensel's lemma with an estimate over a Henselian discrete valuation ring"},{"id":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/local-constancy-of-the-root-count","kind":"node","lean":"TauCeti.exists_card_aroots_eq","parent":3196,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/index.html#local-constancy-of-the-root-count","scope":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local","status":"fully_proved","summary":"For a monic `f` over `\ud835\udcaa[K]` that is separable over `K`, there is a threshold `T > 0` such that every monic `g` over `\ud835\udcaa[K]` with `g.coeff i - f.coeff i \u2208 \ud835\udcc2[K] ^ T` for all `i` has as many roots as `f`, counted with multiplicity, in every finite intermediate field of `\u03a9 / K`. The threshold depends only on `f`.","title":"Local constancy of the root count"},{"id":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/the-scaling-law-for-haar-measure","kind":"node","lean":"MeasureTheory.Measure.addHaar_image_mulVec","parent":3196,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/index.html#the-scaling-law-for-haar-measure","scope":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local","status":"fully_proved","summary":"If the determinant of a square matrix `A` over `\ud835\udcaa[K]` is associated to `\u03d6 ^ k` for a uniformizer `\u03d6`, then for every regular additive Haar measure `\u03bc` on `\u03b9 \u2192 K` and every set `s`, `\u03bc (A '' s) = q ^ (-k) * \u03bc s`, where `q = Nat.card \ud835\udcc0[K]`. This is the nonarchimedean analogue of `MeasureTheory.Measure.addHaar_image_linearMap`.","title":"The scaling law for Haar measure"},{"id":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/infinitely-many-totally-ramified-extensions-in-e","kind":"node","lean":"TauCeti.infinite_totallyRamifiedOfDegree","parent":3196,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/index.html#infinitely-many-totally-ramified-extensions-in-e","scope":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local","status":"fully_proved","summary":"If `0 < n` and `(n : K) = 0`, that is, `K` has characteristic `p > 0` with `p \u2223 n`, then `\u03c3_K(n)` is infinite: its discriminant exponents are unbounded.","title":"Infinitely many totally ramified extensions in equal characteristic"},{"id":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/tauceti-totallyramifiedofdegree","kind":"node","lean":"TauCeti.totallyRamifiedOfDegree","parent":3196,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/index.html#tauceti-totallyramifiedofdegree","scope":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local","status":"fully_proved","summary":"Serre's `\u03c3_K(n)`: the subextensions of `SeparableClosure K` that are totally ramified of degree `n` over `K`.","title":"TauCeti.totallyRamifiedOfDegree"},{"id":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/intermediatefield-wildexponent","kind":"node","lean":"IntermediateField.wildExponent","parent":3196,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/index.html#intermediatefield-wildexponent","scope":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local","status":"fully_proved","summary":"The wild exponent `c(L) = d(L) - [L : K] + 1` of a subextension `L` of `SeparableClosure K`, written `d(L) + 1 - [L : K]` in truncated subtraction. For `L \u2208 \u03c3_K(n)` the truncation is faithful by `IntermediateField.sub_one_le_discriminantExponent`, and `c(L)` is Serre's nonnegative integer `d(L) - n + 1`.","title":"IntermediateField.wildExponent"},{"id":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/the-wild-exponent-is-bounded-on-k-n-away-from-eq","kind":"node","lean":"IntermediateField.wildExponent_le_mul_natCastValuation","parent":3196,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/index.html#the-wild-exponent-is-bounded-on-k-n-away-from-eq","scope":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local","status":"fully_proved","summary":"if `n` is nonzero in `K`, every `L \u2208 \u03c3_K(n)` has `c(L) \u2264 n \u00b7 v_K(n)`. In equal characteristic, where `v_K(n) = 0`, this recovers `c(L) = 0` for `p \u2224 n`.","title":"The wild exponent is bounded on `\u03c3_K(n)` away from equal characteristic `p \u2223 n`"},{"id":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/tauceti-eisensteinset","kind":"node","lean":"TauCeti.eisensteinSet","parent":3196,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/index.html#tauceti-eisensteinset","scope":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local","status":"fully_proved","summary":"The Eisenstein region in the space `Fin n \u2192 K` of coefficient vectors of monic polynomials of degree `n`: every coefficient has valuation below `1`, and the constant term of `monicOfCoeff a` has valuation at least that of every element of valuation below `1`. For `0 < n` the constant term is `a 0`, which then has the valuation of a uniformizer; for `n = 0` it is `1`, and the region is the whole one-point space.\u2026","title":"TauCeti.eisensteinSet"},{"id":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/the-volume-of-the-eisenstein-region","kind":"node","lean":"TauCeti.measure_eisensteinSet","parent":3196,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/index.html#the-volume-of-the-eisenstein-region","scope":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local","status":"fully_proved","summary":"for a left-invariant measure on `Fin n \u2192 K` normalized on the integer box, the Eisenstein region has volume `q ^ (-n) (1 - q\u207b\u00b9)` for `0 < n`, where `q = Nat.card \ud835\udcc0[K]`.","title":"The volume of the Eisenstein region"},{"id":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/intermediatefield-rootcount","kind":"node","lean":"IntermediateField.rootCount","parent":3196,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/index.html#intermediatefield-rootcount","scope":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local","status":"fully_proved","summary":"The root count of a coefficient vector `a : Fin n \u2192 K` in a subextension `L`: the number of roots of the monic polynomial `monicOfCoeff a` lying in `L`, counted with multiplicity.","title":"IntermediateField.rootCount"},{"id":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/the-root-counts-over-k-n-sum-to-n","kind":"node","lean":"TauCeti.tsum_rootCount_eq_of_separable","parent":3196,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/index.html#the-root-counts-over-k-n-sum-to-n","scope":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local","status":"fully_proved","summary":"Let `f` be an Eisenstein polynomial over `\ud835\udcaa[K]` with image `monicOfCoeff a` over `K`, for `a : Fin n \u2192 K`, and assume that `monicOfCoeff a` is separable. Then `\u2211_{L \u2208 \u03c3_K(n)} L.rootCount a = n`: each of the `n` roots of `monicOfCoeff a` in the separable closure lies in exactly one member of `\u03c3_K(n)`, the one it generates.","title":"The root counts over `\u03c3_K(n)` sum to `n`"},{"id":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/polynomial-card-roots-eq-sum-ite-of-addval-lt","kind":"node","lean":"Polynomial.card_roots_eq_sum_ite_of_addVal_lt","parent":3196,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/index.html#polynomial-card-roots-eq-sum-ite-of-addval-lt","scope":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local","status":"fully_proved","summary":"Count roots by a finite disjoint family of balls covering them. If the derivative order at each root is smaller than the ball order, each occupied ball contributes exactly one root. The derivative bound also excludes repeated rational roots, so no separability hypothesis is needed.","title":"Polynomial.card_roots_eq_sum_ite_of_addVal_lt"},{"id":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/serres-finiteness-dichotomy","kind":"node","lean":"TauCeti.infinite_totallyRamifiedOfDegree_iff, TauCeti.infinite_totallyRamifiedOfDegree_iff_or_tsum_eq_top","parent":3196,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/index.html#serres-finiteness-dichotomy","scope":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local","status":"fully_proved","summary":"for `0 < n`, there are infinitely many totally ramified degree-`n` subextensions of a separable closure of `K` exactly when `K` has equal characteristic `p`, the residue characteristic, and `p \u2223 n`.","title":"Serre's finiteness dichotomy"},{"id":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/serres-first-mass-formula","kind":"node","lean":"TauCeti.tsum_one_div_natCard_residueField_pow_wildExponent","parent":3196,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/index.html#serres-first-mass-formula","scope":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local","status":"fully_proved","summary":"the totally ramified degree-`n` subextensions `L` of a separable closure of `K` satisfy `\u2211_L q ^ (-c(L)) = n`, where `q` is the residue cardinality of `K` and `c(L)` is the wild exponent.","title":"Serre's first mass formula"},{"id":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/serres-second-mass-formula","kind":"node","lean":"TauCeti.IsRepresentativeSet.tsum_one_div_natCard_algEquiv_mul_natCard_residueField_pow_wildExponent","parent":3196,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local/index.html#serres-second-mass-formula","scope":"tauceti/number-theory/part-01/counting-totally-ramified-extensions-of-a-local","status":"fully_proved","summary":"for `0 < n` and every set `R` of representatives of the `K`-isomorphism classes in `\u03c3_K(n)`, `\u2211_{M \u2208 R} 1 / (#Aut_K(M) \u00b7 q ^ c(M)) = 1`.","title":"Serre's second mass formula"},{"data_url":"scopes/tauceti/number-theory/part-01/diophantine-approximation-and-transcendence.json","id":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence","kind":"scope","parent":3139,"result":false,"route":"../roadmap/tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/index.html","scope":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Diophantine approximation and transcendence"},{"id":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/roths-theorem-roth-1955","kind":"node","lean":"IsAlgebraic.finite_setOfPred_abs_sub_le_mulHeight\u2081_rpow_neg, IsAlgebraic.exists_pos_mul_mulHeight\u2081_rpow_neg_le_abs_sub, IsAlgebraic.exists_pos_mul_mulHeight\u2081_rpow_neg_le_norm_sub, IsAlgebraic.not_liouvilleWith, IsAlgebraic.exists_pos_mul_max_abs_rpow_le_norm_sub_mul","parent":3217,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/index.html#roths-theorem-roth-1955","scope":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence","status":"fully_proved","summary":"If `\u03b1` is a real algebraic number and `\u03ba > 2`, then only finitely many rational numbers `\u03be` satisfy `|\u03be - \u03b1| \u2264 H(\u03be) ^ (-\u03ba)`, where `H` is the multiplicative height on `\u211a`.","title":"Roth's theorem (Roth, 1955)"},{"id":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/isalgebraic-irrationalityexponent-eq-two","kind":"node","lean":"IsAlgebraic.irrationalityExponent_eq_two","parent":3217,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/index.html#isalgebraic-irrationalityexponent-eq-two","scope":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence","status":"fully_proved","summary":"The irrationality exponent of a real algebraic irrational number is two.","title":"IsAlgebraic.irrationalityExponent_eq_two"},{"id":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/the-lindemann-weierstrass-theorem-in-bakers-form","kind":"node","lean":"TauCeti.linearIndependent_exp","parent":3217,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/index.html#the-lindemann-weierstrass-theorem-in-bakers-form","scope":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence","status":"fully_proved","summary":"The exponentials of pairwise distinct algebraic numbers are linearly independent over the field `integralClosure \u211a \u2102` of algebraic numbers.","title":"The Lindemann\u2013Weierstrass theorem, in Baker's form"},{"id":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/the-lindemann-weierstrass-theorem-algebraic-inde","kind":"node","lean":"TauCeti.algebraicIndependent_exp, TauCeti.LindemannWeierstrass.expEval_injective, TauCeti.LindemannWeierstrass.linearIndependent_exp_algebraMap","parent":3217,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/index.html#the-lindemann-weierstrass-theorem-algebraic-inde","scope":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence","status":"fully_proved","summary":"If the algebraic numbers `u i` are linearly independent over `\u2115` (equivalently, over `\u211a`), then their exponentials are algebraically independent over the field `integralClosure \u211a \u2102` of algebraic numbers.","title":"The Lindemann\u2013Weierstrass theorem, algebraic independence form"},{"id":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/the-thue-roth-inequality-for-binary-forms-everts","kind":"node","lean":"MvPolynomial.IsHomogeneous.exists_pos_mul_max_abs_rpow_le_abs_eval","parent":3217,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/index.html#the-thue-roth-inequality-for-binary-forms-everts","scope":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence","status":"fully_proved","summary":"Let `F \u2208 \u2124[X\u2080, X\u2081]` be a binary form of degree `d` with no squared linear factor over `\u2102`, and let `\u03ba > 2`. Then there is `c > 0` with `|F(x, y)| \u2265 c max(|x|, |y|) ^ (d - \u03ba)` for all integers `x`, `y` with `F(x, y) \u2260 0`. The proof is ineffective: it gives no value of `c`.","title":"The Thue\u2013Roth inequality for binary forms (Evertse, Theorem 6.3)"},{"id":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/kroneckers-approximation-theorem","kind":"node","lean":"TauCeti.kronecker_approximation, TauCeti.kronecker_approximation_nat","parent":3217,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/index.html#kroneckers-approximation-theorem","scope":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence","status":"fully_proved","summary":"If `1` and the real numbers `\u03b1\u1d62` satisfy no nontrivial integer relation, that is, if they are linearly independent over `\u211a`, then for every `\u03b8 \u2208 \u211d^\u03b9` and `\u03b5 > 0` there are infinitely many integer pairs `(x, y)`, `x \u2208 \u2124^\u03b9`, `y \u2208 \u2124`, with `|y \u03b1\u1d62 - x\u1d62 - \u03b8\u1d62| \u2264 \u03b5` for every `i`.","title":"Kronecker's approximation theorem"},{"id":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/the-one-factor-estimate","kind":"node","lean":"TauCeti.exists_log_normalizedValuationWithZero_zpow_sub_one_le","parent":3217,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/index.html#the-one-factor-estimate","scope":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence","status":"fully_proved","summary":"Let `K` be a nonarchimedean local field of characteristic zero, that is, a finite extension of `\u211a_[p]`. For every fixed `\u03b1 \u2208 K` there is a constant `C`, depending on `\u03b1` and `K`, such that","title":"The one-factor estimate"},{"id":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/the-one-factor-estimate-explicit-form","kind":"node","lean":"TauCeti.log_normalizedValuationWithZero_zpow_sub_one_le","parent":3217,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/index.html#the-one-factor-estimate-explicit-form","scope":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence","status":"fully_proved","summary":"Let `x \u2208 K\u02e3` have a power `x ^ M \u2260 1` in a positive-depth step `U(K,i)`, and let `b` be an integer with `(|b| : K) \u2260 0` such that `v_K(\u2113) < (\u2113 - 1) * i` for every prime `\u2113 \u2223 b`. Then","title":"The one-factor estimate, explicit form"},{"id":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/tauceti-mahlermeasure","kind":"node","lean":"TauCeti.mahlerMeasure","parent":3217,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/index.html#tauceti-mahlermeasure","scope":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence","status":"fully_proved","summary":"The *Mahler measure* `M(x)` of `x`: the Mahler measure of its primitive minimal polynomial `F_x`, mapped to `\u2102[X]`. For algebraic `x` this is `a\u2080 \u220f max(1, |x\u207d\u2071\u207e|)` over the complex conjugates of `x` (`mahlerMeasure_eq_leadingCoeff_mul_prod`); for transcendental `x` it is `1`.","title":"TauCeti.mahlerMeasure"},{"id":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/mvpolynomial-weightedindex","kind":"node","lean":"MvPolynomial.weightedIndex","parent":3217,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/index.html#mvpolynomial-weightedindex","scope":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence","status":"fully_proved","summary":"The weighted index is the least reciprocal-weighted order of a nonzero Hasse jet; the infimum of the empty set is `\u22a4`.","title":"MvPolynomial.weightedIndex"},{"id":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/the-structure-of-the-rational-s-units","kind":"node","lean":"TauCeti.sUnitsRatMulEquiv","parent":3217,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/index.html#the-structure-of-the-rational-s-units","scope":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence","status":"fully_proved","summary":"If `S` consists of primes, the map `x \u21a6 (sign x, (v_p(x))_{p \u2208 S})` is an isomorphism from the rational `S`-units to `\u2124\u02e3 \u00d7 Multiplicative (S \u2192 \u2124)`; its inverse sends `(\u03b5, z)` to `\u03b5 \u220f_{p \u2208 S} p ^ z_p` (`TauCeti.coe_sUnitsRatMulEquiv_symm_apply`).","title":"The structure of the rational `S`-units"},{"id":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/hermites-estimate","kind":"node","lean":"Polynomial.norm_hermiteIntegral_le","parent":3217,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/index.html#hermites-estimate","scope":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence","status":"fully_proved","summary":"if `C` bounds `\u2016f\u2016` on the closed disc of radius `\u2016z\u2016`, then `\u2016F_f(z)\u2016 \u2264 \u2016z\u2016 exp \u2016z\u2016 C`.","title":"Hermite's estimate"},{"id":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/hermites-identity","kind":"node","lean":"Polynomial.hermiteIntegral_eq_sumIDeriv","parent":3217,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/index.html#hermites-identity","scope":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence","status":"fully_proved","summary":"integration by parts evaluates Hermite's integral as `F_f(z) = exp z * (sumIDeriv f)(0) - (sumIDeriv f)(z)`.","title":"Hermite's identity"},{"id":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/kroneckers-theorem-density-form","kind":"node","lean":"TauCeti.denseRange_mul_sub_intCast","parent":3217,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/index.html#kroneckers-theorem-density-form","scope":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence","status":"fully_proved","summary":"If `1` and the real numbers `\u03b1\u1d62` satisfy no nontrivial integer relation, that is, if they are linearly independent over `\u211a`, then the points `(y \u03b1\u1d62 - x\u1d62)\u1d62`, for `y \u2208 \u2124` and `x \u2208 \u2124^\u03b9`, are dense in `\u211d^\u03b9`.","title":"Kronecker's theorem, density form"},{"id":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/roths-auxiliary-polynomial-pottmeyer-theorem-2-6","kind":"node","lean":"TauCeti.exists_roth_auxiliary_polynomial","parent":3217,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/index.html#roths-auxiliary-polynomial-pottmeyer-theorem-2-6","scope":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence","status":"fully_proved","summary":"Let `\u03b1` be an algebraic number of degree `d` in a field of characteristic zero, and let `b \u2265 1` be an integer such that `b \u03b1` is integral over `\u2124`. Let `\u03b5 \u2265 0` and let `\u03b9` be a finite set of `n` variables with `2 d \u2264 exp (\u03b5\u00b2 n / 2)`. For all positive integers `r\u1d62`, `i \u2208 \u03b9`, there is a nonzero `P \u2208 \u2124[X\u1d62 : i \u2208 \u03b9]` with `deg_{X\u1d62} P \u2264 r\u1d62`, whose index at `(\u03b1, \u2026, \u03b1)` with weights `r` is at least `(n / 2) (1 - \u03b5)`, and\u2026","title":"Roth's auxiliary polynomial (Pottmeyer, Theorem 2.6.5)"},{"id":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/roths-lemma-pottmeyer-theorem-3-4-1","kind":"node","lean":"TauCeti.weightedIndex_le_of_logHeight_le","parent":3217,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/index.html#roths-lemma-pottmeyer-theorem-3-4-1","scope":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence","status":"fully_proved","summary":"Let `P \u2260 0` be a polynomial in `n \u2265 1` variables over a number field `K`, with partial degrees `deg_{X\u1d62} P \u2264 r\u1d62` for positive weights `r\u1d62` satisfying `r\u1d62\u208a\u2081 \u2264 \u03c3 r\u1d62`, where `0 < \u03c3 \u2264 1 / 2`. If `\u03c3\u207b\u00b9 (h(P) + 4 n r\u2080 [K : \u211a]) \u2264 r\u1d62 h(\u03b2\u1d62)` for every `i`, then the index of `P` at `\u03b2` with weights `r` is at most `2 n \u03c3 ^ (1 / 2 ^ (n - 1))`. Heights are relative logarithmic heights.","title":"Roth's lemma (Pottmeyer, Theorem 3.4.1)"},{"id":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/the-thue-inequality-evertse-corollary-6-4","kind":"node","lean":"MvPolynomial.IsHomogeneous.finite_setOf_eval_ne_zero_and_abs_le_of_three_le_card_roots, MvPolynomial.IsHomogeneous.finite_setOf_eval_ne_zero_and_abs_le","parent":3217,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/index.html#the-thue-inequality-evertse-corollary-6-4","scope":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence","status":"fully_proved","summary":"Let `F \u2208 \u2124[X\u2080, X\u2081]` be a binary form such that `F(X, 1)` has at least three distinct complex zeros. Then for every bound `B`, the inequality `0 < |F(x, y)| \u2264 B` has only finitely many integer solutions. Like Roth's theorem, the proof is ineffective.","title":"The Thue inequality (Evertse, Corollary 6.4)"},{"id":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/thues-theorem-thue-1909-evertse-corollary-6-4","kind":"node","lean":"MvPolynomial.IsHomogeneous.finite_setOf_eval_eq_of_three_le_card_roots, MvPolynomial.IsHomogeneous.finite_setOf_eval_eq","parent":3217,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/index.html#thues-theorem-thue-1909-evertse-corollary-6-4","scope":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence","status":"fully_proved","summary":"Let `F \u2208 \u2124[X\u2080, X\u2081]` be a binary form such that `F(X, 1)` has at least three distinct complex zeros, and let `m \u2260 0`. Then the Thue equation `F(x, y) = m` has only finitely many integer solutions. Like Roth's theorem, the proof is ineffective.","title":"Thue's theorem (Thue 1909; Evertse, Corollary 6.4)"},{"id":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/the-weak-lindemann-weierstrass-theorem","kind":"node","lean":"TauCeti.LindemannWeierstrass.expEval_ne_zero_of_forall_galConj_eq","parent":3217,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/diophantine-approximation-and-transcendence/index.html#the-weak-lindemann-weierstrass-theorem","scope":"tauceti/number-theory/part-01/diophantine-approximation-and-transcendence","status":"fully_proved","summary":"Let `K` be a number field, Galois over `\u211a` and embedded in `\u2102`. If a nonzero formal exponential sum `x = \u2211 \u03b4_\u03b3 [\u03b3] \u2208 K[K]` is fixed by the conjugation `galConj \u03c4` for every `\u03c4 \u2208 Gal(K/\u211a)`, then `\u2211 \u03b4_\u03b3 exp \u03b3 \u2260 0`.","title":"The weak Lindemann\u2013Weierstrass theorem"},{"data_url":"scopes/tauceti/number-theory/part-01/effective-arithmetic-bounds-and-geometry-of-numb.json","id":"tauceti/number-theory/part-01/effective-arithmetic-bounds-and-geometry-of-numb","kind":"scope","parent":3139,"result":false,"route":"../roadmap/tauceti/number-theory/part-01/effective-arithmetic-bounds-and-geometry-of-numb/index.html","scope":"tauceti/number-theory/part-01/effective-arithmetic-bounds-and-geometry-of-numb","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Effective arithmetic bounds and geometry of numbers"},{"id":"tauceti/number-theory/part-01/effective-arithmetic-bounds-and-geometry-of-numb/effective-hermite-minkowski","kind":"node","lean":"NumberField.ncard_setOf_finiteDimensional_abs_discr_le_le","parent":3237,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/effective-arithmetic-bounds-and-geometry-of-numb/index.html#effective-hermite-minkowski","scope":"tauceti/number-theory/part-01/effective-arithmetic-bounds-and-geometry-of-numb","status":"fully_proved","summary":"Inside a fixed extension `A / \u211a`, the number of number fields `K` (finite extensions of `\u211a`) with `|discr K| \u2264 N` is at most `(2 * coeffBoundOfDiscrBdd N + 1) ^ (rankOfDiscrBdd N + 1) * rankOfDiscrBdd N`, an explicit function of `N` alone. This upgrades Mathlib's `NumberField.finite_of_discr_bdd` from finiteness to an explicit count.","title":"Effective Hermite--Minkowski"},{"id":"tauceti/number-theory/part-01/effective-arithmetic-bounds-and-geometry-of-numb/numberfield-card-ideal-absnorm-le","kind":"node","lean":"NumberField.card_ideal_absNorm_le","parent":3237,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/effective-arithmetic-bounds-and-geometry-of-numb/index.html#numberfield-card-ideal-absnorm-le","scope":"tauceti/number-theory/part-01/effective-arithmetic-bounds-and-geometry-of-numb","status":"fully_proved","summary":"In any number field `F`, the set of nonzero integral ideals with norm at most `X` is finite, and for `X \u2265 1` its cardinality is at most `X\u00b2 * 2^[F:\u211a]`.","title":"NumberField.card_ideal_absNorm_le"},{"id":"tauceti/number-theory/part-01/effective-arithmetic-bounds-and-geometry-of-numb/tauceti-polynomial-ncard-biunion-roots-natdegree","kind":"node","lean":"TauCeti.Polynomial.ncard_biUnion_roots_natDegree_le_abs_intCoeff_le","parent":3237,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/effective-arithmetic-bounds-and-geometry-of-numb/index.html#tauceti-polynomial-ncard-biunion-roots-natdegree","scope":"tauceti/number-theory/part-01/effective-arithmetic-bounds-and-geometry-of-numb","status":"fully_proved","summary":"The integer specialisation: the roots in a domain `S` of all integer polynomials of degree at most `d` with every coefficient bounded by `B` in absolute value number at most `(2 \u00b7 B + 1) ^ (d + 1) \u00b7 d`. Each of the `2 \u00b7 B + 1` integers in `[-B, B]` is an allowed coefficient, and a degree-`\u2264 d` polynomial has at most `d` roots.","title":"TauCeti.Polynomial.ncard_biUnion_roots_natDegree_le_abs_intCoeff_le"},{"id":"tauceti/number-theory/part-01/effective-arithmetic-bounds-and-geometry-of-numb/northcotts-theorem-for-the-naive-height-with-eve","kind":"node","lean":"TauCeti.finite_setOf_naiveHeight_le, Submodule.finite_subspaceHeight_le, TauCeti.finite_setOf_absMulHeight\u2081_le, TauCeti.finite_setOf_mahlerMeasure_le","parent":3237,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/effective-arithmetic-bounds-and-geometry-of-numb/index.html#northcotts-theorem-for-the-naive-height-with-eve","scope":"tauceti/number-theory/part-01/effective-arithmetic-bounds-and-geometry-of-numb","status":"fully_proved","summary":"In a field of characteristic zero, the algebraic elements of degree at most `d` and naive height at most `B` form a finite set with at most `d (2\u230aB\u230b + 1)^(d + 1)` elements.","title":"Northcott's theorem for the naive height, with Evertse's count"},{"id":"tauceti/number-theory/part-01/effective-arithmetic-bounds-and-geometry-of-numb/the-regulator-of-a-number-field-of-unit-rank-zer","kind":"node","lean":"NumberField.Units.regulator_eq_one_of_rank_eq_zero","parent":3237,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/effective-arithmetic-bounds-and-geometry-of-numb/index.html#the-regulator-of-a-number-field-of-unit-rank-zer","scope":"tauceti/number-theory/part-01/effective-arithmetic-bounds-and-geometry-of-numb","status":"fully_proved","summary":"When the unit rank is zero, the unit lattice sits inside the zero-dimensional log space, and the regulator \u2014 its covolume, computed by `NumberField.Units.regulator_eq_det'` as a determinant indexed by the infinite places other than the distinguished one \u2014 is the determinant of the empty matrix, `1`.","title":"The regulator of a number field of unit rank zero is `1`"},{"id":"tauceti/number-theory/part-01/effective-arithmetic-bounds-and-geometry-of-numb/numberfield-coeffboundofdiscrbdd","kind":"node","lean":"NumberField.coeffBoundOfDiscrBdd","parent":3237,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/effective-arithmetic-bounds-and-geometry-of-numb/index.html#numberfield-coeffboundofdiscrbdd","scope":"tauceti/number-theory/part-01/effective-arithmetic-bounds-and-geometry-of-numb","status":"fully_proved","summary":"An explicit height bound for the integer minimal polynomial of a primitive generator of a number field of discriminant `\u2264 N`. It is `\u2308M ^ D * (D.choose (D / 2))\u2309\u208a` where `D = rankOfDiscrBdd N` is Mathlib's degree bound and `M = max \u221a(1 + boundOfDiscBdd N ^ 2) 1` dominates the per-conjugate bound of both the real and complex primitive elements of `finite_of_discr_bdd`.","title":"NumberField.coeffBoundOfDiscrBdd"},{"id":"tauceti/number-theory/part-01/effective-arithmetic-bounds-and-geometry-of-numb/numberfield-hermitecountbound","kind":"node","lean":"NumberField.hermiteCountBound","parent":3237,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/effective-arithmetic-bounds-and-geometry-of-numb/index.html#numberfield-hermitecountbound","scope":"tauceti/number-theory/part-01/effective-arithmetic-bounds-and-geometry-of-numb","status":"fully_proved","summary":"The elementary polynomial count appearing in effective Hermite--Minkowski: if every field is generated by a root of an integer polynomial of degree at most `D` and coefficient height at most `C`, then the number of possible generated fields is bounded by `(2 * C + 1) ^ (D + 1) * D`.","title":"NumberField.hermiteCountBound"},{"id":"tauceti/number-theory/part-01/effective-arithmetic-bounds-and-geometry-of-numb/tauceti-index-square-le-of-closure-eq-top","kind":"node","lean":"TauCeti.index_square_le_of_closure_eq_top","parent":3237,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/effective-arithmetic-bounds-and-geometry-of-numb/index.html#tauceti-index-square-le-of-closure-eq-top","scope":"tauceti/number-theory/part-01/effective-arithmetic-bounds-and-geometry-of-numb","status":"fully_proved","summary":"In a commutative group generated by a finite set `S`, the subgroup of squares has index at most `2 ^ S.card`. The quotient by the squares is an elementary abelian `2`-group spanned by the image of `S`.","title":"TauCeti.index_square_le_of_closure_eq_top"},{"data_url":"scopes/tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck.json","id":"tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck","kind":"scope","parent":3139,"result":false,"route":"../roadmap/tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck/index.html","scope":"tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Global number fields, ray classes, adeles, and Hecke characters"},{"id":"tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck/the-ray-class-ideal-count","kind":"node","lean":"TauCeti.GlobalNumberFields.rayClassIdealCount","parent":3246,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck/index.html#the-ray-class-ideal-count","scope":"tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck","status":"fully_proved","summary":"For every modulus `\ud835\udd2a` there is a power saving `\u03b4 > 0` such that, in each ray class `c` of `\ud835\udd2a`, the number of nonzero integral ideals prime to `\ud835\udd2a` of norm at most `x` is `rayClassIdealMainTerm \ud835\udd2a * x + O(x ^ (1 - \u03b4))`. One can take `\u03b4 = 1 / [K : \u211a]`; for that explicit exponent, use `isBigO_rayClassIdealCountingFunction_sub` instead.","title":"The ray class ideal count"},{"id":"tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck/finite-order-hecke-characters-are-exactly-the-ra","kind":"node","lean":"TauCeti.GlobalNumberFields.HeckeCharacter.isFiniteOrder_iff_exists_rayClassCharacter","parent":3246,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck/index.html#finite-order-hecke-characters-are-exactly-the-ra","scope":"tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck","status":"fully_proved","summary":"a Hecke character has finite order exactly when it is the pullback of a ray class character of some modulus.","title":"Finite-order Hecke characters are exactly the ray class characters"},{"id":"tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck/the-infinite-part-of-the-conductor-is-the-set-of","kind":"node","lean":"TauCeti.GlobalNumberFields.HeckeCharacter.mem_conductor_infinitePart_iff","parent":3246,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck/index.html#the-infinite-part-of-the-conductor-is-the-set-of","scope":"tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck","status":"fully_proved","summary":"A real place `w` divides the conductor of a finite-order Hecke character exactly when its real parity at `w` is one, that is, when the character is nontrivial on the negative elements of `K_w`.","title":"The infinite part of the conductor is the set of real signs"},{"id":"tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck/the-ray-class-group-of-q-modulo-n-is-the-galois","kind":"node","lean":"TauCeti.GlobalNumberFields.ratModulusEquivGal","parent":3246,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck/index.html#the-ray-class-group-of-q-modulo-n-is-the-galois","scope":"tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck","status":"fully_proved","summary":"The composite of `ratModulusEquivZMod` with the inverse of Mathlib's `galEquivZMod`: a ray class with residue `a \u2208 (ZMod n)\u02e3` goes to the automorphism acting on the `n`-th roots of unity by `\u03b6 \u21a6 \u03b6 ^ a`. It sends the ray class of a prime `p \u2224 n` to the arithmetic Frobenius at `p` (`isArithFrobAt_iff_eq_ratModulusEquivGal_idealClass`).","title":"The ray class group of `\u211a` modulo `(n)\u00b7\u221e` is the Galois group of `\u211a(\u03b6_n)`"},{"id":"tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck/the-congruence-description-of-the-picard-group","kind":"node","lean":"TauCeti.GlobalNumberFields.NumberFieldOrder.mem_ker_rayClassToPic","parent":3246,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck/index.html#the-congruence-description-of-the-picard-group","scope":"tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck","status":"fully_proved","summary":"A ray class of the conductor maps to the trivial Picard class exactly when it is the ray class of an ideal of `\ud835\udcde K` generated by an element of the order.","title":"The congruence description of the Picard group"},{"id":"tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck/the-congruence-description-of-the-narrow-picard","kind":"node","lean":"TauCeti.GlobalNumberFields.NumberFieldOrder.mem_ker_rayClassToNarrowPic","parent":3246,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck/index.html#the-congruence-description-of-the-narrow-picard","scope":"tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck","status":"fully_proved","summary":"A ray class of the narrow conductor modulus maps to the trivial narrow Picard class exactly when it is the ray class of an ideal of `\ud835\udcde K` generated by a totally positive element of the order.","title":"The congruence description of the narrow Picard group"},{"id":"tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck/an-order-is-gorenstein-exactly-when-all-of-its-p","kind":"node","lean":"TauCeti.GlobalNumberFields.NumberFieldOrder.isGorenstein_iff_forall_isUnit","parent":3246,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck/index.html#an-order-is-gorenstein-exactly-when-all-of-its-p","scope":"tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.GlobalNumberFields.NumberFieldOrder.isGorenstein_iff_forall_isUnit`.","title":"An order is Gorenstein exactly when all of its proper fractional ideals are invertible"},{"id":"tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck/isdedekinddomain-finiteadelering-sintegraladeles","kind":"node","lean":"IsDedekindDomain.FiniteAdeleRing.sIntegralAdeles","parent":3246,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck/index.html#isdedekinddomain-finiteadelering-sintegraladeles","scope":"tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck","status":"fully_proved","summary":"The `S`-integral finite adeles `\u220f_{v \u2208 S} K_v \u00d7 \u220f_{v \u2209 S} \ud835\udcaa_v`: the finite adeles that are integral at every finite place outside `S`, as an `R`-subalgebra of the finite adele ring. For `S = \u2205` these are the integral finite adeles, and every finite adele is `S`-integral for some finite `S`.","title":"IsDedekindDomain.FiniteAdeleRing.sIntegralAdeles"},{"id":"tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck/the-idele-class-group-splits-as-c-k1-r-0","kind":"node","lean":"TauCeti.GlobalNumberFields.normOneProdEquiv","parent":3246,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck/index.html#the-idele-class-group-splits-as-c-k1-r-0","scope":"tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck","status":"fully_proved","summary":"The infinite place `w` determines the isomorphism of topological groups sending an idele class `c` to its norm-one part `c \u00b7 s(\u2016c\u2016)\u207b\u00b9` and its norm `\u2016c\u2016`, where `s = ideleClassNormSection K w`.","title":"The idele class group splits as `C_K\u00b9 \u00d7 \u211d>0`"},{"id":"tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck/base-change-of-adeles-in-a-tower-k-l-m","kind":"node","lean":"TauCeti.GlobalNumberFields.adeleBaseChangeEquiv_tower","parent":3246,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck/index.html#base-change-of-adeles-in-a-tower-k-l-m","scope":"tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck","status":"fully_proved","summary":"the comparison for `M/K` factors through the comparison for `L/K`, followed by the comparison for `M/L`.","title":"Base change of adeles in a tower `K \u2286 L \u2286 M`"},{"id":"tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck/eulers-formula","kind":"node","lean":"TauCeti.GlobalNumberFields.NumberFieldOrder.traceDual_eq_spanSingleton_of_toSubalgebra_eq_adjoin","parent":3246,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck/index.html#eulers-formula","scope":"tauceti/number-theory/part-01/global-number-fields-ray-classes-adeles-and-heck","status":"fully_proved","summary":"the trace dual of a monogenic order `\u2124[x]` is generated by the inverse of the derivative of the minimal polynomial of `x`, evaluated at `x`.","title":"Euler's formula"},{"data_url":"scopes/tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields.json","id":"tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields","kind":"scope","parent":3139,"result":false,"route":"../roadmap/tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields/index.html","scope":"tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Global quadratic forms over number fields"},{"id":"tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields/hilbert-sign-prescription-omeara-71-19a","kind":"node","lean":"TauCeti.NumberField.QuadraticForm.exists_hilbertSymbol_eq_ite","parent":3258,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields/index.html#hilbert-sign-prescription-omeara-71-19a","scope":"tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields","status":"fully_proved","summary":"Let `S` be a finite set of finite places at which `b \u2208 K\u02e3` is a nonsquare and `T` a finite set of real places at which `b` is negative, with `#S + #T` even. Then some `a \u2208 K\u02e3` has Hilbert symbol `(a, b)_v = -1` at the places of `S` and of `T`, and `1` at every other finite and real place.","title":"Hilbert sign prescription (O'Meara 71:19a)"},{"id":"tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields/the-hasse-product-formula","kind":"node","lean":"QuadraticForm.globalInvariants_hasseProduct_eq_one","parent":3258,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields/index.html#the-hasse-product-formula","scope":"tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields","status":"fully_proved","summary":"For every regular form `Q` over a number field `K`, the product of the finite Hasse signs and of the real Hasse signs of `Q` is `1`.","title":"The Hasse product formula"},{"id":"tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields/the-quaternary-case-of-the-hasse-minkowski-theor","kind":"node","lean":"QuadraticForm.not_anisotropic_of_isLocallyIsotropic_quaternary, QuadraticForm.not_anisotropic_of_isLocallyIsotropic_quaternary_of_discr_eq_zero","parent":3258,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields/index.html#the-quaternary-case-of-the-hasse-minkowski-theor","scope":"tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields","status":"fully_proved","summary":"Let `Q` be a regular quaternary form over a number field `K` which is isotropic at every finite and real place. If, over every number field, every regular diagonal ternary form that is isotropic at every finite and real place is isotropic, then `Q` is isotropic over `K`.","title":"The quaternary case of the Hasse\u2013Minkowski theorem"},{"id":"tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields/the-hasse-minkowski-induction-step-in-rank-at-le","kind":"node","lean":"QuadraticForm.not_anisotropic_of_isLocallyIsotropic_of_five_le_finrank","parent":3258,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields/index.html#the-hasse-minkowski-induction-step-in-rank-at-le","scope":"tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields","status":"fully_proved","summary":"Let `Q` be a regular quadratic form of rank `n \u2265 5` over a number field that is isotropic at every finite and real place. If every diagonal regular form of rank `n - 1` that is isotropic at every finite and real place is isotropic over `K`, then `Q` is isotropic over `K`.","title":"The Hasse\u2013Minkowski induction step in rank at least five"},{"id":"tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields/local-equivalence-by-invariants","kind":"node","lean":"QuadraticForm.locallyEquivalent_iff_finrank_eq_and_discr_eq_and_finiteHasse_eq_and_realPositiveIndex_eq","parent":3258,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields/index.html#local-equivalence-by-invariants","scope":"tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields","status":"fully_proved","summary":"Two regular forms over a number field are isometric at every finite and every real place exactly when they have the same rank, the images of their global discriminants agree at every finite place, and they have the same Hasse sign at every finite place and the same positive index at every real place.","title":"Local equivalence by invariants"},{"id":"tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields/the-system-of-local-invariants-of-a-regular-quad","kind":"node","lean":"QuadraticForm.globalInvariants","parent":3258,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields/index.html#the-system-of-local-invariants-of-a-regular-quad","scope":"tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields","status":"fully_proved","summary":"its rank, its plain discriminant `d(Q) \u2208 K\u02e3/(K\u02e3)\u00b2`, its Hasse sign at every finite place and its positive index at every real place.","title":"The system of local invariants of a regular quadratic form `Q` over a number field"},{"id":"tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields/a-system-of-local-invariants-of-a-quadratic-form","kind":"node","lean":"TauCeti.NumberField.QuadraticForm.GlobalFormInvariants","parent":3258,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields/index.html#a-system-of-local-invariants-of-a-quadratic-form","scope":"tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields","status":"fully_proved","summary":"a rank `n`, a global plain discriminant `d \u2208 K\u02e3/(K\u02e3)\u00b2`, a Hasse sign `s_v \u2208 {\u00b11}` at every finite place `v`, and a positive index `p_w` at every real place `w`. No condition relates the data; the systems arising from regular forms of positive rank are those satisfying `IsAdmissible`.","title":"A system of local invariants of a quadratic form over a number field `K`"},{"id":"tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields/the-kernel-of-the-idele-hilbert-character","kind":"node","lean":"TauCeti.NumberField.QuadraticForm.sup_range_ideleNormMap_eq_ker_ideleHilbertCharacter","parent":3258,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields/index.html#the-kernel-of-the-idele-hilbert-character","scope":"tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields","status":"fully_proved","summary":"Let `b \u2208 K\u02e3` be a nonsquare at some finite place or negative at some real place, and let `L` be a number field over `K` in which `b` is a square. Suppose that the idele-class norm group `N_{L/K}(C_L)` has index dividing two in `C_K`, as the norm index theorem gives for `L = K(\u221ab)`. Then the kernel of `\u03c6_b` is `K\u02e3 \u00b7 N_{L/K}(\ud835\udd40_L)`, the subgroup of `\ud835\udd40_K` generated by the principal ideles and the idele norms.","title":"The kernel of the idele Hilbert character"},{"id":"tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields/omeara-72-1-up-to-one-global-representation","kind":"node","lean":"TauCeti.NumberField.QuadraticForm.GlobalFormInvariants.IsLocalRealization.exists_locallyRepresents_prod","parent":3258,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields/index.html#omeara-72-1-up-to-one-global-representation","scope":"tauceti/number-theory/part-01/global-quadratic-forms-over-number-fields","status":"fully_proved","summary":"Let `U_v` and `R_w` be local forms realizing an admissible system `I`. Then there are a regular global form `W` of rank `I.rank` and `\u03b1, \u03b2 \u2208 K\u02e3` such that `\u27e8\u03b1, -\u03b1\u03b2\u27e9` is represented by `\u27e81, -\u03b2\u27e9 \u22a5 W` at every finite and real place, and such that, if `\u27e8\u03b1, -\u03b1\u03b2\u27e9` is represented by `\u27e81, -\u03b2\u27e9 \u22a5 W` over `K`, some regular global form on `K ^ I.rank` has `I` as its system of local invariants.","title":"O'Meara 72:1 up to one global representation"},{"data_url":"scopes/tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields.json","id":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields","kind":"scope","parent":3139,"result":false,"route":"../roadmap/tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/index.html","scope":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Heights, units and S-units of number fields"},{"id":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/a-reduced-fundamental-system-bugeaud-gyory-lemma","kind":"node","lean":"TauCeti.NumberField.Units.exists_fundSystem_prod_absLogHeight\u2081_le","parent":3268,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/index.html#a-reduced-fundamental-system-bugeaud-gyory-lemma","scope":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields","status":"fully_proved","summary":"Some fundamental system `\u03b5` of units has `\u220f i, absLogHeight\u2081 (\u03b5 i) \u2264 (r!) ^ 2 / (2 ^ (r - 1) * [K : \u211a] ^ r) * regulator K`, where `r = rank K`. The constant is attained for `\u211a(\u221a2)`. By Hadamard's bound `regulator_le_prod_absLogHeight\u2081`, every fundamental system also has `regulator K \u2264 (2 * [K : \u211a]) ^ r * \u220f i, absLogHeight\u2081 (\u03b5 i)`.","title":"A reduced fundamental system (Bugeaud\u2013Gy\u0151ry, Lemma 1)"},{"id":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/tauceti-numberfield-units-regulator-le-prod-absl","kind":"node","lean":"TauCeti.NumberField.Units.regulator_le_prod_absLogHeight\u2081","parent":3268,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/index.html#tauceti-numberfield-units-regulator-le-prod-absl","scope":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields","status":"fully_proved","summary":"Every fundamental system bounds the field regulator by `(2d) ^ r` times the product of its absolute logarithmic heights, where `d` is the degree and `r` the unit rank.","title":"TauCeti.NumberField.Units.regulator_le_prod_absLogHeight\u2081"},{"id":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/minkowskis-second-theorem-upper-bound-in-terms-o","kind":"node","lean":"ZLattice.prod_successiveMin_mul_measure_le_two_pow_mul_covolume, Convex.prod_mul_volume_real_le_two_pow, ZLattice.prod_mul_measure_le_two_pow_mul_covolume, ZLattice.prod_successiveMin_mul_measureReal_le_two_pow_mul_covolume, ZLattice.two_pow_div_factorial_mul_covolume_le_prod_mul_measure, ZLattice.two_pow_div_factorial_mul_covolume_le_prod_successiveMin_mul_measure, ZLattice.two_pow_div_factorial_mul_covolume_le_prod_successiveMin_mul_measureReal, ZLattice.prod_successiveMin_le_covolume_of_two_pow_finrank_le","parent":3268,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/index.html#minkowskis-second-theorem-upper-bound-in-terms-o","scope":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields","status":"fully_proved","summary":"For a full lattice `L` in a real normed space `E` of dimension `n`, a Haar measure `\u03bc` on `E`, and a convex body `K` symmetric about `0`, the successive minima `\u03bb i = successiveMin L K i` satisfy `\u03bb\u2080 \u22ef \u03bb\u2099\u208b\u2081 \u00b7 \u03bc K \u2264 2\u207f \u00b7 covolume L \u03bc`. If `K` has empty interior, its measure is zero.","title":"Minkowski's second theorem, upper bound, in terms of the successive minima"},{"id":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/bombieri-and-vaalers-theorem-2-over-z-in-the-sup","kind":"node","lean":"Matrix.exists_linearIndependent_mulVec_eq_zero_prod_norm_le","parent":3268,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/index.html#bombieri-and-vaalers-theorem-2-over-z-in-the-sup","scope":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields","status":"fully_proved","summary":"Let `A` be a `p \u00d7 n` integer matrix with linearly independent rows, and let `g` be the greatest common divisor of its maximal minors. Then `A x = 0` has `n - p` linearly independent integral solutions `x l`, each with coprime coordinates, whose sup norms `\u2016x l\u2016 = max_i |x l i|` satisfy `\u220f l, \u2016x l\u2016 \u2264 \u221adet (A A\u1d40) / g`.","title":"Bombieri and Vaaler's Theorem 2 over `\u2124`, in the sup norm"},{"id":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/vaalers-cube-slicing-theorem","kind":"node","lean":"Submodule.two_pow_finrank_le_volume_setOf_forall_abs_le_one","parent":3268,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/index.html#vaalers-cube-slicing-theorem","scope":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields","status":"fully_proved","summary":"Every central slice of the cube `[-1, 1] ^ \u03b9` by a subspace `V` of `EuclideanSpace \u211d \u03b9` has volume at least `2 ^ finrank \u211d V`, computed in `V`.","title":"Vaaler's cube-slicing theorem"},{"id":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/the-determinant-formula-for-the-s-regulator","kind":"node","lean":"TauCeti.NumberField.SUnit.regulator_eq_abs_det","parent":3268,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/index.html#the-determinant-formula-for-the-s-regulator","scope":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields","status":"fully_proved","summary":"For a family `\u03b5` of `S`-units that, together with the torsion subgroup, generates the `S`-units, an infinite place `w'` and any identification `e` of `\u03b9` with the places of `S` and the infinite places other than `w'`, the `S`-regulator is the absolute value of the determinant of the matrix whose `i`-th row lists `mult w * log (w (\u03b5 i))` at the infinite places `w \u2260 w'` and `log \u2016\u03b5 i\u2016_v` at the places `v \u2208 S`, in the\u2026","title":"The determinant formula for the `S`-regulator"},{"id":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/kroneckers-theorem-for-projective-tuples","kind":"node","lean":"TauCeti.absoluteProjectiveTupleHeight_eq_one_iff, TauCeti.house_eq_one_iff, TauCeti.mahlerMeasure_eq_one_iff","parent":3268,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/index.html#kroneckers-theorem-for-projective-tuples","scope":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields","status":"fully_proved","summary":"A tuple with algebraic coordinates has absolute projective height one if and only if, for every nonzero coordinate `x j`, each ratio `x i / x j` is zero or of finite multiplicative order. This includes the zero tuple.","title":"Kronecker's theorem for projective tuples"},{"id":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/zlattice-successivemin","kind":"node","lean":"ZLattice.successiveMin","parent":3268,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/index.html#zlattice-successivemin","scope":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields","status":"fully_proved","summary":"The successive minima of a `\u2124`-submodule `L` of `E` with respect to a convex body `K`, indexed from zero: `successiveMin L K i` is the infimum of the dilation factors `t \u2265 0` for which `t \u2022 K` contains `i + 1` vectors of `L` that are linearly independent over `\u211d`.","title":"ZLattice.successiveMin"},{"id":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/a-basis-from-the-successive-minima-cassels-chapt","kind":"node","lean":"ZLattice.exists_basis_mem_smul_successiveMin","parent":3268,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/index.html#a-basis-from-the-successive-minima-cassels-chapt","scope":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields","status":"fully_proved","summary":"For a discrete spanning lattice `L` and a symmetric convex body `K` with `0` in its interior, some `\u2124`-basis `b` of `L` has `b i \u2208 (max 1 ((i + 1) / 2) * successiveMin L K i) \u2022 K` for every `i`. The factor is `1` for `i \u2264 1`, and the product of the factors is `n! / 2 ^ (n - 1)` (`TauCeti.prod_max_one_add_one_div_two`).","title":"A basis from the successive minima (Cassels, Chapter V, Lemma 8)"},{"id":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/matrix-minorgcd","kind":"node","lean":"Matrix.minorGcd","parent":3268,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/index.html#matrix-minorgcd","scope":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields","status":"fully_proved","summary":"The gcd of the maximal minors of `A`: the normalized gcd of the determinants `det (A.submatrix id e)` over all embeddings `e : m \u21aa n`. It is `0` exactly when every maximal minor vanishes, in particular when `n` has fewer elements than `m`.","title":"Matrix.minorGcd"},{"id":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/the-height-of-the-kernel-of-an-integer-matrix","kind":"node","lean":"Matrix.minorGcd_mul_subspaceHeight_ker_mulVecLin","parent":3268,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/index.html#the-height-of-the-kernel-of-an-integer-matrix","scope":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields","status":"fully_proved","summary":"The greatest common divisor of the maximal minors of an integer matrix `A`, times the absolute Euclidean height of the space of rational solutions of `A y = 0`, is `\u221adet (A A\u1d40)`. If the rows are linearly dependent, both sides vanish.","title":"The height of the kernel of an integer matrix"},{"id":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/tauceti-numberfield-sunit-logembeddingequiv","kind":"node","lean":"TauCeti.NumberField.SUnit.logEmbeddingEquiv","parent":3268,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/index.html#tauceti-numberfield-sunit-logembeddingequiv","scope":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields","status":"fully_proved","summary":"The `S`-logarithmic embedding induces a `\u2124`-linear equivalence from the `S`-units modulo torsion onto the `S`-unit lattice. This is the `S`-analogue of Mathlib's `NumberField.Units.logEmbeddingEquiv`.","title":"TauCeti.NumberField.SUnit.logEmbeddingEquiv"},{"id":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/bombieri-and-vaalers-siegel-lemma-over-z-in-the","kind":"node","lean":"Matrix.exists_ne_zero_mulVec_eq_zero_norm_le","parent":3268,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/index.html#bombieri-and-vaalers-siegel-lemma-over-z-in-the","scope":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields","status":"fully_proved","summary":"Let `A` be a `p \u00d7 n` integer matrix with linearly independent rows and `p < n`, and let `g` be the greatest common divisor of its maximal minors. Then `A x = 0` has a nonzero integral solution `x` with `\u2016x\u2016 \u2264 (\u221adet (A A\u1d40) / g) ^ (1 / (n - p))`, where `\u2016x\u2016 = max_i |x i|`.","title":"Bombieri and Vaaler's Siegel lemma over `\u2124`, in the sup norm"},{"id":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/minkowskis-first-theorem-on-the-successive-minim","kind":"node","lean":"ZLattice.successiveMin_pow_mul_measureReal_le_two_pow_mul_covolume","parent":3268,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/index.html#minkowskis-first-theorem-on-the-successive-minim","scope":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields","status":"fully_proved","summary":"For a full lattice `L` in a real normed space `E` of positive dimension `n`, a Haar measure `\u03bc` on `E`, and a convex body `K` symmetric about `0`, the first successive minimum `\u03bb\u2080` of `L` with respect to `K` satisfies `\u03bb\u2080\u207f \u00b7 \u03bc.real K \u2264 2\u207f \u00b7 covolume L \u03bc`. It follows from the upper bound of Minkowski's second theorem, since `\u03bb\u2080 \u2264 \u03bb\u1d62` for every `i`.","title":"Minkowski's first theorem on the successive minima"},{"id":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/minkowskis-second-theorem-for-the-unit-lattice","kind":"node","lean":"TauCeti.NumberField.Units.exists_isMaxRank_prod_sum_abs_logEmbedding_le","parent":3268,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/index.html#minkowskis-second-theorem-for-the-unit-lattice","scope":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields","status":"fully_proved","summary":"There are units `u 0, \u2026, u (r - 1)`, where `r = rank K`, with linearly independent logarithmic embeddings and `\u220f i, \u2016log u i\u2016\u2081 \u2264 r! * regulator K`, where `\u2016log x\u2016\u2081` is the `\u2113\u00b9` norm of the logarithmic embedding of `x`. This is Minkowski's second theorem for the unit lattice and the `\u2113\u00b9` unit ball `logBall K`, whose volume is `2 ^ r / r!`.","title":"Minkowski's second theorem for the unit lattice"},{"id":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/siegels-lemma-for-a-rational-subspace-in-the-sup","kind":"node","lean":"Submodule.exists_ne_zero_mem_intPoints_norm_le","parent":3268,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields/index.html#siegels-lemma-for-a-rational-subspace-in-the-sup","scope":"tauceti/number-theory/part-01/heights-units-and-s-units-of-number-fields","status":"fully_proved","summary":"A nonzero subspace `V` of `\u211a^\u03b9` contains a nonzero integral point `z` whose sup norm is at most `H(V) ^ (1 / k)`, where `H(V)` is the absolute Euclidean height of `V` and `k` its dimension.","title":"Siegel's lemma for a rational subspace, in the sup norm"},{"data_url":"scopes/tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten.json","id":"tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten","kind":"scope","parent":3139,"result":false,"route":"../roadmap/tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten/index.html","scope":"tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Imaginary quadratic fields: class numbers, finiteness and class number one"},{"id":"tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten/the-radicand-classification-of-imaginary-quadrat","kind":"node","lean":"NumberField.existsUnique_neg_squarefree_algEquiv","parent":3285,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten/index.html#the-radicand-classification-of-imaginary-quadrat","scope":"tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten","status":"fully_proved","summary":"Every totally complex number field of degree `2` over `\u211a` is isomorphic to `QuadraticAlgebra \u211a d 0` for exactly one negative squarefree integer `d`.","title":"The radicand classification of imaginary quadratic fields"},{"id":"tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten/the-squarefree-radicand-is-an-isomorphism-invari","kind":"node","lean":"NumberField.nonempty_algEquiv_iff_of_squarefree","parent":3285,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten/index.html#the-squarefree-radicand-is-an-isomorphism-invari","scope":"tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten","status":"fully_proved","summary":"Two quadratic fields presented by square roots of squarefree integers `d` and `e` are `\u211a`-isomorphic exactly when `d = e`. Taking `L = K` and the identity shows that the squarefree radicand of a quadratic field does not depend on the presentation.","title":"The squarefree radicand is an isomorphism invariant"},{"id":"tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten/the-two-complex-embeddings-of-an-imaginary-quadr","kind":"node","lean":"QuadraticAlgebra.ringHom_eq_or_eq_conjugate","parent":3285,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten/index.html#the-two-complex-embeddings-of-an-imaginary-quadr","scope":"tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten","status":"fully_proved","summary":"For a nonsquare `a \u2264 0`, every complex embedding of the field `QuadraticAlgebra \u211a a 0` is the distinguished embedding or its complex conjugate.","title":"The two complex embeddings of an imaginary quadratic field"},{"id":"tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten/quadraticalgebra-nrcomplexplaces-eq-one","kind":"node","lean":"QuadraticAlgebra.nrComplexPlaces_eq_one","parent":3285,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten/index.html#quadraticalgebra-nrcomplexplaces-eq-one","scope":"tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten","status":"fully_proved","summary":"An imaginary quadratic field has exactly one complex place, so its signature is `(0, 1)`.","title":"QuadraticAlgebra.nrComplexPlaces_eq_one"},{"id":"tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten/dirichletcharacter-lfunction-apply-one-pos","kind":"node","lean":"DirichletCharacter.LFunction_apply_one_pos","parent":3285,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten/index.html#dirichletcharacter-lfunction-apply-one-pos","scope":"tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten","status":"fully_proved","summary":"The value at `s = 1` of the `L`-function of a nontrivial quadratic Dirichlet character is a positive real number.","title":"DirichletCharacter.LFunction_apply_one_pos"},{"id":"tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten/distinct-proper-classes-of-primitive-positive-de","kind":"node","lean":"TauCeti.BinaryQuadraticForm.posDefProperClassToPic_injective","parent":3285,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten/index.html#distinct-proper-classes-of-primitive-positive-de","scope":"tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten","status":"fully_proved","summary":"This is the injective half of the correspondence between proper classes of primitive positive definite forms of discriminant `-n` and the Picard group of the order of discriminant `-n`.","title":"Distinct proper classes of primitive positive definite forms have distinct Picard classes"},{"id":"tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten/the-distinguished-complex-embedding-of-q-a-for-a","kind":"node","lean":"QuadraticAlgebra.complexEmbedding","parent":3285,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten/index.html#the-distinguished-complex-embedding-of-q-a-for-a","scope":"tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten","status":"fully_proved","summary":"the `\u211a`-algebra map `QuadraticAlgebra \u211a a 0 \u2192\u2090[\u211a] \u2102` sending `\u03c9` to `\u221a(-a) \u00b7 i`, the square root of `a` with nonnegative imaginary part.","title":"The distinguished complex embedding of `\u211a(\u221aa)` for `a \u2264 0`"},{"id":"tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten/the-siegel-function-g-r-attached-to-r-r1-r2-r2","kind":"node","lean":"TauCeti.ModularForm.siegelFunction","parent":3285,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten/index.html#the-siegel-function-g-r-attached-to-r-r1-r2-r2","scope":"tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten","status":"fully_proved","summary":"`g_r(\u03c4) = -e^{\u03c0 i B\u2082(r\u2081) \u03c4} e^{\u03c0 i r\u2082 (r\u2081 - 1)} (1 - w) \u220f_{n \u2265 1} (1 - q\u207f w) (1 - q\u207f w\u207b\u00b9)` with `q = e^{2\u03c0 i \u03c4}`, `w = e^{2\u03c0 i (r\u2081 \u03c4 + r\u2082)}` and `B\u2082 = bernoulliFun 2` the second Bernoulli polynomial `x\u00b2 - x + 1/6`. The product converges for `\u03c4` in the upper half-plane, where `g_r` is holomorphic; elsewhere the value is a junk value.","title":"The Siegel function `g_r(\u03c4)` attached to `r = (r\u2081, r\u2082) \u2208 \u211d\u00b2`"},{"id":"tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten/tauceti-qtheta","kind":"node","lean":"TauCeti.qTheta","parent":3285,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten/index.html#tauceti-qtheta","scope":"tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten","status":"fully_proved","summary":"The theta product `\u03b8_q(w) = (w; q)_\u221e (q w\u207b\u00b9; q)_\u221e`, written as `(1 - w) \u220f_{n \u2265 1} (1 - q\u207f w) (1 - q\u207f w\u207b\u00b9)`. The product converges for `\u2016q\u2016 < 1`; otherwise the value is a junk value. With `q = e^{2\u03c0 i \u03c4}` and `w = e^{2\u03c0 i (r\u2081 \u03c4 + r\u2082)}` it is the product in the Siegel function `g_r(\u03c4)`.","title":"TauCeti.qTheta"},{"id":"tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten/the-picard-class-of-a-proper-class-of-primitive","kind":"node","lean":"TauCeti.BinaryQuadraticForm.posDefProperClassToPic","parent":3285,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten/index.html#the-picard-class-of-a-proper-class-of-primitive","scope":"tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten","status":"fully_proved","summary":"The Picard class of a proper class of primitive positive definite forms, the class of `a\u2124 + ((-b + \u221a(-n))/2)\u2124` in the Picard group of the quadratic order of discriminant `-n`.","title":"The Picard class of a proper class of primitive positive definite forms"},{"id":"tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten/eulers-identity-tauceti-tprod-one-add-pow-mul-tp","kind":"node","lean":"TauCeti.tprod_one_add_pow_mul_tprod_one_sub_pow_odd","parent":3285,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten/index.html#eulers-identity-tauceti-tprod-one-add-pow-mul-tp","scope":"tauceti/number-theory/part-01/imaginary-quadratic-fields-class-numbers-finiten","status":"fully_proved","summary":"Euler's identity `\u220f_{n \u2265 1} (1 + q\u207f) \u220f_{n \u2265 0} (1 - q^{2n+1}) = 1` for `\u2016q\u2016 < 1`. Comparing coefficients, partitions into distinct parts and partitions into odd parts are equinumerous.","title":"Euler's identity (`TauCeti.tprod_one_add_pow_mul_tprod_one_sub_pow_odd`)"},{"data_url":"scopes/tauceti/number-theory/part-01/integral-quadratic-forms-and-lattices.json","id":"tauceti/number-theory/part-01/integral-quadratic-forms-and-lattices","kind":"scope","parent":3139,"result":false,"route":"../roadmap/tauceti/number-theory/part-01/integral-quadratic-forms-and-lattices/index.html","scope":"tauceti/number-theory/part-01/integral-quadratic-forms-and-lattices","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Integral quadratic forms and lattices"},{"id":"tauceti/number-theory/part-01/integral-quadratic-forms-and-lattices/hermites-inequality-for-the-minimum","kind":"node","lean":"TauCeti.IntegralLattice.IsPosDef.minimum_le_rpow_mul_determinant_rpow, TauCeti.IntegralLattice.IsPosDef.three_pow_mul_minimum_pow_le","parent":3297,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/integral-quadratic-forms-and-lattices/index.html#hermites-inequality-for-the-minimum","scope":"tauceti/number-theory/part-01/integral-quadratic-forms-and-lattices","status":"fully_proved","summary":"A positive definite integral lattice of rank `n` satisfies `min L \u2264 (4 / 3) ^ ((n - 1) / 2) \u00b7 (det L) ^ (1 / n)`.","title":"Hermite's inequality for the minimum"},{"id":"tauceti/number-theory/part-01/integral-quadratic-forms-and-lattices/a-positive-definite-genus-contains-finitely-many","kind":"node","lean":"TauCeti.IntegralLattice.IsPosDef.finite_genusClasses","parent":3297,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/integral-quadratic-forms-and-lattices/index.html#a-positive-definite-genus-contains-finitely-many","scope":"tauceti/number-theory/part-01/integral-quadratic-forms-and-lattices","status":"fully_proved","summary":"The lattices of the genus all have the rank and determinant of `L`, so this is the finiteness of positive definite classes of given rank and bounded determinant.","title":"A positive definite genus contains finitely many classes"},{"id":"tauceti/number-theory/part-01/integral-quadratic-forms-and-lattices/e8-e8-u-d16-u","kind":"node","lean":"TauCeti.IntegralLattice.nonempty_isometry_orthogonalSum_hyperbolicPlane_e8PowLattice","parent":3297,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/integral-quadratic-forms-and-lattices/index.html#e8-e8-u-d16-u","scope":"tauceti/number-theory/part-01/integral-quadratic-forms-and-lattices","status":"fully_proved","summary":"Adding the hyperbolic plane `U` makes the two non-isometric even unimodular lattices of rank sixteen isometric.","title":"`E\u2088 \u2295 E\u2088 \u2295 U \u2245 D\u2081\u2086\u207a \u2295 U`"},{"id":"tauceti/number-theory/part-01/integral-quadratic-forms-and-lattices/indefinite-lattices-have-infinite-isometry-group","kind":"node","lean":"TauCeti.IntegralLattice.infinite_isometry_of_isIndefinite","parent":3297,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/integral-quadratic-forms-and-lattices/index.html#indefinite-lattices-have-infinite-isometry-group","scope":"tauceti/number-theory/part-01/integral-quadratic-forms-and-lattices","status":"fully_proved","summary":"A nondegenerate indefinite integral lattice of rank at least three has infinitely many isometries.","title":"Indefinite lattices have infinite isometry groups"},{"id":"tauceti/number-theory/part-01/integral-quadratic-forms-and-lattices/classification-of-odd-unimodular-forms-over-z-2","kind":"node","lean":"LinearMap.BilinForm.IsSymm.equivalent_iff_isSquare_det_mul_det_and_eight_dvd_of_odd","parent":3297,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/integral-quadratic-forms-and-lattices/index.html#classification-of-odd-unimodular-forms-over-z-2","scope":"tauceti/number-theory/part-01/integral-quadratic-forms-and-lattices","status":"fully_proved","summary":"Let `B` and `C` be symmetric forms over `\u2124_[2]` with unit Gram determinants in bases of the same size, each representing a unit, and let `w` and `w'` be characteristic vectors for `B` and `C`. Then `B` and `C` are isometric exactly when the product of their Gram determinants is a square and `B w w \u2261 C w' w'` modulo `8`.","title":"Classification of odd unimodular forms over `\u2124_[2]`"},{"id":"tauceti/number-theory/part-01/integral-quadratic-forms-and-lattices/tauceti-finitequadraticmodule","kind":"node","lean":"TauCeti.FiniteQuadraticModule","parent":3297,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/integral-quadratic-forms-and-lattices/index.html#tauceti-finitequadraticmodule","scope":"tauceti/number-theory/part-01/integral-quadratic-forms-and-lattices","status":"fully_proved","summary":"A finite abelian group equipped with a quadratic map to `\u211a/\u2124`.","title":"TauCeti.FiniteQuadraticModule"},{"id":"tauceti/number-theory/part-01/integral-quadratic-forms-and-lattices/tauceti-finitequadraticmodule-orthogonalgroup","kind":"node","lean":"TauCeti.FiniteQuadraticModule.orthogonalGroup","parent":3297,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/integral-quadratic-forms-and-lattices/index.html#tauceti-finitequadraticmodule-orthogonalgroup","scope":"tauceti/number-theory/part-01/integral-quadratic-forms-and-lattices","status":"fully_proved","summary":"The orthogonal group `O(A)` of a finite quadratic module `A`.","title":"TauCeti.FiniteQuadraticModule.orthogonalGroup"},{"id":"tauceti/number-theory/part-01/integral-quadratic-forms-and-lattices/tauceti-integrallattice-classnumber","kind":"node","lean":"TauCeti.IntegralLattice.classNumber","parent":3297,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/integral-quadratic-forms-and-lattices/index.html#tauceti-integrallattice-classnumber","scope":"tauceti/number-theory/part-01/integral-quadratic-forms-and-lattices","status":"fully_proved","summary":"The class number `h(L)`: the number of isometry classes in the genus of `L`. It is `0` when the genus contains infinitely many classes.","title":"TauCeti.IntegralLattice.classNumber"},{"data_url":"scopes/tauceti/number-theory/part-01/local-fields-and-ramification.json","id":"tauceti/number-theory/part-01/local-fields-and-ramification","kind":"scope","parent":3139,"result":false,"route":"../roadmap/tauceti/number-theory/part-01/local-fields-and-ramification/index.html","scope":"tauceti/number-theory/part-01/local-fields-and-ramification","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Local fields and ramification"},{"id":"tauceti/number-theory/part-01/local-fields-and-ramification/iwasawas-theorem","kind":"node","lean":"TauCeti.nonempty_continuousMulEquiv_iwasawaGroup","parent":3306,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/local-fields-and-ramification/index.html#iwasawas-theorem","scope":"tauceti/number-theory/part-01/local-fields-and-ramification","status":"fully_proved","summary":"the tame quotient of the absolute Galois group of `K` is isomorphic, as a topological group, to the profinite group `\u27e8\u03c3, \u03c4 | \u03c3 \u03c4 \u03c3\u207b\u00b9 = \u03c4 ^ q\u27e9`.","title":"Iwasawa's theorem"},{"id":"tauceti/number-theory/part-01/local-fields-and-ramification/tauceti-localfieldsramification-upperjump-iff-ex","kind":"node","lean":"TauCeti.LocalFieldsRamification.upperJump_iff_exists_isCyclic_intermediateField","parent":3306,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/local-fields-and-ramification/index.html#tauceti-localfieldsramification-upperjump-iff-ex","scope":"tauceti/number-theory/part-01/local-fields-and-ramification","status":"fully_proved","summary":"An upper break of a finite abelian extension is detected in a cyclic subextension. The subextension can be chosen with trivial upper ramification groups at every later index, so the specified break is its last break. Its local-field structures are the canonical ones induced from the base field.","title":"TauCeti.LocalFieldsRamification.upperJump_iff_exists_isCyclic_intermediateField"},{"id":"tauceti/number-theory/part-01/local-fields-and-ramification/herbrands-theorem-in-the-upper-numbering","kind":"node","lean":"TauCeti.LocalFieldsRamification.map_restrictNormalHom_upperRamificationGroup, TauCeti.LocalFieldsRamification.map_restrictNormalHom_lowerRamificationGroupReal, TauCeti.LocalFieldsRamification.natCard_lowerRamificationGroupReal_herbrand_mul","parent":3306,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/local-fields-and-ramification/index.html#herbrands-theorem-in-the-upper-numbering","scope":"tauceti/number-theory/part-01/local-fields-and-ramification","status":"fully_proved","summary":"For a tower `M/L/K` of Galois extensions, the image of the upper ramification group `G^v` of `M/K` under restriction to `L` is the upper ramification group of `L/K` at the same index: `(G/H)^v = G^v H / H` for `H = Gal(M/L)`.","title":"Herbrand's theorem in the upper numbering"},{"id":"tauceti/number-theory/part-01/local-fields-and-ramification/tauceti-dyadiccyclotomiceight-map-restrictsqrttw","kind":"node","lean":"TauCeti.DyadicCyclotomicEight.map_restrictSqrtTwo_lowerRamificationGroup_three_ne","parent":3306,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/local-fields-and-ramification/index.html#tauceti-dyadiccyclotomiceight-map-restrictsqrttw","scope":"tauceti/number-theory/part-01/local-fields-and-ramification","status":"fully_proved","summary":"Lower numbering fails to commute with this restriction: the image at depth three is the whole quadratic Galois group, but the quadratic lower group at depth three is trivial.","title":"TauCeti.DyadicCyclotomicEight.map_restrictSqrtTwo_lowerRamificationGroup_three_ne"},{"id":"tauceti/number-theory/part-01/local-fields-and-ramification/tame-inertia-is-the-prime-to-p-tate-module","kind":"node","lean":"TauCeti.quotientWildInertiaSubgroupEquiv","parent":3306,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/local-fields-and-ramification/index.html#tame-inertia-is-the-prime-to-p-tate-module","scope":"tauceti/number-theory/part-01/local-fields-and-ramification","status":"fully_proved","summary":"the tame character induces an isomorphism of topological groups `I_K / P_K \u2243\u209c* \u2124\u0302^{(p')}(1)` from the tame inertia group, the quotient of the inertia subgroup by the wild inertia subgroup.","title":"Tame inertia is the prime-to-`p` Tate module"},{"id":"tauceti/number-theory/part-01/local-fields-and-ramification/tame-inertia-is-l-p-z-l","kind":"node","lean":"TauCeti.nonempty_continuousMulEquiv_piPadicInt","parent":3306,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/local-fields-and-ramification/index.html#tame-inertia-is-l-p-z-l","scope":"tauceti/number-theory/part-01/local-fields-and-ramification","status":"fully_proved","summary":"the tame inertia group `I_K/P_K` is isomorphic, as a topological group, to the product of the additive groups of `\u2113`-adic integers over the primes `\u2113` different from the residue characteristic `p`. Up to a choice of an isomorphism `\u2124_\u2113(1) \u2243 \u2124_\u2113` for each `\u2113` (`TauCeti.PadicTateTwist.nonempty_continuousAddEquiv`), it is the tame character `I_K/P_K \u2243 \u2124\u0302^{(p')}(1)` (`TauCeti.quotientWildInertiaSubgroupEquiv`) followed\u2026","title":"Tame inertia is `\u220f_{\u2113 \u2260 p} \u2124_\u2113`"},{"id":"tauceti/number-theory/part-01/local-fields-and-ramification/tame-inertia-is-procyclic","kind":"node","lean":"TauCeti.exists_topologicalClosure_zpowers_eq_tameInertia","parent":3306,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/local-fields-and-ramification/index.html#tame-inertia-is-procyclic","scope":"tauceti/number-theory/part-01/local-fields-and-ramification","status":"fully_proved","summary":"it is topologically generated by the class of an element of the inertia subgroup.","title":"Tame inertia is procyclic"},{"id":"tauceti/number-theory/part-01/local-fields-and-ramification/the-galois-group-of-the-maximal-unramified-exten","kind":"node","lean":"TauCeti.maximalUnramifiedGaloisGroupEquivZHat","parent":3306,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/local-fields-and-ramification/index.html#the-galois-group-of-the-maximal-unramified-exten","scope":"tauceti/number-theory/part-01/local-fields-and-ramification","status":"fully_proved","summary":"This continuous multiplicative equivalence sends arithmetic Frobenius to `zHat.gen`.","title":"The Galois group of the maximal unramified extension is the profinite integers"},{"id":"tauceti/number-theory/part-01/local-fields-and-ramification/tauceti-algebraicextensionnormedfield","kind":"node","lean":"TauCeti.algebraicExtensionNormedField","parent":3306,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/local-fields-and-ramification/index.html#tauceti-algebraicextensionnormedfield","scope":"tauceti/number-theory/part-01/local-fields-and-ramification","status":"fully_proved","summary":"The normed-field structure on an algebraic extension `M` of a nonarchimedean local field `K` given by the spectral norm of `M/K` with respect to the normalized absolute value of `K`.","title":"TauCeti.algebraicExtensionNormedField"},{"id":"tauceti/number-theory/part-01/local-fields-and-ramification/the-wild-inertia-subgroup","kind":"node","lean":"TauCeti.wildInertiaSubgroup","parent":3306,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/local-fields-and-ramification/index.html#the-wild-inertia-subgroup","scope":"tauceti/number-theory/part-01/local-fields-and-ramification","status":"fully_proved","summary":"The wild inertia subgroup `P_K` of the absolute Galois group of a nonarchimedean local field `K`: the automorphisms of the algebraic closure fixing the maximal tamely ramified extension `K^{t}`. Through `IntermediateField.fixingSubgroupEquiv` it is `Gal(K^{alg}/K^{t})`.","title":"The wild inertia subgroup"},{"id":"tauceti/number-theory/part-01/local-fields-and-ramification/the-prime-to-p-tate-module-is-the-product-of-its","kind":"node","lean":"TauCeti.PrimeToPTateModule.continuousMulEquivPiPadicTateTwist","parent":3306,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/local-fields-and-ramification/index.html#the-prime-to-p-tate-module-is-the-product-of-its","scope":"tauceti/number-theory/part-01/local-fields-and-ramification","status":"fully_proved","summary":"the `\u2113`-adic components `toPadicTateTwist`, over all primes `\u2113` prime to `p`, form an isomorphism of topological groups `\u2124\u0302^{(p')}(1) \u2243\u209c* \u220f_{\u2113 \u2224 p} \u2124_\u2113(1)`. It holds for every commutative monoid `E`, by the Chinese remainder theorem at each level `\u03bc_m(E)`.","title":"The prime-to-`p` Tate module is the product of its `\u2113`-adic components"},{"id":"tauceti/number-theory/part-01/local-fields-and-ramification/serres-formula-for-the-artin-conductor","kind":"node","lean":"TauCeti.LocalFieldsRamification.artinConductorExponent_mul_card_eq_sum_range, TauCeti.LocalFieldsRamification.artinConductorExponent_mul_card_lowerRamificationGroup_zero, TauCeti.hilbertSymbol_padicTwo, TauCeti.Padic.hilbertSymbol_formula_of_ne_two, TauCeti.Padic.coe_hilbertSymbol_pow_mul_intCast_pow_mul_intCast","parent":3306,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/local-fields-and-ramification/index.html#serres-formula-for-the-artin-conductor","scope":"tauceti/number-theory/part-01/local-fields-and-ramification","status":"fully_proved","summary":"Serre's formula for the Artin conductor, truncated at any index `n` at which `\u03c7` is trivial on `G_n`: for a finite abelian extension of local fields, `f(\u03c7) \u00b7 #G_0 = \u2211_{i < n, \u03c7(G_i) \u2260 1} #G_i`.","title":"Serre's formula for the Artin conductor"},{"id":"tauceti/number-theory/part-01/local-fields-and-ramification/serres-lemma-on-the-lower-index-of-a-quotient","kind":"node","lean":"TauCeti.LocalFieldsRamification.coe_herbrand_lowerIndex_sub_one, TauCeti.LocalFieldsRamification.ramificationIndex_mul_lowerIndex_restrictNormal_eq_sum_card","parent":3306,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/local-fields-and-ramification/index.html#serres-lemma-on-the-lower-index-of-a-quotient","scope":"tauceti/number-theory/part-01/local-fields-and-ramification","status":"fully_proved","summary":"If `\u03c3` has the largest lower index in its coset `\u03c3 H`, and `i_G(\u03c3) = j` is finite, then `\u03c6_{M/L}(j - 1) = i_{G/H}(\u03c3|_L) - 1`.","title":"Serre's lemma on the lower index of a quotient"},{"data_url":"scopes/tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields.json","id":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields","kind":"scope","parent":3139,"result":false,"route":"../roadmap/tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields/index.html","scope":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Local Galois groups of p-adic fields"},{"id":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields/the-generator-rank-of-a-local-absolute-galois-gr","kind":"node","lean":"TauCeti.topologicalGeneratorRankNat_absoluteGaloisGroup","parent":3320,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields/index.html#the-generator-rank-of-a-local-absolute-galois-gr","scope":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields","status":"fully_proved","summary":"For a finite extension `K` of `\u211a_p`, the absolute Galois group `G_K` needs exactly `[K : \u211a_p] + 2` topological generators. A witness of topological finite generation is `TauCeti.isTopologicallyFinitelyGenerated_fieldAbsoluteGaloisGroup`.","title":"The generator rank of a local absolute Galois group (NSW (7.4.1))"},{"id":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields/demushkins-theorem","kind":"node","lean":"TauCeti.isDemushkin_absoluteGaloisGroupProP_of_mu","parent":3320,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields/index.html#demushkins-theorem","scope":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields","status":"fully_proved","summary":"the maximal pro-`p` quotient `G_K(p)` of the absolute Galois group of a nonarchimedean local field `K` containing a primitive `p`th root of unity is a Demushkin group.","title":"Demushkin's theorem"},{"id":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields/the-marked-presentation-of-g-q2-2","kind":"node","lean":"TauCeti.absoluteGaloisGroupProP_two_ratPadic_marked","parent":3320,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields/index.html#the-marked-presentation-of-g-q2-2","scope":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields","status":"fully_proved","summary":"There is an isomorphism `e : G_{\u211a\u2082}(2) \u2243 D\u2080 = \u27e8A, S, Y \u2223 A\u00b2S\u2074(S,Y)\u27e9` along which the standard orientation of `D\u2080` pulls back to the cyclotomic orientation of `G_{\u211a\u2082}(2)`, and the cyclotomic orientation is surjective.","title":"The marked presentation of `G_{\u211a\u2082}(2)`"},{"id":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields/the-marked-presentation-of-g-q2-2-2","kind":"node","lean":"TauCeti.absoluteGaloisGroupProP_two_ratPadicSqrtNegTwo_marked","parent":3320,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields/index.html#the-marked-presentation-of-g-q2-2-2","scope":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields","status":"fully_proved","summary":"The maximal pro-`2` Galois group of `\u211a\u2082(\u221a-2)` is isomorphic to `\u27e8x\u2081, x\u2082, x\u2083, x\u2084 \u2223 x\u2081\u2076 (x\u2081, x\u2082) x\u2083\u2078 (x\u2083, x\u2084)\u27e9`, the even dyadic Demushkin group with `\u03b1 = 4` and `f = 3`, and under this isomorphism the cyclotomic orientation satisfies `\u03c7(x\u2082) \u00b7 (1 + 4) = -1` and `\u03c7(x\u2084) \u00b7 (1 - 8) = 1` and is trivial on `x\u2081` and `x\u2083`.","title":"The marked presentation of `G_{\u211a\u2082(\u221a-2)}(2)`"},{"id":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields/functoriality-of-the-marked-presentation-when-q","kind":"node","lean":"TauCeti.absoluteGaloisGroupProP_marked_of_q_ne_two_congr","parent":3320,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields/index.html#functoriality-of-the-marked-presentation-when-q","scope":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields","status":"fully_proved","summary":"Let `e : K \u2243+* K'` be an isomorphism of fields, and let `m` be a marked presentation of `G_K(p)` with the relator word and the values that `TauCeti.absoluteGaloisGroupProP_marked_of_q_ne_two` records for `K`. Then `(absoluteGaloisGroupProPCongr e).trans m` presents `G_{K'}(p)` by the same relator word and takes the same values of the cyclotomic orientation on the generators.","title":"Functoriality of the marked presentation when `q(K) \u2260 2`"},{"id":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields/functoriality-of-the-marked-presentation-in-odd","kind":"node","lean":"TauCeti.absoluteGaloisGroupProP_two_marked_of_degree_odd_congr","parent":3320,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields/index.html#functoriality-of-the-marked-presentation-in-odd","scope":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields","status":"fully_proved","summary":"Let `e : K \u2243+* K'` be an isomorphism of fields, and let `m` be a marked presentation of `G_K(2)` with the relator word and the values that `TauCeti.absoluteGaloisGroupProP_two_marked_of_degree_odd` records for `K`. Then `(absoluteGaloisGroupProPCongr e).trans m` presents `G_{K'}(2)` by the same relator word `demushkinWordTwoOdd` and takes the same values on the generators `0`, `2` and the remaining ones.","title":"Functoriality of the marked presentation in odd degree"},{"id":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields/functoriality-of-the-marked-presentation-in-even","kind":"node","lean":"TauCeti.absoluteGaloisGroupProP_two_marked_of_degree_even_congr","parent":3320,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields/index.html#functoriality-of-the-marked-presentation-in-even","scope":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields","status":"fully_proved","summary":"The two even dyadic marked theorems `TauCeti.absoluteGaloisGroupProP_two_marked_of_degree_even_plusMinus` and `TauCeti.absoluteGaloisGroupProP_two_marked_of_degree_even_principal` present `G_K(2)` by the same word `demushkinWordTwoEven a f (N + 2)` on `N + 2` generators and record the same values on the second and fourth marked generators, the value `-(1 + a)\u207b\u00b9` and the value `(1 - 2^f)\u207b\u00b9`; they differ only in the\u2026","title":"Functoriality of the marked presentation in even degree"},{"id":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields/the-image-of-the-cyclotomic-character-is-an-inva","kind":"node","lean":"TauCeti.range_localCyclotomicCharacter_eq_of_ringEquiv","parent":3320,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields/index.html#the-image-of-the-cyclotomic-character-is-an-inva","scope":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields","status":"fully_proved","summary":"An isomorphism of fields induces an isomorphism of absolute Galois groups, and the cyclotomic character of the target is the character of the source read through it (`TauCeti.localCyclotomicCharacter_map`); hence the two characters have the same image in `\u2124_p\u02e3`.","title":"The image of the cyclotomic character is an invariant of the field"},{"id":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields/the-cyclotomic-normalization-of-the-local-artin","kind":"node","lean":"TauCeti.ClassFieldTheory.localCyclotomicCharacter_artinMap_unit","parent":3320,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields/index.html#the-cyclotomic-normalization-of-the-local-artin","scope":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields","status":"fully_proved","summary":"Let `F/\u211a_p` be finite. If `\u03c3 \u2208 G_F` represents the absolute Artin symbol of a unit `u` of the valuation ring of `F` (`v(u) = 1`), then the `p`-adic cyclotomic character of `\u03c3` is the inverse of the norm of `u`:","title":"The cyclotomic normalization of the local Artin map of a finite extension of `\u211a_p`"},{"id":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields/local-reciprocity-for-a-l","kind":"node","lean":"TauCeti.padicCompletionUnitsEquivAbelianizationProP","parent":3320,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields/index.html#local-reciprocity-for-a-l","scope":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields","status":"fully_proved","summary":"If `V \u2264 G_K` is the subgroup fixing the image of a finite extension `L/K` under `\u03b9 : L \u2192\u2090[K] K\u02e2`, then the multiplicative `p`-adic completion `A(L)` of `L\u02e3` is canonically isomorphic to `V^ab(p)`, the maximal pro-`p` quotient of the topological abelianization of `V`.","title":"Local reciprocity for `A(L)`"},{"id":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields/the-cokernel-of-the-completed-norm","kind":"node","lean":"TauCeti.padicCompletionUnitsNormCokernelEquiv","parent":3320,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields/index.html#the-cokernel-of-the-completed-norm","scope":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields","status":"fully_proved","summary":"The cokernel of the completed norm `A(L) \u2192 A(K)` is `Gal(L/K)^ab / (Gal(L/K)^ab)^(p^m)`, the maximal `p`-quotient of `Gal(L/K)^ab`, once `p ^ m` contains the full power of `p` dividing `#Gal(L/K)^ab`. The isomorphism is induced by the completed norm residue map (`TauCeti.padicCompletionUnitsNormCokernelEquiv_mk`).","title":"The cokernel of the completed norm"},{"id":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields/existence-of-the-tate-module-of-a-finite-layer-n","kind":"node","lean":"TauCeti.nonempty_layerTateModule","parent":3320,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields/index.html#existence-of-the-tate-module-of-a-finite-layer-n","scope":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields","status":"fully_proved","summary":"For a finite Galois extension `L/K` of `p`-adic fields there is a finitely generated `\u2124_p[Gal(L/K)]`-module of projective dimension at most one which is an extension of the augmentation ideal by `A(L)`.","title":"Existence of the Tate module of a finite layer (NSW (5.6.5) and the proof of (7.4.1))"},{"id":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields/swans-theorem-for-z-p-g-nsw-5-6-10-ii","kind":"node","lean":"TauCeti.nonempty_linearEquiv_of_projective_of_tensorRat","parent":3320,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields/index.html#swans-theorem-for-z-p-g-nsw-5-6-10-ii","scope":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields","status":"fully_proved","summary":"Two finitely generated projective `\u2124_p[G]`-modules with isomorphic rationalizations are isomorphic.","title":"Swan's theorem for `\u2124_p[G]` (NSW (5.6.10)(ii))"},{"id":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields/shafarevichs-theorem","kind":"node","lean":"TauCeti.nonempty_continuousMulEquiv_freeProP_of_not_mu","parent":3320,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields/index.html#shafarevichs-theorem","scope":"tauceti/number-theory/part-01/local-galois-groups-of-p-adic-fields","status":"fully_proved","summary":"If a finite compatible extension `K` of `\u211a_[p]` contains no primitive `p`th root of unity, its maximal pro-`p` Galois group `G_K(p)` is the free pro-`p` group on `[K : \u211a_[p]] + 1` generators.","title":"Shafarevich's theorem"},{"data_url":"scopes/tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi.json","id":"tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi","kind":"scope","parent":3139,"result":false,"route":"../roadmap/tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi/index.html","scope":"tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Modular forms \u2014 Hecke theory, newforms, and L-functions"},{"id":"tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi/the-valence-formula-at-general-level","kind":"node","lean":"TauCeti.ModularForm.valence_formula_finiteIndex","parent":3335,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi/index.html#the-valence-formula-at-general-level","scope":"tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi","status":"fully_proved","summary":"for a nonzero weight-`k` form on a finite-index `\u0393 \u2264 SL(2, \u2124)`, the interior orders \u2014 weighted by `2 / |Stab_\u0393 P|` \u2014 together with the orders indexed by the cusp translation orbits add up to `k \u00b7 [SL(2, \u2124) : \u0393] / 12`.","title":"The valence formula at general level"},{"id":"tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi/a-newform-is-an-eigenvector-of-u-p-at-every-prim","kind":"node","lean":"HeckeRing.GL2.Newform.heckeUCuspNat_eq_qExpansion_coeff_smul","parent":3335,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi/index.html#a-newform-is-an-eigenvector-of-u-p-at-every-prim","scope":"tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi","status":"fully_proved","summary":"A newform is an eigenvector of `U_p` at every prime `p` dividing the level, with eigenvalue `a_p(f)` (Atkin\u2013Lehner; Li; Diamond\u2013Shurman, Theorem 5.8.2; Miyake, Theorem 4.6.13). Together with the good eigensystem this makes a newform a full Hecke eigenform (`Newform.toEigenform`). The statement is on the bad-prime operator `heckeUCuspNat`, matching the `U_p`-spelled API of `HeckeSlash/BadPrime`;\u2026","title":"A newform is an eigenvector of `U_p` at every prime `p` dividing the level, with eigenvalue `a_p(f)`"},{"id":"tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi/heckering-gl2-newform-l-eulerproduct","kind":"node","lean":"HeckeRing.GL2.Newform.L_eulerProduct","parent":3335,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi/index.html#heckering-gl2-newform-l-eulerproduct","scope":"tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi","status":"fully_proved","summary":"Finite Euler products converge to Mathlib's `ModularForm.L` for a newform.","title":"HeckeRing.GL2.Newform.L_eulerProduct"},{"id":"tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi/the-coefficient-field-of-a-newform-of-weight-at","kind":"node","lean":"TauCeti.numberField_CoefficientField","parent":3335,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi/index.html#the-coefficient-field-of-a-newform-of-weight-at","scope":"tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.numberField_CoefficientField`.","title":"The coefficient field of a newform of weight at least two is a number field"},{"id":"tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi/the-cusp-eisenstein-decomposition-of-m-k-n","kind":"node","lean":"TauCeti.isCompl_eisensteinSubspace_range_cuspToModFormCharSpace","parent":3335,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi/index.html#the-cusp-eisenstein-decomposition-of-m-k-n","scope":"tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi","status":"fully_proved","summary":"in weight `k \u2265 3`, the span of the Eisenstein series of nebentypus `\u03c7` is a complement of the image of `S_k(N, \u03c7)` in `M_k(N, \u03c7)`.","title":"The cusp\u2013Eisenstein decomposition of `M_k(N, \u03c7)`"},{"id":"tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi/tauceti-modularsymbols","kind":"node","lean":"TauCeti.ModularSymbols","parent":3335,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi/index.html#tauceti-modularsymbols","scope":"tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi","status":"fully_proved","summary":"The module of modular symbols `\ud835\udd44_w(\u0393; R) = (Div\u2070(\u2119\u00b9(\u211a)) \u2297_R Sym^w(R\u00b2))_\u0393` of weight `w + 2` for a subgroup `\u0393 \u2264 SL(2, \u2124)`, with coefficients in `R`: the `\u0393`-coinvariants of the degree-zero divisors on the cusps tensored with the binary forms of degree `w`.","title":"TauCeti.ModularSymbols"},{"id":"tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi/tauceti-modularsymbols-periodmap","kind":"node","lean":"TauCeti.ModularSymbols.periodMap","parent":3335,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi/index.html#tauceti-modularsymbols-periodmap","scope":"tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi","status":"fully_proved","summary":"The period map `S_k(\u0393) \u2192\u2097[\u2102] (\ud835\udd44_w(\u0393; R) \u2192\u2097[R] \u2102)` for `k = w + 2`: a cusp form `f` on `\u0393` is sent to the `R`-linear functional on the modular symbols induced by its raw period pairing, so that `{\u03b1, \u03b2} \u2297 P \u21a6 \u222b_\u03b2^\u03b1 f(z) P(z, 1) dz`. For `R = \u2124` this is the pairing of cusp forms with the integral modular symbols.","title":"TauCeti.ModularSymbols.periodMap"},{"id":"tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi/periods-determine-a-cusp-form","kind":"node","lean":"TauCeti.ModularSymbols.periodMap_injective","parent":3335,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi/index.html#periods-determine-a-cusp-form","scope":"tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi","status":"fully_proved","summary":"For a finite-index subgroup of `SL(2, \u2124)` and weight `k = w + 2 \u2265 2`, the period map into the dual of the modular symbols over `R` is injective. In particular this applies to the integral modular symbols (`R = \u2124`).","title":"Periods determine a cusp form"},{"id":"tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi/the-atkin-li-pseudo-eigenvalue","kind":"node","lean":"HeckeRing.GL2.Newform.atkinLehnerPseudoEigenvalue","parent":3335,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi/index.html#the-atkin-li-pseudo-eigenvalue","scope":"tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi","status":"fully_proved","summary":"The Atkin\u2013Li pseudo-eigenvalue `\u03bb_Q(f)` of a newform `f` at an exact divisor `Q` of the level: the scalar with `\ud835\udcb2_Q f = \u03bb_Q(f) \u2022 g`, where `\ud835\udcb2_Q` is the slash by the Atkin\u2013Li matrix `atkinLiMatrix N Q` normalized by `(\u221aQ) ^ (2 - k)` and `g = f.atkinLehnerTransform h` is the transported newform. Since `g` has first Fourier coefficient `1`, it is the first Fourier coefficient of `\ud835\udcb2_Q f`.","title":"The Atkin\u2013Li pseudo-eigenvalue"},{"id":"tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi/the-cusps-of-0-n","kind":"node","lean":"TauCeti.gamma0CuspOrbitEquiv","parent":3335,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi/index.html#the-cusps-of-0-n","scope":"tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi","status":"fully_proved","summary":"The cusps of `\u0393\u2080(N)` are in bijection with the pairs `(g, x)` of a divisor `g` of `N` and a unit `x` modulo `gcd(g, N / g)`. The cusp `\u03b3 \u2022 \u221e` corresponds to the label of `\u03b3` (`gamma0CuspOrbitEquiv_cuspOrbitOfSL2Z`).","title":"The cusps of `\u0393\u2080(N)`"},{"id":"tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi/the-main-lemma-per-character-miyake-lemma-4-6-8","kind":"node","lean":"TauCeti.mem_cuspFormsOld_of_forall_coprime_qExpansion_coeff_eq_zero","parent":3335,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi/index.html#the-main-lemma-per-character-miyake-lemma-4-6-8","scope":"tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi","status":"fully_proved","summary":"a cusp form in `S_k(\u0393\u2081(N), \u03c7)` whose Fourier coefficients vanish at every index coprime to `N` lies in the old subspace. It is a sum, over the primes `p \u2223 N`, of forms of the same nebentypus supported on the multiples of `p`, and each of those is old.","title":"The Main Lemma, per character (Miyake, Lemma 4.6.8; Diamond\u2013Shurman, Theorem 5.7.1)"},{"id":"tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi/manins-lemma","kind":"node","lean":"TauCeti.ModularSymbols.degreeZero_eq_span_unimodular, TauCeti.ModularSymbols.span_degreeZeroRep_eq_top","parent":3335,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi/index.html#manins-lemma","scope":"tauceti/number-theory/part-01/modular-forms-hecke-theory-newforms-and-l-functi","status":"fully_proved","summary":"The degree-zero divisors on the cusps are spanned by the unimodular symbols `[g\u221e] - [g0]`, `g \u2208 SL(2, \u2124)`, that is, by the `SL(2, \u2124)`-translates of `[\u221e] - [0]`.","title":"Manin's lemma"},{"data_url":"scopes/tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory.json","id":"tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory","kind":"scope","parent":3139,"result":false,"route":"../roadmap/tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory/index.html","scope":"tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Multiquadratic fields and genus theory"},{"id":"tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory/degree-of-a-multiquadratic-field-over-a-finite-i","kind":"node","lean":"TauCeti.Multiquadratic.finrank_adjoin_range","parent":3348,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory/index.html#degree-of-a-multiquadratic-field-over-a-finite-i","scope":"tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory","status":"fully_proved","summary":"If no nonempty subset product of the radicands `d i` is a square in `K`, then `[K(root\u1d62 : i) : K] = 2^|\u03b9|`.","title":"Degree of a multiquadratic field over a finite index"},{"id":"tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory/residue-degree-one-exactly-at-the-residues","kind":"node","lean":"TauCeti.Multiquadratic.inertiaDeg_eq_one_iff_forall_legendreSym_eq_one","parent":3348,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory/index.html#residue-degree-one-exactly-at-the-residues","scope":"tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory","status":"fully_proved","summary":"Let `K` be generated over `\u211a` by square roots `r i` of integers `d i`, and let `Q` be a prime of `\ud835\udcde K` above an odd prime `p` dividing none of the `d i`. Then `Q` has residue degree `1` iff every `d i` is a quadratic residue mod `p`.","title":"Residue degree one exactly at the residues"},{"id":"tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory/the-number-of-primes-above-2","kind":"node","lean":"TauCeti.Multiquadratic.ncard_primesOver_two_eq_two_pow_sub","parent":3348,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory/index.html#the-number-of-primes-above-2","scope":"tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory","status":"fully_proved","summary":"Let `K` be generated over `\u211a` by square roots of `n` square-class independent integers `d i` (no nonempty subset product is a square) not divisible by `4`. Then there are exactly `2 ^ (n - a - b)` primes of `\ud835\udcde K` above `2`, where the ramification index is `e = 2 ^ a` and the residue degree is `f = 2 ^ b`:","title":"The number of primes above `2`"},{"id":"tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory/the-prime-discriminant-compositum-is-the-genus-f","kind":"node","lean":"TauCeti.Multiquadratic.isGenusField_candidateGenusField","parent":3348,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory/index.html#the-prime-discriminant-compositum-is-the-genus-f","scope":"tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory","status":"fully_proved","summary":"For a squarefree negative integer `d`, `candidateGenusField hd` is abelian over `\u211a`, is unramified at every place over its embedded copy of `\u211a(\u221ad)`, and admits a root-preserving embedding from every other abelian extension with those properties.","title":"The prime-discriminant compositum is the genus field in the imaginary quadratic case"},{"id":"tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory/the-genus-field-isomorphism-gal-k-gen-k-cl-k-cl","kind":"node","lean":"TauCeti.Multiquadratic.autCandidateGenusFieldEquivNarrowElementaryTwoQuotient","parent":3348,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory/index.html#the-genus-field-isomorphism-gal-k-gen-k-cl-k-cl","scope":"tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory","status":"fully_proved","summary":"For a squarefree integer `d` that is not a rational square, the Galois group of the candidate genus field over its embedded quadratic base `K = \u211a(\u221ad)` is isomorphic to the maximal elementary-`2` quotient of the narrow class group of `K`. Both are the space of even-parity sign patterns on the prime discriminants dividing `disc K`: the Galois group by its sign patterns, the class group by its genus characters.","title":"The genus-field isomorphism `Gal(K_gen/K) \u2245 Cl\u207a(K)/Cl\u207a(K)\u00b2`"},{"id":"tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory/the-2-rank-formula-for-the-narrow-class-group-of","kind":"node","lean":"TauCeti.Multiquadratic.narrowTwoRank_eq_ncard_ramifiedPrimes_sub_one","parent":3348,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory/index.html#the-2-rank-formula-for-the-narrow-class-group-of","scope":"tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory","status":"fully_proved","summary":"For `K = \u211a(\u221ad)` with `d` squarefree, of either signature, the `2`-rank of `Cl\u207a(K)` is exactly `t - 1`, where `t` is the number of rational primes ramifying in `K`. For an imaginary `K` the narrow and ordinary class groups coincide, and this recovers `twoRank_eq_ncard_ramifiedPrimes_sub_one`; that identity is also how the Gaussian field `d = -1`, which the ambiguous class number bound excludes, is covered.","title":"The `2`-rank formula for the narrow class group of a quadratic field"},{"id":"tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory/the-genus-character-on-the-narrow-class-group","kind":"node","lean":"TauCeti.Multiquadratic.genusCharFunNarrowClassGroupHom","parent":3348,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory/index.html#the-genus-character-on-the-narrow-class-group","scope":"tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory","status":"fully_proved","summary":"For a factor `t` of a prime-discriminant factorization, this is the homomorphism `Cl\u207a(K) \u2192 \u2124\u02e3` obtained by evaluating a coprime integral representative. Its definition is independent of the representative by `genusCharFunCoprimeIdealHom_eq_of_mk0_eq`.","title":"The genus character on the narrow class group"},{"id":"tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory/numberfield-narrowclassgroup","kind":"node","lean":"NumberField.NarrowClassGroup","parent":3348,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory/index.html#numberfield-narrowclassgroup","scope":"tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory","status":"fully_proved","summary":"The narrow class group `Cl\u207a(K)`: invertible fractional ideals of `\ud835\udcde K` modulo the principal ones with a totally positive generator.","title":"NumberField.NarrowClassGroup"},{"id":"tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory/unique-prime-discriminant-factorization-of-a-fun","kind":"node","lean":"Int.IsFundamentalDiscr.existsUnique_finset_primeDiscriminant","parent":3348,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory/index.html#unique-prime-discriminant-factorization-of-a-fun","scope":"tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory","status":"fully_proved","summary":"Every fundamental discriminant is the product of a unique finite set of prime discriminants.","title":"Unique prime-discriminant factorization of a fundamental discriminant"},{"id":"tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory/the-candidate-genus-field-is-cut-out-of-the-cycl","kind":"node","lean":"TauCeti.Multiquadratic.candidateGenusField_eq_map_characterSubfield_genusCharGroup","parent":3348,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory/index.html#the-candidate-genus-field-is-cut-out-of-the-cycl","scope":"tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory","status":"fully_proved","summary":"For squarefree `d` with fundamental discriminant `D` and a cyclotomic subfield `K \u2286 \u2102` of level `|D|`, the candidate genus field of `\u211a(\u221ad)` is the image in `\u2102` of the intermediate field of `K` that the character correspondence attaches to the genus character group of `D`. The correspondence is taken for the abelian Galois structure of the cyclotomic extension.","title":"The candidate genus field is cut out of the cyclotomic field by the genus characters"},{"id":"tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory/tauceti-multiquadratic-genuschargroup","kind":"node","lean":"TauCeti.Multiquadratic.genusCharGroup","parent":3348,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory/index.html#tauceti-multiquadratic-genuschargroup","scope":"tauceti/number-theory/part-01/multiquadratic-fields-and-genus-theory","status":"fully_proved","summary":"The subgroup of common-level Dirichlet characters obtained by taking products over subsets of the prime-discriminant factors.","title":"TauCeti.Multiquadratic.genusCharGroup"},{"data_url":"scopes/tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf.json","id":"tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf","kind":"scope","parent":3139,"result":false,"route":"../roadmap/tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf/index.html","scope":"tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Number fields, ramification, Frobenius, and the LMFDB invariants"},{"id":"tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf/dedekinds-theorem-with-the-fixed-points-counted","kind":"node","lean":"TauCeti.NumberField.factorizationType_eq_cycleType_isArithFrobAt, TauCeti.NumberField.fullCycleType_galActionHom_restrict_eq_factorDegrees, TauCeti.NumberField.fullCycleType_galActionHom_restrict_minpoly_eq_map_natDegree_monicFactorsMod","parent":3360,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf/index.html#dedekinds-theorem-with-the-fixed-points-counted","scope":"tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf","status":"fully_proved","summary":"The multiset of degrees of the monic irreducible factors of `minpoly \u2124 \u03b8` modulo `p` is the cycle type of a Frobenius at a prime above `p` acting on the roots of `minpoly \u211a \u03b8`, together with one part `1` for each fixed root. This is the form with the cycle type and the fixed points separated; the full cycle type of `fullCycleType_galActionHom_restrict_minpoly_eq_map_natDegree_monicFactorsMod` packages the two\u2026","title":"Dedekind's theorem, with the fixed points counted separately"},{"id":"tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf/the-semi-local-decomposition-k-v-k-l-a-k-v-w-v-l","kind":"node","lean":"TauCeti.semilocalEquiv","parent":3360,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf/index.html#the-semi-local-decomposition-k-v-k-l-a-k-v-w-v-l","scope":"tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf","status":"fully_proved","summary":"completing `L` at the finitely many places above a finite place `v` of `K` decomposes the scalar extension of `L` to `K_v` into the product of those completions.","title":"The semi-local decomposition `K_v \u2297[K] L \u2243\u2090[K_v] \u220f_{w \u2223 v} L_w`"},{"id":"tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf/hilberts-formula-for-a-number-field-different-ex","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.multiplicity_differentIdeal_eq_finsum_card_ramificationGroup_sub_one","parent":3360,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf/index.html#hilberts-formula-for-a-number-field-different-ex","scope":"tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf","status":"fully_proved","summary":"For a finite Galois extension `L/K`, the multiplicity of a prime `w` in the global different is the sum of the orders of the nonnegative-index lower ramification groups, minus one in each degree.","title":"Hilbert's formula for a number-field different exponent"},{"id":"tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf/the-relative-discriminant-coefficient-of-a-fixed","kind":"node","lean":"TauCeti.NumberField.multiplicity_relDiscr_fixedField_eq_sum","parent":3360,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf/index.html#the-relative-discriminant-coefficient-of-a-fixed","scope":"tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf","status":"fully_proved","summary":"Sum the group-theoretic contributions over `H \\ Gal(L/K) / D(Q)`, using any choice of representatives. The formula holds at ramified primes as well as unramified ones, including wild ramification.","title":"The relative discriminant coefficient of a fixed field from double cosets"},{"id":"tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf/soundness-of-unit-candidate-elimination","kind":"node","lean":"TauCeti.NumberField.Units.UnitCandidateEliminationCertificate.sound","parent":3360,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf/index.html#soundness-of-unit-candidate-elimination","scope":"tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf","status":"fully_proved","summary":"A unit expanding at a real place generates modulo torsion if every candidate below its absolute value has been eliminated.","title":"Soundness of unit-candidate elimination"},{"id":"tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf/the-double-coset-law-over-an-integrally-closed-b","kind":"node","lean":"Ideal.doubleCosetQuotientEquivPrimesOverOfIsIntegralClosure","parent":3360,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf/index.html#the-double-coset-law-over-an-integrally-closed-b","scope":"tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf","status":"fully_proved","summary":"For `M / K` finite Galois with group `G`, a subgroup `H` of `G`, the integral closures `B` and `C` of `A` in `M` and in `M ^ H`, and a prime `Q` of `B` above `p` with decomposition group `D`, the primes of `C` above `p` are indexed by the double cosets `H \\ G / D`.","title":"The double coset law over an integrally closed base"},{"id":"tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf/tauceti-reldiscr","kind":"node","lean":"TauCeti.relDiscr","parent":3360,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf/index.html#tauceti-reldiscr","scope":"tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf","status":"fully_proved","summary":"The relative discriminant ideal of the finite torsion-free extension `A \u2192 B`.","title":"TauCeti.relDiscr"},{"id":"tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf/the-ideal-theoretic-artin-map","kind":"node","lean":"TauCeti.NumberFieldArithmetic.artinHomAway","parent":3360,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf/index.html#the-ideal-theoretic-artin-map","scope":"tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf","status":"fully_proved","summary":"The multiplicative extension of `artinElementAway` along the unique factorization of an invertible fractional ideal, on the group of fractional ideals with multiplicity zero at every prime of `S`.","title":"The ideal-theoretic Artin map"},{"id":"tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf/the-polynomial-form-of-dedekinds-theorem","kind":"node","lean":"TauCeti.NumberField.exists_gal_fullCycleType_eq_factorizationType","parent":3360,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf/index.html#the-polynomial-form-of-dedekinds-theorem","scope":"tauceti/number-theory/part-01/number-fields-ramification-frobenius-and-the-lmf","status":"fully_proved","summary":"Let `f` be a monic integer polynomial and `p` a prime not dividing the discriminant of `f`. Then some element of the Galois group of `f` over `\u211a` acts on the complex roots of `f` with full cycle type (cycle lengths counted together with fixed points) equal to the multiset of degrees of the irreducible factors of `f` modulo `p`.","title":"The polynomial form of Dedekind's theorem"},{"data_url":"scopes/tauceti/number-theory/part-01/p-adic-functional-analysis.json","id":"tauceti/number-theory/part-01/p-adic-functional-analysis","kind":"scope","parent":3139,"result":false,"route":"../roadmap/tauceti/number-theory/part-01/p-adic-functional-analysis/index.html","scope":"tauceti/number-theory/part-01/p-adic-functional-analysis","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"P-adic functional analysis"},{"id":"tauceti/number-theory/part-01/p-adic-functional-analysis/serres-theorem-tauceti-residuemodule-isorthonorm","kind":"node","lean":"TauCeti.ResidueModule.isOrthonormalBasis_iff","parent":3370,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/p-adic-functional-analysis/index.html#serres-theorem-tauceti-residuemodule-isorthonorm","scope":"tauceti/number-theory/part-01/p-adic-functional-analysis","status":"fully_proved","summary":"For norms with values in `\u2016\u03d6\u2016 ^ \u2124 \u222a {0}`, a family of the closed unit ball is an orthonormal basis exactly when its residues are a basis of the residue module: they are linearly independent over the residue ring and span.","title":"Serre's theorem (`TauCeti.ResidueModule.isOrthonormalBasis_iff`)"},{"id":"tauceti/number-theory/part-01/p-adic-functional-analysis/tauceti-nonempty-continuouslinearequiv-zeroatinf","kind":"node","lean":"TauCeti.nonempty_continuousLinearEquiv_zeroAtInfty_nat_of_dense_span","parent":3370,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/p-adic-functional-analysis/index.html#tauceti-nonempty-continuouslinearequiv-zeroatinf","scope":"tauceti/number-theory/part-01/p-adic-functional-analysis","status":"fully_proved","summary":"An infinite-dimensional ultrametric Banach space of countable type (some sequence has dense span) over a complete nontrivially normed field is continuously linearly equivalent to `C\u2080(\u2115, K)`.","title":"TauCeti.nonempty_continuousLinearEquiv_zeroAtInfty_nat_of_dense_span"},{"id":"tauceti/number-theory/part-01/p-adic-functional-analysis/mahler-duality","kind":"node","lean":"AbstractMeasure.mahlerMomentsEquiv","parent":3370,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/p-adic-functional-analysis/index.html#mahler-duality","scope":"tauceti/number-theory/part-01/p-adic-functional-analysis","status":"fully_proved","summary":"Over a ring with a multiplicative pseudo-uniformiser, a measure on `\u2124_p` is determined by its moments `\u03bc (x \u21a6 x choose n)`, the coefficients of its Amice transform, and every bounded sequence is the moment sequence of a measure. The inverse sends a bounded sequence `b` to the measure `f \u21a6 \u2211 n, \u0394^n f (0) * b n`.","title":"Mahler duality"},{"id":"tauceti/number-theory/part-01/p-adic-functional-analysis/mahler-duality-for-field-coefficients","kind":"node","lean":"AbstractMeasure.norm_toCLMEquiv_eq_iSup_coeff","parent":3370,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/p-adic-functional-analysis/index.html#mahler-duality-for-field-coefficients","scope":"tauceti/number-theory/part-01/p-adic-functional-analysis","status":"fully_proved","summary":"Over a complete ultrametric nontrivially normed field with a contractive `\u2124_[p]`-algebra structure, such as a complete extension of `\u211a_p`, the operator norm of a measure on `\u2124_p` is the supremum of the norms of the coefficients of its Amice transform.","title":"Mahler duality for field coefficients"},{"id":"tauceti/number-theory/part-01/p-adic-functional-analysis/the-integral-dual-unit-ball","kind":"node","lean":"AbstractMeasure.existsUnique_integral_iff_norm_le_one","parent":3370,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/p-adic-functional-analysis/index.html#the-integral-dual-unit-ball","scope":"tauceti/number-theory/part-01/p-adic-functional-analysis","status":"fully_proved","summary":"A `\u211a_p`-valued measure on `\u2124_p` has operator norm at most `1` exactly when there is a unique `\u2124_p`-valued measure `\u03bc` with `\u03bd (f) = \u03bc (f)` for every `\u2124_p`-valued test function `f`.","title":"The integral dual unit ball"},{"id":"tauceti/number-theory/part-01/p-adic-functional-analysis/s-h-d-h","kind":"node","lean":"TauCeti.gaussNorm_amiceRadius_le_amiceDualNorm","parent":3370,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/p-adic-functional-analysis/index.html#s-h-d-h","scope":"tauceti/number-theory/part-01/p-adic-functional-analysis","status":"fully_proved","summary":"The Gauss norm at the radius `r_h` is at most the dual stage norm at level `h`, provided the latter is the supremum of a bounded family.","title":"`S_h \u2264 D_h`"},{"id":"tauceti/number-theory/part-01/p-adic-functional-analysis/d-h-p-s-h-1","kind":"node","lean":"TauCeti.amiceDualNorm_le_mul_gaussNorm_amiceRadius_succ","parent":3370,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/p-adic-functional-analysis/index.html#d-h-p-s-h-1","scope":"tauceti/number-theory/part-01/p-adic-functional-analysis","status":"fully_proved","summary":"The dual stage norm at level `h` is at most `p` times the Gauss norm at the radius `r_{h+1}`, provided the latter is the supremum of a bounded family.","title":"`D_h \u2264 p S_{h+1}`"},{"id":"tauceti/number-theory/part-01/p-adic-functional-analysis/tauceti-huber-exists-tateseminorm","kind":"node","lean":"TauCeti.Huber.exists_tateSeminorm","parent":3370,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/p-adic-functional-analysis/index.html#tauceti-huber-exists-tateseminorm","scope":"tauceti/number-theory/part-01/p-adic-functional-analysis","status":"fully_proved","summary":"A ring with separately continuous multiplication in which the sets `\u03d6 ^ n A\u2080`, `n \u2265 0`, form a neighbourhood basis of zero carries an ultrametric ring seminorm, equal to the Tate gauge, whose open balls about zero form a neighbourhood basis of zero.","title":"TauCeti.Huber.exists_tateSeminorm"},{"id":"tauceti/number-theory/part-01/p-adic-functional-analysis/the-universal-property-of-c0-i-r","kind":"node","lean":"ZeroAtInftyContinuousMap.linearCombinationEquiv","parent":3370,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/p-adic-functional-analysis/index.html#the-universal-property-of-c0-i-r","scope":"tauceti/number-theory/part-01/p-adic-functional-analysis","status":"fully_proved","summary":"Over a commutative ring admitting a multiplicative pseudo-uniformiser, bounded families `I \u2192 M` in a complete ultrametric module correspond to continuous linear maps `C\u2080(I, R) \u2192L[R] M`, isometrically for the sup norm and the operator norm. A family `e` is sent to its linear combination map `a \u21a6 \u2211' i, a i \u2022 e i`, and a map `u` to its values `u (single i 1)` at the unit coordinate vectors.","title":"The universal property of `C\u2080(I, R)`"},{"id":"tauceti/number-theory/part-01/p-adic-functional-analysis/tauceti-analyticstage-transition","kind":"node","lean":"TauCeti.AnalyticStage.transition","parent":3370,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/p-adic-functional-analysis/index.html#tauceti-analyticstage-transition","scope":"tauceti/number-theory/part-01/p-adic-functional-analysis","status":"fully_proved","summary":"The transition `LA_h(E) \u2192 LA_{h+1}(E)`. The disc of `a \u2208 ZMod (p\u02b0)` is the union of the discs of the classes `a' = a + p\u02b0 b`, `0 \u2264 b < p`, in `ZMod (p ^ (h + 1))`, where `b = \u230aa'.val / p\u02b0\u230b`. On the disc of `a'` the coordinates are related by `w = b + p v`, and the transition re-expands `\u2211\u2096 c(a, k) w\u1d4f` in powers of `v`:","title":"TauCeti.AnalyticStage.transition"},{"id":"tauceti/number-theory/part-01/p-adic-functional-analysis/amices-basis-theorem","kind":"node","lean":"TauCeti.AnalyticStage.isOrthonormalBasis_amiceTerm","parent":3370,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/p-adic-functional-analysis/index.html#amices-basis-theorem","scope":"tauceti/number-theory/part-01/p-adic-functional-analysis","status":"fully_proved","summary":"The disc expansions `amiceTerm p h n 1` of Amice's polynomials `e_{h,n}(x) = \u230an / p\u02b0\u230b! (x choose n)` form an orthonormal basis of `LA_h(\u2124_p)`.","title":"Amice's basis theorem"},{"id":"tauceti/number-theory/part-01/p-adic-functional-analysis/amices-theorem","kind":"node","lean":"TauCeti.AnalyticStage.amiceEquiv, TauCeti.AnalyticStage.norm_amiceMap, TauCeti.AnalyticStage.amiceMap_surjective","parent":3370,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/p-adic-functional-analysis/index.html#amices-theorem","scope":"tauceti/number-theory/part-01/p-adic-functional-analysis","status":"fully_proved","summary":"Amice's map `(c\u2099) \u21a6 \u2211\u2099 e_{h,n} \u2022 c\u2099`, where `e_{h,n}(x) = \u230an / p\u02b0\u230b! (x choose n)`, is an isometric isomorphism from the null sequences in `E` onto the analytic stage `LA_h(E)`.","title":"Amice's theorem"},{"data_url":"scopes/tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants.json","id":"tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants","kind":"scope","parent":3139,"result":false,"route":"../roadmap/tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants/index.html","scope":"tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Quadratic forms and cohomological invariants"},{"id":"tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants/the-splitting-criterion-for-quaternion-algebras","kind":"node","lean":"TauCeti.QuaternionAlgebra.nonempty_algEquiv_matrix_tfae","parent":3383,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants/index.html#the-splitting-criterion-for-quaternion-algebras","scope":"tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants","status":"fully_proved","summary":"The splitting criterion for quaternion algebras (Lam III.2.7 and III.4.2, Serre III.1, Gille-Szamuely 1.1.9). For units `a` and `b` of a field in which `2` is invertible, the following are equivalent: 1. `\u210d[K,a,b]` is isomorphic to `M\u2082(K)`; 2. `b` is a norm from `K[\u221aa] = QuadraticAlgebra K a 0`; 3. `b = x\u00b2 - a y\u00b2` for some `x, y \u2208 K`; 4. the ternary form `\u27e81, -a, -b\u27e9` is isotropic; 5. the norm form of `\u210d[K,a,b]`,\u2026","title":"The splitting criterion for quaternion algebras"},{"id":"tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants/the-classification-of-regular-forms-over-a-local","kind":"node","lean":"TauCeti.RegularFormClass.localInvariantsEquiv","parent":3383,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants/index.html#the-classification-of-regular-forms-over-a-local","scope":"tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants","status":"fully_proved","summary":"Taking rank, discriminant and local Hasse invariant is a bijection from regular-form classes over a nonarchimedean local field onto admissible triples. Its inverse is the realization map.","title":"The classification of regular forms over a local field, as an equivalence"},{"id":"tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants/hilberts-product-formula","kind":"node","lean":"TauCeti.hilbertSymbol_productFormula, TauCeti.finprod_hilbertSymbol_padic_mul_hilbertSymbol_real_eq_one","parent":3383,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants/index.html#hilberts-product-formula","scope":"tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants","status":"fully_proved","summary":"For `a, b \u2208 K\u02e3` in a number field `K`, the product of the norm-equation Hilbert symbols `(a_v, b_v)_v` over the finite places and of the real Hilbert symbols `(a_w, b_w)_w` over the real places of `K` is `1`.","title":"Hilbert's product formula"},{"id":"tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants/the-comparison-isomorphism","kind":"node","lean":"TauCeti.brauerCohomologyEquiv","parent":3383,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants/index.html#the-comparison-isomorphism","scope":"tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants","status":"fully_proved","summary":"The comparison isomorphism `Br(K) \u2243 H\u00b2_cont(G_K, (K\u02e2)\u02e3)` with Mathlib's continuous cohomology, for `K` in any universe: the explicit comparison `brauerExplicitH2Equiv K` followed by `explicitH2AddEquivContinuousCohomology`. It is characterized by `brauerCohomologyEquiv_crossedProductClass`: the class of the crossed product of a cocycle of a finite Galois `L \u2286 K\u02e2` goes to the continuous class of the inflation of\u2026","title":"The comparison isomorphism"},{"id":"tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants/tauceti-brauer2equivh2-basechange","kind":"node","lean":"TauCeti.brauer2EquivH2_baseChange","parent":3383,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants/index.html#tauceti-brauer2equivh2-basechange","scope":"tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants","status":"fully_proved","summary":"Scalar extension of any two-torsion Brauer class corresponds under `\u03b9` to restriction of its mod-two Galois cohomology class.","title":"TauCeti.brauer2EquivH2_baseChange"},{"id":"tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants/the-relative-stiefel-whitney-formula-in-degree-t","kind":"node","lean":"TauCeti.sw2Class_formClass_traceTransfer_smul_sq, TauCeti.sw1Class_traceTransfer_mk_rankOne, TauCeti.sw1_eq_add_galoisCor_of_equivalent_scharlauTransfer, TauCeti.sw1_eq_add_galoisCor_of_equivalent_traceTransfer, TauCeti.sw1Class_formClass_traceTransfer_smul_sq, TauCeti.sw2_eq_add_galoisEvens_add_cup_of_equivalent_traceTransfer","parent":3383,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants/index.html#the-relative-stiefel-whitney-formula-in-degree-t","scope":"tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants","status":"fully_proved","summary":"for a quadratic extension `L/K`, which is separable as `2` is invertible, and `a \u2208 L\u02e3` with Kummer class `(a) \u2208 H\u00b9(G_L, \ud835\udd3d\u2082)`,","title":"The relative Stiefel\u2013Whitney formula in degree two, on the trace forms themselves (Kahn, Th\u00e9or\u00e8me 2 in degree two)"},{"id":"tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants/the-evens-norm-of-a-kummer-class-over-a-local-fi","kind":"node","lean":"TauCeti.galoisEvens_kummerClass_eq_hilbertSymbol_smul","parent":3383,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants/index.html#the-evens-norm-of-a-kummer-class-over-a-local-fi","scope":"tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants","status":"fully_proved","summary":"if `L` is generated over a local field `K` by a square root `x \u2209 K` of `d`, and `a \u2208 L\u02e3` has nonzero trace `t`, then `N^{Ev}((a)) = \u03b5((t, \u2212d \u00b7 N a)_K \u00b7 (2, d)_K) \u2022 ((c) \u222a (e))` for all `c, e` with `(c, e)_K = -1`, where `\u03b5` sends the signs `+1`, `-1` to `0`, `1`.","title":"The Evens norm of a Kummer class over a local field, as an element of `H\u00b2(G_K, \ud835\udd3d\u2082)`"},{"id":"tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants/lams-comparison-of-the-clifford-and-hasse-invari","kind":"node","lean":"TauCeti.RegularFormClass.cliffordInvariant_eq_hasseInvariant_mul","parent":3383,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants/index.html#lams-comparison-of-the-clifford-and-hasse-invari","scope":"tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants","status":"fully_proved","summary":"For a class of rank `n` with discriminant `d`, `c(q) = s(q) \u00b7 [(-1, d)]^C(n-1,2) \u00b7 [(-1,-1)]^C(n+1,4)`, where `C(n-1,2) = (n-1)(n-2)/2` and `C(n+1,4) = (n+1)n(n-1)(n-2)/24`.","title":"Lam's comparison of the Clifford and Hasse invariants (Lam V.3.20)"},{"id":"tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants/the-hilbert-symbol-as-a-bilinear-form-over-f2","kind":"node","lean":"TauCeti.hilbertSymbolForm","parent":3383,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants/index.html#the-hilbert-symbol-as-a-bilinear-form-over-f2","scope":"tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants","status":"fully_proved","summary":"Over a nonarchimedean local field in which `2` is invertible, the Hilbert symbol on square classes, read in `ZMod 2` through the dictionary `+1 \u21a6 0`, `-1 \u21a6 1`, is a `ZMod 2`-bilinear form on `K\u02e3/(K\u02e3)\u00b2`. Its sign is the symbol (`TauCeti.hilbertSign_hilbertSymbolForm`).","title":"The Hilbert symbol as a bilinear form over `\ud835\udd3d\u2082`"},{"id":"tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants/a-kummer-cup-over-a-local-field-is-its-hilbert-s","kind":"node","lean":"TauCeti.cup_kummerClass_eq_hilbertSymbol_smul","parent":3383,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants/index.html#a-kummer-cup-over-a-local-field-is-its-hilbert-s","scope":"tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants","status":"fully_proved","summary":"if `(c, e)_K = -1`, then `(a) \u222a (b) = \u03b5 \u2022 ((c) \u222a (e))`, where `\u03b5 \u2208 ZMod 2` is `0` if `(a, b)_K = 1` and `1` if `(a, b)_K = -1`.","title":"A Kummer cup over a local field is its Hilbert symbol times a fixed nonzero cup"},{"id":"tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants/local-classification-omeara-63-20-serre-iv-thm-7","kind":"node","lean":"TauCeti.RegularFormClass.localInvariants_injective","parent":3383,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants/index.html#local-classification-omeara-63-20-serre-iv-thm-7","scope":"tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants","status":"fully_proved","summary":"A regular-form class over a nonarchimedean local field is determined by its rank, discriminant and local Hasse invariant.","title":"Local classification (O'Meara 63:20, Serre IV Thm 7)"},{"id":"tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants/the-relative-stiefel-whitney-formula-over-a-loca","kind":"node","lean":"TauCeti.galoisEvens_kummerClass_eq_zero_iff_localHasse_eq","parent":3383,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants/index.html#the-relative-stiefel-whitney-formula-over-a-loca","scope":"tauceti/number-theory/part-01/quadratic-forms-and-cohomological-invariants","status":"fully_proved","summary":"for a quadratic extension `L` of a local field `K` and `a \u2208 L\u02e3`, the Evens norm `N^{Ev}((a)) \u2208 H\u00b2(G_K, \ud835\udd3d\u2082)`, taken along any `K`-embedding `\u03c3` of `L` into a separable closure of `K`, vanishes exactly when the twisted trace form `Tr_*\u27e8a\u27e9` and the trace form `Tr_*\u27e81\u27e9` have the same local Hasse invariant.","title":"The relative Stiefel\u2013Whitney formula over a local field"},{"data_url":"scopes/tauceti/number-theory/part-01/theta-series-of-lattices.json","id":"tauceti/number-theory/part-01/theta-series-of-lattices","kind":"scope","parent":3139,"result":false,"route":"../roadmap/tauceti/number-theory/part-01/theta-series-of-lattices/index.html","scope":"tauceti/number-theory/part-01/theta-series-of-lattices","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Theta series of lattices"},{"id":"tauceti/number-theory/part-01/theta-series-of-lattices/poissons-summation-formula","kind":"node","lean":"SchwartzMap.tsum_eq_tsum_dual_fourier, SchwartzMap.tsum_eq_tsum_fourier_span_orthonormalBasis","parent":3396,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/theta-series-of-lattices/index.html#poissons-summation-formula","scope":"tauceti/number-theory/part-01/theta-series-of-lattices","status":"fully_proved","summary":"Poisson's summation formula over a full lattice `L` of a finite-dimensional real inner product space: for a Schwartz function `f` and `v : E`, `\u2211' \u2113 : L, f (v + \u2113) = (covolume L)\u207b\u00b9 * \u2211' m : L^\u2228, \ud835\udcd5 f m * exp (2\u03c0i \u27eav, m\u27eb)`. The sign of the phase is the one forced by Mathlib's `\ud835\udcd5 f w = \u222b x, exp (-2\u03c0i \u27eax, w\u27eb) f x`.","title":"Poisson's summation formula"},{"id":"tauceti/number-theory/part-01/theta-series-of-lattices/zlattice-iseven-qexpansion-thetaseries-coeff","kind":"node","lean":"ZLattice.IsEven.qExpansion_thetaSeries_coeff","parent":3396,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/theta-series-of-lattices/index.html#zlattice-iseven-qexpansion-thetaseries-coeff","scope":"tauceti/number-theory/part-01/theta-series-of-lattices","status":"fully_proved","summary":"The coefficients of the `q`-expansion of the theta series of an even lattice: the `m`-th coefficient in `q = exp (2 \u03c0 i \u03c4)` is the number `r_L(2m)` of vectors of half-norm `m`.","title":"ZLattice.IsEven.qExpansion_thetaSeries_coeff"},{"id":"tauceti/number-theory/part-01/theta-series-of-lattices/zlattice-iseven-thetaseries-slash-t","kind":"node","lean":"ZLattice.IsEven.thetaSeries_slash_T","parent":3396,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/theta-series-of-lattices/index.html#zlattice-iseven-thetaseries-slash-t","scope":"tauceti/number-theory/part-01/theta-series-of-lattices","status":"fully_proved","summary":"The theta series of an even lattice is invariant under `T`, in every weight: `\u0398_L \u2223[k] T = \u0398_L`.","title":"ZLattice.IsEven.thetaSeries_slash_T"},{"id":"tauceti/number-theory/part-01/theta-series-of-lattices/zlattice-iseven-thetacoset-one-vadd","kind":"node","lean":"ZLattice.IsEven.thetaCoset_one_vadd","parent":3396,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/theta-series-of-lattices/index.html#zlattice-iseven-thetacoset-one-vadd","scope":"tauceti/number-theory/part-01/theta-series-of-lattices","status":"fully_proved","summary":"The translation law of the theta series of a coset of an even lattice: for `\u03b3 \u2208 L^\u2228`, `\u03b8_{\u03b3+L}(\u03c4 + 1) = e(q_L(\u03b3)) \u03b8_{\u03b3+L}(\u03c4)` with `e(q_L(\u03b3)) = exp (\u03c0 i \u2016\u03b3\u2016\u00b2)`.","title":"ZLattice.IsEven.thetaCoset_one_vadd"},{"id":"tauceti/number-theory/part-01/theta-series-of-lattices/zlattice-isintegral-det-eq-card-discriminantgrou","kind":"node","lean":"ZLattice.IsIntegral.det_eq_card_discriminantGroup","parent":3396,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/theta-series-of-lattices/index.html#zlattice-isintegral-det-eq-card-discriminantgrou","scope":"tauceti/number-theory/part-01/theta-series-of-lattices","status":"fully_proved","summary":"For an integral lattice `L`, the determinant is the order of the discriminant group `L^\u2228 \u29f8 L`. In particular it is a positive integer.","title":"ZLattice.IsIntegral.det_eq_card_discriminantGroup"},{"id":"tauceti/number-theory/part-01/theta-series-of-lattices/zlattice-isintegral-thetaseries-dual-eq-sum","kind":"node","lean":"ZLattice.IsIntegral.thetaSeries_dual_eq_sum","parent":3396,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/theta-series-of-lattices/index.html#zlattice-isintegral-thetaseries-dual-eq-sum","scope":"tauceti/number-theory/part-01/theta-series-of-lattices","status":"fully_proved","summary":"For an integral lattice `L`, the theta series of the dual lattice is the sum of the coset theta series over the finite discriminant group `L^\u2228 \u29f8 L`: `\u0398_{L^\u2228} = \u2211_{\u03b3 \u2208 A_L} \u03b8_\u03b3`. The discriminant group of an integral lattice is finite (`IsIntegral.finite_discriminantGroup`), so a `Fintype` instance is always available.","title":"ZLattice.IsIntegral.thetaSeries_dual_eq_sum"},{"id":"tauceti/number-theory/part-01/theta-series-of-lattices/zlattice-thetaseries","kind":"node","lean":"ZLattice.thetaSeries","parent":3396,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/theta-series-of-lattices/index.html#zlattice-thetaseries","scope":"tauceti/number-theory/part-01/theta-series-of-lattices","status":"fully_proved","summary":"The theta series `\u0398_L(\u03c4) = \u2211_{v \u2208 L} exp (\u03c0 i \u2016v\u2016\u00b2 \u03c4)` of `L : Submodule \u2124 E`. It converges on `\u210d` when `L` is discrete and `E` is finite-dimensional (`ZLattice.summable_thetaSeries`).","title":"ZLattice.thetaSeries"},{"id":"tauceti/number-theory/part-01/theta-series-of-lattices/zlattice-thetacoset","kind":"node","lean":"ZLattice.thetaCoset","parent":3396,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/theta-series-of-lattices/index.html#zlattice-thetacoset","scope":"tauceti/number-theory/part-01/theta-series-of-lattices","status":"fully_proved","summary":"The theta series `\u03b8_{\u03b3+L}(\u03c4) = \u2211_{v \u2208 L} exp (\u03c0 i \u2016\u03b3 + v\u2016\u00b2 \u03c4)` of the translate `\u03b3 + L` of `L : Submodule \u2124 E`. For `\u03b3` in the dual lattice this is the coset theta series of the coset `\u03b3 + L`.","title":"ZLattice.thetaCoset"},{"id":"tauceti/number-theory/part-01/theta-series-of-lattices/zlattice-representationnumber","kind":"node","lean":"ZLattice.representationNumber","parent":3396,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/theta-series-of-lattices/index.html#zlattice-representationnumber","scope":"tauceti/number-theory/part-01/theta-series-of-lattices","status":"fully_proved","summary":"The representation number `r_L(t)`: the number of vectors of `L` of squared norm `t`. For a discrete `L` in a proper space every shell is finite, so this is a genuine count.","title":"ZLattice.representationNumber"},{"id":"tauceti/number-theory/part-01/theta-series-of-lattices/zlattice-dual","kind":"node","lean":"ZLattice.dual","parent":3396,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/theta-series-of-lattices/index.html#zlattice-dual","scope":"tauceti/number-theory/part-01/theta-series-of-lattices","status":"fully_proved","summary":"The dual lattice of `L : Submodule \u2124 E`: the vectors whose inner product with every vector of `L` is an integer. It is Mathlib's `BilinForm.dualSubmodule` of the inner product.","title":"ZLattice.dual"},{"id":"tauceti/number-theory/part-01/theta-series-of-lattices/zlattice-iseven","kind":"node","lean":"ZLattice.IsEven","parent":3396,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/theta-series-of-lattices/index.html#zlattice-iseven","scope":"tauceti/number-theory/part-01/theta-series-of-lattices","status":"fully_proved","summary":"A lattice is even if the squared norm of each of its vectors is an even integer.","title":"ZLattice.IsEven"},{"id":"tauceti/number-theory/part-01/theta-series-of-lattices/milgrams-formula","kind":"node","lean":"ZLattice.IsEven.dualGaussSum_one_one","parent":3396,"result":true,"route":"../roadmap/tauceti/number-theory/part-01/theta-series-of-lattices/index.html#milgrams-formula","scope":"tauceti/number-theory/part-01/theta-series-of-lattices","status":"fully_proved","summary":"If `L` is an even lattice and `dim E = 2k`, then `\u2211_{\u03b3 \u2208 L^\u2228/L} e(\u2016\u03b3\u2016\u00b2 / 2) = \u221a|L^\u2228/L| \u00b7 i^k`, where `i^k = e(n/8)` for `n = 2k`. The left side is written as a sum over `L^\u2228 / 1L`; `ZLattice.dualGaussSum_one` rewrites it as a sum over the discriminant group `L^\u2228/L`.","title":"Milgram's formula"},{"data_url":"scopes/tauceti/number-theory/part-02.json","id":"tauceti/number-theory/part-02","kind":"scope","navigation":"transparent","parent":3138,"result":false,"route":"../roadmap/tauceti/number-theory/part-02/index.html","scope":"tauceti/number-theory/part-02","status":"fully_proved","title":"Arithmetic Dirichlet Series to Other results"},{"data_url":"scopes/tauceti/number-theory/part-02/arithmetic-dirichlet-series.json","id":"tauceti/number-theory/part-02/arithmetic-dirichlet-series","kind":"scope","parent":3409,"result":false,"route":"../roadmap/tauceti/number-theory/part-02/arithmetic-dirichlet-series/index.html","scope":"tauceti/number-theory/part-02/arithmetic-dirichlet-series","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.NumberTheory.ArithmeticDirichletSeries` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Arithmetic Dirichlet Series"},{"id":"tauceti/number-theory/part-02/arithmetic-dirichlet-series/a-negative-power-abel-bound","kind":"node","lean":"TauCeti.summatory_mul_rpow_neg_le","parent":3410,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/arithmetic-dirichlet-series/index.html#a-negative-power-abel-bound","scope":"tauceti/number-theory/part-02/arithmetic-dirichlet-series","status":"fully_proved","summary":"Suppose every index has `N`-value at least `1`, and the partial sums of a real weight `w` are at most `C * t ^ \u03b2` on `[1, x]`. Twisting the weight by `(N i) ^ (-\u03b1)` with `\u03b1 \u2265 0` gives partial sums at most `C x ^ (\u03b2 - \u03b1) + \u03b1 C (x ^ (\u03b2 - \u03b1) - 1) / (\u03b2 - \u03b1)`.","title":"A negative-power Abel bound"},{"id":"tauceti/number-theory/part-02/arithmetic-dirichlet-series/a-one-sided-abel-bound","kind":"node","lean":"TauCeti.summatory_mul_le_of_summatory_le, TauCeti.tsum_mul_le_of_summatory_le","parent":3410,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/arithmetic-dirichlet-series/index.html#a-one-sided-abel-bound","scope":"tauceti/number-theory/part-02/arithmetic-dirichlet-series","status":"fully_proved","summary":"Let every index have `N`-value at least the cutoff `a \u2265 0`, and let `g` be nonincreasing on `[a, x]` with `0 \u2264 g x`. If the summatory function of a real weight `w` is at most `C` at every cutoff in `[a, x]`, then the summatory function at `x` of the twisted weight `i \u21a6 w i * g (N i)` is at most `C * g a`.","title":"A one-sided Abel bound"},{"id":"tauceti/number-theory/part-02/arithmetic-dirichlet-series/an-imaginary-power-abel-bound","kind":"node","lean":"TauCeti.norm_summatory_mul_cpow_le_of_summatory_le","parent":3410,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/arithmetic-dirichlet-series/index.html#an-imaginary-power-abel-bound","scope":"tauceti/number-theory/part-02/arithmetic-dirichlet-series","status":"fully_proved","summary":"Suppose every index has `N`-value at least `1`, and the partial sums of `w` are bounded by `C * t ^ \u03b8` on `[1, x]` for a positive exponent `\u03b8`. Twisting the weight by `(N i) ^ (-z)` with `Re z = 0` preserves that exponent, at the cost of the explicit factor `1 + \u2016z\u2016 / \u03b8`.","title":"An imaginary-power Abel bound"},{"id":"tauceti/number-theory/part-02/arithmetic-dirichlet-series/the-arithmetic-perron-formula-for-ideal-sums","kind":"node","lean":"TauCeti.perronFormula, TauCeti.tendsto_truncatedPerron_LSeries_natCast","parent":3410,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/arithmetic-dirichlet-series/index.html#the-arithmetic-perron-formula-for-ideal-sums","scope":"tauceti/number-theory/part-02/arithmetic-dirichlet-series","status":"fully_proved","summary":"Where the `L`-series of the norm coefficients of `f` converges absolutely on the line `Re s = c`, the truncated Perron integral tends, as the height grows, to the ideal sum `\u2211_{N(I) \u2264 x} f(I)` in which each ideal of norm exactly `x` carries half weight. The two readings of this limit are `TauCeti.perronFormula_of_forall_absNorm_ne` and `TauCeti.perronFormula_natCast`.","title":"The arithmetic Perron formula for ideal sums"},{"id":"tauceti/number-theory/part-02/arithmetic-dirichlet-series/the-limiting-arithmetic-perron-formula","kind":"node","lean":"TauCeti.tendsto_truncatedPerron_LSeries","parent":3410,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/arithmetic-dirichlet-series/index.html#the-limiting-arithmetic-perron-formula","scope":"tauceti/number-theory/part-02/arithmetic-dirichlet-series","status":"fully_proved","summary":"As the truncation height grows the integral tends to the series of sharp steps `\u2211' n, f n * perronStep (x / n)`. This holds at every positive `x`, integer endpoints included; `TauCeti.tsum_mul_perronStep_natCast` evaluates the limit there.","title":"The limiting arithmetic Perron formula"},{"id":"tauceti/number-theory/part-02/arithmetic-dirichlet-series/the-off-norm-arithmetic-perron-formula","kind":"node","lean":"TauCeti.norm_truncatedPerron_LSeries_sub_sum_le","parent":3410,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/arithmetic-dirichlet-series/index.html#the-off-norm-arithmetic-perron-formula","scope":"tauceti/number-theory/part-02/arithmetic-dirichlet-series","status":"fully_proved","summary":"When `x` is not a natural number at which `f` is nonzero, the truncated integral differs from the sharp partial sum `\u2211_{n < x} f n` by at most the series of the smoothed-step kernel errors, a series that absolute convergence on the line makes summable.","title":"The off-norm arithmetic Perron formula"},{"id":"tauceti/number-theory/part-02/arithmetic-dirichlet-series/the-perron-formula-off-the-norms","kind":"node","lean":"TauCeti.perronFormula_of_forall_absNorm_ne, TauCeti.perronFormula_natCast","parent":3410,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/arithmetic-dirichlet-series/index.html#the-perron-formula-off-the-norms","scope":"tauceti/number-theory/part-02/arithmetic-dirichlet-series","status":"fully_proved","summary":"If `x > 0` is not the absolute norm of a nonzero ideal, the truncated Perron integral tends to the inclusive ideal sum `\u2211_{N(I) \u2264 x} f(I)`.","title":"The Perron formula off the norms"},{"id":"tauceti/number-theory/part-02/arithmetic-dirichlet-series/the-truncated-perron-formula-off-the-norms","kind":"node","lean":"TauCeti.norm_truncatedPerron_LSeries_normCoeff_sub_idealSummatory_le, TauCeti.truncatedPerron_LSeries","parent":3410,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/arithmetic-dirichlet-series/index.html#the-truncated-perron-formula-off-the-norms","scope":"tauceti/number-theory/part-02/arithmetic-dirichlet-series","status":"fully_proved","summary":"If `x > 0` is not the absolute norm of a nonzero ideal, then at every positive height `T` the truncated Perron integral differs from the inclusive ideal sum `\u2211_{N(I) \u2264 x} f(I)` by at most the series of the smoothed-step kernel errors at the ratios `x / n`.","title":"The truncated Perron formula off the norms"},{"data_url":"scopes/tauceti/number-theory/part-02/class-field-theory.json","id":"tauceti/number-theory/part-02/class-field-theory","kind":"scope","parent":3409,"result":false,"route":"../roadmap/tauceti/number-theory/part-02/class-field-theory/index.html","scope":"tauceti/number-theory/part-02/class-field-theory","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.NumberTheory.ClassFieldTheory` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Class Field Theory: other results"},{"id":"tauceti/number-theory/part-02/class-field-theory/the-cyclic-hasse-norm-theorem","kind":"node","lean":"TauCeti.ClassFieldTheory.cyclicHasseNorm, TauCeti.ClassFieldTheory.isGlobalNorm_iff_isLocalNormEverywhere","parent":3419,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/class-field-theory/index.html#the-cyclic-hasse-norm-theorem","scope":"tauceti/number-theory/part-02/class-field-theory","status":"fully_proved","summary":"For a cyclic extension `L/K` of number fields, an element `a \u2208 K\u02e3` lies in `N_{L/K}(L\u02e3)` exactly when its principal idele is the norm of an idele of `L`.","title":"The cyclic Hasse norm theorem"},{"id":"tauceti/number-theory/part-02/class-field-theory/the-equivariant-kummer-isomorphism-on-kummer-cla","kind":"node","lean":"TauCeti.ClassFieldTheory.dualTensorHom_kummerH1ConjRepresentationEquiv","parent":3419,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/class-field-theory/index.html#the-equivariant-kummer-isomorphism-on-kummer-cla","scope":"tauceti/number-theory/part-02/class-field-theory","status":"fully_proved","summary":"Under `kummerH1ConjRepresentationEquiv`, the Kummer class of a power class `x`, read on `V` through `restrictSubgroupH1Equiv`, corresponds to the homomorphism `M \u2192 L\u02e3 \u29f8 (L\u02e3)\u207f` sending `m` to `log(m) \u2022 x`, where `log : M \u2243 \u03bc\u2099 \u2243 \u2124/n` is `e` followed by `kummerCoeffAddEquivZMod`.","title":"The equivariant Kummer isomorphism on Kummer classes (`TauCeti.ClassFieldTheory.dualTensorHom_kummerH1ConjRepresentationEquiv`)"},{"id":"tauceti/number-theory/part-02/class-field-theory/the-euler-characteristic-formula-and-g0","kind":"node","lean":"TauCeti.ClassFieldTheory.forall_localEulerCharacteristic_eq_localCardNorm_iff, TauCeti.ClassFieldTheory.localEulerCharacteristicK0_eq_localCardNormK0_iff","parent":3419,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/class-field-theory/index.html#the-euler-characteristic-formula-and-g0","scope":"tauceti/number-theory/part-02/class-field-theory","status":"fully_proved","summary":"Tate's local Euler characteristic formula `\u03c7_F(X) = \u03c6_F(X)` holds for every finite smooth discrete Galois representation `X` exactly when, for every open normal subgroup `V` of `G_F`, the descended Euler characteristic and the descended normalized absolute value of the order agree on the Grothendieck group of `(ZMod n)[G_F \u29f8 V]`.","title":"The Euler characteristic formula and `G\u2080`"},{"id":"tauceti/number-theory/part-02/class-field-theory/global-existence-milne-vii-theorem-9-5","kind":"node","lean":"TauCeti.ClassFieldTheory.globalAbelianExistence","parent":3419,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/class-field-theory/index.html#global-existence-milne-vii-theorem-9-5","scope":"tauceti/number-theory/part-02/class-field-theory","status":"fully_proved","summary":"For a number field `K`, every open subgroup of finite index of the idele class group `C_K` is the norm subgroup of a finite abelian extension of `K`.","title":"Global existence (Milne VII, Theorem 9.5)"},{"id":"tauceti/number-theory/part-02/class-field-theory/the-hasse-principle-for-brauer-classes-from-idel","kind":"node","lean":"TauCeti.ClassFieldTheory.brLocalization_injective_of_subsingleton_groupCohomology_one, TauCeti.ClassFieldTheory.brLocalization_injective, TauCeti.ClassFieldTheory.brLocalization_injective_of_subsingleton_h1_globalFormation","parent":3419,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/class-field-theory/index.html#the-hasse-principle-for-brauer-classes-from-idel","scope":"tauceti/number-theory/part-02/class-field-theory","status":"fully_proved","summary":"If `H\u00b9(Gal(L/K), C_L)` vanishes for every finite Galois extension `L/K`, then a Brauer class of `K` whose localizations at all places vanish is zero: `Br K \u2192 \u2a01_v Br K_v` is injective.","title":"The Hasse principle for Brauer classes, from idele-class `H\u00b9`"},{"id":"tauceti/number-theory/part-02/class-field-theory/the-kronecker-weber-theorem","kind":"node","lean":"TauCeti.ClassFieldTheory.kroneckerWeber","parent":3419,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/class-field-theory/index.html#the-kronecker-weber-theorem","scope":"tauceti/number-theory/part-02/class-field-theory","status":"fully_proved","summary":"every finite abelian extension of `\u211a` embeds in a cyclotomic field `\u211a(\u03b6_n)`.","title":"The Kronecker\u2013Weber theorem"},{"id":"tauceti/number-theory/part-02/class-field-theory/the-kummer-case-of-the-second-fundamental-inequa","kind":"node","lean":"TauCeti.ClassFieldTheory.index_range_ideleClassNormMap_le_of_finrank_prime","parent":3419,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/class-field-theory/index.html#the-kummer-case-of-the-second-fundamental-inequa","scope":"tauceti/number-theory/part-02/class-field-theory","status":"fully_proved","summary":"Let `L/K` be a Galois extension of number fields of prime degree `p`, and suppose that `K` contains the `p`-th roots of unity. Then the idele-class norm group has index at most `p`: `[C_K : N_{L/K}(C_L)] \u2264 p`.","title":"The Kummer case of the second fundamental inequality"},{"id":"tauceti/number-theory/part-02/class-field-theory/local-tate-duality-in-degree-one","kind":"node","lean":"TauCeti.ClassFieldTheory.dualityMap1_kummerCoeff_bijective, TauCeti.ClassFieldTheory.tateDualityPairing_flip_perfect, TauCeti.ClassFieldTheory.dualityMap2_kummerCoeff_bijective, TauCeti.ClassFieldTheory.dualityMap0_kummerCoeff_bijective","parent":3419,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/class-field-theory/index.html#local-tate-duality-in-degree-one","scope":"tauceti/number-theory/part-02/class-field-theory","status":"fully_proved","summary":"Let `K` be a nonarchimedean local field and `n` a natural number invertible in `K`. For every finite discrete `G_K`-module `A` killed by `n`, Tate's duality map `H\u00b9(G_K, A) \u2192 Hom(H\u00b9(G_K, Hom(A, \u03bc\u2099)), H\u00b2(G_K, \u03bc\u2099))` is bijective.","title":"Local Tate duality in degree one"},{"id":"tauceti/number-theory/part-02/class-field-theory/modular-artin-reduction-of-the-local-euler-chara","kind":"node","lean":"TauCeti.ClassFieldTheory.localEulerCharacteristic_eq_localCardNorm_of_indFDRep","parent":3419,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/class-field-theory/index.html#modular-artin-reduction-of-the-local-euler-chara","scope":"tauceti/number-theory/part-02/class-field-theory","status":"fully_proved","summary":"If `\u03c7_F = \u03c6_F` holds for the inflation to `G_F` of every representation of a finite Galois quotient `G_F \u29f8 V` induced from a cyclic subgroup of order prime to `\u2113`, then it holds for every finite smooth discrete `ZMod \u2113`-representation of `G_F`.","title":"Modular-Artin reduction of the local Euler characteristic formula"},{"id":"tauceti/number-theory/part-02/class-field-theory/perfect-local-tate-duality-in-mixed-characterist","kind":"node","lean":"TauCeti.ClassFieldTheory.tateDualityPairing_perfect_mixed, TauCeti.ClassFieldTheory.tateDualityPairing_perfect","parent":3419,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/class-field-theory/index.html#perfect-local-tate-duality-in-mixed-characterist","scope":"tauceti/number-theory/part-02/class-field-theory","status":"fully_proved","summary":"For a characteristic-zero nonarchimedean local field `F`, every finite smooth discrete `ZMod n`-representation has a perfect local Tate-duality pairing in complementary degrees.","title":"Perfect local Tate duality in mixed characteristic"},{"id":"tauceti/number-theory/part-02/class-field-theory/the-principal-ideal-theorem-for-a-class-formatio","kind":"node","lean":"TauCeti.ClassFieldTheory.ClassFormation.groundInclusion_mem_normSubgroup_of_range_eq_commutator","parent":3419,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/class-field-theory/index.html#the-principal-ideal-theorem-for-a-class-formatio","scope":"tauceti/number-theory/part-02/class-field-theory","status":"fully_proved","summary":"Let `V \u25c1 U` be a layer, `K/F` in field notation, and restrict it to the ground subgroup `U'` whose image `U'/V` is the commutator subgroup of `U/V`, the layer `K/E` over the maximal abelian subextension `E`. Then every element of the ground level `A^U` is, in `A^{U'}`, a norm from `A^V`.","title":"The principal ideal theorem for a class formation"},{"id":"tauceti/number-theory/part-02/class-field-theory/shapiros-lemma-for-the-normalized-order","kind":"node","lean":"TauCeti.ClassFieldTheory.localCardNorm_galRepOfQuotient_ind, NumberField.InfinitePlace.infiniteSemilocalUnitsTateCohomologyIsoPos, IsDedekindDomain.HeightOneSpectrum.semilocalUnitsTateCohomologyIsoPos, TauCeti.ClassFieldTheory.localEulerCharacteristic_galRepOfQuotient_ind, TauCeti.ClassFieldTheory.semilocalUnitsTateCohomologyIso","parent":3419,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/class-field-theory/index.html#shapiros-lemma-for-the-normalized-order","scope":"tauceti/number-theory/part-02/class-field-theory","status":"fully_proved","summary":"For `L` the fixed field of `C \u2264 G_K \u29f8 V`, the invariant `\u03c6_K(A) = \u2016#A\u2016_K = |#A|_p ^ [K : \u211a_p]` of the inflation of `Ind_C^{G_K \u29f8 V} B` is that of `B` over `L`: `#Ind B = #B ^ [L : K]` and `[L : \u211a_p] = [K : \u211a_p] [L : K]`.","title":"Shapiro's lemma for the normalized order"},{"id":"tauceti/number-theory/part-02/class-field-theory/tates-duality-on-an-open-subgroup-acting-trivial","kind":"node","lean":"TauCeti.ClassFieldTheory.dualityMap_bijective_of_smul_kummerCoeff_eq_self, TauCeti.ClassFieldTheory.dualityMap_bijective_discreteCoind_kummerCoeff","parent":3419,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/class-field-theory/index.html#tates-duality-on-an-open-subgroup-acting-trivial","scope":"tauceti/number-theory/part-02/class-field-theory","status":"fully_proved","summary":"Let `V` be an open subgroup of `G_K` acting trivially on `\u03bc\u2099`, for `n` invertible in the local field `K`. For every finite discrete `V`-module `A` killed by `n` on which `V` acts trivially, the three duality maps `H\u2071(V, A) \u2192 Hom(H\u00b2\u207b\u2071(V, Hom(A, \u03bc\u2099)), H\u00b2(V, \u03bc\u2099))` are bijective.","title":"Tate's duality on an open subgroup acting trivially on `\u03bc\u2099`"},{"id":"tauceti/number-theory/part-02/class-field-theory/tates-local-euler-characteristic-formula","kind":"node","lean":"TauCeti.ClassFieldTheory.localEulerCharacteristic_eq_localCardNorm, TauCeti.ClassFieldTheory.localEulerCharacteristic_eq_localCardNorm_of_index_coprime","parent":3419,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/class-field-theory/index.html#tates-local-euler-characteristic-formula","scope":"tauceti/number-theory/part-02/class-field-theory","status":"fully_proved","summary":"The Euler characteristic of a finite smooth discrete Galois module equals the normalized absolute value of its order.","title":"Tate's local Euler characteristic formula"},{"data_url":"scopes/tauceti/number-theory/part-02/diophantine-approximation.json","id":"tauceti/number-theory/part-02/diophantine-approximation","kind":"scope","parent":3409,"result":false,"route":"../roadmap/tauceti/number-theory/part-02/diophantine-approximation/index.html","scope":"tauceti/number-theory/part-02/diophantine-approximation","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.NumberTheory.DiophantineApproximation` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Diophantine Approximation"},{"id":"tauceti/number-theory/part-02/diophantine-approximation/dirichlets-lower-bound-for-mahlers-exponent","kind":"node","lean":"TauCeti.natCast_le_mahlerExponent, TauCeti.card_le_linearFormExponent, TauCeti.inv_card_le_simultaneousExponent","parent":3434,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/diophantine-approximation/index.html#dirichlets-lower-bound-for-mahlers-exponent","scope":"tauceti/number-theory/part-02/diophantine-approximation","status":"fully_proved","summary":"If `\u03be` is not algebraic of degree at most `n`, then `w_n(\u03be) \u2265 n`.","title":"Dirichlet's lower bound for Mahler's exponent"},{"id":"tauceti/number-theory/part-02/diophantine-approximation/dirichlets-theorem-nearest-integer-form","kind":"node","lean":"Real.infinite_setOf_mul_abs_mul_sub_round_le_one, Matrix.exists_int_abs_sum_mul_sub_le, TauCeti.dirichlet_linearForm_infinite, TauCeti.infinite_linearFormApproximations_of_lt_card, Matrix.dirichlet_linearForms, Matrix.dirichlet_linearForms_infinite, TauCeti.infinite_simultaneousApproximationDenominators_inv_card","parent":3434,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/diophantine-approximation/index.html#dirichlets-theorem-nearest-integer-form","scope":"tauceti/number-theory/part-02/diophantine-approximation","status":"fully_proved","summary":"For every real `\u03b1`, infinitely many natural numbers `q \u2265 1` satisfy `q \u2016q \u03b1\u2016 \u2264 1`.","title":"Dirichlet's theorem, nearest-integer form"},{"id":"tauceti/number-theory/part-02/diophantine-approximation/liouvilles-bound-for-mahlers-exponent","kind":"node","lean":"IsAlgebraic.mahlerExponent_le","parent":3434,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/diophantine-approximation/index.html#liouvilles-bound-for-mahlers-exponent","scope":"tauceti/number-theory/part-02/diophantine-approximation","status":"fully_proved","summary":"If `\u03be` is a real algebraic number of degree `D` over `\u211a`, then `w\u2099(\u03be) \u2264 D - 1` for every `n`.","title":"Liouville's bound for Mahler's exponent"},{"id":"tauceti/number-theory/part-02/diophantine-approximation/liouvilles-inequality","kind":"node","lean":"Polynomial.inv_le_norm_ratCast_sub, IsAlgebraic.inv_le_norm_ratCast_sub, IsAlgebraic.exists_pos_le_norm_aeval_mul_supNorm_pow","parent":3434,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/diophantine-approximation/index.html#liouvilles-inequality","scope":"tauceti/number-theory/part-02/diophantine-approximation","status":"fully_proved","summary":"Let `f \u2208 \u2124[X]` have degree `d` and a complex root `\u03b1`. Every rational number `\u03be` with `f(\u03be) \u2260 0` satisfies `|\u03be - \u03b1| \u2265 (2 ^ (d - 1) M(f) H(\u03be) ^ d)\u207b\u00b9`, where `M(f)` is the Mahler measure of `f` and `H(\u03be)` the height of `\u03be`.","title":"Liouville's inequality"},{"id":"tauceti/number-theory/part-02/diophantine-approximation/liouvilles-theorem-in-degree-two","kind":"node","lean":"Irrational.badlyApproximable_of_aeval_eq_zero, TauCeti.badlyApproximable_of_natDegree_minpoly_eq_two","parent":3434,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/diophantine-approximation/index.html#liouvilles-theorem-in-degree-two","scope":"tauceti/number-theory/part-02/diophantine-approximation","status":"fully_proved","summary":"An irrational root of a nonzero integer polynomial of degree at most `2` is badly approximable.","title":"Liouville's theorem in degree two"},{"id":"tauceti/number-theory/part-02/diophantine-approximation/thues-approximation-theorem-for-real","kind":"node","lean":"TauCeti.thue_approximation_real, TauCeti.thue_approximation","parent":3434,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/diophantine-approximation/index.html#thues-approximation-theorem-for-real","scope":"tauceti/number-theory/part-02/diophantine-approximation","status":"fully_proved","summary":"Let `\u03b1` be a real algebraic number of degree `d \u2265 2` and let `\u03ba > d / 2 + 1`. Then only finitely many rationals `\u03be` satisfy `|\u03be - \u03b1| \u2264 H(\u03be) ^ (-\u03ba)`. This is a variant of Evertse, Theorem 6.9, rather than that statement verbatim: every degree `d \u2265 2` is allowed.","title":"Thue's approximation theorem for real `\u03b1`"},{"id":"tauceti/number-theory/part-02/diophantine-approximation/thues-gap-principle-evertse-theorem-6-14","kind":"node","lean":"TauCeti.thue_gap_principle","parent":3434,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/diophantine-approximation/index.html#thues-gap-principle-evertse-theorem-6-14","scope":"tauceti/number-theory/part-02/diophantine-approximation","status":"fully_proved","summary":"Let `\u03b1` be a complex algebraic number of degree `d \u2265 2` and let `\u03ba > d / 2 + 1`. There are constants `C \u2265 1` and `\u03bb \u2265 1` such that for all rationals `\u03be\u2081` and `\u03be` with `|\u03be\u2081 - \u03b1| \u2264 H(\u03be\u2081) ^ (-\u03ba)`, `H(\u03be\u2081) \u2265 C` and `|\u03be - \u03b1| \u2264 H(\u03be) ^ (-\u03ba)`, the heights satisfy `H(\u03be) \u2264 H(\u03be\u2081) ^ \u03bb`. The degree hypothesis already forces `\u03b1` to be algebraic.","title":"Thue's gap principle (Evertse, Theorem 6.14)"},{"data_url":"scopes/tauceti/number-theory/part-02/hecke-ring.json","id":"tauceti/number-theory/part-02/hecke-ring","kind":"scope","parent":3409,"result":false,"route":"../roadmap/tauceti/number-theory/part-02/hecke-ring/index.html","scope":"tauceti/number-theory/part-02/hecke-ring","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.NumberTheory.HeckeRing` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Hecke Ring"},{"id":"tauceti/number-theory/part-02/hecke-ring/the-generators-generate-r-p-shimura-theorem-3-20","kind":"node","lean":"HeckeRing.GLn.range_evalHom","parent":3442,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/hecke-ring/index.html#the-generators-generate-r-p-shimura-theorem-3-20","scope":"tauceti/number-theory/part-02/hecke-ring","status":"fully_proved","summary":"for `p` prime, the polynomials in the generators `heckeGen k` are exactly the elements of the `p`-local Hecke ring.","title":"The generators generate `R_p` (Shimura, Theorem 3.20)"},{"id":"tauceti/number-theory/part-02/hecke-ring/the-prime-power-degree-shimura-theorem-3-24-degr","kind":"node","lean":"HeckeRing.GL2.degree_diagCoset_of_ratio_eq_prime_pow","parent":3442,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/hecke-ring/index.html#the-prime-power-degree-shimura-theorem-3-24-degr","scope":"tauceti/number-theory/part-02/hecke-ring","status":"fully_proved","summary":"for prime `p` and `k \u2265 1`, a divisibility chain `a` of ratio `a\u2081 / a\u2080 = p\u1d4f` has `deg T(a\u2080, a\u2081) = p\u1d4f\u207b\u00b9 (p + 1)`. The archetype is `a = (p\u2071, p\u2071\u207a\u1d4f)`.","title":"The prime-power degree (Shimura, Theorem 3.24, degree count)"},{"id":"tauceti/number-theory/part-02/hecke-ring/shimuras-commutativity-criterion-proposition-3-8","kind":"node","lean":"HeckeCosetModule.mul_comm_of_antiInvolution","parent":3442,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/hecke-ring/index.html#shimuras-commutativity-criterion-proposition-3-8","scope":"tauceti/number-theory/part-02/hecke-ring","status":"fully_proved","summary":"the Hecke ring over a commutative semiring is commutative when an anti-involution fixes every double coset.","title":"Shimura's commutativity criterion (Proposition 3.8 of [Shimura][shimura1971])"},{"id":"tauceti/number-theory/part-02/hecke-ring/shimuras-covering-identity-for-a-product-of-doub","kind":"node","lean":"DoubleCoset.doubleCoset_mul_doubleCoset_eq_iUnion_rightCosets","parent":3442,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/hecke-ring/index.html#shimuras-covering-identity-for-a-product-of-doub","scope":"tauceti/number-theory/part-02/hecke-ring","status":"fully_proved","summary":"If `\u0393\u2081 \u03b4\u2081 \u0393\u2082` is the union of the right cosets `\u0393\u2081 a\u1d62` and `\u0393\u2082 \u03b4\u2082 \u0393\u2083` is the union of the right cosets `\u0393\u2082 b\u2c7c`, then the pointwise product `\u0393\u2081 \u03b4\u2081 \u0393\u2082 \u00b7 \u0393\u2082 \u03b4\u2082 \u0393\u2083` is the union of the right cosets `\u0393\u2081 a\u1d62 b\u2c7c`.","title":"Shimura's covering identity for a product of double cosets (\u00a73.4)"},{"id":"tauceti/number-theory/part-02/hecke-ring/shimuras-decomposition-of-a-double-coset-into-ri","kind":"node","lean":"DoubleCoset.doubleCoset_eq_iUnion_rightCosets, DoubleCoset.doubleCoset_eq_iUnion_rightCosetRep","parent":3442,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/hecke-ring/index.html#shimuras-decomposition-of-a-double-coset-into-ri","scope":"tauceti/number-theory/part-02/hecke-ring","status":"fully_proved","summary":"`\u0393\u2081g\u0393\u2082` is the union of the right cosets `\u0393\u2081 \u00b7 (g \u03c4\u1d65\u207b\u00b9)`, where `\u03c4\u1d65` runs over representatives of `\u0393\u2082 \u29f8 (\u0393\u2082 \u2229 g\u207b\u00b9\u0393\u2081g)` \u2014 the mirror of `DoubleCoset.doubleCoset_eq_iUnion_leftCosets`, and of Mathlib's `doubleCoset_union_rightCoset`, which is indexed by all of `\u0393\u2082` and so repeats each coset.","title":"Shimura's decomposition of a double coset into right cosets"},{"data_url":"scopes/tauceti/number-theory/part-02/height.json","id":"tauceti/number-theory/part-02/height","kind":"scope","parent":3409,"result":false,"route":"../roadmap/tauceti/number-theory/part-02/height/index.html","scope":"tauceti/number-theory/part-02/height","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.NumberTheory.Height` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Height"},{"id":"tauceti/number-theory/part-02/height/evertses-lower-bound","kind":"node","lean":"TauCeti.one_le_natDenominator_pow_mul_house_pow_mul_norm, TauCeti.one_le_natDenominator_pow_mul_house_pow_mul_norm_of_mem_rootSet","parent":3448,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/height/index.html#evertses-lower-bound","scope":"tauceti/number-theory/part-02/height","status":"fully_proved","summary":"A nonzero algebraic number `z \u2208 \u2102` of degree `d` satisfies `den(z) ^ d * house(z) ^ (d - 1) * \u2016z\u2016 \u2265 1`.","title":"Evertse's lower bound"},{"id":"tauceti/number-theory/part-02/height/liouvilles-inequality-in-exponential-form","kind":"node","lean":"NumberField.eq_zero_of_norm_le_exp_of_house_le_exp, NumberField.one_le_norm_mul_house_pow, IsDedekindDomain.HeightOneSpectrum.neg_log_valuation_aeval_mul_log_absNorm_le","parent":3448,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/height/index.html#liouvilles-inequality-in-exponential-form","scope":"tauceti/number-theory/part-02/height","status":"fully_proved","summary":"Let `K` be a number field of degree `d`, `\u03c3 : K \u2192+* \u2102`, `y \u2208 K` and `m \u2260 0` an integer with `m ^ N y` an algebraic integer. If `\u2016\u03c3 y\u2016 \u2264 exp A`, `\u2308y\u2309 \u2264 exp B` and `|m| ^ N \u2264 exp C` with `C + A + (d - 1) (C + B) < 0`, then `y = 0`.","title":"Liouville's inequality in exponential form"},{"id":"tauceti/number-theory/part-02/height/roths-lemma-in-one-variable-index-form","kind":"node","lean":"MvPolynomial.weightedIndex_toMvPolynomial_le, Polynomial.rootMultiplicity_mul_logHeight\u2081_le","parent":3448,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/height/index.html#roths-lemma-in-one-variable-index-form","scope":"tauceti/number-theory/part-02/height","status":"fully_proved","summary":"Let `P \u2260 0` be placed in the coordinate `i`, with `deg P \u2264 r i`. If `r i * h(a i) \u2265 s\u207b\u00b9 * (h(P) + [K : \u211a] * r i * log 2)` for some `s > 0`, then the index of `P` at `a` with weights `r` is at most `s`. Heights are relative. Replacing `log 2` by a larger constant, such as Pottmeyer's `4`, only strengthens the hypothesis.","title":"Roth's lemma in one variable, index form"},{"data_url":"scopes/tauceti/number-theory/part-02/l-functions.json","id":"tauceti/number-theory/part-02/l-functions","kind":"scope","parent":3409,"result":false,"route":"../roadmap/tauceti/number-theory/part-02/l-functions/index.html","scope":"tauceti/number-theory/part-02/l-functions","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.NumberTheory.LFunctions` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"L Functions"},{"id":"tauceti/number-theory/part-02/l-functions/chebyshevs-identity-for-a-number-field","kind":"node","lean":"TauCeti.LFunctions.idealSummatory_log_absNorm_eq","parent":3452,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/l-functions/index.html#chebyshevs-identity-for-a-number-field","scope":"tauceti/number-theory/part-02/l-functions","status":"fully_proved","summary":"Summing `log N\ud835\udd1e = \u2211_{\ud835\udd1f \u2223 \ud835\udd1e} \u039b_K(\ud835\udd1f)` over the nonzero integral ideals of norm at most `x` and exchanging the order of summation, `\u2211_{N\ud835\udd1e \u2264 x} log N\ud835\udd1e = \u2211_{N(\ud835\udd2d ^ k) \u2264 x} log N\ud835\udd2d \u00b7 #{\ud835\udd1e : N\ud835\udd1e \u2264 x / N(\ud835\udd2d ^ k)}`.","title":"Chebyshev's identity for a number field"},{"id":"tauceti/number-theory/part-02/l-functions/mertens-first-theorem-for-a-number-field","kind":"node","lean":"TauCeti.LFunctions.isBigO_primeSummatory_log_absNorm_div_sub_log, TauCeti.LFunctions.isBigO_primePowerSummatory_div_sub_log","parent":3452,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/l-functions/index.html#mertens-first-theorem-for-a-number-field","scope":"tauceti/number-theory/part-02/l-functions","status":"fully_proved","summary":"The sum of `log N\ud835\udd2d / N\ud835\udd2d` over the nonzero prime ideals `\ud835\udd2d` of `\ud835\udcde K` of norm at most `x` is `log x + O(1)`.","title":"Mertens' first theorem for a number field"},{"id":"tauceti/number-theory/part-02/l-functions/mertens-second-theorem-for-a-number-field","kind":"node","lean":"TauCeti.LFunctions.isBigO_primeSummatory_inv_absNorm_sub_log_log_sub_mertensConstant, TauCeti.LFunctions.tendsto_primeSummatory_inv_absNorm_sub_log_log","parent":3452,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/l-functions/index.html#mertens-second-theorem-for-a-number-field","scope":"tauceti/number-theory/part-02/l-functions","status":"fully_proved","summary":"The sum of `1 / N\ud835\udd2d` over the nonzero prime ideals `\ud835\udd2d` of `\ud835\udcde K` of norm at most `x` is `log log x + B_K + O(1 / log x)`, where `B_K = mertensConstant K`.","title":"Mertens' second theorem for a number field"},{"data_url":"scopes/tauceti/number-theory/part-02/l-series.json","id":"tauceti/number-theory/part-02/l-series","kind":"scope","parent":3409,"result":false,"route":"../roadmap/tauceti/number-theory/part-02/l-series/index.html","scope":"tauceti/number-theory/part-02/l-series","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.NumberTheory.LSeries` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"L Series"},{"id":"tauceti/number-theory/part-02/l-series/the-chebyshev-bound","kind":"node","lean":"TauCeti.LSeries.isBigO_sum_Icc_norm_id_of_boundary","parent":3456,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/l-series/index.html#the-chebyshev-bound","scope":"tauceti/number-theory/part-02/l-series","status":"fully_proved","summary":"Nonnegative coefficients whose Dirichlet series is summable on `Re s > 1` and has a boundary remainder `G = LSeries a - A / (s - 1)` continuous on `Re s \u2265 1` satisfy `\u2211_{1 \u2264 n \u2264 t} \u2016a n\u2016 = O(t)`.","title":"The Chebyshev bound"},{"id":"tauceti/number-theory/part-02/l-series/landaus-theorem","kind":"node","lean":"TauCeti.LSeries.landau, TauCeti.LSeries.landau_of_abscissaOfConv, TauCeti.LSeries.abscissaOfAbsConv_le_of_differentiableOn","parent":3456,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/l-series/index.html#landaus-theorem","scope":"tauceti/number-theory/part-02/l-series","status":"fully_proved","summary":"If `a` has nonnegative values and its abscissa of absolute convergence is the real number `\u03c3`, then `LSeries a` has no analytic continuation across `\u03c3`.","title":"Landau's theorem"},{"id":"tauceti/number-theory/part-02/l-series/meromorphic-form-of-landaus-theorem","kind":"node","lean":"TauCeti.LSeries.meromorphicOrderAt_lt_zero_of_eq_LSeries","parent":3456,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/l-series/index.html#meromorphic-form-of-landaus-theorem","scope":"tauceti/number-theory/part-02/l-series","status":"fully_proved","summary":"Let `F` be a meromorphic continuation of a Dirichlet series with nonnegative coefficients to a neighborhood of its finite, actual abscissa of absolute convergence. Then the meromorphic order of `F` at that abscissa is negative. In particular, the singularity forced by `landau` is a pole, not a removable singularity.","title":"Meromorphic form of Landau's theorem"},{"data_url":"scopes/tauceti/number-theory/part-02/local-field.json","id":"tauceti/number-theory/part-02/local-field","kind":"scope","parent":3409,"result":false,"route":"../roadmap/tauceti/number-theory/part-02/local-field/index.html","scope":"tauceti/number-theory/part-02/local-field","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.NumberTheory.LocalField` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Local Field"},{"id":"tauceti/number-theory/part-02/local-field/dedekinds-different-theorem-the-tame-case","kind":"node","lean":"TauCeti.differentExponent_eq_ramificationIndex_sub_one_iff, TauCeti.discriminantExponent_eq_inertiaDegree_mul_ramificationIndex_sub_one_iff","parent":3460,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/local-field/index.html#dedekinds-different-theorem-the-tame-case","scope":"tauceti/number-theory/part-02/local-field","status":"fully_proved","summary":"`d(L/K) = e(L/K) - 1` exactly when `L/K` is tamely ramified.","title":"Dedekind's different theorem, the tame case"},{"id":"tauceti/number-theory/part-02/local-field/the-finite-level-of-the-generator-theorem-nsw-pr","kind":"node","lean":"TauCeti.exists_generating_tuple_quotient","parent":3460,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/local-field/index.html#the-finite-level-of-the-generator-theorem-nsw-pr","scope":"tauceti/number-theory/part-02/local-field","status":"fully_proved","summary":"For a finite extension `K` of `\u211a_p`, every finite quotient of `G_K` through which the commutator subgroup of wild inertia dies is generated by `[K : \u211a_p] + 2` elements. The generators form a tuple, in which repetitions are allowed: the quotient may have fewer than `[K : \u211a_p] + 2` elements.","title":"The finite level of the generator theorem (NSW, proof of (7.4.1))"},{"id":"tauceti/number-theory/part-02/local-field/the-first-half-of-dedekinds-different-theorem","kind":"node","lean":"TauCeti.ramificationIndex_sub_one_le_differentExponent, TauCeti.inertiaDegree_mul_ramificationIndex_sub_one_le_discriminantExponent","parent":3460,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/local-field/index.html#the-first-half-of-dedekinds-different-theorem","scope":"tauceti/number-theory/part-02/local-field","status":"fully_proved","summary":"`e(L/K) - 1 \u2264 d(L/K)`.","title":"The first half of Dedekind's different theorem"},{"id":"tauceti/number-theory/part-02/local-field/the-gram-formula-for-the-kummer-cup-over-q2","kind":"node","lean":"TauCeti.kummerCup_padicTwo","parent":3460,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/local-field/index.html#the-gram-formula-for-the-kummer-cup-over-q2","scope":"tauceti/number-theory/part-02/local-field","status":"fully_proved","summary":"In square-class coordinates along `-1`, `2`, `5`, the cup is `x\u2080 y\u2080 + x\u2081 y\u2082 + x\u2082 y\u2081` times the nonzero class `(-1) \u222a (-1)`. The scalar lies in `ZMod 2`.","title":"The Gram formula for the Kummer cup over `\u211a\u2082`"},{"id":"tauceti/number-theory/part-02/local-field/hasse-arf-in-prime-degree","kind":"node","lean":"TauCeti.LocalFieldsRamification.UpperJump.exists_eq_intCast_of_finrank_prime, TauCeti.LocalFieldsRamification.hasseArf, TauCeti.LocalFieldsRamification.UpperJump.exists_eq_intCast_of_isPGroup, TauCeti.LocalFieldsRamification.UpperJump.exists_eq_intCast_of_exponent_prime","parent":3460,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/local-field/index.html#hasse-arf-in-prime-degree","scope":"tauceti/number-theory/part-02/local-field","status":"fully_proved","summary":"Every upper ramification break of a finite Galois extension of prime degree is an integer.","title":"Hasse--Arf in prime degree"},{"id":"tauceti/number-theory/part-02/local-field/hilberts-formula-truncated","kind":"node","lean":"TauCeti.sum_range_card_lowerRamificationGroup_sub_one_le_differentExponent, TauCeti.differentExponent_eq_finsum_lowerRamificationGroup, TauCeti.differentExponent_eq_sum_range_lowerRamificationGroup","parent":3460,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/local-field/index.html#hilberts-formula-truncated","scope":"tauceti/number-theory/part-02/local-field","status":"fully_proved","summary":"for a finite Galois extension `L/K` of nonarchimedean local fields, `\u2211_{i < m} (#G_i - 1) \u2264 d(L/K)` for every `m`.","title":"Hilbert's formula, truncated"},{"id":"tauceti/number-theory/part-02/local-field/the-local-square-theorem-bounds-the-defect-of-a","kind":"node","lean":"TauCeti.defectExponent_le_two_mul_natCastValuation","parent":3460,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/local-field/index.html#the-local-square-theorem-bounds-the-defect-of-a","scope":"tauceti/number-theory/part-02/local-field","status":"fully_proved","summary":"A unit of `\ud835\udcaa[K]` approximated by a square to order `2 v_K(2) + 1` is a square, so the defect exponent of a nonsquare unit is at most `2 v_K(2)`. In odd residue characteristic the bound is `0`.","title":"The local square theorem bounds the defect of a unit"},{"id":"tauceti/number-theory/part-02/local-field/serres-quotient-formula-for-the-lower-index","kind":"node","lean":"TauCeti.LocalFieldsRamification.ramificationIndex_mul_lowerIndex_eq_sum, TauCeti.LocalFieldsRamification.ramificationIndex_mul_lowerIndex_restrictNormal_eq_sum","parent":3460,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/local-field/index.html#serres-quotient-formula-for-the-lower-index","scope":"tauceti/number-theory/part-02/local-field","status":"fully_proved","summary":"For an intermediate field `L` of `M/K` normal over `K` and `\u03c3' : Gal(L/K)`, `e(M/L) \u00b7 i_{L/K}(\u03c3') = \u2211_{\u03c3|_L = \u03c3'} i_{M/K}(\u03c3)`, the sum running over the automorphisms of `M/K` restricting to `\u03c3'`.","title":"Serre's quotient formula for the lower index"},{"id":"tauceti/number-theory/part-02/local-field/the-upper-bound-of-dedekinds-different-theorem","kind":"node","lean":"TauCeti.differentExponent_le_ramificationIndex_sub_one_add_natCastValuation","parent":3460,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/local-field/index.html#the-upper-bound-of-dedekinds-different-theorem","scope":"tauceti/number-theory/part-02/local-field","status":"fully_proved","summary":"If the ramification index `e(L/K)` is nonzero in `L`, the different exponent of `L/K` is at most `e(L/K) \u2212 1 + v_L(e(L/K))`.","title":"The upper bound of Dedekind's different theorem"},{"data_url":"scopes/tauceti/number-theory/part-02/modular-forms.json","id":"tauceti/number-theory/part-02/modular-forms","kind":"scope","parent":3409,"result":false,"route":"../roadmap/tauceti/number-theory/part-02/modular-forms/index.html","scope":"tauceti/number-theory/part-02/modular-forms","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.NumberTheory.ModularForms` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Modular Forms"},{"id":"tauceti/number-theory/part-02/modular-forms/the-choie-zagier-criterion-on-k-mn-popa-zagier-3","kind":"node","lean":"TauCeti.TraceFormulaMatrixModule.one_sub_S_apply_mem_range_one_sub_T_iff","parent":3470,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/modular-forms/index.html#the-choie-zagier-criterion-on-k-mn-popa-zagier-3","scope":"tauceti/number-theory/part-02/modular-forms","status":"fully_proved","summary":"for `n \u2260 0` and `2`, `3` invertible in `k`, an element `x \u2208 k[\u2133\u2099]` lies in `(1 + S) k[\u2133\u2099] + (1 + U + U\u00b2) k[\u2133\u2099]`, with `U = T * S`, if and only if `(1 - S) x \u2208 (1 - T) k[\u2133\u2099]`, all products being left multiplications.","title":"The Choie\u2013Zagier criterion on `k[\u2133\u2099]` (Popa\u2013Zagier, \u00a73, Lemma 1)"},{"id":"tauceti/number-theory/part-02/modular-forms/classical-sturm-bound-for-finite-index-subgroups","kind":"node","lean":"ModularForm.sturm_bound_finiteIndex_SL2Z","parent":3470,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/modular-forms/index.html#classical-sturm-bound-for-finite-index-subgroups","scope":"tauceti/number-theory/part-02/modular-forms","status":"fully_proved","summary":"Formalized in Tau Ceti as `ModularForm.sturm_bound_finiteIndex_SL2Z`.","title":"Classical Sturm bound for finite-index subgroups of `SL(2, \u2124)`"},{"id":"tauceti/number-theory/part-02/modular-forms/dedekinds-formula-for-the-index-of-0-n","kind":"node","lean":"CongruenceSubgroup.Gamma0_index_mul_prod_primeFactors","parent":3470,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/modular-forms/index.html#dedekinds-formula-for-the-index-of-0-n","scope":"tauceti/number-theory/part-02/modular-forms","status":"fully_proved","summary":"`[SL\u2082(\u2124) : \u0393\u2080(N)] = N \u220f_{p \u2223 N} (1 + 1/p)`, with the denominators cleared.","title":"Dedekind's formula for the index of `\u0393\u2080(N)`"},{"id":"tauceti/number-theory/part-02/modular-forms/the-eichler-selberg-trace-formula-at-level-one","kind":"node","lean":"TauCeti.trace_heckeTCuspLevelOne_eq_neg_hurwitzClassNumber_sub_min","parent":3470,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/modular-forms/index.html#the-eichler-selberg-trace-formula-at-level-one","scope":"tauceti/number-theory/part-02/modular-forms","status":"fully_proved","summary":"For even `k = w + 2 \u2265 4` and `n \u2265 1`, the trace of `T\u2099` on the cusp forms of weight `k` on `SL(2, \u2124)` is","title":"The Eichler\u2013Selberg trace formula at level one"},{"id":"tauceti/number-theory/part-02/modular-forms/heckes-abscissa-bound-for-modular-forms","kind":"node","lean":"ModularForm.abscissaOfAbsConv_qExpansion_coeff_le","parent":3470,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/modular-forms/index.html#heckes-abscissa-bound-for-modular-forms","scope":"tauceti/number-theory/part-02/modular-forms","status":"fully_proved","summary":"for weight `k \u2265 0`, the Dirichlet series of the `q`-expansion coefficients converges absolutely for `Re s > k + 1` (from `a\u2099 = O(n\u1d4f)`).","title":"Hecke's abscissa bound for modular forms"},{"id":"tauceti/number-theory/part-02/modular-forms/the-lipschitz-formula-along-residue-classes","kind":"node","lean":"TauCeti.EisensteinSeries.qExpansion_identity_zmod","parent":3470,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/modular-forms/index.html#the-lipschitz-formula-along-residue-classes","scope":"tauceti/number-theory/part-02/modular-forms","status":"fully_proved","summary":"For a function `f` of residues modulo `v`, a weight `k \u2265 2` and `z` in the upper half-plane, `\u2211_{n \u2208 \u2124} f(n) (z + n)^(-k) = (-2\u03c0i)^k / ((k-1)! v^k) \u2211_{m \u2265 1} \ud835\udcd5f(-m) m^(k-1) e^(2\u03c0i m z / v)`, where `\ud835\udcd5f(-m) = \u2211_{r mod v} f(r) e^(2\u03c0i r m / v)` is the discrete Fourier transform of `f`. For `v = 1` this is `f 0` times Mathlib's `EisensteinSeries.qExpansion_identity_pnat`.","title":"The Lipschitz formula along residue classes"},{"id":"tauceti/number-theory/part-02/modular-forms/the-petersson-adjoint-formula-in-the-second-argu","kind":"node","lean":"HeckeRing.GL2.peterssonInnerCosets_heckeTCuspNat_right","parent":3470,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/modular-forms/index.html#the-petersson-adjoint-formula-in-the-second-argu","scope":"tauceti/number-theory/part-02/modular-forms","status":"fully_proved","summary":"For `n` coprime to `N`,","title":"The Petersson adjoint formula in the second argument"},{"id":"tauceti/number-theory/part-02/modular-forms/popa-zagiers-ambient-trace-identity-popa-zagier","kind":"node","lean":"TauCeti.TraceFormulaMatrixModule.PeriodRelation.trace_periodAction_eq","parent":3470,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/modular-forms/index.html#popa-zagiers-ambient-trace-identity-popa-zagier","scope":"tauceti/number-theory/part-02/modular-forms","status":"fully_proved","summary":"for positive even `w`, `k = w + 2`, `n \u2265 1`, and any `\u03be \u2208 \u2102[\u2133\u2099]` satisfying both the period relation (A) and the exchange relations (B), the trace of `\u03be` on the ambient space `V_w` of degree-`w` binary forms is `2 Tr(T\u2099 | S_k(SL(2, \u2124))) + \u03c3_{w+1}(n)`.","title":"Popa\u2013Zagier's ambient trace identity (Popa\u2013Zagier, \u00a72, Propositions 2 and 3)"},{"id":"tauceti/number-theory/part-02/modular-forms/popa-zagiers-criterion-for-1-t-rn","kind":"node","lean":"TauCeti.TraceFormulaMatrixModule.mem_range_one_sub_T_iff","parent":3470,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/modular-forms/index.html#popa-zagiers-criterion-for-1-t-rn","scope":"tauceti/number-theory/part-02/modular-forms","status":"fully_proved","summary":"An element of `k[\u2133\u2099]` lies in the range of left multiplication by `1 - T` if and only if its coefficients sum to zero along every orbit of `\u0393_\u221e = \u27e8T\u27e9`.","title":"Popa\u2013Zagier's criterion for `(1 - T)\u00b7\u211b\u2099`"},{"id":"tauceti/number-theory/part-02/modular-forms/strong-multiplicity-one-miyake-theorem-4-6-12-fi","kind":"node","lean":"HeckeRing.GL2.Newform.eq_of_forall_notMem_eigenvalue_eq","parent":3470,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/modular-forms/index.html#strong-multiplicity-one-miyake-theorem-4-6-12-fi","scope":"tauceti/number-theory/part-02/modular-forms","status":"fully_proved","summary":"two newforms of level `N` and weight `k` whose eigenvalues agree at every index coprime to `N` outside a finite set are equal. No assumption is made on the nebentypus.","title":"Strong multiplicity one (Miyake, Theorem 4.6.12, fixed level)"},{"id":"tauceti/number-theory/part-02/modular-forms/sturm-bound-for-subgroups-of-gl-2-r-of-finite-re","kind":"node","lean":"ModularForm.sturm_bound_finiteIndex","parent":3470,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/modular-forms/index.html#sturm-bound-for-subgroups-of-gl-2-r-of-finite-re","scope":"tauceti/number-theory/part-02/modular-forms","status":"fully_proved","summary":"A modular form of weight `k` whose `q`-expansion at the cusp `\u221e` has order strictly greater than `k \u00b7 [\ud835\udcae\u2112 : \ud835\udca2 \u2293 \ud835\udcae\u2112] / 12` is identically zero.","title":"Sturm bound for subgroups of `GL(2, \u211d)` of finite relative index in `SL(2, \u2124)` with discrete strict periods"},{"id":"tauceti/number-theory/part-02/modular-forms/webers-identity","kind":"node","lean":"TauCeti.ModularForm.siegelFunction_pow_four_add_add","parent":3470,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/modular-forms/index.html#webers-identity","scope":"tauceti/number-theory/part-02/modular-forms","status":"fully_proved","summary":"Weber's identity `g\u2080\u2074 + g\u2081\u2074 + g\u2082\u2074 = 0` for the level-two Siegel functions `g\u2080 = g_{(0, 1/2)}`, `g\u2081 = g_{(1/2, 0)}` and `g\u2082 = g_{(1/2, 1/2)}`. In Weber's notation it reads `\ud835\udd23\u2081\u2078 + \ud835\udd23\u2082\u2078 = \ud835\udd23\u2078`.","title":"Weber's identity"},{"data_url":"scopes/tauceti/number-theory/part-02/multiquadratic.json","id":"tauceti/number-theory/part-02/multiquadratic","kind":"scope","parent":3409,"result":false,"route":"../roadmap/tauceti/number-theory/part-02/multiquadratic/index.html","scope":"tauceti/number-theory/part-02/multiquadratic","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.NumberTheory.Multiquadratic` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Multiquadratic"},{"id":"tauceti/number-theory/part-02/multiquadratic/the-converse-of-the-multiquadratic-degree-theore","kind":"node","lean":"TauCeti.Multiquadratic.exists_root_adjoin_range_eq_top","parent":3483,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/multiquadratic/index.html#the-converse-of-the-multiquadratic-degree-theore","scope":"tauceti/number-theory/part-02/multiquadratic","status":"fully_proved","summary":"Let `2 \u2260 0` in `K` and let `L / K` be a finite Galois extension whose Galois group has exponent dividing two. Then `L = K(\u221ad\u2081, \u2026, \u221ad\u2099)` for radicands `d\u2081, \u2026, d\u2099 \u2208 K` that are square-class independent \u2014 no nonempty subset product is a square \u2014 and consequently `[L : K] = 2\u207f`.","title":"The converse of the multiquadratic degree theorem"},{"id":"tauceti/number-theory/part-02/multiquadratic/dirichlets-theorem-for-prime-discriminant-charac","kind":"node","lean":"TauCeti.Multiquadratic.exists_prime_gt_forall_primeDiscriminantCharFun_eq","parent":3483,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/multiquadratic/index.html#dirichlets-theorem-for-prime-discriminant-charac","scope":"tauceti/number-theory/part-02/multiquadratic","status":"fully_proved","summary":"Let `s` be a finite set of prime discriminants, at most one of them even, let `\u03b5` assign a sign to each, and let `N` be any bound. Then some odd prime `q > N` has `\u03c7_P(q) = \u03b5 P` for every `P \u2208 s`. In particular there are infinitely many such primes.","title":"Dirichlet's theorem for prime-discriminant characters"},{"id":"tauceti/number-theory/part-02/multiquadratic/the-multiquadratic-degree-theorem-restated-via-l","kind":"node","lean":"TauCeti.Multiquadratic.finrank_adjoin_range_of_linearIndependent","parent":3483,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/multiquadratic/index.html#the-multiquadratic-degree-theorem-restated-via-l","scope":"tauceti/number-theory/part-02/multiquadratic","status":"fully_proved","summary":"If the square classes of the radicands `d : \u03b9 \u2192 K\u02e3` are `ZMod 2`-linearly independent, then the field generated over `K` by their square roots `root i` has degree `2^|\u03b9|`. This is `finrank_adjoin_range` consuming the structural form of square-class independence.","title":"The multiquadratic degree theorem, restated via linear independence"},{"id":"tauceti/number-theory/part-02/multiquadratic/the-principal-genus-theorem","kind":"node","lean":"TauCeti.Multiquadratic.isSquare_iff_forall_genusCharFunNarrowClassGroupHom_eq_one","parent":3483,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/multiquadratic/index.html#the-principal-genus-theorem","scope":"tauceti/number-theory/part-02/multiquadratic","status":"fully_proved","summary":"Let `K = \u211a(\u221ad)` with `d` squarefree and let `\u220f P \u2208 s, P = fundamentalDiscriminant d` be the prime-discriminant factorization of its discriminant. A narrow ideal class of `K` is a square exactly when every genus character is trivial on it; that is, the principal genus is the group of squares.","title":"The principal genus theorem"},{"data_url":"scopes/tauceti/number-theory/part-02/number-field.json","id":"tauceti/number-theory/part-02/number-field","kind":"scope","parent":3409,"result":false,"route":"../roadmap/tauceti/number-theory/part-02/number-field/index.html","scope":"tauceti/number-theory/part-02/number-field","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.NumberTheory.NumberField` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Number Field"},{"id":"tauceti/number-theory/part-02/number-field/brills-theorem-parity-form","kind":"node","lean":"NumberField.InfinitePlace.even_nrComplexPlaces_iff_zero_lt_discr","parent":3488,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/number-field/index.html#brills-theorem-parity-form","scope":"tauceti/number-theory/part-02/number-field","status":"fully_proved","summary":"The number of complex places is even exactly when the discriminant is positive.","title":"Brill's theorem, parity form"},{"id":"tauceti/number-theory/part-02/number-field/dedekinds-criterion","kind":"node","lean":"TauCeti.NumberField.IntegralPrimitiveElement.not_dvd_index_iff, TauCeti.NumberField.IntegralPrimitiveElement.dvd_index_of_ne_one_of_dvd_map, TauCeti.NumberField.IntegralPrimitiveElement.not_dvd_index_of_forall_eq_one_or_not_dvd_map","parent":3488,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/number-field/index.html#dedekinds-criterion","scope":"tauceti/number-theory/part-02/number-field","status":"fully_proved","summary":"Let `\u03b8` be an integral primitive element of `K` with minimal polynomial `f = minpoly \u2124 \u03b8`, let `p` be a prime, and factor `f mod p = \u220f i, \u03c6 i ^ e i` into distinct monic irreducible polynomials `\u03c6 i` over `ZMod p`, each with multiplicity `e i > 0`. Choose lifts `\u03a6 i : \u2124[X]` of the `\u03c6 i` and let `H : \u2124[X]` satisfy `p H = f - \u220f i, \u03a6 i ^ e i` (which records the factorisation, by `map_minpoly_eq_prod_of_C_mul_eq`). Then\u2026","title":"Dedekind's criterion"},{"id":"tauceti/number-theory/part-02/number-field/dirichlets-unit-lattice-theorem-with-all-places","kind":"node","lean":"TauCeti.isZLattice_fullUnitLattice","parent":3488,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/number-field/index.html#dirichlets-unit-lattice-theorem-with-all-places","scope":"tauceti/number-theory/part-02/number-field","status":"fully_proved","summary":"The logarithmic image of the units is a full integral lattice in the sum-zero hyperplane.","title":"Dirichlet's unit lattice theorem, with all places retained"},{"id":"tauceti/number-theory/part-02/number-field/dirichlets-unit-theorem-numberfield-fg-units-int","kind":"node","lean":"NumberField.fg_units_integralClosure, NumberField.unitsMulEquivTorsionProdMultiplicative","parent":3488,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/number-field/index.html#dirichlets-unit-theorem-numberfield-fg-units-int","scope":"tauceti/number-theory/part-02/number-field","status":"fully_proved","summary":"Dirichlet's unit theorem (finite generation) for the integral closure of `\ud835\udcde K` in a finite extension `L` of the number field `K`: its unit group is finitely generated.","title":"Dirichlet's unit theorem (`NumberField.fg_units_integralClosure`)"},{"id":"tauceti/number-theory/part-02/number-field/evertses-lemma-evertse-bombieri-gubler-lemma-7-5","kind":"node","lean":"NumberField.exists_evertse, NumberField.exists_evertse_unweighted","parent":3488,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/number-field/index.html#evertses-lemma-evertse-bombieri-gubler-lemma-7-5","scope":"tauceti/number-theory/part-02/number-field","status":"fully_proved","summary":"There is a constant `C`, depending only on `K` and `m`, with the following property. Let `S\u2080` be a finite set of finite places, `x` a basis of a `K`-vector space `V` indexed by `Fin m`, and at every infinite place and every place of `S\u2080` let the forms `L v k` be linearly independent, the weights `\u03bd v k` positive and `\u03bc v` nondecreasing, with `|L v k (x j)|_v \u2264 \u03bd v k * \u03bc v j`. Then there are `\u03be j l \u2208 K`, integral\u2026","title":"Evertse's lemma (Evertse; Bombieri\u2013Gubler, Lemma 7.5.29)"},{"id":"tauceti/number-theory/part-02/number-field/the-galois-action-on-the-semi-local-decompositio","kind":"node","lean":"TauCeti.GlobalNumberFields.infiniteAdeleSemilocalEquiv_infiniteAdeleGaloisAction","parent":3488,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/number-field/index.html#the-galois-action-on-the-semi-local-decompositio","scope":"tauceti/number-theory/part-02/number-field","status":"fully_proved","summary":"Under `infiniteAdeleSemilocalEquiv`, the Galois action of `\u03c3 \u2208 Aut(L/K)` on infinite adeles applies the base change `id \u2297 \u03c3` to each semi-local factor `K_v \u2297[K] L`.","title":"The Galois action on the semi-local decomposition"},{"id":"tauceti/number-theory/part-02/number-field/hadamards-bound-at-q-2","kind":"node","lean":"TauCeti.NumberField.Sqrt2.regulator_le_four_mul_absLogHeight\u2081_one_add, TauCeti.NumberField.Units.regulator_le_prod_sum_abs_logEmbedding, TauCeti.NumberField.Units.regOfFamily_le_prod_sum_abs_logEmbedding, TauCeti.NumberField.Units.regOfFamily_le_prod_sum_mult_mul_abs_log, TauCeti.NumberField.Units.regulator_le_prod_sum_mult_mul_abs_log","parent":3488,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/number-field/index.html#hadamards-bound-at-q-2","scope":"tauceti/number-theory/part-02/number-field","status":"fully_proved","summary":"The regulator is at most `(2d) ^ r = 4` times the absolute logarithmic height of the fundamental unit; here `log (1 + \u221a2) \u2264 4 \u00b7 log (1 + \u221a2) / 2`.","title":"Hadamard's bound at `\u211a(\u221a2)`"},{"id":"tauceti/number-theory/part-02/number-field/hermites-theorem-finitely-many-extensions-of-bou","kind":"node","lean":"TauCeti.NumberField.finite_of_finrank_le_of_ramifiedSupport_subset","parent":3488,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/number-field/index.html#hermites-theorem-finitely-many-extensions-of-bou","scope":"tauceti/number-theory/part-02/number-field","status":"fully_proved","summary":"Inside a field `A` containing the number field `K`, only finitely many finite extensions `L / K` have degree at most `d` and are ramified only at places in the finite set `S`.","title":"Hermite's theorem: finitely many extensions of bounded degree are unramified outside a finite set"},{"id":"tauceti/number-theory/part-02/number-field/hilberts-different-formula-checked-in-q-i","kind":"node","lean":"TauCeti.NumberField.GaussianRationals.finsum_card_ramificationGroup_sub_one_eq_two","parent":3488,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/number-field/index.html#hilberts-different-formula-checked-in-q-i","scope":"tauceti/number-theory/part-02/number-field","status":"fully_proved","summary":"the sum `\u03a3_{i \u2265 0} (#G_i \u2212 1)` over the ramification groups of the prime above `2` is the different exponent `v_\ud835\udd2d(\ud835\udd21) = 2`, as the general `multiplicity_differentIdeal_eq_finsum_card_ramificationGroup_sub_one` predicts.","title":"Hilbert's different formula, checked in `\u211a(i)`"},{"id":"tauceti/number-theory/part-02/number-field/hilberts-theorem-90-for-quadratic-conjugation","kind":"node","lean":"NumberField.exists_ne_zero_mul_eq_mul_ringOfIntegersQuadraticConj","parent":3488,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/number-field/index.html#hilberts-theorem-90-for-quadratic-conjugation","scope":"tauceti/number-theory/part-02/number-field","status":"fully_proved","summary":"Let `\u03c3` be the quadratic conjugation of a quadratic number field and let `x y : \ud835\udcde K` with `x \u2260 0` have equal norms, `x \u03c3x = y \u03c3y`. Then there is a nonzero `\u03b5 : \ud835\udcde K` with `x \u03b5 = y \u03c3\u03b5`; equivalently `y / x = \u03b5 / \u03c3\u03b5` is a \"coboundary\". The construction is explicit: `\u03b5 = \u03c3x (x + y)` works unless `x + y = 0`, and then `\u03b5 = \u03c3x \u03b8 x` does.","title":"Hilbert's Theorem 90 for quadratic conjugation"},{"id":"tauceti/number-theory/part-02/number-field/the-kummer-dedekind-counting-bound","kind":"node","lean":"TauCeti.NumberField.ncard_primesOverOfInertiaDeg_le","parent":3488,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/number-field/index.html#the-kummer-dedekind-counting-bound","scope":"tauceti/number-theory/part-02/number-field","status":"fully_proved","summary":"For a `\u03b8` whose conductor exponent is prime to `p`, the primes of `\ud835\udcde K` above `p` with residue degree `d` are no more numerous than the degree-`d` monic irreducible factors of `minpoly \u2124 \u03b8` modulo `p`.","title":"The Kummer\u2013Dedekind counting bound"},{"id":"tauceti/number-theory/part-02/number-field/liouvilles-inequality-at-a-finite-place","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.abs_log_valuation_mul_log_absNorm_le_logHeight\u2081","parent":3488,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/number-field/index.html#liouvilles-inequality-at-a-finite-place","scope":"tauceti/number-theory/part-02/number-field","status":"fully_proved","summary":"The absolute value of the order `-WithZero.log (v.valuation K x)` of an element `x` of a number field at a finite place `v`, weighted by the logarithm of the norm of `v`, is at most the logarithmic height of `x`. Both sides vanish at `x = 0`.","title":"Liouville's inequality at a finite place"},{"id":"tauceti/number-theory/part-02/number-field/siegels-lemma-over-a-number-field-with-rational","kind":"node","lean":"NumberField.exists_ne_zero_int_vec_abs_le_of_house_le","parent":3488,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/number-field/index.html#siegels-lemma-over-a-number-field-with-rational","scope":"tauceti/number-theory/part-02/number-field","status":"fully_proved","summary":"Let `K` be a number field of degree `d` and `a` a matrix over `\ud835\udcde K` with `M` rows and `N` columns, whose entries have house at most `A \u2265 1`. If `d * M < N`, there is a nonzero `x \u2208 \u2124^N` with `a x = 0` and `|x\u2c7c| \u2264 (3 N A) ^ (d M / (N - d M))` for every `j`.","title":"Siegel's lemma over a number field, with rational integer unknowns (Evertse, Theorem 3.20)"},{"data_url":"scopes/tauceti/number-theory/part-02/padics.json","id":"tauceti/number-theory/part-02/padics","kind":"scope","parent":3409,"result":false,"route":"../roadmap/tauceti/number-theory/part-02/padics/index.html","scope":"tauceti/number-theory/part-02/padics","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.NumberTheory.Padics` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Padics"},{"id":"tauceti/number-theory/part-02/padics/labutes-classification-of-the-closed-subgroups-o","kind":"node","lean":"TauCeti.closedSubgroup_units_two_classification","parent":3502,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/padics/index.html#labutes-classification-of-the-closed-subgroups-o","scope":"tauceti/number-theory/part-02/padics","status":"fully_proved","summary":"Every nontrivial closed subgroup of `\u2124_2\u02e3` is a principal unit group `U^(f)` with `f \u2265 2`, a subgroup `V^(f) = {\u00b11} \u00d7 U^(f)` with `f \u2265 2`, the subgroup `{\u00b11}`, or a twisted subgroup `U^[f]` with `f \u2265 2`, generated by `-1 + 2 ^ f`. The four families are pairwise distinct and faithfully indexed by `f`: `unitsPrincipal_inj`, `unitsPlusMinus_inj`, `topologicalClosure_zpowers_two_eq_iff` and the `ne` lemmas above.","title":"Labute's classification of the closed subgroups of `\u2124_2\u02e3`"},{"id":"tauceti/number-theory/part-02/padics/legendres-formula-for-pk","kind":"node","lean":"TauCeti.sub_one_mul_padicValNat_factorial_pow, TauCeti.padicValNat_factorial_eq_add_sum","parent":3502,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/padics/index.html#legendres-formula-for-pk","scope":"tauceti/number-theory/part-02/padics","status":"fully_proved","summary":"Since `p\u1d4f` has base-`p` digit sum one, `(p - 1) v_p((p\u1d4f)!) = p\u1d4f - 1`.","title":"Legendre's formula for `(p\u1d4f)!`"},{"id":"tauceti/number-theory/part-02/padics/mahlers-theorem-scalar-linear-form","kind":"node","lean":"PadicInt.mahlerLinearIsometryEquiv","parent":3502,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/padics/index.html#mahlers-theorem-scalar-linear-form","scope":"tauceti/number-theory/part-02/padics","status":"fully_proved","summary":"Mathlib's Mahler isometry `mahlerEquiv E : C(\u2124_[p], E) \u2243\u2097\u1d62[\u2124_[p]] C\u2080(\u2115, E)` is linear over any semiring `S` acting on the coefficient module `E`.","title":"Mahler's theorem, scalar-linear form"},{"id":"tauceti/number-theory/part-02/padics/the-schwarz-lemma-on-a-grid-of-p-adic-integers","kind":"node","lean":"PowerSeries.IsRestricted.norm_evalAt_le_gaussNorm_mul_norm_prod, PowerSeries.IsRestricted.norm_evalAt_le_gaussNorm_mul_norm_factorial, PowerSeries.IsRestricted.norm_evalAt_le_gaussNorm_mul_norm_factorial_pow","parent":3502,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/padics/index.html#the-schwarz-lemma-on-a-grid-of-p-adic-integers","scope":"tauceti/number-theory/part-02/padics","status":"fully_proved","summary":"Let `K` be a complete normed field extension of `\u211a_p` and `f` a power series over `K` restricted at a radius `c \u2265 1`. If the Hasse derivatives `(D\u207d\u1d57\u207e f)(b)` vanish for `t < T` at every point `b` of a finite set `s` of `p`-adic integers, then at every `p`-adic integer `x`","title":"The Schwarz lemma on a grid of `p`-adic integers"},{"id":"tauceti/number-theory/part-02/padics/van-der-puts-theorem","kind":"node","lean":"PadicInt.hasSum_vanDerPut, PadicInt.vanDerPutEquiv","parent":3502,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/padics/index.html#van-der-puts-theorem","scope":"tauceti/number-theory/part-02/padics","status":"fully_proved","summary":"a continuous function on `\u2124_[p]` with values in an ultrametric normed group is the uniformly convergent sum of its van der Put terms, without assuming completeness.","title":"Van der Put's theorem"},{"data_url":"scopes/tauceti/number-theory/part-02/transcendental.json","id":"tauceti/number-theory/part-02/transcendental","kind":"scope","parent":3409,"result":false,"route":"../roadmap/tauceti/number-theory/part-02/transcendental/index.html","scope":"tauceti/number-theory/part-02/transcendental","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.NumberTheory.Transcendental` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Transcendental"},{"id":"tauceti/number-theory/part-02/transcendental/bakers-denominator-bound","kind":"node","lean":"TauCeti.pow_mul_eval_hasseDeriv_multichoose_X_add_one_pow_mem_range_intCast, TauCeti.lcm_pow_mul_eval_hasseDeriv_multichoose_X_add_one_pow_mem_range_intCast, TauCeti.pow_mul_eval_hasseDeriv_map_multichoose_X_add_one_comp_C_mul_X_mem_range_intCast","parent":3508,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/transcendental/index.html#bakers-denominator-bound","scope":"tauceti/number-theory/part-02/transcendental","status":"fully_proved","summary":"If each of `1, \u2026, k` divides the integer `L`, then for every integer `x` and all `l, m`,","title":"Baker's denominator bound"},{"id":"tauceti/number-theory/part-02/transcendental/the-gelfond-schneider-theorem-for-real-numbers-e","kind":"node","lean":"TauCeti.transcendental_rpow","parent":3508,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/transcendental/index.html#the-gelfond-schneider-theorem-for-real-numbers-e","scope":"tauceti/number-theory/part-02/transcendental","status":"fully_proved","summary":"If `\u03b1 > 0` and `\u03b2` are real algebraic numbers with `\u03b1 \u2260 1` and `\u03b2` irrational, then `\u03b1 ^ \u03b2` is transcendental.","title":"The Gelfond\u2013Schneider theorem for real numbers (Evertse, Theorem 4.21)"},{"id":"tauceti/number-theory/part-02/transcendental/liouvilles-lower-bound-for-gelfonds-values-karat","kind":"node","lean":"TauCeti.exists_one_le_pow_mul_norm_gelfond_value","parent":3508,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/transcendental/index.html#liouvilles-lower-bound-for-gelfonds-values-karat","scope":"tauceti/number-theory/part-02/transcendental","status":"fully_proved","summary":"Let `K` be a number field of degree `h`, `\u03b1, \u03b2, \u03b3 \u2208 K`, `m \u2208 \u2115` and `c \u2208 \u211d`. There is `c\u2082 \u2265 1` with the following property. Let `q ^ 2 = 2 m n`, `r \u2265 n` and `t\u2080 \u2264 m`, and let `\u03b7_{ab} \u2208 \ud835\udcde K`, `a, b < q`, have house at most `c ^ n n ^ ((n + 1) / 2)`, as for Gelfond's auxiliary function. If `\u03c1 = \u2211_{a, b} \u03b7_{ab} ((a + 1) + (b + 1) \u03b2) ^ r \u03b1 ^ ((a + 1) t\u2080) \u03b3 ^ ((b + 1) t\u2080)` is nonzero, then\u2026","title":"Liouville's lower bound for Gelfond's values (Karatarakis\u2013Wiedijk, \u00a73)"},{"id":"tauceti/number-theory/part-02/transcendental/schwarzs-lemma-for-schneiders-auxiliary-function","kind":"node","lean":"TauCeti.norm_schneider_auxiliary_le","parent":3508,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/transcendental/index.html#schwarzs-lemma-for-schneiders-auxiliary-function","scope":"tauceti/number-theory/part-02/transcendental","status":"fully_proved","summary":"Let `\u03b2 \u2208 \u2102` be irrational and let `F(z) = \u2211_{i < D\u2081, j < D\u2082} a_{ij} z ^ i exp (j \u03bb z)` vanish at `u + v \u03b2` for `1 \u2264 u, v \u2264 L`. If `\u2016a_{ij}\u2016 \u2264 A`, `L (1 + \u2016\u03b2\u2016) \u2264 R`, `3 R \u2264 T` and `1 \u2264 T`, then on the disc `\u2016z\u2016 \u2264 R` `\u2016F z\u2016 \u2264 D\u2081 D\u2082 A T ^ D\u2081 exp (D\u2082 \u2016\u03bb\u2016 T) (3 R / T) ^ (L ^ 2)`.","title":"Schwarz's lemma for Schneider's auxiliary function"},{"data_url":"scopes/tauceti/number-theory/part-02/other-results.json","id":"tauceti/number-theory/part-02/other-results","kind":"scope","parent":3409,"result":false,"route":"../roadmap/tauceti/number-theory/part-02/other-results/index.html","scope":"tauceti/number-theory/part-02/other-results","status":"fully_proved","summary":"Results named in the rest of Tau Ceti's `TauCeti.NumberTheory` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Other results"},{"id":"tauceti/number-theory/part-02/other-results/chos-exponent-identity","kind":"node","lean":"TauCeti.DyadicJordanProfile.nQ_eq_nM_add_densityExponent","parent":3513,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/other-results/index.html#chos-exponent-identity","scope":"tauceti/number-theory/part-02/other-results","status":"fully_proved","summary":"The two deformation exponents differ by the density exponent: `N_Q = N_M + N`.","title":"Cho's exponent identity"},{"id":"tauceti/number-theory/part-02/other-results/dedekinds-criterion-for-a-quadratic-order","kind":"node","lean":"QuadraticAlgebra.span_natCast_omega_sub_mul_span_natCast_omega_sub_eq_iff","parent":3513,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/other-results/index.html#dedekinds-criterion-for-a-quadratic-order","scope":"tauceti/number-theory/part-02/other-results","status":"fully_proved","summary":"Let `r` be a root of `X\u00b2 - b X - a` modulo `p`, so that `b - r` is the second root. Then `p` factors as `(p) = (p, \u03c9 - r) (p, \u03c9 - (b - r))` in `\u2124[\u03c9]` if and only if `p` does not divide both the derivative `2r - b` and the quotient `(r\u00b2 - b r - a)/p`, that is, unless `p \u2223 2r - b` and `p\u00b2 \u2223 r\u00b2 - b r - a`.","title":"Dedekind's criterion for a quadratic order"},{"id":"tauceti/number-theory/part-02/other-results/distinct-slopes-splitting-theorem","kind":"node","lean":"Polynomial.splits_of_nodup_newtonSlopes","parent":3513,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/other-results/index.html#distinct-slopes-splitting-theorem","scope":"tauceti/number-theory/part-02/other-results","status":"fully_proved","summary":"Over a complete ultrametric normed field, any polynomial whose finite Newton slopes have multiplicity one splits into linear factors over the base field.","title":"Distinct-slopes splitting theorem"},{"id":"tauceti/number-theory/part-02/other-results/the-exponent-in-siegels-lemma-is-optimal","kind":"node","lean":"TauCeti.Int.Matrix.not_forall_exists_ne_zero_int_vec_norm_le","parent":3513,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/other-results/index.html#the-exponent-in-siegels-lemma-is-optimal","scope":"tauceti/number-theory/part-02/other-results","status":"fully_proved","summary":"For an `m \u00d7 (m + 1)` integer system, Siegel's lemma `Int.Matrix.exists_ne_zero_int_vec_norm_le` gives a nonzero integer solution of sup norm at most `((m + 1) \u00b7 max 1 \u2016A\u2016) ^ m`. For `0 < m` no exponent `e < m` works, with any constant `C`: the system `b x\u1d62 = x\u1d62\u208a\u2081` with `b` large has no nonzero integer solution of sup norm at most `C \u00b7 max 1 \u2016A\u2016 ^ e`.","title":"The exponent in Siegel's lemma is optimal"},{"id":"tauceti/number-theory/part-02/other-results/markoffs-theorem","kind":"node","lean":"TauCeti.exists_foldr_markoffVieta_eq, TauCeti.mem_positiveMarkoffTriples_iff_exists_foldr","parent":3513,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/other-results/index.html#markoffs-theorem","scope":"tauceti/number-theory/part-02/other-results","status":"fully_proved","summary":"Every positive Markoff triple `x` is obtained from `(1, 1, 1)` by a finite word in the Vieta moves: `x = R_{i\u2081} (R_{i\u2082} \u22ef (R_{i_k} (1, 1, 1)))`. No coordinate permutation is needed.","title":"Markoff's theorem"},{"id":"tauceti/number-theory/part-02/other-results/minkowskis-linear-forms-theorem","kind":"node","lean":"Matrix.exists_ne_zero_int_abs_sum_mul_le","parent":3513,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/other-results/index.html#minkowskis-linear-forms-theorem","scope":"tauceti/number-theory/part-02/other-results","status":"fully_proved","summary":"Let `A` be an invertible real square matrix indexed by a nonempty finite type `\u03b9`, and let `a i \u2265 0` be bounds whose product is at least `|det A|`. Then some nonzero integer vector `z` satisfies `|\u2211 j, A i j * z j| \u2264 a i` for every `i`.","title":"Minkowski's linear forms theorem"},{"id":"tauceti/number-theory/part-02/other-results/newton-polygon-irreducibility-criterion","kind":"node","lean":"Polynomial.irreducible_of_forall_mem_newtonSlopes_eq, Polynomial.irreducible_of_forall_mem_newtonSlopes_eq_div","parent":3513,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/other-results/index.html#newton-polygon-irreducibility-criterion","scope":"tauceti/number-theory/part-02/other-results","status":"fully_proved","summary":"If the valuation takes integer values on nonzero elements, a polynomial `f` with nonzero constant coefficient whose finite slopes all equal a rational number `q`, and whose degree is the denominator of `q`, is irreducible.","title":"Newton polygon irreducibility criterion"},{"id":"tauceti/number-theory/part-02/other-results/sylvesters-law-of-inertia-at-a-real-place","kind":"node","lean":"QuadraticForm.equivalent_atRealPlace_iff_realSignature_eq","parent":3513,"result":true,"route":"../roadmap/tauceti/number-theory/part-02/other-results/index.html#sylvesters-law-of-inertia-at-a-real-place","scope":"tauceti/number-theory/part-02/other-results","status":"fully_proved","summary":"Two regular quadratic forms become isometric at a real place exactly when their signatures there agree.","title":"Sylvester's law of inertia at a real place"},{"data_url":"scopes/tauceti/probability.json","id":"tauceti/probability","kind":"scope","parent":1095,"result":false,"route":"../roadmap/tauceti/probability/index.html","scope":"tauceti/probability","status":"fully_proved","summary":"Results in Tau Ceti's `TauCeti.Probability` modules, and the roadmaps whose articles are mostly there: by roadmap, then by area.","title":"Probability"},{"data_url":"scopes/tauceti/probability/exchangeability-and-de-finetti.json","id":"tauceti/probability/exchangeability-and-de-finetti","kind":"scope","parent":3522,"result":false,"route":"../roadmap/tauceti/probability/exchangeability-and-de-finetti/index.html","scope":"tauceti/probability/exchangeability-and-de-finetti","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Exchangeability and de Finetti"},{"id":"tauceti/probability/exchangeability-and-de-finetti/jointly-exchangeable-arrays-are-exactly-the-aldo","kind":"node","lean":"TauCeti.Probability.jointlyExchangeable_iff_exists_map_jointArray_eq","parent":3523,"result":true,"route":"../roadmap/tauceti/probability/exchangeability-and-de-finetti/index.html#jointly-exchangeable-arrays-are-exactly-the-aldo","scope":"tauceti/probability/exchangeability-and-de-finetti","status":"fully_proved","summary":"An array process with a.e.-measurable entries in a standard Borel space, on a probability space, is jointly exchangeable if and only if its law is the law of a measurable joint Aldous--Hoover coding `X i j = F(U, U_i, U_j, U_{i,j})` under independent uniform noise.","title":"Jointly exchangeable arrays are exactly the Aldous--Hoover codings in law"},{"id":"tauceti/probability/exchangeability-and-de-finetti/the-diaconis-freedman-theorem","kind":"node","lean":"TauCeti.Probability.markovExchangeable_iff_mixedMarkovChain, TauCeti.Probability.MarkovExchangeable.mixedMarkovChain, TauCeti.Probability.MarkovExchangeable.mixedMarkovChain_of_ae_initial_eq","parent":3523,"result":true,"route":"../roadmap/tauceti/probability/exchangeability-and-de-finetti/index.html#the-diaconis-freedman-theorem","scope":"tauceti/probability/exchangeability-and-de-finetti","status":"fully_proved","summary":"A recurrent process is Markov exchangeable if and only if it is a mixture of Markov chains.","title":"The Diaconis--Freedman theorem"},{"id":"tauceti/probability/exchangeability-and-de-finetti/the-de-finetti-ryll-nardzewski-equivalence","kind":"node","lean":"TauCeti.Probability.deFinetti_RyllNardzewski_equivalence, TauCeti.Probability.contractable_iff_exchangeable_and_mixedIID","parent":3523,"result":true,"route":"../roadmap/tauceti/probability/exchangeability-and-de-finetti/index.html#the-de-finetti-ryll-nardzewski-equivalence","scope":"tauceti/probability/exchangeability-and-de-finetti","status":"fully_proved","summary":"A process with a.e. measurable coordinates, valued in a nonempty standard Borel space, is contractable iff it is both exchangeable and conditionally i.i.d. Derived from the two-way `contractable_iff_conditionallyIID`, with the exchangeability conjunct supplied by `ConditionallyIID.exchangeable` (the conjunct is redundant given conditional i.i.d.-ness).","title":"The de Finetti\u2013Ryll-Nardzewski equivalence"},{"id":"tauceti/probability/exchangeability-and-de-finetti/finite-de-finetti-theorem-total-variation-form","kind":"node","lean":"TauCeti.Probability.ExchangeableAt.etvDist_prefixLaw_sampleWithReplacement_le, TauCeti.Probability.ExchangeableAt.finiteDeFinetti","parent":3523,"result":true,"route":"../roadmap/tauceti/probability/exchangeability-and-de-finetti/index.html#finite-de-finetti-theorem-total-variation-form","scope":"tauceti/probability/exchangeability-and-de-finetti","status":"fully_proved","summary":"If the first `n` coordinates of a process are exchangeable and `m \u2264 n`, then the total variation distance between the `m`-prefix law and the with-replacement sample of the `n`-prefix law (for `n > 0`, the empirical-product mixture) is at most `2 * choose m 2 / n`.","title":"Finite de Finetti theorem, total-variation form"},{"id":"tauceti/probability/exchangeability-and-de-finetti/hewitt-savage-in-ergodic-form","kind":"node","lean":"TauCeti.Probability.ergodicSMul_infinitePi_const","parent":3523,"result":true,"route":"../roadmap/tauceti/probability/exchangeability-and-de-finetti/index.html#hewitt-savage-in-ergodic-form","scope":"tauceti/probability/exchangeability-and-de-finetti","status":"fully_proved","summary":"The finitely supported permutations of the time index act ergodically on an i.i.d. product law `P^{\u2297\u2115}`.","title":"Hewitt\u2013Savage in ergodic form"},{"id":"tauceti/probability/exchangeability-and-de-finetti/conditional-independence-as-a-functional-represe","kind":"node","lean":"ProbabilityTheory.CondIndepFun.exists_independent_coding","parent":3523,"result":true,"route":"../roadmap/tauceti/probability/exchangeability-and-de-finetti/index.html#conditional-independence-as-a-functional-represe","scope":"tauceti/probability/exchangeability-and-de-finetti","status":"fully_proved","summary":"If `X` and `Y` are conditionally independent given `Z`, then their joint law with `Z` is obtained by keeping `Z` and applying two measurable coding functions to separate uniform coordinates. When `\u03bc` is a probability measure, the noises are independent both of one another and of the original sample carrying `Z`.","title":"Conditional independence as a functional representation"},{"id":"tauceti/probability/exchangeability-and-de-finetti/measuretheory-measure-definettibarycenter","kind":"node","lean":"MeasureTheory.Measure.deFinettiBarycenter","parent":3523,"result":true,"route":"../roadmap/tauceti/probability/exchangeability-and-de-finetti/index.html#measuretheory-measure-definettibarycenter","scope":"tauceti/probability/exchangeability-and-de-finetti","status":"fully_proved","summary":"The de Finetti barycenter of a mixing law `\u03c0` on `ProbabilityMeasure \u03b1`: the measure `\u222b P^{\u2297\u2115} d\u03c0(P)` on `\u2115 \u2192 \u03b1` mixing the countable powers against `\u03c0`. For `\u03c0` a probability measure this is the law of a sequence drawn i.i.d. from a `\u03c0`-random probability measure.","title":"MeasureTheory.Measure.deFinettiBarycenter"},{"id":"tauceti/probability/exchangeability-and-de-finetti/the-de-finetti-correspondence","kind":"node","lean":"TauCeti.Probability.deFinettiEquiv","parent":3523,"result":true,"route":"../roadmap/tauceti/probability/exchangeability-and-de-finetti/index.html#the-de-finetti-correspondence","scope":"tauceti/probability/exchangeability-and-de-finetti","status":"fully_proved","summary":"Over a standard Borel state space, the de Finetti barycenter is a bijection from mixing laws \u2014 probability measures on `ProbabilityMeasure \u03b1` \u2014 onto exchangeable probability measures on `\u2115 \u2192 \u03b1`.","title":"The de Finetti correspondence"},{"id":"tauceti/probability/exchangeability-and-de-finetti/the-path-law-of-a-recurrent-process-started-at-a","kind":"node","lean":"TauCeti.Probability.Recurrent.pathLaw_eq_map_pathOfExcursions","parent":3523,"result":true,"route":"../roadmap/tauceti/probability/exchangeability-and-de-finetti/index.html#the-path-law-of-a-recurrent-process-started-at-a","scope":"tauceti/probability/exchangeability-and-de-finetti","status":"fully_proved","summary":"Only the base state's singleton needs to be measurable.","title":"The path law of a recurrent process started at `a\u2080` is the image of its excursion law"},{"id":"tauceti/probability/exchangeability-and-de-finetti/de-finettis-theorem","kind":"node","lean":"TauCeti.Probability.deFinetti, TauCeti.Probability.deFinetti_equivalence, TauCeti.Probability.conditionallyIID_of_exchangeableFamily, TauCeti.Probability.deFinetti_tendsto_empiricalMeasure_apply, TauCeti.Probability.deFinetti_empiricalMeasure, TauCeti.Probability.deFinetti_viaL2, TauCeti.Probability.deFinetti_viaKoopman, TauCeti.Probability.conditionallyIID_of_exchangeable, TauCeti.Probability.mixedIIDWith_of_contractable, TauCeti.Probability.exchangeable_iff_mixedIID, TauCeti.Probability.mixedIID_of_contractable, TauCeti.Probability.mixedIID_of_exchangeable, TauCeti.Probability.deFinetti_mixture","parent":3523,"result":true,"route":"../roadmap/tauceti/probability/exchangeability-and-de-finetti/index.html#de-finettis-theorem","scope":"tauceti/probability/exchangeability-and-de-finetti","status":"fully_proved","summary":"An exchangeable process valued in a nonempty standard Borel space is conditionally i.i.d. This is the conventional theorem-name handle for `conditionallyIID_of_exchangeable`.","title":"De Finetti's theorem"},{"id":"tauceti/probability/exchangeability-and-de-finetti/de-finetti-ryll-nardzewski-two-way-mixture-form","kind":"node","lean":"TauCeti.Probability.contractable_iff_mixedIID, TauCeti.Probability.contractable_iff_conditionallyIID","parent":3523,"result":true,"route":"../roadmap/tauceti/probability/exchangeability-and-de-finetti/index.html#de-finetti-ryll-nardzewski-two-way-mixture-form","scope":"tauceti/probability/exchangeability-and-de-finetti","status":"fully_proved","summary":"A process with a.e. measurable coordinates, valued in a nonempty standard Borel space, under a finite measure, is contractable iff it is mixed i.i.d. Forward: the reverse-martingale de Finetti chain (`mixedIID_of_contractable`), run on a measurable version; converse: `MixedIID.contractable`, which needs no side hypotheses.","title":"De Finetti\u2013Ryll-Nardzewski, two-way mixture form"},{"id":"tauceti/probability/exchangeability-and-de-finetti/exact-finite-de-finetti-bound-exchangeable-law-t","kind":"node","lean":"TauCeti.Probability.ExchangeableAt.prefixLaw_le_sampleWithReplacement_add_exact_collision, TauCeti.Probability.ExchangeableAt.sampleWithReplacement_le_prefixLaw_add_exact_collision","parent":3523,"result":true,"route":"../roadmap/tauceti/probability/exchangeability-and-de-finetti/index.html#exact-finite-de-finetti-bound-exchangeable-law-t","scope":"tauceti/probability/exchangeability-and-de-finetti","status":"fully_proved","summary":"Under finite exchangeability and `m \u2264 n`, the prefix law exceeds its with-replacement sample on a measurable event by at most the exact index-collision error.","title":"Exact finite de Finetti bound, exchangeable law to empirical mixture"},{"id":"tauceti/probability/exchangeability-and-de-finetti/quantitative-finite-de-finetti-bound-exchangeabl","kind":"node","lean":"TauCeti.Probability.ExchangeableAt.prefixLaw_le_sampleWithReplacement_add, TauCeti.Probability.ExchangeableAt.sampleWithReplacement_le_prefixLaw_add","parent":3523,"result":true,"route":"../roadmap/tauceti/probability/exchangeability-and-de-finetti/index.html#quantitative-finite-de-finetti-bound-exchangeabl","scope":"tauceti/probability/exchangeability-and-de-finetti","status":"fully_proved","summary":"If the first `n` coordinates of a process are exchangeable and `m \u2264 n`, then every measurable event under the `m`-prefix law has mass at most its mass under sampling `m` entries with replacement from the first `n` coordinates, plus `choose m 2 / n`. For `n > 0` that sample is the mixture of `m`-fold products of the empirical distribution of the first `n` coordinates.","title":"Quantitative finite de Finetti bound, exchangeable law to empirical mixture"},{"data_url":"scopes/tauceti/probability/financial-mathematics-one-factor-markets-and-rep.json","id":"tauceti/probability/financial-mathematics-one-factor-markets-and-rep","kind":"scope","parent":3522,"result":false,"route":"../roadmap/tauceti/probability/financial-mathematics-one-factor-markets-and-rep/index.html","scope":"tauceti/probability/financial-mathematics-one-factor-markets-and-rep","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Financial mathematics: one-factor markets and replication"},{"id":"tauceti/probability/financial-mathematics-one-factor-markets-and-rep/the-black-scholes-call-formula","kind":"node","lean":"TauCeti.BlackScholes.riskNeutralPrice_call","parent":3537,"result":true,"route":"../roadmap/tauceti/probability/financial-mathematics-one-factor-markets-and-rep/index.html#the-black-scholes-call-formula","scope":"tauceti/probability/financial-mathematics-one-factor-markets-and-rep","status":"fully_proved","summary":"The discounted risk-neutral expectation of the call payoff `(S_\u03c4 - K)\u207a` is `s \u03a6(d\u208a) - K e^{-r\u03c4} \u03a6(d\u208b)`.","title":"The Black\u2013Scholes call formula"},{"id":"tauceti/probability/financial-mathematics-one-factor-markets-and-rep/the-black-scholes-put-formula","kind":"node","lean":"TauCeti.BlackScholes.riskNeutralPrice_put","parent":3537,"result":true,"route":"../roadmap/tauceti/probability/financial-mathematics-one-factor-markets-and-rep/index.html#the-black-scholes-put-formula","scope":"tauceti/probability/financial-mathematics-one-factor-markets-and-rep","status":"fully_proved","summary":"The discounted risk-neutral expectation of the put payoff `(K - S_\u03c4)\u207a` is `K e^{-r\u03c4} \u03a6(-d\u208b) - s \u03a6(-d\u208a)`.","title":"The Black\u2013Scholes put formula"},{"id":"tauceti/probability/financial-mathematics-one-factor-markets-and-rep/the-black-scholes-equation-for-the-call","kind":"node","lean":"TauCeti.BlackScholes.callPrice_pde","parent":3537,"result":true,"route":"../roadmap/tauceti/probability/financial-mathematics-one-factor-markets-and-rep/index.html#the-black-scholes-equation-for-the-call","scope":"tauceti/probability/financial-mathematics-one-factor-markets-and-rep","status":"fully_proved","summary":"In calendar time `t < T`, the call price `C(s, t) = callPrice s r \u03c3 K (T - t)` satisfies `\u2202C/\u2202t + \u03c3\u00b2 s\u00b2 / 2 \u2202\u00b2C/\u2202s\u00b2 + r (s \u2202C/\u2202s - C) = 0` at every spot `s > 0`.","title":"The Black\u2013Scholes equation for the call"},{"id":"tauceti/probability/financial-mathematics-one-factor-markets-and-rep/the-black-scholes-equation-for-the-put","kind":"node","lean":"TauCeti.BlackScholes.putPrice_pde","parent":3537,"result":true,"route":"../roadmap/tauceti/probability/financial-mathematics-one-factor-markets-and-rep/index.html#the-black-scholes-equation-for-the-put","scope":"tauceti/probability/financial-mathematics-one-factor-markets-and-rep","status":"fully_proved","summary":"In calendar time `t < T`, the put price `P(s, t) = putPrice s r \u03c3 K (T - t)` satisfies `\u2202P/\u2202t + \u03c3\u00b2 s\u00b2 / 2 \u2202\u00b2P/\u2202s\u00b2 + r (s \u2202P/\u2202s - P) = 0` at every spot `s > 0`.","title":"The Black\u2013Scholes equation for the put"},{"id":"tauceti/probability/financial-mathematics-one-factor-markets-and-rep/put-call-parity-for-the-closed-form-prices","kind":"node","lean":"TauCeti.BlackScholes.callPrice_sub_putPrice","parent":3537,"result":true,"route":"../roadmap/tauceti/probability/financial-mathematics-one-factor-markets-and-rep/index.html#put-call-parity-for-the-closed-form-prices","scope":"tauceti/probability/financial-mathematics-one-factor-markets-and-rep","status":"fully_proved","summary":"`C - P = s - K e^{-r\u03c4}`.","title":"Put\u2013call parity for the closed-form prices"},{"id":"tauceti/probability/financial-mathematics-one-factor-markets-and-rep/tauceti-blackscholes-deriv-callprice-spot-mem-io","kind":"node","lean":"TauCeti.BlackScholes.deriv_callPrice_spot_mem_Ioo","parent":3537,"result":true,"route":"../roadmap/tauceti/probability/financial-mathematics-one-factor-markets-and-rep/index.html#tauceti-blackscholes-deriv-callprice-spot-mem-io","scope":"tauceti/probability/financial-mathematics-one-factor-markets-and-rep","status":"fully_proved","summary":"The call delta lies strictly between `0` and `1`.","title":"TauCeti.BlackScholes.deriv_callPrice_spot_mem_Ioo"},{"id":"tauceti/probability/financial-mathematics-one-factor-markets-and-rep/tauceti-blackscholes-riskneutralprice","kind":"node","lean":"TauCeti.BlackScholes.riskNeutralPrice","parent":3537,"result":true,"route":"../roadmap/tauceti/probability/financial-mathematics-one-factor-markets-and-rep/index.html#tauceti-blackscholes-riskneutralprice","scope":"tauceti/probability/financial-mathematics-one-factor-markets-and-rep","status":"fully_proved","summary":"The discounted risk-neutral expectation `exp (-r \u03c4) * E[h (S_\u03c4)]` of a payoff `h` of the terminal spot `S_\u03c4 = terminalSpot s r \u03c3 \u03c4 Z`, `Z \u223c N(0, 1)`. For a non-integrable payoff the Bochner integral, and hence this value, is zero.","title":"TauCeti.BlackScholes.riskNeutralPrice"},{"id":"tauceti/probability/financial-mathematics-one-factor-markets-and-rep/tauceti-probability-setintegral-exp-mul-gaussian","kind":"node","lean":"TauCeti.Probability.setIntegral_exp_mul_gaussianReal","parent":3537,"result":true,"route":"../roadmap/tauceti/probability/financial-mathematics-one-factor-markets-and-rep/index.html#tauceti-probability-setintegral-exp-mul-gaussian","scope":"tauceti/probability/financial-mathematics-one-factor-markets-and-rep","status":"fully_proved","summary":"Integrating the exponential weight `exp (t * x)` against `N(m, v)` over a set `s` gives the moment-generating function `exp (m t + v t\u00b2 / 2)` times the mass of `s` under the tilted law `N(m + v t, v)`.","title":"TauCeti.Probability.setIntegral_exp_mul_gaussianReal"},{"id":"tauceti/probability/financial-mathematics-one-factor-markets-and-rep/tauceti-blackscholes-mul-gaussianpdfreal-dplus","kind":"node","lean":"TauCeti.BlackScholes.mul_gaussianPDFReal_dPlus","parent":3537,"result":true,"route":"../roadmap/tauceti/probability/financial-mathematics-one-factor-markets-and-rep/index.html#tauceti-blackscholes-mul-gaussianpdfreal-dplus","scope":"tauceti/probability/financial-mathematics-one-factor-markets-and-rep","status":"fully_proved","summary":"The Black\u2013Scholes density identity `s \u03c6(d\u208a) = K e^{-r\u03c4} \u03c6(d\u208b)`, where `\u03c6` is the standard Gaussian density. It expresses that `d\u208a\u00b2 - d\u208b\u00b2 = 2 (log (s / K) + r \u03c4)`.","title":"TauCeti.BlackScholes.mul_gaussianPDFReal_dPlus"},{"data_url":"scopes/tauceti/probability/free-probability-free-products-and-random-matrix.json","id":"tauceti/probability/free-probability-free-products-and-random-matrix","kind":"scope","parent":3522,"result":false,"route":"../roadmap/tauceti/probability/free-probability-free-products-and-random-matrix/index.html","scope":"tauceti/probability/free-probability-free-products-and-random-matrix","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Free probability, free products, and random matrix models"},{"id":"tauceti/probability/free-probability-free-products-and-random-matrix/tauceti-apply-prod-offn-eq-sum-prod-freecumulant","kind":"node","lean":"TauCeti.apply_prod_ofFn_eq_sum_prod_freeCumulant","parent":3547,"result":true,"route":"../roadmap/tauceti/probability/free-probability-free-products-and-random-matrix/index.html#tauceti-apply-prod-offn-eq-sum-prod-freecumulant","scope":"tauceti/probability/free-probability-free-products-and-random-matrix","status":"fully_proved","summary":"The *moment\u2013cumulant formula* for a word `a\u2081, \u2026, a\u2099` with `n \u2265 1` in multiplicative form: `\u03c6 (a\u2081 * \u22ef * a\u2099) = \u2211_{\u03c0 \u2208 NC(n)} \u220f_{B \u2208 \u03c0} \u03ba_B(a\u2081, \u2026, a\u2099)`.","title":"TauCeti.apply_prod_ofFn_eq_sum_prod_freeCumulant"},{"id":"tauceti/probability/free-probability-free-products-and-random-matrix/tauceti-freecumulant-eq-sum-mu-mul-partitionmome","kind":"node","lean":"TauCeti.freeCumulant_eq_sum_mu_mul_partitionMoment","parent":3547,"result":true,"route":"../roadmap/tauceti/probability/free-probability-free-products-and-random-matrix/index.html#tauceti-freecumulant-eq-sum-mu-mul-partitionmome","scope":"tauceti/probability/free-probability-free-products-and-random-matrix","status":"fully_proved","summary":"The free cumulant of a word is the sum over the noncrossing partitions `\u03c0` of its block moments weighted by the M\u00f6bius function, `\u03ba_s(a) = \u2211_{\u03c0 \u2208 NC(s)} \u03bc(\u03c0, 1_s) \u03c6_\u03c0(a)`.","title":"TauCeti.freeCumulant_eq_sum_mu_mul_partitionMoment"},{"id":"tauceti/probability/free-probability-free-products-and-random-matrix/tauceti-freecumulant-fiberprod-eq-sum-prod","kind":"node","lean":"TauCeti.freeCumulant_fiberProd_eq_sum_prod","parent":3547,"result":true,"route":"../roadmap/tauceti/probability/free-probability-free-products-and-random-matrix/index.html#tauceti-freecumulant-fiberprod-eq-sum-prod","scope":"tauceti/probability/free-probability-free-products-and-random-matrix","status":"fully_proved","summary":"*Free cumulants with products as arguments*, in multiplicative form: `\u03ba(fiberProd a s f) = \u2211_{\u03c0 \u2294 \u03c3_f = 1_s} \u220f_{B \u2208 \u03c0} \u03ba_B(a)`, where `\u03c3_f` is the partition of `s` into the fibres of `f`.","title":"TauCeti.freeCumulant_fiberProd_eq_sum_prod"},{"id":"tauceti/probability/free-probability-free-products-and-random-matrix/tauceti-noncrossingpartition-card-fin-eq-catalan","kind":"node","lean":"TauCeti.NoncrossingPartition.card_fin_eq_catalan","parent":3547,"result":true,"route":"../roadmap/tauceti/probability/free-probability-free-products-and-random-matrix/index.html#tauceti-noncrossingpartition-card-fin-eq-catalan","scope":"tauceti/probability/free-probability-free-products-and-random-matrix","status":"fully_proved","summary":"There are `catalan n` noncrossing partitions of the `n` ordered positions of `Fin n`.","title":"TauCeti.NoncrossingPartition.card_fin_eq_catalan"},{"id":"tauceti/probability/free-probability-free-products-and-random-matrix/tauceti-noncrossingpartition-card-ispairing-fin","kind":"node","lean":"TauCeti.NoncrossingPartition.card_isPairing_fin_eq_catalan","parent":3547,"result":true,"route":"../roadmap/tauceti/probability/free-probability-free-products-and-random-matrix/index.html#tauceti-noncrossingpartition-card-ispairing-fin","scope":"tauceti/probability/free-probability-free-products-and-random-matrix","status":"fully_proved","summary":"There are `catalan n` noncrossing pairings of the `2 * n` ordered positions of `Fin (2 * n)`.","title":"TauCeti.NoncrossingPartition.card_isPairing_fin_eq_catalan"},{"id":"tauceti/probability/free-probability-free-products-and-random-matrix/positivelinearmap-law-eq-law-iff-forall-apply-po","kind":"node","lean":"PositiveLinearMap.law_eq_law_iff_forall_apply_pow","parent":3547,"result":true,"route":"../roadmap/tauceti/probability/free-probability-free-products-and-random-matrix/index.html#positivelinearmap-law-eq-law-iff-forall-apply-po","scope":"tauceti/probability/free-probability-free-products-and-random-matrix","status":"fully_proved","summary":"Two selfadjoint elements, possibly in different C\u22c6-algebras and with respect to different positive functionals, have the same law exactly when they have the same moments: `\u03bc_a = \u03bc_b \u2194 \u2200 n, \u03c6 (a ^ n) = \u03c8 (b ^ n)`.","title":"PositiveLinearMap.law_eq_law_iff_forall_apply_pow"},{"id":"tauceti/probability/free-probability-free-products-and-random-matrix/realization-of-compactly-supported-laws","kind":"node","lean":"PositiveLinearMap.law_integralPositiveLinearMap_comap_restrict_id","parent":3547,"result":true,"route":"../roadmap/tauceti/probability/free-probability-free-products-and-random-matrix/index.html#realization-of-compactly-supported-laws","scope":"tauceti/probability/free-probability-free-products-and-random-matrix","status":"fully_proved","summary":"A finite measure `\u03bc` on `\u211d` concentrated on a compact set `K` is the law of the coordinate function `x \u21a6 x` in the commutative C\u22c6-algebra `C(K, \u2102)`, with respect to integration against `\u03bc` on `K`. This functional is a state when `\u03bc` is a probability measure (`ContinuousMap.integralPositiveLinearMap_one`).","title":"Realization of compactly supported laws"},{"id":"tauceti/probability/free-probability-free-products-and-random-matrix/tauceti-isfreelyindependent-eqon-isup","kind":"node","lean":"TauCeti.IsFreelyIndependent.eqOn_iSup","parent":3547,"result":true,"route":"../roadmap/tauceti/probability/free-probability-free-products-and-random-matrix/index.html#tauceti-isfreelyindependent-eqon-isup","scope":"tauceti/probability/free-probability-free-products-and-random-matrix","status":"fully_proved","summary":"Freeness determines the joint distribution from the individual ones: two normalised functionals for which `B` is free and which agree on each `B i` agree on the subalgebra generated by the `B i`.","title":"TauCeti.IsFreelyIndependent.eqOn_iSup"},{"id":"tauceti/probability/free-probability-free-products-and-random-matrix/tauceti-freecumulant","kind":"node","lean":"TauCeti.freeCumulant","parent":3547,"result":true,"route":"../roadmap/tauceti/probability/free-probability-free-products-and-random-matrix/index.html#tauceti-freecumulant","scope":"tauceti/probability/free-probability-free-products-and-random-matrix","status":"fully_proved","summary":"The *free cumulant* `\u03ba_s(a)` of the word `(a_i)_{i \u2208 s}`: the cumulant of the partition of `s` into one part. For `s = Finset.univ : Finset (Fin n)` with `n \u2265 1` this is the `n`-th free cumulant `\u03ba_n(a\u2081, \u2026, a\u2099)`; for `s = \u2205` it is the meaningless value `1`.","title":"TauCeti.freeCumulant"},{"id":"tauceti/probability/free-probability-free-products-and-random-matrix/tauceti-partitioncumulant-eq-prod-freecumulant","kind":"node","lean":"TauCeti.partitionCumulant_eq_prod_freeCumulant","parent":3547,"result":true,"route":"../roadmap/tauceti/probability/free-probability-free-products-and-random-matrix/index.html#tauceti-partitioncumulant-eq-prod-freecumulant","scope":"tauceti/probability/free-probability-free-products-and-random-matrix","status":"fully_proved","summary":"The free cumulants are *multiplicative*: the free cumulant of a noncrossing partition is the product of the free cumulants of the words read along its parts, `\u03ba_\u03c0(a) = \u220f_{B \u2208 \u03c0} \u03ba_B(a)`.","title":"TauCeti.partitionCumulant_eq_prod_freeCumulant"},{"id":"tauceti/probability/free-probability-free-products-and-random-matrix/tauceti-noncrossingpartition-iicorderiso","kind":"node","lean":"TauCeti.NoncrossingPartition.iicOrderIso","parent":3547,"result":true,"route":"../roadmap/tauceti/probability/free-probability-free-products-and-random-matrix/index.html#tauceti-noncrossingpartition-iicorderiso","scope":"tauceti/probability/free-probability-free-products-and-random-matrix","status":"fully_proved","summary":"The lower interval `[0_s, \u03c0]` of the lattice of noncrossing partitions of `s` is isomorphic to the product of the lattices of noncrossing partitions of the parts of `\u03c0`: a noncrossing partition `\u03c3 \u2264 \u03c0` corresponds to its restrictions to the parts of `\u03c0`, and a family of noncrossing partitions of the parts corresponds to their juxtaposition.","title":"TauCeti.NoncrossingPartition.iicOrderIso"},{"id":"tauceti/probability/free-probability-free-products-and-random-matrix/positivelinearmap-law","kind":"node","lean":"PositiveLinearMap.law","parent":3547,"result":true,"route":"../roadmap/tauceti/probability/free-probability-free-products-and-random-matrix/index.html#positivelinearmap-law","scope":"tauceti/probability/free-probability-free-products-and-random-matrix","status":"fully_proved","summary":"The law `\u03bc_a` of an element `a` of a C\u22c6-algebra with respect to a positive linear functional `\u03c6`: for selfadjoint `a`, the finite Borel measure on `\u211d`, concentrated on `spectrum \u211d a` (`law_compl_spectrum`), with `\u222b f d\u03bc_a = \u03c6 (cfc f a)` for every `f` continuous on the spectrum (`ofReal_integral_law`), and characterized by this property (`eq_law_iff`). For `a` not selfadjoint it is the zero measure\u2026","title":"PositiveLinearMap.law"},{"id":"tauceti/probability/free-probability-free-products-and-random-matrix/tauceti-isfreelyindependent","kind":"node","lean":"TauCeti.IsFreelyIndependent","parent":3547,"result":true,"route":"../roadmap/tauceti/probability/free-probability-free-products-and-random-matrix/index.html#tauceti-isfreelyindependent","scope":"tauceti/probability/free-probability-free-products-and-random-matrix","status":"fully_proved","summary":"Subalgebras `B i` of `A` are *freely independent* with respect to a linear functional `\u03c6` if `\u03c6 (a\u2081 * \u22ef * a\u2099) = 0` for every nonempty word `[(i\u2081, a\u2081), \u2026, (i\u2099, a\u2099)]` whose consecutive labels are distinct and whose letters satisfy `a\u2c7c \u2208 B i\u2c7c` and `\u03c6 a\u2c7c = 0`. Labels that are not adjacent may coincide.","title":"TauCeti.IsFreelyIndependent"},{"id":"tauceti/probability/free-probability-free-products-and-random-matrix/the-weak-free-central-limit-theorem","kind":"node","lean":"TauCeti.IsFreelyIndependentElements.tendsto_law_inv_sqrt_smul_sum_semicircleReal, TauCeti.IsFreelyIndependentElements.tendsto_law_inv_sqrt_smul_sum_of_apply_pow_eq","parent":3547,"result":true,"route":"../roadmap/tauceti/probability/free-probability-free-products-and-random-matrix/index.html#the-weak-free-central-limit-theorem","scope":"tauceti/probability/free-probability-free-products-and-random-matrix","status":"fully_proved","summary":"Centered free selfadjoint elements whose moments are uniformly bounded in each degree and whose average variance tends to `v \u2265 0` have normalized-sum laws converging weakly to `semicircleReal 0 v`. No common compact support of the approximating laws is required.","title":"The weak free central limit theorem"},{"data_url":"scopes/tauceti/probability/markov-chains-and-renewal-processes.json","id":"tauceti/probability/markov-chains-and-renewal-processes","kind":"scope","parent":3522,"result":false,"route":"../roadmap/tauceti/probability/markov-chains-and-renewal-processes/index.html","scope":"tauceti/probability/markov-chains-and-renewal-processes","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Markov chains and renewal processes"},{"id":"tauceti/probability/markov-chains-and-renewal-processes/nonexplosion-of-a-renewal-process","kind":"node","lean":"TauCeti.Probability.ae_tendsto_arrival_atTop","parent":3562,"result":true,"route":"../roadmap/tauceti/probability/markov-chains-and-renewal-processes/index.html#nonexplosion-of-a-renewal-process","scope":"tauceti/probability/markov-chains-and-renewal-processes","status":"fully_proved","summary":"If the increments are independent, identically distributed and not almost surely zero, then almost surely the arrival times tend to infinity.","title":"Nonexplosion of a renewal process"},{"id":"tauceti/probability/markov-chains-and-renewal-processes/finiteness-of-the-renewal-counts","kind":"node","lean":"TauCeti.Probability.ae_renewalCount_lt_top","parent":3562,"result":true,"route":"../roadmap/tauceti/probability/markov-chains-and-renewal-processes/index.html#finiteness-of-the-renewal-counts","scope":"tauceti/probability/markov-chains-and-renewal-processes","status":"fully_proved","summary":"For independent, identically distributed increments that are not almost surely zero, almost surely every count `renewalCount T t` is finite. The almost-sure quantifier is outside the quantifier over all times.","title":"Finiteness of the renewal counts"},{"id":"tauceti/probability/markov-chains-and-renewal-processes/finite-moments-of-the-renewal-counts","kind":"node","lean":"TauCeti.Probability.lintegral_renewalCount_pow_lt_top","parent":3562,"result":true,"route":"../roadmap/tauceti/probability/markov-chains-and-renewal-processes/index.html#finite-moments-of-the-renewal-counts","scope":"tauceti/probability/markov-chains-and-renewal-processes","status":"fully_proved","summary":"For independent, identically distributed increments that are not almost surely zero, every count `renewalCount T t` has a finite moment of every order `p`. No assumption is made on the mean of the increments.","title":"Finite moments of the renewal counts"},{"id":"tauceti/probability/markov-chains-and-renewal-processes/chernoff-bound-for-the-arrival-times","kind":"node","lean":"TauCeti.Probability.measureReal_arrival_le_le_exp_mul_mgf_pow","parent":3562,"result":true,"route":"../roadmap/tauceti/probability/markov-chains-and-renewal-processes/index.html#chernoff-bound-for-the-arrival-times","scope":"tauceti/probability/markov-chains-and-renewal-processes","status":"fully_proved","summary":"For independent, identically distributed increments and every rate `s \u2265 0`, the probability that the `n`-th arrival is at most `t` is at most `exp (s t)` times the `n`-th power of the Laplace transform `mgf (T 0) P (-s)` of the increment law. By `natCast_le_renewalCount_iff` this is also a bound on `P(n \u2264 renewalCount T t)`.","title":"Chernoff bound for the arrival times"},{"id":"tauceti/probability/markov-chains-and-renewal-processes/uniqueness-of-the-path-law","kind":"node","lean":"MeasureTheory.Measure.eq_markovChainLaw","parent":3562,"result":true,"route":"../roadmap/tauceti/probability/markov-chains-and-renewal-processes/index.html#uniqueness-of-the-path-law","scope":"tauceti/probability/markov-chains-and-renewal-processes","status":"fully_proved","summary":"For a finite initial measure `\u03bd`, a measure on path space whose time-zero marginal is `\u03bd` and whose prefix laws satisfy the Markov recursion of `markovChainLaw_map_prefix_succ_eq_compProd` is the path law `markovChainLaw \u03bd \u03ba`.","title":"Uniqueness of the path law"},{"id":"tauceti/probability/markov-chains-and-renewal-processes/the-time-n-marginal-of-the-chain","kind":"node","lean":"MeasureTheory.Measure.markovChainLaw_map_eval","parent":3562,"result":true,"route":"../roadmap/tauceti/probability/markov-chains-and-renewal-processes/index.html#the-time-n-marginal-of-the-chain","scope":"tauceti/probability/markov-chains-and-renewal-processes","status":"fully_proved","summary":"the state at time `n` has law `(\u03ba ^ n) \u2218\u2098 \u03bd`, the initial law propagated by the `n`-step transition kernel. This holds for every initial measure `\u03bd`.","title":"The time-`n` marginal of the chain"},{"id":"tauceti/probability/markov-chains-and-renewal-processes/the-path-law-kernel-of-a-homogeneous-markov-chai","kind":"node","lean":"ProbabilityTheory.Kernel.markovChainKernel","parent":3562,"result":true,"route":"../roadmap/tauceti/probability/markov-chains-and-renewal-processes/index.html#the-path-law-kernel-of-a-homogeneous-markov-chai","scope":"tauceti/probability/markov-chains-and-renewal-processes","status":"fully_proved","summary":"the kernel sending a state `x` to the law `markovChainLaw (Measure.dirac x) \u03ba` of the chain started at `x` (`markovChainKernel_apply`). Being a kernel, it records that this law depends measurably on the starting state, and every path law of the chain is its mixture over the initial law (`markovChainLaw_eq_comp`).","title":"The path-law kernel of a homogeneous Markov chain"},{"id":"tauceti/probability/markov-chains-and-renewal-processes/tauceti-probability-renewalcount","kind":"node","lean":"TauCeti.Probability.renewalCount","parent":3562,"result":true,"route":"../roadmap/tauceti/probability/markov-chains-and-renewal-processes/index.html#tauceti-probability-renewalcount","scope":"tauceti/probability/markov-chains-and-renewal-processes","status":"fully_proved","summary":"The number `#{n \u2265 1 | arrival T n \u03c9 \u2264 t}` of arrivals in `[0, t]`, counted with multiplicity, as an extended nonnegative number. The artificial arrival `arrival T 0 = 0` is not counted.","title":"TauCeti.Probability.renewalCount"},{"id":"tauceti/probability/markov-chains-and-renewal-processes/tauceti-probability-renewalcount-eq-natcast-iff","kind":"node","lean":"TauCeti.Probability.renewalCount_eq_natCast_iff","parent":3562,"result":true,"route":"../roadmap/tauceti/probability/markov-chains-and-renewal-processes/index.html#tauceti-probability-renewalcount-eq-natcast-iff","scope":"tauceti/probability/markov-chains-and-renewal-processes","status":"fully_proved","summary":"The count at time `t` equals `n` exactly when `arrival T n \u03c9 \u2264 t < arrival T (n + 1) \u03c9`.","title":"TauCeti.Probability.renewalCount_eq_natCast_iff"},{"data_url":"scopes/tauceti/probability/standard-probability-distributions-and-their-ele.json","id":"tauceti/probability/standard-probability-distributions-and-their-ele","kind":"scope","parent":3522,"result":false,"route":"../roadmap/tauceti/probability/standard-probability-distributions-and-their-ele/index.html","scope":"tauceti/probability/standard-probability-distributions-and-their-ele","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Standard probability distributions and their elementary theory"},{"id":"tauceti/probability/standard-probability-distributions-and-their-ele/the-bartlett-decomposition-of-the-standard-wisha","kind":"node","lean":"TauCeti.Probability.map_choleskyLowerCoordinates_comap_nonsingularWishartMeasure, TauCeti.Probability.bartlett_nonsingularWishartMeasure","parent":3572,"result":true,"route":"../roadmap/tauceti/probability/standard-probability-distributions-and-their-ele/index.html#the-bartlett-decomposition-of-the-standard-wisha","scope":"tauceti/probability/standard-probability-distributions-and-their-ele","status":"fully_proved","summary":"Lift `TauCeti.Probability.nonsingularWishartMeasure n 1` to the positive-definite cone, where Cholesky factorization is defined, and read the on-or-below-diagonal entries of the factor: the resulting law is the product of the Bartlett coordinate laws. Cholesky factorization inverts the Gram map of `TauCeti.Probability.map_lowerTriangleGram_pi_bartlettCoordinateMeasure`, and the product law lives on the\u2026","title":"The Bartlett decomposition of the standard Wishart law"},{"id":"tauceti/probability/standard-probability-distributions-and-their-ele/the-diagonal-entries-of-an-inverse-wishart-matri","kind":"node","lean":"TauCeti.Probability.map_coe_apply_inverseWishartMeasure","parent":3572,"result":true,"route":"../roadmap/tauceti/probability/standard-probability-distributions-and-their-ele/index.html#the-diagonal-entries-of-an-inverse-wishart-matri","scope":"tauceti/probability/standard-probability-distributions-and-their-ele","status":"fully_proved","summary":"For a positive-definite scale `S`, the entry `(i, i)` of an inverse-Wishart matrix of degree `n` has the inverse-gamma law of shape `(n - p + 1) / 2` and scale `S i i / 2`. For degrees at most `p - 1` both sides are the zero measure.","title":"The diagonal entries of an inverse-Wishart matrix are inverse gamma"},{"id":"tauceti/probability/standard-probability-distributions-and-their-ele/order-statistics-of-uniform-samples-are-beta-dis","kind":"node","lean":"TauCeti.Probability.map_orderStatistic_pi_uniformMeasure","parent":3572,"result":true,"route":"../roadmap/tauceti/probability/standard-probability-distributions-and-their-ele/index.html#order-statistics-of-uniform-samples-are-beta-dis","scope":"tauceti/probability/standard-probability-distributions-and-their-ele","status":"fully_proved","summary":"The `k`-th order statistic (counted from `0`) of `n` independent uniform variables on the unit interval has the beta law with parameters `k + 1` and `n - k`.","title":"Order statistics of uniform samples are beta distributed"},{"id":"tauceti/probability/standard-probability-distributions-and-their-ele/tauceti-probability-map-sum-sq-pi-gaussianreal-n","kind":"node","lean":"TauCeti.Probability.map_sum_sq_pi_gaussianReal_noncentral","parent":3572,"result":true,"route":"../roadmap/tauceti/probability/standard-probability-distributions-and-their-ele/index.html#tauceti-probability-map-sum-sq-pi-gaussianreal-n","scope":"tauceti/probability/standard-probability-distributions-and-their-ele","status":"fully_proved","summary":"The sum of squares of independent unit-variance Gaussian coordinates has noncentral chi-squared law, including the empty family.","title":"TauCeti.Probability.map_sum_sq_pi_gaussianReal_noncentral"},{"id":"tauceti/probability/standard-probability-distributions-and-their-ele/tauceti-probability-noncentralfpdfreal","kind":"node","lean":"TauCeti.Probability.noncentralFPDFReal","parent":3572,"result":true,"route":"../roadmap/tauceti/probability/standard-probability-distributions-and-their-ele/index.html#tauceti-probability-noncentralfpdfreal","scope":"tauceti/probability/standard-probability-distributions-and-their-ele","status":"fully_proved","summary":"The noncentral F density as a Poisson series of scaled central F densities. Invalid degrees give zero.","title":"TauCeti.Probability.noncentralFPDFReal"},{"id":"tauceti/probability/standard-probability-distributions-and-their-ele/measuretheory-probabilitymeasure-tendsto-haaradd","kind":"node","lean":"MeasureTheory.ProbabilityMeasure.tendsto_haarAddCircle_iff","parent":3572,"result":true,"route":"../roadmap/tauceti/probability/standard-probability-distributions-and-their-ele/index.html#measuretheory-probabilitymeasure-tendsto-haaradd","scope":"tauceti/probability/standard-probability-distributions-and-their-ele","status":"fully_proved","summary":"Circle probability measures tend weakly to Haar probability exactly when every nonzero integer Fourier moment tends to zero.","title":"MeasureTheory.ProbabilityMeasure.tendsto_haarAddCircle_iff"},{"id":"tauceti/probability/standard-probability-distributions-and-their-ele/the-multivariate-gamma-integral","kind":"node","lean":"TauCeti.integral_posDef_multivariateGamma","parent":3572,"result":true,"route":"../roadmap/tauceti/probability/standard-probability-distributions-and-their-ele/index.html#the-multivariate-gamma-integral","scope":"tauceti/probability/standard-probability-distributions-and-their-ele","status":"fully_proved","summary":"For `((p : \u211d) - 1) / 2 < a`, the integral of `(det A) ^ (a - (p + 1) / 2) * exp (-trace A)` over the cone of positive-definite symmetric `p \u00d7 p` matrices, against `TauCeti.symmetricLebesgue p`, is `\u0393_p(a)`.","title":"The multivariate Gamma integral"},{"id":"tauceti/probability/standard-probability-distributions-and-their-ele/tauceti-inverseregularizedgamma","kind":"node","lean":"TauCeti.inverseRegularizedGamma","parent":3572,"result":true,"route":"../roadmap/tauceti/probability/standard-probability-distributions-and-their-ele/index.html#tauceti-inverseregularizedgamma","scope":"tauceti/probability/standard-probability-distributions-and-their-ele","status":"fully_proved","summary":"The inverse `P\u207b\u00b9(s, u)` of the regularized lower incomplete gamma function in its argument.","title":"TauCeti.inverseRegularizedGamma"},{"id":"tauceti/probability/standard-probability-distributions-and-their-ele/tauceti-probability-vonmisesfishernaturalmeasure","kind":"node","lean":"TauCeti.Probability.vonMisesFisherNaturalMeasure","parent":3572,"result":true,"route":"../roadmap/tauceti/probability/standard-probability-distributions-and-their-ele/index.html#tauceti-probability-vonmisesfishernaturalmeasure","scope":"tauceti/probability/standard-probability-distributions-and-their-ele","status":"fully_proved","summary":"The von Mises\u2013Fisher law with natural parameter `\u03b7`, relative to uniform sphere probability. In dimension zero the sphere and this measure are empty.","title":"TauCeti.Probability.vonMisesFisherNaturalMeasure"},{"data_url":"scopes/tauceti/probability/distributions.json","id":"tauceti/probability/distributions","kind":"scope","parent":3522,"result":false,"route":"../roadmap/tauceti/probability/distributions/index.html","scope":"tauceti/probability/distributions","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Probability.Distributions` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Distributions"},{"id":"tauceti/probability/distributions/lebesgue-decomposition","kind":"node","lean":"TauCeti.Probability.multivariateGaussianMixtureMeasure_eq_add_withDensity","parent":3582,"result":true,"route":"../roadmap/tauceti/probability/distributions/index.html#lebesgue-decomposition","scope":"tauceti/probability/distributions","status":"fully_proved","summary":"A multivariate Gaussian mixture is the sum of its weighted components whose covariance is not positive definite, each singular with respect to Lebesgue measure by `mutuallySingular_multivariateGaussian_volume`, and of Lebesgue measure weighted by the mixture density.","title":"Lebesgue decomposition"},{"id":"tauceti/probability/distributions/the-levy-khintchine-formula-for-stable-exponents","kind":"node","lean":"TauCeti.Probability.integral_exp_sub_one_sub_sin_stableLevyMeasure","parent":3582,"result":true,"route":"../roadmap/tauceti/probability/distributions/index.html#the-levy-khintchine-formula-for-stable-exponents","scope":"tauceti/probability/distributions","status":"fully_proved","summary":"For `0 < \u03b1 < 2` and `|\u03b2| \u2264 1`, integrating the sine-compensated characteristic kernel against the stable L\u00e9vy measure gives the stable characteristic exponent in Nolan's `S\u2080` parameterization.","title":"The L\u00e9vy\u2013Khintchine formula for stable exponents"},{"id":"tauceti/probability/distributions/the-maurey-pisier-inequality","kind":"node","lean":"ProbabilityTheory.IsGaussian.lintegral_exp_mul_sub_le","parent":3582,"result":true,"route":"../roadmap/tauceti/probability/distributions/index.html#the-maurey-pisier-inequality","scope":"tauceti/probability/distributions","status":"fully_proved","summary":"For a centred Gaussian measure `\u03bc` and a `C\u00b9` function `f`, `\u222b\u222b exp (t (f y - f x)) d\u03bc(x) d\u03bc(y) \u2264 \u222b exp (\u03c0\u00b2 t\u00b2 / 8 \u00b7 Var[f' x; \u03bc]) d\u03bc(x)`.","title":"The Maurey\u2013Pisier inequality"},{"id":"tauceti/probability/distributions/parsevals-identity-for-the-gaussian-hermite-basi","kind":"node","lean":"TauCeti.tsum_norm_sq_integral_hermite_mul_gaussianReal, TauCeti.tsum_inner_mul_inner_L2piMul_gaussianHermiteHilbertBasis, TauCeti.tsum_norm_sq_integral_prod_hermite_mul_pi_gaussianReal","parent":3582,"result":true,"route":"../roadmap/tauceti/probability/distributions/index.html#parsevals-identity-for-the-gaussian-hermite-basi","scope":"tauceti/probability/distributions","status":"fully_proved","summary":"The squared norms of the Gaussian Hermite coefficients of `f` sum to `\u2016f\u2016\u00b2`.","title":"Parseval's identity for the Gaussian Hermite basis"},{"id":"tauceti/probability/distributions/rademacher-convexity-inequality","kind":"node","lean":"Finset.norm_le_integral_norm_sum_smul_rademacher","parent":3582,"result":true,"route":"../roadmap/tauceti/probability/distributions/index.html#rademacher-convexity-inequality","scope":"tauceti/probability/distributions","status":"fully_proved","summary":"Every summand's norm is bounded by the expected norm of the signed sum. Pairwise independence of the Rademacher signs suffices.","title":"Rademacher convexity inequality"},{"data_url":"scopes/tauceti/probability/moments.json","id":"tauceti/probability/moments","kind":"scope","parent":3522,"result":false,"route":"../roadmap/tauceti/probability/moments/index.html","scope":"tauceti/probability/moments","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Probability.Moments` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Moments"},{"id":"tauceti/probability/moments/azuma-hoeffding-inequality","kind":"node","lean":"TauCeti.Probability.measureReal_sum_range_ge_le_of_bounded_martingale_differences, TauCeti.Probability.measureReal_sum_range_ge_le_of_mem_Icc_martingale_differences","parent":3588,"result":true,"route":"../roadmap/tauceti/probability/moments/index.html#azuma-hoeffding-inequality","scope":"tauceti/probability/moments","status":"fully_proved","summary":"Let `Y i` be a strongly adapted sequence of martingale differences with `|Y i| \u2264 b i` almost surely, mean zero at time zero, and conditional mean zero at each later step. Then the upper tail of the sum of its first `n` terms is bounded by `exp (-\u03b5 ^ 2 / (2 * \u2211 i < n, b i ^ 2))`.","title":"Azuma-Hoeffding inequality"},{"id":"tauceti/probability/moments/bernsteins-inequality","kind":"node","lean":"ProbabilityTheory.iIndepFun.measureReal_sum_ge_le_of_ae_le, ProbabilityTheory.iIndepFun.measureReal_abs_sum_ge_le_of_abs_le","parent":3588,"result":true,"route":"../roadmap/tauceti/probability/moments/index.html#bernsteins-inequality","scope":"tauceti/probability/moments","status":"fully_proved","summary":"Let the `X\u1d62`, `i \u2208 s`, be independent with mean zero, finite second moments and `X\u1d62 \u2264 b` almost surely, and let `v = \u2211 i \u2208 s, E X\u1d62\u00b2`. Then for `0 \u2264 \u03b5`, `P(\u03b5 \u2264 \u2211 i \u2208 s, X\u1d62) \u2264 exp (-\u03b5 ^ 2 / (2 * (v + b * \u03b5 / 3)))`.","title":"Bernstein's inequality"},{"id":"tauceti/probability/moments/bernsteins-moment-generating-function-bound","kind":"node","lean":"MeasureTheory.MemLp.mgf_le_exp_of_ae_le, ProbabilityTheory.iIndepFun.mgf_sum_le_exp_of_ae_le","parent":3588,"result":true,"route":"../roadmap/tauceti/probability/moments/index.html#bernsteins-moment-generating-function-bound","scope":"tauceti/probability/moments","status":"fully_proved","summary":"If `X` has mean zero, a finite second moment and `X \u2264 b` almost surely, then for `0 \u2264 t` with `t * b < 3`, `E exp (t X) \u2264 exp (E X\u00b2 * t ^ 2 / (2 * (1 - t * b / 3)))`.","title":"Bernstein's moment-generating-function bound"},{"id":"tauceti/probability/moments/bienaymes-identity-for-random-vectors","kind":"node","lean":"TauCeti.integral_norm_sum_sq_of_pairwise_indepFun","parent":3588,"result":true,"route":"../roadmap/tauceti/probability/moments/index.html#bienaymes-identity-for-random-vectors","scope":"tauceti/probability/moments","status":"fully_proved","summary":"If the random vectors `X i`, `i \u2208 s`, in a real Hilbert space are square-integrable, centered and pairwise independent, then the expected squared norm of their sum is the sum of their expected squared norms.","title":"Bienaym\u00e9's identity for random vectors"},{"id":"tauceti/probability/moments/chebyshevs-inequality-for-the-second-moment","kind":"node","lean":"TauCeti.meas_ge_le_integral_norm_sq_div_sq","parent":3588,"result":true,"route":"../roadmap/tauceti/probability/moments/index.html#chebyshevs-inequality-for-the-second-moment","scope":"tauceti/probability/moments","status":"fully_proved","summary":"For any measure, if `\u2016Y\u2016\u00b2` is integrable and `0 < c`, the set where `\u2016Y\u2016` is at least `c` has measure at most `(\u222b \u2016Y\u2016\u00b2) / c\u00b2`. Unlike `ProbabilityTheory.meas_ge_le_variance_div_sq`, `Y` is not centered at its mean.","title":"Chebyshev's inequality for the second moment"},{"id":"tauceti/probability/moments/entrywise-chebyshev-union-bound","kind":"node","lean":"TauCeti.measure_exists_lt_norm_sub_le","parent":3588,"result":true,"route":"../roadmap/tauceti/probability/moments/index.html#entrywise-chebyshev-union-bound","scope":"tauceti/probability/moments","status":"fully_proved","summary":"For any measure `\u03bc`, if every squared entrywise deviation of a matrix-valued function `S` from a fixed matrix `A` is integrable with integral at most `v`, then the measure of the event where some entry deviates by more than `\u03b7 > 0` is at most `ENNReal.ofReal (card m * card n * v / \u03b7\u00b2)`.","title":"Entrywise Chebyshev union bound"},{"id":"tauceti/probability/moments/one-sided-sub-exponential-bernstein-inequality","kind":"node","lean":"ProbabilityTheory.iIndepFun.measureReal_sum_ge_le_of_hasSubexponentialMGF","parent":3588,"result":true,"route":"../roadmap/tauceti/probability/moments/index.html#one-sided-sub-exponential-bernstein-inequality","scope":"tauceti/probability/moments","status":"fully_proved","summary":"For independent centered variables with common parameter `K`, this bounds the upper tail of a finite sum.","title":"One-sided sub-exponential Bernstein inequality"},{"id":"tauceti/probability/moments/the-quadratic-fenchel-young-inequality-in-mean","kind":"node","lean":"TauCeti.integral_inner_sub_integral_le","parent":3588,"result":true,"route":"../roadmap/tauceti/probability/moments/index.html#the-quadratic-fenchel-young-inequality-in-mean","scope":"tauceti/probability/moments","status":"fully_proved","summary":"Let `X` and `Y` be square-integrable random vectors, where the law of `X` has covariance matrix `A * T * A` for a positive-semidefinite matrix `A`. Then the correlation of the centred variable `X - \ud835\udd3c X` with `Y` is at most `(\ud835\udd3c \u27eaA Y, Y\u27eb + tr (A * T)) / 2`.","title":"The quadratic Fenchel\u2013Young inequality in mean"},{"id":"tauceti/probability/moments/sub-exponential-bernstein-inequality","kind":"node","lean":"ProbabilityTheory.iIndepFun.measureReal_abs_sum_ge_le_of_hasSubexponentialMGF","parent":3588,"result":true,"route":"../roadmap/tauceti/probability/moments/index.html#sub-exponential-bernstein-inequality","scope":"tauceti/probability/moments","status":"fully_proved","summary":"If the `X\u1d62`, `i \u2208 s`, are independent, centered and sub-exponential with common parameter `K`, then for `t > 0`","title":"Sub-exponential Bernstein inequality"},{"id":"tauceti/probability/moments/symmetrization-by-jensens-inequality","kind":"node","lean":"TauCeti.ofReal_mgf_sub_integral_le_lintegral","parent":3588,"result":true,"route":"../roadmap/tauceti/probability/moments/index.html#symmetrization-by-jensens-inequality","scope":"tauceti/probability/moments","status":"fully_proved","summary":"For a probability measure `\u03bc`, the moment generating function of `f - \ud835\udd3c f` at `t` is at most `\u222b\u222b exp (t (f x - f y)) d\u03bc(x) d\u03bc(y)`.","title":"Symmetrization by Jensen's inequality"},{"id":"tauceti/probability/moments/two-sided-azuma-hoeffding-inequality","kind":"node","lean":"TauCeti.Probability.measureReal_abs_sum_range_ge_le_of_bounded_martingale_differences, TauCeti.Probability.measureReal_abs_sum_range_ge_le_of_mem_Icc_martingale_differences","parent":3588,"result":true,"route":"../roadmap/tauceti/probability/moments/index.html#two-sided-azuma-hoeffding-inequality","scope":"tauceti/probability/moments","status":"fully_proved","summary":"Under the assumptions of `measureReal_sum_range_ge_le_of_bounded_martingale_differences`, the absolute value of the sum exceeds `\u03b5` with probability at most twice the one-sided bound.","title":"Two-sided Azuma-Hoeffding inequality"},{"id":"tauceti/probability/moments/the-uncentered-chebyshev-inequality","kind":"node","lean":"TauCeti.meas_gt_le_div_sq_of_integral_sq_le","parent":3588,"result":true,"route":"../roadmap/tauceti/probability/moments/index.html#the-uncentered-chebyshev-inequality","scope":"tauceti/probability/moments","status":"fully_proved","summary":"If `Y\u00b2` is integrable with `\u222b Y\u00b2 \u2264 v` and `0 < \u03b7`, then the set where `Y` exceeds `\u03b7` has measure at most `v / \u03b7\u00b2`.","title":"The uncentered Chebyshev inequality"},{"data_url":"scopes/tauceti/probability/other-results.json","id":"tauceti/probability/other-results","kind":"scope","parent":3522,"result":false,"route":"../roadmap/tauceti/probability/other-results/index.html","scope":"tauceti/probability/other-results","status":"fully_proved","summary":"Results named in the rest of Tau Ceti's `TauCeti.Probability` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Other results"},{"id":"tauceti/probability/other-results/chapman-kolmogorov-lower-bound","kind":"node","lean":"ProbabilityTheory.Kernel.pow_apply_singleton_mul_le_pow_add_apply","parent":3601,"result":true,"route":"../roadmap/tauceti/probability/other-results/index.html#chapman-kolmogorov-lower-bound","scope":"tauceti/probability/other-results","status":"fully_proved","summary":"The `m + n`-step mass of a measurable set `s` is at least the mass of the paths that sit at the intermediate state `y` at time `m`.","title":"Chapman\u2013Kolmogorov lower bound"},{"id":"tauceti/probability/other-results/the-contractable-form-of-de-finettis-theorem-via","kind":"node","lean":"TauCeti.Probability.conditionallyIID_of_contractable_viaL2","parent":3601,"result":true,"route":"../roadmap/tauceti/probability/other-results/index.html#the-contractable-form-of-de-finettis-theorem-via","scope":"tauceti/probability/other-results","status":"fully_proved","summary":"A contractable process valued in a nonempty standard Borel space is conditionally i.i.d.","title":"The contractable form of de Finetti's theorem via `L\u00b2` averaging"},{"id":"tauceti/probability/other-results/de-finettis-theorem-for-the-rows-of-a-block-of-p","kind":"node","lean":"TauCeti.Probability.JointlyExchangeable.conditionallyIID_arrayRow_arrayBlockPair, TauCeti.Probability.SeparatelyExchangeable.conditionallyIID_arrayCol, TauCeti.Probability.JointlyExchangeable.conditionallyIID_arrayRow_arrayBlock, TauCeti.Probability.SeparatelyExchangeable.conditionallyIID_arrayRow","parent":3601,"result":true,"route":"../roadmap/tauceti/probability/other-results/index.html#de-finettis-theorem-for-the-rows-of-a-block-of-p","scope":"tauceti/probability/other-results","status":"fully_proved","summary":"The conclusion simultaneously describes both orientations of the selected rectangular cross-block.","title":"De Finetti's theorem for the rows of a block of pairs"},{"id":"tauceti/probability/other-results/de-finettis-theorem-transported-along-an-explici","kind":"node","lean":"TauCeti.Probability.conditionallyIID_of_exchangeableFamily_of_equiv_nat","parent":3601,"result":true,"route":"../roadmap/tauceti/probability/other-results/index.html#de-finettis-theorem-transported-along-an-explici","scope":"tauceti/probability/other-results","status":"fully_proved","summary":"Under a finite measure, an exchangeable family with measurable coordinates whose index type is equivalent to `\u2115` is conditionally i.i.d.","title":"De Finetti's theorem transported along an explicit enumeration"},{"id":"tauceti/probability/other-results/doobs-maximal-inequality-in-continuous-time","kind":"node","lean":"MeasureTheory.Submartingale.mul_measure_exists_le_lt_le, MeasureTheory.Submartingale.mul_measure_exists_mem_le_le","parent":3601,"result":true,"route":"../roadmap/tauceti/probability/other-results/index.html#doobs-maximal-inequality-in-continuous-time","scope":"tauceti/probability/other-results","status":"fully_proved","summary":"Let `f` be a nonnegative submartingale indexed by a densely ordered separable time set, such as `\u211d\u22650`, whose paths are almost surely right-continuous at every time before `T`. Then `\u03b5 P(\u2203 t \u2264 T, \u03b5 < f_t) \u2264 E[f_T]`.","title":"Doob's maximal inequality in continuous time"},{"id":"tauceti/probability/other-results/the-efron-stein-inequality-on-measure-pi-over-a","kind":"node","lean":"TauCeti.Probability.efronStein","parent":3601,"result":true,"route":"../roadmap/tauceti/probability/other-results/index.html#the-efron-stein-inequality-on-measure-pi-over-a","scope":"tauceti/probability/other-results","status":"fully_proved","summary":"for a square integrable measurable function `f` of independent coordinates, the variance of `f` is at most the sum over `i` of the average variance of `f` in coordinate `i` with the other coordinates frozen.","title":"The Efron\u2013Stein inequality on `Measure.pi` over a finite index type"},{"id":"tauceti/probability/other-results/the-elementary-renewal-theorem","kind":"node","lean":"MeasureTheory.Measure.tendsto_renewalFunction_div, TauCeti.Probability.tendsto_integral_renewalCountNat_div","parent":3601,"result":true,"route":"../roadmap/tauceti/probability/other-results/index.html#the-elementary-renewal-theorem","scope":"tauceti/probability/other-results","status":"fully_proved","summary":"For a probability measure `\u03bc` on `\u211d\u22650` other than the point mass at zero, the renewal function satisfies `m t / t \u2192 (\ud835\udd3c \u03bc)\u207b\u00b9` as `t \u2192 \u221e`, where the reciprocal of the extended mean is taken before conversion to a real number, so that the limit is zero for infinite mean.","title":"The elementary renewal theorem"},{"id":"tauceti/probability/other-results/girsanovs-theorem-for-a-constant-drift","kind":"node","lean":"TauCeti.Probability.IsFilteredPreBrownianReal.sub_mul_min_tilted_mul_eval, TauCeti.Probability.IsFilteredPreBrownianReal.isBrownianReal_sub_mul_min_tilted_mul_eval","parent":3601,"result":true,"route":"../roadmap/tauceti/probability/other-results/index.html#girsanovs-theorem-for-a-constant-drift","scope":"tauceti/probability/other-results","status":"fully_proved","summary":"If `W` is a Brownian motion relative to `\u2131` under `P`, then `t \u21a6 W t - \u03b8 min(t, T)` is a Brownian motion relative to `\u2131` under the measure `P.tilted (\u03b8 * W T \u00b7)`, whose density with respect to `P` is `exp (\u03b8 W_T - \u03b8\u00b2 T / 2)`.","title":"Girsanov's theorem for a constant drift"},{"id":"tauceti/probability/other-results/levys-downward-theorem-specializes-correctly-to","kind":"node","lean":"TauCeti.Probability.tendsto_ae_condExp_of_eventually_const","parent":3601,"result":true,"route":"../roadmap/tauceti/probability/other-results/index.html#levys-downward-theorem-specializes-correctly-to","scope":"tauceti/probability/other-results","status":"fully_proved","summary":"Run the flagship `MeasureTheory.tendsto_ae_condExp_iInf` on an antitone filtration that is constant from index `N` on, then collapse its limit through `condExp_iInf_eq_of_eventually_const`. The a.e. limit it delivers is `\u03bc[f | \ud835\udd3d N]` \u2014 exactly the value the sequence already reaches by the elementary `tendsto_condExp_of_eventually_const`. The general and elementary computations agree.","title":"L\u00e9vy's downward theorem specializes correctly to an eventually constant filtration"},{"id":"tauceti/probability/other-results/the-mean-ergodic-theorem-for-conditional-expecta","kind":"node","lean":"TauCeti.Probability.tendsto_eLpNorm_birkhoffAverage_sub_condExp","parent":3601,"result":true,"route":"../roadmap/tauceti/probability/other-results/index.html#the-mean-ergodic-theorem-for-conditional-expecta","scope":"tauceti/probability/other-results","status":"fully_proved","summary":"The Birkhoff time averages of an integrable square-integrable observable converge in `L\u00b2` to its conditional expectation given the invariant \u03c3-algebra of the transformation.","title":"The mean ergodic theorem for conditional expectations"},{"id":"tauceti/probability/other-results/the-renewal-central-limit-theorem","kind":"node","lean":"TauCeti.Probability.tendstoInDistribution_renewalCountNat_sub_div, TauCeti.Probability.tendstoInDistribution_renewalCountNat_sub_div_toReal_lintegral, TauCeti.Probability.tendstoInDistribution_renewalCountNat_toNNReal_sub_div","parent":3601,"result":true,"route":"../roadmap/tauceti/probability/other-results/index.html#the-renewal-central-limit-theorem","scope":"tauceti/probability/other-results","status":"fully_proved","summary":"Let `T 0, T 1, \u2026` be independent, identically distributed nonnegative waiting times with nonzero variance `\u03c3\u00b2 = Var[T 0; P]` and expectation `\u03bc = P[T 0]`. As `t \u2192 \u221e` in `\u211d\u22650`, the standardized renewal count `((N_t : \u211d) - t / \u03bc) / (\u03c3 * \u221a(t / \u03bc ^ 3))` converges in distribution to any standard normal random variable `Z`. Precomposing with `MeasureTheory.TendstoInDistribution.comp_tendsto` gives convergence along any\u2026","title":"The renewal central limit theorem"},{"data_url":"scopes/tauceti/quantum-groups.json","id":"tauceti/quantum-groups","kind":"scope","parent":1095,"result":false,"route":"../roadmap/tauceti/quantum-groups/index.html","scope":"tauceti/quantum-groups","status":"fully_proved","summary":"Results in Tau Ceti's `TauCeti.QuantumGroups` modules, and the roadmaps whose articles are mostly there: by roadmap, then by area.","title":"Quantum Groups"},{"data_url":"scopes/tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi.json","id":"tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi","kind":"scope","parent":3613,"result":false,"route":"../roadmap/tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi/index.html","scope":"tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Quantum groups: compact reconstruction and generic rank-one representations"},{"id":"tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi/complete-reducibility","kind":"node","lean":"TauCeti.UqSl2.isSemisimpleModule_of_finiteDimensional","parent":3614,"result":true,"route":"../roadmap/tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi/index.html#complete-reducibility","scope":"tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi","status":"fully_proved","summary":"At a generic parameter, every finite-dimensional `UqSl2 q`-module is semisimple.","title":"Complete reducibility"},{"id":"tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi/tauceti-uqsl2-islocallyfinitemodule-iff-exists-l","kind":"node","lean":"TauCeti.UqSl2.isLocallyFiniteModule_iff_exists_linearEquiv_dfinsupp_ladderModule","parent":3614,"result":true,"route":"../roadmap/tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi/index.html#tauceti-uqsl2-islocallyfinitemodule-iff-exists-l","scope":"tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi","status":"fully_proved","summary":"At a generic parameter, a `UqSl2 q`-module is locally finite exactly when it is isomorphic to a direct sum of signed ladder modules `L_m^\u03b5`.","title":"TauCeti.UqSl2.isLocallyFiniteModule_iff_exists_linearEquiv_dfinsupp_ladderModule"},{"id":"tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi/type-1-locally-finite-modules","kind":"node","lean":"TauCeti.UqSl2.isTypeOne_and_isLocallyFiniteModule_iff_exists_linearEquiv_dfinsupp_ladderModule","parent":3614,"result":true,"route":"../roadmap/tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi/index.html#type-1-locally-finite-modules","scope":"tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi","status":"fully_proved","summary":"At a generic parameter, a module is of type 1 and locally finite exactly when it is isomorphic to a direct sum of positive signed ladder modules `L_m^+`.","title":"Type-1 locally finite modules"},{"id":"tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi/classification-of-finite-dimensional-simple-u-q","kind":"node","lean":"TauCeti.UqSl2.existsUnique_linearEquiv_ladderModule","parent":3614,"result":true,"route":"../roadmap/tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi/index.html#classification-of-finite-dimensional-simple-u-q","scope":"tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi","status":"fully_proved","summary":"At a generic parameter, every finite-dimensional simple `UqSl2 q`-module is isomorphic to `L_m^\u03b5` for exactly one natural number `m` and sign `\u03b5`.","title":"Classification of finite-dimensional simple `U_q(sl\u2082)`-modules"},{"id":"tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi/tauceti-uqsl2-polynomialalgequivcenter","kind":"node","lean":"TauCeti.UqSl2.polynomialAlgEquivCenter","parent":3614,"result":true,"route":"../roadmap/tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi/index.html#tauceti-uqsl2-polynomialalgequivcenter","scope":"tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi","status":"fully_proved","summary":"At a generic parameter, the center of `U_q(sl\u2082)` is a polynomial algebra in one variable: evaluation at the Casimir element is an isomorphism `\u2102[X] \u2243\u2090[\u2102] Z(U_q(sl\u2082))`.","title":"TauCeti.UqSl2.polynomialAlgEquivCenter"},{"id":"tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi/tauceti-uqsl2-pbwbasis","kind":"node","lean":"TauCeti.UqSl2.pbwBasis","parent":3614,"result":true,"route":"../roadmap/tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi/index.html#tauceti-uqsl2-pbwbasis","scope":"tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi","status":"fully_proved","summary":"The Poincar\u00e9--Birkhoff--Witt basis of `UqSl2 q`, indexed by the exponents `(a, b, c)` of the ordered monomial `F ^ a * K ^ b * E ^ c`.","title":"TauCeti.UqSl2.pbwBasis"},{"id":"tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi/triangular-decomposition-of-u-q-sl2","kind":"node","lean":"TauCeti.UqSl2.triangularEquiv","parent":3614,"result":true,"route":"../roadmap/tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi/index.html#triangular-decomposition-of-u-q-sl2","scope":"tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi","status":"fully_proved","summary":"Multiplication `p \u2297 r \u2297 s \u21a6 p(F) * r(K) * s(E)` is a linear isomorphism `\u2102[X] \u2297 \u2102[T, T\u207b\u00b9] \u2297 \u2102[X] \u2243 UqSl2 q`, that is, `U\u207b \u2297 U\u2070 \u2297 U\u207a \u2243 U_q(sl\u2082)`. It sends the tensor product of monomial bases to the PBW basis.","title":"Triangular decomposition of `U_q(sl\u2082)`"},{"id":"tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi/tauceti-uqsl2-instisdomain","kind":"node","lean":"TauCeti.UqSl2.instIsDomain","parent":3614,"result":true,"route":"../roadmap/tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi/index.html#tauceti-uqsl2-instisdomain","scope":"tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi","status":"fully_proved","summary":"For a nonzero parameter, `UqSl2 q` is a domain: both twists of its Ore tower are injective, and an Ore extension of a domain by an injective twist is a domain. In particular the presentation is nontrivial.","title":"TauCeti.UqSl2.instIsDomain"},{"id":"tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi/tauceti-oreextension","kind":"node","lean":"TauCeti.OreExtension","parent":3614,"result":true,"route":"../roadmap/tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi/index.html#tauceti-oreextension","scope":"tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi","status":"fully_proved","summary":"The Ore extension `R[X; \u03c3, \u03b4]`, represented by its finitely supported left coefficients.","title":"TauCeti.OreExtension"},{"id":"tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi/tauceti-uqsl2-laddermodule","kind":"node","lean":"TauCeti.UqSl2.LadderModule","parent":3614,"result":true,"route":"../roadmap/tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi/index.html#tauceti-uqsl2-laddermodule","scope":"tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi","status":"fully_proved","summary":"The underlying complex vector space `\u2102^(m + 1)` of the signed ladder module `L_m^\u03b5` of `UqSl2 q`. Its `UqSl2 q`-module structure, defined for nonzero `q`, is `LadderModule.instModule`.","title":"TauCeti.UqSl2.LadderModule"},{"id":"tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi/the-simple-quotient-of-m-q-m","kind":"node","lean":"TauCeti.UqSl2.VermaModule.ladderModuleEquivQuotient","parent":3614,"result":true,"route":"../roadmap/tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi/index.html#the-simple-quotient-of-m-q-m","scope":"tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi","status":"fully_proved","summary":"At a generic parameter, the quotient of `M(\u03b5 q^m)` by the submodule generated by `v_(m + 1)` is isomorphic to the signed ladder module `L_m^\u03b5`, by the module map sending `v\u2080` to the class of `v\u2080`.","title":"The simple quotient of `M(\u03b5 q^m)`"},{"id":"tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi/the-evaluation-isomorphism","kind":"node","lean":"TauCeti.UqSl2.ladderModuleTensorHomEquiv","parent":3614,"result":true,"route":"../roadmap/tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi/index.html#the-evaluation-isomorphism","scope":"tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi","status":"fully_proved","summary":"At a generic parameter, a locally finite `UqSl2 q`-module `V` is the direct sum over `(m, \u03b5)` of `L_m^\u03b5 \u2297 Hom_U(L_m^\u03b5, V)`, by `v \u2297 \u03c6 \u21a6 \u03c6 v`. Here `UqSl2 q` acts on the left tensor factors, and the multiplicity spaces `Hom_U(L_m^\u03b5, V)` may be infinite-dimensional.","title":"The evaluation isomorphism"},{"id":"tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi/the-isotypic-decomposition-of-a-semisimple-modul","kind":"node","lean":"TauCeti.directSumIsotypicTensorEquiv","parent":3614,"result":true,"route":"../roadmap/tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi/index.html#the-isotypic-decomposition-of-a-semisimple-modul","scope":"tauceti/quantum-groups/quantum-groups-compact-reconstruction-and-generi","status":"fully_proved","summary":"Let `S j` be pairwise non-isomorphic simple modules, finite-dimensional over the algebraically closed field `k`, such that every simple submodule of the semisimple module `M` is isomorphic to one of them. Then evaluation `s \u2297 f \u21a6 f s` is an isomorphism `\u2a01\u2c7c S j \u2297[k] (S j \u2192\u2097[A] M) \u2243 M` of `A`-modules.","title":"The isotypic decomposition of a semisimple module"},{"data_url":"scopes/tauceti/quantum-groups/other-results.json","id":"tauceti/quantum-groups/other-results","kind":"scope","parent":3613,"result":false,"route":"../roadmap/tauceti/quantum-groups/other-results/index.html","scope":"tauceti/quantum-groups/other-results","status":"fully_proved","summary":"Results named in the rest of Tau Ceti's `TauCeti.QuantumGroups` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Other results"},{"id":"tauceti/quantum-groups/other-results/the-biedenharn-elliott-pentagon-identity","kind":"node","lean":"TauCeti.UqSl2.sixJ_pentagon","parent":3628,"result":true,"route":"../roadmap/tauceti/quantum-groups/other-results/index.html#the-biedenharn-elliott-pentagon-identity","scope":"tauceti/quantum-groups/other-results","status":"fully_proved","summary":"The Biedenharn\u2013Elliott pentagon identity for the `\u03b1\u2080 = 1` change-of-decomposition (`F`-matrix) entries `sixJ` of `clebschGordanInclusion` across all signed ladder modules `L_a^\u03b5 \u2297 L_b^\u03b4 \u2297 L_c^\u03b6 \u2297 L_d^\u03b8` at generic `q`.","title":"The Biedenharn\u2013Elliott pentagon identity"},{"data_url":"scopes/tauceti/representation-theory.json","id":"tauceti/representation-theory","kind":"scope","parent":1095,"result":false,"route":"../roadmap/tauceti/representation-theory/index.html","scope":"tauceti/representation-theory","status":"fully_proved","summary":"Results in Tau Ceti's `TauCeti.RepresentationTheory` modules, and the roadmaps whose articles are mostly there: by roadmap, then by area.","title":"Representation Theory"},{"data_url":"scopes/tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr.json","id":"tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr","kind":"scope","parent":3630,"result":false,"route":"../roadmap/tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr/index.html","scope":"tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Complex tori: varying lattices and logarithmic transforms"},{"id":"tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr/the-integral-connecting-class","kind":"node","lean":"TauCeti.\u03b4_cyclicRepShortComplex_groupCohomology\u03c0Odd_one","parent":3631,"result":true,"route":"../roadmap/tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr/index.html#the-integral-connecting-class","scope":"tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr","status":"fully_proved","summary":"For the short exact sequence `0 \u2192 \u039b \u2192 E \u2192 T \u2192 0` of cyclic representations, let `t` be killed by the norm `\u2211 j < m, AT^j`, lift it to `s : E`, and write the norm `\u2211 j < m, AE^j s` as the image of a fixed point `v` of `A\u039b`. Then the connecting homomorphism sends the degree-`1` class of `t` to the degree-`2` class of `v`.","title":"The integral connecting class"},{"id":"tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr/the-connecting-isomorphism-h1-c-m-t-h2-c-m","kind":"node","lean":"TauCeti.cyclicConnectingEquiv","parent":3631,"result":true,"route":"../roadmap/tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr/index.html#the-connecting-isomorphism-h1-c-m-t-h2-c-m","scope":"tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr","status":"fully_proved","summary":"When multiplication by `m` is bijective on `E`, for instance when `E` is a real vector space, the cohomology of `E` vanishes in positive degrees and the connecting homomorphism of `0 \u2192 \u039b \u2192 E \u2192 T \u2192 0` is an isomorphism.","title":"The connecting isomorphism `H\u00b9(C_m, T) \u2243 H\u00b2(C_m, \u039b)`"},{"id":"tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr/naturality-under-homomorphisms-of-cyclic-groups","kind":"node","lean":"TauCeti.map_\u03b4_cyclicRepShortComplex","parent":3631,"result":true,"route":"../roadmap/tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr/index.html#naturality-under-homomorphisms-of-cyclic-groups","scope":"tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr","status":"fully_proved","summary":"Suppose `\u03c7 : C_l \u2192 C_m` sends the selected generator to its `d`-th power. Maps of exact sequences intertwining the `d`-th iterates of the source automorphisms with the target automorphisms commute with the connecting homomorphisms. This includes restriction, inflation, and coefficient maps.","title":"Naturality under homomorphisms of cyclic groups"},{"id":"tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr/odd-degree-classes-of-the-cyclic-representation","kind":"node","lean":"AddEquiv.groupCohomology\u03c0Odd_cyclicRep_eq_iff","parent":3631,"result":true,"route":"../roadmap/tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr/index.html#odd-degree-classes-of-the-cyclic-representation","scope":"tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr","status":"fully_proved","summary":"Two norm-zero elements `x` and `y` have the same class in odd degree exactly when `y = x + (b - A b)` for some `b : T`.","title":"Odd-degree classes of the cyclic representation"},{"id":"tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr/odd-degree-cohomology-of-the-trivial-action","kind":"node","lean":"TauCeti.groupCohomologyOddEquivTorsionBy","parent":3631,"result":true,"route":"../roadmap/tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr/index.html#odd-degree-cohomology-of-the-trivial-action","scope":"tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr","status":"fully_proved","summary":"When `C_m` acts trivially on `T`, its cohomology in odd degree is the `m`-torsion `T[m]`: the class of a norm-zero element `x` corresponds to `x` itself.","title":"Odd-degree cohomology of the trivial action"},{"id":"tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr/uniform-bound-for-inverse-period-maps","kind":"node","lean":"TauCeti.exists_pos_mul_norm_le_norm_sum_smul_period_of_isCompact","parent":3631,"result":true,"route":"../roadmap/tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr/index.html#uniform-bound-for-inverse-period-maps","scope":"tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr","status":"fully_proved","summary":"For a basis `b` of `\u039b`, the real linear maps `x \u21a6 \u2211 i, x i \u2022 P y (b i)` are bounded below by one constant `c > 0` as `y` ranges over a compact set: `c * \u2016x\u2016 \u2264 \u2016\u2211 i, x i \u2022 P y (b i)\u2016`.","title":"Uniform bound for inverse period maps"},{"id":"tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr/the-global-trivialization-of-a-continuous-family","kind":"node","lean":"TauCeti.torusFamilyHomeomorphProd","parent":3631,"result":true,"route":"../roadmap/tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr/index.html#the-global-trivialization-of-a-continuous-family","scope":"tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr","status":"fully_proved","summary":"For a `\u2124`-basis `b` of `\u039b` indexed by `\u03b9`, the map `[y, v] \u21a6 (y, [F_y\u207b\u00b9 v])`, with `F_y\u207b\u00b9` the coordinate map of the period basis at `y`, is a homeomorphism from the orbit space of the period translation action to the product of `Y` with the fixed torus `\u211d^\u03b9/\u2124^\u03b9`. It lies over `Y` (`TauCeti.fst_torusFamilyHomeomorphProd`), and on each fibre it is induced by the real linear isomorphism `F_y\u207b\u00b9`.","title":"The global trivialization of a continuous family of full lattices"},{"id":"tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr/addequiv-cyclicrep","kind":"node","lean":"AddEquiv.cyclicRep","parent":3631,"result":true,"route":"../roadmap/tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr/index.html#addequiv-cyclicrep","scope":"tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr","status":"fully_proved","summary":"The cyclic representation of `A` as an object of `Rep \u2124 C_m`.","title":"AddEquiv.cyclicRep"},{"id":"tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr/tauceti-torusfamilyprojection","kind":"node","lean":"TauCeti.torusFamilyProjection","parent":3631,"result":true,"route":"../roadmap/tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr/index.html#tauceti-torusfamilyprojection","scope":"tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr","status":"fully_proved","summary":"The projection `(Y \u00d7 E)/\u039b \u2192 Y` of the family of tori to its parameter space, descended from `Prod.fst` along the period translation action.","title":"TauCeti.torusFamilyProjection"},{"id":"tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr/tauceti-torusfamilyfiberhomeomorph","kind":"node","lean":"TauCeti.torusFamilyFiberHomeomorph","parent":3631,"result":true,"route":"../roadmap/tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr/index.html#tauceti-torusfamilyfiberhomeomorph","scope":"tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr","status":"fully_proved","summary":"The canonical identification of the translation quotient at `y` with the literal fibre of the total period quotient. It sends `[v]` to `[y, v]` and needs no lattice basis.","title":"TauCeti.torusFamilyFiberHomeomorph"},{"id":"tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr/tauceti-periodbasis","kind":"node","lean":"TauCeti.periodBasis","parent":3631,"result":true,"route":"../roadmap/tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr/index.html#tauceti-periodbasis","scope":"tauceti/representation-theory/complex-tori-varying-lattices-and-logarithmic-tr","status":"fully_proved","summary":"The period basis at `y`: the periods `P y (b i)` of a `\u2124`-basis `b` of `\u039b`, viewed as a real basis of `E`. They form a `\u2124`-basis of the full lattice `LinearMap.range (P y)`, hence a real basis of `E`. Its inverse coordinate map `x \u21a6 \u2211 i, x i \u2022 P y (b i)` is the real linear isomorphism `\u211d^\u03b9 \u2243 E` carrying the standard lattice `\u2124^\u03b9` onto the periods at `y`.","title":"TauCeti.periodBasis"},{"data_url":"scopes/tauceti/representation-theory/continuous-cohomology-of-profinite-groups.json","id":"tauceti/representation-theory/continuous-cohomology-of-profinite-groups","kind":"scope","parent":3630,"result":false,"route":"../roadmap/tauceti/representation-theory/continuous-cohomology-of-profinite-groups/index.html","scope":"tauceti/representation-theory/continuous-cohomology-of-profinite-groups","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Continuous cohomology of profinite groups"},{"id":"tauceti/representation-theory/continuous-cohomology-of-profinite-groups/hilbert-90-for-the-absolute-galois-group-stated","kind":"node","lean":"TauCeti.hilbert90","parent":3643,"result":true,"route":"../roadmap/tauceti/representation-theory/continuous-cohomology-of-profinite-groups/index.html#hilbert-90-for-the-absolute-galois-group-stated","scope":"tauceti/representation-theory/continuous-cohomology-of-profinite-groups","status":"fully_proved","summary":"`H\u00b9(G_K, (K\u02e2)\u02e3) = 0` (NSW (6.2.1)). This is the form in which the vanishing feeds the canonical all-degree theory, where it makes the Kummer map `K\u02e3 \u2192 H\u00b9(G_K, \u03bc\u2099)` surjective for `n` positive and invertible in `K`.","title":"Hilbert 90 for the absolute Galois group, stated for Mathlib's canonical continuous cohomology"},{"id":"tauceti/representation-theory/continuous-cohomology-of-profinite-groups/the-kummer-isomorphism","kind":"node","lean":"TauCeti.kummerIso","parent":3643,"result":true,"route":"../roadmap/tauceti/representation-theory/continuous-cohomology-of-profinite-groups/index.html#the-kummer-isomorphism","scope":"tauceti/representation-theory/continuous-cohomology-of-profinite-groups","status":"fully_proved","summary":"The Kummer isomorphism `K\u02e3 \u29f8 (K\u02e3)\u207f \u2243* H\u00b9(G_K, \u03bc\u2099)` (NSW (6.2.1) and the display following it), for `n` invertible in `K`. It is `TauCeti.kummerClassMap`, which is injective since the kernel of the Kummer map is `(K\u02e3)\u207f` and surjective by Hilbert 90.","title":"The Kummer isomorphism"},{"id":"tauceti/representation-theory/continuous-cohomology-of-profinite-groups/the-strict-cohomological-dimension-through-integ","kind":"node","lean":"TauCeti.strictCohomologicalDimensionLE_iff_forall_openSubgroup","parent":3643,"result":true,"route":"../roadmap/tauceti/representation-theory/continuous-cohomology-of-profinite-groups/index.html#the-strict-cohomological-dimension-through-integ","scope":"tauceti/representation-theory/continuous-cohomology-of-profinite-groups","status":"fully_proved","summary":"For a profinite group `G` and a prime `p`, `StrictCohomologicalDimensionLE p G n` holds if and only if `CohomologicalDimensionLE p G n` holds and the `p`-primary component of `H\u207f\u207a\u00b9(U, \u2124)` vanishes for every open subgroup `U`, the integers carrying the trivial action.","title":"The strict cohomological dimension through integral coefficients (NSW (3.3.4)), as vanishing predicates"},{"id":"tauceti/representation-theory/continuous-cohomology-of-profinite-groups/the-p-cohomological-dimension-of-the-profinite-i","kind":"node","lean":"TauCeti.zHat.cohomologicalDimensionAt_eq_one","parent":3643,"result":true,"route":"../roadmap/tauceti/representation-theory/continuous-cohomology-of-profinite-groups/index.html#the-p-cohomological-dimension-of-the-profinite-i","scope":"tauceti/representation-theory/continuous-cohomology-of-profinite-groups","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.zHat.cohomologicalDimensionAt_eq_one`.","title":"The `p`-cohomological dimension of the profinite integers is one, for every prime `p`"},{"id":"tauceti/representation-theory/continuous-cohomology-of-profinite-groups/tauceti-zhat-h2equivratcircle","kind":"node","lean":"TauCeti.zHat.h2EquivRatCircle","parent":3643,"result":true,"route":"../roadmap/tauceti/representation-theory/continuous-cohomology-of-profinite-groups/index.html#tauceti-zhat-h2equivratcircle","scope":"tauceti/representation-theory/continuous-cohomology-of-profinite-groups","status":"fully_proved","summary":"Second integral continuous cohomology of the profinite integers is the rational circle. The inverse sends a rational-circle value to the connecting class of the unique continuous character taking that value at the canonical generator.","title":"TauCeti.zHat.h2EquivRatCircle"},{"id":"tauceti/representation-theory/continuous-cohomology-of-profinite-groups/the-index-two-corestriction-cup-sequence-is-exac","kind":"node","lean":"OpenSubgroup.exact_cor_cup_of_index_two","parent":3643,"result":true,"route":"../roadmap/tauceti/representation-theory/continuous-cohomology-of-profinite-groups/index.html#the-index-two-corestriction-cup-sequence-is-exac","scope":"tauceti/representation-theory/continuous-cohomology-of-profinite-groups","status":"fully_proved","summary":"The kernel of cupping with the subgroup character consists exactly of the corestrictions from `U`.","title":"The index-two corestriction--cup sequence is exact in every degree"},{"id":"tauceti/representation-theory/continuous-cohomology-of-profinite-groups/tauceti-cyclictwocohompolynomialequiv","kind":"node","lean":"TauCeti.cyclicTwoCohomPolynomialEquiv","parent":3643,"result":true,"route":"../roadmap/tauceti/representation-theory/continuous-cohomology-of-profinite-groups/index.html#tauceti-cyclictwocohompolynomialequiv","scope":"tauceti/representation-theory/continuous-cohomology-of-profinite-groups","status":"fully_proved","summary":"The canonical graded mod-two cohomology algebra of an order-two discrete group is a polynomial algebra, with `X` sent to its nontrivial character in degree one.","title":"TauCeti.cyclicTwoCohomPolynomialEquiv"},{"id":"tauceti/representation-theory/continuous-cohomology-of-profinite-groups/tauceti-contcohomology-smoothtensorinduceddiagon","kind":"node","lean":"TauCeti.ContCohomology.smoothTensorInducedDiagonalClass_eq_graphClass","parent":3643,"result":true,"route":"../roadmap/tauceti/representation-theory/continuous-cohomology-of-profinite-groups/index.html#tauceti-contcohomology-smoothtensorinduceddiagon","scope":"tauceti/representation-theory/continuous-cohomology-of-profinite-groups","status":"fully_proved","summary":"The tensor-induced class of the homogeneous degree-one cocycle of a continuous character is the index-two graph class. The transversal has representatives `1` and `s`, and the coefficient identification multiplies the two tensor factors. The equality includes only the transport of degrees from `2` to `card (G \u29f8 U) * 1`.","title":"TauCeti.ContCohomology.smoothTensorInducedDiagonalClass_eq_graphClass"},{"id":"tauceti/representation-theory/continuous-cohomology-of-profinite-groups/dimension-shifting-in-every-positive-degree","kind":"node","lean":"TauCeti.ContCohomology.dimensionShiftIso","parent":3643,"result":true,"route":"../roadmap/tauceti/representation-theory/continuous-cohomology-of-profinite-groups/index.html#dimension-shifting-in-every-positive-degree","scope":"tauceti/representation-theory/continuous-cohomology-of-profinite-groups","status":"fully_proved","summary":"Dimension shifting in every positive degree, `H\u2071\u207a\u00b9(G, M) \u2245 H\u2071(G, Coind_1^G M \u29f8 M)` for `i \u2265 1` and a compact group `G`, as an isomorphism of Mathlib's canonical continuous cohomology. Its inverse is the connecting map of `TauCeti.ContCohomology.coindShortExact G \u22a5 M` (`dimensionShiftIso_inv`), an isomorphism by `isIso_coindShortExact_bot_delta`. The analogous statement on the explicit low-degree model, for a\u2026","title":"Dimension shifting in every positive degree"},{"id":"tauceti/representation-theory/continuous-cohomology-of-profinite-groups/the-index-two-degree-one-evens-norm","kind":"node","lean":"TauCeti.ContCohomology.evensNormIndexTwo","parent":3643,"result":true,"route":"../roadmap/tauceti/representation-theory/continuous-cohomology-of-profinite-groups/index.html#the-index-two-degree-one-evens-norm","scope":"tauceti/representation-theory/continuous-cohomology-of-profinite-groups","status":"fully_proved","summary":"The index-two degree-one Evens norm `N\u1d31\u1d5b : H\u00b9(U, \ud835\udd3d\u2082) \u2192 H\u00b2(G, \ud835\udd3d\u2082)` for an open subgroup `U` of index two: the graph class of a continuous homomorphism representing the class, which `homClass_surjective` provides and `graphClass_representative_independent` makes irrelevant. On the class of `\u03b1` it is `graphClass U hU \u03b1` (`evensNormIndexTwo_homClass`). It is a function and not an additive map.","title":"The index-two degree-one Evens norm"},{"id":"tauceti/representation-theory/continuous-cohomology-of-profinite-groups/the-evens-norm-of-a-cocycle","kind":"node","lean":"TauCeti.smoothTensorInducedDiagonalClass","parent":3643,"result":true,"route":"../roadmap/tauceti/representation-theory/continuous-cohomology-of-profinite-groups/index.html#the-evens-norm-of-a-cocycle","scope":"tauceti/representation-theory/continuous-cohomology-of-profinite-groups","status":"fully_proved","summary":"for a homogeneous cocycle `z` of degree `n` of the total coinduced resolution of `A`, fixed by `U`, the class in `H^{l n}(G, Ind\u2297 A)` of its diagonal tensor `z \u2297 \u22ef \u2297 z`, one factor for each of the `l` cosets of `U`. Here `A` has characteristic two, so that the tensor-induced resolution is a complex, and the class is computed through `TauCeti.smoothTensorInducedGradedResolutionContraction` for some order on the\u2026","title":"The Evens norm of a cocycle"},{"id":"tauceti/representation-theory/continuous-cohomology-of-profinite-groups/tauceti-smoothtensorinduceddiagonalclass-trivial","kind":"node","lean":"TauCeti.smoothTensorInducedDiagonalClass_trivialF2_changeTransversal","parent":3643,"result":true,"route":"../roadmap/tauceti/representation-theory/continuous-cohomology-of-profinite-groups/index.html#tauceti-smoothtensorinduceddiagonalclass-trivial","scope":"tauceti/representation-theory/continuous-cohomology-of-profinite-groups","status":"fully_proved","summary":"After identifying tensor-induced trivial `\ud835\udd3d\u2082` coefficients with the ambient trivial coefficient object, the class of an invariant diagonal cocycle does not depend on the transversal.","title":"TauCeti.smoothTensorInducedDiagonalClass_trivialF2_changeTransversal"},{"id":"tauceti/representation-theory/continuous-cohomology-of-profinite-groups/artins-theorem-for-a-finite-group-of-automorphis","kind":"node","lean":"TauCeti.galEquivOfInjective","parent":3643,"result":true,"route":"../roadmap/tauceti/representation-theory/continuous-cohomology-of-profinite-groups/index.html#artins-theorem-for-a-finite-group-of-automorphis","scope":"tauceti/representation-theory/continuous-cohomology-of-profinite-groups","status":"fully_proved","summary":"An injective homomorphism `f : H \u2192* Gal(L/K)` from a finite group identifies `H` with the Galois group of `L` over the fixed field of the image of `f` (`FixedPoints.toAlgAutMulEquiv`). No finiteness of `L/K` is needed.","title":"Artin's theorem for a finite group of automorphisms"},{"data_url":"scopes/tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex.json","id":"tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex","kind":"scope","parent":3630,"result":false,"route":"../roadmap/tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex/index.html","scope":"tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Finite reflection invariants, degrees, and Weyl exponents"},{"id":"tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex/moliens-formula","kind":"node","lean":"Representation.mk_finrank_symmetricAlgebraInvariants_inf_homogeneousSubmodule_eq_sum_inv_reverse_charpoly, Representation.mk_finrank_symmetricAlgebraInvariants_inf_homogeneousSubmodule, Representation.finrank_symmetricAlgebraInvariants_inf_homogeneousSubmodule","parent":3657,"result":true,"route":"../roadmap/tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex/index.html#moliens-formula","scope":"tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex","status":"fully_proved","summary":"the image in `K\u27e6X\u27e7` of the Hilbert series of the invariant subalgebra is the average over the group of the inverses of `det(1 - t g | M)`, the reverse of the characteristic polynomial of `\u03c1 g`: `Hilb(Sym(M)^G, t) = #G\u207b\u00b9 \u2211_{g \u2208 G} det(1 - t g | M)\u207b\u00b9`.","title":"Molien's formula"},{"id":"tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex/noethers-finiteness-theorem-tauceti-fixedpoints","kind":"node","lean":"TauCeti.FixedPoints.finiteType_subalgebra","parent":3657,"result":true,"route":"../roadmap/tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex/index.html#noethers-finiteness-theorem-tauceti-fixedpoints","scope":"tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex","status":"fully_proved","summary":"Over a Noetherian base ring `R`, the invariants of a finite group acting on a finite-type `R`-algebra form a finite-type `R`-algebra.","title":"Noether's finiteness theorem (`TauCeti.FixedPoints.finiteType_subalgebra`)"},{"id":"tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex/noethers-degree-bound","kind":"node","lean":"TauCeti.FixedPoints.subalgebra_eq_adjoin_sum_smul_prod_pow","parent":3657,"result":true,"route":"../roadmap/tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex/index.html#noethers-degree-bound","scope":"tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex","status":"fully_proved","summary":"If a family `x`, finite or not, generates a commutative algebra `A` over a `\u211a`-algebra `R`, then the invariants of a finite group `G` acting on `A` are generated by the orbit sums `\u2211_{g \u2208 G} g \u2022 x^d` of the monomials `x^d` of total degree at most `|G|`.","title":"Noether's degree bound"},{"id":"tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex/invariant-polynomials-detect-orbits","kind":"node","lean":"Representation.forall_mem_symmetricAlgebraInvariants_lift_eval_eq_iff","parent":3657,"result":true,"route":"../roadmap/tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex/index.html#invariant-polynomials-detect-orbits","scope":"tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex","status":"fully_proved","summary":"Two vectors have the same value on every invariant polynomial function exactly when they lie in the same orbit.","title":"Invariant polynomials detect orbits"},{"id":"tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex/the-fibre-over-the-vertex-is-the-origin","kind":"node","lean":"Representation.forall_homogeneous_mem_symmetricAlgebraInvariants_lift_eval_eq_zero_iff","parent":3657,"result":true,"route":"../roadmap/tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex/index.html#the-fibre-over-the-vertex-is-the-origin","scope":"tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex","status":"fully_proved","summary":"Every homogeneous invariant polynomial function of positive degree vanishes at `v` exactly when `v = 0`.","title":"The fibre over the vertex is the origin"},{"id":"tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex/representation-radical-symmetricalgebrahilbertid","kind":"node","lean":"Representation.radical_symmetricAlgebraHilbertIdeal_of_isReduced","parent":3657,"result":true,"route":"../roadmap/tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex/index.html#representation-radical-symmetricalgebrahilbertid","scope":"tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex","status":"fully_proved","summary":"For a finite group over a reduced base, the radical of the Hilbert ideal is the irrelevant ideal.","title":"Representation.radical_symmetricAlgebraHilbertIdeal_of_isReduced"},{"id":"tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex/finiteness-of-the-coinvariant-algebra","kind":"node","lean":"Representation.moduleFinite_quotient_symmetricAlgebraHilbertIdeal","parent":3657,"result":true,"route":"../roadmap/tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex/index.html#finiteness-of-the-coinvariant-algebra","scope":"tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex","status":"fully_proved","summary":"For a finite group acting on a finitely generated module, the coinvariant algebra `Sym M \u29f8 A\u208a \u00b7 Sym M` is a finite `R`-module: it is of finite type, and generated by nilpotent, hence integral, elements.","title":"Finiteness of the coinvariant algebra"},{"id":"tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex/representation-symmetricalgebrainvariants","kind":"node","lean":"Representation.symmetricAlgebraInvariants","parent":3657,"result":true,"route":"../roadmap/tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex/index.html#representation-symmetricalgebrainvariants","scope":"tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex","status":"fully_proved","summary":"The invariant subalgebra `Sym(M)^G`: the elements of `Sym M` fixed by the induced action of every element of `G`, that is, the fixed subalgebra `FixedPoints.subalgebra` of the bundled action.","title":"Representation.symmetricAlgebraInvariants"},{"id":"tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex/representation-symmetricalgebrareynolds","kind":"node","lean":"Representation.symmetricAlgebraReynolds","parent":3657,"result":true,"route":"../roadmap/tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex/index.html#representation-symmetricalgebrareynolds","scope":"tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex","status":"fully_proved","summary":"The Reynolds operator `p \u21a6 #G\u207b\u00b9 \u2022 \u2211 g, g \u2022 p`, from `Sym M` onto the invariant subalgebra. It is linear over the invariant subalgebra and restricts to the identity on it, but it is generally not multiplicative.","title":"Representation.symmetricAlgebraReynolds"},{"id":"tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex/the-graded-trace-formula-for-an-endomorphism-of","kind":"node","lean":"Module.End.mk_trace_homogeneousSubmoduleMap\u209b\u2097_mul_reverse_charpoly_of_pow_eq_one","parent":3657,"result":true,"route":"../roadmap/tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex/index.html#the-graded-trace-formula-for-an-endomorphism-of","scope":"tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex","status":"fully_proved","summary":"The graded trace formula for an endomorphism of finite order of a finite-dimensional vector space over an arbitrary field: if `f ^ n = 1` with `n` nonzero in `K`, then `(\u2211\u2098 tr(f | Sym\u1d50 V) t\u1d50) * det(1 - t f) = 1`.","title":"The graded trace formula for an endomorphism of finite order"},{"id":"tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex/symmetricalgebra-divideddifference","kind":"node","lean":"SymmetricAlgebra.dividedDifference","parent":3657,"result":true,"route":"../roadmap/tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex/index.html#symmetricalgebra-divideddifference","scope":"tauceti/representation-theory/finite-reflection-invariants-degrees-and-weyl-ex","status":"fully_proved","summary":"The divided-difference operator `p \u21a6 (p - \u03c3 p) / \u2113` on the symmetric algebra, for a linear endomorphism `\u03c3` of `M` such that `m - \u03c3 m` is a multiple of `\u2113` for every `m`, and an element `\u2113` that is a nonzerodivisor in `Sym M`. It is characterized by `\u03b9 \u2113 * dividedDifference \u03c3 \u2113 h\u03c3 h\u2113 p = p - \u03c3 p` (`SymmetricAlgebra.dividedDifference_eq_iff`).","title":"SymmetricAlgebra.dividedDifference"},{"data_url":"scopes/tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras.json","id":"tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras","kind":"scope","parent":3630,"result":false,"route":"../roadmap/tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras/index.html","scope":"tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Zigzag, preprojective, and Ginzburg algebras"},{"id":"tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras/the-koszulity-dichotomy-for-zigzag-algebras","kind":"node","lean":"TauCeti.forall_exists_isLinear_zigzagGradedSimple_iff","parent":3669,"result":true,"route":"../roadmap/tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras/index.html#the-koszulity-dichotomy-for-zigzag-algebras","scope":"tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras","status":"fully_proved","summary":"for a connected finite simple graph `G` on at least two vertices, every graded vertex simple of `ZigzagAlgebra k G` admits a linear graded projective resolution if and only if `G` is not isomorphic to the diagram of any valid Dynkin type.","title":"The Koszulity dichotomy for zigzag algebras"},{"id":"tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras/the-finite-infinite-dichotomy-for-preprojective","kind":"node","lean":"TauCeti.finiteDimensional_preprojectiveAlgebra_orientedQuiver_iff","parent":3669,"result":true,"route":"../roadmap/tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras/index.html#the-finite-infinite-dichotomy-for-preprojective","scope":"tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras","status":"fully_proved","summary":"for a connected finite simple graph `G` and any orientation `o` of `G`, the preprojective algebra `\u03a0_k(OrientedQuiver G o)` is finite-dimensional over `k` if and only if `G` is isomorphic to the diagram of a valid Dynkin type.","title":"The finite/infinite dichotomy for preprojective algebras of oriented graphs"},{"id":"tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras/the-koszulity-dichotomy-for-preprojective-algebr","kind":"node","lean":"TauCeti.forall_preprojectiveKoszulLeft_injective_iff","parent":3669,"result":true,"route":"../roadmap/tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras/index.html#the-koszulity-dichotomy-for-preprojective-algebr","scope":"tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras","status":"fully_proved","summary":"for a connected finite simple graph `G` and any orientation `o` of `G`, the left Koszul differential of `\u03a0_k(OrientedQuiver G o)` is injective at every vertex if and only if `G` is not isomorphic to the diagram of any valid Dynkin type.","title":"The Koszulity dichotomy for preprojective algebras of oriented graphs"},{"id":"tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras/the-braid-group-action-of-a-zigzag-algebra","kind":"node","lean":"TauCeti.zigzagBraidArtinAction","parent":3669,"result":true,"route":"../roadmap/tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras/index.html#the-braid-group-action-of-a-zigzag-algebra","scope":"tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras","status":"fully_proved","summary":"The Artin group of the graph `G` acts on the homotopy category of graded `Z`-modules, by sending the standard generator at a vertex `i` to the class of tensoring with the braid complex `B\u1d62`.","title":"The braid group action of a zigzag algebra"},{"id":"tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras/the-action-of-the-graphs-artin-group-on-k0-gr-pr","kind":"node","lean":"TauCeti.zigzagBraidK0Action","parent":3669,"result":true,"route":"../roadmap/tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras/index.html#the-action-of-the-graphs-artin-group-on-k0-gr-pr","scope":"tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras","status":"fully_proved","summary":"The standard generator at `i` acts by the deformed simple reflection at `q`, in the basis of vertex-projective classes; by `TauCeti.zigzagBraidK0Action_gen_zigzagAlgebraGradedProjectiveClass` it sends `[P_j]` to the Euler class of the braid complex `B_i \u2297_Z P_j`.","title":"The action of the graph's Artin group on `K\u2080^gr(proj Z)`"},{"id":"tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras/liu-wangs-a-algebra-is-formal-for-every-scalar-c","kind":"node","lean":"TauCeti.isFormal_liuWangAffineD4AInfinityAlgebra","parent":3669,"result":true,"route":"../roadmap/tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras/index.html#liu-wangs-a-algebra-is-formal-for-every-scalar-c","scope":"tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras","status":"fully_proved","summary":"the displayed quartic operation `m\u2084` is a Hochschild coboundary of the strict zigzag algebra, so it defines the trivial `A\u221e` deformation.","title":"Liu--Wang's `A\u221e` algebra is formal, for every scalar `c`"},{"id":"tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras/the-quartic-hochschild-cohomology-hh-4-2-of-a-zi","kind":"node","lean":"TauCeti.not_hochschildVanishesAt_four_zigzagStrictAInfinityAlgebra","parent":3669,"result":true,"route":"../roadmap/tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras/index.html#the-quartic-hochschild-cohomology-hh-4-2-of-a-zi","scope":"tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras","status":"fully_proved","summary":"The quartic Hochschild cohomology `HH^{4, -2}` of a zigzag algebra is nonzero when four distinct darts of `G` have a common head, that is when some vertex has four distinct neighbours, over every nontrivial commutative ring: some Hochschild cocycle of the strict zigzag algebra of suspended degree one, concentrated on words of length four, is not the Hochschild differential of a cochain of degree zero.","title":"The quartic Hochschild cohomology `HH^{4, -2}` of a zigzag algebra is nonzero"},{"id":"tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras/coutures-classification","kind":"node","lean":"TauCeti.SkewZigzagParameter.cohomologyClass_eq_iff_exists_vertexFixing_algEquiv, TauCeti.SkewZigzagParameter.exists_firstCohomologyRelabel_eq_iff_exists_algEquiv_mem_span, TauCeti.SkewZigzagParameter.exists_firstCohomologyRelabel_eq_iff_exists_vertexPermuting_algEquiv","parent":3669,"result":true,"route":"../roadmap/tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras/index.html#coutures-classification","scope":"tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras","status":"fully_proved","summary":"over a commutative ring, two skew-zigzag parameters of a finite graph have the same cohomology class in `H\u00b9(G, k\u02e3)` exactly when their relation quotients are isomorphic by an algebra isomorphism fixing every vertex idempotent.","title":"Couture's classification"},{"id":"tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras/tauceti-gradedcartanmatrix-zigzagalgebra","kind":"node","lean":"TauCeti.gradedCartanMatrix_zigzagAlgebra","parent":3669,"result":true,"route":"../roadmap/tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras/index.html#tauceti-gradedcartanmatrix-zigzagalgebra","scope":"tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras","status":"fully_proved","summary":"The categorical graded Cartan map of the public algebra has matrix `(1 + q\u00b2) I + q A_G` in the vertex-projective and vertex-head class bases, including isolated vertices.","title":"TauCeti.gradedCartanMatrix_zigzagAlgebra"},{"id":"tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras/tauceti-zigzagbraidtensorlocalizermorphism","kind":"node","lean":"TauCeti.zigzagBraidTensorLocalizerMorphism","parent":3669,"result":true,"route":"../roadmap/tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras/index.html#tauceti-zigzagbraidtensorlocalizermorphism","scope":"tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras","status":"fully_proved","summary":"The positive braid tensor functor defines a morphism of the quasi-isomorphism localizer.","title":"TauCeti.zigzagBraidTensorLocalizerMorphism"},{"id":"tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras/tauceti-zigzagalgebragradedprojectivehomtracepai","kind":"node","lean":"TauCeti.zigzagAlgebraGradedProjectiveHomTracePairing_isPerfPair","parent":3669,"result":true,"route":"../roadmap/tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras/index.html#tauceti-zigzagalgebragradedprojectivehomtracepai","scope":"tauceti/representation-theory/zigzag-preprojective-and-ginzburg-algebras","status":"fully_proved","summary":"Complementary shifted Hom spaces of actual componentwise vertex projectives are in perfect trace pairing over every commutative coefficient ring.","title":"TauCeti.zigzagAlgebraGradedProjectiveHomTracePairing_isPerfPair"},{"data_url":"scopes/tauceti/representation-theory/character-table.json","id":"tauceti/representation-theory/character-table","kind":"scope","parent":3630,"result":false,"route":"../roadmap/tauceti/representation-theory/character-table/index.html","scope":"tauceti/representation-theory/character-table","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.RepresentationTheory.CharacterTable` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Character Table"},{"id":"tauceti/representation-theory/character-table/burnsides-paqb-theorem","kind":"node","lean":"TauCeti.isSolvable_of_card_eq_prime_pow_mul_prime_pow, TauCeti.isSolvable_of_card_dvd_prime_pow_mul_prime_pow","parent":3681,"result":true,"route":"../roadmap/tauceti/representation-theory/character-table/index.html#burnsides-paqb-theorem","scope":"tauceti/representation-theory/character-table","status":"fully_proved","summary":"a finite group of order `p\u1d43q\u1d47`, for primes `p` and `q`, is solvable.","title":"Burnside's `p\u1d43q\u1d47` theorem"},{"id":"tauceti/representation-theory/character-table/burnsides-vanishing-theorem","kind":"node","lean":"Representation.char_eq_zero_or_norm_char_eq_finrank, FDRep.char_eq_zero_or_norm_char_eq_finrank, FDRep.char_eq_zero_or_exists_apply_eq_smul, Representation.char_eq_zero_or_exists_apply_eq_smul","parent":3681,"result":true,"route":"../roadmap/tauceti/representation-theory/character-table/index.html#burnsides-vanishing-theorem","scope":"tauceti/representation-theory/character-table","status":"fully_proved","summary":"If the conjugacy class of `g` has size coprime to the degree of an irreducible complex representation `\u03c1`, then its character either vanishes at `g` or attains there the maximum absolute value `\u03c7(1)` that any character value can have.","title":"Burnside's vanishing theorem"},{"id":"tauceti/representation-theory/character-table/burnside-brauer-conjugacy-class-bound","kind":"node","lean":"FDRep.exists_pos_finrank_hom_tensorPower_lt_card_conjClasses","parent":3681,"result":true,"route":"../roadmap/tauceti/representation-theory/character-table/index.html#burnside-brauer-conjugacy-class-bound","scope":"tauceti/representation-theory/character-table","status":"fully_proved","summary":"Every nonzero finite-dimensional complex representation has a nonzero map into a tensor power of a faithful representation, with exponent strictly smaller than the number of conjugacy classes.","title":"Burnside\u2013Brauer, conjugacy-class bound"},{"id":"tauceti/representation-theory/character-table/dixons-size-bound-in-the-shape-the-residue-windo","kind":"node","lean":"TauCeti.IsGoodDixonPrime.two_mul_natAbs_lt_of_natAbs_le_sqrt","parent":3681,"result":true,"route":"../roadmap/tauceti/representation-theory/character-table/index.html#dixons-size-bound-in-the-shape-the-residue-windo","scope":"tauceti/representation-theory/character-table","status":"fully_proved","summary":"An integer of absolute value at most `\u230a\u221a|G|\u230b` lies strictly inside the window of half-width `p / 2`, since `2\u230a\u221a|G|\u230b < p`.","title":"Dixon's size bound in the shape the residue window asks for"},{"id":"tauceti/representation-theory/character-table/the-frobenius-formula","kind":"node","lean":"TauCeti.card_productOneTriples_eq_sum_characterTable","parent":3681,"result":true,"route":"../roadmap/tauceti/representation-theory/character-table/index.html#the-frobenius-formula","scope":"tauceti/representation-theory/character-table","status":"fully_proved","summary":"The number of triples `(x, y, z)` with `x \u2208 C\u2080`, `y \u2208 C\u2081`, `z \u2208 C\u221e` and `z * y * x = 1` is","title":"The Frobenius formula"},{"id":"tauceti/representation-theory/character-table/the-frobenius-schur-trichotomy","kind":"node","lean":"TauCeti.Representation.frobeniusSchurIndicator_eq_one_or_eq_zero_or_eq_neg_one","parent":3681,"result":true,"route":"../roadmap/tauceti/representation-theory/character-table/index.html#the-frobenius-schur-trichotomy","scope":"tauceti/representation-theory/character-table","status":"fully_proved","summary":"The indicator of an irreducible representation over an algebraically closed field of characteristic zero takes only the values `1`, `0` and `-1`.","title":"The Frobenius-Schur trichotomy"},{"id":"tauceti/representation-theory/character-table/the-good-prime-structure-theorem","kind":"node","lean":"TauCeti.IsGoodDixonPrime.nonempty_center_algEquiv_conjClasses","parent":3681,"result":true,"route":"../roadmap/tauceti/representation-theory/character-table/index.html#the-good-prime-structure-theorem","scope":"tauceti/representation-theory/character-table","status":"fully_proved","summary":"At a good Dixon prime the centre of `ZMod p [G]` is the algebra of `ZMod p`-valued functions on the conjugacy classes of `G`.","title":"The good-prime structure theorem"},{"id":"tauceti/representation-theory/character-table/the-good-splitting-structure-theorem-for-a-finit","kind":"node","lean":"TauCeti.nonempty_center_algEquiv_conjClasses","parent":3681,"result":true,"route":"../roadmap/tauceti/representation-theory/character-table/index.html#the-good-splitting-structure-theorem-for-a-finit","scope":"tauceti/representation-theory/character-table","status":"fully_proved","summary":"The centre of `K[G]` is `K`-algebra isomorphic to the functions on the conjugacy classes of `G`.","title":"The good-splitting structure theorem for a finite coefficient field"},{"id":"tauceti/representation-theory/character-table/maschkes-theorem-at-a-good-dixon-prime","kind":"node","lean":"TauCeti.IsGoodDixonPrime.isSemisimpleRing","parent":3681,"result":true,"route":"../roadmap/tauceti/representation-theory/character-table/index.html#maschkes-theorem-at-a-good-dixon-prime","scope":"tauceti/representation-theory/character-table","status":"fully_proved","summary":"the modular group algebra `ZMod p [G]` is a semisimple ring.","title":"Maschke's theorem at a good Dixon prime"},{"data_url":"scopes/tauceti/representation-theory/classical-groups.json","id":"tauceti/representation-theory/classical-groups","kind":"scope","parent":3630,"result":false,"route":"../roadmap/tauceti/representation-theory/classical-groups/index.html","scope":"tauceti/representation-theory/classical-groups","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.RepresentationTheory.ClassicalGroups` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Classical Groups"},{"id":"tauceti/representation-theory/classical-groups/brauer-duality-on-the-tensor-square","kind":"node","lean":"TauCeti.range_brauerActionOrth_two_eq_centralizer_of_sq_add_sq_eq_one, TauCeti.range_brauerActionOrth_two_eq_centralizer_of_noZeroDivisors, TauCeti.adjoin_symplecticCrossing_symplecticCupCap_eq_centralizer, TauCeti.adjoin_symplecticCrossing_symplecticCupCap_eq_centralizer_of_charZero, TauCeti.range_brauerActionOrth_two_eq_centralizer","parent":3691,"result":true,"route":"../roadmap/tauceti/representation-theory/classical-groups/index.html#brauer-duality-on-the-tensor-square","scope":"tauceti/representation-theory/classical-groups","status":"fully_proved","summary":"Over a field `K` with `2 \u2260 0` that has a point `(c, s)` of the unit circle with `c \u2260 0` and `s \u2260 0`, the image of the Brauer algebra `B\u2082(n)` in the endomorphisms of `(K\u207f)^{\u22972}` is the commutant of the diagonal action of the orthogonal group: an operator commuting with every `g \u2297 g` is a combination of the identity, the flip of the two tensor factors and the cup-cap generator.","title":"Brauer duality on the tensor square"},{"id":"tauceti/representation-theory/classical-groups/the-gelfand-tsetlin-decomposition","kind":"node","lean":"TauCeti.isInternal_gtLine","parent":3691,"result":true,"route":"../roadmap/tauceti/representation-theory/classical-groups/index.html#the-gelfand-tsetlin-decomposition","scope":"tauceti/representation-theory/classical-groups","status":"fully_proved","summary":"The rational Weyl module `V_\u03bb` of `GL n \u2102` is the internal direct sum of the Gelfand\u2013Tsetlin lines of the patterns with top weight `\u03bb`.","title":"The Gelfand\u2013Tsetlin decomposition"},{"id":"tauceti/representation-theory/classical-groups/the-gelfand-tsetlin-dimension-formula","kind":"node","lean":"TauCeti.GTPattern.card_topRow_eq_weylDimension, TauCeti.GTPattern.card_topWeight_eq_weylDimension","parent":3691,"result":true,"route":"../roadmap/tauceti/representation-theory/classical-groups/index.html#the-gelfand-tsetlin-dimension-formula","scope":"tauceti/representation-theory/classical-groups","status":"fully_proved","summary":"The Gelfand-Tsetlin patterns with top row a weakly decreasing sequence `\u03bb` are counted by the Weyl dimension formula `\u220f_{i < j} (\u03bb\u1d62 - \u03bb\u2c7c + j - i) / (j - i)`.","title":"The Gelfand-Tsetlin dimension formula"},{"id":"tauceti/representation-theory/classical-groups/the-schur-weyl-decomposition-of-the-character-of","kind":"node","lean":"TauCeti.sum_finrank_spechtModule_mul_char_weylRepOfShape_ofPartition, TauCeti.sum_finrank_spechtModule_mul_char_weylFDRepOfShape_ofPartition","parent":3691,"result":true,"route":"../roadmap/tauceti/representation-theory/classical-groups/index.html#the-schur-weyl-decomposition-of-the-character-of","scope":"tauceti/representation-theory/classical-groups","status":"fully_proved","summary":"For every `g \u2208 GL n k`, the character of `(k\u207f)^{\u2297d}` at `g` is the sum over the partitions `\u03bc` of `d` of the characters of the Weyl modules `\ud835\udd4a^\u03bc(k\u207f)` at `g`, each weighted by the dimension `f^\u03bc` of the Specht module `S^\u03bc`.","title":"The Schur-Weyl decomposition of the character of a tensor power"},{"id":"tauceti/representation-theory/classical-groups/the-weyl-dimension-formula-for-gl-n","kind":"node","lean":"TauCeti.finrank_rationalWeylFDRep, TauCeti.oddOrthogonalWeylDimension_eq_prod_prod_div, TauCeti.evenOrthogonalWeylDimension_eq_prod_prod_div, TauCeti.symplecticWeylDimension_eq_prod_prod_div, TauCeti.weylDimension_eq_prod_prod_div","parent":3691,"result":true,"route":"../roadmap/tauceti/representation-theory/classical-groups/index.html#the-weyl-dimension-formula-for-gl-n","scope":"tauceti/representation-theory/classical-groups","status":"fully_proved","summary":"over a field of characteristic zero, the rational Weyl module of a dominant weight `\u03bb` has dimension `\u220f_{i < j} (\u03bb\u1d62 - \u03bb\u2c7c + j - i) / (j - i)`, the Weyl dimension of `\u03bb`.","title":"The Weyl dimension formula for `GL n`"},{"data_url":"scopes/tauceti/representation-theory/compact.json","id":"tauceti/representation-theory/compact","kind":"scope","parent":3630,"result":false,"route":"../roadmap/tauceti/representation-theory/compact/index.html","scope":"tauceti/representation-theory/compact","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.RepresentationTheory.Compact` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Compact"},{"id":"tauceti/representation-theory/compact/the-frobenius-schur-reality-trichotomy-for-compa","kind":"node","lean":"ContRepresentation.frobeniusSchurIndicator_trichotomy","parent":3697,"result":true,"route":"../roadmap/tauceti/representation-theory/compact/index.html#the-frobenius-schur-reality-trichotomy-for-compa","scope":"tauceti/representation-theory/compact","status":"fully_proved","summary":"The indicator of an irreducible unitary representation of a compact group on a finite-dimensional complex inner product space is `1`, `0` or `-1`.","title":"The Frobenius-Schur reality trichotomy for compact groups"},{"id":"tauceti/representation-theory/compact/maschkes-theorem-in-orthogonal-form","kind":"node","lean":"ContRepresentation.exists_orthogonal_irreducible_decomposition_of_finite","parent":3697,"result":true,"route":"../roadmap/tauceti/representation-theory/compact/index.html#maschkes-theorem-in-orthogonal-form","scope":"tauceti/representation-theory/compact","status":"fully_proved","summary":"A finite-dimensional representation `\u03c0` of a finite group on an inner product space is an internal direct sum of finitely many irreducible subrepresentations, of dimensions adding up to `dim V`, which are moreover pairwise *orthogonal* for an invariant inner product `\u27eae \u00b7, e \u00b7\u27eb` obtained from the given one by a continuous linear automorphism `e` of the carrier.","title":"Maschke's theorem in orthogonal form"},{"id":"tauceti/representation-theory/compact/the-spectral-theorem-for-a-symmetric-convolution","kind":"node","lean":"TauCeti.orthogonalComplement_iSup_eigenspaces_convolutionOperator_eq_bot","parent":3697,"result":true,"route":"../roadmap/tauceti/representation-theory/compact/index.html#the-spectral-theorem-for-a-symmetric-convolution","scope":"tauceti/representation-theory/compact","status":"fully_proved","summary":"its eigenspaces span a dense subspace of `L\u00b2(G)`, in the sense that their supremum has trivial orthogonal complement. Together with `finiteDimensional_eigenspace_convolutionOperator` this says that `L\u00b2(G)` is exhausted, up to the kernel of the operator, by finite-dimensional translation-invariant subspaces.","title":"The spectral theorem for a symmetric convolution operator"},{"data_url":"scopes/tauceti/representation-theory/grothendieck-group.json","id":"tauceti/representation-theory/grothendieck-group","kind":"scope","parent":3630,"result":false,"route":"../roadmap/tauceti/representation-theory/grothendieck-group/index.html","scope":"tauceti/representation-theory/grothendieck-group","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.RepresentationTheory.GrothendieckGroup` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Grothendieck Group"},{"id":"tauceti/representation-theory/grothendieck-group/artins-identity-in-g0-k-g-over-every-field-writt","kind":"node","lean":"TauCeti.natCard_nsmul_one_eq_sum_artinCoeff_indK0","parent":3701,"result":true,"route":"../roadmap/tauceti/representation-theory/grothendieck-group/index.html#artins-identity-in-g0-k-g-over-every-field-writt","scope":"tauceti/representation-theory/grothendieck-group","status":"fully_proved","summary":"No division by the group order is required.","title":"Artin's identity in `G\u2080(k[G])` over every field, written as induction of the unit from subgroups"},{"id":"tauceti/representation-theory/grothendieck-group/artins-induction-identity-for-every-virtual-clas","kind":"node","lean":"TauCeti.natCard_nsmul_eq_sum_artinCoeff_indK0_resK0","parent":3701,"result":true,"route":"../roadmap/tauceti/representation-theory/grothendieck-group/index.html#artins-induction-identity-for-every-virtual-clas","scope":"tauceti/representation-theory/grothendieck-group","status":"fully_proved","summary":"Multiplying by `|G|` expresses a class as the Artin-coefficient sum of its induced restrictions.","title":"Artin's induction identity for every virtual class over every field"},{"id":"tauceti/representation-theory/grothendieck-group/artins-signed-permutation-identity-in-the-exact","kind":"node","lean":"TauCeti.natCard_nsmul_permK0_quotient_top_eq_sum_artinCoeff","parent":3701,"result":true,"route":"../roadmap/tauceti/representation-theory/grothendieck-group/index.html#artins-signed-permutation-identity-in-the-exact","scope":"tauceti/representation-theory/grothendieck-group","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.natCard_nsmul_permK0_quotient_top_eq_sum_artinCoeff`.","title":"Artin's signed permutation identity in the exact Grothendieck group over every field"},{"id":"tauceti/representation-theory/grothendieck-group/the-cartan-path-count-formula","kind":"node","lean":"TauCeti.jordanHolderMultiplicity_vertexProjectiveModuleFG_eq_card_path","parent":3701,"result":true,"route":"../roadmap/tauceti/representation-theory/grothendieck-group/index.html#the-cartan-path-count-formula","scope":"tauceti/representation-theory/grothendieck-group","status":"fully_proved","summary":"The multiplicity of `S\u1d62` in the vertex projective `P\u2c7c` is the number of oriented paths from `j` to `i`. With simples in rows and projectives in columns, these are the entries of the Cartan matrix of a finite acyclic path algebra.","title":"The Cartan path-count formula"},{"id":"tauceti/representation-theory/grothendieck-group/the-general-field-cartan-ext-euler-identity","kind":"node","lean":"TauCeti.cartanMatrix_transpose_mul_extEulerMatrix_eq_diagonal","parent":3701,"result":true,"route":"../roadmap/tauceti/representation-theory/grothendieck-group/index.html#the-general-field-cartan-ext-euler-identity","scope":"tauceti/representation-theory/grothendieck-group","status":"fully_proved","summary":"Let `P i \u27f6 S i` be projective covers of simple modules. If every pair of finitely generated modules is Euler-admissible, then `C\u1d40 * E = diagonal (fun i \u21a6 dim_k End_A(S i))`, where `C i j = [P j : S i]` is the Cartan matrix and `E` is the Ext-Euler matrix in the simple-class basis.","title":"The general-field Cartan/Ext-Euler identity"},{"data_url":"scopes/tauceti/representation-theory/homological.json","id":"tauceti/representation-theory/homological","kind":"scope","parent":3630,"result":false,"route":"../roadmap/tauceti/representation-theory/homological/index.html","scope":"tauceti/representation-theory/homological","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.RepresentationTheory.Homological` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Homological"},{"id":"tauceti/representation-theory/homological/corestriction-through-shapiros-lemma","kind":"node","lean":"TauCeti.groupCohomology.coindIso_hom_comp_corestriction, TauCeti.groupHomology.indIso_inv_comp_map_counit, TauCeti.groupHomology.indIso_inv_comp_map_counit_assoc","parent":3707,"result":true,"route":"../roadmap/tauceti/representation-theory/homological/index.html#corestriction-through-shapiros-lemma","scope":"tauceti/representation-theory/homological","status":"fully_proved","summary":"Precomposed with Shapiro's isomorphism `H\u207f(G, Coind_S^G Res_S A) \u2245 H\u207f(S, Res_S A)`, corestriction is the map induced by the trace `Coind_S^G Res_S A \u27f6 A`, the counit of `Rep.coindResAdjunction`.","title":"Corestriction through Shapiro's lemma"},{"id":"tauceti/representation-theory/homological/the-degree-one-finite-quotient-colimit-theorem","kind":"node","lean":"TauCeti.ContCohomology.explicitFiniteQuotientColimit1","parent":3707,"result":true,"route":"../roadmap/tauceti/representation-theory/homological/index.html#the-degree-one-finite-quotient-colimit-theorem","scope":"tauceti/representation-theory/homological","status":"fully_proved","summary":"`H\u00b9(G, M)` is the colimit of the finite-level first cohomology groups `H\u00b9(G \u29f8 U, M^U)`, through the inflation maps.","title":"The degree-one finite-quotient colimit theorem"},{"id":"tauceti/representation-theory/homological/the-degree-two-finite-quotient-colimit-theorem","kind":"node","lean":"TauCeti.ContCohomology.explicitFiniteQuotientColimit2","parent":3707,"result":true,"route":"../roadmap/tauceti/representation-theory/homological/index.html#the-degree-two-finite-quotient-colimit-theorem","scope":"tauceti/representation-theory/homological","status":"fully_proved","summary":"`H\u00b2(G, M)` is the colimit of the finite-level second cohomology groups `H\u00b2(G \u29f8 U, M^U)`, through the inflation maps.","title":"The degree-two finite-quotient colimit theorem"},{"id":"tauceti/representation-theory/homological/the-degree-zero-finite-quotient-colimit-theorem","kind":"node","lean":"TauCeti.ContCohomology.explicitFiniteQuotientColimit0","parent":3707,"result":true,"route":"../roadmap/tauceti/representation-theory/homological/index.html#the-degree-zero-finite-quotient-colimit-theorem","scope":"tauceti/representation-theory/homological","status":"fully_proved","summary":"`H\u2070(G, M)` is the colimit of `H\u2070(G \u29f8 U, M^U)` over the open normal subgroups, through the inflation maps.","title":"The degree-zero finite-quotient colimit theorem"},{"id":"tauceti/representation-theory/homological/the-finite-quotient-colimit-theorem-in-every-deg","kind":"node","lean":"TauCeti.ContCohomology.continuousFiniteQuotientColimit","parent":3707,"result":true,"route":"../roadmap/tauceti/representation-theory/homological/index.html#the-finite-quotient-colimit-theorem-in-every-deg","scope":"tauceti/representation-theory/homological","status":"fully_proved","summary":"For a compact group `G` and a discrete module `M` with continuous action, the inflation-and-inclusion cocone exhibits the canonical continuous cohomology `H\u207f(G, M)` as the colimit, in `TopModuleCat \u2124`, of the cohomology groups `H\u207f(G \u29f8 U, M^U)` of the finite quotients over the open normal subgroups `U` of `G`.","title":"The finite-quotient colimit theorem in every degree"},{"id":"tauceti/representation-theory/homological/the-mackey-double-coset-formula-in-degree-one-ns","kind":"node","lean":"TauCeti.ContCohomology.explicitCor1_mackey, TauCeti.ContCohomology.explicitCor2_mackey, TauCeti.ContCohomology.explicitCor0_mackey, TauCeti.ContinuousCohomology.corestriction_mackey","parent":3707,"result":true,"route":"../roadmap/tauceti/representation-theory/homological/index.html#the-mackey-double-coset-formula-in-degree-one-ns","scope":"tauceti/representation-theory/homological","status":"fully_proved","summary":"for an open subgroup `U` of finite index and any subgroup `V`, `res^G_V \u2218 cor^G_U = \u2211_{VsU} cor^V_{V \u2293 sUs\u207b\u00b9} \u2218 (s)_* \u2218 res` on `H\u00b9(U, M)`, for any choice `r` of representatives `s = r D` of the double cosets.","title":"The Mackey double-coset formula in degree one (NSW (1.5.6))"},{"id":"tauceti/representation-theory/homological/the-mackey-formula-for-the-degree-one-corestrict","kind":"node","lean":"TauCeti.ContCohomology.cochainsCor1_mackeyTransversal, TauCeti.ContCohomology.explicitMap2_explicitCor2_eq_sum, TauCeti.ContCohomology.cochainsCor2_mackeyTransversal","parent":3707,"result":true,"route":"../roadmap/tauceti/representation-theory/homological/index.html#the-mackey-formula-for-the-degree-one-corestrict","scope":"tauceti/representation-theory/homological","status":"fully_proved","summary":"At the adapted transversal `Subgroup.mackeyTransversal`, the restriction to `V` of the corestriction of a `1`-cochain `f` on `U` is, on the nose, the sum over the double cosets `V \\ G / U` of the corestrictions, from the Mackey subgroups to `V` at the canonical transversal, of the conjugates `y \u21a6 s \u2022 f (s\u207b\u00b9 y s)` of `f`.","title":"The Mackey formula for the degree-one corestriction cochain"},{"id":"tauceti/representation-theory/homological/nakayama-rim-sylow-form","kind":"node","lean":"Rep.projective_of_isZero_res_sylow, Rep.projective_of_isZero_res, Rep.projective_ker_of_isZero_res","parent":3707,"result":true,"route":"../roadmap/tauceti/representation-theory/homological/index.html#nakayama-rim-sylow-form","scope":"tauceti/representation-theory/homological","status":"fully_proved","summary":"Nakayama\u2013Rim, Sylow form (Serre, *Local Fields*, IX \u00a7\u00a73\u20135; Rim, Ann. of Math. 69 (1959)). Let `k` be a commutative ring, `G` a finite group, and `A` a representation of `G` on a projective `k`-module. Suppose that for each prime `p` dividing `|G|` some Sylow `p`-subgroup `P` satisfies: `p` is a unit in `k`, or `pk` is a maximal ideal, multiplication by `p` is injective on `A`, and\u2026","title":"Nakayama\u2013Rim, Sylow form"},{"id":"tauceti/representation-theory/homological/restriction-through-shapiros-lemma","kind":"node","lean":"TauCeti.groupCohomology.map_unit_comp_coindIso_hom","parent":3707,"result":true,"route":"../roadmap/tauceti/representation-theory/homological/index.html#restriction-through-shapiros-lemma","scope":"tauceti/representation-theory/homological","status":"fully_proved","summary":"For a `G`-representation `B`, the map induced by the unit `B \u27f6 Coind_S^G Res_S B` of restriction\u2013coinduction, followed by Shapiro's isomorphism, is restriction `H\u207f(G, B) \u27f6 H\u207f(S, Res_S B)`.","title":"Restriction through Shapiro's lemma"},{"id":"tauceti/representation-theory/homological/shapiros-isomorphism-on-an-adjunct","kind":"node","lean":"TauCeti.groupCohomology.map_resCoindToHom_comp_coindIso_hom","parent":3707,"result":true,"route":"../roadmap/tauceti/representation-theory/homological/index.html#shapiros-isomorphism-on-an-adjunct","scope":"tauceti/representation-theory/homological","status":"fully_proved","summary":"For a `G`-representation `B` and a morphism `f : Res_S B \u27f6 A`, the map induced by the adjunct `B \u27f6 Coind_S^G A` of `f`, followed by Shapiro's isomorphism, is the change-of-group map `H\u207f(G, B) \u27f6 H\u207f(S, A)` induced by `f` along `S \u2264 G`.","title":"Shapiro's isomorphism on an adjunct"},{"id":"tauceti/representation-theory/homological/shapiros-lemma-in-degree-one","kind":"node","lean":"TauCeti.ContCohomology.explicitShapiro1, TauCeti.TateCohomology.herbrandQuotient_coind, TauCeti.ContinuousCohomology.indInflationShapiroIso, TauCeti.TateCohomology.coindIsoPos, TauCeti.TateCohomology.coindIso, TauCeti.ContinuousCohomology.bijective_shapiroMap_one, TauCeti.ContCohomology.explicitShapiro2, TauCeti.ContinuousCohomology.bijective_shapiroMap_two, TauCeti.ContCohomology.explicitShapiro0, TauCeti.ContinuousCohomology.bijective_shapiroMap_zero, TauCeti.ContinuousCohomology.bijective_shapiroMap_of_le_two, TauCeti.ContinuousCohomology.bijective_shapiroMap, TauCeti.ContinuousCohomology.isIso_shapiroMap, TauCeti.ContinuousCohomology.quasiIso_shapiroCochainMap, TauCeti.ContinuousCohomology.quasiIso_shapiroCochainMapTopRep, groupCohomology.map_id_eq_zero_iff_of_comp_eq_resCoindToHom","parent":3707,"result":true,"route":"../roadmap/tauceti/representation-theory/homological/index.html#shapiros-lemma-in-degree-one","scope":"tauceti/representation-theory/homological","status":"fully_proved","summary":"Shapiro's lemma in degree one, `H\u00b9(G, Coind_U^G A) \u2245 H\u00b9(U, A)`, for a profinite `G` and a closed subgroup `U`. The forward map is restriction to `U` followed by evaluation at `1`, and it involves no choice; the continuous section of `G \u2192 G \u29f8 U` is used only to prove it bijective.","title":"Shapiro's lemma in degree one"},{"id":"tauceti/representation-theory/homological/the-sylow-reduction-of-tates-triviality-criterio","kind":"node","lean":"TauCeti.groupCohomology.isZero_of_isZero_sylow","parent":3707,"result":true,"route":"../roadmap/tauceti/representation-theory/homological/index.html#the-sylow-reduction-of-tates-triviality-criterio","scope":"tauceti/representation-theory/homological","status":"fully_proved","summary":"if, for every prime `p` dividing the order of `G`, `H\u207f\u207a\u00b9(P, A) = 0` for some Sylow `p`-subgroup `P`, then `H\u207f\u207a\u00b9(G, A) = 0`.","title":"The Sylow reduction of Tate's triviality criterion (Milne II 3.10, last paragraph)"},{"id":"tauceti/representation-theory/homological/tates-cohomological-triviality-criterion-milne-i","kind":"node","lean":"TauCeti.TateCohomology.isZero_of_forall_isPGroup, TauCeti.groupCohomology.isZero_of_forall_isPGroup","parent":3707,"result":true,"route":"../roadmap/tauceti/representation-theory/homological/index.html#tates-cohomological-triviality-criterion-milne-i","scope":"tauceti/representation-theory/homological","status":"fully_proved","summary":"Let `G` be a finite group and `A` a representation of `G`. If its Tate cohomology vanishes in degrees `q` and `q + 1` for every subgroup `S` of `G` of prime-power order, then it vanishes in degree `n` for every subgroup `S` of `G` and every integer `n`.","title":"Tate's cohomological triviality criterion (Milne II 3.10)"},{"id":"tauceti/representation-theory/homological/tates-isomorphism-criterion","kind":"node","lean":"TauCeti.TateCohomology.map_bijective_of_forall_isPGroup, TauCeti.TateCohomology.map_res_bijective_of_forall_isPGroup","parent":3707,"result":true,"route":"../roadmap/tauceti/representation-theory/homological/index.html#tates-isomorphism-criterion","scope":"tauceti/representation-theory/homological","status":"fully_proved","summary":"Let `G` be a finite group and `f : A \u27f6 B` a morphism of representations of `G`. If, for every subgroup `S` of `G` of prime-power order, the map induced by `f` on the Tate cohomology of `S` is surjective in degree `q - 1`, bijective in degree `q` and injective in degree `q + 1`, then `f` induces a bijection `H\u0302\u207f(G, A) \u2192 H\u0302\u207f(G, B)` in every degree `n`.","title":"Tate's isomorphism criterion"},{"id":"tauceti/representation-theory/homological/tates-lattice-lemma","kind":"node","lean":"TauCeti.TateCohomology.herbrandQuotient_eq_of_isBaseChange","parent":3707,"result":true,"route":"../roadmap/tauceti/representation-theory/homological/index.html#tates-lattice-lemma","scope":"tauceti/representation-theory/homological","status":"fully_proved","summary":"Let `G` be a finite cyclic group, `K` a nontrivial commutative `\u211a`-algebra (classically `\u211d`) and `\u03c1` a representation of `G` on a `K`-module `V`. If `M` and `N` are integral representations of `G`, free over `\u2124` with `M` of finite rank, and `G`-equivariant maps `i : M \u2192 V` and `j : N \u2192 V` exhibit `V` as the base change of `M` and of `N` to `K`, then `M` and `N` have the same Herbrand quotient. The rank of `N` is\u2026","title":"Tate's lattice lemma"},{"id":"tauceti/representation-theory/homological/tates-theorem","kind":"node","lean":"TauCeti.TateCohomology.cup_bijective_of_forall_isPGroup, TauCeti.TateCohomology.cup_bijective_of_cupTrivialInt_injective","parent":3707,"result":true,"route":"../roadmap/tauceti/representation-theory/homological/index.html#tates-theorem","scope":"tauceti/representation-theory/homological","status":"fully_proved","summary":"Let `G` be a finite group, `N` a representation of `G` over `\u2124` and `u \u2208 H\u00b2(G, N)`. Suppose that","title":"Tate's theorem"},{"id":"tauceti/representation-theory/homological/the-tate-nakayama-theorem","kind":"node","lean":"TauCeti.TateCohomology.cup_bijective_of_forall_isPGroup_of_lTensor_injective","parent":3707,"result":true,"route":"../roadmap/tauceti/representation-theory/homological/index.html#the-tate-nakayama-theorem","scope":"tauceti/representation-theory/homological","status":"fully_proved","summary":"The Tate\u2013Nakayama theorem (Nakayama, Ann. of Math. 65 (1957); Serre, *Local Fields*, IX \u00a78). Let `G` be a finite group, `C` a representation of `G` over `\u2124` and `u \u2208 H\u00b2(G, C)`. Suppose that for every subgroup `S` of `G` of prime-power order","title":"The Tate\u2013Nakayama theorem"},{"data_url":"scopes/tauceti/representation-theory/induction.json","id":"tauceti/representation-theory/induction","kind":"scope","parent":3630,"result":false,"route":"../roadmap/tauceti/representation-theory/induction/index.html","scope":"tauceti/representation-theory/induction","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.RepresentationTheory.Induction` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Induction"},{"id":"tauceti/representation-theory/induction/an-induction-theorem-for-a-family-of-subgroups-m","kind":"node","lean":"TauCeti.mem_virtualCharacters_iff_forall_comp_subtype","parent":3725,"result":true,"route":"../roadmap/tauceti/representation-theory/induction/index.html#an-induction-theorem-for-a-family-of-subgroups-m","scope":"tauceti/representation-theory/induction","status":"fully_proved","summary":"For a class function on `G`, membership in the virtual-character lattice is equivalent to membership of all its restrictions to the family.","title":"An induction theorem for a family of subgroups makes restriction to that family detect the virtual characters"},{"id":"tauceti/representation-theory/induction/an-induction-theorem-is-a-single-membership","kind":"node","lean":"TauCeti.ClassFunction.indVirtualCharacters_eq_virtualCharacters_iff","parent":3725,"result":true,"route":"../roadmap/tauceti/representation-theory/induction/index.html#an-induction-theorem-is-a-single-membership","scope":"tauceti/representation-theory/induction","status":"fully_proved","summary":"The virtual characters induced from a family of subgroups exhaust `R(G)` exactly when the constant function `1`, the character of the trivial representation, is one of them.","title":"An induction theorem is a single membership"},{"id":"tauceti/representation-theory/induction/an-induction-theorem-one-prime-at-a-time-and-wit","kind":"node","lean":"TauCeti.ClassFunction.indVirtualCharacters_eq_virtualCharacters_iff_forall_prime","parent":3725,"result":true,"route":"../roadmap/tauceti/representation-theory/induction/index.html#an-induction-theorem-one-prime-at-a-time-and-wit","scope":"tauceti/representation-theory/induction","status":"fully_proved","summary":"Let `A` be a subring of `k` admitting an additive map `t : A \u2192 \u2124` with `t 1 = 1`; for example `\u2124[\u03b6]` for a root of unity `\u03b6` in characteristic zero (`PowerBasis.exists_linearMap_apply_one`). The virtual characters induced from a family of subgroups exhaust `R(G)` exactly when, for every prime `p`, some multiple `m \u2022 1` with `p \u2224 m` is an `A`-linear combination of them.","title":"An induction theorem, one prime at a time and with coefficients in a subring"},{"id":"tauceti/representation-theory/induction/artins-induction-theorem-with-its-induced-charac","kind":"node","lean":"TauCeti.ClassFunction.natCard_nsmul_mem_indCharacterSpanInt_isCyclic, TauCeti.ClassFunction.natCard_nsmul_mem_indVirtualCharacters_isCyclic","parent":3725,"result":true,"route":"../roadmap/tauceti/representation-theory/induction/index.html#artins-induction-theorem-with-its-induced-charac","scope":"tauceti/representation-theory/induction","status":"fully_proved","summary":"Over a algebraically closed field in which `|G|` is invertible (in particular, in characteristic zero), `|G| \u2022 f` belongs to the additive subgroup generated by characters induced from irreducible characters of cyclic subgroups.","title":"Artin's induction theorem with its induced-character-span target"},{"id":"tauceti/representation-theory/induction/artins-sharp-induction-theorem-read-for-the-elem","kind":"node","lean":"TauCeti.ClassFunction.natCard_nsmul_mem_indVirtualCharacters_isElementary","parent":3725,"result":true,"route":"../roadmap/tauceti/representation-theory/induction/index.html#artins-sharp-induction-theorem-read-for-the-elem","scope":"tauceti/representation-theory/induction","status":"fully_proved","summary":"Every cyclic subgroup is elementary, so the sharp cyclic theorem gives the explicit multiplier `|G|` here too.","title":"Artin's sharp induction theorem, read for the elementary subgroups"},{"id":"tauceti/representation-theory/induction/brauers-induction-theorem","kind":"node","lean":"TauCeti.ClassFunction.indVirtualCharacters_eq_virtualCharacters_isElementary, TauCeti.ClassFunction.character_mem_indCharacterSpanInt_isElementary","parent":3725,"result":true,"route":"../roadmap/tauceti/representation-theory/induction/index.html#brauers-induction-theorem","scope":"tauceti/representation-theory/induction","status":"fully_proved","summary":"Over an algebraically closed field of characteristic zero, every virtual character of a finite group is an integral combination of virtual characters induced from elementary subgroups.","title":"Brauer's induction theorem"},{"id":"tauceti/representation-theory/induction/brauers-local-lemma","kind":"node","lean":"TauCeti.exists_mem_span_indVirtualCharacters_isPElementary_not_dvd","parent":3725,"result":true,"route":"../roadmap/tauceti/representation-theory/induction/index.html#brauers-local-lemma","scope":"tauceti/representation-theory/induction","status":"fully_proved","summary":"For every prime `p` there is a function `\u03c8 : G \u2192 \u2115`, none of whose values is divisible by `p`, whose cast to `k` is an `A`-combination of virtual characters induced from `p`-elementary subgroups, for every subring `A` of `k` containing the `|G|`-th roots of unity. It is the sum, over the conjugacy classes of `p`-regular elements `s`, of `orderOf s` times the induced `p`-section indicator of `s`: at `y` only the\u2026","title":"Brauer's local lemma"},{"id":"tauceti/representation-theory/induction/burnsides-lemma-for-two-permutation-characters","kind":"node","lean":"TauCeti.sum_character_ofMulAction_mul_character_ofMulAction_eq_card_orbits_mul_card_group","parent":3725,"result":true,"route":"../roadmap/tauceti/representation-theory/induction/index.html#burnsides-lemma-for-two-permutation-characters","scope":"tauceti/representation-theory/induction","status":"fully_proved","summary":"The unnormalized sum `\u2211 g, \u03c7_X(g) \u00b7 \u03c7_Y(g\u207b\u00b9)` is the number of orbits of `G` on `X \u00d7 Y` times the order of `G`.","title":"Burnside's lemma for two permutation characters"},{"id":"tauceti/representation-theory/induction/cliffords-theorem-dimension-form","kind":"node","lean":"FDRep.clifford_restrict_finrank, FDRep.clifford_restrict_iso_of_isAtom, TauCeti.clifford_restrict_character, Representation.finrank_linearMap_eq_of_isAtom, Representation.finrank_linearMap_eq_of_isAtom_of_isAtom, FDRep.clifford_restrict_iso, TauCeti.Representation.exists_nonempty_linearEquiv_conjSubrep, TauCeti.Representation.exists_isAtom_forall_nonempty_linearEquiv_conjSubrep, Representation.exists_forall_finrank_linearMap_eq","parent":3725,"result":true,"route":"../roadmap/tauceti/representation-theory/induction/index.html#cliffords-theorem-dimension-form","scope":"tauceti/representation-theory/induction","status":"fully_proved","summary":"The dimension of an irreducible representation of `G` is the common multiplicity `e` of the constituents of its restriction to `N`, times the number `[G : inertia V]` of those constituents, times the dimension of one of them.","title":"Clifford's theorem, dimension form"},{"id":"tauceti/representation-theory/induction/the-divisibilities-in-cliffords-theorem","kind":"node","lean":"FDRep.clifford_restrict_dvd_finrank","parent":3725,"result":true,"route":"../roadmap/tauceti/representation-theory/induction/index.html#the-divisibilities-in-cliffords-theorem","scope":"tauceti/representation-theory/induction","status":"fully_proved","summary":"For an irreducible constituent `V` of the restriction of an irreducible `W` to a normal subgroup, both the index of the inertia group of `V` and the dimension of `V` divide the dimension of `W`; the index divides `[G : N]`, and it divides the quotient `dim W / (e * dim V)` that the dimension identity leaves.","title":"The divisibilities in Clifford's theorem"},{"id":"tauceti/representation-theory/induction/frobenius-reciprocity-for-class-functions","kind":"node","lean":"TauCeti.frobenius_reciprocity_classFunction, TauCeti.frobenius_reciprocity, TauCeti.frobenius_reciprocity_resFDRep","parent":3725,"result":true,"route":"../roadmap/tauceti/representation-theory/induction/index.html#frobenius-reciprocity-for-class-functions","scope":"tauceti/representation-theory/induction","status":"fully_proved","summary":"The normalized pairing over `G` of an induced class function with a class function of `G` is the normalized pairing over `S` of the original class function with the restricted one.","title":"Frobenius reciprocity for class functions"},{"id":"tauceti/representation-theory/induction/the-mackey-criterion-for-a-linear-character-of-a","kind":"node","lean":"TauCeti.simple_indFDRep_ofLinearCharacter_iff_of_conj_eq_inv","parent":3725,"result":true,"route":"../roadmap/tauceti/representation-theory/induction/index.html#the-mackey-criterion-for-a-linear-character-of-a","scope":"tauceti/representation-theory/induction","status":"fully_proved","summary":"the induced representation is irreducible exactly when some value of `\u03c7` is not a square root of `1`. Conjugation by any element outside `N` inverts it, by `TauCeti.conj_eq_inv_of_notMem_of_index_two`, so `{}^s \u03c7 = \u03c7\u207b\u00b9` for every such `s` and the criterion becomes the single condition `\u03c7 \u2260 \u03c7\u207b\u00b9`. The inversion hypothesis is asked of the one outside element `s` only.","title":"The Mackey criterion for a linear character of an inverted subgroup of index two"},{"id":"tauceti/representation-theory/induction/the-mackey-decomposition-formula-character-form","kind":"node","lean":"TauCeti.character_resFDRep_indFDRep_mackey, Subgroup.indClassFun_mackey","parent":3725,"result":true,"route":"../roadmap/tauceti/representation-theory/induction/index.html#the-mackey-decomposition-formula-character-form","scope":"tauceti/representation-theory/induction","status":"fully_proved","summary":"For a finite-dimensional representation `A` of a finite-index subgroup `H` and an element `x` of a subgroup `K`, the character of `Res_K (Ind_H^G A)` at `x` is the sum, over the double cosets `K \\ G / H`, of the characters of the Mackey summands.","title":"The Mackey decomposition formula, character form"},{"id":"tauceti/representation-theory/induction/the-mackey-decomposition-of-a-restricted-induced","kind":"node","lean":"Subgroup.comap_subtype_indClassFunction_mackey","parent":3725,"result":true,"route":"../roadmap/tauceti/representation-theory/induction/index.html#the-mackey-decomposition-of-a-restricted-induced","scope":"tauceti/representation-theory/induction","status":"fully_proved","summary":"Restricting to `K` a class function induced from `H` gives the sum, over the double cosets `K \\ G / H`, of the class functions induced to `K` from the Mackey subgroups.","title":"The Mackey decomposition of a restricted induced class function"},{"id":"tauceti/representation-theory/induction/the-mackey-irreducibility-criterion","kind":"node","lean":"TauCeti.simple_indFDRep_iff_doubleCoset, TauCeti.simple_indFDRep_iff_of_normal, TauCeti.simple_indFDRep_ofLinearCharacter_iff, TauCeti.simple_indFDRep_iff","parent":3725,"result":true,"route":"../roadmap/tauceti/representation-theory/induction/index.html#the-mackey-irreducibility-criterion","scope":"tauceti/representation-theory/induction","status":"fully_proved","summary":"The representation induced from `A` is irreducible exactly when `A` is irreducible and every double coset `HsH` other than `H` itself, read at its chosen representative, is Mackey disjoint. Algebraic closure supplies scalar Schur.","title":"The Mackey irreducibility criterion"},{"id":"tauceti/representation-theory/induction/the-modular-artin-induction-theorem-nsw-7-3-4-mi","kind":"node","lean":"TauCeti.modularArtin_exists_nsmul_mem_indCyclicCoprimeK0","parent":3725,"result":true,"route":"../roadmap/tauceti/representation-theory/induction/index.html#the-modular-artin-induction-theorem-nsw-7-3-4-mi","scope":"tauceti/representation-theory/induction","status":"fully_proved","summary":"In characteristic `p`, every class in `G\u2080(k[G])` has a positive multiple in `TauCeti.indCyclicCoprimeK0`, the subgroup generated by the classes induced from cyclic subgroups of order prime to `p`; the multiplier `|G|` works for every class.","title":"The modular Artin induction theorem (NSW (7.3.4), Milne ADT I Lemma 2.10)"},{"id":"tauceti/representation-theory/induction/the-modular-artin-theorem-with-multiplier-g","kind":"node","lean":"TauCeti.modularArtin_natCard_nsmul_mem_indCyclicCoprimeK0","parent":3725,"result":true,"route":"../roadmap/tauceti/representation-theory/induction/index.html#the-modular-artin-theorem-with-multiplier-g","scope":"tauceti/representation-theory/induction","status":"fully_proved","summary":"In characteristic `p`, the group-order multiple of every class in `G\u2080(k[G])` lies in `TauCeti.indCyclicCoprimeK0`, the subgroup generated by the classes induced from cyclic subgroups of order prime to `p`; for the prime `p`, order prime to `p` is non-divisibility of the order by `p` (`Nat.Prime.coprime_iff_not_dvd`).","title":"The modular Artin theorem with multiplier `|G|`"},{"data_url":"scopes/tauceti/representation-theory/lie.json","id":"tauceti/representation-theory/lie","kind":"scope","parent":3630,"result":false,"route":"../roadmap/tauceti/representation-theory/lie/index.html","scope":"tauceti/representation-theory/lie","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.RepresentationTheory.Lie` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Lie"},{"id":"tauceti/representation-theory/lie/ados-theorem-in-characteristic-zero","kind":"node","lean":"TauCeti.adoCharZero, TauCeti.exists_faithful_nilrepresentation_charZero, TauCeti.adoCharP","parent":3743,"result":true,"route":"../roadmap/tauceti/representation-theory/lie/index.html#ados-theorem-in-characteristic-zero","scope":"tauceti/representation-theory/lie","status":"fully_proved","summary":"Every finite-dimensional Lie algebra over a field of characteristic zero admits a faithful finite-dimensional representation.","title":"Ado's theorem in characteristic zero"},{"id":"tauceti/representation-theory/lie/the-ado-iwasawa-theorem","kind":"node","lean":"TauCeti.adoIwasawa, TauCeti.exists_faithful_nilrepresentation","parent":3743,"result":true,"route":"../roadmap/tauceti/representation-theory/lie/index.html#the-ado-iwasawa-theorem","scope":"tauceti/representation-theory/lie","status":"fully_proved","summary":"Every finite-dimensional Lie algebra over a field admits a faithful finite-dimensional representation.","title":"The Ado\u2013Iwasawa theorem"},{"id":"tauceti/representation-theory/lie/birkhoffs-theorem-on-a-weighted-pbw-quotient","kind":"node","lean":"TauCeti.exists_faithful_weightedPBWRepresentation","parent":3743,"result":true,"route":"../roadmap/tauceti/representation-theory/lie/index.html#birkhoffs-theorem-on-a-weighted-pbw-quotient","scope":"tauceti/representation-theory/lie","status":"fully_proved","summary":"A finite-dimensional nilpotent Lie algebra has a faithful left-multiplication representation on a finite-dimensional weighted enveloping quotient, with one positive nilpotence bound for all acting operators. The basis, weights, and cutoff are supplied by the conclusion, in every characteristic.","title":"Birkhoff's theorem on a weighted PBW quotient"},{"id":"tauceti/representation-theory/lie/hochschilds-nilpotence-theorem-in-characteristic","kind":"node","lean":"LieSubalgebra.isNilpotent_toEnd_of_isNilpotent_ad_of_isCompl_radical","parent":3743,"result":true,"route":"../roadmap/tauceti/representation-theory/lie/index.html#hochschilds-nilpotence-theorem-in-characteristic","scope":"tauceti/representation-theory/lie","status":"fully_proved","summary":"if the nilradical of a finite-dimensional Lie algebra `L` acts nilpotently on a finite-dimensional `L`-module `M`, then so does every `ad`-nilpotent element of `L`. The Lie subalgebra `S` is any Levi complement, that is, any complement of the solvable radical; the conclusion does not depend on it.","title":"Hochschild's nilpotence theorem in characteristic zero"},{"id":"tauceti/representation-theory/lie/hochschilds-strengthening-of-ados-theorem-in-cha","kind":"node","lean":"TauCeti.exists_faithful_preserving_ad_nilpotence_charZero","parent":3743,"result":true,"route":"../roadmap/tauceti/representation-theory/lie/index.html#hochschilds-strengthening-of-ados-theorem-in-cha","scope":"tauceti/representation-theory/lie","status":"fully_proved","summary":"A finite-dimensional Lie algebra over a field of characteristic zero has a faithful finite-dimensional representation that preserves nilpotence of the adjoint action: every `ad`-nilpotent element acts nilpotently.","title":"Hochschild's strengthening of Ado's theorem in characteristic zero"},{"id":"tauceti/representation-theory/lie/hochschilds-strengthening-of-the-ado-iwasawa-the","kind":"node","lean":"TauCeti.exists_faithful_preserving_ad_nilpotence","parent":3743,"result":true,"route":"../roadmap/tauceti/representation-theory/lie/index.html#hochschilds-strengthening-of-the-ado-iwasawa-the","scope":"tauceti/representation-theory/lie","status":"fully_proved","summary":"A finite-dimensional Lie algebra over an arbitrary field has a faithful finite-dimensional representation that preserves nilpotence of the adjoint action: every `ad`-nilpotent element acts nilpotently.","title":"Hochschild's strengthening of the Ado\u2013Iwasawa theorem"},{"id":"tauceti/representation-theory/lie/the-matrix-form-of-the-ado-iwasawa-theorem","kind":"node","lean":"TauCeti.exists_injective_lieHom_matrix","parent":3743,"result":true,"route":"../roadmap/tauceti/representation-theory/lie/index.html#the-matrix-form-of-the-ado-iwasawa-theorem","scope":"tauceti/representation-theory/lie","status":"fully_proved","summary":"Every finite-dimensional Lie algebra over a field embeds in `\ud835\udd24\ud835\udd29_n = Matrix (Fin n) (Fin n) K` for some `n`. Over `\u211d` this is the embedding consumed by the integration of a real Lie algebra to a Lie group.","title":"The matrix form of the Ado\u2013Iwasawa theorem"},{"data_url":"scopes/tauceti/representation-theory/quiver.json","id":"tauceti/representation-theory/quiver","kind":"scope","parent":3630,"result":false,"route":"../roadmap/tauceti/representation-theory/quiver/index.html","scope":"tauceti/representation-theory/quiver","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.RepresentationTheory.Quiver` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Quiver"},{"id":"tauceti/representation-theory/quiver/coutures-graded-classification-by-aut-g-orbits-o","kind":"node","lean":"TauCeti.SkewZigzagParameter.exists_firstCohomologyRelabel_eq_iff_exists_graded_algEquiv","parent":3751,"result":true,"route":"../roadmap/tauceti/representation-theory/quiver/index.html#coutures-graded-classification-by-aut-g-orbits-o","scope":"tauceti/representation-theory/quiver","status":"fully_proved","summary":"Over a nontrivial commutative ring with only trivial idempotents, the cohomology classes of two parameters lie in the same graph-automorphism orbit exactly when their skew-zigzag relation quotients are graded isomorphic. Together with `cohomologyClass_surjective`, this describes all graded isomorphism classes of the relation quotients arising from skew-zigzag parameters.","title":"Couture's graded classification by `Aut(G)`-orbits on `H\u00b9(G, k\u02e3)`"},{"id":"tauceti/representation-theory/quiver/the-entrywise-graded-cartan-formula","kind":"node","lean":"TauCeti.zigzagGradedCartanMatrix_apply","parent":3751,"result":true,"route":"../roadmap/tauceti/representation-theory/quiver/index.html#the-entrywise-graded-cartan-formula","scope":"tauceti/representation-theory/quiver","status":"fully_proved","summary":"The diagonal entries are `1 + q\u00b2`, the entries over an edge are `q`, and all the others vanish.","title":"The entrywise graded Cartan formula"},{"id":"tauceti/representation-theory/quiver/gabriels-presentation-theorem","kind":"node","lean":"TauCeti.exists_quiver_admissibleIdeal_morita, TauCeti.IsBasic.exists_surjective_pathAlgebra_isAdmissibleIdeal_ker","parent":3751,"result":true,"route":"../roadmap/tauceti/representation-theory/quiver/index.html#gabriels-presentation-theorem","scope":"tauceti/representation-theory/quiver","status":"fully_proved","summary":"Every finite-dimensional algebra `A` over an algebraically closed field `k` is Morita equivalent to a bound quiver algebra: there are a finite quiver `Q` and an admissible ideal `I` of the path algebra `kQ` such that `A` is Morita equivalent to `kQ \u29f8 I`. The algebra `kQ \u29f8 I` is isomorphic to the basic corner `eAe` of `A` (`TauCeti.IsSemiperfectRing.exists_isIdempotentElem_full_isBasic_corner`), presented by\u2026","title":"Gabriel's presentation theorem"},{"id":"tauceti/representation-theory/quiver/the-hilbert-series-identity-for-a-quiver-without","kind":"node","lean":"TauCeti.mul_preprojectiveHilbertSeries_eq_one, TauCeti.mul_preprojectiveHilbertSeries_eq_one_of_two_mul_le_sum","parent":3751,"result":true,"route":"../roadmap/tauceti/representation-theory/quiver/index.html#the-hilbert-series-identity-for-a-quiver-without","scope":"tauceti/representation-theory/quiver","status":"fully_proved","summary":"If every vertex of `Q` is the tail of an arrow and `A_{v,w} = #(w \u27f6 v)` is the arrow-count matrix of the doubled quiver, then `((1 + X\u00b2) I - X A) H = I` for the matrix Hilbert series `H` of `\u03a0`.","title":"The Hilbert series identity for a quiver without sinks"},{"id":"tauceti/representation-theory/quiver/the-koszul-defect-formula","kind":"node","lean":"TauCeti.finrank_preprojectiveCorner_add_two_add_finrank_preprojectiveCorner","parent":3751,"result":true,"route":"../roadmap/tauceti/representation-theory/quiver/index.html#the-koszul-defect-formula","scope":"tauceti/representation-theory/quiver","status":"fully_proved","summary":"For vertices `i` and `v` and every degree `m`,","title":"The Koszul defect formula"},{"id":"tauceti/representation-theory/quiver/the-krull-schmidt-theorem-for-quiver-representat","kind":"node","lean":"TauCeti.QuiverRep.exists_equiv_iso_of_iso_biproduct, TauCeti.QuiverRep.exists_equiv_iso_of_biproduct_iso","parent":3751,"result":true,"route":"../roadmap/tauceti/representation-theory/quiver/index.html#the-krull-schmidt-theorem-for-quiver-representat","scope":"tauceti/representation-theory/quiver","status":"fully_proved","summary":"If a pointwise finite-dimensional representation of a finite quiver is isomorphic to a finite biproduct of indecomposable representations in two ways, the two families of summands are matched by a bijection of their index sets under which corresponding summands are isomorphic.","title":"The Krull-Schmidt theorem for quiver representations"},{"data_url":"scopes/tauceti/representation-theory/spin.json","id":"tauceti/representation-theory/spin","kind":"scope","parent":3630,"result":false,"route":"../roadmap/tauceti/representation-theory/spin/index.html","scope":"tauceti/representation-theory/spin","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.RepresentationTheory.Spin` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Spin"},{"id":"tauceti/representation-theory/spin/the-even-structure-theorem","kind":"node","lean":"TauCeti.SpinPolarizationData.evenCliffordEquivProdEnd","parent":3758,"result":true,"route":"../roadmap/tauceti/representation-theory/spin/index.html#the-even-structure-theorem","scope":"tauceti/representation-theory/spin","status":"fully_proved","summary":"for an even-dimensional polarized quadratic space the even Clifford subalgebra is the product of the endomorphism algebras of the two half-spin summands.","title":"The even structure theorem"},{"id":"tauceti/representation-theory/spin/the-odd-structure-theorem-in-operator-form","kind":"node","lean":"TauCeti.SpinPolarizationData.evenCliffordEquivEnd, TauCeti.SpinPolarizationData.cliffordEquivEndProd","parent":3758,"result":true,"route":"../roadmap/tauceti/representation-theory/spin/index.html#the-odd-structure-theorem-in-operator-form","scope":"tauceti/representation-theory/spin","status":"fully_proved","summary":"for a polarized quadratic space of odd dimension over a field of characteristic not two, the Fock action is an isomorphism of `F`-algebras from the even Clifford subalgebra onto the endomorphism algebra of the spinor module `S = \u22c0\u00b7W`.","title":"The odd structure theorem in operator form"},{"id":"tauceti/representation-theory/spin/the-structure-theorem","kind":"node","lean":"TauCeti.SpinPolarizationData.cliffordEquivMatrix","parent":3758,"result":true,"route":"../roadmap/tauceti/representation-theory/spin/index.html#the-structure-theorem","scope":"tauceti/representation-theory/spin","status":"fully_proved","summary":"the Clifford algebra of a polarized quadratic space of dimension `2 * l` is the matrix algebra `M_{2^l}(K)`, read in a basis of the spinor module `\u22c0\u00b7W`, which has dimension `2 ^ l`.","title":"The structure theorem"},{"id":"tauceti/representation-theory/spin/the-structure-theorem-in-odd-dimension","kind":"node","lean":"CliffordAlgebra.nonempty_algEquiv_matrix_prod_of_finrank_eq_two_mul_add_one","parent":3758,"result":true,"route":"../roadmap/tauceti/representation-theory/spin/index.html#the-structure-theorem-in-odd-dimension","scope":"tauceti/representation-theory/spin","status":"fully_proved","summary":"over a separably closed field of characteristic not two, the Clifford algebra of a nondegenerate quadratic form on a space of dimension `2 * l + 1` is the product of two copies of the matrix algebra `M_{2^l}(F)`.","title":"The structure theorem in odd dimension"},{"id":"tauceti/representation-theory/spin/the-structure-theorem-in-operator-form","kind":"node","lean":"TauCeti.SpinPolarizationData.cliffordEquivEnd","parent":3758,"result":true,"route":"../roadmap/tauceti/representation-theory/spin/index.html#the-structure-theorem-in-operator-form","scope":"tauceti/representation-theory/spin","status":"fully_proved","summary":"for an even-dimensional polarized quadratic space, the Fock action is an isomorphism of `K`-algebras from `CliffordAlgebra Q` onto the endomorphism algebra of the spinor module `S = \u22c0\u00b7W`.","title":"The structure theorem in operator form"},{"id":"tauceti/representation-theory/spin/the-structure-theorem-over-a-separably-closed-fi","kind":"node","lean":"CliffordAlgebra.nonempty_algEquiv_matrix_of_finrank_eq_two_mul","parent":3758,"result":true,"route":"../roadmap/tauceti/representation-theory/spin/index.html#the-structure-theorem-over-a-separably-closed-fi","scope":"tauceti/representation-theory/spin","status":"fully_proved","summary":"the Clifford algebra of a nondegenerate quadratic form on a `2l`-dimensional space is the matrix algebra `M_{2^l}(F)`.","title":"The structure theorem over a separably closed field"},{"data_url":"scopes/tauceti/representation-theory/su2.json","id":"tauceti/representation-theory/su2","kind":"scope","parent":3630,"result":false,"route":"../roadmap/tauceti/representation-theory/su2/index.html","scope":"tauceti/representation-theory/su2","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.RepresentationTheory.SU2` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"SU2"},{"id":"tauceti/representation-theory/su2/the-weyl-character-formula-for-su-2","kind":"node","lean":"TauCeti.SU2.character_symPower_torusHom_eq_div, TauCeti.SU2.character_symPower_torusExp_eq_sin_div_sin","parent":3765,"result":true,"route":"../roadmap/tauceti/representation-theory/su2/index.html#the-weyl-character-formula-for-su-2","scope":"tauceti/representation-theory/su2","status":"fully_proved","summary":"Away from the two points `z = \u00b11` of the maximal torus, where the Weyl denominator vanishes, the character of `Sym\u1d48(\u2102\u00b2)` at `diag (z, z\u207b\u00b9)` is the quotient of the Weyl numerator by the Weyl denominator.","title":"The Weyl character formula for `SU(2)`"},{"id":"tauceti/representation-theory/su2/the-weyl-integration-formula-for-su-2","kind":"node","lean":"TauCeti.SU2.weyl_integration_formula","parent":3765,"result":true,"route":"../roadmap/tauceti/representation-theory/su2/index.html#the-weyl-integration-formula-for-su-2","scope":"tauceti/representation-theory/su2","status":"fully_proved","summary":"For a continuous class function `f` on `SU(2)`, the integral of `f` against the Haar probability measure is the integral over the Weyl chamber `[0, \u03c0]` of the maximal torus against the Weyl density `(2\u03c0)\u207b\u00b9 \u00b7 4 sin\u00b2\u03b8 d\u03b8`:","title":"The Weyl integration formula for `SU(2)`"},{"id":"tauceti/representation-theory/su2/the-weyl-numerator-identity","kind":"node","lean":"TauCeti.SU2.sub_inv_mul_character_symPower_torusHom, TauCeti.SU2.sin_mul_character_symPower_torusExp","parent":3765,"result":true,"route":"../roadmap/tauceti/representation-theory/su2/index.html#the-weyl-numerator-identity","scope":"tauceti/representation-theory/su2","status":"fully_proved","summary":"Multiplying the weight string of `Sym\u1d48(\u2102\u00b2)` by the Weyl denominator `z - z\u207b\u00b9` telescopes to `z^{d+1} - z^{-(d+1)}`.","title":"The Weyl numerator identity"},{"data_url":"scopes/tauceti/representation-theory/symmetric.json","id":"tauceti/representation-theory/symmetric","kind":"scope","parent":3630,"result":false,"route":"../roadmap/tauceti/representation-theory/symmetric/index.html","scope":"tauceti/representation-theory/symmetric","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.RepresentationTheory.Symmetric` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Symmetric"},{"id":"tauceti/representation-theory/symmetric/frobeniuss-formula","kind":"node","lean":"TauCeti.psumPart_eq_sum_spechtCharValue_smul_schurPoly","parent":3769,"result":true,"route":"../roadmap/tauceti/representation-theory/symmetric/index.html#frobeniuss-formula","scope":"tauceti/representation-theory/symmetric","status":"fully_proved","summary":"the power-sum product of a partition `\u03c1` of `n` expands in the Schur polynomials as `p_\u03c1 = \u2211_{\u03bb \u22a2 n} \u03c7^\u03bb(\u03c1) s_\u03bb`, the coefficients being the column of `\u03c1` in the character table of `S\u2099`. This holds in every finite alphabet and over every commutative ring; the Schur polynomials of the partitions with more parts than the alphabet has letters vanish.","title":"Frobenius's formula"},{"id":"tauceti/representation-theory/symmetric/the-inverse-of-frobeniuss-formula","kind":"node","lean":"TauCeti.sum_spechtChar_smul_psumPart","parent":3769,"result":true,"route":"../roadmap/tauceti/representation-theory/symmetric/index.html#the-inverse-of-frobeniuss-formula","scope":"tauceti/representation-theory/symmetric","status":"fully_proved","summary":"weighting the power-sum products over the cycle types by a Specht character recovers the Schur polynomial, `\u2211_{\u03c0 \u2208 S\u2099} \u03c7^\u03bc(\u03c0) p_{\u03c1(\u03c0)} = n! s_\u03bc`. Over a `\u211a`-algebra this says that `(1 / n!) \u2211_\u03c0 \u03c7^\u03bc(\u03c0) p_{\u03c1(\u03c0)}` is `s_\u03bc`.","title":"The inverse of Frobenius's formula"},{"id":"tauceti/representation-theory/symmetric/jamess-dominance-lemma","kind":"node","lean":"TauCeti.dominates_of_asAlgebraHom_columnAntisymmetrizer_ne_zero, TauCeti.YoungTableau.dominates_of_asAlgebraHom_columnAntisymmetrizer_apply_ne_zero","parent":3769,"result":true,"route":"../roadmap/tauceti/representation-theory/symmetric/index.html#jamess-dominance-lemma","scope":"tauceti/representation-theory/symmetric","status":"fully_proved","summary":"If the column antisymmetrizer of a `lam`-tableau `t` does not annihilate the `\u03bc`-tabloid `q`, then the shape of `t` dominates `\u03bc`.","title":"James's dominance lemma"},{"id":"tauceti/representation-theory/symmetric/jamess-submodule-theorem","kind":"node","lean":"TauCeti.spechtSubrepresentation_le_or_le_orthogonal","parent":3769,"result":true,"route":"../roadmap/tauceti/representation-theory/symmetric/index.html#jamess-submodule-theorem","scope":"tauceti/representation-theory/symmetric","status":"fully_proved","summary":"A subrepresentation `U` of the Young permutation module `M^\u03bc` either contains the Specht module `S^\u03bc` or is orthogonal to it for the tabloid form.","title":"James's submodule theorem"},{"id":"tauceti/representation-theory/symmetric/schur-weyl-duality-the-commutant-of-the-image-of","kind":"node","lean":"TauCeti.centralizer_range_permTensorActionAlgHom_eq_range_tensorPowerRep_asAlgebraHom, TauCeti.centralizer_span_range_map_const_eq_range_permTensorActionAlgHom, TauCeti.centralizer_range_permTensorAction_eq_range_tensorPowerRep_asAlgebraHom, TauCeti.centralizer_range_tensorPowerRep_eq_range_permTensorActionAlgHom, TauCeti.mem_range_permTensorActionAlgHom_iff_forall_commute, TauCeti.mem_range_permTensorActionAlgHom_iff_forall_commute_tensorPowerRep, TauCeti.mem_range_tensorPowerRep_asAlgebraHom_iff_forall_commute_permTensorAction, TauCeti.centralizer_range_tensorPowerRep_asAlgebraHom_eq_range_permTensorActionAlgHom","parent":3769,"result":true,"route":"../roadmap/tauceti/representation-theory/symmetric/index.html#schur-weyl-duality-the-commutant-of-the-image-of","scope":"tauceti/representation-theory/symmetric","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.centralizer_range_permTensorActionAlgHom_eq_range_tensorPowerRep_asAlgebraHom`.","title":"Schur-Weyl duality: the commutant of the image of `k[S_d]` is the image of `k[GL\u2099]`"},{"id":"tauceti/representation-theory/symmetric/the-standard-basis-theorem","kind":"node","lean":"TauCeti.standardPolytabloidBasis","parent":3769,"result":true,"route":"../roadmap/tauceti/representation-theory/symmetric/index.html#the-standard-basis-theorem","scope":"tauceti/representation-theory/symmetric","status":"fully_proved","summary":"The polytabloids of the standard `\u03bc`-tableaux are a basis of the Specht module `S^\u03bc`: they are linearly independent by `TauCeti.linearIndependent_polytabloid` and they span by `TauCeti.spechtSubrepresentation_eq_span_standard`.","title":"The standard basis theorem"},{"data_url":"scopes/tauceti/representation-theory/symmetric-algebra.json","id":"tauceti/representation-theory/symmetric-algebra","kind":"scope","parent":3630,"result":false,"route":"../roadmap/tauceti/representation-theory/symmetric-algebra/index.html","scope":"tauceti/representation-theory/symmetric-algebra","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.RepresentationTheory.SymmetricAlgebra` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Symmetric Algebra"},{"id":"tauceti/representation-theory/symmetric-algebra/chevalleys-theorem-the-coinvariant-algebra-of-a","kind":"node","lean":"Representation.nonempty_equiv_coinvariantAlgebra_dual_leftRegular_of_pseudoreflectionSubgroup_eq_top","parent":3776,"result":true,"route":"../roadmap/tauceti/representation-theory/symmetric-algebra/index.html#chevalleys-theorem-the-coinvariant-algebra-of-a","scope":"tauceti/representation-theory/symmetric-algebra","status":"fully_proved","summary":"If a finite group `G` acts faithfully on a finite-dimensional vector space `V` over a field of characteristic zero and is generated by pseudoreflections, then the coinvariant algebra of the polynomial functions `Sym(V*)`, as an ungraded representation, is equivalent to the regular representation `K[G]`.","title":"Chevalley's theorem: the coinvariant algebra of a reflection group is the regular representation"},{"id":"tauceti/representation-theory/symmetric-algebra/the-chevalley-shephard-todd-theorem","kind":"node","lean":"Representation.exists_bijective_aeval_symmetricAlgebraInvariants_dual_of_pseudoreflectionSubgroup_eq_top, Representation.pseudoreflectionSubgroup_eq_top_iff_exists_bijective_aeval_symmetricAlgebraInvariants_dual","parent":3776,"result":true,"route":"../roadmap/tauceti/representation-theory/symmetric-algebra/index.html#the-chevalley-shephard-todd-theorem","scope":"tauceti/representation-theory/symmetric-algebra","status":"fully_proved","summary":"Let a finite group `G`, whose order is invertible in the field `K`, act linearly on a finite-dimensional vector space `V`, and suppose that `G` is generated by pseudoreflections, the elements whose fixed space is a hyperplane. Then the polynomial invariants `Sym(V*)^G` form a polynomial algebra on `dim V` homogeneous invariants `f\u2081, \u2026, f\u2099` of positive degree: `K[X\u2081, \u2026, X\u2099] \u2192 Sym(V*)^G`, `X\u1d62 \u21a6 f\u1d62`, is an isomorphism.","title":"The Chevalley\u2013Shephard\u2013Todd theorem"},{"id":"tauceti/representation-theory/symmetric-algebra/the-converse-of-the-chevalley-shephard-todd-theo","kind":"node","lean":"Representation.pseudoreflectionSubgroup_eq_top_of_bijective_aeval_symmetricAlgebraInvariants_dual","parent":3776,"result":true,"route":"../roadmap/tauceti/representation-theory/symmetric-algebra/index.html#the-converse-of-the-chevalley-shephard-todd-theo","scope":"tauceti/representation-theory/symmetric-algebra","status":"fully_proved","summary":"Let a finite group `G` act faithfully and linearly on a finite-dimensional vector space `V` over a field `K` of characteristic zero. If the polynomial invariants `Sym(V*)^G` form a polynomial algebra on homogeneous invariants, then `G` is generated by its pseudoreflections, the elements whose fixed space is a hyperplane.","title":"The converse of the Chevalley\u2013Shephard\u2013Todd theorem"},{"id":"tauceti/representation-theory/symmetric-algebra/graded-nakayama-lemma-for-polynomial-invariants","kind":"node","lean":"Representation.adjoin_range_eq_top_of_le_span_sup_sq, Representation.span_range_eq_top_of_span_range_mk_eq_top","parent":3776,"result":true,"route":"../roadmap/tauceti/representation-theory/symmetric-algebra/index.html#graded-nakayama-lemma-for-polynomial-invariants","scope":"tauceti/representation-theory/symmetric-algebra","status":"fully_proved","summary":"Homogeneous invariants of positive degree whose classes span `A\u208a \u29f8 A\u208a\u00b2`, where `A\u208a` is the ideal of invariants with zero constant term, generate the invariant algebra `Sym(M)^G` as an `R`-algebra.","title":"Graded Nakayama lemma for polynomial invariants"},{"id":"tauceti/representation-theory/symmetric-algebra/the-homogeneous-noether-bound","kind":"node","lean":"Representation.symmetricAlgebraInvariants_eq_adjoin_homogeneous","parent":3776,"result":true,"route":"../roadmap/tauceti/representation-theory/symmetric-algebra/index.html#the-homogeneous-noether-bound","scope":"tauceti/representation-theory/symmetric-algebra","status":"fully_proved","summary":"The invariants of a finite linear group over a rational algebra are generated by homogeneous invariants of positive degree at most the group order. This holds without finite generation or freeness of the underlying module.","title":"The homogeneous Noether bound"},{"id":"tauceti/representation-theory/symmetric-algebra/noethers-finiteness-theorem-representation-finit","kind":"node","lean":"Representation.finiteType_symmetricAlgebraInvariants","parent":3776,"result":true,"route":"../roadmap/tauceti/representation-theory/symmetric-algebra/index.html#noethers-finiteness-theorem-representation-finit","scope":"tauceti/representation-theory/symmetric-algebra","status":"fully_proved","summary":"Noether's finiteness theorem for the invariants of a finite group acting on a finitely generated module over a Noetherian ring: the invariant subalgebra is of finite type.","title":"Noether's finiteness theorem (`Representation.finiteType_symmetricAlgebraInvariants`)"},{"data_url":"scopes/tauceti/representation-theory/other-results.json","id":"tauceti/representation-theory/other-results","kind":"scope","parent":3630,"result":false,"route":"../roadmap/tauceti/representation-theory/other-results/index.html","scope":"tauceti/representation-theory/other-results","status":"fully_proved","summary":"Results named in the rest of Tau Ceti's `TauCeti.RepresentationTheory` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Other results"},{"id":"tauceti/representation-theory/other-results/burnside-density-theorem","kind":"node","lean":"Representation.asAlgebraHom_surjective_of_isIrreducible","parent":3783,"result":true,"route":"../roadmap/tauceti/representation-theory/other-results/index.html#burnside-density-theorem","scope":"tauceti/representation-theory/other-results","status":"fully_proved","summary":"The monoid algebra of a finite-dimensional irreducible representation over an algebraically closed field exhausts the full endomorphism algebra.","title":"Burnside density theorem"},{"id":"tauceti/representation-theory/other-results/frobenius-reciprocity-for-coind1-uniqueness","kind":"node","lean":"ContIntertwiningMap.coind\u2081_ext","parent":3783,"result":true,"route":"../roadmap/tauceti/representation-theory/other-results/index.html#frobenius-reciprocity-for-coind1-uniqueness","scope":"tauceti/representation-theory/other-results","status":"fully_proved","summary":"Two intertwining maps into a coinduced representation that agree after evaluation at `1` are equal.","title":"Frobenius reciprocity for `coind\u2081`, uniqueness"},{"id":"tauceti/representation-theory/other-results/frobeniuss-theorem-returns-the-normal-complement","kind":"node","lean":"TauCeti.frobeniusKernelSubgroup_eq_of_isComplement'","parent":3783,"result":true,"route":"../roadmap/tauceti/representation-theory/other-results/index.html#frobeniuss-theorem-returns-the-normal-complement","scope":"tauceti/representation-theory/other-results","status":"fully_proved","summary":"If `G = N \u22ca H` with `N` normal and `H` a trivial-intersection subgroup, then the subgroup that the exceptional-character argument above constructs -- out of the irreducible characters of `H`, with no reference to `N` -- is `N` itself.","title":"Frobenius's theorem returns the normal complement it was handed"},{"id":"tauceti/representation-theory/other-results/frobeniuss-theorem-the-frobenius-kernel-is-norma","kind":"node","lean":"TauCeti.frobeniusKernelSubgroup_normal","parent":3783,"result":true,"route":"../roadmap/tauceti/representation-theory/other-results/index.html#frobeniuss-theorem-the-frobenius-kernel-is-norma","scope":"tauceti/representation-theory/other-results","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.frobeniusKernelSubgroup_normal`.","title":"Frobenius's theorem: the Frobenius kernel is normal"},{"id":"tauceti/representation-theory/other-results/maschkes-theorem-for-intertwining-maps","kind":"node","lean":"Representation.IntertwiningMap.exists_leftInverse_of_injective","parent":3783,"result":true,"route":"../roadmap/tauceti/representation-theory/other-results/index.html#maschkes-theorem-for-intertwining-maps","scope":"tauceti/representation-theory/other-results","status":"fully_proved","summary":"Over a field in which the order of the finite group `G` is invertible, an injective intertwining map `f : \u03c1 \u2192 \u03c3` has an intertwining left inverse `p : \u03c3 \u2192 \u03c1`, that is, `p \u2218 f = id`.","title":"Maschke's theorem for intertwining maps"},{"id":"tauceti/representation-theory/other-results/schurs-lemma-for-contravariant-forms","kind":"node","lean":"Representation.exists_forall_apply_eq_mul_apply, Representation.eq_zero_of_isEmpty_equiv","parent":3783,"result":true,"route":"../roadmap/tauceti/representation-theory/other-results/index.html#schurs-lemma-for-contravariant-forms","scope":"tauceti/representation-theory/other-results","status":"fully_proved","summary":"Over an algebraically closed field, a contravariant form on a finite-dimensional irreducible representation is a scalar multiple of any separating contravariant form.","title":"Schur's lemma for contravariant forms"},{"id":"tauceti/representation-theory/other-results/schurs-lemma-scalar-form","kind":"node","lean":"Representation.IsIrreducible.exists_forall_apply_eq_smul","parent":3783,"result":true,"route":"../roadmap/tauceti/representation-theory/other-results/index.html#schurs-lemma-scalar-form","scope":"tauceti/representation-theory/other-results","status":"fully_proved","summary":"Over an algebraically closed field, a linear endomorphism of a finite-dimensional irreducible representation commuting with the action is multiplication by a scalar. This is `Representation.IsIrreducible.algebraMap_intertwiningMap_bijective_of_isAlgClosed` read pointwise on an unbundled endomorphism.","title":"Schur's lemma, scalar form"},{"id":"tauceti/representation-theory/other-results/schurs-lemma-scalar-half","kind":"node","lean":"ContRepresentation.exists_eq_smul_one_of_isIrreducible, ContRepresentation.eq_zero_of_isEmpty_equiv","parent":3783,"result":true,"route":"../roadmap/tauceti/representation-theory/other-results/index.html#schurs-lemma-scalar-half","scope":"tauceti/representation-theory/other-results","status":"fully_proved","summary":"Every continuous self-intertwiner of an irreducible finite-dimensional representation over an algebraically closed field is a scalar multiple of the identity.","title":"Schur's lemma, scalar half"},{"data_url":"scopes/tauceti/ring-theory.json","id":"tauceti/ring-theory","kind":"scope","parent":1095,"result":false,"route":"../roadmap/tauceti/ring-theory/index.html","scope":"tauceti/ring-theory","status":"fully_proved","summary":"Results in Tau Ceti's `TauCeti.RingTheory` modules, and the roadmaps whose articles are mostly there: by roadmap, then by area.","title":"Ring Theory"},{"data_url":"scopes/tauceti/ring-theory/part-01.json","id":"tauceti/ring-theory/part-01","kind":"scope","navigation":"transparent","parent":3792,"result":false,"route":"../roadmap/tauceti/ring-theory/part-01/index.html","scope":"tauceti/ring-theory/part-01","status":"fully_proved","title":"Analytic adic geometry for\u2026 to Power Series"},{"data_url":"scopes/tauceti/ring-theory/part-01/analytic-adic-geometry-for-diamonds.json","id":"tauceti/ring-theory/part-01/analytic-adic-geometry-for-diamonds","kind":"scope","parent":3793,"result":false,"route":"../roadmap/tauceti/ring-theory/part-01/analytic-adic-geometry-for-diamonds/index.html","scope":"tauceti/ring-theory/part-01/analytic-adic-geometry-for-diamonds","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Analytic adic geometry for diamonds"},{"id":"tauceti/ring-theory/part-01/analytic-adic-geometry-for-diamonds/morphisms-out-of-the-huber-pair-of-a-finite-alge","kind":"node","lean":"TauCeti.Huber.Pair.finiteAlgebra.homEquiv","parent":3794,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/analytic-adic-geometry-for-diamonds/index.html#morphisms-out-of-the-huber-pair-of-a-finite-alge","scope":"tauceti/ring-theory/part-01/analytic-adic-geometry-for-diamonds","status":"fully_proved","summary":"Morphisms of Huber pairs `(B, B\u207a) \u2192 (D, D\u207a)` under `(A, A\u207a)` are exactly the `A`-algebra maps `B \u2192 D`.","title":"Morphisms out of the Huber pair of a finite algebra"},{"id":"tauceti/ring-theory/part-01/analytic-adic-geometry-for-diamonds/a-finite-algebra-over-a-tate-ring-is-a-tate-ring","kind":"node","lean":"TauCeti.Huber.IsTateRing.of_isModuleTopology","parent":3794,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/analytic-adic-geometry-for-diamonds/index.html#a-finite-algebra-over-a-tate-ring-is-a-tate-ring","scope":"tauceti/ring-theory/part-01/analytic-adic-geometry-for-diamonds","status":"fully_proved","summary":"If `B` is a commutative `A`-algebra, finite as an `A`-module and carrying its natural topology, then `B` is a Tate ring; the image of a pseudouniformiser of `A` is a pseudouniformiser of `B`.","title":"A finite algebra over a Tate ring is a Tate ring"},{"id":"tauceti/ring-theory/part-01/analytic-adic-geometry-for-diamonds/tauceti-isfiniteetalesplit-iff-exists-section","kind":"node","lean":"TauCeti.isFiniteEtaleSplit_iff_exists_section","parent":3794,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/analytic-adic-geometry-for-diamonds/index.html#tauceti-isfiniteetalesplit-iff-exists-section","scope":"tauceti/ring-theory/part-01/analytic-adic-geometry-for-diamonds","status":"fully_proved","summary":"A ring has no nonsplit finite \u00e9tale covers if and only if every faithfully flat finite \u00e9tale algebra over it admits a section. Faithful flatness is needed: over a nonzero ring the zero algebra is finite \u00e9tale and totally split, but has no section.","title":"TauCeti.isFiniteEtaleSplit_iff_exists_section"},{"id":"tauceti/ring-theory/part-01/analytic-adic-geometry-for-diamonds/tauceti-isfiniteetalesplit-of-henselianlocalring","kind":"node","lean":"TauCeti.IsFiniteEtaleSplit.of_henselianLocalRing","parent":3794,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/analytic-adic-geometry-for-diamonds/index.html#tauceti-isfiniteetalesplit-of-henselianlocalring","scope":"tauceti/ring-theory/part-01/analytic-adic-geometry-for-diamonds","status":"fully_proved","summary":"A Henselian local ring with separably closed residue field, that is, a strictly Henselian local ring, has no nonsplit finite \u00e9tale covers: a faithfully flat finite \u00e9tale algebra has a point over the separably closed residue field, which lifts to a section.","title":"TauCeti.IsFiniteEtaleSplit.of_henselianLocalRing"},{"id":"tauceti/ring-theory/part-01/analytic-adic-geometry-for-diamonds/henselian-pairs-are-stable-under-directed-colimi","kind":"node","lean":"Ring.DirectLimit.henselianRing","parent":3794,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/analytic-adic-geometry-for-diamonds/index.html#henselian-pairs-are-stable-under-directed-colimi","scope":"tauceti/ring-theory/part-01/analytic-adic-geometry-for-diamonds","status":"fully_proved","summary":"If every `(G i, I i)` is a Henselian pair and the transition maps carry `I i` into `I j`, then the directed colimit `Ring.DirectLimit G f` is Henselian along the colimit ideal `\u2a06 i, (I i).map (of G f i)`.","title":"Henselian pairs are stable under directed colimits"},{"id":"tauceti/ring-theory/part-01/analytic-adic-geometry-for-diamonds/tauceti-isfiniteetalesplit","kind":"node","lean":"TauCeti.IsFiniteEtaleSplit","parent":3794,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/analytic-adic-geometry-for-diamonds/index.html#tauceti-isfiniteetalesplit","scope":"tauceti/ring-theory/part-01/analytic-adic-geometry-for-diamonds","status":"fully_proved","summary":"A ring `R` *has no nonsplit finite \u00e9tale covers* if every finite \u00e9tale `R`-algebra is totally split. This is the conclusion of Scholze's Lemma 15.3 for the perfectoid tower `A_\u221e`.","title":"TauCeti.IsFiniteEtaleSplit"},{"id":"tauceti/ring-theory/part-01/analytic-adic-geometry-for-diamonds/tauceti-huber-ispseudocoherentover","kind":"node","lean":"TauCeti.Huber.IsPseudoCoherentOver","parent":3794,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/analytic-adic-geometry-for-diamonds/index.html#tauceti-huber-ispseudocoherentover","scope":"tauceti/ring-theory/part-01/analytic-adic-geometry-for-diamonds","status":"fully_proved","summary":"`M` is *`m`-pseudocoherent* over `R`: it has a resolution `\u22ef \u2192 F\u2081 \u2192 F\u2080 \u2192 M \u2192 0` by finite free modules `F\u1d62 = Fin (n i) \u2192 R`, with augmentation `\u03b5` and differentials `d i : F\u1d62\u208a\u2081 \u2192 F\u1d62`, which is exact at `M, F\u2080, \u2026, F_{m-1}`. Thus `0`-pseudocoherent means finitely generated, `1`-pseudocoherent means finitely presented, and for `m = \u22a4` the whole resolution is exact (Kedlaya\u2013Liu II, Definition 1.1.1; Stacks, Tag 064T,\u2026","title":"TauCeti.Huber.IsPseudoCoherentOver"},{"id":"tauceti/ring-theory/part-01/analytic-adic-geometry-for-diamonds/analytic-points-of-a-pre-adic-space-wedhorn-prop","kind":"node","lean":"TauCeti.PreAdicSpace.analyticPoints","parent":3794,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/analytic-adic-geometry-for-diamonds/index.html#analytic-points-of-a-pre-adic-space-wedhorn-prop","scope":"tauceti/ring-theory/part-01/analytic-adic-geometry-for-diamonds","status":"fully_proved","summary":"the points `x` having an open neighbourhood `U` such that `\ud835\udcaa_X(U)` contains a pseudouniformiser, that is, a topologically nilpotent unit. On `Spa(A, A\u207a)` these are the points whose support is not open (`TauCeti.ValuationSpectrum.analyticPoints_presentationLimitPreAdicSpace`).","title":"Analytic points of a pre-adic space (Wedhorn, Proposition and Definition 8.36)"},{"data_url":"scopes/tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations.json","id":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations","kind":"scope","parent":3793,"result":false,"route":"../roadmap/tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations/index.html","scope":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Bordism, formal groups and chromatic operations"},{"id":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations/the-symmetric-2-cocycle-lemma-with-uniqueness","kind":"node","lean":"TauCeti.existsUnique_eq_smul_lazardCocycle","parent":3803,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations/index.html#the-symmetric-2-cocycle-lemma-with-uniqueness","scope":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations","status":"fully_proved","summary":"for `n \u2265 2`, a homogeneous symmetric additive 2-cocycle of degree `n` over any commutative ring is `r \u2022 C_n` for a unique `r`.","title":"The symmetric 2-cocycle lemma, with uniqueness"},{"id":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations/the-symmetric-2-cocycle-lemma-when-2-0-at-powers","kind":"node","lean":"TauCeti.eq_zero_of_two_eq_zero_of_aeval_X_X_eq_zero","parent":3803,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations/index.html#the-symmetric-2-cocycle-lemma-when-2-0-at-powers","scope":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations","status":"fully_proved","summary":"If `2 = 0` in `R`, a homogeneous symmetric additive 2-cocycle `f` of degree `2\u1d4f` over `R` with zero diagonal `f(X, X) = 0` is zero.","title":"The symmetric 2-cocycle lemma when `2 = 0`, at powers of two"},{"id":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations/lazards-computation-of-the-indecomposables-of-th","kind":"node","lean":"TauCeti.LazardRing.cotangentBasis","parent":3803,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations/index.html#lazards-computation-of-the-indecomposables-of-th","scope":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations","status":"fully_proved","summary":"The `\u2124`-module `I \u29f8 I\u00b2` of indecomposables of `L`, for `I` the augmentation ideal, is free with basis the classes of the generators `x\u2099 \u2208 L\u2082\u2099`, `n \u2265 1`.","title":"Lazard's computation of the indecomposables of the Lazard ring"},{"id":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations/the-graded-universal-property-of-the-lazard-ring","kind":"node","lean":"TauCeti.LazardRing.gradedLift","parent":3803,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations/index.html#the-graded-universal-property-of-the-lazard-ring","scope":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations","status":"fully_proved","summary":"for a commutative ring `A` graded by `\ud835\udc9c`, graded ring homomorphisms `L \u2192+*\u1d4d A` correspond to commutative formal group laws over `A` whose coefficient of every monomial `X^(d 0) Y^(d 1)` lies in `\ud835\udc9c (2 (d 0 + d 1 - 1))`, that is, whose coefficient `a\u1d62\u2c7c` lies in `\ud835\udc9c (2 (i + j - 1))` for all `i, j \u2265 1`. A formal group law corresponds to the homomorphism `TauCeti.LazardRing.lift` classifying it.","title":"The graded universal property of the Lazard ring"},{"id":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations/strict-coordinate-changes-of-commutative-formal","kind":"node","lean":"TauCeti.LazardRing.coordinateChangeLift","parent":3803,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations/index.html#strict-coordinate-changes-of-commutative-formal","scope":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations","status":"fully_proved","summary":"pairs of a commutative formal group law `F` and a strict coordinate `\u03c6` over `R` correspond to ring homomorphisms `L[b\u2081, b\u2082, \u2026] \u2192+* R`. The homomorphism attached to `(F, \u03c6)` restricts to the homomorphism `L \u2192+* R` classifying `F` and sends `b\u1d62` to the coefficient of `X^(i + 1)` in `\u03c6`.","title":"Strict coordinate changes of commutative formal group laws are representable"},{"id":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations/coassociativity-id-id-of-the-coproduct","kind":"node","lean":"TauCeti.LazardRing.coproduct_coassoc","parent":3803,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations/index.html#coassociativity-id-id-of-the-coproduct","scope":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations","status":"fully_proved","summary":"composition of strict isomorphisms is associative. Both sides take values in the triple tensor product `\u0393 \u2297_L \u0393 \u2297_L \u0393 = MvPolynomial \u2115+ (\u0393 \u2297_L \u0393)`, the pushout of `\u0393 \u2297_L \u0393` along the target of its right factor and of `\u0393` along `\u03b7_L`, whose three factors are `C \u2218 C`, `C \u2218 MvPolynomial.map \u03b7_R` and `MvPolynomial.map (MvPolynomial.map \u03b7_R \u2218 \u03b7_R)`. In it, `\u0394 \u2297 id` is `tensorLift (C \u2218 \u0394, \u03b9\u2083)` with `\u03b9\u2083` the third factor,\u2026","title":"Coassociativity `(\u0394 \u2297 id) \u0394 = (id \u2297 \u0394) \u0394` of the coproduct"},{"id":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations/tauceti-formalgrouphom","kind":"node","lean":"TauCeti.FormalGroupHom","parent":3803,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations/index.html#tauceti-formalgrouphom","scope":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations","status":"fully_proved","summary":"A homomorphism of formal group laws is a series `f` with zero constant term such that `f(F(X,Y)) = G(f(X),f(Y))`.","title":"TauCeti.FormalGroupHom"},{"id":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations/tauceti-formalgroupiso","kind":"node","lean":"TauCeti.FormalGroupIso","parent":3803,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations/index.html#tauceti-formalgroupiso","scope":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations","status":"fully_proved","summary":"An isomorphism of formal group laws with specified mutually inverse homomorphisms. The inverse laws are equations of substitution operators on one-variable power series.","title":"TauCeti.FormalGroupIso"},{"id":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations/formalgroup-logiso","kind":"node","lean":"FormalGroup.logIso","parent":3803,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations/index.html#formalgroup-logiso","scope":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations","status":"fully_proved","summary":"The normalized logarithm is a strict isomorphism from `F` to the additive formal group, with the formal exponential as its inverse.","title":"FormalGroup.logIso"},{"id":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations/formalgroup-height","kind":"node","lean":"FormalGroup.height","parent":3803,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations/index.html#formalgroup-height","scope":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations","status":"fully_proved","summary":"The height of a formal group law `F` over a field `K` of characteristic `p`: `\u22a4` if multiplication by `p` on `F` is zero, and otherwise the exponent `h` such that the first nonzero term of `[p]_F(T)` has degree `p ^ h` (see `FormalGroup.height_eq_natCast_iff`).","title":"FormalGroup.height"},{"id":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations/the-p-series-has-order-a-power-of-p-in-character","kind":"node","lean":"FormalGroup.exists_order_nsmulSeries_eq_pow","parent":3803,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations/index.html#the-p-series-has-order-a-power-of-p-in-character","scope":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations","status":"fully_proved","summary":"if `p` is a prime with `p = 0` in `R` and `F` is commutative, then either `[p]_F = 0` or the first nonzero term of `[p]_F` has degree `p ^ h` for some `h \u2265 1`.","title":"The `p`-series has order a power of `p` in characteristic `p`"},{"id":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations/formalgroup-topnilpotentpoint","kind":"node","lean":"FormalGroup.TopNilpotentPoint","parent":3803,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations/index.html#formalgroup-topnilpotentpoint","scope":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations","status":"fully_proved","summary":"The points of a formal group law `F` in `A`: the elements of a topological ring `A` satisfying `PowerSeries.HasEval`, that is, the topologically nilpotent elements. Evaluation of power series over `R` at these elements converges when `A` is a complete, Hausdorff, linearly topologised `R`-algebra; under those assumptions they form a group under `x +_F y = F(x, y)`. This is the analogue for `A` of the power-series\u2026","title":"FormalGroup.TopNilpotentPoint"},{"id":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations/lazards-classification-of-formal-group-laws-by-h","kind":"node","lean":"FormalGroup.height_eq_height_iff_nonempty_formalGroupIso","parent":3803,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations/index.html#lazards-classification-of-formal-group-laws-by-h","scope":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations","status":"fully_proved","summary":"Over a separably closed field `K` of characteristic `p`, two commutative formal group laws are isomorphic exactly when they have the same height. In height `\u22a4` both are isomorphic to the additive formal group law.","title":"Lazard's classification of formal group laws by height"},{"id":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations/lazards-theorem","kind":"node","lean":"TauCeti.LazardRing.mvPolynomialEquiv, TauCeti.LazardRing.adjoin_range_generator, TauCeti.LazardRing.mvPolynomialEquiv_mem_grading_iff, TauCeti.LazardRing.algebraicIndependent_generator","parent":3803,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations/index.html#lazards-theorem","scope":"tauceti/ring-theory/part-01/bordism-formal-groups-and-chromatic-operations","status":"fully_proved","summary":"the Lazard ring is the polynomial ring `\u2124[x\u2081, x\u2082, \u2026]` on the generators `x\u2099 = TauCeti.LazardRing.generator n`. The isomorphism is evaluation at the generators.","title":"Lazard's theorem"},{"data_url":"scopes/tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut.json","id":"tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut","kind":"scope","parent":3793,"result":false,"route":"../roadmap/tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut/index.html","scope":"tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Cyclic cohomology, spectral triples and noncommutative tori"},{"id":"tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut/tauceti-quasiiso-cycliccochaincomplextototal","kind":"node","lean":"TauCeti.quasiIso_cyclicCochainComplexToTotal","parent":3818,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut/index.html#tauceti-quasiiso-cycliccochaincomplextototal","scope":"tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut","status":"fully_proved","summary":"Over a ring containing `\u211a`, the inclusion of Connes' cyclic cochain complex into the total complex of the cyclic bicomplex is a quasi-isomorphism: cyclic cohomology defined by cyclic cochains is the cohomology of the total cyclic bicomplex.","title":"TauCeti.quasiIso_cyclicCochainComplexToTotal"},{"id":"tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut/tauceti-cycliccohomology-selfequiv","kind":"node","lean":"TauCeti.cyclicCohomology.selfEquiv","parent":3818,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut/index.html#tauceti-cycliccohomology-selfequiv","scope":"tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut","status":"fully_proved","summary":"If multiplication by `2` is injective on `M`, then `HC\u207f(R; M) \u2245 M` in every even degree `n`, by the class of the product cochain. In particular `HC\u00b2\u1d4f(\u2102) \u2245 \u2102`.","title":"TauCeti.cyclicCohomology.selfEquiv"},{"id":"tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut/tauceti-iszero-cycliccohomology-self-of-odd","kind":"node","lean":"TauCeti.isZero_cyclicCohomology_self_of_odd","parent":3818,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut/index.html#tauceti-iszero-cycliccohomology-self-of-odd","scope":"tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut","status":"fully_proved","summary":"The cyclic cohomology of the ground ring vanishes in odd degree.","title":"TauCeti.isZero_cyclicCohomology_self_of_odd"},{"id":"tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut/conjugation-invariance-of-the-even-pairing","kind":"node","lean":"TauCeti.cyclicCohomology.evenPairing_units_conj","parent":3818,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut/index.html#conjugation-invariance-of-the-even-pairing","scope":"tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut","status":"fully_proved","summary":"Conjugating an idempotent by a unit does not change its even pairing with cyclic cohomology.","title":"Conjugation invariance of the even pairing"},{"id":"tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut/connes-odd-pairing","kind":"node","lean":"TauCeti.cyclicCohomology.oddPairing","parent":3818,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut/index.html#connes-odd-pairing","scope":"tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut","status":"fully_proved","summary":"Connes' odd pairing `\u27e8[u], [\u03c6]\u27e9 = (1 + i)\u207b\u00b9 2\u207b\u207f \u0393(n/2 + 1)\u207b\u00b9 \u03c6(u\u207b\u00b9 - 1, u - 1, \u2026, u - 1)` of an invertible `u` with cyclic cohomology classes of odd degree `n = 2m + 1`.","title":"Connes' odd pairing"},{"id":"tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut/stabilization-of-the-odd-pairing","kind":"node","lean":"TauCeti.cyclicCohomology.oddPairing_matrixExtension_of_val_eq_fromBlocks_one","parent":3818,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut/index.html#stabilization-of-the-odd-pairing","scope":"tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut","status":"fully_proved","summary":"enlarging an invertible matrix `u` to the block diagonal matrix `u \u2295 1` does not change its odd pairing.","title":"Stabilization of the odd pairing"},{"id":"tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut/tauceti-cyclecocycle","kind":"node","lean":"TauCeti.cycleCocycle","parent":3818,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut/index.html#tauceti-cyclecocycle","scope":"tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut","status":"fully_proved","summary":"The character of a cycle, as a cyclic cocycle.","title":"TauCeti.cycleCocycle"},{"id":"tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut/tauceti-relativecycliccohomology","kind":"node","lean":"TauCeti.relativeCyclicCohomology","parent":3818,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut/index.html#tauceti-relativecycliccohomology","scope":"tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut","status":"fully_proved","summary":"The relative cyclic cohomology of a nonunital algebra: the reduced cyclic cohomology of its forced unitization, the cohomology of the kernel of scalar restriction on the cyclic cochain complex of the forced unitization.","title":"TauCeti.relativeCyclicCohomology"},{"id":"tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut/tauceti-cycliccohomology-matrixextension","kind":"node","lean":"TauCeti.cyclicCohomology.matrixExtension","parent":3818,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut/index.html#tauceti-cycliccohomology-matrixextension","scope":"tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut","status":"fully_proved","summary":"The map `HC\u207f(A; M) \u2192 HC\u207f(Matrix \u03b9 \u03b9 A; M)` on cyclic cohomology induced by matrix extension of cyclic cocycles.","title":"TauCeti.cyclicCohomology.matrixExtension"},{"id":"tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut/tauceti-continuouscycliccochains","kind":"node","lean":"TauCeti.continuousCyclicCochains","parent":3818,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut/index.html#tauceti-continuouscycliccochains","scope":"tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut","status":"fully_proved","summary":"The submodule of jointly continuous cochains satisfying the signed cyclic relation.","title":"TauCeti.continuousCyclicCochains"},{"id":"tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut/tauceti-continuous-continuousconnesb","kind":"node","lean":"TauCeti.continuous_continuousConnesB","parent":3818,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut/index.html#tauceti-continuous-continuousconnesb","scope":"tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut","status":"fully_proved","summary":"Connes' operator is continuous for bounded convergence of cochains.","title":"TauCeti.continuous_continuousConnesB"},{"id":"tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut/connes-formula-for-the-relative-odd-pairing","kind":"node","lean":"TauCeti.relativeAlgebraicK1.oddPairing_mk_mk","parent":3818,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut/index.html#connes-formula-for-the-relative-odd-pairing","scope":"tauceti/ring-theory/part-01/cyclic-cohomology-spectral-triples-and-noncommut","status":"fully_proved","summary":"an invertible matrix `u` over `A\u207a` with a relative class pairs with the class of a relative cyclic cocycle `\u03c6` of degree `n = 2m + 1` as `(1 + i)\u207b\u00b9 2\u207b\u207f \u0393(n/2 + 1)\u207b\u00b9 (\u03c6#Tr)(u\u207b\u00b9 - 1, u - 1, \u2026, u\u207b\u00b9 - 1, u - 1)`.","title":"Connes' formula for the relative odd pairing"},{"data_url":"scopes/tauceti/ring-theory/part-01/derived-module-algebra.json","id":"tauceti/ring-theory/part-01/derived-module-algebra","kind":"scope","parent":3793,"result":false,"route":"../roadmap/tauceti/ring-theory/part-01/derived-module-algebra/index.html","scope":"tauceti/ring-theory/part-01/derived-module-algebra","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Derived module algebra"},{"id":"tauceti/ring-theory/part-01/derived-module-algebra/roby-representability","kind":"node","lean":"DividedPowerAlgebra.homogeneousLawEquiv","parent":3831,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/derived-module-algebra/index.html#roby-representability","scope":"tauceti/ring-theory/part-01/derived-module-algebra","status":"fully_proved","summary":"composition with the universal law `\u03b3_d` is a linear equivalence between linear maps `\u0393^d_R(M) \u2192 N` and homogeneous polynomial laws `M \u2192 N` of degree `d`.","title":"Roby representability"},{"id":"tauceti/ring-theory/part-01/derived-module-algebra/base-change-of-divided-power-algebras","kind":"node","lean":"DividedPowerAlgebra.baseChangeEquiv","parent":3831,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/derived-module-algebra/index.html#base-change-of-divided-power-algebras","scope":"tauceti/ring-theory/part-01/derived-module-algebra","status":"fully_proved","summary":"for a commutative `R`-algebra `S`, the canonical map `S \u2297[R] \u0393_R(M) \u2192 \u0393_S(S \u2297[R] M)` is an isomorphism of `S`-algebras. No flatness is assumed.","title":"Base change of divided power algebras"},{"id":"tauceti/ring-theory/part-01/derived-module-algebra/base-change-of-homogeneous-divided-powers","kind":"node","lean":"DividedPowerAlgebra.baseChangeHomogeneousEquiv","parent":3831,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/derived-module-algebra/index.html#base-change-of-homogeneous-divided-powers","scope":"tauceti/ring-theory/part-01/derived-module-algebra","status":"fully_proved","summary":"for a commutative `R`-algebra `S`, the base change isomorphism restricts to an `S`-linear isomorphism `S \u2297[R] \u0393^d_R(M) \u2243\u2097[S] \u0393^d_S(S \u2297[R] M)` taking `1 \u2297 dp R d m` to `dp S d (1 \u2297 m)` (`baseChangeHomogeneousEquiv_one_tmul_dp`). No flatness is assumed.","title":"Base change of homogeneous divided powers"},{"id":"tauceti/ring-theory/part-01/derived-module-algebra/dividedpoweralgebra-isinternal-homogeneoussubmod","kind":"node","lean":"DividedPowerAlgebra.isInternal_homogeneousSubmodule","parent":3831,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/derived-module-algebra/index.html#dividedpoweralgebra-isinternal-homogeneoussubmod","scope":"tauceti/ring-theory/part-01/derived-module-algebra","status":"fully_proved","summary":"The homogeneous pieces of the divided power algebra form an internal direct sum: the sum map `\u2a01 d, \u0393^d_R(M) \u2192 \u0393_R(M)` is bijective.","title":"DividedPowerAlgebra.isInternal_homogeneousSubmodule"},{"id":"tauceti/ring-theory/part-01/derived-module-algebra/the-koszul-complex-of-a-weakly-regular-sequence","kind":"node","lean":"TauCeti.koszulComplex.exactAt_of_isWeaklyRegular","parent":3831,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/derived-module-algebra/index.html#the-koszul-complex-of-a-weakly-regular-sequence","scope":"tauceti/ring-theory/part-01/derived-module-algebra","status":"fully_proved","summary":"if `x = (x\u2080, \u2026, x\u2099\u208b\u2081)` is a weakly regular sequence in `R`, then `H\u2096\u208a\u2081(K(x)) = 0` for every `k`. With `homologyZeroEquivSpan`, the complex of finite free modules `K(x)` resolves `R \u29f8 (x\u2080, \u2026, x\u2099\u208b\u2081)`.","title":"The Koszul complex of a weakly regular sequence is acyclic in positive degrees"},{"id":"tauceti/ring-theory/part-01/derived-module-algebra/zeroth-koszul-homology-of-a-finite-sequence","kind":"node","lean":"TauCeti.koszulComplex.homologyZeroEquivSpan","parent":3831,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/derived-module-algebra/index.html#zeroth-koszul-homology-of-a-finite-sequence","scope":"tauceti/ring-theory/part-01/derived-module-algebra","status":"fully_proved","summary":"for `x = (x\u1d62)` in `R`, the Koszul complex of the functional `v \u21a6 \u2211\u1d62 v\u1d62 x\u1d62` has `H\u2080 \u2243 R \u29f8 (x\u1d62)`.","title":"Zeroth Koszul homology of a finite sequence"},{"id":"tauceti/ring-theory/part-01/derived-module-algebra/mittag-leffler-towers-have-vanishing-lim1-stacks","kind":"node","lean":"TauCeti.ModuleTower.subsingleton_limOne_of_isMittagLeffler","parent":3831,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/derived-module-algebra/index.html#mittag-leffler-towers-have-vanishing-lim1-stacks","scope":"tauceti/ring-theory/part-01/derived-module-algebra","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.ModuleTower.subsingleton_limOne_of_isMittagLeffler`.","title":"Mittag-Leffler towers have vanishing `lim\u00b9` (Stacks, Tag 07KW)"},{"id":"tauceti/ring-theory/part-01/derived-module-algebra/lim1-of-the-multiplication-by-p-tower-is-z-p-z","kind":"node","lean":"TauCeti.ModuleTower.limOneMultiplicationTowerEquiv","parent":3831,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/derived-module-algebra/index.html#lim1-of-the-multiplication-by-p-tower-is-z-p-z","scope":"tauceti/ring-theory/part-01/derived-module-algebra","status":"fully_proved","summary":"`lim\u00b9` of the multiplication-by-`p` tower is `\u2124_[p] \u29f8 \u2124`, the isomorphism being induced by the `p`-adic summation map `(a\u2099) \u21a6 \u2211\u2099 p\u207f a\u2099` (`limOneMultiplicationTowerEquiv_mk`).","title":"`lim\u00b9` of the multiplication-by-`p` tower is `\u2124_[p] \u29f8 \u2124`"},{"id":"tauceti/ring-theory/part-01/derived-module-algebra/dividedpoweralgebra-gamma","kind":"node","lean":"DividedPowerAlgebra.gamma","parent":3831,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/derived-module-algebra/index.html#dividedpoweralgebra-gamma","scope":"tauceti/ring-theory/part-01/derived-module-algebra","status":"fully_proved","summary":"The universal homogeneous polynomial law `\u03b3_d : M \u2192 \u0393^d_R(M)` of degree `d`. After scalar extension to a commutative `R`-algebra `S` it sends `\u2211\u2c7c s\u2c7c \u2297 m\u2c7c` to `\u2211_{\u2211\u2c7c n\u2c7c = d} (\u220f\u2c7c s\u2c7c ^ n\u2c7c) \u2297 \u220f\u2c7c dp R n\u2c7c m\u2c7c` (see `gamma_toFun_sum_tmul`), and its ground value is `m \u21a6 dp R d m` (see `ground_gamma_apply`).","title":"DividedPowerAlgebra.gamma"},{"id":"tauceti/ring-theory/part-01/derived-module-algebra/polynomiallaw-coeff","kind":"node","lean":"PolynomialLaw.coeff","parent":3831,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/derived-module-algebra/index.html#polynomiallaw-coeff","scope":"tauceti/ring-theory/part-01/derived-module-algebra","status":"fully_proved","summary":"The coefficients of polynomial laws `M \u2192\u209a\u2097[R] N` at the generic combination `\u2211\u2c7c X\u2c7c \u2297 m\u2c7c`: `coeff m f k` is the coefficient of the monomial `X\u1d4f` in `f (\u2211\u2c7c X\u2c7c \u2297 m\u2c7c)`, computed over the polynomial ring `R[X\u2c7c]`.","title":"PolynomialLaw.coeff"},{"id":"tauceti/ring-theory/part-01/derived-module-algebra/polynomiallaw-ismultiplicative","kind":"node","lean":"PolynomialLaw.IsMultiplicative","parent":3831,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/derived-module-algebra/index.html#polynomiallaw-ismultiplicative","scope":"tauceti/ring-theory/part-01/derived-module-algebra","status":"fully_proved","summary":"A polynomial law `f : A \u2192\u209a\u2097[R] B` between `R`-algebras is multiplicative if `f 1 = 1` and `f (x * y) = f x * f y` for every commutative `R`-algebra `S` and all `x y : S \u2297[R] A`. See `IsMultiplicative.toFun_one` and `IsMultiplicative.toFun_mul` for algebras in other universes.","title":"PolynomialLaw.IsMultiplicative"},{"id":"tauceti/ring-theory/part-01/derived-module-algebra/polynomiallaw-alghomequiv","kind":"node","lean":"PolynomialLaw.algHomEquiv","parent":3831,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/derived-module-algebra/index.html#polynomiallaw-alghomequiv","scope":"tauceti/ring-theory/part-01/derived-module-algebra","status":"fully_proved","summary":"Algebra maps `A \u2192\u2090[R] B` are identified with the multiplicative polynomial laws `A \u2192\u209a\u2097[R] B` homogeneous of degree one, by `\u03c6 \u21a6 \u03c6.toLinearMap.toPolynomialLaw` with inverse `IsHomogeneousOfDegree.toAlgHom`.","title":"PolynomialLaw.algHomEquiv"},{"id":"tauceti/ring-theory/part-01/derived-module-algebra/existence-of-injective-envelopes","kind":"node","lean":"TauCeti.exists_isInjectiveEnvelope","parent":3831,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/derived-module-algebra/index.html#existence-of-injective-envelopes","scope":"tauceti/ring-theory/part-01/derived-module-algebra","status":"fully_proved","summary":"Every module has an injective envelope, in its own universe.","title":"Existence of injective envelopes"},{"data_url":"scopes/tauceti/ring-theory/part-01/finite-etale-covers-of-punctured-spectra-and-loc.json","id":"tauceti/ring-theory/part-01/finite-etale-covers-of-punctured-spectra-and-loc","kind":"scope","parent":3793,"result":false,"route":"../roadmap/tauceti/ring-theory/part-01/finite-etale-covers-of-punctured-spectra-and-loc/index.html","scope":"tauceti/ring-theory/part-01/finite-etale-covers-of-punctured-spectra-and-loc","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Finite \u00e9tale covers of punctured spectra and local purity"},{"id":"tauceti/ring-theory/part-01/finite-etale-covers-of-punctured-spectra-and-loc/the-discriminant-criterion","kind":"node","lean":"TauCeti.Algebra.etale_iff_isUnit_discr","parent":3845,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/finite-etale-covers-of-punctured-spectra-and-loc/index.html#the-discriminant-criterion","scope":"tauceti/ring-theory/part-01/finite-etale-covers-of-punctured-spectra-and-loc","status":"fully_proved","summary":"A finite free algebra is \u00e9tale iff the discriminant of a basis is a unit. Stacks, Lemma 49.3.1 (0BJF), affine form.","title":"The discriminant criterion"},{"id":"tauceti/ring-theory/part-01/finite-etale-covers-of-punctured-spectra-and-loc/tauceti-algebra-etale-iff-traceform-nondegenerat","kind":"node","lean":"TauCeti.Algebra.etale_iff_traceForm_nondegenerate","parent":3845,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/finite-etale-covers-of-punctured-spectra-and-loc/index.html#tauceti-algebra-etale-iff-traceform-nondegenerat","scope":"tauceti/ring-theory/part-01/finite-etale-covers-of-punctured-spectra-and-loc","status":"fully_proved","summary":"A finite-dimensional algebra over a field is \u00e9tale iff its trace form is nondegenerate. The fibre case in the proof of Stacks, Lemma 49.3.1 (0BJF).","title":"TauCeti.Algebra.etale_iff_traceForm_nondegenerate"},{"id":"tauceti/ring-theory/part-01/finite-etale-covers-of-punctured-spectra-and-loc/tauceti-algebra-etale-iff-etale-fiber-of-height","kind":"node","lean":"TauCeti.Algebra.etale_iff_etale_fiber_of_height_le_one","parent":3845,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/finite-etale-covers-of-punctured-spectra-and-loc/index.html#tauceti-algebra-etale-iff-etale-fiber-of-height","scope":"tauceti/ring-theory/part-01/finite-etale-covers-of-punctured-spectra-and-loc","status":"fully_proved","summary":"A finite free algebra over a Noetherian ring is \u00e9tale if and only if its fibres over all primes of height at most one are \u00e9tale. Minimal primes must be included; no generic \u00e9taleness assumption is imposed.","title":"TauCeti.Algebra.etale_iff_etale_fiber_of_height_le_one"},{"id":"tauceti/ring-theory/part-01/finite-etale-covers-of-punctured-spectra-and-loc/tauceti-algebra-etale-of-isunramifiedat-of-two-l","kind":"node","lean":"TauCeti.Algebra.etale_of_isUnramifiedAt_of_two_le_ringKrullDim","parent":3845,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/finite-etale-covers-of-punctured-spectra-and-loc/index.html#tauceti-algebra-etale-of-isunramifiedat-of-two-l","scope":"tauceti/ring-theory/part-01/finite-etale-covers-of-punctured-spectra-and-loc","status":"fully_proved","summary":"A finite flat algebra over a Noetherian local ring of dimension at least two is \u00e9tale everywhere if it is unramified at each prime not lying over the closed point. This is the finite flat case of local purity, with no regularity or normality assumptions.","title":"TauCeti.Algebra.etale_of_isUnramifiedAt_of_two_le_ringKrullDim"},{"id":"tauceti/ring-theory/part-01/finite-etale-covers-of-punctured-spectra-and-loc/a-regular-local-ring-is-cohen-macaulay","kind":"node","lean":"TauCeti.IsRegularLocalRing.isCohenMacaulay","parent":3845,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/finite-etale-covers-of-punctured-spectra-and-loc/index.html#a-regular-local-ring-is-cohen-macaulay","scope":"tauceti/ring-theory/part-01/finite-etale-covers-of-punctured-spectra-and-loc","status":"fully_proved","summary":"Stacks, Lemma 10.106.3 (00NQ).","title":"A regular local ring is Cohen\u2013Macaulay"},{"id":"tauceti/ring-theory/part-01/finite-etale-covers-of-punctured-spectra-and-loc/tauceti-algebra-isunit-discr-localization-iff","kind":"node","lean":"TauCeti.Algebra.isUnit_discr_localization_iff","parent":3845,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/finite-etale-covers-of-punctured-spectra-and-loc/index.html#tauceti-algebra-isunit-discr-localization-iff","scope":"tauceti/ring-theory/part-01/finite-etale-covers-of-punctured-spectra-and-loc","status":"fully_proved","summary":"The discriminant of a basis of a finite free algebra is a unit at a prime `\ud835\udd2d` iff the base change to the localization `A_\ud835\udd2d` is \u00e9tale over `A_\ud835\udd2d`.","title":"TauCeti.Algebra.isUnit_discr_localization_iff"},{"id":"tauceti/ring-theory/part-01/finite-etale-covers-of-punctured-spectra-and-loc/tauceti-isregularlocalring-isregular-pair-of-fin","kind":"node","lean":"TauCeti.IsRegularLocalRing.isRegular_pair_of_finite_normal","parent":3845,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/finite-etale-covers-of-punctured-spectra-and-loc/index.html#tauceti-isregularlocalring-isregular-pair-of-fin","scope":"tauceti/ring-theory/part-01/finite-etale-covers-of-punctured-spectra-and-loc","status":"fully_proved","summary":"Over a two-dimensional regular local ring, every pair generating the maximal ideal is regular on any finite injective normal local algebra, considered as a module over the base.","title":"TauCeti.IsRegularLocalRing.isRegular_pair_of_finite_normal"},{"data_url":"scopes/tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con.json","id":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con","kind":"scope","parent":3793,"result":false,"route":"../roadmap/tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/index.html","scope":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Formal Laurent fields, Puiseux expansions and continuous differential calculus"},{"id":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/newton-puiseux-theorem","kind":"node","lean":"TauCeti.puiseuxField.exists_isRoot_levelEmbedding, TauCeti.LaurentSeries.exists_isRoot_map_expand","parent":3853,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/index.html#newton-puiseux-theorem","scope":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con","status":"fully_proved","summary":"Over an algebraically closed field `K` of characteristic zero, every nonconstant polynomial over the Puiseux field has a root lying in a single ramified level.","title":"Newton\u2013Puiseux theorem"},{"id":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/tauceti-puiseuxfield-instisalgclosure","kind":"node","lean":"TauCeti.puiseuxField.instIsAlgClosure","parent":3853,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/index.html#tauceti-puiseuxfield-instisalgclosure","scope":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con","status":"fully_proved","summary":"Over an algebraically closed field `K` of characteristic zero, the Puiseux field is an algebraic closure of the Laurent-series field `K\u2e28X\u2e29`.","title":"TauCeti.puiseuxField.instIsAlgClosure"},{"id":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/finite-extensions-of-kx","kind":"node","lean":"TauCeti.LaurentSeries.Level.algEquivOfFinrankEq","parent":3853,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/index.html#finite-extensions-of-kx","scope":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con","status":"fully_proved","summary":"If `K` is algebraically closed of characteristic zero, then every extension `L` of `K\u2e28X\u2e29` of degree `e` is isomorphic over `K\u2e28X\u2e29` to the ramified level `K\u2e28u\u2e29`, `u ^ e = X`. The isomorphism is a chosen one: it is built from a chosen embedding of `L` into the Puiseux field, whose image is the level-`e` subfield.","title":"Finite extensions of `K\u2e28X\u2e29`"},{"id":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/tauceti-puiseuxfield-mem-level-factorial-of-aeva","kind":"node","lean":"TauCeti.puiseuxField.mem_level_factorial_of_aeval_eq_zero","parent":3853,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/index.html#tauceti-puiseuxfield-mem-level-factorial-of-aeva","scope":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con","status":"fully_proved","summary":"If `K` contains all roots of unity, then a Puiseux series that is a root of a nonzero polynomial of degree `d` over `K\u2e28X\u2e29` lies in level `d!`. Thus all roots of such a polynomial lie in a common level.","title":"TauCeti.puiseuxField.mem_level_factorial_of_aeval_eq_zero"},{"id":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/tauceti-puiseuxfield-levelgalequivrootsofunity","kind":"node","lean":"TauCeti.puiseuxField.levelGalEquivRootsOfUnity","parent":3853,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/index.html#tauceti-puiseuxfield-levelgalequivrootsofunity","scope":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con","status":"fully_proved","summary":"If the coefficient field has a primitive `d`-th root of unity, then the Galois group of the level-`d` subfield of the Puiseux field over `K\u2e28X\u2e29` is the group of `d`-th roots of unity of `K`. The inverse is `levelKummer` (`TauCeti.puiseuxField.levelGalEquivRootsOfUnity_symm_apply`).","title":"TauCeti.puiseuxField.levelGalEquivRootsOfUnity"},{"id":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/faithfulness","kind":"node","lean":"TauCeti.puiseuxField.kummer_injective","parent":3853,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/index.html#faithfulness","scope":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con","status":"fully_proved","summary":"Distinct points of the Tate module act by distinct automorphisms of the Puiseux field: the automorphism attached to `\u03b6` sends `t ^ (1 / n)` to `\u03b6\u2099 t ^ (1 / n)`.","title":"Faithfulness"},{"id":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/finite-map-residue-formula-for-laurent-series-su","kind":"node","lean":"TauCeti.LaurentSeries.coeff_neg_one_substAlgHom_mul_derivative_eq_nsmul","parent":3853,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/index.html#finite-map-residue-formula-for-laurent-series-su","scope":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con","status":"fully_proved","summary":"Substituting a nonzero power series `\u03c6` of zero constant coefficient multiplies the residue of `f dX` by `\u03c6.order.toNat`, in all characteristics: `Res(f(\u03c6) D\u03c6 dX) = \u03c6.order.toNat \u2022 Res(f dX)`.","title":"Finite-map residue formula for Laurent series substitution"},{"id":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/the-continuous-derivations-of-kx-form-a-free-mod","kind":"node","lean":"TauCeti.continuousDerivationsEquiv","parent":3853,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/index.html#the-continuous-derivations-of-kx-form-a-free-mod","scope":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con","status":"fully_proved","summary":"the equivalence sends `a` to `a \u2022 D`, and its inverse evaluates a derivation at the parameter `X`.","title":"The continuous derivations of `K\u2e28X\u2e29` form a free module of rank one with basis the formal derivative `D`"},{"id":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/formal-coordinate-changes-act-invertibly-on-one","kind":"node","lean":"TauCeti.ContinuousKaehlerDifferential.substEquiv","parent":3853,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/index.html#formal-coordinate-changes-act-invertibly-on-one","scope":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con","status":"fully_proved","summary":"For a formal coordinate `\u03c6`, substitution `f dX \u21a6 f (\u03c6) d\u03c6` into continuous one-forms, as an equivalence that is semilinear over the substitution automorphism `TauCeti.LaurentSeries.substAlgEquiv \u03c6` of `K\u2e28X\u2e29`. Its inverse is substitution of the compositional inverse `\u03c6\u207b\u00b9`.","title":"Formal coordinate changes act invertibly on one-forms"},{"id":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/tauceti-puiseuxfield-levelcoconeiscolimit","kind":"node","lean":"TauCeti.puiseuxField.levelCoconeIsColimit","parent":3853,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/index.html#tauceti-puiseuxfield-levelcoconeiscolimit","scope":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con","status":"fully_proved","summary":"The Puiseux field is the colimit of the ramified Laurent levels in `CommAlgCat K`.","title":"TauCeti.puiseuxField.levelCoconeIsColimit"},{"id":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/the-puiseux-field-is-not-complete","kind":"node","lean":"TauCeti.puiseuxField.not_completeSpace","parent":3853,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/index.html#the-puiseux-field-is-not-complete","scope":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con","status":"fully_proved","summary":"The partial sums of `\u2211 j, t ^ ((j + 1) + 1 / (j + 2))` form a Cauchy sequence without a limit.","title":"The Puiseux field is not complete"},{"id":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/tauceti-formalcoordinate","kind":"node","lean":"TauCeti.FormalCoordinate","parent":3853,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/index.html#tauceti-formalcoordinate","scope":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con","status":"fully_proved","summary":"A formal coordinate over `R`: a power series `\u03c6 = u X + \u22ef` with zero constant coefficient and unit linear coefficient `u`. These are the changes of formal parameter `X \u21a6 \u03c6`.","title":"TauCeti.FormalCoordinate"},{"id":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/tauceti-formalcoordinate-jetlimitequiv","kind":"node","lean":"TauCeti.FormalCoordinate.jetLimitEquiv","parent":3853,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/index.html#tauceti-formalcoordinate-jetlimitequiv","scope":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con","status":"fully_proved","summary":"The formal coordinates are the inverse limit of their jet groups: passing to all jets is an isomorphism of topological groups onto the compatible families of jets.","title":"TauCeti.FormalCoordinate.jetLimitEquiv"},{"id":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/hahnmodule-instliering","kind":"node","lean":"HahnModule.instLieRing","parent":3853,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/index.html#hahnmodule-instliering","scope":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con","status":"fully_proved","summary":"Hahn series with coefficients in a Lie ring form a Lie ring under the convolution bracket.","title":"HahnModule.instLieRing"},{"id":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/liealgebra-loopalgebra-tohahnmodule","kind":"node","lean":"LieAlgebra.LoopAlgebra.toHahnModule","parent":3853,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con/index.html#liealgebra-loopalgebra-tohahnmodule","scope":"tauceti/ring-theory/part-01/formal-laurent-fields-puiseux-expansions-and-con","status":"fully_proved","summary":"The embedding of polynomial loops into `L`-valued Hahn series, sending `t ^ a \u2297 x` to the monomial `x t ^ a`.","title":"LieAlgebra.LoopAlgebra.toHahnModule"},{"data_url":"scopes/tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and.json","id":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and","kind":"scope","parent":3793,"result":false,"route":"../roadmap/tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and/index.html","scope":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Foundations of adic spaces, part II \u2014 morphisms and analytification"},{"id":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and/exactness-of-the-cech-complex-of-localizations","kind":"node","lean":"TauCeti.cechComplexLocalization_exact","parent":3869,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and/index.html#exactness-of-the-cech-complex-of-localizations","scope":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and","status":"fully_proved","summary":"If `\u03b9` is finite and the `f i` generate the unit ideal, the \u010cech complex of localizations of `M` is exact in every positive degree.","title":"Exactness of the \u010cech complex of localizations"},{"id":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and/completion-preserves-strict-exact-sequences","kind":"node","lean":"AddMonoidHom.exact_completion","parent":3869,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and/index.html#completion-preserves-strict-exact-sequences","scope":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and","status":"fully_proved","summary":"If `M \u2192 N \u2192 P` is exact with strict maps, where `M` is a first-countable nonarchimedean group and `N` is a nonarchimedean group, then the sequence of separated completions is exact. Its maps are again strict by `AddMonoidHom.isStrictMap_completion` when `M` and `N` are first countable. No Hausdorff hypothesis is needed.","title":"Completion preserves strict exact sequences"},{"id":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and/linear-maps-of-finite-modules-over-a-huber-ring","kind":"node","lean":"TauCeti.Huber.PairOfDefinition.isStrictMap_of_isNoetherianRing","parent":3869,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and/index.html#linear-maps-of-finite-modules-over-a-huber-ring","scope":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and","status":"fully_proved","summary":"If `A\u2080` is noetherian, every `A`-linear map between finitely generated `A`-modules with their natural topologies is continuous and open onto its image.","title":"Linear maps of finite modules over a Huber ring with a noetherian ring of definition are strict (Huber, Lemma 2.3(i))"},{"id":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and/submodules-of-finite-modules-over-a-complete-hub","kind":"node","lean":"TauCeti.Huber.PairOfDefinition.isClosed_of_isNoetherianRing","parent":3869,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and/index.html#submodules-of-finite-modules-over-a-complete-hub","scope":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and","status":"fully_proved","summary":"If `A` is complete and Hausdorff with a noetherian ring of definition, every `A`-submodule of a finite `A`-module is closed in its natural topology.","title":"Submodules of finite modules over a complete Huber ring with a noetherian ring of definition are closed (Huber, Lemma 2.3(ii))"},{"id":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and/completion-of-a-finite-module-is-base-change-to","kind":"node","lean":"TauCeti.Huber.PairOfDefinition.ofTensorProduct_bijective","parent":3869,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and/index.html#completion-of-a-finite-module-is-base-change-to","scope":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and","status":"fully_proved","summary":"If `A` is a Huber ring with a noetherian ring of definition and `M` is a finitely generated `A`-module with its natural topology, the canonical map `\u00c2 \u2297[A] M \u2192 M\u0302` is bijective.","title":"Completion of a finite module is base change to the completed ring (Huber, Lemma 2.3(iii))"},{"id":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and/the-blowup-of-a-finitely-generated-submodule-is","kind":"node","lean":"TauCeti.AlgebraicGeometry.submoduleBlowup.isProper_toSpec","parent":3869,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and/index.html#the-blowup-of-a-finitely-generated-submodule-is","scope":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and","status":"fully_proved","summary":"If `F` is finite, the structure morphism `Proj R(J) \u27f6 Spec C` is proper.","title":"The blowup of a finitely generated submodule is proper"},{"id":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and/the-affine-charts-of-the-blowup-of-a-submodule","kind":"node","lean":"TauCeti.AlgebraicGeometry.submoduleBlowup.basicOpenIsoSpec","parent":3869,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and/index.html#the-affine-charts-of-the-blowup-of-a-submodule","scope":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and","status":"fully_proved","summary":"For a nonempty finite subset `S \u2286 F` with product `t`, the standard open `D\u208a(t T^{#S})` of `Proj R(J)` is isomorphic to `Spec C[F/S]`, the spectrum of the `C`-subalgebra of `B_t` generated by the fractions `f/f'` with `f \u2208 F` and `f' \u2208 S`. It is Mathlib's `Proj.basicOpenIsoSpec` followed by the spectrum of `awayEquivFractionSubalgebra`.","title":"The affine charts of the blowup of a submodule"},{"id":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and/noetherianity-of-complete-rings","kind":"node","lean":"Ideal.isNoetherianRing_of_isAdicComplete","parent":3869,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and/index.html#noetherianity-of-complete-rings","scope":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and","status":"fully_proved","summary":"A ring that is `I`-adically complete for a finitely generated ideal `I` is Noetherian as soon as `R \u29f8 I` is Noetherian.","title":"Noetherianity of complete rings"},{"id":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and/tauceti-huber-isnoetheriantype","kind":"node","lean":"TauCeti.Huber.IsNoetherianType","parent":3869,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and/index.html#tauceti-huber-isnoetheriantype","scope":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and","status":"fully_proved","summary":"A Huber ring is of noetherian type if it has a noetherian ring of definition, or if it is a strongly noetherian Tate ring.","title":"TauCeti.Huber.IsNoetherianType"},{"id":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and/hubers-filtration-k-i-p-of-the-term-of-degree-p","kind":"node","lean":"TauCeti.Huber.NoetherianCech.filtration","parent":3869,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and/index.html#hubers-filtration-k-i-p-of-the-term-of-degree-p","scope":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and","status":"fully_proved","summary":"for a set `N \u2286 M`, the product over the faces `S`, `|S| = p + 1`, of the subgroups `I\u2071 \u00b7 C[F_p / f_S] \u00b7 N` of `M_{f_S}`.","title":"Huber's filtration `K_i^p` of the term of degree `p`"},{"id":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and/tauceti-huber-noetheriancech-completedterm","kind":"node","lean":"TauCeti.Huber.NoetherianCech.completedTerm","parent":3869,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and/index.html#tauceti-huber-noetheriancech-completedterm","scope":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and","status":"fully_proved","summary":"The separated completion of the degree-`p` term of the rational-localization \u010cech complex, for the product of the natural module topologies on its faces.","title":"TauCeti.Huber.NoetherianCech.completedTerm"},{"id":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and/the-tensor-topology-on-b-a-c-for-adic-structure","kind":"node","lean":"TauCeti.Huber.tensorTopology","parent":3869,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and/index.html#the-tensor-topology-on-b-a-c-for-adic-structure","scope":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-ii-morphisms-and","status":"fully_proved","summary":"for any compatible pairs of definition `(A\u2080, I)`, `(B\u2080, I B\u2080)` and `(C\u2080, I C\u2080)`, its neighbourhoods of zero are generated by the images of `I\u207f F`, where `F` is the image of `B\u2080 \u2297[A\u2080] C\u2080` (`TauCeti.Huber.hasBasis_tensorTopology`). It is not an instance, since `B \u2297[A] C` may carry other topologies.","title":"The tensor topology on `B \u2297[A] C`, for adic structure maps `A \u2192 B` and `A \u2192 C`"},{"data_url":"scopes/tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f.json","id":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f","kind":"scope","parent":3793,"result":false,"route":"../roadmap/tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f/index.html","scope":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Foundations of adic spaces, part IV \u2014 \u00e9tale sites, families and dagger geometry"},{"id":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f/units-of-the-monsky-washnitzer-algebra","kind":"node","lean":"TauCeti.Dagger.isUnit_washnitzerAlgebra_iff","parent":3882,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f/index.html#units-of-the-monsky-washnitzer-algebra","scope":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f","status":"fully_proved","summary":"Over a complete ultrametric normed field, an overconvergent series is a unit of `W` iff its constant coefficient is nonzero and strictly dominates the norm of every other coefficient.","title":"Units of the Monsky\u2013Washnitzer algebra"},{"id":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f/tauceti-dagger-washnitzeralgebra-isunit-totate-i","kind":"node","lean":"TauCeti.Dagger.washnitzerAlgebra.isUnit_toTate_iff","parent":3882,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f/index.html#tauceti-dagger-washnitzeralgebra-isunit-totate-i","scope":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f","status":"fully_proved","summary":"An overconvergent series is a unit of the Tate algebra iff it is a unit of the Monsky\u2013Washnitzer algebra: the inverse of an overconvergent series is overconvergent.","title":"TauCeti.Dagger.washnitzerAlgebra.isUnit_toTate_iff"},{"id":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f/primitives-of-overconvergent-series-are-overconv","kind":"node","lean":"TauCeti.Dagger.primitive_mem_washnitzerAlgebra","parent":3882,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f/index.html#primitives-of-overconvergent-series-are-overconv","scope":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f","status":"fully_proved","summary":"Over an ultrametric normed field of characteristic zero, the termwise primitive with respect to `X\u1d62` of an overconvergent series is overconvergent. See `MvPowerSeries.IsRestricted.primitive` for the radius-by-radius version.","title":"Primitives of overconvergent series are overconvergent"},{"id":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f/the-gauss-norm-bound-for-primitives","kind":"node","lean":"MvPowerSeries.exists_gaussNorm_primitive_le","parent":3882,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f/index.html#the-gauss-norm-bound-for-primitives","scope":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f","status":"fully_proved","summary":"Over a normed field of characteristic zero, for polyradii `0 \u2264 c' \u2264 c` with `c'\u1d62 < c\u1d62`, there is `C \u2265 0` such that for every `f` with a Gauss norm at `c`, `\u222b f dX\u1d62` has a Gauss norm at `c'` and `gaussNorm c' (\u222b f dX\u1d62) \u2264 C * gaussNorm c f`.","title":"The Gauss norm bound for primitives"},{"id":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f/distinguishing-automorphisms-of-monsky-washnitze","kind":"node","lean":"TauCeti.Dagger.washnitzerAlgebra.exists_triangularEquiv_isDistinguished_isUnit","parent":3882,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f/index.html#distinguishing-automorphisms-of-monsky-washnitze","scope":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f","status":"fully_proved","summary":"Over a complete nonarchimedean field `K`, every nonzero overconvergent series `f` in the variables `Option \u03c3`, for finite `\u03c3`, is carried by a triangular automorphism `Y \u21a6 Y`, `X\u1d62 \u21a6 X\u1d62 + Y ^ \u03b1 i` of `K\u27e8X, Y\u27e9\u2020` to a series which, as a restricted series in the variable `Y = none` over the Tate algebra in the variables `\u03c3`, is distinguished of some degree `s` at radius `1` with a unit coefficient in degree `s`.","title":"Distinguishing automorphisms of Monsky\u2013Washnitzer algebras"},{"id":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f/radius-selection-for-distinguished-overconvergen","kind":"node","lean":"TauCeti.Dagger.exists_isDistinguished_isUnit_of_isDistinguished_one","parent":3882,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f/index.html#radius-selection-for-distinguished-overconvergen","scope":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f","status":"fully_proved","summary":"Let `g` be an overconvergent series in the variables `Option \u03c3` which, as a series in `Y = X none` over the Tate algebra in the variables `\u03c3`, is distinguished of degree `s` at radius `1` with a unit coefficient `g_s`. Then for every `\u03c1\u2080 > 1` there are radii `1 < \u03c1 \u2264 r \u2264 \u03c1\u2080` such that `g` is restricted at `r`, and its coefficient slices `g\u2098`, which are restricted at `\u03c1`, form a series `G = \u2211\u2098 g\u2098 Y\u1d50` over the series\u2026","title":"Radius selection for distinguished overconvergent series"},{"id":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f/tauceti-dagger-washnitzeralgebra","kind":"node","lean":"TauCeti.Dagger.washnitzerAlgebra","parent":3882,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f/index.html#tauceti-dagger-washnitzeralgebra","scope":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f","status":"fully_proved","summary":"The Monsky\u2013Washnitzer algebra `R\u27e8X\u27e9\u2020` of overconvergent power series: the series restricted at some constant radius `\u03c1 > 1`.","title":"TauCeti.Dagger.washnitzerAlgebra"},{"id":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f/tauceti-dagger-washnitzeralgebra-totate","kind":"node","lean":"TauCeti.Dagger.washnitzerAlgebra.toTate","parent":3882,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f/index.html#tauceti-dagger-washnitzeralgebra-totate","scope":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f","status":"fully_proved","summary":"The inclusion of the Monsky\u2013Washnitzer algebra into the unit-radius Tate algebra `R{X/1}`, through which `W` carries the Gauss norm.","title":"TauCeti.Dagger.washnitzerAlgebra.toTate"},{"id":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f/isomorphisms-of-monsky-washnitzer-algebras-from","kind":"node","lean":"TauCeti.Dagger.washnitzerAlgebra.substEquiv","parent":3882,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f/index.html#isomorphisms-of-monsky-washnitzer-algebras-from","scope":"tauceti/ring-theory/part-01/foundations-of-adic-spaces-part-iv-etale-sites-f","status":"fully_proved","summary":"If two families of unit-ball polynomials without constant term are inverse to each other under substitution, the substitutions are mutually inverse `R`-algebra isomorphisms `R\u27e8X\u27e9\u2020 \u2243 R\u27e8Y\u27e9\u2020`.","title":"Isomorphisms of Monsky\u2013Washnitzer algebras from inverse substitutions"},{"data_url":"scopes/tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste.json","id":"tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste","kind":"scope","parent":3793,"result":false,"route":"../roadmap/tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste/index.html","scope":"tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Morse\u2013Novikov homology and dynamical Reidemeister torsion"},{"id":"tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste/tauceti-novikovseries-completionpkg","kind":"node","lean":"TauCeti.NovikovSeries.completionPkg","parent":3892,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste/index.html#tauceti-novikovseries-completionpkg","scope":"tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste","status":"fully_proved","summary":"The Novikov series are the completion of the group ring `H \u2192\u2080 R` for the uniform structure of its period filtration: they are complete and separated, and the inclusion `ofFinsupp` is a uniform inducing map with dense range.","title":"TauCeti.NovikovSeries.completionPkg"},{"id":"tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste/tauceti-novikovseries-isdomain-of-fg","kind":"node","lean":"TauCeti.NovikovSeries.isDomain_of_fg","parent":3892,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste/index.html#tauceti-novikovseries-isdomain-of-fg","scope":"tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste","status":"fully_proved","summary":"Over a domain, the Novikov ring of a finitely generated cancellative torsion-free commutative monoid, for instance of a finitely generated torsion-free abelian group, is a domain for every period homomorphism.","title":"TauCeti.NovikovSeries.isDomain_of_fg"},{"id":"tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste/exponential-and-logarithm-of-novikov-series","kind":"node","lean":"TauCeti.NovikovSeries.expMulEquiv","parent":3892,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste/index.html#exponential-and-logarithm-of-novikov-series","scope":"tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste","status":"fully_proved","summary":"over a `\u211a`-algebra `K`, the exponential is an isomorphism from the additive group of the positive ideal onto its principal unit group in the nonnegative subring, with inverse the logarithm. This is `Ideal.expMulEquiv` for the positive ideal, whose series are topologically nilpotent.","title":"Exponential and logarithm of Novikov series"},{"id":"tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste/multiplicativity-of-the-torsion","kind":"node","lean":"HomologicalComplex.reidemeisterTorsion_eq_mul_of_shortExact","parent":3892,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste/index.html#multiplicativity-of-the-torsion","scope":"tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste","status":"fully_proved","summary":"Let `0 \u2192 C\u2081 \u2192 C\u2082 \u2192 C\u2083 \u2192 0` be a short exact sequence of chain complexes of vector spaces, with `C\u2081` and `C\u2083` acyclic, and with compatible bases: the basis `c\u2082 i` of `C\u2082\u1d62` is the image of the basis `c\u2081 i` of `C\u2081\u1d62` followed by lifts of the basis `c\u2083 i` of `C\u2083\u1d62`. Then `\u03c4(C\u2082) = \u03b5 \u00b7 \u03c4(C\u2081) \u00b7 \u03c4(C\u2083)`, with the sign `\u03b5 = \u220f\u2c7c (-1) ^ (dim B\u2083,\u2c7c\u208a\u2081 \u00b7 dim B\u2081,\u2c7c)` given by the dimensions of the boundaries `B\u2096,\u2c7c \u2286 C\u2096,\u2c7c`. This also\u2026","title":"Multiplicativity of the torsion"},{"id":"tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste/homologicalcomplex-reidemeistertorsion-eq-zero-i","kind":"node","lean":"HomologicalComplex.reidemeisterTorsion_eq_zero_iff","parent":3892,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste/index.html#homologicalcomplex-reidemeistertorsion-eq-zero-i","scope":"tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste","status":"fully_proved","summary":"The torsion of a bounded based complex vanishes if and only if the complex is not acyclic.","title":"HomologicalComplex.reidemeisterTorsion_eq_zero_iff"},{"id":"tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste/homologicalcomplex-reidemeistertorsion-eq-mul-fi","kind":"node","lean":"HomologicalComplex.reidemeisterTorsion_eq_mul_finprod_det","parent":3892,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste/index.html#homologicalcomplex-reidemeistertorsion-eq-mul-fi","scope":"tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste","status":"fully_proved","summary":"The basis-change formula for the torsion: replacing the bases `c i` of `C` by `c' i` multiplies the torsion by `\u220f\u1d62 [c'\u1d62 / c\u1d62] ^ ((-1) ^ i)`, where `[c'\u1d62 / c\u1d62]` is the determinant of `c' i` with respect to `c i`. This holds for every complex, both sides being `0` when `C` is not acyclic or is unbounded.","title":"HomologicalComplex.reidemeisterTorsion_eq_mul_finprod_det"},{"id":"tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste/tauceti-novikovseries-mapringhom","kind":"node","lean":"TauCeti.NovikovSeries.mapRingHom","parent":3892,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste/index.html#tauceti-novikovseries-mapringhom","scope":"tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste","status":"fully_proved","summary":"The pushforward `P_(\u03c1,\u03c6)` of Novikov series along a ring homomorphism `\u03c1` of coefficients and an exponent homomorphism `\u03c6` rescaling periods by a positive constant, as a ring homomorphism: the coefficient at `h'` is the sum of the images under `\u03c1` of the coefficients at the exponents `h` with `\u03c6 h = h'`.","title":"TauCeti.NovikovSeries.mapRingHom"},{"id":"tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste/tauceti-novikovseries-tolaurentseries","kind":"node","lean":"TauCeti.NovikovSeries.toLaurentSeries","parent":3892,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste/index.html#tauceti-novikovseries-tolaurentseries","scope":"tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste","status":"fully_proved","summary":"Novikov series on `\u2124` for the period map `k \u21a6 k` are the Laurent series: the ring isomorphism which keeps every coefficient.","title":"TauCeti.NovikovSeries.toLaurentSeries"},{"id":"tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste/matrix-continuous-mulvec-extension-iff","kind":"node","lean":"Matrix.continuous_mulVec_extension_iff","parent":3892,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste/index.html#matrix-continuous-mulvec-extension-iff","scope":"tauceti/ring-theory/part-01/morse-novikov-homology-and-dynamical-reidemeiste","status":"fully_proved","summary":"A matrix over the group ring has a unique continuous extension of its action to completed coordinate modules. Its entries are included coefficientwise in the Novikov ring. No topology on the group ring is needed to state this uniqueness property.","title":"Matrix.continuous_mulVec_extension_iff"},{"data_url":"scopes/tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha.json","id":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha","kind":"scope","parent":3793,"result":false,"route":"../roadmap/tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/index.html","scope":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"p-adic measures, completed group algebras, and characteristic ideals"},{"data_url":"scopes/tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01.json","id":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01","kind":"scope","navigation":"transparent","parent":3902,"result":false,"route":"../roadmap/tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/index.html","scope":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01","status":"fully_proved","title":"Integral measures are the\u2026 to The Weierstrass quotient bound\u2026"},{"id":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/integral-measures-are-the-completed-group-algebr","kind":"node","lean":"AbstractMeasure.completedGroupAlgebraEquiv","parent":3903,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/index.html#integral-measures-are-the-completed-group-algebr","scope":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01","status":"fully_proved","summary":"The canonical algebra equivalence between integral measures on a profinite group and its completed group algebra.","title":"Integral measures are the completed group algebra"},{"id":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/integral-measures-are-the-compatible-families-al","kind":"node","lean":"AbstractMeasure.towerCoordinatesEquiv","parent":3903,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/index.html#integral-measures-are-the-compatible-families-al","scope":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01","status":"fully_proved","summary":"For a compact group with a tower of finite quotients, sending an integral measure to its finite projections at all levels is a linear equivalence onto the compatible families.","title":"Integral measures are the compatible families along a tower"},{"id":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/integral-measures-on-z-px-from-unit-coordinates","kind":"node","lean":"AbstractMeasure.existsUnique_finiteProjection_unitsToZModPow_eq","parent":3903,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/index.html#integral-measures-on-z-px-from-unit-coordinates","scope":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01","status":"fully_proved","summary":"For every family of coefficients `c n : (\u2124/p^n\u2124)\u02e3 \u2192\u2080 \u2124_p` compatible under the reductions `(\u2124/p^n\u2124)\u02e3 \u2192 (\u2124/p^m\u2124)\u02e3`, there is a unique integral measure on `\u2124_p\u02e3` whose finite projection along the reduction `\u2124_p\u02e3 \u2192 (\u2124/p^n\u2124)\u02e3` is `c n` for every `n`.","title":"Integral measures on `\u2124_p\u02e3` from unit coordinates"},{"id":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/abstractmeasure-amicetransform-weight-id","kind":"node","lean":"AbstractMeasure.amiceTransform_weight_id","parent":3903,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/index.html#abstractmeasure-amicetransform-weight-id","scope":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01","status":"fully_proved","summary":"For `\u2124_p`-valued measures, multiplication by `x` acts on the Amice transform as the Mahler derivation: `A (x \u03bc) = (1 + T) d/dT A_\u03bc`.","title":"AbstractMeasure.amiceTransform_weight_id"},{"id":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/the-height-one-primes-of-ox","kind":"node","lean":"PowerSeries.height_eq_one_iff_of_isDiscreteValuationRing","parent":3903,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/index.html#the-height-one-primes-of-ox","scope":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01","status":"fully_proved","summary":"Over a discrete valuation ring `O` with uniformiser `\u03d6`, complete for its maximal ideal, a prime ideal of `O\u27e6X\u27e7` has height one if and only if it is generated by `C \u03d6` or by an irreducible polynomial distinguished at the maximal ideal.","title":"The height-one primes of `O\u27e6X\u27e7`"},{"id":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/the-kernel-of-the-level-coordinate","kind":"node","lean":"TauCeti.Iwasawa.ker_levelCoordinate","parent":3903,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/index.html#the-kernel-of-the-level-coordinate","scope":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01","status":"fully_proved","summary":"The kernel of the level coordinate is the principal ideal generated by `\u03c9_n = (1 + T) ^ p\u207f - 1`.","title":"The kernel of the level coordinate"},{"id":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/the-level-coordinates-separate-points","kind":"node","lean":"TauCeti.Iwasawa.iwasawaCoordinate_injective","parent":3903,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/index.html#the-level-coordinates-separate-points","scope":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01","status":"fully_proved","summary":"over a noetherian `O`, the product `O\u27e6T\u27e7 \u2192 \u220f\u2099 O[\u2124/p\u207f]` of the level maps is injective.","title":"The level coordinates separate points"},{"id":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/tauceti-finitequotienttower","kind":"node","lean":"TauCeti.FiniteQuotientTower","parent":3903,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/index.html#tauceti-finitequotienttower","scope":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01","status":"fully_proved","summary":"A tower of finite quotients of a topological group `\u0393`: continuous surjective homomorphisms `q n : \u0393 \u2192 A n` onto discrete groups, with transition homomorphisms `t h : A n \u2192 A m` for `m \u2264 n` compatible with the projections, such that the projections separate the points of `\u0393`.","title":"TauCeti.FiniteQuotientTower"},{"id":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/tauceti-finitequotienttower-compatiblefamilies","kind":"node","lean":"TauCeti.FiniteQuotientTower.compatibleFamilies","parent":3903,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/index.html#tauceti-finitequotienttower-compatiblefamilies","scope":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01","status":"fully_proved","summary":"The compatible families along a tower: the families of finitely supported coefficients `c n : A n \u2192\u2080 R` such that pushing `c n` forward along the transition map to a level `m \u2264 n`, which sums the coefficients over each fibre, gives `c m`.","title":"TauCeti.FiniteQuotientTower.compatibleFamilies"},{"id":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/powerseries-isweierstrassfactorizationat-powerba","kind":"node","lean":"PowerSeries.IsWeierstrassFactorizationAt.powerBasis","parent":3903,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/index.html#powerseries-isweierstrassfactorizationat-powerba","scope":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01","status":"fully_proved","summary":"For a Weierstrass factorization `g = f * h`, the power basis of `A\u27e6X\u27e7 \u29f8 (g)` over `A` generated by the class of `X`: the classes of `1, X, \u2026, X ^ (deg f - 1)` form a basis. It is the monic power basis of `A[X] \u29f8 (f)` transported along `algEquivQuotient`.","title":"PowerSeries.IsWeierstrassFactorizationAt.powerBasis"},{"id":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/extension-of-coefficients-from-the-unit-ball","kind":"node","lean":"AbstractMeasure.extendCoefficients","parent":3903,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/index.html#extension-of-coefficients-from-the-unit-ball","scope":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01","status":"fully_proved","summary":"For a ring homomorphism `j : O \u2192+* K` into the closed unit ball of a complete ultrametric field, `extendCoefficients j hj \u03bc` is the `K`-valued measure on the profinite space `X` whose coefficient on each fibre of a map to a finite discrete space is the image under `j` of the coefficient of `\u03bc` (`finiteProjection_extendCoefficients`). When `j` is continuous it is the unique `K`-valued measure with\u2026","title":"Extension of coefficients from the unit ball"},{"id":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/factor-theorem-for-restricted-series","kind":"node","lean":"PowerSeries.IsRestricted.evalAt_eq_zero_iff","parent":3903,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/index.html#factor-theorem-for-restricted-series","scope":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01","status":"fully_proved","summary":"Over a complete ultrametric normed commutative ring, a series restricted at a radius `c > 0` vanishes at a point `a` of the closed disc of radius `c` exactly when it is `X - a` times a series restricted at `c`.","title":"Factor theorem for restricted series"},{"id":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/the-schwarz-lemma-for-restricted-series","kind":"node","lean":"PowerSeries.IsRestricted.norm_evalAt_le_gaussNorm_mul_prod, PowerSeries.IsRestricted.norm_evalAt_le_gaussNorm_mul_prod_pow","parent":3903,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/index.html#the-schwarz-lemma-for-restricted-series","scope":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01","status":"fully_proved","summary":"Let `f` be a power series over a complete ultrametric normed field `K`, restricted at a radius `c > 0`. Let `Z` be a finite multiset of points of the closed disc of radius `c` such that `f` vanishes at each `b \u2208 Z` to at least the multiplicity of `b` in `Z`, in the sense that the Hasse derivatives `(D\u207d\u1d57\u207e f)(b)` vanish for `t` below that multiplicity. Then at every point `x` of the closed disc of radius `c` in a\u2026","title":"The Schwarz lemma for restricted series"},{"id":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/strassmanns-theorem","kind":"node","lean":"TauCeti.PowerSeries.IsDistinguished.encard_setOf_evalAt_eq_zero_le","parent":3903,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/index.html#strassmanns-theorem","scope":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01","status":"fully_proved","summary":"Let `f` be a power series over a complete ultrametric normed field `K`, restricted at a radius `c > 0` and distinguished of degree `s` there, that is, `s` is the largest index at which the weighted coefficient norm `\u2016a\u2099\u2016 c\u207f` attains the Gauss norm. Then `f` has at most `s` zeros of norm at most `c` in any complete normed field extension `L` of `K`.","title":"Strassmann's theorem"},{"id":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/the-weierstrass-division-estimate-bosch-guntzer","kind":"node","lean":"TauCeti.PowerSeries.IsDistinguished.gaussNorm_mul_add_eq_max, TauCeti.PowerSeries.IsMulDistinguished.gaussNorm_mul_add_eq_max","parent":3903,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/index.html#the-weierstrass-division-estimate-bosch-guntzer","scope":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01","status":"fully_proved","summary":"If `f` is distinguished of degree `s` at the radius `c`, right multiplication by its coefficient in degree `s` scales norms exactly, `q` is restricted, and `r` vanishes in every degree `\u2265 s`, then","title":"The Weierstrass division estimate (Bosch\u2013G\u00fcntzer\u2013Remmert \u00a75.2.1, Theorem 2)"},{"id":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/the-weierstrass-division-identity","kind":"node","lean":"TauCeti.PowerSeries.IsMulDistinguished.mul_weierstrassDiv_add_weierstrassMod","parent":3903,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/index.html#the-weierstrass-division-identity","scope":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01","status":"fully_proved","summary":"`f = g * q + r` for the Weierstrass quotient `q` and remainder `r` of `f`.","title":"The Weierstrass division identity"},{"id":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/the-weierstrass-lower-bound-for-the-quotient","kind":"node","lean":"TauCeti.PowerSeries.IsDistinguished.le_gaussNorm_mul_add, TauCeti.PowerSeries.IsDistinguished.gaussNorm_le_gaussNorm_mul_add","parent":3903,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/index.html#the-weierstrass-lower-bound-for-the-quotient","scope":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01","status":"fully_proved","summary":"In a decomposition `q * f + r` by a distinguished series `f` of degree `s`, with `r` vanishing in every degree `\u2265 s`, the Gauss norm of the sum is at least the product of the Gauss norms of `q` and `f`, provided right multiplication by the coefficient of `f` in degree `s` scales norms exactly.","title":"The Weierstrass lower bound for the quotient"},{"id":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/the-weierstrass-norm-identity","kind":"node","lean":"TauCeti.PowerSeries.IsMulDistinguished.gaussNorm_eq_max","parent":3903,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/index.html#the-weierstrass-norm-identity","scope":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01","status":"fully_proved","summary":"The Gauss norm of `f` at `c` is the larger of `\u2016g\u2016 * \u2016q\u2016` and `\u2016r\u2016`, for its Weierstrass quotient `q` and remainder `r`.","title":"The Weierstrass norm identity"},{"id":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/weierstrass-preparation","kind":"node","lean":"TauCeti.PowerSeries.IsDistinguished.existsUnique_isUnit_isMonicOfDegree_mul_eq, TauCeti.PowerSeries.IsDistinguished.exists_isUnit_isMonicOfDegree_mul_eq","parent":3903,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/index.html#weierstrass-preparation","scope":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01","status":"fully_proved","summary":"Weierstrass preparation (Bosch\u2013G\u00fcntzer\u2013Remmert \u00a75.2.2, Theorem 1), in its unique-existence form: over a complete ultrametric normed commutative ring with multiplicative norm, a restricted series distinguished of degree `s` at a positive radius, whose coefficient in degree `s` is a unit, is in exactly one way a unit of the ring of restricted series times a monic polynomial of degree `s`.","title":"Weierstrass preparation"},{"id":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/the-weierstrass-quotient-bound-q-f-g-for-the-gau","kind":"node","lean":"TauCeti.PowerSeries.IsMulDistinguished.gaussNorm_weierstrassDiv_le","parent":3903,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01/index.html#the-weierstrass-quotient-bound-q-f-g-for-the-gau","scope":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-01","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.PowerSeries.IsMulDistinguished.gaussNorm_weierstrassDiv_le`.","title":"The Weierstrass quotient bound `\u2016q\u2016 \u2264 \u2016f\u2016 / \u2016g\u2016`, for the Gauss norm at `c`"},{"data_url":"scopes/tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-02.json","id":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-02","kind":"scope","navigation":"transparent","parent":3902,"result":false,"route":"../roadmap/tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-02/index.html","scope":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-02","status":"fully_proved","title":"The Weierstrass remainder bound\u2026"},{"id":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-02/the-weierstrass-remainder-bound-r-f-for-the-gaus","kind":"node","lean":"TauCeti.PowerSeries.IsMulDistinguished.gaussNorm_weierstrassMod_le","parent":3924,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-02/index.html#the-weierstrass-remainder-bound-r-f-for-the-gaus","scope":"tauceti/ring-theory/part-01/p-adic-measures-completed-group-algebras-and-cha/part-02","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.PowerSeries.IsMulDistinguished.gaussNorm_weierstrassMod_le`.","title":"The Weierstrass remainder bound `\u2016r\u2016 \u2264 \u2016f\u2016`, for the Gauss norm at `c`"},{"data_url":"scopes/tauceti/ring-theory/part-01/rigid-analytic-geometry.json","id":"tauceti/ring-theory/part-01/rigid-analytic-geometry","kind":"scope","parent":3793,"result":false,"route":"../roadmap/tauceti/ring-theory/part-01/rigid-analytic-geometry/index.html","scope":"tauceti/ring-theory/part-01/rigid-analytic-geometry","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Rigid analytic geometry"},{"id":"tauceti/ring-theory/part-01/rigid-analytic-geometry/the-maximum-modulus-principle","kind":"node","lean":"TauCeti.IsAffinoidAlgebra.exists_pointNorm_eq_supSeminorm","parent":3926,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/rigid-analytic-geometry/index.html#the-maximum-modulus-principle","scope":"tauceti/ring-theory/part-01/rigid-analytic-geometry","status":"fully_proved","summary":"On a nonzero affinoid algebra the supremum seminorm is attained at a classical point: for every `f : A` there is a maximal ideal `x` of `A` with `|f(x)| = |f|sup`.","title":"The maximum modulus principle"},{"id":"tauceti/ring-theory/part-01/rigid-analytic-geometry/tauceti-mvpowerseries-supseminorm-eq-norm","kind":"node","lean":"TauCeti.MvPowerSeries.supSeminorm_eq_norm","parent":3926,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/rigid-analytic-geometry/index.html#tauceti-mvpowerseries-supseminorm-eq-norm","scope":"tauceti/ring-theory/part-01/rigid-analytic-geometry","status":"fully_proved","summary":"On a Tate algebra the supremum seminorm is the Gauss norm: `|f|sup = \u2016f\u2016`.","title":"TauCeti.MvPowerSeries.supSeminorm_eq_norm"},{"id":"tauceti/ring-theory/part-01/rigid-analytic-geometry/affinoid-noether-normalisation","kind":"node","lean":"TauCeti.IsAffinoidAlgebra.exists_finite_injective","parent":3926,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/rigid-analytic-geometry/index.html#affinoid-noether-normalisation","scope":"tauceti/ring-theory/part-01/rigid-analytic-geometry","status":"fully_proved","summary":"Every nonzero affinoid algebra over a complete ultrametric field is finite over an injectively embedded finite-variable Tate algebra. Finiteness uses the algebra structure induced by the displayed map.","title":"Affinoid Noether normalisation"},{"id":"tauceti/ring-theory/part-01/rigid-analytic-geometry/tauceti-isaffinoidalgebra-continuous-alghom","kind":"node","lean":"TauCeti.IsAffinoidAlgebra.continuous_algHom","parent":3926,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/rigid-analytic-geometry/index.html#tauceti-isaffinoidalgebra-continuous-alghom","scope":"tauceti/ring-theory/part-01/rigid-analytic-geometry","status":"fully_proved","summary":"Every homomorphism from a noetherian ultrametric Banach algebra to a completely normed affinoid algebra is continuous. The affinoid presentation is purely algebraic.","title":"TauCeti.IsAffinoidAlgebra.continuous_algHom"},{"id":"tauceti/ring-theory/part-01/rigid-analytic-geometry/equivalence-of-complete-affinoid-norms","kind":"node","lean":"TauCeti.IsAffinoidAlgebra.toContinuousAlgEquiv","parent":3926,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/rigid-analytic-geometry/index.html#equivalence-of-complete-affinoid-norms","scope":"tauceti/ring-theory/part-01/rigid-analytic-geometry","status":"fully_proved","summary":"An algebra isomorphism between completely normed affinoid algebras is a homeomorphism, whatever complete ultrametric algebra norms they carry. Applied to two such norms on one algebra, this says that they define the same topology.","title":"Equivalence of complete affinoid norms"},{"id":"tauceti/ring-theory/part-01/rigid-analytic-geometry/heights-of-maximal-ideals-of-affinoid-domains","kind":"node","lean":"TauCeti.IsAffinoidAlgebra.height_eq_ringKrullDim_of_isDomain","parent":3926,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/rigid-analytic-geometry/index.html#heights-of-maximal-ideals-of-affinoid-domains","scope":"tauceti/ring-theory/part-01/rigid-analytic-geometry","status":"fully_proved","summary":"Every maximal ideal of an affinoid domain `A` has height `dim A`.","title":"Heights of maximal ideals of affinoid domains"},{"id":"tauceti/ring-theory/part-01/rigid-analytic-geometry/the-supremum-seminorm-is-the-spectral-value-of-t","kind":"node","lean":"TauCeti.IsAffinoidAlgebra.supSeminorm_eq_iSup_minpoly_coeff_rpow","parent":3926,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/rigid-analytic-geometry/index.html#the-supremum-seminorm-is-the-spectral-value-of-t","scope":"tauceti/ring-theory/part-01/rigid-analytic-geometry","status":"fully_proved","summary":"Let `B` be an affinoid algebra which is integral and torsion-free over an integrally closed domain `A`. For `f : B` with minimal polynomial `X ^ d + \u2211_{i < d} a\u1d62 X ^ i` over `A`, `|f|sup = max_{i < d} |a\u1d62|sup ^ (1 / (d - i))`.","title":"The supremum seminorm is the spectral value of the minimal polynomial"},{"id":"tauceti/ring-theory/part-01/rigid-analytic-geometry/tauceti-supseminorm","kind":"node","lean":"TauCeti.supSeminorm","parent":3926,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/rigid-analytic-geometry/index.html#tauceti-supseminorm","scope":"tauceti/ring-theory/part-01/rigid-analytic-geometry","status":"fully_proved","summary":"The supremum seminorm `|f|sup` of `f : A`: the supremum of its absolute values `|f(x)|` at the maximal ideals `x` of `A`. It is zero on the zero algebra, which has no maximal ideals.","title":"TauCeti.supSeminorm"},{"id":"tauceti/ring-theory/part-01/rigid-analytic-geometry/restricted-series-over-an-affinoid-algebra-form","kind":"node","lean":"TauCeti.IsAffinoidAlgebra.isRestricted_subring","parent":3926,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/rigid-analytic-geometry/index.html#restricted-series-over-an-affinoid-algebra-form","scope":"tauceti/ring-theory/part-01/rigid-analytic-geometry","status":"fully_proved","summary":"For an affinoid algebra `A` with a complete ultrametric algebra norm, the Gauss-normed algebra `A\u27e8X\u2081, \u2026, X\u2098\u27e9` of unit-radius restricted series is affinoid: a Tate presentation `K\u27e8Y\u2081, \u2026, Y\u2099\u27e9 \u2192 A` induces a presentation `K\u27e8X\u2081, \u2026, X\u2098, Y\u2081, \u2026, Y\u2099\u27e9 \u2192 A\u27e8X\u2081, \u2026, X\u2098\u27e9`.","title":"Restricted series over an affinoid algebra form an affinoid algebra"},{"id":"tauceti/ring-theory/part-01/rigid-analytic-geometry/the-universal-property-of-restricted-series-over","kind":"node","lean":"TauCeti.IsAffinoidAlgebra.existsUnique_algHom_isRestricted_subring","parent":3926,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/rigid-analytic-geometry/index.html#the-universal-property-of-restricted-series-over","scope":"tauceti/ring-theory/part-01/rigid-analytic-geometry","status":"fully_proved","summary":"Let `A` and `B` be affinoid algebras with complete ultrametric algebra norms. A `K`-algebra map `\u03c6 : A \u2192 B` and power-bounded `b\u2081, \u2026, b\u2096 \u2208 B` extend to exactly one `K`-algebra map `A\u27e8X\u2081, \u2026, X\u2096\u27e9 \u2192 B` sending each `X\u1d62` to `b\u1d62`, namely `TauCeti.IsAffinoidAlgebra.restrictedLift`. It is continuous, as is every algebra map between such algebras (`TauCeti.IsAffinoidAlgebra.continuous_algHom`).","title":"The universal property of restricted series over an affinoid algebra"},{"id":"tauceti/ring-theory/part-01/rigid-analytic-geometry/tauceti-gtopology","kind":"node","lean":"TauCeti.GTopology","parent":3926,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/rigid-analytic-geometry/index.html#tauceti-gtopology","scope":"tauceti/ring-theory/part-01/rigid-analytic-geometry","status":"fully_proved","summary":"Admissible covers on a family of subsets closed under finite intersections and containing the whole space. The family is represented by an order embedding `\u03b9`, preserving intersections and the largest open. Covering families are sets, so repetitions and indexing choices carry no additional data.","title":"TauCeti.GTopology"},{"id":"tauceti/ring-theory/part-01/rigid-analytic-geometry/tauceti-gtopology-issheaf-iff","kind":"node","lean":"TauCeti.GTopology.isSheaf_iff","parent":3926,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/rigid-analytic-geometry/index.html#tauceti-gtopology-issheaf-iff","scope":"tauceti/ring-theory/part-01/rigid-analytic-geometry","status":"fully_proved","summary":"The site sheaf condition is precisely unique gluing of matching families on all admissible covers. `Presieve.IsSheafFor` is Mathlib's existence-and-uniqueness condition for amalgamations.","title":"TauCeti.GTopology.isSheaf_iff"},{"data_url":"scopes/tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat.json","id":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat","kind":"scope","parent":3793,"result":false,"route":"../roadmap/tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat/index.html","scope":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Scheme, stack, cohomology and intersection foundations"},{"id":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat/the-henselization-of-a-pair-is-henselian","kind":"node","lean":"TauCeti.Henselization.henselian","parent":3939,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat/index.html#the-henselization-of-a-pair-is-henselian","scope":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat","status":"fully_proved","summary":"The pair formed by the henselization of `(R, I)` and the extended ideal is a henselian pair, with no hypothesis on `R` or `I`.","title":"The henselization of a pair is henselian"},{"id":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat/the-universal-property-of-the-henselization","kind":"node","lean":"TauCeti.Henselization.existsUnique_lift","parent":3939,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat/index.html#the-universal-property-of-the-henselization","scope":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat","status":"fully_proved","summary":"For a henselian pair `(S, J)` receiving a map of pairs from `(R, I)`, there is a unique `R`-algebra map from the henselization of `(R, I)` to `S` carrying the extended ideal into `J`.","title":"The universal property of the henselization"},{"id":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat/the-universal-property-of-the-henselization-of-a","kind":"node","lean":"TauCeti.Henselization.existsUnique_isLocalHom","parent":3939,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat/index.html#the-universal-property-of-the-henselization-of-a","scope":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat","status":"fully_proved","summary":"Every local homomorphism from a local ring `R` to a henselian local ring `S` factors through a unique local `R`-algebra map from the henselization of `R` along its maximal ideal. Every `R`-algebra map from the henselization to `S` is in fact local and equal to this one (`eq_lift`).","title":"The universal property of the henselization of a local ring"},{"id":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat/resolution-of-curves-regularity-of-the-normaliza","kind":"node","lean":"TauCeti.AlgebraicGeometry.isRegular_normalization_fromSpecStalk_genericPoint","parent":3939,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat/index.html#resolution-of-curves-regularity-of-the-normaliza","scope":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat","status":"fully_proved","summary":"Let `Y` be an integral locally Noetherian scheme of dimension at most one, and suppose that the normalization of `Y` in its function field, along `Spec k(Y) \u27f6 Y`, is finite over `Y`. Then this normalization is a regular scheme.","title":"Resolution of curves: regularity of the normalization"},{"id":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat/resolution-of-curves-the-normalization-is-a-modi","kind":"node","lean":"TauCeti.AlgebraicGeometry.isModification_fromNormalization_fromSpecStalk_genericPoint","parent":3939,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat/index.html#resolution-of-curves-the-normalization-is-a-modi","scope":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat","status":"fully_proved","summary":"If the normalization of an integral scheme `Y` in its function field, along `Spec k(Y) \u27f6 Y`, is finite over `Y`, then it is a modification of `Y`: it is proper, and an isomorphism over a dense open of `Y` with dense preimage. Stacks, Lemma 33.41.1 (0C45) over a field.","title":"Resolution of curves: the normalization is a modification"},{"id":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat/zariskis-main-theorem-global-form","kind":"node","lean":"Algebra.QuasiFinite.exists_fg_subalgebra_and_span_eq_top","parent":3939,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat/index.html#zariskis-main-theorem-global-form","scope":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat","status":"fully_proved","summary":"A quasi-finite algebra `A` of finite type over `R` has a module-finite subalgebra `A'` and finitely many elements `r \u2208 A'` generating the unit ideal of `A`, such that for every `x \u2208 A` and every such `r`, all large enough powers of `r` move `x` into `A'`. Geometrically, `Spec A` is covered by finitely many basic opens `D(r)` on which it agrees with the finite `R`-scheme `Spec A'`.","title":"Zariski's main theorem, global form"},{"id":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat/the-absolute-frobenius-is-a-universal-homeomorph","kind":"node","lean":"AlgebraicGeometry.Scheme.universally_isHomeomorph_frobenius","parent":3939,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat/index.html#the-absolute-frobenius-is-a-universal-homeomorph","scope":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat","status":"fully_proved","summary":"every base change of `F_X` is a homeomorphism of underlying spaces.","title":"The absolute Frobenius is a universal homeomorphism"},{"id":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat/the-absolute-frobenius-of-a-scheme-locally-of-fi","kind":"node","lean":"AlgebraicGeometry.Scheme.isFinite_frobenius","parent":3939,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat/index.html#the-absolute-frobenius-of-a-scheme-locally-of-fi","scope":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat","status":"fully_proved","summary":"For a scheme `X` locally of finite type over `Spec R`, with `R` a perfect ring of characteristic `p`, the absolute Frobenius `F_X : X \u27f6 X` is a finite morphism.","title":"The absolute Frobenius of a scheme locally of finite type over a perfect ring is finite"},{"id":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat/tauceti-ring-isjapanese","kind":"node","lean":"TauCeti.Ring.IsJapanese","parent":3939,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat/index.html#tauceti-ring-isjapanese","scope":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat","status":"fully_proved","summary":"A domain `R` is *Japanese* (or *N-2*) if for every finite extension `L` of its fraction field, the integral closure of `R` in `L` is a finite `R`-module; inseparable extensions are included. Stacks, Definition 10.161.1 (032F). See `IsJapanese.finite_integralClosure` for an arbitrary model of the fraction field and an extension in an arbitrary universe.","title":"TauCeti.Ring.IsJapanese"},{"id":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat/tauceti-ring-isnagata","kind":"node","lean":"TauCeti.Ring.IsNagata","parent":3939,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat/index.html#tauceti-ring-isnagata","scope":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat","status":"fully_proved","summary":"A commutative ring `R` is *Nagata* if it is Noetherian and the quotient `R \u29f8 p` by every prime ideal `p` is a Japanese domain. Stacks, Definition 10.162.1 (032R).","title":"TauCeti.Ring.IsNagata"},{"id":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat/module-satisfiesserres","kind":"node","lean":"Module.SatisfiesSerreS","parent":3939,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat/index.html#module-satisfiesserres","scope":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat","status":"fully_proved","summary":"An `R`-module `M` satisfies *Serre's condition* `(S_n)` if for every prime `p` of `R` the depth of `M_p` over `R_p` is at least `min (n, dim Supp M_p)`. This is meant for finite modules over a Noetherian ring. Since the zero module has depth `\u22a4`, primes outside the support impose nothing. Stacks, Definition 10.157.1 (031P).","title":"Module.SatisfiesSerreS"},{"id":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat/ring-satisfiesserrer","kind":"node","lean":"Ring.SatisfiesSerreR","parent":3939,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat/index.html#ring-satisfiesserrer","scope":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat","status":"fully_proved","summary":"A ring `R` satisfies *Serre's condition* `(R_n)` if every prime of height at most `n` lies in the regular locus, that is, `R_p` is a regular local ring whenever `ht p \u2264 n`. Stacks, Definition 10.157.1 (031P).","title":"Ring.SatisfiesSerreR"},{"id":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat/tauceti-excellence-geometricallyregular","kind":"node","lean":"TauCeti.Excellence.GeometricallyRegular","parent":3939,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat/index.html#tauceti-excellence-geometricallyregular","scope":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat","status":"fully_proved","summary":"An algebra `B` over a field `k` is *geometrically regular* if `B` is Noetherian and `L \u2297[k] B` is a regular ring for every finite purely inseparable field extension `L / k` (Stacks, Definition 10.166.2, in the form of Lemma 10.166.1). The extensions `L` range over the universe of `k`; `GeometricallyRegular.isRegularRing_tensorProduct'` removes that restriction.","title":"TauCeti.Excellence.GeometricallyRegular"},{"id":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat/tauceti-excellence-regularalgebramap","kind":"node","lean":"TauCeti.Excellence.RegularAlgebraMap","parent":3939,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat/index.html#tauceti-excellence-regularalgebramap","scope":"tauceti/ring-theory/part-01/scheme-stack-cohomology-and-intersection-foundat","status":"fully_proved","summary":"An `R`-algebra `B` is a *regular algebra*, or `R \u2192 B` is a *regular ring map*, if `B` is flat over `R` and for every prime `p` of `R` the fibre `\u03ba(p) \u2297[R] B` is geometrically regular over the residue field `\u03ba(p)` (Stacks, Definition 15.42.1).","title":"TauCeti.Excellence.RegularAlgebraMap"},{"data_url":"scopes/tauceti/ring-theory/part-01/ideal.json","id":"tauceti/ring-theory/part-01/ideal","kind":"scope","parent":3793,"result":false,"route":"../roadmap/tauceti/ring-theory/part-01/ideal/index.html","scope":"tauceti/ring-theory/part-01/ideal","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.RingTheory.Ideal` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Ideal"},{"id":"tauceti/ring-theory/part-01/ideal/artin-rees-for-a-finitely-generated-submodule","kind":"node","lean":"Ideal.exists_pow_add_smul_inf_eq_pow_smul","parent":3954,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/ideal/index.html#artin-rees-for-a-finitely-generated-submodule","scope":"tauceti/ring-theory/part-01/ideal","status":"fully_proved","summary":"If `N\u2080` is a finitely generated submodule over a noetherian ring, there is a shift `k` such that `I ^ (n + k) \u2022 N\u2080 \u2293 L` equals `I ^ n \u2022 (I ^ k \u2022 N\u2080 \u2293 L)` for every submodule `L` and every `n`. The ambient module need not be finite: only `N\u2080` is, so Mathlib's `Ideal.exists_pow_inf_eq_pow_smul` is applied inside `N\u2080`. In particular `I ^ (n + k) \u2022 N\u2080 \u2293 L \u2264 I ^ n \u2022 L`; when `L \u2264 N\u2080`, this shows that the filtration `N\u2080`\u2026","title":"Artin\u2013Rees for a finitely generated submodule"},{"id":"tauceti/ring-theory/part-01/ideal/the-golod-shafarevich-inequality-for-finite-dime","kind":"node","lean":"TauCeti.card_sq_lt_four_mul_card_of_relations_le_span","parent":3954,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/ideal/index.html#the-golod-shafarevich-inequality-for-finite-dime","scope":"tauceti/ring-theory/part-01/ideal","status":"fully_proved","summary":"Let `A` be a finite-dimensional algebra over a field `k` and `I` a proper two-sided ideal of `A`, generated as a left ideal by a nonempty finite family `x`. If every relation `\u2211 a\u1d62 x\u1d62 = 0` among the generators is an `A`-linear combination of a finite family `y` of vectors in `A^\u03b9` with all entries in `I`, then `(#x)\u00b2 < 4 \u00b7 #y`.","title":"The Golod\u2013Shafarevich inequality for finite-dimensional algebras"},{"id":"tauceti/ring-theory/part-01/ideal/the-krull-intersection-theorem-for-an-extended-i","kind":"node","lean":"Ideal.mem_iInf_map_algebraMap_pow_iff","parent":3954,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/ideal/index.html#the-krull-intersection-theorem-for-an-extended-i","scope":"tauceti/ring-theory/part-01/ideal","status":"fully_proved","summary":"If `R` is Noetherian and `A` is a finite `R`-module, an element of every power of the two-sided ideal of `A` generated by `I` is fixed by multiplication by some scalar in `I`.","title":"The Krull intersection theorem for an extended ideal"},{"id":"tauceti/ring-theory/part-01/ideal/serres-criterion","kind":"node","lean":"TauCeti.Ideal.mem_inertia_iff_of_adjoin_eq_top","parent":3954,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/ideal/index.html#serres-criterion","scope":"tauceti/ring-theory/part-01/ideal","status":"fully_proved","summary":"When `S` is generated over `R` by a set `s` and `G` acts by `R`-algebra automorphisms, membership in the inertia subgroup of a two-sided ideal `I` is decided on `s` alone: the elements moved into `I` form an `R`-subalgebra.","title":"Serre's criterion"},{"data_url":"scopes/tauceti/ring-theory/part-01/iwasawa.json","id":"tauceti/ring-theory/part-01/iwasawa","kind":"scope","parent":3793,"result":false,"route":"../roadmap/tauceti/ring-theory/part-01/iwasawa/index.html","scope":"tauceti/ring-theory/part-01/iwasawa","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.RingTheory.Iwasawa` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Iwasawa"},{"id":"tauceti/ring-theory/part-01/iwasawa/the-structure-theorem-for-iwasawa-modules","kind":"node","lean":"TauCeti.Iwasawa.exists_isPseudoIsomorphism_elementary","parent":3959,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/iwasawa/index.html#the-structure-theorem-for-iwasawa-modules","scope":"tauceti/ring-theory/part-01/iwasawa","status":"fully_proved","summary":"Let `O` be a discrete valuation ring with uniformiser `\u03d6`, complete for its maximal ideal. Every finitely generated `O\u27e6X\u27e7`-module admits a pseudo-isomorphism onto an elementary module `O\u27e6X\u27e7\u02b3 \u00d7 \u220f\u1d62 O\u27e6X\u27e7 \u29f8 (\u03d6 ^ m\u1d62) \u00d7 \u220f\u2c7c O\u27e6X\u27e7 \u29f8 (F\u2c7c ^ n\u2c7c)` with every `m\u1d62` and `n\u2c7c` positive and every `F\u2c7c` irreducible and distinguished. Neukirch\u2013Schmidt\u2013Wingberg, Theorem (5.3.8).","title":"The structure theorem for Iwasawa modules"},{"id":"tauceti/ring-theory/part-01/iwasawa/the-structure-theorem-for-torsion-modules","kind":"node","lean":"TauCeti.Iwasawa.exists_isPseudoIsomorphism_pi_quotient_pow","parent":3959,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/iwasawa/index.html#the-structure-theorem-for-torsion-modules","scope":"tauceti/ring-theory/part-01/iwasawa","status":"fully_proved","summary":"Over a noetherian integrally closed domain `A`, a finitely generated torsion module `M` admits a pseudo-isomorphism `M \u2192 \u220f\u1d62 A \u29f8 \ud835\udd2d\u1d62 ^ n\u1d62` onto a finite product, with each `\ud835\udd2d\u1d62` a prime of height one and each `n\u1d62` positive. Neukirch\u2013Schmidt\u2013Wingberg, (5.1.10).","title":"The structure theorem for torsion modules"},{"id":"tauceti/ring-theory/part-01/iwasawa/the-structure-theorem-over-a-regular-local-ring","kind":"node","lean":"TauCeti.Iwasawa.exists_isPseudoIsomorphism_prod_pi_quotient_pow","parent":3959,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/iwasawa/index.html#the-structure-theorem-over-a-regular-local-ring","scope":"tauceti/ring-theory/part-01/iwasawa","status":"fully_proved","summary":"Let `A` be a regular local ring of dimension at most two, such as the Iwasawa algebra `O\u27e6T\u27e7`. A finitely generated `A`-module `M` admits a pseudo-isomorphism `M \u2192 A\u02b3 \u00d7 \u220f\u1d62 A \u29f8 \ud835\udd2d\u1d62 ^ n\u1d62`, with each `\ud835\udd2d\u1d62` a prime of height one and each `n\u1d62` positive. Neukirch\u2013Schmidt\u2013Wingberg, Theorem (5.1.10).","title":"The structure theorem over a regular local ring of dimension at most two"},{"data_url":"scopes/tauceti/ring-theory/part-01/krull-schmidt.json","id":"tauceti/ring-theory/part-01/krull-schmidt","kind":"scope","parent":3793,"result":false,"route":"../roadmap/tauceti/ring-theory/part-01/krull-schmidt/index.html","scope":"tauceti/ring-theory/part-01/krull-schmidt","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.RingTheory.KrullSchmidt` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Krull\u2013Schmidt"},{"id":"tauceti/ring-theory/part-01/krull-schmidt/azumayas-exchange-lemma","kind":"node","lean":"TauCeti.exists_linearEquiv_and_isCompl_biSup_ne","parent":3963,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/krull-schmidt/index.html#azumayas-exchange-lemma","scope":"tauceti/ring-theory/part-01/krull-schmidt","status":"fully_proved","summary":"Let `M` be the internal direct sum of a finite family `Q` of indecomposable submodules, and let `N` be a direct summand of `M` whose endomorphism ring is local (so in particular `N` is nonzero). Then `N` is isomorphic to one of the `Q i\u2080`, and can be exchanged for it: `N` and the remaining summands `\u2a06 j \u2260 i\u2080, Q j` are complementary in `M`.","title":"Azumaya's exchange lemma"},{"id":"tauceti/ring-theory/part-01/krull-schmidt/fittings-lemma","kind":"node","lean":"TauCeti.IsIndecomposableModule.isNilpotent_or_bijective, TauCeti.IsIndecomposableModule.isNilpotent_or_isUnit","parent":3963,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/krull-schmidt/index.html#fittings-lemma","scope":"tauceti/ring-theory/part-01/krull-schmidt","status":"fully_proved","summary":"an endomorphism of an indecomposable module that is both Noetherian and Artinian is either nilpotent or bijective.","title":"Fitting's lemma"},{"id":"tauceti/ring-theory/part-01/krull-schmidt/the-krull-schmidt-theorem","kind":"node","lean":"TauCeti.exists_equiv_linearEquiv_of_finset, TauCeti.exists_equiv_linearEquiv_of_iSupIndep, TauCeti.exists_equiv_linearEquiv_of_isLocalRing_end, TauCeti.exists_equiv_linearEquiv_of_directSum, TauCeti.exists_equiv_linearEquiv_of_directSum_of_isLocalRing_end, TauCeti.exists_equiv_linearEquiv_of_directSumEquiv","parent":3963,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/krull-schmidt/index.html#the-krull-schmidt-theorem","scope":"tauceti/ring-theory/part-01/krull-schmidt","status":"fully_proved","summary":"The Krull-Schmidt theorem, in the `Finset` packaging that `TauCeti.exists_indecomposable_decomposition` produces: two `Finset`s of indecomposable submodules each presenting the module as an internal direct sum are matched by a bijection under which corresponding summands are isomorphic.","title":"The Krull-Schmidt theorem"},{"data_url":"scopes/tauceti/ring-theory/part-01/local-ring.json","id":"tauceti/ring-theory/part-01/local-ring","kind":"scope","parent":3793,"result":false,"route":"../roadmap/tauceti/ring-theory/part-01/local-ring/index.html","scope":"tauceti/ring-theory/part-01/local-ring","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.RingTheory.LocalRing` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Local Ring"},{"id":"tauceti/ring-theory/part-01/local-ring/cohen-structure-theorem-with-the-minimal-number","kind":"node","lean":"TauCeti.exists_surjective_mvPowerSeries_spanFinrank, TauCeti.exists_surjective_mvPowerSeries_of_span_eq_maximalIdeal","parent":3967,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/local-ring/index.html#cohen-structure-theorem-with-the-minimal-number","scope":"tauceti/ring-theory/part-01/local-ring","status":"fully_proved","summary":"A complete local ring containing a field whose maximal ideal `\ud835\udd2a` is finitely generated has a coefficient field `s` and is a quotient of the power series ring over its residue field in `spanFinrank \ud835\udd2a` variables, by a surjection which is `s` on constants and sends the variables to generators of `\ud835\udd2a`.","title":"Cohen structure theorem with the minimal number of variables"},{"id":"tauceti/ring-theory/part-01/local-ring/cohens-theorem-a-complete-local-ring-containing","kind":"node","lean":"TauCeti.exists_residueFieldSection","parent":3967,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/local-ring/index.html#cohens-theorem-a-complete-local-ring-containing","scope":"tauceti/ring-theory/part-01/local-ring","status":"fully_proved","summary":"If the complete local ring `R` is an algebra over some field `k`, then reduction `R \u2192 ResidueField R` has a ring homomorphism section. When `k` is perfect, `TauCeti.residueFieldSection k R` is even a `k`-algebra section. In general a `k`-linear section need not exist.","title":"Cohen's theorem: a complete local ring containing a field has a coefficient field"},{"id":"tauceti/ring-theory/part-01/local-ring/hilberts-counting-identity","kind":"node","lean":"TauCeti.IsLocalRing.sum_addVal_smul_sub_eq_finsum_card_ramificationGroup_sub_one","parent":3967,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/local-ring/index.html#hilberts-counting-identity","scope":"tauceti/ring-theory/part-01/local-ring","status":"fully_proved","summary":"Let a finite group `G` act faithfully on a discrete valuation ring `S` generated over a base `R` fixed by `G` by a single element `\u03be`. Then the sum over the nontrivial `\u03c3 \u2208 G` of the valuations `v (\u03c3 \u2022 \u03be - \u03be)` is `\u2211_{i \u2265 0} (#G_i - 1)`, where `G_i` is the `i`-th ramification group: each `\u03c3 \u2260 1` lies in exactly `v (\u03c3 \u2022 \u03be - \u03be)` of the groups `G_0, G_1, \u2026`, and the sum is finite because the filtration is eventually\u2026","title":"Hilbert's counting identity"},{"id":"tauceti/ring-theory/part-01/local-ring/nakayamas-lemma-in-coordinates","kind":"node","lean":"TauCeti.maximalIdeal_eq_span_range_of_span_eq_top","parent":3967,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/local-ring/index.html#nakayamas-lemma-in-coordinates","scope":"tauceti/ring-theory/part-01/local-ring","status":"fully_proved","summary":"Elements of `\ud835\udd2a` whose gradients span `\u03ba\u207f` generate `\ud835\udd2a`.","title":"Nakayama's lemma in coordinates"},{"id":"tauceti/ring-theory/part-01/local-ring/serres-criterion-for-the-ramification-filtration","kind":"node","lean":"TauCeti.IsLocalRing.mem_ramificationGroup_iff_of_adjoin_eq_top","parent":3967,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/local-ring/index.html#serres-criterion-for-the-ramification-filtration","scope":"tauceti/ring-theory/part-01/local-ring","status":"fully_proved","summary":"when `S` is generated over `R` by a set `s` and `G` acts by `R`-algebra automorphisms, membership in `G_i` is decided on `s` alone. This is `TauCeti.Ideal.mem_inertia_iff_of_adjoin_eq_top` at a power of the maximal ideal.","title":"Serre's criterion for the ramification filtration"},{"data_url":"scopes/tauceti/ring-theory/part-01/multivariate-polynomials.json","id":"tauceti/ring-theory/part-01/multivariate-polynomials","kind":"scope","parent":3793,"result":false,"route":"../roadmap/tauceti/ring-theory/part-01/multivariate-polynomials/index.html","scope":"tauceti/ring-theory/part-01/multivariate-polynomials","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.RingTheory.MvPolynomial` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Multivariate Polynomials"},{"id":"tauceti/ring-theory/part-01/multivariate-polynomials/buchbergers-criterion","kind":"node","lean":"MonomialOrder.isGroebnerBasis_iff_span_eq_and_isRemainder_sPolynomial_zero, MonomialOrder.isGroebnerBasis_span_iff_isRemainder_sPolynomial_zero, MonomialOrder.isGroebnerBasis_span_of_isRemainder_sPolynomial_zero","parent":3973,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/multivariate-polynomials/index.html#buchbergers-criterion","scope":"tauceti/ring-theory/part-01/multivariate-polynomials","status":"fully_proved","summary":"A set `G` whose nonzero members have unit leading coefficients is a Gr\u00f6bner basis of `I` if and only if it generates `I` and the S-polynomial of any two members of `G` has remainder `0` on division by `G`.","title":"Buchberger's criterion"},{"id":"tauceti/ring-theory/part-01/multivariate-polynomials/the-cauchy-identity","kind":"node","lean":"TauCeti.mk_sum_aeval_schurPoly_mul_aeval_schurPoly, TauCeti.sum_aeval_schurPoly_mul_aeval_schurPoly","parent":3973,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/multivariate-polynomials/index.html#the-cauchy-identity","scope":"tauceti/ring-theory/part-01/multivariate-polynomials","status":"fully_proved","summary":"for families `f : \u03c3 \u2192 S` and `g : \u03c4 \u2192 S`,","title":"The Cauchy identity"},{"id":"tauceti/ring-theory/part-01/multivariate-polynomials/the-dimension-theorem-at-the-origin","kind":"node","lean":"MvPolynomial.exists_ne_zero_forall_aeval_eq_zero","parent":3973,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/multivariate-polynomials/index.html#the-dimension-theorem-at-the-origin","scope":"tauceti/ring-theory/part-01/multivariate-polynomials","status":"fully_proved","summary":"let `K` be an algebraically closed extension of `k`. Fewer than `#\u03c3` polynomials in `k[X_\u03c3]` vanishing at the origin have a common zero in `K^\u03c3` other than the origin.","title":"The dimension theorem at the origin"},{"id":"tauceti/ring-theory/part-01/multivariate-polynomials/the-dual-jacobi-trudi-identity","kind":"node","lean":"TauCeti.diagramSchurPoly_eq_det_esymmInt, Nat.Partition.schurPoly_eq_det_esymmInt","parent":3973,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/multivariate-polynomials/index.html#the-dual-jacobi-trudi-identity","scope":"tauceti/ring-theory/part-01/multivariate-polynomials","status":"fully_proved","summary":"The Schur polynomial of a Young diagram `\u03bc` in the alphabet `Fin N` is the determinant `det (e_{\u03bc'\u1d62 - i + j})_{0 \u2264 i, j < r}` of elementary symmetric polynomials, where `\u03bc'\u1d62` is the length of the `i`-th column, for every `r` at least the number of columns of `\u03bc`. The index `\u03bc'\u1d62 - i + j` is an integer, and `e_m = 0` for `m < 0` (`TauCeti.esymmInt`).","title":"The dual Jacobi--Trudi identity"},{"id":"tauceti/ring-theory/part-01/multivariate-polynomials/the-elimination-theorem","kind":"node","lean":"MonomialOrder.IsGroebnerBasis.comap_rename","parent":3973,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/multivariate-polynomials/index.html#the-elimination-theorem","scope":"tauceti/ring-theory/part-01/multivariate-polynomials","status":"fully_proved","summary":"Let `m` be an elimination order for the variables `e \u03c4`: every monomial below a monomial in these variables is again a monomial in these variables. If `G` is a Gr\u00f6bner basis of `I` for `m`, then the members of `G` in the variables `e \u03c4` form a Gr\u00f6bner basis of the elimination ideal `I.comap (rename e)` for every monomial order `m'` on `\u03c4` which `e` carries to `m`.","title":"The elimination theorem"},{"id":"tauceti/ring-theory/part-01/multivariate-polynomials/the-elimination-theorem-for-the-lexicographic-or","kind":"node","lean":"MonomialOrder.IsGroebnerBasis.comap_rename_lex","parent":3973,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/multivariate-polynomials/index.html#the-elimination-theorem-for-the-lexicographic-or","scope":"tauceti/ring-theory/part-01/multivariate-polynomials","status":"fully_proved","summary":"If `e : \u03c4 \u2192 \u03c3` is strictly monotone with image an upper set of variables, then the members of a lexicographic Gr\u00f6bner basis of `I` in the variables `e \u03c4` form a lexicographic Gr\u00f6bner basis of the elimination ideal.","title":"The elimination theorem for the lexicographic order"},{"id":"tauceti/ring-theory/part-01/multivariate-polynomials/eulers-identity-tauceti-jacobimatrixat-mulvec-se","kind":"node","lean":"TauCeti.jacobiMatrixAt_mulVec_self","parent":3973,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/multivariate-polynomials/index.html#eulers-identity-tauceti-jacobimatrixat-mulvec-se","scope":"tauceti/ring-theory/part-01/multivariate-polynomials","status":"fully_proved","summary":"Euler's identity for the Jacobian matrix of a family of homogeneous polynomials: applied to the point itself, it returns each equation's value scaled by its degree.","title":"Euler's identity (`TauCeti.jacobiMatrixAt_mulVec_self`)"},{"id":"tauceti/ring-theory/part-01/multivariate-polynomials/the-fischer-decomposition","kind":"node","lean":"MvPolynomial.harmonicSubmodule_sup_map_mulLeft_normSq, MvPolynomial.disjoint_ker_laplacian_map_mulLeft_normSq","parent":3973,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/multivariate-polynomials/index.html#the-fischer-decomposition","scope":"tauceti/ring-theory/part-01/multivariate-polynomials","status":"fully_proved","summary":"every homogeneous polynomial of degree `e + 2` is the sum of a harmonic one and `r\u00b2` times a homogeneous polynomial of degree `e`. The sum is direct by `MvPolynomial.disjoint_ker_laplacian_map_mulLeft_normSq`.","title":"The Fischer decomposition"},{"id":"tauceti/ring-theory/part-01/multivariate-polynomials/the-generalized-wronskian-criterion","kind":"node","lean":"MvPolynomial.linearIndependent_iff_exists_genWronskian_ne_zero","parent":3973,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/multivariate-polynomials/index.html#the-generalized-wronskian-criterion","scope":"tauceti/ring-theory/part-01/multivariate-polynomials","status":"fully_proved","summary":"Over a field of characteristic zero, a family of multivariate polynomials is linearly independent if and only if some generalized Wronskian with admissible rows `|D k| \u2264 k` is nonzero.","title":"The generalized Wronskian criterion"},{"id":"tauceti/ring-theory/part-01/multivariate-polynomials/jacobis-bialternant-formula","kind":"node","lean":"TauCeti.diagramSchurPoly_mul_alternant, TauCeti.schurPoly_mul_alternant","parent":3973,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/multivariate-polynomials/index.html#jacobis-bialternant-formula","scope":"tauceti/ring-theory/part-01/multivariate-polynomials","status":"fully_proved","summary":"For a Young diagram `\u03bc` with at most `N` rows, the Schur polynomial `s_\u03bc` in the alphabet `Fin N` times the Vandermonde alternant `a_\u03b4`, `\u03b4_j = N - 1 - j`, is the alternant `a_{\u03bb+\u03b4}` of the row lengths `\u03bb_j = \u03bc.rowLen j` shifted by the staircase: `s_\u03bc \u00b7 a_\u03b4 = a_{\u03bb+\u03b4}`. This is the division-free form of `s_\u03bc = a_{\u03bb+\u03b4} / a_\u03b4`.","title":"Jacobi's bialternant formula"},{"id":"tauceti/ring-theory/part-01/multivariate-polynomials/jacobis-complementary-minor-identity-for-the-two","kind":"node","lean":"TauCeti.det_submatrix_hsymmToeplitz","parent":3973,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/multivariate-polynomials/index.html#jacobis-complementary-minor-identity-for-the-two","scope":"tauceti/ring-theory/part-01/multivariate-polynomials","status":"fully_proved","summary":"A minor of `(h_{j - i})` on the rows `\u03c1 (p)` and the columns `\u03c0 (p)` is the complementary minor of `((-1)^{j - i} e_{j - i})`, on the rows `\u03c0 (m)` and the columns `\u03c1 (m)`, signed by `\u03c0` and `\u03c1`.","title":"Jacobi's complementary-minor identity for the two Toeplitz matrices of symmetric polynomials"},{"id":"tauceti/ring-theory/part-01/multivariate-polynomials/the-jacobi-trudi-identity","kind":"node","lean":"TauCeti.diagramSchurPoly_eq_det_hsymmInt, TauCeti.schurPoly_eq_det_hsymmInt","parent":3973,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/multivariate-polynomials/index.html#the-jacobi-trudi-identity","scope":"tauceti/ring-theory/part-01/multivariate-polynomials","status":"fully_proved","summary":"The Schur polynomial of a Young diagram `\u03bc` in the alphabet `Fin N` is the determinant `det (h_{\u03bc\u1d62 - i + j})_{0 \u2264 i, j < r}` of complete homogeneous symmetric polynomials, for every `r` at least the number of rows of `\u03bc`. The index `\u03bc\u1d62 - i + j` is an integer, and `h_m = 0` for `m < 0` (`TauCeti.hsymmInt`).","title":"The Jacobi--Trudi identity"},{"id":"tauceti/ring-theory/part-01/multivariate-polynomials/the-jacobian-criterion-in-a-ratio-chart","kind":"node","lean":"TauCeti.rank_jacobiMatrixAt_dehomogenize_div, TauCeti.mulVec_surjective_jacobiMatrixAt_iff_exists_eval_pderiv_ne_zero","parent":3973,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/multivariate-polynomials/index.html#the-jacobian-criterion-in-a-ratio-chart","scope":"tauceti/ring-theory/part-01/multivariate-polynomials","status":"fully_proved","summary":"at a common zero `b` of a family of homogeneous equations with `b i \u2260 0`, the Jacobian matrix of the dehomogenized equations at the coordinate ratios `b (i.succAbove j) / b i` has the same rank as the Jacobian matrix of the homogeneous equations at `b`.","title":"The Jacobian criterion in a ratio chart"},{"id":"tauceti/ring-theory/part-01/multivariate-polynomials/newtons-identities-for-a-multiset","kind":"node","lean":"Multiset.mul_esymm_eq_sum, TauCeti.esymm_mem_adjoin_psum","parent":3973,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/multivariate-polynomials/index.html#newtons-identities-for-a-multiset","scope":"tauceti/ring-theory/part-01/multivariate-polynomials","status":"fully_proved","summary":"`k` times the `k`-th elementary symmetric function of the elements of `s` is an explicit combination of the lower elementary symmetric functions and the power sums `(s.map (\u00b7 ^ j)).sum`. This is `MvPolynomial.mul_esymm_eq_sum` evaluated at the elements of `s`.","title":"Newton's identities for a multiset"},{"data_url":"scopes/tauceti/ring-theory/part-01/mv-power-series.json","id":"tauceti/ring-theory/part-01/mv-power-series","kind":"scope","parent":3793,"result":false,"route":"../roadmap/tauceti/ring-theory/part-01/mv-power-series/index.html","scope":"tauceti/ring-theory/part-01/mv-power-series","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.RingTheory.MvPowerSeries` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Mv Power Series"},{"id":"tauceti/ring-theory/part-01/mv-power-series/the-cauchy-estimate-for-partial-derivatives","kind":"node","lean":"TauCeti.MvPowerSeries.norm_restrictedPDeriv_le, MvPowerSeries.norm_restrictedHasseDeriv_le","parent":3988,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/mv-power-series/index.html#the-cauchy-estimate-for-partial-derivatives","scope":"tauceti/ring-theory/part-01/mv-power-series","status":"fully_proved","summary":"`\u2016\u2202\u1d62 f\u2016 \u2264 c\u1d62\u207b\u00b9 \u2016f\u2016` for the Gauss norm at the polyradius `c`.","title":"The Cauchy estimate for partial derivatives"},{"id":"tauceti/ring-theory/part-01/mv-power-series/kedlayas-hilbert-basis-theorem-for-restricted-po","kind":"node","lean":"TauCeti.MvPowerSeries.isNoetherianRing_isRestricted_subring_of_division, TauCeti.WittVector.RadiusRing.isNoetherianRing_isRestricted_subring","parent":3988,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/mv-power-series/index.html#kedlayas-hilbert-basis-theorem-for-restricted-po","scope":"tauceti/ring-theory/part-01/mv-power-series","status":"fully_proved","summary":"Kedlaya's Hilbert basis theorem for restricted power series, the argument of Kedlaya's Theorem 3.2 for an abstract Euclidean normed ring. Let `R` be a complete ring with a multiplicative ultrametric norm, and `deg : R \u2192 \u2115` a degree such that `deg y = deg x` whenever `\u2016x - y\u2016 < \u2016x\u2016`, and such that for `x \u2260 0` every `y` can be divided as `y = z * x + w` with `\u2016w\u2016 \u2264 \u2016y\u2016` and either `w = 0` or `deg w < deg x`. Then for\u2026","title":"Kedlaya's Hilbert basis theorem for restricted power series"},{"id":"tauceti/ring-theory/part-01/mv-power-series/neumann-series-criterion-mvpowerseries-isnormmul","kind":"node","lean":"MvPowerSeries.isNormMulUnit_of_norm_sub_C_lt","parent":3988,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/mv-power-series/index.html#neumann-series-criterion-mvpowerseries-isnormmul","scope":"tauceti/ring-theory/part-01/mv-power-series","status":"fully_proved","summary":"Over a complete ultrametric norm-one ring, a restricted series whose distance from a constant multiplicative unit `a` is less than `\u2016a\u2016` is a multiplicative unit.","title":"Neumann series criterion (`MvPowerSeries.isNormMulUnit_of_norm_sub_C_lt`)"},{"id":"tauceti/ring-theory/part-01/mv-power-series/the-weierstrass-division-estimate-in-several-var","kind":"node","lean":"TauCeti.MvPowerSeries.norm_mul_add_eq_max_of_isMulDistinguished","parent":3988,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/mv-power-series/index.html#the-weierstrass-division-estimate-in-several-var","scope":"tauceti/ring-theory/part-01/mv-power-series","status":"fully_proved","summary":"Over an ultrametric normed commutative ring, let the iterated series of `g` in the separated variable `X none` be multiplicatively distinguished of degree `s`. If `r` has degree less than `s` in `X none`, then","title":"The Weierstrass division estimate in several variables"},{"data_url":"scopes/tauceti/ring-theory/part-01/polynomial.json","id":"tauceti/ring-theory/part-01/polynomial","kind":"scope","parent":3793,"result":false,"route":"../roadmap/tauceti/ring-theory/part-01/polynomial/index.html","scope":"tauceti/ring-theory/part-01/polynomial","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.RingTheory.Polynomial` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Polynomial"},{"id":"tauceti/ring-theory/part-01/polynomial/gausss-lemma-at-a-nonarchimedean-absolute-value","kind":"node","lean":"Polynomial.IsPrimitive.gaussNorm_map_eq_one","parent":3993,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/polynomial/index.html#gausss-lemma-at-a-nonarchimedean-absolute-value","scope":"tauceti/ring-theory/part-01/polynomial","status":"fully_proved","summary":"A primitive polynomial with integer coefficients has Gauss norm of radius one equal to `1` at every nonarchimedean absolute value, in any characteristic.","title":"Gauss's lemma at a nonarchimedean absolute value"},{"id":"tauceti/ring-theory/part-01/polynomial/structure-theorem-for-subresultants-degree-gap","kind":"node","lean":"Polynomial.subresultant_eq_zero_of_getElem?_sturmSeq, Polynomial.exists_subresultant_natDegree_eq_C_mul_of_getElem?_sturmSeq, Polynomial.exists_subresultant_natDegree_sub_one_eq_C_mul_of_getElem?_sturmSeq","parent":3993,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/polynomial/index.html#structure-theorem-for-subresultants-degree-gap","scope":"tauceti/ring-theory/part-01/polynomial","status":"fully_proved","summary":"For consecutive entries `s, t` of the signed remainder sequence of `p` and `q` after the first, the subresultants of `p` and `q` at the indices strictly between `deg t` and `deg s - 1` vanish.","title":"Structure theorem for subresultants, degree gap"},{"id":"tauceti/ring-theory/part-01/polynomial/vietas-formulas-on-the-symmetric-power","kind":"node","lean":"TauCeti.Sym.coeff_toMonic, MvPolynomial.aeval_esymm_eq_coeff_prod_X_sub_C, Polynomial.IsRootEnumeration.aeval_esymm_eq_coeff","parent":3993,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/polynomial/index.html#vietas-formulas-on-the-symmetric-power","scope":"tauceti/ring-theory/part-01/polynomial","status":"fully_proved","summary":"the `k`-th coefficient of the monic polynomial of an unordered `n`-tuple is the `(n - k)`-th elementary symmetric function of the tuple, up to the sign `(-1) ^ (n - k)`.","title":"Vieta's formulas on the symmetric power"},{"id":"tauceti/ring-theory/part-01/polynomial/the-wronskian-criterion","kind":"node","lean":"Polynomial.linearIndependent_iff_hasseWronskian_ne_zero","parent":3993,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/polynomial/index.html#the-wronskian-criterion","scope":"tauceti/ring-theory/part-01/polynomial","status":"fully_proved","summary":"Over a field of characteristic zero, a family of polynomials is linearly independent if and only if its Hasse Wronskian is nonzero.","title":"The Wronskian criterion"},{"data_url":"scopes/tauceti/ring-theory/part-01/power-series.json","id":"tauceti/ring-theory/part-01/power-series","kind":"scope","parent":3793,"result":false,"route":"../roadmap/tauceti/ring-theory/part-01/power-series/index.html","scope":"tauceti/ring-theory/part-01/power-series","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.RingTheory.PowerSeries` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Power Series"},{"id":"tauceti/ring-theory/part-01/power-series/the-bezout-identity-for-the-entire-resultant","kind":"node","lean":"Polynomial.exists_C_entireResultant_eq","parent":3998,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/power-series/index.html#the-bezout-identity-for-the-entire-resultant","scope":"tauceti/ring-theory/part-01/power-series","status":"fully_proved","summary":"For monic `P` and entire `F` there are an entire series `G` and a polynomial `H` of degree less than `deg P` with `Res(P, F) = P * G + H * F`.","title":"The B\u00e9zout identity for the entire resultant"},{"id":"tauceti/ring-theory/part-01/power-series/cauchy-hadamard-for-entire-series","kind":"node","lean":"PowerSeries.IsEntire.tendsto_norm_coeff_rpow_inv_natCast","parent":3998,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/power-series/index.html#cauchy-hadamard-for-entire-series","scope":"tauceti/ring-theory/part-01/power-series","status":"fully_proved","summary":"The `n`-th roots `\u2016a\u2099\u2016 ^ (1 / n)` of the coefficient norms of an entire series tend to zero.","title":"Cauchy\u2013Hadamard for entire series"},{"id":"tauceti/ring-theory/part-01/power-series/cauchy-hadamard-formula","kind":"node","lean":"PowerSeries.radius_inv_eq_limsup, PowerSeries.radius_eq_liminf","parent":3998,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/power-series/index.html#cauchy-hadamard-formula","scope":"tauceti/ring-theory/part-01/power-series","status":"fully_proved","summary":"the inverse of the radius of `\u2211 a\u2099 X\u207f` is `limsup \u2016a\u2099\u2016 ^ (1 / n)`.","title":"Cauchy\u2013Hadamard formula"},{"id":"tauceti/ring-theory/part-01/power-series/factor-theorem-for-entire-series","kind":"node","lean":"PowerSeries.IsEntire.evalAt_eq_zero_iff","parent":3998,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/power-series/index.html#factor-theorem-for-entire-series","scope":"tauceti/ring-theory/part-01/power-series","status":"fully_proved","summary":"An entire series vanishes at `a` exactly when it is `X - a` times an entire series.","title":"Factor theorem for entire series"},{"id":"tauceti/ring-theory/part-01/power-series/hensels-lemma-for-kt","kind":"node","lean":"PowerSeries.henselianLocalRing","parent":3998,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/power-series/index.html#hensels-lemma-for-kt","scope":"tauceti/ring-theory/part-01/power-series","status":"fully_proved","summary":"Over a field `K`, the power series ring `K\u27e6t\u27e7` is a Henselian local ring. If `a\u2080` is a root modulo `t` of a monic polynomial `f` with `f' (a\u2080)` a unit, its constant coefficient `c` is a simple root of the reduction of `f`, which therefore factors as `(X - c) * h` with `X - c` coprime to `h`; lifting this factorization (`Polynomial.Monic.exists_monic_mul_eq_of_isCoprime`) gives a linear factor `X - a` of `f` with\u2026","title":"Hensel's lemma for `K\u27e6t\u27e7`"},{"id":"tauceti/ring-theory/part-01/power-series/the-taylor-identity","kind":"node","lean":"PowerSeries.IsEntire.evalAt_taylor, PowerSeries.IsRestricted.evalAt_taylor, PowerSeries.evalAt_taylor_of_summable","parent":3998,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/power-series/index.html#the-taylor-identity","scope":"tauceti/ring-theory/part-01/power-series","status":"fully_proved","summary":"For an entire series `f`, the Taylor translate at `a` evaluated at `b` is `f(b + a)`.","title":"The Taylor identity"},{"id":"tauceti/ring-theory/part-01/power-series/the-taylor-remainder-estimate","kind":"node","lean":"PowerSeries.IsRestricted.norm_evalAt_add_le_max","parent":3998,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/power-series/index.html#the-taylor-remainder-estimate","scope":"tauceti/ring-theory/part-01/power-series","status":"fully_proved","summary":"Over a complete ultrametric normed ring, let `f` be restricted at a radius `c > 0` and let `\u2016a\u2016 \u2264 c`, `\u2016y\u2016 \u2264 c`. If `\u03b5` bounds the first `T` terms `\u2016(\u0394\u1d57 f)(a)\u2016 \u2016y\u2016\u1d57`, `t < T`, of the expansion of `f` at `a`, then","title":"The Taylor remainder estimate"},{"id":"tauceti/ring-theory/part-01/power-series/the-whitney-formula-for-roots","kind":"node","lean":"PowerSeries.aeval_multiset_esymm_add_multiplicativeSequence","parent":3998,"result":true,"route":"../roadmap/tauceti/ring-theory/part-01/power-series/index.html#the-whitney-formula-for-roots","scope":"tauceti/ring-theory/part-01/power-series","status":"fully_proved","summary":"on the union of a family `s` of `n` roots and a family `t` of `m` roots, `K\u2096` is the convolution of the values of the multiplicative sequence on `s` and on `t`.","title":"The Whitney formula for roots"},{"data_url":"scopes/tauceti/ring-theory/part-02.json","id":"tauceti/ring-theory/part-02","kind":"scope","navigation":"transparent","parent":3792,"result":false,"route":"../roadmap/tauceti/ring-theory/part-02/index.html","scope":"tauceti/ring-theory/part-02","status":"fully_proved","title":"Regular Local Ring to Other results"},{"data_url":"scopes/tauceti/ring-theory/part-02/regular-local-ring.json","id":"tauceti/ring-theory/part-02/regular-local-ring","kind":"scope","parent":4007,"result":false,"route":"../roadmap/tauceti/ring-theory/part-02/regular-local-ring/index.html","scope":"tauceti/ring-theory/part-02/regular-local-ring","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.RingTheory.RegularLocalRing` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Regular Local Ring"},{"id":"tauceti/ring-theory/part-02/regular-local-ring/the-jacobian-criterion-for-a-hypersurface","kind":"node","lean":"Ideal.isRegularLocalRing_localization_of_derivation_notMem","parent":4008,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/regular-local-ring/index.html#the-jacobian-criterion-for-a-hypersurface","scope":"tauceti/ring-theory/part-02/regular-local-ring","status":"fully_proved","summary":"Let `f \u2208 A` and let `P` be a prime of `A \u29f8 (f)` lying over the prime `Q` of `A`, with `A_Q` a regular local ring. If some derivation `D` of `A` maps `f` outside `Q`, then the localization of `A \u29f8 (f)` at `P` is a regular local ring.","title":"The Jacobian criterion for a hypersurface"},{"id":"tauceti/ring-theory/part-02/regular-local-ring/kunzs-theorem","kind":"node","lean":"TauCeti.IsRegularRing.flat_frobenius","parent":4008,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/regular-local-ring/index.html#kunzs-theorem","scope":"tauceti/ring-theory/part-02/regular-local-ring","status":"fully_proved","summary":"The Frobenius endomorphism of a regular ring of exponential characteristic `p` is flat. Stacks, Lemma 51.17.6 (0EC0), the implication from regularity to flatness.","title":"Kunz's theorem"},{"id":"tauceti/ring-theory/part-02/regular-local-ring/serres-theorem-tauceti-isregularlocalring-of-has","kind":"node","lean":"TauCeti.IsRegularLocalRing.of_hasProjectiveDimensionLE, TauCeti.IsRegularLocalRing.of_globalDimension_lt_top","parent":4008,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/regular-local-ring/index.html#serres-theorem-tauceti-isregularlocalring-of-has","scope":"tauceti/ring-theory/part-02/regular-local-ring","status":"fully_proved","summary":"A Noetherian local ring whose residue field has finite projective dimension is a regular local ring. Its dimension is then that projective dimension, by `TauCeti.IsRegularLocalRing.projectiveDimension_quotient_maximalIdeal_eq_ringKrullDim`.","title":"Serre's theorem (`TauCeti.IsRegularLocalRing.of_hasProjectiveDimensionLE`)"},{"data_url":"scopes/tauceti/ring-theory/part-02/semisimple.json","id":"tauceti/ring-theory/part-02/semisimple","kind":"scope","parent":4007,"result":false,"route":"../roadmap/tauceti/ring-theory/part-02/semisimple/index.html","scope":"tauceti/ring-theory/part-02/semisimple","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.RingTheory.Semisimple` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Semisimple"},{"id":"tauceti/ring-theory/part-02/semisimple/the-double-centralizer-theorem","kind":"node","lean":"TauCeti.toModuleEnd_moduleEnd_bijective, TauCeti.algEquivEndEndOfIsSimpleRing, TauCeti.toModuleEnd_moduleEnd_bijective_of_isSimpleRing, AlgHom.centralizer_centralizer_range, Subalgebra.centralizer_centralizer_of_isSemisimpleModule","parent":4012,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/semisimple/index.html#the-double-centralizer-theorem","scope":"tauceti/ring-theory/part-02/semisimple","status":"fully_proved","summary":"Let `M` be a faithful semisimple `R`-module which is finite over its endomorphism ring `D = Module.End R M`. Then the natural map `R \u2192 Module.End D M` is bijective: `R` is exactly the ring of `D`-linear endomorphisms of `M`.","title":"The double centralizer theorem"},{"id":"tauceti/ring-theory/part-02/semisimple/the-multiplicity-theorem","kind":"node","lean":"TauCeti.finrank_linearMap_eq_natCard_of_linearEquiv_pi","parent":4012,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/semisimple/index.html#the-multiplicity-theorem","scope":"tauceti/ring-theory/part-02/semisimple","status":"fully_proved","summary":"If `M` decomposes as a finite direct sum of simple modules `N i`, then the dimension of the space of `A`-linear maps from a simple module `S` into `M` is the number of factors isomorphic to `S`.","title":"The multiplicity theorem"},{"id":"tauceti/ring-theory/part-02/semisimple/the-multiplicity-theorem-for-a-product-of-simple","kind":"node","lean":"TauCeti.finrank_linearMap_pi_eq_natCard","parent":4012,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/semisimple/index.html#the-multiplicity-theorem-for-a-product-of-simple","scope":"tauceti/ring-theory/part-02/semisimple","status":"fully_proved","summary":"The dimension of the space of `A`-linear maps from a simple module `S` into a finite product of simple modules counts the factors isomorphic to `S`.","title":"The multiplicity theorem for a product of simple modules"},{"id":"tauceti/ring-theory/part-02/semisimple/schurs-lemma-vanishing-form","kind":"node","lean":"TauCeti.hom_eq_zero_of_isEmpty_linearEquiv, TauCeti.endAlgEquivSelfOfIsSimpleModule, LinearEquiv.finrank_linearMap_eq_one, TauCeti.finrank_linearMap_eq_zero_of_isEmpty_linearEquiv","parent":4012,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/semisimple/index.html#schurs-lemma-vanishing-form","scope":"tauceti/ring-theory/part-02/semisimple","status":"fully_proved","summary":"A linear map between simple modules is zero if the two modules are not linearly equivalent.","title":"Schur's lemma, vanishing form"},{"data_url":"scopes/tauceti/ring-theory/part-02/other-results.json","id":"tauceti/ring-theory/part-02/other-results","kind":"scope","parent":4007,"result":false,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/index.html","scope":"tauceti/ring-theory/part-02/other-results","status":"fully_proved","summary":"Results named in the rest of Tau Ceti's `TauCeti.RingTheory` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Other results"},{"data_url":"scopes/tauceti/ring-theory/part-02/other-results/part-01.json","id":"tauceti/ring-theory/part-02/other-results/part-01","kind":"scope","navigation":"transparent","parent":4017,"result":false,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-01/index.html","scope":"tauceti/ring-theory/part-02/other-results/part-01","status":"fully_proved","title":"Algebraic Hartogs principle for\u2026 to Krull\u2013Akizuki for Mathlib's\u2026"},{"id":"tauceti/ring-theory/part-02/other-results/part-01/algebraic-hartogs-principle-for-fractions","kind":"node","lean":"IsLocalRing.mem_range_algebraMap_of_two_le_depth","parent":4018,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-01/index.html#algebraic-hartogs-principle-for-fractions","scope":"tauceti/ring-theory/part-02/other-results/part-01","status":"fully_proved","summary":"Let `R` be a Noetherian local ring of depth at least two with total ring of fractions `K`. An element of `K` that lies in the image of `R_p` for every prime `p` other than the maximal ideal lies in `R`: if for every such `p` some `t \u2209 p` has `t x \u2208 R`, then `x \u2208 R`.","title":"Algebraic Hartogs principle for fractions"},{"id":"tauceti/ring-theory/part-02/other-results/part-01/the-auslander-buchsbaum-formula","kind":"node","lean":"ModuleCat.projectiveDimension_add_depth_eq_depth","parent":4018,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-01/index.html#the-auslander-buchsbaum-formula","scope":"tauceti/ring-theory/part-02/other-results/part-01","status":"fully_proved","summary":"For a nonzero finite module `M` of finite projective dimension over a Noetherian local ring `R`, `projectiveDimension M + depth M = depth R`. Stacks, Proposition 10.111.1 (090V).","title":"The Auslander\u2013Buchsbaum formula"},{"id":"tauceti/ring-theory/part-02/other-results/part-01/the-cauchy-estimate-at-unit-radius","kind":"node","lean":"TauCeti.Dagger.washnitzerAlgebra.norm_toTate_hasseDeriv_le","parent":4018,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-01/index.html#the-cauchy-estimate-at-unit-radius","scope":"tauceti/ring-theory/part-02/other-results/part-01","status":"fully_proved","summary":"Hasse derivatives do not increase the Gauss norm of an overconvergent series.","title":"The Cauchy estimate at unit radius"},{"id":"tauceti/ring-theory/part-02/other-results/part-01/chase-harrison-rosenberg-tauceti-algebra-isfinit","kind":"node","lean":"TauCeti.Algebra.isFiniteEtaleTorsor_iff","parent":4018,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-01/index.html#chase-harrison-rosenberg-tauceti-algebra-isfinit","scope":"tauceti/ring-theory/part-02/other-results/part-01","status":"fully_proved","summary":"`B` is a finite \u00e9tale `G`-torsor over `A` if and only if `A \u2192 B` is injective, `A` is the ring of invariants, and the action is free: for each `g \u2260 1` the elements `g \u2022 b - b` generate the unit ideal of `B`.","title":"Chase\u2013Harrison\u2013Rosenberg (`TauCeti.Algebra.isFiniteEtaleTorsor_iff`)"},{"id":"tauceti/ring-theory/part-02/other-results/part-01/chase-harrison-rosenberg-tauceti-mulsemiringacti","kind":"node","lean":"TauCeti.MulSemiringAction.tensorToPi_bijective, TauCeti.MulSemiringAction.tensorToPi_surjective_iff","parent":4018,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-01/index.html#chase-harrison-rosenberg-tauceti-mulsemiringacti","scope":"tauceti/ring-theory/part-02/other-results/part-01","status":"fully_proved","summary":"For a free action of a finite group with invariant ring `R`, `tensorToPi : A \u2297[R] A \u2192 (G \u2192 A)` is bijective.","title":"Chase\u2013Harrison\u2013Rosenberg (`TauCeti.MulSemiringAction.tensorToPi_bijective`)"},{"id":"tauceti/ring-theory/part-02/other-results/part-01/chinese-remainder-theorem-for-adjoinroot","kind":"node","lean":"AdjoinRoot.equivPiFactors","parent":4018,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-01/index.html#chinese-remainder-theorem-for-adjoinroot","scope":"tauceti/ring-theory/part-02/other-results/part-01","status":"fully_proved","summary":"for `f` nonzero and squarefree, `K[X] \u29f8 (f)` is the product of the fields `K[X] \u29f8 (p)` over the monic irreducible factors `p` of `f`.","title":"Chinese Remainder Theorem for `AdjoinRoot`"},{"id":"tauceti/ring-theory/part-02/other-results/part-01/cramers-bound-for-subalgebras-of-integral-elemen","kind":"node","lean":"TauCeti.IntegralOrder.det_smul_mem_span","parent":4018,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-01/index.html#cramers-bound-for-subalgebras-of-integral-elemen","scope":"tauceti/ring-theory/part-02/other-results/part-01","status":"fully_proved","summary":"Let `e` be an `F`-basis of `E` lying in an `R`-subalgebra `\u0393` all of whose elements are integral over the integrally closed domain `R`, and let `r = det (Tr_F (e i * e j))` be the Gram determinant of the regular trace form. Then `r \u2022 x` lies in the `R`-span of `e` for every `x \u2208 \u0393`.","title":"Cramer's bound for subalgebras of integral elements"},{"id":"tauceti/ring-theory/part-02/other-results/part-01/dedekinds-different-theorem-the-tame-case-as-an","kind":"node","lean":"TauCeti.multiplicity_differentIdeal_eq_ramificationIdx_sub_one_iff, TauCeti.pow_dvd_differentIdeal_iff_of_isCoprime, TauCeti.ramificationIdx_sub_one_le_multiplicity_differentIdeal, TauCeti.pow_ramificationIdx_dvd_differentIdeal_iff, TauCeti.ramificationIdx_le_multiplicity_differentIdeal_iff","parent":4018,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-01/index.html#dedekinds-different-theorem-the-tame-case-as-an","scope":"tauceti/ring-theory/part-02/other-results/part-01","status":"fully_proved","summary":"the multiplicity of `P` in the different ideal is exactly `e(P \u2223 p) - 1` when the residue extension is separable and `e(P \u2223 p)` is nonzero in the residue field `A \u29f8 p`.","title":"Dedekind's different theorem, the tame case, as an exponent"},{"id":"tauceti/ring-theory/part-02/other-results/part-01/eulers-criterion-in-a-finite-residue-domain","kind":"node","lean":"Ideal.exists_pow_eq_mk_iff_pow_sub_one_mem","parent":4018,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-01/index.html#eulers-criterion-in-a-finite-residue-domain","scope":"tauceti/ring-theory/part-02/other-results/part-01","status":"fully_proved","summary":"Let `P` be a prime of `R` with finite quotient and `n \u2223 #(R \u29f8 P) - 1`. For `a \u2209 P`, the class of `a` is an `n`-th power in `R \u29f8 P` exactly when `a ^ ((#(R \u29f8 P) - 1) / n) \u2261 1 (mod P)`.","title":"Euler's criterion in a finite residue domain"},{"id":"tauceti/ring-theory/part-02/other-results/part-01/the-frobenius-surjectivity-criterion","kind":"node","lean":"TauCeti.tfae_forall_exists_dvd_sub_pow","parent":4018,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-01/index.html#the-frobenius-surjectivity-criterion","scope":"tauceti/ring-theory/part-02/other-results/part-01","status":"fully_proved","summary":"The Frobenius surjectivity criterion (Bhatt\u2013Morrow\u2013Scholze, *Integral p-adic Hodge theory*, Lemma 3.9). In a `\u03c0`-adically complete and separated ring `A` with `\u03c0 ^ p \u2223 p`, the following are equivalent: every element of `A` is a `p`-th power modulo `\u03c0 ^ p`, modulo `p`, and modulo `\u03c0 p`.","title":"The Frobenius surjectivity criterion"},{"id":"tauceti/ring-theory/part-02/other-results/part-01/the-graded-nakayama-lemma-for-generators","kind":"node","lean":"TauCeti.GradedAlgebra.adjoin_eq_top_of_irrelevant_le","parent":4018,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-01/index.html#the-graded-nakayama-lemma-for-generators","scope":"tauceti/ring-theory/part-02/other-results/part-01","status":"fully_proved","summary":"If a set `s` of homogeneous elements of positive degree generates the irrelevant ideal `A\u208a` of an `\u2115`-graded commutative semiring modulo `A\u208a\u00b2`, then `s` generates the semiring as an algebra over its degree-zero part.","title":"The graded Nakayama lemma for generators"},{"id":"tauceti/ring-theory/part-02/other-results/part-01/the-hadamard-three-circles-inequality-on-w-o-1-p","kind":"node","lean":"TauCeti.WittVector.gaussValuationAway_le_max, TauCeti.WittVector.gaussValuation_mul_inv_pow_le_max","parent":4018,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-01/index.html#the-hadamard-three-circles-inequality-on-w-o-1-p","scope":"tauceti/ring-theory/part-02/other-results/part-01","status":"fully_proved","summary":"For radii `\u03c1\u2081 \u2264 \u03c1 \u2264 \u03c1\u2082`, the Gauss valuation `\u03bb_\u03c1` is bounded by the larger of `\u03bb_{\u03c1\u2081}` and `\u03bb_{\u03c1\u2082}`, so the interval norm of an interval dominates every Gauss valuation at a radius inside it.","title":"The Hadamard three circles inequality on `\ud835\udd4e O[1/(p [\u03d6])]`"},{"id":"tauceti/ring-theory/part-02/other-results/part-01/hazewinkels-functional-equation-lemma","kind":"node","lean":"TauCeti.coeff_inv_smul_\ud835\udd3e\u2090_toPowerSeries_mem","parent":4018,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-01/index.html#hazewinkels-functional-equation-lemma","scope":"tauceti/ring-theory/part-02/other-results/part-01","status":"fully_proved","summary":"Let `f = b\u2081 X + \u22ef` be a formal coordinate over `K` whose linear coefficient `b\u2081` is a unit of `A`, and suppose that `f` satisfies the functional equation. Then the formal group law `f\u207b\u00b9(f(X) + f(Y))` with logarithm `f`, the transport `f\u207b\u00b9 \u2022 \ud835\udd3e\u2090` of the additive formal group law, has all its coefficients in `A`.","title":"Hazewinkel's functional equation lemma"},{"id":"tauceti/ring-theory/part-02/other-results/part-01/hensels-lemma-for-the-norm","kind":"node","lean":"TauCeti.Algebra.exists_norm_eq_of_norm_sub_mem","parent":4018,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-01/index.html#hensels-lemma-for-the-norm","scope":"tauceti/ring-theory/part-02/other-results/part-01","status":"fully_proved","summary":"Let `S` be a finite free algebra over a ring `R` Henselian at an ideal `I`, containing a unit `w` whose trace is a unit. If a unit `v` of `R` is congruent to the norm of `a` modulo `I`, then `v` is the norm of some `y` congruent to `a` modulo `IS`.","title":"Hensel's lemma for the norm"},{"id":"tauceti/ring-theory/part-02/other-results/part-01/hensels-lemma-with-a-derivative-dividing-k","kind":"node","lean":"TauCeti.IsAdicComplete.existsUnique_eval_eq_zero_of_dvd_pow","parent":4018,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-01/index.html#hensels-lemma-with-a-derivative-dividing-k","scope":"tauceti/ring-theory/part-02/other-results/part-01","status":"fully_proved","summary":"Let `R` be `\u03c0`-adically complete with `\u03c0` a non-zero-divisor, and let `f` be a polynomial with `f'(t) \u2223 \u03c0 ^ k` and `f(t) \u2261 0 mod \u03c0 ^ (2k + 1)`. Then `f` has exactly one root congruent to `t` modulo `\u03c0 ^ (k + 1)`.","title":"Hensel's lemma with a derivative dividing `\u03c0 ^ k`"},{"id":"tauceti/ring-theory/part-02/other-results/part-01/hensels-lemma-without-monicity","kind":"node","lean":"Polynomial.exists_isRoot_sub_mem","parent":4018,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-01/index.html#hensels-lemma-without-monicity","scope":"tauceti/ring-theory/part-02/other-results/part-01","status":"fully_proved","summary":"Over a Henselian pair `(R, J)`, a simple root of a polynomial modulo `J` lifts to a root with the same image in `R \u29f8 J`, whether or not the polynomial is monic.","title":"Hensel's lemma without monicity"},{"id":"tauceti/ring-theory/part-02/other-results/part-01/jacobis-formula","kind":"node","lean":"Derivation.leibniz_det","parent":4018,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-01/index.html#jacobis-formula","scope":"tauceti/ring-theory/part-02/other-results/part-01","status":"fully_proved","summary":"a derivation `D` differentiates the determinant of `M` as the trace of the adjugate of `M` times the entrywise derivative of `M`.","title":"Jacobi's formula"},{"id":"tauceti/ring-theory/part-02/other-results/part-01/the-jacobian-criterion-for-etaleness","kind":"node","lean":"TauCeti.etale_aeval_of_basis","parent":4018,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-01/index.html#the-jacobian-criterion-for-etaleness","scope":"tauceti/ring-theory/part-02/other-results/part-01","status":"fully_proved","summary":"If `T` is smooth over `R` and the differentials `d a\u1d62` of finitely many elements `a\u1d62` of `T` form a basis of `\u03a9[T\u2044R]`, then `T` is \u00e9tale over the polynomial ring `R[X\u1d62]` via `X\u1d62 \u21a6 a\u1d62`.","title":"The Jacobian criterion for \u00e9taleness"},{"id":"tauceti/ring-theory/part-02/other-results/part-01/kedlayas-estimate-4-9-1-in-the-normalisation-p","kind":"node","lean":"TauCeti.WittVector.IntervalLocalization.exists_sum_eq_mul_inv_pow","parent":4018,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-01/index.html#kedlayas-estimate-4-9-1-in-the-normalisation-p","scope":"tauceti/ring-theory/part-02/other-results/part-01","status":"fully_proved","summary":"Every `d \u2208 \ud835\udd4e O[1/(p [\u03d6])]` is a polynomial `\u2211_{i \u2264 m} z\u1d62 p^(-(m - i))` in `p\u207b\u00b9` whose coefficients `z\u1d62` lie in `\ud835\udd4e O[1/[\u03d6]]` and satisfy `\u03bb_{\u03c1\u2082}(z\u1d62) \u03c1\u2081^(-(m - i)) \u2264 \u03bb_{[\u03c1\u2081, \u03c1\u2082]}(d)`. Writing `d = y / (p [\u03d6]) ^ m` and splitting `y = \u2211_{i < m} [y\u1d62] p\u2071 + p\u1d50 b`, one takes `z\u1d62 = [y\u1d62] / [\u03d6] ^ m` for `i < m` and `z\u2098 = b / [\u03d6] ^ m`.","title":"Kedlaya's estimate (4.9.1) in the normalisation `\u03bb_\u03c1(p) = \u03c1`"},{"id":"tauceti/ring-theory/part-02/other-results/part-01/krull-akizuki-for-mathlibs-integralclosure","kind":"node","lean":"TauCeti.integralClosure.isNoetherianRing, TauCeti.IsIntegralClosure.isNoetherianRing","parent":4018,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-01/index.html#krull-akizuki-for-mathlibs-integralclosure","scope":"tauceti/ring-theory/part-02/other-results/part-01","status":"fully_proved","summary":"The specialisation of `TauCeti.IsIntegralClosure.isNoetherianRing` to the integral closure of `A` in `L` as a subalgebra.","title":"Krull\u2013Akizuki for Mathlib's `integralClosure`"},{"data_url":"scopes/tauceti/ring-theory/part-02/other-results/part-02.json","id":"tauceti/ring-theory/part-02/other-results/part-02","kind":"scope","navigation":"transparent","parent":4017,"result":false,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-02/index.html","scope":"tauceti/ring-theory/part-02/other-results/part-02","status":"fully_proved","title":"Krull\u2013Akizuki's length bound to Wedhorn's degree decomposition\u2026"},{"id":"tauceti/ring-theory/part-02/other-results/part-02/krull-akizukis-length-bound","kind":"node","lean":"TauCeti.length_quotient_lsmul_le_finrank_mul_ord","parent":4039,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-02/index.html#krull-akizukis-length-bound","scope":"tauceti/ring-theory/part-02/other-results/part-02","status":"fully_proved","summary":"For any `A`-submodule `M` of a finite-dimensional `K`-vector space `V` \u2014 finitely generated or not \u2014 and any nonzero `a : A`, `length (M \u29f8 aM) \u2264 dim_K V * ord_A a`.","title":"Krull\u2013Akizuki's length bound"},{"id":"tauceti/ring-theory/part-02/other-results/part-02/the-kummer-dedekind-irreducibility-criterion","kind":"node","lean":"TauCeti.KummerDedekind.Ideal.irreducible_map_iff_irreducible_minpoly","parent":4039,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-02/index.html#the-kummer-dedekind-irreducibility-criterion","scope":"tauceti/ring-theory/part-02/other-results/part-02","status":"fully_proved","summary":"`p` stays prime in `S` exactly when `minpoly R x` is irreducible modulo `p`.","title":"The Kummer\u2013Dedekind irreducibility criterion"},{"id":"tauceti/ring-theory/part-02/other-results/part-02/the-leibniz-kernel-identity-from-a-closed-produc","kind":"node","lean":"TauCeti.NovikovModule.kernelComp_eq_add_of_functionalBoundary_eq_zero","parent":4039,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-02/index.html#the-leibniz-kernel-identity-from-a-closed-produc","scope":"tauceti/ring-theory/part-02/other-results/part-02","status":"fully_proved","summary":"Let the product kernel `n` have cochain degree `0`. If the readout of `n`, a functional on `C\u2081 \u2297 C\u2082 \u2297 C\u2080^\u2228`, is annihilated by the total differential built from the Koszul tensor differential of `C\u2081 \u2297 C\u2082`, then `n` obeys the Leibniz identity between kernels `\u2211 n(p\u2081, p\u2082, p) \u03b4\u2080(p, p\u2080) = \u2211 \u03b4\u2081(p\u2081, p) n(p, p\u2082, p\u2080) + (-1)^(d\u2081 p\u2081) \u2211 \u03b4\u2082(p\u2082, p) n(p\u2081, p, p\u2080)` of `TauCeti.NovikovModule.kernelMap_kernelProduct_eq_add`.","title":"The Leibniz kernel identity from a closed product functional"},{"id":"tauceti/ring-theory/part-02/other-results/part-02/maschkes-theorem-after-base-change","kind":"node","lean":"MonoidAlgebra.isSemisimpleRing_tensor","parent":4039,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-02/index.html#maschkes-theorem-after-base-change","scope":"tauceti/ring-theory/part-02/other-results/part-02","status":"fully_proved","summary":"If `k` is a field in which the order of the finite group `G` is nonzero, then the base change `R[G] \u2297[R] k` of the group algebra over any commutative ring `R` mapping to `k` is a semisimple ring, being isomorphic to `k[G]`.","title":"Maschke's theorem after base change"},{"id":"tauceti/ring-theory/part-02/other-results/part-02/nakayamas-lemma-for-an-idempotent","kind":"node","lean":"IsIdempotentElem.exists_smul_eq_mul_of_mul_sub_smul_mem","parent":4039,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-02/index.html#nakayamas-lemma-for-an-idempotent","scope":"tauceti/ring-theory/part-02/other-results/part-02","status":"fully_proved","summary":"Let `T` be a module-finite `S`-algebra and `J` an ideal of `S` in the Jacobson radical. If an idempotent `e` of `T` satisfies `e T \u2286 S e + JT`, then `e T = S e`.","title":"Nakayama's lemma for an idempotent"},{"id":"tauceti/ring-theory/part-02/other-results/part-02/nakayamas-lemma-over-z","kind":"node","lean":"Submodule.eq_bot_of_forall_prime_le_smul, Module.exists_finset_card_le_finrank_and_smul_mem_span, Submodule.exists_sub_prod_mem_and_smul_eq_zero, Submodule.eq_bot_of_forall_isMaximal_le_smul","parent":4039,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-02/index.html#nakayamas-lemma-over-z","scope":"tauceti/ring-theory/part-02/other-results/part-02","status":"fully_proved","summary":"A finitely generated subgroup `N` of an abelian group with `N \u2264 p \u2022 N` for every prime `p` is zero.","title":"Nakayama's lemma over `\u2124`"},{"id":"tauceti/ring-theory/part-02/other-results/part-02/the-open-mapping-theorem-over-a-tate-ring","kind":"node","lean":"LinearMap.isOpenMap_of_isTateRing","parent":4039,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-02/index.html#the-open-mapping-theorem-over-a-tate-ring","scope":"tauceti/ring-theory/part-02/other-results/part-02","status":"fully_proved","summary":"A surjective linear map continuous at zero, from a complete first-countable nonarchimedean module to a Hausdorff Baire module, is open. The target needs only continuous constant scalar actions.","title":"The open mapping theorem over a Tate ring"},{"id":"tauceti/ring-theory/part-02/other-results/part-02/serres-criterion-for-normality","kind":"node","lean":"TauCeti.forall_isDomain_and_isIntegrallyClosed_iff_satisfiesSerreR_one_and_satisfiesSerreS_two","parent":4039,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-02/index.html#serres-criterion-for-normality","scope":"tauceti/ring-theory/part-02/other-results/part-02","status":"fully_proved","summary":"A Noetherian ring is normal, that is, every localization at a prime is an integrally closed domain, if and only if it satisfies `(R_1)` and `(S_2)`. Stacks, Lemma 10.157.4 (031S).","title":"Serre's criterion for normality"},{"id":"tauceti/ring-theory/part-02/other-results/part-02/serres-lemma","kind":"node","lean":"TauCeti.eq_one_of_isOfFinOrder_of_pow_dvd_sub_one","parent":4039,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-02/index.html#serres-lemma","scope":"tauceti/ring-theory/part-02/other-results/part-02","status":"fully_proved","summary":"Let `p` be a prime and `A` a ring with no `p`-torsion in which only `0` is divisible by every power of `p`. An element of finite order of `A` that is congruent to `1` modulo `p ^ k`, with `p ^ k \u2265 3`, is equal to `1`.","title":"Serre's lemma"},{"id":"tauceti/ring-theory/part-02/other-results/part-02/taylors-formula-for-polynomials","kind":"node","lean":"Polynomial.taylorExp_smul_derivative'","parent":4039,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-02/index.html#taylors-formula-for-polynomials","scope":"tauceti/ring-theory/part-02/other-results/part-02","status":"fully_proved","summary":"Over a `\u211a`-algebra, the Taylor exponential of the scaled derivative `r \u2022 d / dX` is the Taylor shift `p \u21a6 p (X + r)`.","title":"Taylor's formula for polynomials"},{"id":"tauceti/ring-theory/part-02/other-results/part-02/universality-of-hazewinkels-formal-group-law-ove","kind":"node","lean":"TauCeti.existsUnique_eq_hazewinkelFormalGroup","parent":4039,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-02/index.html#universality-of-hazewinkels-formal-group-law-ove","scope":"tauceti/ring-theory/part-02/other-results/part-02","status":"fully_proved","summary":"a formal group law over a `\u211a`-algebra whose logarithm has no terms outside the degrees `p\u207f` is Hazewinkel's formal group law of a unique family `v`.","title":"Universality of Hazewinkel's formal group law over `\u211a`-algebras"},{"id":"tauceti/ring-theory/part-02/other-results/part-02/wedhorns-degree-decomposition-lemma-8-33-i","kind":"node","lean":"TauCeti.Huber.twoSidedRestrictedSubmodule_eq_sup","parent":4039,"result":true,"route":"../roadmap/tauceti/ring-theory/part-02/other-results/part-02/index.html#wedhorns-degree-decomposition-lemma-8-33-i","scope":"tauceti/ring-theory/part-02/other-results/part-02","status":"fully_proved","summary":"A two-sided restricted family is the sum of its non-negative part and its negative part: `A\u27e8z, z\u207b\u00b9\u27e9 = A\u27e8z\u27e9 + z\u207b\u00b9A\u27e8z\u207b\u00b9\u27e9` at the level of coefficients. This is `twoSidedRestrictedSubmodule_eq_sup_compl` at the sign partition, which is where the restrictedness of each summand is established.","title":"Wedhorn's degree decomposition, Lemma 8.33(i)"},{"data_url":"scopes/tauceti/topology.json","id":"tauceti/topology","kind":"scope","parent":1095,"result":false,"route":"../roadmap/tauceti/topology/index.html","scope":"tauceti/topology","status":"fully_proved","summary":"Results in Tau Ceti's `TauCeti.Topology` modules, and the roadmaps whose articles are mostly there: by roadmap, then by area.","title":"Topology"},{"data_url":"scopes/tauceti/topology/peripheral-actions-on-free-pro-p-groups.json","id":"tauceti/topology/peripheral-actions-on-free-pro-p-groups","kind":"scope","parent":4052,"result":false,"route":"../roadmap/tauceti/topology/peripheral-actions-on-free-pro-p-groups/index.html","scope":"tauceti/topology/peripheral-actions-on-free-pro-p-groups","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Peripheral actions on free pro-`p` groups"},{"id":"tauceti/topology/peripheral-actions-on-free-pro-p-groups/the-peripheral-product-identity","kind":"node","lean":"TauCeti.Peripheral.exists_defect_eq_one","parent":4053,"result":true,"route":"../roadmap/tauceti/topology/peripheral-actions-on-free-pro-p-groups/index.html#the-peripheral-product-identity","scope":"tauceti/topology/peripheral-actions-on-free-pro-p-groups","status":"fully_proved","summary":"For every unit `u \u2208 \u2124_p\u02e3`, the basis `x = basis e` of a free pro-`p` group admits conjugators `c` on the basis, with `c 0 = 1` in positive rank, and `d` on the cusp such that `(\u220f_i (c i)\u207b\u00b9 * x i ^ u * c i) * (d\u207b\u00b9 * cusp x ^ u * d) = 1`, the product taken in order and `^ u` the `p`-adic power.","title":"The peripheral product identity"},{"id":"tauceti/topology/peripheral-actions-on-free-pro-p-groups/solutions-at-every-level","kind":"node","lean":"TauCeti.Peripheral.exists_mem_level","parent":4053,"result":true,"route":"../roadmap/tauceti/topology/peripheral-actions-on-free-pro-p-groups/index.html#solutions-at-every-level","scope":"tauceti/topology/peripheral-actions-on-free-pro-p-groups","status":"fully_proved","summary":"For a unit `u`, every peripheral level of the basis `x = basis e` of a free pro-`p` group contains a pair of conjugators whose conjugator at the first basis element is trivial when the rank is positive.","title":"Solutions at every level"},{"id":"tauceti/topology/peripheral-actions-on-free-pro-p-groups/the-compactness-step","kind":"node","lean":"TauCeti.Peripheral.exists_defect_eq_one_of_forall_exists_mem_level","parent":4053,"result":true,"route":"../roadmap/tauceti/topology/peripheral-actions-on-free-pro-p-groups/index.html#the-compactness-step","scope":"tauceti/topology/peripheral-actions-on-free-pro-p-groups","status":"fully_proved","summary":"If every peripheral level of a family `x` in a pro-`p` group contains a pair of conjugators whose conjugator at the first basis element is trivial when the rank is positive, then some such pair has trivial defect. The exponent `u` need not be a unit.","title":"The compactness step"},{"id":"tauceti/topology/peripheral-actions-on-free-pro-p-groups/tauceti-peripheral-not-exists-peripheral-identit","kind":"node","lean":"TauCeti.Peripheral.not_exists_peripheral_identity_two","parent":4053,"result":true,"route":"../roadmap/tauceti/topology/peripheral-actions-on-free-pro-p-groups/index.html#tauceti-peripheral-not-exists-peripheral-identit","scope":"tauceti/topology/peripheral-actions-on-free-pro-p-groups","status":"fully_proved","summary":"At exponent `2`, no normalized peripheral product in a rank-two free pro-`2` group is trivial even modulo the second closed lower-central-series term.","title":"TauCeti.Peripheral.not_exists_peripheral_identity_two"},{"id":"tauceti/topology/peripheral-actions-on-free-pro-p-groups/the-peripheral-power-theorem","kind":"node","lean":"TauCeti.Peripheral.exists_peripheralAut","parent":4053,"result":true,"route":"../roadmap/tauceti/topology/peripheral-actions-on-free-pro-p-groups/index.html#the-peripheral-power-theorem","scope":"tauceti/topology/peripheral-actions-on-free-pro-p-groups","status":"fully_proved","summary":"For every unit `u \u2208 \u2124_p\u02e3` there are a continuous automorphism `\u03c6` of `F`, conjugators `c` on the basis `x = basis e`, with `c 0 = 1` in positive rank, and a conjugator `d` on the cusp, such that `\u03c6 (x i) = (c i)\u207b\u00b9 * x i ^ u * c i` for every `i` and `\u03c6 (cusp x) = d\u207b\u00b9 * cusp x ^ u * d`, where `^ u` is the `p`-adic power.","title":"The peripheral-power theorem"},{"id":"tauceti/topology/peripheral-actions-on-free-pro-p-groups/the-peripheral-conjugators-are-not-all-equal","kind":"node","lean":"TauCeti.Peripheral.exists_ne_of_apply_eq_conj","parent":4053,"result":true,"route":"../roadmap/tauceti/topology/peripheral-actions-on-free-pro-p-groups/index.html#the-peripheral-conjugators-are-not-all-equal","scope":"tauceti/topology/peripheral-actions-on-free-pro-p-groups","status":"fully_proved","summary":"For a free pro-`p` group of rank at least two and `u \u2208 \u2124_p` other than `0` and `1`, if an endomorphism `\u03c6` satisfies `\u03c6 (x i) = (c i)\u207b\u00b9 * x i ^ u * c i` for every basis element and `\u03c6 z = d\u207b\u00b9 * z ^ u * d` on the cusp, then some `c i` differs from `d`.","title":"The peripheral conjugators are not all equal"},{"id":"tauceti/topology/peripheral-actions-on-free-pro-p-groups/the-permutation-and-the-exponent-are-determined","kind":"node","lean":"TauCeti.Peripheral.IsPeripheralPermAut.unique","parent":4053,"result":true,"route":"../roadmap/tauceti/topology/peripheral-actions-on-free-pro-p-groups/index.html#the-permutation-and-the-exponent-are-determined","scope":"tauceti/topology/peripheral-actions-on-free-pro-p-groups","status":"fully_proved","summary":"For the basis of a free pro-`p` group of rank `r \u2265 2`, an automorphism peripheral for `(\u03c3, u)` and for `(\u03c3', u')` has `\u03c3 = \u03c3'` and `u = u'`.","title":"The permutation and the exponent are determined in rank at least two"},{"id":"tauceti/topology/peripheral-actions-on-free-pro-p-groups/the-permutation-and-the-exponent-of-a-permutatio","kind":"node","lean":"TauCeti.Peripheral.permData","parent":4053,"result":true,"route":"../roadmap/tauceti/topology/peripheral-actions-on-free-pro-p-groups/index.html#the-permutation-and-the-exponent-of-a-permutatio","scope":"tauceti/topology/peripheral-actions-on-free-pro-p-groups","status":"fully_proved","summary":"The permutation and the exponent of a permutation-peripheral automorphism, for the basis of a free pro-`p` group of rank at least two, as a homomorphism.","title":"The permutation and the exponent of a permutation-peripheral automorphism"},{"id":"tauceti/topology/peripheral-actions-on-free-pro-p-groups/the-data-are-not-determined-in-rank-one","kind":"node","lean":"TauCeti.Peripheral.exists_isPeripheralPermAut_swap_rank_one","parent":4053,"result":true,"route":"../roadmap/tauceti/topology/peripheral-actions-on-free-pro-p-groups/index.html#the-data-are-not-determined-in-rank-one","scope":"tauceti/topology/peripheral-actions-on-free-pro-p-groups","status":"fully_proved","summary":"In rank one the peripheral tuple is `(x, x\u207b\u00b9)`, and an automorphism with `x \u21a6 x ^ u` is peripheral both for the trivial permutation with exponent `u` and for the transposition with exponent `-u`.","title":"The data are not determined in rank one"},{"id":"tauceti/topology/peripheral-actions-on-free-pro-p-groups/the-section-of-the-exponent-over-the-principal-u","kind":"node","lean":"TauCeti.Peripheral.exists_section_principalUnits","parent":4053,"result":true,"route":"../roadmap/tauceti/topology/peripheral-actions-on-free-pro-p-groups/index.html#the-section-of-the-exponent-over-the-principal-u","scope":"tauceti/topology/peripheral-actions-on-free-pro-p-groups","status":"fully_proved","summary":"Let `F` be a marked free pro-`p` group of positive rank, and let `k \u2265 1`, with `k \u2265 2` when `p = 2`. Then the exponent character of the peripheral automorphism group has a continuous homomorphic section over the principal units `U^(k) = 1 + p ^ k \u2124_p`: a continuous homomorphism `s : U^(k) \u2192\u209c* peripheralAut hF (basis e)` with `exponent (s v) = v` for every `v`.","title":"The section of the exponent over the principal units"},{"id":"tauceti/topology/peripheral-actions-on-free-pro-p-groups/the-peripheral-automorphisms-form-a-closed-subgr","kind":"node","lean":"TauCeti.Peripheral.isClosed_peripheralAut","parent":4053,"result":true,"route":"../roadmap/tauceti/topology/peripheral-actions-on-free-pro-p-groups/index.html#the-peripheral-automorphisms-form-a-closed-subgr","scope":"tauceti/topology/peripheral-actions-on-free-pro-p-groups","status":"fully_proved","summary":"The peripheral automorphisms form a closed subgroup of `ContinuousAut F` for the congruence topology, when `F` is topologically finitely generated.","title":"The peripheral automorphisms form a closed subgroup"},{"id":"tauceti/topology/peripheral-actions-on-free-pro-p-groups/the-exponent-character-is-continuous","kind":"node","lean":"TauCeti.Peripheral.continuous_exponent","parent":4053,"result":true,"route":"../roadmap/tauceti/topology/peripheral-actions-on-free-pro-p-groups/index.html#the-exponent-character-is-continuous","scope":"tauceti/topology/peripheral-actions-on-free-pro-p-groups","status":"fully_proved","summary":"Its graph is closed, being the peripheral graph of the basis, and its target `\u2124_[p]\u02e3` is compact.","title":"The exponent character is continuous"},{"id":"tauceti/topology/peripheral-actions-on-free-pro-p-groups/tauceti-peripheral-peripheralaut","kind":"node","lean":"TauCeti.Peripheral.peripheralAut","parent":4053,"result":true,"route":"../roadmap/tauceti/topology/peripheral-actions-on-free-pro-p-groups/index.html#tauceti-peripheral-peripheralaut","scope":"tauceti/topology/peripheral-actions-on-free-pro-p-groups","status":"fully_proved","summary":"The continuous automorphisms peripheral for `x` with some common unit exponent.","title":"TauCeti.Peripheral.peripheralAut"},{"id":"tauceti/topology/peripheral-actions-on-free-pro-p-groups/tauceti-peripheral-exponent","kind":"node","lean":"TauCeti.Peripheral.exponent","parent":4053,"result":true,"route":"../roadmap/tauceti/topology/peripheral-actions-on-free-pro-p-groups/index.html#tauceti-peripheral-exponent","scope":"tauceti/topology/peripheral-actions-on-free-pro-p-groups","status":"fully_proved","summary":"The unique common unit exponent of a peripheral automorphism of a marked free pro-`p` group of positive rank.","title":"TauCeti.Peripheral.exponent"},{"id":"tauceti/topology/peripheral-actions-on-free-pro-p-groups/tauceti-peripheral-peripheralaut-eq-top-rank-one","kind":"node","lean":"TauCeti.Peripheral.peripheralAut_eq_top_rank_one","parent":4053,"result":true,"route":"../roadmap/tauceti/topology/peripheral-actions-on-free-pro-p-groups/index.html#tauceti-peripheral-peripheralaut-eq-top-rank-one","scope":"tauceti/topology/peripheral-actions-on-free-pro-p-groups","status":"fully_proved","summary":"In rank one the peripheral automorphism group is the whole group of continuous automorphisms.","title":"TauCeti.Peripheral.peripheralAut_eq_top_rank_one"},{"id":"tauceti/topology/peripheral-actions-on-free-pro-p-groups/tauceti-peripheral-level","kind":"node","lean":"TauCeti.Peripheral.level","parent":4053,"result":true,"route":"../roadmap/tauceti/topology/peripheral-actions-on-free-pro-p-groups/index.html#tauceti-peripheral-level","scope":"tauceti/topology/peripheral-actions-on-free-pro-p-groups","status":"fully_proved","summary":"The `n`-th peripheral level is the set of conjugators whose defect lies in the `n`-th term of the closed lower central series.","title":"TauCeti.Peripheral.level"},{"id":"tauceti/topology/peripheral-actions-on-free-pro-p-groups/tauceti-peripheral-exists-mem-level-succ","kind":"node","lean":"TauCeti.Peripheral.exists_mem_level_succ","parent":4053,"result":true,"route":"../roadmap/tauceti/topology/peripheral-actions-on-free-pro-p-groups/index.html#tauceti-peripheral-exists-mem-level-succ","scope":"tauceti/topology/peripheral-actions-on-free-pro-p-groups","status":"fully_proved","summary":"A pair of conjugators whose defect lies in `\u03b3_n`, for positive `n`, can be corrected so that its defect lies in `\u03b3_{n+1}`. The corrections lie in `\u03b3_{n-1}`, and the correction at the first basis element is trivial. The unit hypothesis on `u` is used to rescale the required graded correction.","title":"TauCeti.Peripheral.exists_mem_level_succ"},{"id":"tauceti/topology/peripheral-actions-on-free-pro-p-groups/tauceti-peripheral-forall-mem-level-iff","kind":"node","lean":"TauCeti.Peripheral.forall_mem_level_iff","parent":4053,"result":true,"route":"../roadmap/tauceti/topology/peripheral-actions-on-free-pro-p-groups/index.html#tauceti-peripheral-forall-mem-level-iff","scope":"tauceti/topology/peripheral-actions-on-free-pro-p-groups","status":"fully_proved","summary":"A pair of conjugators lies in every peripheral level exactly when its defect is trivial: in a pro-`p` group the terms of the closed lower central series have trivial intersection.","title":"TauCeti.Peripheral.forall_mem_level_iff"},{"id":"tauceti/topology/peripheral-actions-on-free-pro-p-groups/the-principal-units-are-a-copy-of-z-p","kind":"node","lean":"TauCeti.principalUnitsEquiv","parent":4053,"result":true,"route":"../roadmap/tauceti/topology/peripheral-actions-on-free-pro-p-groups/index.html#the-principal-units-are-a-copy-of-z-p","scope":"tauceti/topology/peripheral-actions-on-free-pro-p-groups","status":"fully_proved","summary":"For `f \u2265 1`, with `f \u2265 2` when `p = 2`, and a unit `w` of exact level `f`, the `p`-adic power map `l \u21a6 w ^ l` is a topological group isomorphism from the additive group of `\u2124_[p]` onto `U^(f) = 1 + p ^ f \u2124_p`. It is onto because `w` topologically generates `U^(f)`, and injective because `w` has infinite order.","title":"The principal units are a copy of `\u2124_p`"},{"id":"tauceti/topology/peripheral-actions-on-free-pro-p-groups/the-peripheral-power-theorem-for-the-conjugates","kind":"node","lean":"TauCeti.Peripheral.exists_peripheralAut_conj_cusp","parent":4053,"result":true,"route":"../roadmap/tauceti/topology/peripheral-actions-on-free-pro-p-groups/index.html#the-peripheral-power-theorem-for-the-conjugates","scope":"tauceti/topology/peripheral-actions-on-free-pro-p-groups","status":"fully_proved","summary":"For every unit `u \u2208 \u2124_p\u02e3` there are a continuous automorphism `\u03c6` of `F`, conjugators `c` on the basis `x = basis e`, with `c 0 = 1` in positive rank, and `d : F`, such that `\u03c6 (x i) = (c i)\u207b\u00b9 * x i ^ u * c i` for every `i` and, for every `q : F`, `\u03c6` carries the conjugate `y = q * cusp x * q\u207b\u00b9` of the cusp to `d'\u207b\u00b9 * y ^ u * d'` with `d' = q * d * (\u03c6 q)\u207b\u00b9`. At `q = 1` the conjugator is `d`.","title":"The peripheral-power theorem for the conjugates of the cusp"},{"data_url":"scopes/tauceti/topology/profinite-and-pro-p-groups.json","id":"tauceti/topology/profinite-and-pro-p-groups","kind":"scope","parent":4052,"result":false,"route":"../roadmap/tauceti/topology/profinite-and-pro-p-groups/index.html","scope":"tauceti/topology/profinite-and-pro-p-groups","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Profinite and pro-`p` groups"},{"data_url":"scopes/tauceti/topology/profinite-and-pro-p-groups/part-01.json","id":"tauceti/topology/profinite-and-pro-p-groups/part-01","kind":"scope","navigation":"transparent","parent":4074,"result":false,"route":"../roadmap/tauceti/topology/profinite-and-pro-p-groups/part-01/index.html","scope":"tauceti/topology/profinite-and-pro-p-groups/part-01","status":"fully_proved","title":"The classification of Demushkin\u2026 to The necessity half of the\u2026"},{"id":"tauceti/topology/profinite-and-pro-p-groups/part-01/the-classification-of-demushkin-groups-labute","kind":"node","lean":"TauCeti.IsDemushkin.nonempty_continuousMulEquiv_iff","parent":4075,"result":true,"route":"../roadmap/tauceti/topology/profinite-and-pro-p-groups/part-01/index.html#the-classification-of-demushkin-groups-labute","scope":"tauceti/topology/profinite-and-pro-p-groups/part-01","status":"fully_proved","summary":"two Demushkin groups are topologically isomorphic if and only if they have the same rank and their canonical characters have the same image. The forward direction is the invariance of both invariants under isomorphism; the converse is `TauCeti.IsDemushkin.nonempty_continuousMulEquiv_of_range_demushkinCharacter_eq`.","title":"The classification of Demushkin groups (Labute)"},{"id":"tauceti/topology/profinite-and-pro-p-groups/part-01/uniqueness-of-the-dyadic-demushkin-groups-of-eve","kind":"node","lean":"TauCeti.IsDemushkin.nonempty_continuousMulEquiv_of_even_demushkinRank_of_range_eq_unitsPlusMinus","parent":4075,"result":true,"route":"../roadmap/tauceti/topology/profinite-and-pro-p-groups/part-01/index.html#uniqueness-of-the-dyadic-demushkin-groups-of-eve","scope":"tauceti/topology/profinite-and-pro-p-groups/part-01","status":"fully_proved","summary":"Two Demushkin groups at `p = 2` of the same even rank `n \u2265 4` whose canonical characters have the same image `V^(f) = {\u00b11} \u00d7 U^(f)`, `f \u2265 2` finite, are topologically isomorphic.","title":"Uniqueness of the dyadic Demushkin groups of even rank with image `{\u00b11} \u00d7 U^(f)` (Labute, Theorem 6)"},{"id":"tauceti/topology/profinite-and-pro-p-groups/part-01/tates-theorem-an-infinite-demushkin-group-has-p","kind":"node","lean":"TauCeti.IsDemushkin.cohomologicalDimensionAt_eq_two","parent":4075,"result":true,"route":"../roadmap/tauceti/topology/profinite-and-pro-p-groups/part-01/index.html#tates-theorem-an-infinite-demushkin-group-has-p","scope":"tauceti/topology/profinite-and-pro-p-groups/part-01","status":"fully_proved","summary":"The lower bound is `H\u00b2(G, \ud835\udd3d_p) \u2260 0`; the upper bound is the right exactness of `H\u00b2` on the finite `\ud835\udd3d_p[G]`-modules, which Tate's perfect duality provides.","title":"Tate's theorem: an infinite Demushkin group has `p`-cohomological dimension exactly `2`"},{"id":"tauceti/topology/profinite-and-pro-p-groups/part-01/tates-duality-map-1-of-an-infinite-demushkin-gro","kind":"node","lean":"TauCeti.IsDemushkin.dualityMap1_zModTwist_bijective","parent":4075,"result":true,"route":"../roadmap/tauceti/topology/profinite-and-pro-p-groups/part-01/index.html#tates-duality-map-1-of-an-infinite-demushkin-gro","scope":"tauceti/topology/profinite-and-pro-p-groups/part-01","status":"fully_proved","summary":"`H\u00b9(G, M) \u00d7 H\u00b9(G, Hom(M, I(\u03c7)/p\u2071)) \u2192 H\u00b2(G, I(\u03c7)/p\u2071)` is a perfect pairing.","title":"Tate's duality map `\u03b1\u2081` of an infinite Demushkin group with coefficients `I(\u03c7)/p\u2071` is bijective on every finite discrete `G`-module `M` killed by `p\u2071`"},{"id":"tauceti/topology/profinite-and-pro-p-groups/part-01/an-open-subgroup-of-an-infinite-demushkin-group","kind":"node","lean":"TauCeti.IsDemushkin.openSubgroup","parent":4075,"result":true,"route":"../roadmap/tauceti/topology/profinite-and-pro-p-groups/part-01/index.html#an-open-subgroup-of-an-infinite-demushkin-group","scope":"tauceti/topology/profinite-and-pro-p-groups/part-01","status":"fully_proved","summary":"The subgroup is pro-`p` and topologically finitely generated with `G`; its `H\u00b2(U, \ud835\udd3d_p)` is one-dimensional by `TauCeti.IsDemushkin.finrank_cohomFp_two_openSubgroup`; and its cup product is nondegenerate because a nonzero class of `H\u00b9(U, \ud835\udd3d_p) \u2245 H\u00b9(G, Coind_U^G \ud835\udd3d_p)` is paired nontrivially by Tate's duality map `\u03b1\u2081` of `G`, a pairing which is the corestriction of a cup product of `U`.","title":"An open subgroup of an infinite Demushkin group is Demushkin (Serre's expos\u00e9, \u00a79.2)"},{"id":"tauceti/topology/profinite-and-pro-p-groups/part-01/the-isomorphism-type-of-an-open-subgroup-of-a-de","kind":"node","lean":"TauCeti.IsDemushkin.nonempty_continuousMulEquiv_openSubgroup_iff","parent":4075,"result":true,"route":"../roadmap/tauceti/topology/profinite-and-pro-p-groups/part-01/index.html#the-isomorphism-type-of-an-open-subgroup-of-a-de","scope":"tauceti/topology/profinite-and-pro-p-groups/part-01","status":"fully_proved","summary":"an open subgroup `U` of an infinite Demushkin group `G` is topologically isomorphic to a Demushkin group `H` if and only if `H` has rank `2 + [G : U] (n(G) - 2)` and the canonical character of `H` has image `\u03c7(U)`, where `\u03c7` is the canonical character of `G`.","title":"The isomorphism type of an open subgroup of a Demushkin group"},{"id":"tauceti/topology/profinite-and-pro-p-groups/part-01/serres-theorem-at-arbitrary-rank-as-an-equivalen","kind":"node","lean":"TauCeti.IsProP.cohomologicalDimensionAt_le_one_iff_exists_convergesToOne_continuousMulEquiv_presentation, TauCeti.IsProP.nonempty_continuousMulEquiv_freeProP_of_cohomologicalDimensionAt_le_one, TauCeti.IsProP.exists_convergesToOne_continuousMulEquiv_presentation_of_cohomologicalDimensionAt_le_one","parent":4075,"result":true,"route":"../roadmap/tauceti/topology/profinite-and-pro-p-groups/part-01/index.html#serres-theorem-at-arbitrary-rank-as-an-equivalen","scope":"tauceti/topology/profinite-and-pro-p-groups/part-01","status":"fully_proved","summary":"A pro-`p` group `G` has `cd_p G \u2264 1` if and only if it is free pro-`p` on a pointed profinite space: some subset `s` of `G` converging to `1` has a presentation `F_p(insert 1 s, 1) \u2192 G` that is a topological isomorphism.","title":"Serre's theorem at arbitrary rank, as an equivalence"},{"id":"tauceti/topology/profinite-and-pro-p-groups/part-01/the-golod-shafarevich-inequality","kind":"node","lean":"IsPGroup.sq_topologicalGeneratorRankNat_lt_four_mul_finrank_cohomFp_two","parent":4075,"result":true,"route":"../roadmap/tauceti/topology/profinite-and-pro-p-groups/part-01/index.html#the-golod-shafarevich-inequality","scope":"tauceti/topology/profinite-and-pro-p-groups/part-01","status":"fully_proved","summary":"A nontrivial finite `p`-group `G`, with the discrete topology, satisfies `d(G)\u00b2 < 4 r(G)`, where `d(G)` is its generator rank and `r(G) = dim_{\ud835\udd3d_p} H\u00b2(G, \ud835\udd3d_p)` is its relation rank. Finite generation follows from `TauCeti.isTopologicallyFinitelyGenerated_of_fg`.","title":"The Golod\u2013Shafarevich inequality"},{"id":"tauceti/topology/profinite-and-pro-p-groups/part-01/the-demushkin-predicate","kind":"node","lean":"TauCeti.IsDemushkin","parent":4075,"result":true,"route":"../roadmap/tauceti/topology/profinite-and-pro-p-groups/part-01/index.html#the-demushkin-predicate","scope":"tauceti/topology/profinite-and-pro-p-groups/part-01","status":"fully_proved","summary":"The Demushkin predicate (Labute, p. 106), for a prime `p` and a topological group `G`: a pro-`p` group whose `H\u00b9(G, \ud835\udd3d_p)` is finite-dimensional, whose `H\u00b2(G, \ud835\udd3d_p)` is one-dimensional, and on which the cup product `H\u00b9(G, \ud835\udd3d_p) \u00d7 H\u00b9(G, \ud835\udd3d_p) \u2192 H\u00b2(G, \ud835\udd3d_p)` is nondegenerate on both sides. The primality of `p` is a hypothesis of the predicate. Profiniteness is not: no field mentions `CompactSpace G` or\u2026","title":"The Demushkin predicate"},{"id":"tauceti/topology/profinite-and-pro-p-groups/part-01/the-unconstrained-successive-approximation-theor","kind":"node","lean":"TauCeti.freeProP.exists_continuousMulEquiv_apply_eq_of_range_basisModificationDelta_eq_top","parent":4075,"result":true,"route":"../roadmap/tauceti/topology/profinite-and-pro-p-groups/part-01/index.html#the-unconstrained-successive-approximation-theor","scope":"tauceti/topology/profinite-and-pro-p-groups/part-01","status":"fully_proved","summary":"Let `r, w \u2208 \u03bb_1(F)` be relators of the free pro-`p` group `F` on a finite linearly ordered type with the same class `\u03c1 \u2208 gr_1(F)`, and suppose the basis-modification map `\u03b4_\u03c1` is onto `gr_{m+1}(F)` for every `m \u2265 1`. Then a continuous automorphism of `F` carries `r` to `w`.","title":"The unconstrained successive-approximation theorem"},{"id":"tauceti/topology/profinite-and-pro-p-groups/part-01/tauceti-profinitefrattini","kind":"node","lean":"TauCeti.profiniteFrattini","parent":4075,"result":true,"route":"../roadmap/tauceti/topology/profinite-and-pro-p-groups/part-01/index.html#tauceti-profinitefrattini","scope":"tauceti/topology/profinite-and-pro-p-groups/part-01","status":"fully_proved","summary":"The Frattini subgroup `\u03a6(G)` of a topological group: the intersection of its maximal open subgroups. For a profinite group this is the Frattini subgroup of Ribes\u2013Zalesskii; with the discrete topology it is Mathlib's `frattini` (`TauCeti.profiniteFrattini_eq_frattini`).","title":"TauCeti.profiniteFrattini"},{"id":"tauceti/topology/profinite-and-pro-p-groups/part-01/generation-modulo-the-frattini-subgroup","kind":"node","lean":"TauCeti.topologicalClosure_closure_eq_top_iff_quotient_profiniteFrattini","parent":4075,"result":true,"route":"../roadmap/tauceti/topology/profinite-and-pro-p-groups/part-01/index.html#generation-modulo-the-frattini-subgroup","scope":"tauceti/topology/profinite-and-pro-p-groups/part-01","status":"fully_proved","summary":"A set topologically generates a profinite group if and only if its image topologically generates the quotient by the Frattini subgroup.","title":"Generation modulo the Frattini subgroup"},{"id":"tauceti/topology/profinite-and-pro-p-groups/part-01/the-constrained-successive-approximation-theorem","kind":"node","lean":"TauCeti.freeProP.exists_continuousMulEquiv_apply_eq","parent":4075,"result":true,"route":"../roadmap/tauceti/topology/profinite-and-pro-p-groups/part-01/index.html#the-constrained-successive-approximation-theorem","scope":"tauceti/topology/profinite-and-pro-p-groups/part-01","status":"fully_proved","summary":"Let `r, w \u2208 \u03bb_1(F)` be relators of the free pro-`p` group `F` on a finite linearly ordered type with the same class `\u03c1 \u2208 gr_1(F)`, let `Z \u2286 F` and let `C : X \u2192 Subgroup F` be closed admissible corrections, with every admissible basis modification `\u03b8_\u03c9`, `\u03c9 : X \u2192 \u03bb_1(F)` with `\u03c9_i \u2208 C i` for every `i`, preserving each `C i`, and every admissible endomorphism `\u03c6` of `F`, one with `x_i\u207b\u00b9 * \u03c6(x_i) \u2208 C i` for every `i`,\u2026","title":"The constrained successive-approximation theorem"},{"id":"tauceti/topology/profinite-and-pro-p-groups/part-01/the-existence-theorem-of-the-classification-for","kind":"node","lean":"TauCeti.exists_isDemushkin_range_demushkinCharacter_eq_iff","parent":4075,"result":true,"route":"../roadmap/tauceti/topology/profinite-and-pro-p-groups/part-01/index.html#the-existence-theorem-of-the-classification-for","scope":"tauceti/topology/profinite-and-pro-p-groups/part-01","status":"fully_proved","summary":"Let `A \u2264 \u2124_p\u02e3` be a closed pro-`p` subgroup, contained in `1 + 4\u2124_2` when `p = 2`; these are the possible images of the canonical characters of the Demushkin groups with `q \u2260 2` (`TauCeti.demushkinQ_ne_two_iff_range_demushkinCharacter_le`). Then `(n, A)` is the pair of invariants of a Demushkin group, presented on `n` generators by one relator, exactly when `n` is even and `(A : A^p) < p ^ n`. Together with\u2026","title":"The existence theorem of the classification for `q \u2260 2`, as an equivalence (Labute, Theorem 1 and Remark 2; Demushkin and Serre)"},{"id":"tauceti/topology/profinite-and-pro-p-groups/part-01/the-existence-theorem-of-the-classification-of-d","kind":"node","lean":"TauCeti.exists_isDemushkin_one_range_eq_zpowers_neg_one, TauCeti.exists_isDemushkin_range_eq_of_even_of_lt, TauCeti.exists_isDemushkin_range_eq_unitsPlusMinus_of_odd, TauCeti.exists_isDemushkin_range_eq_zpowers_neg_one_of_odd","parent":4075,"result":true,"route":"../roadmap/tauceti/topology/profinite-and-pro-p-groups/part-01/index.html#the-existence-theorem-of-the-classification-of-d","scope":"tauceti/topology/profinite-and-pro-p-groups/part-01","status":"fully_proved","summary":"There is a Demushkin group of rank `1`, presented on one generator by the single relator `x\u2081\u00b2`, namely `\u2124/2`, which has exactly one continuous character with the prescription property, and that character has image `{\u00b11}`. This is the instance `n = 1` of the odd-rank statement at level `f = \u221e`, whose relator word reads `x\u2081\u00b2` at rank one.","title":"The existence theorem of the classification of Demushkin groups, rank one (Labute, Remark 2 (iii); NSW (3.9.10))"},{"id":"tauceti/topology/profinite-and-pro-p-groups/part-01/labutes-criterion","kind":"node","lean":"TauCeti.isDemushkin_iff_nondegenerate_degreeOneForm, TauCeti.isDemushkin_of_nondegenerate_degreeOneForm","parent":4075,"result":true,"route":"../roadmap/tauceti/topology/profinite-and-pro-p-groups/part-01/index.html#labutes-criterion","scope":"tauceti/topology/profinite-and-pro-p-groups/part-01","status":"fully_proved","summary":"Let `F` be the free pro-`p` group on a finite type `X`, let `r \u2208 \u03a6(F)`, and let `G \u2245 \u27e8X \u2223 r\u27e9`. Then `G` is a Demushkin group exactly when `X` is nonempty and the degree-one form of the class of `r` in `gr_1(F)` is nondegenerate.","title":"Labute's criterion"},{"id":"tauceti/topology/profinite-and-pro-p-groups/part-01/labutes-theorem-2","kind":"node","lean":"TauCeti.freeProP.exists_continuousMulEquiv_apply_eq_of_range_demushkinCharacter_eq","parent":4075,"result":true,"route":"../roadmap/tauceti/topology/profinite-and-pro-p-groups/part-01/index.html#labutes-theorem-2","scope":"tauceti/topology/profinite-and-pro-p-groups/part-01","status":"fully_proved","summary":"Let `r, r'` be relators of the free pro-`p` group `F` on `n` generators whose presented groups are Demushkin groups whose canonical characters have the same image. Then a continuous automorphism of `F` carries `r` to `r'`; in particular the closed normal closures of `r` and `r'` are carried to each other, so two Demushkin relators with the same invariants differ by a change of basis of `F`. The relators lie in\u2026","title":"Labute's Theorem 2"},{"id":"tauceti/topology/profinite-and-pro-p-groups/part-01/labutes-theorem-2-for-q-2","kind":"node","lean":"TauCeti.freeProP.exists_continuousMulEquiv_apply_eq_of_demushkinQ_eq","parent":4075,"result":true,"route":"../roadmap/tauceti/topology/profinite-and-pro-p-groups/part-01/index.html#labutes-theorem-2-for-q-2","scope":"tauceti/topology/profinite-and-pro-p-groups/part-01","status":"fully_proved","summary":"Let `r, r' \u2208 \u03a6(F)` be relators of the free pro-`p` group on `n` generators whose presented groups are Demushkin groups with the same `q`-invariant `q \u2260 2`. Then a continuous automorphism of `F` carries `r` to `r'`; in particular the closed normal closures of `r` and `r'` are carried to each other.","title":"Labute's Theorem 2 for `q \u2260 2`"},{"id":"tauceti/topology/profinite-and-pro-p-groups/part-01/labutes-theorem-2-for-odd-rank","kind":"node","lean":"TauCeti.freeProP.exists_continuousMulEquiv_apply_eq_of_range_demushkinCharacter_eq_of_odd","parent":4075,"result":true,"route":"../roadmap/tauceti/topology/profinite-and-pro-p-groups/part-01/index.html#labutes-theorem-2-for-odd-rank","scope":"tauceti/topology/profinite-and-pro-p-groups/part-01","status":"fully_proved","summary":"Let `r, r' \u2208 \u03a6(F)` be relators of the free pro-`2` group on an odd number `n` of generators whose presented groups are Demushkin groups whose canonical characters have the same image. Then a continuous automorphism of `F` carries `r` to `r'`; in particular the closed normal closures of `r` and `r'` are carried to each other.","title":"Labute's Theorem 2 for odd rank"},{"id":"tauceti/topology/profinite-and-pro-p-groups/part-01/the-necessity-half-of-the-existence-theorem-for","kind":"node","lean":"TauCeti.IsDemushkin.profiniteIndex_subgroupOf_map_powMonoidHom_range_lt_of_demushkinQ_ne_two, TauCeti.IsDemushkin.profiniteIndex_subgroupOf_map_powMonoidHom_range_lt_of_even_demushkinRank, TauCeti.IsDemushkin.range_demushkinCharacter_eq_zpowers_neg_one_or_unitsPlusMinus_of_odd_demushkinRank, TauCeti.freeProP.range_demushkinCharacter_eq_zpowers_neg_one_or_unitsPlusMinus_of_odd","parent":4075,"result":true,"route":"../roadmap/tauceti/topology/profinite-and-pro-p-groups/part-01/index.html#the-necessity-half-of-the-existence-theorem-for","scope":"tauceti/topology/profinite-and-pro-p-groups/part-01","status":"fully_proved","summary":"For a Demushkin group `G` of rank `n` with `q(G) \u2260 2`, the image `A` of its canonical character has `(A : A^p) < p ^ n`. Indeed `A` is trivial, with `(A : A^p) = 1`, or the principal unit group `1 + q\u2124_p`, with `(A : A^p) = p`, while `n \u2265 2`. Together with the parity of the rank (`TauCeti.IsDemushkin.even_demushkinRank_of_demushkinQ_ne_two`), this places `(n, A)` in the first situation of the existence theorem.","title":"The necessity half of the existence theorem for `q \u2260 2` (Labute, Theorem 1)"},{"data_url":"scopes/tauceti/topology/profinite-and-pro-p-groups/part-02.json","id":"tauceti/topology/profinite-and-pro-p-groups/part-02","kind":"scope","navigation":"transparent","parent":4074,"result":false,"route":"../roadmap/tauceti/topology/profinite-and-pro-p-groups/part-02/index.html","scope":"tauceti/topology/profinite-and-pro-p-groups/part-02","status":"fully_proved","title":"The relative\u2026"},{"id":"tauceti/topology/profinite-and-pro-p-groups/part-02/the-relative-successive-approximation-theorem","kind":"node","lean":"TauCeti.freeProP.exists_continuousMulEquiv_apply_eq_of_exponentSum_eq","parent":4096,"result":true,"route":"../roadmap/tauceti/topology/profinite-and-pro-p-groups/part-02/index.html#the-relative-successive-approximation-theorem","scope":"tauceti/topology/profinite-and-pro-p-groups/part-02","status":"fully_proved","summary":"Let `r, w \u2208 \u03bb_1(F)` be relators of the free pro-`p` group `F` on a finite linearly ordered type with the same class `\u03c1 \u2208 gr_1(F)` and the same exponent vector `v = exponentSum r`, and suppose that for every `m \u2265 1` each element of `\u03bb_{m+1}(F)` lying in the closed commutator subgroup `K` has class `\u03b4_\u03c1(\u27e6\u03c9\u27e7)` for a family `\u03c9 : X \u2192 \u03bb_m(F)` with `\u03c9_i \u2208 K` at every generator `i` with `v_i \u2260 0`. Then a continuous\u2026","title":"The relative successive-approximation theorem"},{"data_url":"scopes/tauceti/topology/profinite-integers-profinite-powers-and-continuo.json","id":"tauceti/topology/profinite-integers-profinite-powers-and-continuo","kind":"scope","parent":4052,"result":false,"route":"../roadmap/tauceti/topology/profinite-integers-profinite-powers-and-continuo/index.html","scope":"tauceti/topology/profinite-integers-profinite-powers-and-continuo","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names, and the other results named in the modules where only this status names declarations, or in folders below the areas where only it names 2 or more, as Tau Ceti has them at commit ace60277.","title":"Profinite integers, profinite powers, and continuous automorphisms"},{"id":"tauceti/topology/profinite-integers-profinite-powers-and-continuo/the-continuous-automorphisms-of-a-pro-p-group-ac","kind":"node","lean":"TauCeti.IsProP.isProP_ker_mapQuotient_proPFrattini","parent":4098,"result":true,"route":"../roadmap/tauceti/topology/profinite-integers-profinite-powers-and-continuo/index.html#the-continuous-automorphisms-of-a-pro-p-group-ac","scope":"tauceti/topology/profinite-integers-profinite-powers-and-continuo","status":"fully_proved","summary":"For a compact pro-`p` group `G`, the kernel of `ContinuousAut G \u2192* MulAut (G \u29f8 proPFrattini p G)` is pro-`p` for the congruence topology.","title":"The continuous automorphisms of a pro-`p` group acting trivially on its Frattini quotient form a pro-`p` group"},{"id":"tauceti/topology/profinite-integers-profinite-powers-and-continuo/tauceti-continuousaut-isopen-ker-mapquotient-pro","kind":"node","lean":"TauCeti.ContinuousAut.isOpen_ker_mapQuotient_proPFrattini","parent":4098,"result":true,"route":"../roadmap/tauceti/topology/profinite-integers-profinite-powers-and-continuo/index.html#tauceti-continuousaut-isopen-ker-mapquotient-pro","scope":"tauceti/topology/profinite-integers-profinite-powers-and-continuo","status":"fully_proved","summary":"For a topologically finitely generated compact group, the continuous automorphisms acting trivially on the pro-`p` Frattini quotient form an open subgroup for the congruence topology: the pro-`p` Frattini subgroup is open and topologically characteristic.","title":"TauCeti.ContinuousAut.isOpen_ker_mapQuotient_proPFrattini"},{"id":"tauceti/topology/profinite-integers-profinite-powers-and-continuo/the-kernel-on-the-frattini-quotient-is-the-inver","kind":"node","lean":"TauCeti.IsProP.range_pi_mapQuotient_ker_proPFrattini","parent":4098,"result":true,"route":"../roadmap/tauceti/topology/profinite-integers-profinite-powers-and-continuo/index.html#the-kernel-on-the-frattini-quotient-is-the-inver","scope":"tauceti/topology/profinite-integers-profinite-powers-and-continuo","status":"fully_proved","summary":"For a topologically finitely generated pro-`p` group `G`, the coordinates on the quotients `G \u29f8 N` by the topologically characteristic open normal subgroups `N \u2264 proPFrattini p G` identify the kernel of `ContinuousAut G \u2192* MulAut (G \u29f8 proPFrattini p G)` with the families of automorphisms `\u03c3 N` of the finite `p`-groups `G \u29f8 N` that act trivially on the Frattini quotients of `G \u29f8 N` and are compatible with the\u2026","title":"The kernel on the Frattini quotient is the inverse limit of the finite kernels"},{"id":"tauceti/topology/profinite-integers-profinite-powers-and-continuo/gr-1-of-a-free-pro-p-group-is-the-exterior-squar","kind":"node","lean":"TauCeti.freeProP.exteriorSquareEquivLcsGradedPieceOne","parent":4098,"result":true,"route":"../roadmap/tauceti/topology/profinite-integers-profinite-powers-and-continuo/index.html#gr-1-of-a-free-pro-p-group-is-the-exterior-squar","scope":"tauceti/topology/profinite-integers-profinite-powers-and-continuo","status":"fully_proved","summary":"For the free pro-`p` group `F` on a finite type, the graded bracket induces an isomorphism of `\u2124_p`-modules `\u22c0\u00b2gr_0(F) \u2243 gr_1(F)`, `x \u2227 y \u21a6 [x, y]` (`exteriorSquareEquivLcsGradedPieceOne_\u03b9Multi`).","title":"`gr_1` of a free pro-`p` group is the exterior square of `gr_0`"},{"id":"tauceti/topology/profinite-integers-profinite-powers-and-continuo/the-z-p-basis-of-gr-1-of-a-free-pro-p-group-of-f","kind":"node","lean":"TauCeti.freeProP.lcsDegreeOneBasis","parent":4098,"result":true,"route":"../roadmap/tauceti/topology/profinite-integers-profinite-powers-and-continuo/index.html#the-z-p-basis-of-gr-1-of-a-free-pro-p-group-of-f","scope":"tauceti/topology/profinite-integers-profinite-powers-and-continuo","status":"fully_proved","summary":"The `\u2124_p`-basis of `gr_1` of a free pro-`p` group of finite rank, formed by the brackets `[x_i, x_j]` of the generator classes for `i < j` (`TauCeti.lcsGradedPiece_one_freeProP_bijective`).","title":"The `\u2124_p`-basis of `gr_1` of a free pro-`p` group of finite rank"},{"id":"tauceti/topology/profinite-integers-profinite-powers-and-continuo/tauceti-freeprop-finrank-lcsgradedpiece-one","kind":"node","lean":"TauCeti.freeProP.finrank_lcsGradedPiece_one","parent":4098,"result":true,"route":"../roadmap/tauceti/topology/profinite-integers-profinite-powers-and-continuo/index.html#tauceti-freeprop-finrank-lcsgradedpiece-one","scope":"tauceti/topology/profinite-integers-profinite-powers-and-continuo","status":"fully_proved","summary":"`gr_1` of the free pro-`p` group on `X` has rank `#X choose 2` over `\u2124_p`.","title":"TauCeti.freeProP.finrank_lcsGradedPiece_one"},{"id":"tauceti/topology/profinite-integers-profinite-powers-and-continuo/tauceti-outeraction","kind":"node","lean":"TauCeti.outerAction","parent":4098,"result":true,"route":"../roadmap/tauceti/topology/profinite-integers-profinite-powers-and-continuo/index.html#tauceti-outeraction","scope":"tauceti/topology/profinite-integers-profinite-powers-and-continuo","status":"fully_proved","summary":"The outer action of `E \u29f8 N` on a normal subgroup `N`: the class of `e` acts by the outer class of conjugation by `e` (`outerAction_mk`). It is well defined because conjugation by an element of `N` is inner.","title":"TauCeti.outerAction"},{"id":"tauceti/topology/profinite-integers-profinite-powers-and-continuo/tauceti-continuous-outeraction","kind":"node","lean":"TauCeti.continuous_outerAction","parent":4098,"result":true,"route":"../roadmap/tauceti/topology/profinite-integers-profinite-powers-and-continuo/index.html#tauceti-continuous-outeraction","scope":"tauceti/topology/profinite-integers-profinite-powers-and-continuo","status":"fully_proved","summary":"For compact `N`, the outer action `E \u29f8 N \u2192 ContinuousOut N` is continuous for the quotient topologies.","title":"TauCeti.continuous_outerAction"},{"id":"tauceti/topology/profinite-integers-profinite-powers-and-continuo/the-product-decomposition-of-the-profinite-integ","kind":"node","lean":"TauCeti.zHat.ringEquivPiPadicInt","parent":4098,"result":true,"route":"../roadmap/tauceti/topology/profinite-integers-profinite-powers-and-continuo/index.html#the-product-decomposition-of-the-profinite-integ","scope":"tauceti/topology/profinite-integers-profinite-powers-and-continuo","status":"fully_proved","summary":"The ring homomorphism from `Additive zHat` to `\u2200 \u2113 : Nat.Primes, \u2124_[\u2113]` with components the `\u2113`-adic components `zHat.component \u2113` is a ring isomorphism: injective by `zHat.ext_of_component` and surjective by `zHat.exists_forall_component_eq`. It is a homeomorphism (`zHat.continuous_ringEquivPiPadicInt`, `zHat.continuous_ringEquivPiPadicInt_symm`), so an isomorphism of topological rings.","title":"The product decomposition of the profinite integers"},{"id":"tauceti/topology/profinite-integers-profinite-powers-and-continuo/the-unit-group-decomposition-of-the-profinite-in","kind":"node","lean":"TauCeti.zHat.unitsEquivPiPadicInt","parent":4098,"result":true,"route":"../roadmap/tauceti/topology/profinite-integers-profinite-powers-and-continuo/index.html#the-unit-group-decomposition-of-the-profinite-in","scope":"tauceti/topology/profinite-integers-profinite-powers-and-continuo","status":"fully_proved","summary":"The product decomposition `Additive zHat \u2243+* \u2200 \u2113, \u2124_[\u2113]` restricts to an isomorphism of topological groups on units.","title":"The unit-group decomposition of the profinite integers"},{"id":"tauceti/topology/profinite-integers-profinite-powers-and-continuo/every-invertible-matrix-over-f-p-comes-from-a-co","kind":"node","lean":"TauCeti.freeProP.continuousAutToGL_surjective","parent":4098,"result":true,"route":"../roadmap/tauceti/topology/profinite-integers-profinite-powers-and-continuo/index.html#every-invertible-matrix-over-f-p-comes-from-a-co","scope":"tauceti/topology/profinite-integers-profinite-powers-and-continuo","status":"fully_proved","summary":"the automorphism of the Frattini quotient it defines lifts.","title":"Every invertible matrix over `\ud835\udd3d_p` comes from a continuous automorphism of the free pro-`p` group of finite rank"},{"id":"tauceti/topology/profinite-integers-profinite-powers-and-continuo/the-profinite-power","kind":"node","lean":"TauCeti.zpowHat","parent":4098,"result":true,"route":"../roadmap/tauceti/topology/profinite-integers-profinite-powers-and-continuo/index.html#the-profinite-power","scope":"tauceti/topology/profinite-integers-profinite-powers-and-continuo","status":"fully_proved","summary":"The profinite power `x ^\u1dbb a` of an element `x` of a profinite group by a profinite integer `a`: the value at `a`, read multiplicatively, of the continuous homomorphism `zHat.lift x` from the profinite integers to `G` that sends the generator to `x`.","title":"The profinite power"},{"id":"tauceti/topology/profinite-integers-profinite-powers-and-continuo/the-l-adic-comparison","kind":"node","lean":"TauCeti.zpowHat_eq_padicPow_component","parent":4098,"result":true,"route":"../roadmap/tauceti/topology/profinite-integers-profinite-powers-and-continuo/index.html#the-l-adic-comparison","scope":"tauceti/topology/profinite-integers-profinite-powers-and-continuo","status":"fully_proved","summary":"On a pro-`\u2113` group, the profinite power by `a` is Tau Ceti's `\u2113`-adic power by the `\u2113`-adic component of `a`.","title":"The `\u2113`-adic comparison"},{"id":"tauceti/topology/profinite-integers-profinite-powers-and-continuo/the-graded-lie-homomorphism-of-a-continuous-homo","kind":"node","lean":"TauCeti.gradedLieHom","parent":4098,"result":true,"route":"../roadmap/tauceti/topology/profinite-integers-profinite-powers-and-continuo/index.html#the-graded-lie-homomorphism-of-a-continuous-homo","scope":"tauceti/topology/profinite-integers-profinite-powers-and-continuo","status":"fully_proved","summary":"the graded maps `TauCeti.gradedMap p f hf k` of all degrees, assembled into a homomorphism of graded Lie algebras `\u2a01 k, gr_k(G) \u2192\u2097\u2045ZMod p\u2046 \u2a01 k, gr_k(H)`. Its value on homogeneous elements is `TauCeti.gradedLieHom_of`.","title":"The graded Lie homomorphism of a continuous homomorphism `f : G \u2192* H`"},{"id":"tauceti/topology/profinite-integers-profinite-powers-and-continuo/the-z-l-lie-algebra-of-a-pro-l-group","kind":"node","lean":"TauCeti.IsProP.gradedLieAlgebra","parent":4098,"result":true,"route":"../roadmap/tauceti/topology/profinite-integers-profinite-powers-and-continuo/index.html#the-z-l-lie-algebra-of-a-pro-l-group","scope":"tauceti/topology/profinite-integers-profinite-powers-and-continuo","status":"fully_proved","summary":"For a pro-`\u2113` group `G`, the graded Lie ring `\u2a01 k, gr_k(G)` of the lower `p`-series is a Lie algebra over `\u2124_\u2113`, for the `\u2124_\u2113`-module structure which in each degree is `TauCeti.IsProP.gradedPieceModule`: the graded bracket is `\u2124_\u2113`-bilinear. For `p = 0` this is the `\u2124_\u2113`-Lie algebra of the closed lower central series.","title":"The `\u2124_\u2113`-Lie algebra of a pro-`\u2113` group"},{"id":"tauceti/topology/profinite-integers-profinite-powers-and-continuo/tauceti-continuousaut-compactspace","kind":"node","lean":"TauCeti.ContinuousAut.compactSpace","parent":4098,"result":true,"route":"../roadmap/tauceti/topology/profinite-integers-profinite-powers-and-continuo/index.html#tauceti-continuousaut-compactspace","scope":"tauceti/topology/profinite-integers-profinite-powers-and-continuo","status":"fully_proved","summary":"For a topologically finitely generated profinite group, the congruence topology on the continuous automorphisms is compact: `ContinuousAut G` is a closed subspace of the product of the finite automorphism groups of the characteristic open quotients.","title":"TauCeti.ContinuousAut.compactSpace"},{"id":"tauceti/topology/profinite-integers-profinite-powers-and-continuo/tauceti-closedsubgroupconjclasses","kind":"node","lean":"TauCeti.ClosedSubgroupConjClasses","parent":4098,"result":true,"route":"../roadmap/tauceti/topology/profinite-integers-profinite-powers-and-continuo/index.html#tauceti-closedsubgroupconjclasses","scope":"tauceti/topology/profinite-integers-profinite-powers-and-continuo","status":"fully_proved","summary":"The closed subgroups of `G` up to conjugacy: the orbits of the conjugation action of `G` on `ClosedSubgroup G`.","title":"TauCeti.ClosedSubgroupConjClasses"},{"id":"tauceti/topology/profinite-integers-profinite-powers-and-continuo/the-jacobi-identity","kind":"node","lean":"TauCeti.gradedBracket_jacobi","parent":4098,"result":true,"route":"../roadmap/tauceti/topology/profinite-integers-profinite-powers-and-continuo/index.html#the-jacobi-identity","scope":"tauceti/topology/profinite-integers-profinite-powers-and-continuo","status":"fully_proved","summary":"The Jacobi identity, with the three terms transported to the degree `i + j + k + 2`.","title":"The Jacobi identity"},{"id":"tauceti/topology/profinite-integers-profinite-powers-and-continuo/the-pro-p-nielsen-schreier-theorem","kind":"node","lean":"TauCeti.freeProP.exists_convergesToOne_continuousMulEquiv_of_isClosed, TauCeti.freeProP.nonempty_continuousMulEquiv_freeProP_openSubgroup_index","parent":4098,"result":true,"route":"../roadmap/tauceti/topology/profinite-integers-profinite-powers-and-continuo/index.html#the-pro-p-nielsen-schreier-theorem","scope":"tauceti/topology/profinite-integers-profinite-powers-and-continuo","status":"fully_proved","summary":"A closed subgroup `U` of the free pro-`p` group on any type `X` is free pro-`p` on a pointed profinite space: some subset `s` of `U` converging to `1` has a topological isomorphism `F_p(insert 1 s, 1) \u2243\u209c* U` sending the generator attached to each point of `insert 1 s` to that point, so that `s` is a basis of `U` converging to `1`.","title":"The pro-`p` Nielsen\u2013Schreier theorem"},{"id":"tauceti/topology/profinite-integers-profinite-powers-and-continuo/the-schreier-index-formula-for-open-subgroups","kind":"node","lean":"TauCeti.freeProP.topologicalGeneratorRankNat_openSubgroup, TauCeti.freeProP.topologicalGeneratorRankNat_add_index","parent":4098,"result":true,"route":"../roadmap/tauceti/topology/profinite-integers-profinite-powers-and-continuo/index.html#the-schreier-index-formula-for-open-subgroups","scope":"tauceti/topology/profinite-integers-profinite-powers-and-continuo","status":"fully_proved","summary":"For an open subgroup `U` of index `m` in the free pro-`p` group on a nonempty finite type `X` of cardinality `n`, `d(U) = 1 + m * (n - 1)`.","title":"The Schreier index formula for open subgroups"},{"data_url":"scopes/tauceti/topology/restricted-products-of-topological-groups-and-ra.json","id":"tauceti/topology/restricted-products-of-topological-groups-and-ra","kind":"scope","parent":4052,"result":false,"route":"../roadmap/tauceti/topology/restricted-products-of-topological-groups-and-ra/index.html","scope":"tauceti/topology/restricted-products-of-topological-groups-and-ra","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Restricted products of topological groups and rational diagonals"},{"id":"tauceti/topology/restricted-products-of-topological-groups-and-ra/tauceti-awaydecomposition","kind":"node","lean":"TauCeti.awayDecomposition","parent":4119,"result":true,"route":"../roadmap/tauceti/topology/restricted-products-of-topological-groups-and-ra/index.html#tauceti-awaydecomposition","scope":"tauceti/topology/restricted-products-of-topological-groups-and-ra","status":"fully_proved","summary":"For a finite set `S` of indices, the decomposition of a restricted product as the plain product of the factors at `S` times the restricted product away from `S`, coordinatewise the identity in both directions: `x` goes to `((x i)_{i \u2208 S}, (x i)_{i \u2209 S})`, and `(y, z)` goes to the element with coordinate `y \u27e8i, _\u27e9` at `i \u2208 S` and `z \u27e8i, _\u27e9` at `i \u2209 S`.","title":"TauCeti.awayDecomposition"},{"id":"tauceti/topology/restricted-products-of-topological-groups-and-ra/tauceti-continuous-awaydecomposition-symm","kind":"node","lean":"TauCeti.continuous_awayDecomposition_symm","parent":4119,"result":true,"route":"../roadmap/tauceti/topology/restricted-products-of-topological-groups-and-ra/index.html#tauceti-continuous-awaydecomposition-symm","scope":"tauceti/topology/restricted-products-of-topological-groups-and-ra","status":"fully_proved","summary":"The inverse of the decomposition is continuous when the reference subgroups away from `S` are open; the reference subgroups at `S` play no role, since the finite factor carries no integrality condition. Together with `continuous_awayDecomposition` this makes the decomposition a homeomorphism for such families.","title":"TauCeti.continuous_awayDecomposition_symm"},{"id":"tauceti/topology/restricted-products-of-topological-groups-and-ra/tauceti-not-istopologicalgroup-restrictedproduct","kind":"node","lean":"TauCeti.not_isTopologicalGroup_restrictedProduct_rat_bot","parent":4119,"result":true,"route":"../roadmap/tauceti/topology/restricted-products-of-topological-groups-and-ra/index.html#tauceti-not-istopologicalgroup-restrictedproduct","scope":"tauceti/topology/restricted-products-of-topological-groups-and-ra","status":"fully_proved","summary":"The restricted product `\u03a0\u02b3 n : \u2115, [Multiplicative \u211a, \u22a5]` is not a topological group: the hypothesis `Fact (\u2200 i, IsOpen (A i))` of Mathlib's `RestrictedProduct.isTopologicalGroup` cannot be dropped.","title":"TauCeti.not_isTopologicalGroup_restrictedProduct_rat_bot"},{"id":"tauceti/topology/restricted-products-of-topological-groups-and-ra/tauceti-not-continuous-restrictedproductsum-symm","kind":"node","lean":"TauCeti.not_continuous_restrictedProductSum_symm","parent":4119,"result":true,"route":"../roadmap/tauceti/topology/restricted-products-of-topological-groups-and-ra/index.html#tauceti-not-continuous-restrictedproductsum-symm","scope":"tauceti/topology/restricted-products-of-topological-groups-and-ra","status":"fully_proved","summary":"The inverse of the splitting `restrictedProductSum` of a restricted product over a sum of index types is not continuous in general: for `\u03a0\u02b3 k : \u2115 \u2295 \u2115, [Multiplicative \u211a, \u22a5]`, with the trivial, non-open, reference subgroup at every index, it is a discontinuous bijection. The openness hypothesis of `continuous_restrictedProductSum_symm` therefore cannot be dropped.","title":"TauCeti.not_continuous_restrictedProductSum_symm"},{"id":"tauceti/topology/restricted-products-of-topological-groups-and-ra/tauceti-restrictedproductsum","kind":"node","lean":"TauCeti.restrictedProductSum","parent":4119,"result":true,"route":"../roadmap/tauceti/topology/restricted-products-of-topological-groups-and-ra/index.html#tauceti-restrictedproductsum","scope":"tauceti/topology/restricted-products-of-topological-groups-and-ra","status":"fully_proved","summary":"A restricted product over `\u03b9\u2081 \u2295 \u03b9\u2082` with respect to a filter `\ud835\udcd5` is the product of the restricted products over the two summands with respect to the comaps of `\ud835\udcd5` along `Sum.inl` and `Sum.inr`, coordinatewise the identity in both directions.","title":"TauCeti.restrictedProductSum"},{"id":"tauceti/topology/restricted-products-of-topological-groups-and-ra/tauceti-continuous-restrictedproductsum-symm-of","kind":"node","lean":"TauCeti.continuous_restrictedProductSum_symm_of_principal","parent":4119,"result":true,"route":"../roadmap/tauceti/topology/restricted-products-of-topological-groups-and-ra/index.html#tauceti-continuous-restrictedproductsum-symm-of","scope":"tauceti/topology/restricted-products-of-topological-groups-and-ra","status":"fully_proved","summary":"For a principal filter on `\u03b9\u2081 \u2295 \u03b9\u2082` the inverse of the splitting is continuous for every reference family, so that the splitting is a homeomorphism: the target is a principal stage, whose topology is induced from the product.","title":"TauCeti.continuous_restrictedProductSum_symm_of_principal"},{"id":"tauceti/topology/restricted-products-of-topological-groups-and-ra/tauceti-continuous-rationaldiagonal","kind":"node","lean":"TauCeti.continuous_rationalDiagonal","parent":4119,"result":true,"route":"../roadmap/tauceti/topology/restricted-products-of-topological-groups-and-ra/index.html#tauceti-continuous-rationaldiagonal","scope":"tauceti/topology/restricted-products-of-topological-groups-and-ra","status":"fully_proved","summary":"The diagonal is continuous when the coordinate maps are continuous and there is one cofinite set of indices at which every `\u03b3` is integral. Pointwise eventual integrality alone builds the map but is not enough for continuity, because the restricted-product topology is finer than the topology induced from the full product.","title":"TauCeti.continuous_rationalDiagonal"},{"id":"tauceti/topology/restricted-products-of-topological-groups-and-ra/tauceti-continuous-eval-and-not-continuous-ratio","kind":"node","lean":"TauCeti.continuous_eval_and_not_continuous_rationalDiagonal_range_coeMonoidHom","parent":4119,"result":true,"route":"../roadmap/tauceti/topology/restricted-products-of-topological-groups-and-ra/index.html#tauceti-continuous-eval-and-not-continuous-ratio","scope":"tauceti/topology/restricted-products-of-topological-groups-and-ra","status":"fully_proved","summary":"The uniform integrality set in `continuous_rationalDiagonal` cannot be dropped. Take discrete groups with trivial reference subgroups, infinitely many of them nontrivial, and for `\u0393` the range of the coercion from the restricted product to the full product, that is, the finitely supported elements, with the topology induced from the full product. The coordinate maps are continuous and every element of `\u0393` is\u2026","title":"TauCeti.continuous_eval_and_not_continuous_rationalDiagonal_range_coeMonoidHom"},{"id":"tauceti/topology/restricted-products-of-topological-groups-and-ra/tauceti-restrictedproductmap-surjective-iff","kind":"node","lean":"TauCeti.restrictedProductMap_surjective_iff","parent":4119,"result":true,"route":"../roadmap/tauceti/topology/restricted-products-of-topological-groups-and-ra/index.html#tauceti-restrictedproductmap-surjective-iff","scope":"tauceti/topology/restricted-products-of-topological-groups-and-ra","status":"fully_proved","summary":"A componentwise restricted-product homomorphism is surjective exactly when every coordinate map is surjective and, at all but finitely many indices, the coordinate map carries the source reference subgroup onto the target one. Surjectivity of each coordinate map alone is not enough: an element of the target may lie in the target reference subgroups at infinitely many indices where no preimage lies in the source\u2026","title":"TauCeti.restrictedProductMap_surjective_iff"},{"id":"tauceti/topology/restricted-products-of-topological-groups-and-ra/tauceti-restrictedproductmap","kind":"node","lean":"TauCeti.restrictedProductMap","parent":4119,"result":true,"route":"../roadmap/tauceti/topology/restricted-products-of-topological-groups-and-ra/index.html#tauceti-restrictedproductmap","scope":"tauceti/topology/restricted-products-of-topological-groups-and-ra","status":"fully_proved","summary":"The componentwise homomorphism of restricted products induced by a family that eventually maps each reference subgroup into the corresponding target subgroup.","title":"TauCeti.restrictedProductMap"},{"id":"tauceti/topology/restricted-products-of-topological-groups-and-ra/tauceti-restrictedproductcongr","kind":"node","lean":"TauCeti.restrictedProductCongr","parent":4119,"result":true,"route":"../roadmap/tauceti/topology/restricted-products-of-topological-groups-and-ra/index.html#tauceti-restrictedproductcongr","scope":"tauceti/topology/restricted-products-of-topological-groups-and-ra","status":"fully_proved","summary":"The multiplicative equivalence between the restricted products with respect to two reference families that agree at all but finitely many indices. It is the identity in every coordinate.","title":"TauCeti.restrictedProductCongr"},{"id":"tauceti/topology/restricted-products-of-topological-groups-and-ra/tauceti-doublecosetcongr","kind":"node","lean":"TauCeti.doubleCosetCongr","parent":4119,"result":true,"route":"../roadmap/tauceti/topology/restricted-products-of-topological-groups-and-ra/index.html#tauceti-doublecosetcongr","scope":"tauceti/topology/restricted-products-of-topological-groups-and-ra","status":"fully_proved","summary":"Transport of a double-coset space along a change of reference family. The subgroups on the right are the images of those on the left under `restrictedProductCongr`, not the integral subgroups of the new family; see `exists_map_integralSubgroup_ne`.","title":"TauCeti.doubleCosetCongr"},{"id":"tauceti/topology/restricted-products-of-topological-groups-and-ra/tauceti-addawaydecomposition","kind":"node","lean":"TauCeti.addAwayDecomposition","parent":4119,"result":true,"route":"../roadmap/tauceti/topology/restricted-products-of-topological-groups-and-ra/index.html#tauceti-addawaydecomposition","scope":"tauceti/topology/restricted-products-of-topological-groups-and-ra","status":"fully_proved","summary":"For a finite set `S` of indices, the decomposition of a restricted product of additive groups as the plain product of the factors at `S` times the restricted product away from `S`, coordinatewise the identity in both directions: `x` goes to `((x i)_{i \u2208 S}, (x i)_{i \u2209 S})`, and `(y, z)` goes to the element with coordinate `y \u27e8i, _\u27e9` at `i \u2208 S` and `z \u27e8i, _\u27e9` at `i \u2209 S`.","title":"TauCeti.addAwayDecomposition"},{"id":"tauceti/topology/restricted-products-of-topological-groups-and-ra/tauceti-addrationaldiagonal","kind":"node","lean":"TauCeti.addRationalDiagonal","parent":4119,"result":true,"route":"../roadmap/tauceti/topology/restricted-products-of-topological-groups-and-ra/index.html#tauceti-addrationaldiagonal","scope":"tauceti/topology/restricted-products-of-topological-groups-and-ra","status":"fully_proved","summary":"The diagonal additive homomorphism into a restricted product induced by a family of additive homomorphisms whose values are eventually in the reference subgroups. The eventual-integrality evidence `h` is an argument, not a consequence of the construction.","title":"TauCeti.addRationalDiagonal"},{"data_url":"scopes/tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun.json","id":"tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun","kind":"scope","parent":4052,"result":false,"route":"../roadmap/tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun/index.html","scope":"tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun","status":"fully_proved","summary":"The declarations the status of Tau Ceti's roadmap names as Tau Ceti has them at commit ace60277.","title":"Sato hyperfunctions and analytic generalized functions"},{"id":"tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun/support-truncation","kind":"node","lean":"CategoryTheory.Sheaf.subsingleton_supportedH_of_isZero_supportedHSheaf","parent":4134,"result":true,"route":"../roadmap/tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun/index.html#support-truncation","scope":"tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun","status":"fully_proved","summary":"If the local cohomology sheaves `\ud835\udcd7_S^q(F)` vanish for all `q < n`, then the cohomology groups `H_S^q(W, F)` with supports in `S` vanish for all `q < n` and every open `W`.","title":"Support truncation"},{"id":"tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun/topcat-sheaf-map-surjective-of-locally-flasque","kind":"node","lean":"TopCat.Sheaf.map_surjective_of_locally_flasque","parent":4134,"result":true,"route":"../roadmap/tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun/index.html#topcat-sheaf-map-surjective-of-locally-flasque","scope":"tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun","status":"fully_proved","summary":"If every restriction map of a sheaf inside some neighbourhood of each point is surjective, then every restriction map is surjective.","title":"TopCat.Sheaf.map_surjective_of_locally_flasque"},{"id":"tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun/categorytheory-sheaf-supportedhexcisionequiv","kind":"node","lean":"CategoryTheory.Sheaf.supportedHExcisionEquiv","parent":4134,"result":true,"route":"../roadmap/tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun/index.html#categorytheory-sheaf-supportedhexcisionequiv","scope":"tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun","status":"fully_proved","summary":"Excision for cohomology with supports: if `W' \u2264 W` and `S \u2229 W \u2286 W'` (that is, `W \u2264 W' \u2294 (X \u2216 S)`), restriction is an isomorphism `H_S\u207f(W, F) \u2243+ H_S\u207f(W', F)`.","title":"CategoryTheory.Sheaf.supportedHExcisionEquiv"},{"id":"tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun/iscompact-ispolynomiallyconvex-of-forall-im-eq-z","kind":"node","lean":"IsCompact.isPolynomiallyConvex_of_forall_im_eq_zero","parent":4134,"result":true,"route":"../roadmap/tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun/index.html#iscompact-ispolynomiallyconvex-of-forall-im-eq-z","scope":"tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun","status":"fully_proved","summary":"Every compact subset of the real locus `\u211d\u207f \u2286 \u2102\u207f` is polynomially convex.","title":"IsCompact.isPolynomiallyConvex_of_forall_im_eq_zero"},{"id":"tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun/tauceti-ispolynomiallyconvex-exists-subset-inter","kind":"node","lean":"TauCeti.IsPolynomiallyConvex.exists_subset_interior_polynomialPolyhedron","parent":4134,"result":true,"route":"../roadmap/tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun/index.html#tauceti-ispolynomiallyconvex-exists-subset-inter","scope":"tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun","status":"fully_proved","summary":"A compact polynomially convex set `K \u2286 \u2102\u207f` lies in the interior of a polynomial polyhedron contained in any given open neighbourhood `U` of `K`.","title":"TauCeti.IsPolynomiallyConvex.exists_subset_interior_polynomialPolyhedron"},{"id":"tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun/the-presheaf-of-one-dimensional-hyperfunctions-i","kind":"node","lean":"TauCeti.Hyperfunction.oneDimBoundaryPresheaf_isSeparated","parent":4134,"result":true,"route":"../roadmap/tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun/index.html#the-presheaf-of-one-dimensional-hyperfunctions-i","scope":"tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun","status":"fully_proved","summary":"a section that vanishes on the members of an open cover vanishes.","title":"The presheaf of one-dimensional hyperfunctions is separated"},{"id":"tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun/categorytheory-sheaf-supportedh","kind":"node","lean":"CategoryTheory.Sheaf.supportedH","parent":4134,"result":true,"route":"../roadmap/tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun/index.html#categorytheory-sheaf-supportedh","scope":"tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun","status":"fully_proved","summary":"The cohomology `H_S\u207f(W, F)` of `F` on `W` with supports in the closed subset `S`: the relative cohomology of the inclusion `W \u2216 S \u27f6 W`.","title":"CategoryTheory.Sheaf.supportedH"},{"id":"tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun/categorytheory-sheaf-supportedhsheaf","kind":"node","lean":"CategoryTheory.Sheaf.supportedHSheaf","parent":4134,"result":true,"route":"../roadmap/tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun/index.html#categorytheory-sheaf-supportedhsheaf","scope":"tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun","status":"fully_proved","summary":"The local cohomology sheaf `\ud835\udcd7_S\u207f(F)` of `F` with supports in the closed subset `S`: the sheafification of the presheaf `W \u21a6 H_S\u207f(W, F)` of cohomology with supports in `S`.","title":"CategoryTheory.Sheaf.supportedHSheaf"},{"id":"tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun/tauceti-hyperfunction-onedimboundaryquotient","kind":"node","lean":"TauCeti.Hyperfunction.OneDimBoundaryQuotient","parent":4134,"result":true,"route":"../roadmap/tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun/index.html#tauceti-hyperfunction-onedimboundaryquotient","scope":"tauceti/topology/sato-hyperfunctions-and-analytic-generalized-fun","status":"fully_proved","summary":"The space `O(\u2102 \u2216 \u211d) / O(W(U))` of one-dimensional hyperfunctions on an open set `U \u2286 \u211d`, represented by holomorphic functions on `\u2102 \u2216 \u211d` modulo those extending holomorphically across `U`.","title":"TauCeti.Hyperfunction.OneDimBoundaryQuotient"},{"data_url":"scopes/tauceti/topology/algebra.json","id":"tauceti/topology/algebra","kind":"scope","parent":4052,"result":false,"route":"../roadmap/tauceti/topology/algebra/index.html","scope":"tauceti/topology/algebra","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Topology.Algebra` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Algebra"},{"data_url":"scopes/tauceti/topology/algebra/part-01.json","id":"tauceti/topology/algebra/part-01","kind":"scope","navigation":"transparent","parent":4144,"result":false,"route":"../roadmap/tauceti/topology/algebra/part-01/index.html","scope":"tauceti/topology/algebra/part-01","status":"fully_proved","title":"The abelianization structure\u2026 to Structure theorem for\u2026"},{"id":"tauceti/topology/algebra/part-01/the-abelianization-structure-theorem-for-one-rel","kind":"node","lean":"TauCeti.presentedProP.oneRelatorAbelianizationEquiv, TauCeti.IsDemushkin.nonempty_continuousMulEquiv_topologicalAbelianization","parent":4145,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-01/index.html#the-abelianization-structure-theorem-for-one-rel","scope":"tauceti/topology/algebra/part-01","status":"fully_proved","summary":"Let `G = presentedProP p X {r}` be the pro-`p` group on a finite type `X` with the single relator `r`, and write the exponent vector of `r` as `exponentSum r = q \u2022 w` with `w x\u2080 = 1`. Then","title":"The abelianization structure theorem for one-relator pro-`p` groups"},{"id":"tauceti/topology/algebra/part-01/burnsides-basis-theorem-cardinal-form","kind":"node","lean":"TauCeti.IsProP.topologicalGeneratorRank_eq_rank_continuousZModDual, TauCeti.IsProP.rank_H1_eq_topologicalGeneratorRank, TauCeti.IsProP.finrank_H1_eq_topologicalGeneratorRankNat, TauCeti.IsProP.topologicalClosure_normalClosure_eq_iff_quotient_pLowerCentralStep, TauCeti.topologicalGeneratorRank_le_rank_continuousZModDual, TauCeti.topologicallyGenerates_iff_frattiniQuotient_span_topologicalClosure_eq_top, TauCeti.topologicallyGenerates_iff_frattiniQuotient, TauCeti.IsProP.finrank_continuousZModDual_eq_topologicalGeneratorRankNat, TauCeti.topologicallyGenerates_iff_frattiniQuotient_span_eq_top","parent":4145,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-01/index.html#burnsides-basis-theorem-cardinal-form","scope":"tauceti/topology/algebra/part-01","status":"fully_proved","summary":"The topological generator rank of a profinite pro-`p` group is the dimension of its continuous `\ud835\udd3d_p`-dual over `\ud835\udd3d_p`. No finiteness hypothesis is needed, and the statement is an identity of cardinals.","title":"Burnside's basis theorem, cardinal form"},{"id":"tauceti/topology/algebra/part-01/burnsides-criterion-modulo-1","kind":"node","lean":"TauCeti.IsProP.surjective_of_forall_inv_mul_mem_pLowerCentralSeries_one","parent":4145,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-01/index.html#burnsides-criterion-modulo-1","scope":"tauceti/topology/algebra/part-01","status":"fully_proved","summary":"A continuous endomorphism of a pro-`p` group congruent to the identity modulo `\u03bb_1 = \u03a6` is surjective.","title":"Burnside's criterion modulo `\u03bb_1`"},{"id":"tauceti/topology/algebra/part-01/burnsides-surjectivity-criterion","kind":"node","lean":"TauCeti.IsProP.surjective_iff_surjective_quotient_index_eq","parent":4145,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-01/index.html#burnsides-surjectivity-criterion","scope":"tauceti/topology/algebra/part-01","status":"fully_proved","summary":"A continuous homomorphism from a compact group to a profinite pro-`p` group is surjective exactly when its composites with all index-`p` quotient maps are surjective.","title":"Burnside's surjectivity criterion"},{"id":"tauceti/topology/algebra/part-01/the-existence-theorem-of-the-classification-for","kind":"node","lean":"TauCeti.exists_isDemushkin_range_demushkinCharacter_eq_iff_of_odd","parent":4145,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-01/index.html#the-existence-theorem-of-the-classification-for","scope":"tauceti/topology/algebra/part-01","status":"fully_proved","summary":"Let `n` be odd and let `A \u2264 \u2124\u2082\u02e3` be a subgroup. Then `(n, A)` is the pair of invariants of a Demushkin group, presented on `n` generators by one relator, exactly when `A = {\u00b11}`, or `n \u2265 3` and `A = {\u00b11} \u00d7 U^(f)` for some `f \u2265 2`. At `n = 1` the only admissible image is `{\u00b11}`, the image of the canonical character of `\u2124/2`. Every such `A` is closed and pro-`2`, so no hypothesis on `A` is needed.","title":"The existence theorem of the classification for odd rank, as an equivalence (Labute, Theorem 1 and Remark 2; Serre, Theorem 3.2)"},{"id":"tauceti/topology/algebra/part-01/the-existence-theorem-of-the-classification-of-d","kind":"node","lean":"TauCeti.exists_isDemushkin_range_demushkinCharacter_eq_iff_of_even","parent":4145,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-01/index.html#the-existence-theorem-of-the-classification-of-d","scope":"tauceti/topology/algebra/part-01","status":"fully_proved","summary":"Let `n` be even and let `A \u2264 \u2124_p\u02e3` be a closed pro-`p` subgroup. Then `(n, A)` is the pair of invariants `(rank, Im \u03c7)` of a Demushkin group, presented on `n` generators by one relator, exactly when `(A : A^p) < p ^ n` as supernatural numbers. At `p = 2` the condition excludes exactly `n = 0` and the non-procyclic images `{\u00b11} \u00d7 U^(f)` in rank two; at an odd prime it excludes exactly `n = 0`. Together with\u2026","title":"The existence theorem of the classification of Demushkin groups for even rank, as an equivalence (Labute, Theorem 1 and Remark 2; Serre, Theorem 3.2)"},{"id":"tauceti/topology/algebra/part-01/the-first-isomorphism-theorem-for-a-surjective-s","kind":"node","lean":"ContinuousMulEquiv.quotientKerEquivOfSurjective, ContinuousAddEquiv.quotientKerEquivOfSurjective","parent":4145,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-01/index.html#the-first-isomorphism-theorem-for-a-surjective-s","scope":"tauceti/topology/algebra/part-01","status":"fully_proved","summary":"`G \u29f8 f.ker \u2243\u209c* H`.","title":"The first isomorphism theorem for a surjective strict homomorphism of topological groups"},{"id":"tauceti/topology/algebra/part-01/the-frattini-criterion-for-a-kernel","kind":"node","lean":"ContinuousMonoidHom.ker_le_proPFrattini_of_forall_exists_comp_eq","parent":4145,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-01/index.html#the-frattini-criterion-for-a-kernel","scope":"tauceti/topology/algebra/part-01","status":"fully_proved","summary":"If every continuous `\ud835\udd3d_p`-valued character of `G` factors through a continuous homomorphism `\u03c6 : G \u2192 H`, then the kernel of `\u03c6` lies in the pro-`p` Frattini subgroup of `G`.","title":"The Frattini criterion for a kernel"},{"id":"tauceti/topology/algebra/part-01/gaschutzs-lifting-lemma-for-profinite-groups","kind":"node","lean":"TauCeti.exists_comp_eq_and_topologicalClosure_closure_range_eq_top, TauCeti.exists_comp_eq_and_topologicalClosure_closure_range_eq_top_of_topologicalGeneratorRankNat_le","parent":4145,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-01/index.html#gaschutzs-lifting-lemma-for-profinite-groups","scope":"tauceti/topology/algebra/part-01","status":"fully_proved","summary":"Along a continuous surjection `f : G \u2192* H` from a profinite group onto a Hausdorff group, an `\u03b9`-tuple that topologically generates `H` lifts to an `\u03b9`-tuple that topologically generates `G`, provided `G` is topologically generated by some `\u03b9`-tuple, with `\u03b9` finite.","title":"Gasch\u00fctz's lifting lemma for profinite groups"},{"id":"tauceti/topology/algebra/part-01/henkels-open-mapping-theorem","kind":"node","lean":"TauCeti.HasZeroSequenceOfUnits.isOpenMap","parent":4145,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-01/index.html#henkels-open-mapping-theorem","scope":"tauceti/topology/algebra/part-01","status":"fully_proved","summary":"A surjective `A`-equivariant additive map from a complete first-countable nonarchimedean group to a Baire space is open, when `A` has a zero sequence of units.","title":"Henkel's open mapping theorem"},{"id":"tauceti/topology/algebra/part-01/henkels-theorem-in-quotient-form","kind":"node","lean":"TauCeti.HasZeroSequenceOfUnits.isQuotientMap","parent":4145,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-01/index.html#henkels-theorem-in-quotient-form","scope":"tauceti/topology/algebra/part-01","status":"fully_proved","summary":"Under exactly the hypotheses of `TauCeti.HasZeroSequenceOfUnits.isOpenMap`, the map does not merely carry open sets to open sets: the topology of `N` is the one coinduced from `M`, so a map out of `N` is continuous exactly when its composite with `f` is.","title":"Henkel's theorem in quotient form"},{"id":"tauceti/topology/algebra/part-01/labutes-theorem-2-at-the-even-rank-dyadic-endpoi","kind":"node","lean":"TauCeti.freeProP.exists_continuousMulEquiv_apply_eq_of_range_eq_zpowers_neg_one","parent":4145,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-01/index.html#labutes-theorem-2-at-the-even-rank-dyadic-endpoi","scope":"tauceti/topology/algebra/part-01","status":"fully_proved","summary":"Let `r, r' \u2208 \u03a6(F)` be relators of the free pro-`2` group on an even number `n` of generators presenting Demushkin groups whose canonical characters both have image `{\u00b11}`. Then a continuous automorphism of `F` carries `r` to `r'`; in particular the closed normal closures of `r` and `r'` are carried to each other.","title":"Labute's Theorem 2 at the even-rank dyadic endpoint"},{"id":"tauceti/topology/algebra/part-01/labutes-theorem-4-a-demushkin-group-has-exactly","kind":"node","lean":"TauCeti.IsDemushkin.existsUnique_hasPrescriptionProperty","parent":4145,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-01/index.html#labutes-theorem-4-a-demushkin-group-has-exactly","scope":"tauceti/topology/algebra/part-01","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.IsDemushkin.existsUnique_hasPrescriptionProperty`.","title":"Labute's Theorem 4: a Demushkin group has exactly one continuous character with the prescription property"},{"id":"tauceti/topology/algebra/part-01/lagranges-theorem-for-profinite-groups","kind":"node","lean":"Subgroup.profiniteOrder_eq_mul_profiniteIndex","parent":4145,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-01/index.html#lagranges-theorem-for-profinite-groups","scope":"tauceti/topology/algebra/part-01","status":"fully_proved","summary":"the supernatural order of the ambient group is the order of a closed subgroup times its supernatural index.","title":"Lagrange's theorem for profinite groups"},{"id":"tauceti/topology/algebra/part-01/lyndons-theorem-for-the-relation-module-nsw-5-6","kind":"node","lean":"TauCeti.freeProfiniteGroup.abelianizationProPEquivRelationModule","parent":4145,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-01/index.html#lyndons-theorem-for-the-relation-module-nsw-5-6","scope":"tauceti/topology/algebra/part-01","status":"fully_proved","summary":"For a finite group `G` generated by the family `g : \u03b9 \u2192 G`, the maximal abelian pro-`p` quotient `R^ab(p)` of the kernel `R` of `lift g : F \u2192 G` is isomorphic to the relation module `relationModule \u2124_[p] G g`. The isomorphism sends the class of `r \u2208 R` to its Fox derivative (`coe_abelianizationProPEquivRelationModule_ofMul`), it is `\u2124_p`-linear (`abelianizationProPEquivRelationModule_padicPow`), and it is\u2026","title":"Lyndon's theorem for the relation module (NSW (5.6.6))"},{"id":"tauceti/topology/algebra/part-01/macbeaths-theorem","kind":"node","lean":"TauCeti.macbeathEquiv, TauCeti.ker_lift_eq_normalClosure_macbeathRels, TauCeti.ker_lift_eq_normalClosure_macbeathRels_of_locallyFinite, TauCeti.macbeathHom_injective_of_locallyFinite","parent":4145,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-01/index.html#macbeaths-theorem","scope":"tauceti/topology/algebra/part-01","status":"fully_proved","summary":"If the translates of a path-connected open set `U` cover a simply connected space `X`, then `G` is presented by the Macbeath generators and relations of `U`.","title":"Macbeath's theorem"},{"id":"tauceti/topology/algebra/part-01/nakayamas-lemma-for-pro-p-groups","kind":"node","lean":"TauCeti.IsProP.eq_bot_of_le_pLowerCentralStep, TauCeti.IsProP.le_of_le_topologicalClosure_sup_pLowerCentralStep","parent":4145,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-01/index.html#nakayamas-lemma-for-pro-p-groups","scope":"tauceti/topology/algebra/part-01","status":"fully_proved","summary":"In a profinite pro-`p` group a subgroup `K` with `K \u2264 closure (K\u1d56 \u2b1d [K, G])` is trivial: it lies in every term of the lower `p`-series.","title":"Nakayama's lemma for pro-`p` groups"},{"id":"tauceti/topology/algebra/part-01/the-open-frattini-criterion","kind":"node","lean":"TauCeti.IsProP.isTopologicallyFinitelyGenerated_iff_isOpen_proPFrattini","parent":4145,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-01/index.html#the-open-frattini-criterion","scope":"tauceti/topology/algebra/part-01","status":"fully_proved","summary":"A profinite pro-`p` group is topologically finitely generated exactly when its Frattini subgroup is open.","title":"The open Frattini criterion"},{"id":"tauceti/topology/algebra/part-01/the-schreier-bound","kind":"node","lean":"TauCeti.topologicalGeneratorRankNat_le_of_openSubgroup_of_finiteIndex, TauCeti.topologicalGeneratorRankNat_le_of_openSubgroup","parent":4145,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-01/index.html#the-schreier-bound","scope":"tauceti/topology/algebra/part-01","status":"fully_proved","summary":"Let `U` be an open subgroup of finite index in a topologically finitely generated topological group `G`. Then `U` is topologically finitely generated (`TauCeti.IsTopologicallyFinitelyGenerated.of_openSubgroup_of_finiteIndex`), and `d(U) \u2264 1 + [G : U] * (d(G) - 1)`, with subtraction in `\u2115`. This is the topological form of Schreier's index formula `Subgroup.rank_le_one_add_index_mul_rank_sub_one`. The bound is sharp:\u2026","title":"The Schreier bound"},{"id":"tauceti/topology/algebra/part-01/structure-theorem-for-topologically-finitely-gen","kind":"node","lean":"TauCeti.IsProP.exists_continuousMulEquiv_pi_padicInt_prod_pi_zmod, TauCeti.IsProP.exists_continuousLinearEquiv_pi_padicInt_prod_torsion","parent":4145,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-01/index.html#structure-theorem-for-topologically-finitely-gen","scope":"tauceti/topology/algebra/part-01","status":"fully_proved","summary":"Such a group is topologically isomorphic to `\u2124_p ^ r \u00d7 \u220f i : Fin m, \u2124/p^(e i)` with every `e i` positive, where the finite factor carries the discrete topology.","title":"Structure theorem for topologically finitely generated abelian pro-`p` groups"},{"data_url":"scopes/tauceti/topology/algebra/part-02.json","id":"tauceti/topology/algebra/part-02","kind":"scope","navigation":"transparent","parent":4144,"result":false,"route":"../roadmap/tauceti/topology/algebra/part-02/index.html","scope":"tauceti/topology/algebra/part-02","status":"fully_proved","title":"Structure theorem for\u2026 to The two-term Euler formula"},{"id":"tauceti/topology/algebra/part-02/structure-theorem-for-torsion-free-topologically","kind":"node","lean":"TauCeti.IsProP.exists_continuousMulEquiv_pi_padicInt","parent":4166,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-02/index.html#structure-theorem-for-torsion-free-topologically","scope":"tauceti/topology/algebra/part-02","status":"fully_proved","summary":"Such a group is topologically isomorphic to `\u2124_p ^ r`.","title":"Structure theorem for torsion-free topologically finitely generated abelian pro-`p` groups"},{"id":"tauceti/topology/algebra/part-02/the-successive-approximation-theorem-with-tails","kind":"node","lean":"TauCeti.freeProP.exists_continuousMulEquiv_apply_eq_of_range_sup_gradedPowIterSpan_eq_top","parent":4166,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-02/index.html#the-successive-approximation-theorem-with-tails","scope":"tauceti/topology/algebra/part-02","status":"fully_proved","summary":"Let `r, w \u2208 \u03bb_1(F)` be relators of the free pro-`p` group `F` on a finite linearly ordered type with the same class `\u03c1 \u2208 gr_1(F)`, let `S` be a set of generators and `l\u2081, l\u2082` disjoint duplicate-free lists of generators containing `S`, and suppose `gr_{m+1}(F) = Im \u03b4_\u03c1 + \u27e8\u03c0^{m+1} \u03be_i : i \u2208 S\u27e9` for every `m \u2265 1`. Then a continuous automorphism `e` of `F` carries `r` to `(\u220f_{i \u2208 l\u2081} t_i) * w * (\u220f_{i \u2208 l\u2082} t_i)` for\u2026","title":"The successive-approximation theorem with tails"},{"id":"tauceti/topology/algebra/part-02/tates-criterion-for-the-cohomological-dimension","kind":"node","lean":"TauCeti.IsProP.cohomologicalDimensionAt_le_of_forall_coeffMap_proj_surjective","parent":4166,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-02/index.html#tates-criterion-for-the-cohomological-dimension","scope":"tauceti/topology/algebra/part-02","status":"fully_proved","summary":"Let `G` be a profinite pro-`p` group. If for every short exact sequence `0 \u2192 A \u2192 B \u2192 C \u2192 0` of finite discrete `G`-modules killed by `p` the map `H\u207f(G, B) \u2192 H\u207f(G, C)` is surjective, then `cd_p G \u2264 n`: right exactness kills `H\u207f\u207a\u00b9` on those modules, and d\u00e9vissage extends the vanishing to every finite discrete `p`-primary module.","title":"Tate's criterion for the cohomological dimension of a pro-`p` group"},{"id":"tauceti/topology/algebra/part-02/tates-duality-criterion-for-cd-p-g-2","kind":"node","lean":"TauCeti.IsProP.cohomologicalDimensionAt_le_two_of_forall_dualityMap2_bijective","parent":4166,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-02/index.html#tates-duality-criterion-for-cd-p-g-2","scope":"tauceti/topology/algebra/part-02","status":"fully_proved","summary":"Let `G` be a profinite pro-`p` group and `N` a discrete `G`-module. If Tate's duality map in degree two, `\u03b1\u2082 : H\u00b2(G, M) \u2192 Hom(H\u2070(G, InternalHom G M N), H\u00b2(G, N))`, is bijective for every finite discrete `G`-module `M` killed by `p`, then `cd_p G \u2264 2`: duality makes `H\u00b2(G, -)` right exact on those modules (`TauCeti.ContCohomology.DiscreteShortExact.coeffMap_proj_surjective_of_dualityMap2`), and right exactness\u2026","title":"Tate's duality criterion for `cd_p G \u2264 2`"},{"id":"tauceti/topology/algebra/part-02/the-three-term-euler-formula","kind":"node","lean":"TauCeti.IsProP.finrank_cohomFp_zero_sub_finrank_cohomFp_one_add_finrank_cohomFp_two_openSubgroup, TauCeti.IsProP.one_sub_topologicalGeneratorRankNat_add_finrank_H2, TauCeti.CohomologicalDimensionLE.finrank_cohomFp_zero_sub_finrank_cohomFp_one_add_finrank_cohomFp_two_openSubgroup, TauCeti.CohomologicalDimensionLE.one_sub_topologicalGeneratorRankNat_add_finrank_H2","parent":4166,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-02/index.html#the-three-term-euler-formula","scope":"tauceti/topology/algebra/part-02","status":"fully_proved","summary":"Let `G` be a topologically finitely generated profinite pro-`p` group whose canonical `H\u00b3` vanishes on the discrete modules of order `p` with trivial action, with `H\u00b2(G, \ud835\udd3d_p)` finite, and let `U` be an open subgroup. Then, in `\u2124`,","title":"The three-term Euler formula"},{"id":"tauceti/topology/algebra/part-02/the-three-term-euler-identity-under-cd-p-g-2","kind":"node","lean":"TauCeti.CohomologicalDimensionLE.natCard_H0_mul_natCard_H2_mul_pow, TauCeti.CohomologicalDimensionLE.natCard_H2_openSubgroup_mul_pow, TauCeti.IsProP.natCard_H2_openSubgroup_mul_pow","parent":4166,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-02/index.html#the-three-term-euler-identity-under-cd-p-g-2","scope":"tauceti/topology/algebra/part-02","status":"fully_proved","summary":"For a topologically finitely generated profinite pro-`p` group `G` with `cd_p G \u2264 2`, the trivial action on `\ud835\udd3d_p`, and a finite discrete `G`-module `M` of order `p ^ k`, `|H\u2070(G, M)| * |H\u00b2(G, M)| * p ^ (k * d(G)) = |H\u00b9(G, M)| * p ^ k * |H\u00b2(G, \ud835\udd3d_p)| ^ k`.","title":"The three-term Euler identity under `cd_p G \u2264 2`"},{"id":"tauceti/topology/algebra/part-02/the-topological-first-isomorphism-theorem-for-ri","kind":"node","lean":"RingHom.isHomeomorph_kerLift, LinearMap.quotKerContinuousLinearEquivOfSurjective","parent":4166,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-02/index.html#the-topological-first-isomorphism-theorem-for-ri","scope":"tauceti/topology/algebra/part-02","status":"fully_proved","summary":"A ring homomorphism that is a quotient map presents its target as the quotient of its source by its kernel, topologically as well as algebraically.","title":"The topological first isomorphism theorem for rings"},{"id":"tauceti/topology/algebra/part-02/the-topological-nakayama-lemma","kind":"node","lean":"Submodule.le_of_le_smul_of_isCompact, Submodule.eq_bot_of_le_smul_of_isCompact","parent":4166,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-02/index.html#the-topological-nakayama-lemma","scope":"tauceti/topology/algebra/part-02","status":"fully_proved","summary":"Let `I` be a finitely generated ideal of topologically nilpotent elements. If `P` is a compact submodule and `N` a closed submodule of a topological module with `P \u2264 N \u2294 I \u2022 P`, then `P \u2264 N`. Compactness of `P` replaces the finite generation hypothesis of `Submodule.le_of_le_smul_of_le_jacobson`.","title":"The topological Nakayama lemma"},{"id":"tauceti/topology/algebra/part-02/the-torsion-free-case-of-the-structure-theorem","kind":"node","lean":"TauCeti.presentedProP.oneRelator_q_eq_zero_iff_mem_topologicalClosure_commutator","parent":4166,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-02/index.html#the-torsion-free-case-of-the-structure-theorem","scope":"tauceti/topology/algebra/part-02","status":"fully_proved","summary":"The coordinate `q` of the exponent vector `exponentSum r = q \u2022 w`, `w x\u2080 = 1`, of the relator vanishes exactly when `r` lies in the closed commutator subgroup of the free pro-`p` group; then the factor `\u2124_p \u29f8 q \u2124_p` of `oneRelatorAbelianizationEquiv` is `\u2124_p` and `G^{ab} \u2245 \u2124_p^X` is torsion-free.","title":"The torsion-free case of the structure theorem"},{"id":"tauceti/topology/algebra/part-02/the-trivial-filtration-theorem","kind":"node","lean":"TauCeti.exists_filtration_of_isProP, TauCeti.exists_ne_zero_invariant_of_isProP, TauCeti.exists_notMem_nsmul_mem_smul_sub_mem_of_isProP","parent":4166,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-02/index.html#the-trivial-filtration-theorem","scope":"tauceti/topology/algebra/part-02","status":"fully_proved","summary":"A finite discrete `p`-primary additive group `M` with a continuous action of a pro-`p` group `G` has a `G`-stable increasing chain of subgroups `N 0 = \u22a5 \u2264 N 1 \u2264 \u2026` reaching `\u22a4` after `padicValNat p (Nat.card M)` steps, the composition length of `M`, and constant at `\u22a4` from there on, with `|N i| = p ^ i` along the way. Each step adjoins an element `x` with `x \u2209 N i`, `p \u2022 x \u2208 N i` and `g \u2022 x - x \u2208 N i` for every\u2026","title":"The trivial-filtration theorem"},{"id":"tauceti/topology/algebra/part-02/the-two-term-euler-formula","kind":"node","lean":"TauCeti.IsProP.finrank_cohomFp_zero_sub_finrank_cohomFp_one_openSubgroup, TauCeti.IsProP.topologicalGeneratorRankNat_add_index, TauCeti.CohomologicalDimensionLE.finrank_cohomFp_zero_sub_finrank_cohomFp_one_openSubgroup, TauCeti.CohomologicalDimensionLE.topologicalGeneratorRankNat_add_index, TauCeti.CohomologicalDimensionLE.one_sub_topologicalGeneratorRankNat_eq, TauCeti.IsProP.one_sub_topologicalGeneratorRankNat_eq","parent":4166,"result":true,"route":"../roadmap/tauceti/topology/algebra/part-02/index.html#the-two-term-euler-formula","scope":"tauceti/topology/algebra/part-02","status":"fully_proved","summary":"Let `G` be a topologically finitely generated profinite pro-`p` group whose explicit `H\u00b2` vanishes on the discrete modules of order `p` with trivial action, and let `U` be an open subgroup. Then, in `\u2124`,","title":"The two-term Euler formula"},{"data_url":"scopes/tauceti/topology/baire.json","id":"tauceti/topology/baire","kind":"scope","parent":4052,"result":false,"route":"../roadmap/tauceti/topology/baire/index.html","scope":"tauceti/topology/baire","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Topology.Baire` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Baire"},{"id":"tauceti/topology/baire/the-kuratowski-ulam-theorem","kind":"node","lean":"BaireMeasurableSet.eventually_eventually_left_iff_mem_residual, BaireMeasurableSet.eventually_eventually_right_iff_mem_residual, BaireMeasurableSet.eventually_isMeagre_curry_left_iff_isMeagre, BaireMeasurableSet.eventually_isMeagre_curry_right_iff_isMeagre","parent":4178,"result":true,"route":"../roadmap/tauceti/topology/baire/index.html#the-kuratowski-ulam-theorem","scope":"tauceti/topology/baire","status":"fully_proved","summary":"If `Y` is second countable, a set `s` with the Baire property in `X \u00d7 Y` is comeagre if and only if comeagrely many sections `(x, \u00b7) \u207b\u00b9' s` are comeagre, that is, if and only if it belongs to the curried residual filter `(residual X).curry (residual Y)`.","title":"The Kuratowski\u2013Ulam theorem"},{"id":"tauceti/topology/baire/mycielskis-theorem","kind":"node","lean":"TauCeti.mycielski, TauCeti.mycielski_prod","parent":4178,"result":true,"route":"../roadmap/tauceti/topology/baire/index.html#mycielskis-theorem","scope":"tauceti/topology/baire","status":"fully_proved","summary":"Let `X` be a nonempty perfect completely metrizable space, for example a nonempty perfect Polish space, and let `R n` be a meagre set of `k n`-tuples of points of `X` for each `n`. There is a continuous injection `f` of Cantor space into `X` such that for every `n` and every injective tuple `a` of `k n` points of Cantor space, `f \u2218 a \u2209 R n`.","title":"Mycielski's theorem"},{"id":"tauceti/topology/baire/one-half-of-the-kuratowski-ulam-theorem","kind":"node","lean":"TauCeti.curry_le_residual, TauCeti.eventually_eventually_left_of_mem_residual, TauCeti.eventually_eventually_right_of_mem_residual","parent":4178,"result":true,"route":"../roadmap/tauceti/topology/baire/index.html#one-half-of-the-kuratowski-ulam-theorem","scope":"tauceti/topology/baire","status":"fully_proved","summary":"If `Y` is second countable, every comeagre subset of `X \u00d7 Y` has comeagre sections `(x, \u00b7) \u207b\u00b9' s` over a comeagre set of `x`.","title":"One half of the Kuratowski\u2013Ulam theorem"},{"data_url":"scopes/tauceti/topology/homotopy.json","id":"tauceti/topology/homotopy","kind":"scope","parent":4052,"result":false,"route":"../roadmap/tauceti/topology/homotopy/index.html","scope":"tauceti/topology/homotopy","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Topology.Homotopy` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Homotopy"},{"id":"tauceti/topology/homotopy/alexanders-lemma","kind":"node","lean":"Homeomorph.alexanderIsotopyClosedBall","parent":4182,"result":true,"route":"../roadmap/tauceti/topology/homotopy/index.html#alexanders-lemma","scope":"tauceti/topology/homotopy","status":"fully_proved","summary":"A homeomorphism of the closed unit ball fixing the unit sphere is joined to the identity through homeomorphisms of the ball fixing the sphere.","title":"Alexander's lemma"},{"id":"tauceti/topology/homotopy/dolds-local-to-global-homotopy-lifting-theorem","kind":"node","lean":"TauCeti.HasHomotopyLiftingProperty.of_partitionOfUnity","parent":4182,"result":true,"route":"../roadmap/tauceti/topology/homotopy/index.html#dolds-local-to-global-homotopy-lifting-theorem","scope":"tauceti/topology/homotopy","status":"fully_proved","summary":"Let `p : E \u2192 B` be a map and `(U i)_{i \u2208 \u03b9}` a numerable cover of `B`, witnessed by a partition of unity `\u03c1 : PartitionOfUnity \u03b9 B` with `support (\u03c1 i) \u2286 U i` for every `i`. If each restriction `p \u207b\u00b9' U i \u2192 U i` has the homotopy lifting property with respect to every open subset of `A`, then `p` has the homotopy lifting property with respect to `A`.","title":"Dold's local-to-global homotopy lifting theorem"},{"id":"tauceti/topology/homotopy/dolds-local-to-global-theorem-for-hurewicz-fibra","kind":"node","lean":"TauCeti.isHurewiczFibration_of_partitionOfUnity","parent":4182,"result":true,"route":"../roadmap/tauceti/topology/homotopy/index.html#dolds-local-to-global-theorem-for-hurewicz-fibra","scope":"tauceti/topology/homotopy","status":"fully_proved","summary":"A continuous map `p : E \u2192 B` whose restrictions `p \u207b\u00b9' U i \u2192 U i` over a numerable cover `(U i)_{i \u2208 \u03b9}` of `B` (witnessed by a partition of unity `\u03c1` with `support (\u03c1 i) \u2286 U i`) are Hurewicz fibrations is itself a Hurewicz fibration.","title":"Dold's local-to-global theorem for Hurewicz fibrations"},{"id":"tauceti/topology/homotopy/strms-product-theorem","kind":"node","lean":"TauCeti.HasHomotopyExtensionProperty.prod_univ_union_univ_prod","parent":4182,"result":true,"route":"../roadmap/tauceti/topology/homotopy/index.html#strms-product-theorem","scope":"tauceti/topology/homotopy","status":"fully_proved","summary":"if closed subsets `A \u2286 X` and `B \u2286 Y` have the homotopy extension property, then so does `A \u00d7 Y \u222a X \u00d7 B \u2286 X \u00d7 Y`.","title":"Str\u00f8m's product theorem"},{"id":"tauceti/topology/homotopy/strms-theorem","kind":"node","lean":"TauCeti.NDRPair.ofRetraction","parent":4182,"result":true,"route":"../roadmap/tauceti/topology/homotopy/index.html#strms-theorem","scope":"tauceti/topology/homotopy","status":"fully_proved","summary":"A retraction `r` of the cylinder `I \u00d7 X` onto `{0} \u00d7 X \u222a I \u00d7 A`, for a closed subset `A`, gives NDR data for `A`: the deformation is the second coordinate of `r`, and the control function is `x \u21a6 \u2a06 t, t - (r (t, x)).1`, the largest amount by which `r` lowers a point of the cylinder over `x`.","title":"Str\u00f8m's theorem"},{"data_url":"scopes/tauceti/topology/other-results.json","id":"tauceti/topology/other-results","kind":"scope","parent":4052,"result":false,"route":"../roadmap/tauceti/topology/other-results/index.html","scope":"tauceti/topology/other-results","status":"fully_proved","summary":"Results named in the rest of Tau Ceti's `TauCeti.Topology` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Other results"},{"id":"tauceti/topology/other-results/a-refined-arzela-ascoli-theorem-ambrosio-gigli-s","kind":"node","lean":"TauCeti.tendsto_subseq_of_limsup_edist_le","parent":4188,"result":true,"route":"../roadmap/tauceti/topology/other-results/index.html#a-refined-arzela-ascoli-theorem-ambrosio-gigli-s","scope":"tauceti/topology/other-results","status":"fully_proved","summary":"Let `l y` be the filter of an auxiliary convergence `y_n \u21c0 y` on a weak extended metric space `X`, for which the distance is jointly sequentially lower semicontinuous. The topology of `X` need not be induced by its extended distance; the auxiliary convergence is specified only by `l`. Let `v k : T \u2192 X` be maps on a separable space `T` such that","title":"A refined Arzel\u00e0\u2013Ascoli theorem (Ambrosio\u2013Gigli\u2013Savar\u00e9, Proposition 3.3.1)"},{"id":"tauceti/topology/other-results/alexandrovs-theorem-and-its-converse","kind":"node","lean":"TauCeti.isG\u03b4_iff_isCompletelyMetrizableSpace","parent":4188,"result":true,"route":"../roadmap/tauceti/topology/other-results/index.html#alexandrovs-theorem-and-its-converse","scope":"tauceti/topology/other-results","status":"fully_proved","summary":"a subset of a completely metrizable space is completely metrizable in the subspace topology if and only if it is a G\u03b4 set.","title":"Alexandrov's theorem and its converse"},{"id":"tauceti/topology/other-results/the-arzela-ascoli-theorem-in-c-x-y","kind":"node","lean":"ArzelaAscoli.isCompact_closure_of_equicontinuous, ArzelaAscoli.isCompact_closure_of_equicontinuous_of_totallyBounded","parent":4188,"result":true,"route":"../roadmap/tauceti/topology/other-results/index.html#the-arzela-ascoli-theorem-in-c-x-y","scope":"tauceti/topology/other-results","status":"fully_proved","summary":"On a space `X` whose topology is coherent with its compact sets (for instance a locally compact space), an equicontinuous set of continuous maps whose values at each point lie in a common compact set has compact closure in the compact-open topology.","title":"The Arzel\u00e0\u2013Ascoli theorem in `C(X, Y)`"},{"id":"tauceti/topology/other-results/cartans-criterion","kind":"node","lean":"TauCeti.Topology.subsingleton_H'_succ_of_isCechAcyclicOn","parent":4188,"result":true,"route":"../roadmap/tauceti/topology/other-results/index.html#cartans-criterion","scope":"tauceti/topology/other-results","status":"fully_proved","summary":"Let `B` be a set of quasi-compact opens of `X` which is closed under binary intersections and contains, inside each of its members, a basis of the topology of that member. A sheaf of abelian groups which is \u010cech acyclic for the finite covers of members of `B` by members of `B` has no cohomology in positive degrees on the members of `B`.","title":"Cartan's criterion"},{"id":"tauceti/topology/other-results/the-cellular-approximation-theorem-hatcher-theor","kind":"node","lean":"TauCeti.exists_isCellular_homotopicRel, TauCeti.exists_isCellular_homotopic","parent":4188,"result":true,"route":"../roadmap/tauceti/topology/other-results/index.html#the-cellular-approximation-theorem-hatcher-theor","scope":"tauceti/topology/other-results","status":"fully_proved","summary":"Every map `f : C \u2192 C'` of relative CW complexes sending the base `D` into the base `E` is homotopic, relative to `D`, to a cellular map.","title":"The cellular approximation theorem (Hatcher, Theorem 4.8)"},{"id":"tauceti/topology/other-results/ekelands-variational-principle","kind":"node","lean":"LowerSemicontinuous.exists_add_mul_dist_le_and_lt_add_mul_dist","parent":4188,"result":true,"route":"../roadmap/tauceti/topology/other-results/index.html#ekelands-variational-principle","scope":"tauceti/topology/other-results","status":"fully_proved","summary":"Let `\u03c6` be a lower semicontinuous function, bounded below, on a complete metric space, and let `\u03b5 > 0`. For every `x` there is a point `y` such that `\u03c6 y + \u03b5 * dist y x \u2264 \u03c6 x` and `y` is the strict global minimum of `z \u21a6 \u03c6 z + \u03b5 * dist z y`.","title":"Ekeland's variational principle"},{"id":"tauceti/topology/other-results/grothendiecks-inequality","kind":"node","lean":"LinearMap.abs_sum_apply_apply_le_eight_mul, LinearMap.norm_sum_apply_apply_le_sixteen_mul","parent":4188,"result":true,"route":"../roadmap/tauceti/topology/other-results/index.html#grothendiecks-inequality","scope":"tauceti/topology/other-results","status":"fully_proved","summary":"Grothendieck's inequality for bounded real bilinear forms on spaces of continuous functions, with constant `\u03ba_\u211d = 8`. Let `K` and `L` be compact normal spaces and `B` a bilinear form on `C(K, \u211d) \u00d7 C(L, \u211d)` with `|B f g| \u2264 C * \u2016f\u2016 * \u2016g\u2016`. If `\u2211 i, f i x ^ 2 \u2264 1` and `\u2211 i, g i y ^ 2 \u2264 1` everywhere, then `|\u2211 i, B (f i) (g i)| \u2264 8 * C`.","title":"Grothendieck's inequality"},{"id":"tauceti/topology/other-results/lerays-theorem","kind":"node","lean":"TauCeti.Topology.homologyCechComplexIsoH","parent":4188,"result":true,"route":"../roadmap/tauceti/topology/other-results/index.html#lerays-theorem","scope":"tauceti/topology/other-results","status":"fully_proved","summary":"Let `U` be a finite family of opens covering a topological space `X`, and let `F` be a sheaf of abelian groups on `X` with no cohomology in positive degrees on any nonempty finite intersection of members of `U`. Then the \u010cech cohomology `H\u207f(U, F)`, the cohomology of the \u010cech complex of `F` for `U`, is isomorphic to the sheaf cohomology `H\u207f(X, F)`, for every `n`, naturally in `F`\u2026","title":"Leray's theorem"},{"data_url":"scopes/tauceti/other-areas.json","id":"tauceti/other-areas","kind":"scope","parent":1095,"result":false,"route":"../roadmap/tauceti/other-areas/index.html","scope":"tauceti/other-areas","status":"fully_proved","summary":"The areas of Tau Ceti with too few articles for a part of their own.","title":"Other Areas"},{"data_url":"scopes/tauceti/other-areas/computability.json","id":"tauceti/other-areas/computability","kind":"scope","parent":4197,"result":false,"route":"../roadmap/tauceti/other-areas/computability/index.html","scope":"tauceti/other-areas/computability","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Computability` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Computability"},{"id":"tauceti/other-areas/computability/degree-form-of-the-friedberg-muchnik-theorem","kind":"node","lean":"TauCeti.exists_incomparable_ce_turingDegrees_mem_Ioo","parent":4198,"result":true,"route":"../roadmap/tauceti/other-areas/computability/index.html#degree-form-of-the-friedberg-muchnik-theorem","scope":"tauceti/other-areas/computability","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.exists_incomparable_ce_turingDegrees_mem_Ioo`.","title":"Degree form of the Friedberg\u2013Muchnik theorem"},{"id":"tauceti/other-areas/computability/eilenbergs-theorem","kind":"node","lean":"TauCeti.isAutomatic_iff_finite_kernel","parent":4198,"result":true,"route":"../roadmap/tauceti/other-areas/computability/index.html#eilenbergs-theorem","scope":"tauceti/other-areas/computability","status":"fully_proved","summary":"for `2 \u2264 k`, a sequence is `k`-automatic if and only if its `k`-kernel is finite.","title":"Eilenberg's theorem"},{"id":"tauceti/other-areas/computability/the-friedberg-muchnik-theorem-s-vii-2-1","kind":"node","lean":"TauCeti.friedberg_muchnik","parent":4198,"result":true,"route":"../roadmap/tauceti/other-areas/computability/index.html#the-friedberg-muchnik-theorem-s-vii-2-1","scope":"tauceti/other-areas/computability","status":"fully_proved","summary":"Formalized in Tau Ceti as `TauCeti.friedberg_muchnik`.","title":"The Friedberg\u2013Muchnik theorem [S, VII.2.1]"},{"id":"tauceti/other-areas/computability/soares-lemma-1","kind":"node","lean":"TauCeti.Oracle.Single.Code.computablePred_mem_stageLimit_of_eval_eq, TauCeti.Oracle.Single.Code.mem_stageLimit_iff_of_lt_agreementLength","parent":4198,"result":true,"route":"../roadmap/tauceti/other-areas/computability/index.html#soares-lemma-1","scope":"tauceti/other-areas/computability","status":"fully_proved","summary":"Let `A` and `C` be the limits of monotone stages whose binary codes are computable, and let requirement `e` hold, at every stage `s`, a restraint at least the agreement restraint of the program `c` against `A s` with target `C s` and fuel `s`. If only finitely many numbers injure `e`, and `c` relative to the limit of `A` computes the characteristic function of the limit of `C`, then the limit of `C` is computable.","title":"Soare's Lemma 1"},{"data_url":"scopes/tauceti/other-areas/data.json","id":"tauceti/other-areas/data","kind":"scope","parent":4197,"result":false,"route":"../roadmap/tauceti/other-areas/data/index.html","scope":"tauceti/other-areas/data","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Data` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Data"},{"id":"tauceti/other-areas/data/birkhoffs-contraction-theorem","kind":"node","lean":"Matrix.hilbertProjectiveDist_mulVec_le","parent":4203,"result":true,"route":"../roadmap/tauceti/other-areas/data/index.html#birkhoffs-contraction-theorem","scope":"tauceti/other-areas/data","status":"fully_proved","summary":"A matrix `K` with strictly positive entries contracts Hilbert's projective metric between strictly positive vectors by the factor `tanh (K.projectiveDiameter / 4)`, which is strictly less than `1`.","title":"Birkhoff's contraction theorem"},{"id":"tauceti/other-areas/data/the-chinese-remainder-theorem-for-units","kind":"node","lean":"ZMod.unitsEquivPi","parent":4203,"result":true,"route":"../roadmap/tauceti/other-areas/data/index.html#the-chinese-remainder-theorem-for-units","scope":"tauceti/other-areas/data","status":"fully_proved","summary":"`(\u2124/n)\u02e3` is the product of the unit groups modulo the prime powers exactly dividing `n`.","title":"The Chinese remainder theorem for units"},{"id":"tauceti/other-areas/data/fines-theorem","kind":"node","lean":"Nat.card_filter_not_dvd_choose","parent":4203,"result":true,"route":"../roadmap/tauceti/other-areas/data/index.html#fines-theorem","scope":"tauceti/other-areas/data","status":"fully_proved","summary":"The number of `k \u2264 n` for which `p \u2224 n choose k` is `\u220f\u2c7c (n\u2c7c + 1)`, the product over the base-`p` digits `n\u2c7c` of `n`.","title":"Fine's theorem"},{"id":"tauceti/other-areas/data/hensels-lemma-for-n-th-roots-as-a-count","kind":"node","lean":"ZMod.natCard_nthRoots_prime_pow_eq_natCard_nthRoots","parent":4203,"result":true,"route":"../roadmap/tauceti/other-areas/data/index.html#hensels-lemma-for-n-th-roots-as-a-count","scope":"tauceti/other-areas/data","status":"fully_proved","summary":"For `n` coprime to the prime `p`, an integer `a` prime to `p` and `k \u2260 0`, reduction modulo `p` is a bijection from the `n`-th roots of `a` modulo `p ^ k` to those modulo `p`; so there are equally many.","title":"Hensel's lemma for `n`-th roots, as a count"},{"id":"tauceti/other-areas/data/legendres-identity-over-a-fixed-range","kind":"node","lean":"Nat.sub_one_mul_sum_div_pow_succ_eq_sub_sum_digits","parent":4203,"result":true,"route":"../roadmap/tauceti/other-areas/data/index.html#legendres-identity-over-a-fixed-range","scope":"tauceti/other-areas/data","status":"fully_proved","summary":"For `a < b ^ f`, `(b - 1) \u2211_{i < f} \u230aa / b ^ (i + 1)\u230b = a - s(a)`, where `s(a)` is the base-`b` digit sum of `a`. This is `Nat.sub_one_mul_sum_log_div_pow_eq_sub_sum_digits` with the sum extended by zero terms from `log b a + 1` to `f`.","title":"Legendre's identity over a fixed range"},{"id":"tauceti/other-areas/data/lucas-theorem-for-divisibility-one-digit-at-a-ti","kind":"node","lean":"Nat.dvd_choose_add_mul_iff","parent":4203,"result":true,"route":"../roadmap/tauceti/other-areas/data/index.html#lucas-theorem-for-divisibility-one-digit-at-a-ti","scope":"tauceti/other-areas/data","status":"fully_proved","summary":"For last digits `r, a < p`, the prime `p` divides `(r + p q) choose (a + p b)` exactly when `r < a` or `p` divides `q choose b`.","title":"Lucas' theorem for divisibility, one digit at a time"},{"id":"tauceti/other-areas/data/the-sinkhorn-knopp-theorem-for-a-strictly-positi","kind":"node","lean":"Matrix.exists_sinkhorn_scaling","parent":4203,"result":true,"route":"../roadmap/tauceti/other-areas/data/index.html#the-sinkhorn-knopp-theorem-for-a-strictly-positi","scope":"tauceti/other-areas/data","status":"fully_proved","summary":"a strictly positive matrix `K` and strictly positive row sums `a` and column sums `b` of equal total mass admit strictly positive row factors `u` and column factors `v` such that the diagonal scaling `u i * K i j * v j` has row sums `a` and column sums `b`.","title":"The Sinkhorn\u2013Knopp theorem for a strictly positive kernel"},{"id":"tauceti/other-areas/data/vandermondes-identity-for-finitely-many-groups","kind":"node","lean":"Finset.sum_piAntidiag_prod_choose, Finset.sum_filter_piAntidiag_prod_choose_fiberwise","parent":4203,"result":true,"route":"../roadmap/tauceti/other-areas/data/index.html#vandermondes-identity-for-finitely-many-groups","scope":"tauceti/other-areas/data","status":"fully_proved","summary":"the ways of choosing `n` objects from groups of sizes `K i`, `i \u2208 s`, sorted by how many come from each group.","title":"Vandermonde's identity for finitely many groups"},{"data_url":"scopes/tauceti/other-areas/examples.json","id":"tauceti/other-areas/examples","kind":"scope","parent":4197,"result":false,"route":"../roadmap/tauceti/other-areas/examples/index.html","scope":"tauceti/other-areas/examples","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.Examples` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Examples"},{"id":"tauceti/other-areas/examples/cassels-basis-lemma-on-l","kind":"node","lean":"TauCeti.GaugeBasisExample.exists_basis_norm_le","parent":4212,"result":true,"route":"../roadmap/tauceti/other-areas/examples/index.html#cassels-basis-lemma-on-l","scope":"tauceti/other-areas/examples","status":"fully_proved","summary":"Applied to the minima witnesses `e 0, \u2026, e 4`, it gives a basis `b` of `L` with `\u2016b i\u2016 \u2264 max 1 ((i + 1) / 2)`.","title":"Cassels' basis lemma on `L`"},{"data_url":"scopes/tauceti/other-areas/information-theory.json","id":"tauceti/other-areas/information-theory","kind":"scope","parent":4197,"result":false,"route":"../roadmap/tauceti/other-areas/information-theory/index.html","scope":"tauceti/other-areas/information-theory","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.InformationTheory` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Information Theory"},{"id":"tauceti/other-areas/information-theory/csiszars-pythagorean-identity","kind":"node","lean":"TauCeti.klDiv_eq_klDiv_add_klDiv","parent":4214,"result":true,"route":"../roadmap/tauceti/other-areas/information-theory/index.html#csiszars-pythagorean-identity","scope":"tauceti/other-areas/information-theory","status":"fully_proved","summary":"Let `\u03b3 \u226a \u03c0 \u226a R` be finite measures such that `llr \u03c0 R` is integrable against both `\u03b3` and `\u03c0`, with the same integral. Then `klDiv \u03b3 R = klDiv \u03b3 \u03c0 + klDiv \u03c0 R`.","title":"Csisz\u00e1r's Pythagorean identity"},{"id":"tauceti/other-areas/information-theory/pinskers-inequality","kind":"node","lean":"TauCeti.etvDist_sq_le_two_mul_klDiv, TauCeti.etvDist_sq_le_two_mul_mass_mul_klDiv, TauCeti.two_mul_sq_measureReal_sub_le_klDiv","parent":4214,"result":true,"route":"../roadmap/tauceti/other-areas/information-theory/index.html#pinskers-inequality","scope":"tauceti/other-areas/information-theory","status":"fully_proved","summary":"Pinsker's inequality for probability measures, with natural logarithms and no absolute-continuity assumption. Mathlib's `etvDist` on the left is twice the usual distance `dTV(\u03bc, \u03bd) = sup_s |\u03bc s - \u03bd s|`, so the bound is equivalently `2 * dTV(\u03bc, \u03bd)\u00b2 \u2264 klDiv \u03bc \u03bd`.","title":"Pinsker's inequality"},{"data_url":"scopes/tauceti/other-areas/low-dim-topology.json","id":"tauceti/other-areas/low-dim-topology","kind":"scope","parent":4197,"result":false,"route":"../roadmap/tauceti/other-areas/low-dim-topology/index.html","scope":"tauceti/other-areas/low-dim-topology","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.LowDimTopology` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Low Dim Topology"},{"id":"tauceti/other-areas/low-dim-topology/lis-theorem-refutes-waldhausens-conjecture","kind":"node","lean":"TauCeti.LiRankGenusTheorem.not_rankGenusConjecture","parent":4217,"result":true,"route":"../roadmap/tauceti/other-areas/low-dim-topology/index.html#lis-theorem-refutes-waldhausens-conjecture","scope":"tauceti/other-areas/low-dim-topology","status":"fully_proved","summary":"some closed orientable `3`-manifold has Heegaard genus different from the rank of its fundamental group.","title":"Li's theorem refutes Waldhausen's conjecture"},{"id":"tauceti/other-areas/low-dim-topology/under-gabais-theorem-a-generator-of-h2-s1-s2-z-h","kind":"node","lean":"TauCeti.GabaiZeroSurgeryGenusTheorem.minimalGenus_circle_prod_sphere","parent":4217,"result":true,"route":"../roadmap/tauceti/other-areas/low-dim-topology/index.html#under-gabais-theorem-a-generator-of-h2-s1-s2-z-h","scope":"tauceti/other-areas/low-dim-topology","status":"fully_proved","summary":"`S\u00b9 \u00d7 S\u00b2` is the `0`-surgery on the unknot, which has Seifert genus `0`.","title":"Under Gabai's theorem, a generator of `H\u2082(S\u00b9 \u00d7 S\u00b2; \u2124)` has minimal genus `0`"},{"id":"tauceti/other-areas/low-dim-topology/yasuis-theorem-spelled-out","kind":"node","lean":"TauCeti.yasuiTheorem_iff","parent":4217,"result":true,"route":"../roadmap/tauceti/other-areas/low-dim-topology/index.html#yasuis-theorem-spelled-out","scope":"tauceti/other-areas/low-dim-topology","status":"fully_proved","summary":"there are knots `K K' : S\u00b9 \u2192 S\u00b3` with `0`-surgeries `Y`, `Y'`, oriented compatibly with an orientation of `S\u00b3`, and an orientation-preserving homeomorphism `Y \u2243\u209c Y'`, such that `K` is smoothly concordant neither to `K'` nor to its reverse.","title":"Yasui's theorem spelled out"},{"data_url":"scopes/tauceti/other-areas/set-theory.json","id":"tauceti/other-areas/set-theory","kind":"scope","parent":4197,"result":false,"route":"../roadmap/tauceti/other-areas/set-theory/index.html","scope":"tauceti/other-areas/set-theory","status":"fully_proved","summary":"Results named in Tau Ceti's `TauCeti.SetTheory` modules and in no roadmap's section, as Tau Ceti has them at commit ace60277.","title":"Set Theory"},{"id":"tauceti/other-areas/set-theory/the-cardinality-form-of-silvers-theorem","kind":"node","lean":"Setoid.mk_quotient_le_aleph0_or_eq_continuum","parent":4221,"result":true,"route":"../roadmap/tauceti/other-areas/set-theory/index.html#the-cardinality-form-of-silvers-theorem","scope":"tauceti/other-areas/set-theory","status":"fully_proved","summary":"A Borel equivalence relation on a standard Borel space has at most `\u2135\u2080` or exactly `\ud835\udd20` classes.","title":"The cardinality form of Silver's theorem"},{"id":"tauceti/other-areas/set-theory/the-category-route-to-silvers-theorem","kind":"node","lean":"Setoid.hasCantorAntichainOn_range_of_forall_gSGraph_adj","parent":4221,"result":true,"route":"../roadmap/tauceti/other-areas/set-theory/index.html#the-category-route-to-silvers-theorem","scope":"tauceti/other-areas/set-theory","status":"fully_proved","summary":"Let `E` be a Borel equivalence relation and `\u03c6` a continuous map from Cantor space sending the edges of `gSGraph S` to `E`-inequivalent pairs. If `S` is dense, the range of `\u03c6` carries a Cantor antichain for `E`.","title":"The category route to Silver's theorem"},{"id":"tauceti/other-areas/set-theory/the-kechris-solecki-todorcevic-g0-dichotomy","kind":"node","lean":"SimpleGraph.g0_dichotomy","parent":4221,"result":true,"route":"../roadmap/tauceti/other-areas/set-theory/index.html#the-kechris-solecki-todorcevic-g0-dichotomy","scope":"tauceti/other-areas/set-theory","status":"fully_proved","summary":"A simple graph on a Polish space whose edge relation is analytic has a measurable coloring by `\u2115`, or receives a continuous homomorphism from `gZero`, and not both.","title":"The Kechris\u2013Solecki\u2013Todorcevic G\u2080 dichotomy"},{"id":"tauceti/other-areas/set-theory/the-kechris-solecki-todorcevic-lemma","kind":"node","lean":"SetRel.exists_measurableSet_isIndependent_superset","parent":4221,"result":true,"route":"../roadmap/tauceti/other-areas/set-theory/index.html#the-kechris-solecki-todorcevic-lemma","scope":"tauceti/other-areas/set-theory","status":"fully_proved","summary":"In a Hausdorff space with its Borel \u03c3-algebra, an analytic independent set of an analytic relation is contained in a Borel independent set.","title":"The Kechris\u2013Solecki\u2013Todorcevic lemma"},{"id":"tauceti/other-areas/set-theory/lusin-sierpinski-kunen-martin-boundedness-theore","kind":"node","lean":"MeasureTheory.AnalyticSet.exists_treeRank_lt_omega_one, MeasureTheory.AnalyticSet.exists_treeRank_lt_omega_one_of_isClosed","parent":4221,"result":true,"route":"../roadmap/tauceti/other-areas/set-theory/index.html#lusin-sierpinski-kunen-martin-boundedness-theore","scope":"tauceti/other-areas/set-theory","status":"fully_proved","summary":"Let `s` be an analytic subset of a topological space `X` and `F : X \u2192 Descriptive.tree A` a family of descriptive trees over a countable alphabet `A : Type u`. Suppose `F x` is well-founded for every `x \u2208 s` and `{x : s | u \u2208 F x.1}` is closed in the subspace `s` for every finite sequence `u : List A`. Then some countable ordinal `\u03b2 < \u03c9\u2081` strictly bounds `rank (F x)` across all `x \u2208 s`.","title":"Lusin\u2013Sierpi\u0144ski / Kunen\u2013Martin boundedness theorem (relative form)"},{"id":"tauceti/other-areas/set-theory/millers-independence-lemma","kind":"node","lean":"TauCeti.exists_mem_orientedGS_of_not_isMeagre, TauCeti.exists_gSGraph_adj_of_not_isMeagre","parent":4221,"result":true,"route":"../roadmap/tauceti/other-areas/set-theory/index.html#millers-independence-lemma","scope":"tauceti/other-areas/set-theory","status":"fully_proved","summary":"If every word extends to a word of `S`, then every non-meagre subset of Cantor space with the Baire property contains both ends of an oriented edge of `orientedGS S`.","title":"Miller's independence lemma"},{"id":"tauceti/other-areas/set-theory/the-perfect-set-theorem-for-analytic-sets-sousli","kind":"node","lean":"MeasureTheory.AnalyticSet.exists_nat_bool_injection_of_not_countable","parent":4221,"result":true,"route":"../roadmap/tauceti/other-areas/set-theory/index.html#the-perfect-set-theorem-for-analytic-sets-sousli","scope":"tauceti/other-areas/set-theory","status":"fully_proved","summary":"An uncountable analytic subset of a Hausdorff space contains the range of a continuous injection of Cantor space.","title":"The perfect set theorem for analytic sets (Souslin)"},{"id":"tauceti/other-areas/set-theory/silvers-theorem","kind":"node","lean":"Setoid.silver_dichotomy, Setoid.borelReducible_bot_iff_not_countable_quotient, Setoid.silver_dichotomy_standardBorel","parent":4221,"result":true,"route":"../roadmap/tauceti/other-areas/set-theory/index.html#silvers-theorem","scope":"tauceti/other-areas/set-theory","status":"fully_proved","summary":"A Borel equivalence relation on a Polish space has countably many classes, or a Cantor antichain on the whole space: a continuous map of Cantor space sending distinct points to inequivalent points.","title":"Silver's theorem"},{"id":"tauceti/other-areas/set-theory/silvers-theorem-relative-to-a-borel-subset","kind":"node","lean":"Setoid.countable_or_hasCantorAntichainOn_of_measurableSet","parent":4221,"result":true,"route":"../roadmap/tauceti/other-areas/set-theory/index.html#silvers-theorem-relative-to-a-borel-subset","scope":"tauceti/other-areas/set-theory","status":"fully_proved","summary":"Let `A` be a Borel subset of a Polish space `X`, `F` a Borel equivalence relation on `A`, and `R` any equivalence relation on `X` whose restriction to `A` refines `F`. Then `F` has countably many classes, or `A` carries a Cantor antichain for `R`, continuous for the topology of `X`.","title":"Silver's theorem relative to a Borel subset"},{"data_url":"chapters/flt.json","id":"flt","kind":"scope","parent":0,"result":false,"route":"../roadmap/flt/index.html","scope":"flt","status":"stated","summary":"The blueprint of Fermat's Last Theorem (Imperial), read at commit a3210204 (2026-10-22): one article for each of its 172 nodes whose Lean declarations it names and marks formalized, or in Mathlib. Its other 69 nodes have no article, and a section of only those is left out: the map records what the project contains, not what it plans. A statement for which the blueprint marks no proof `\\leanok` says so. Dependencies\u2026","title":"Fermat's Last Theorem (Imperial)"},{"data_url":"scopes/flt/introduction.json","id":"flt/introduction","kind":"scope","parent":4231,"result":false,"route":"../roadmap/flt/introduction/index.html","scope":"flt/introduction","status":"planned","summary":"Fermat's Last Theorem is the statement that if $a,b,c,n$ are positive whole numbers with $n\\geq 3$, then $a^n+b^n\\not=c^n$. It is thus the claim that a family of *Diophantine equations* ($a^3+b^3=c^3, a^4+b^4=c^4,\\ldots$) has no positive integer solutions. Diophantus was a Greek mathematician who lived around 1800 years ago, and he would have been able to understand the statement of the theorem (he knew about\u2026","title":"Introduction"},{"id":"flt/introduction/which-proof-is-being-formalised","kind":"node","parent":4232,"result":false,"route":"../roadmap/flt/introduction/which-proof-is-being-formalised/index.html","scope":"flt/introduction","status":"can_state","summary":"At the time of writing, these notes do not contain anywhere near a proof of FLT, or even a sketch proof, although we are currently actively working on fixing this.","title":"Which proof is being formalised?"},{"id":"flt/introduction/the-structure-of-this-blueprint","kind":"node","parent":4232,"result":false,"route":"../roadmap/flt/introduction/the-structure-of-this-blueprint/index.html","scope":"flt/introduction","status":"can_state","summary":"This blueprint is a *nonlinear* document, comprising many chapters. The chapters are not designed to be read in order. Each chapter is self-contained and has a well-defined goal, typically stated at the top of the chapter.","title":"The structure of this blueprint"},{"id":"flt/introduction/remarks","kind":"node","parent":4232,"result":false,"route":"../roadmap/flt/introduction/remarks/index.html","scope":"flt/introduction","status":"can_state","summary":"The actual blueprint currently also contains a lot of disorganised ideas. Currently these should be disregarded.","title":"Remarks"},{"data_url":"scopes/flt/first-reductions-of-the-problem.json","id":"flt/first-reductions-of-the-problem","kind":"scope","parent":4231,"result":false,"route":"../roadmap/flt/first-reductions-of-the-problem/index.html","scope":"flt/first-reductions-of-the-problem","status":"stated","title":"First reductions of the problem"},{"id":"flt/first-reductions-of-the-problem/goal","kind":"node","parent":4236,"result":false,"route":"../roadmap/flt/first-reductions-of-the-problem/goal/index.html","scope":"flt/first-reductions-of-the-problem","status":"can_state","summary":"The goal of this chapter is to reduce FLT to a deep theorem of Mazur and a deep theorem of Wiles about a Galois representation.","title":"Goal"},{"id":"flt/first-reductions-of-the-problem/overview","kind":"node","parent":4236,"result":false,"route":"../roadmap/flt/first-reductions-of-the-problem/overview/index.html","scope":"flt/first-reductions-of-the-problem","status":"can_state","summary":"The proof of Fermat's Last Theorem is by contradiction. We assume that we have a counterexample $a^n+b^n=c^n$, and manipulate it until it satisfies the axioms of a \u201cFrey package\u201d, a basic concept which we will explain below. From the Frey package we build a Frey curve \u2013 an elliptic curve defined over the rationals. We then look at a certain representation of a Galois group coming from this elliptic curve, and\u2026","title":"Overview"},{"data_url":"scopes/flt/first-reductions-of-the-problem/reduction-to-n-geq-5-and-prime.json","id":"flt/first-reductions-of-the-problem/reduction-to-n-geq-5-and-prime","kind":"scope","parent":4236,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reduction to "],["math","\\(n\\geq5\\)"],["text"," and prime"]],"source":"Reduction to $n\\geq5$ and prime"},"result":false,"route":"../roadmap/flt/first-reductions-of-the-problem/reduction-to-n-geq-5-and-prime/index.html","scope":"flt/first-reductions-of-the-problem/reduction-to-n-geq-5-and-prime","status":"fully_proved","summary":"Euler proved Fermat's Last Theorem for $p=3$;","title":"Reduction to $n\\geq5$ and prime"},{"id":"flt/first-reductions-of-the-problem/reduction-to-n-geq-5-and-prime/fermatlasttheorem-of-odd-primes","kind":"node","lean":"FermatLastTheorem.of_odd_primes","parent":4239,"result":true,"route":"../roadmap/flt/first-reductions-of-the-problem/reduction-to-n-geq-5-and-prime/index.html#fermatlasttheorem-of-odd-primes","scope":"flt/first-reductions-of-the-problem/reduction-to-n-geq-5-and-prime","status":"fully_proved","summary":"If there is a counterexample to Fermat's Last Theorem, then there is a counterexample $a^p+b^p=c^p$ with $p$ an odd prime.","title":"FermatLastTheorem.of_odd_primes"},{"id":"flt/first-reductions-of-the-problem/reduction-to-n-geq-5-and-prime/fermatlasttheoremthree","kind":"node","lean":"fermatLastTheoremThree","parent":4239,"result":true,"route":"../roadmap/flt/first-reductions-of-the-problem/reduction-to-n-geq-5-and-prime/index.html#fermatlasttheoremthree","scope":"flt/first-reductions-of-the-problem/reduction-to-n-geq-5-and-prime","status":"fully_proved","summary":"There are no solutions in positive integers to $a^3+b^3=c^3$.","title":"fermatLastTheoremThree"},{"id":"flt/first-reductions-of-the-problem/reduction-to-n-geq-5-and-prime/fermatlasttheorem-of-p-ge-5","kind":"node","lean":"FermatLastTheorem.of_p_ge_5","parent":4239,"result":true,"route":"../roadmap/flt/first-reductions-of-the-problem/reduction-to-n-geq-5-and-prime/index.html#fermatlasttheorem-of-p-ge-5","scope":"flt/first-reductions-of-the-problem/reduction-to-n-geq-5-and-prime","status":"fully_proved","summary":"If there is a counterexample to Fermat's Last Theorem, then there is a counterexample $a^p+b^p=c^p$ with $p$ prime and $p\\geq 5$.","title":"FermatLastTheorem.of_p_ge_5"},{"data_url":"scopes/flt/first-reductions-of-the-problem/frey-packages.json","id":"flt/first-reductions-of-the-problem/frey-packages","kind":"scope","parent":4236,"result":false,"route":"../roadmap/flt/first-reductions-of-the-problem/frey-packages/index.html","scope":"flt/first-reductions-of-the-problem/frey-packages","status":"fully_proved","summary":"For convenience we make the following definition.","title":"Frey packages"},{"id":"flt/first-reductions-of-the-problem/frey-packages/freypackage","kind":"node","lean":"FreyPackage","parent":4243,"result":true,"route":"../roadmap/flt/first-reductions-of-the-problem/frey-packages/index.html#freypackage","scope":"flt/first-reductions-of-the-problem/frey-packages","status":"fully_proved","summary":"A *Frey package* $(a,b,c,p)$ is three nonzero pairwise-coprime integers $a$, $b$, $c$, with $a\\equiv3\\pmod4$ and $b\\equiv0\\pmod2$, and a prime $p\\geq5$, such that $a^p+b^p=c^p$.","title":"FreyPackage"},{"id":"flt/first-reductions-of-the-problem/frey-packages/freypackage-of-not-fermatlasttheorem-p-ge-5","kind":"node","lean":"FreyPackage.of_not_FermatLastTheoremFor_p_ge_5","parent":4243,"result":true,"route":"../roadmap/flt/first-reductions-of-the-problem/frey-packages/index.html#freypackage-of-not-fermatlasttheorem-p-ge-5","scope":"flt/first-reductions-of-the-problem/frey-packages","status":"fully_proved","summary":"If Fermat's Last Theorem is false for $p \\ge 5$ and prime, then there exists a Frey package.","title":"FreyPackage.of_not_FermatLastTheorem_p_ge_5"},{"id":"flt/first-reductions-of-the-problem/galois-representations-and-elliptic-curves","kind":"node","parent":4236,"result":false,"route":"../roadmap/flt/first-reductions-of-the-problem/galois-representations-and-elliptic-curves/index.html","scope":"flt/first-reductions-of-the-problem","status":"can_state","summary":"To continue, we need some of the theory of elliptic curves over $\\mathbb{Q}$. So let $f(X)$ denote any monic cubic polynomial with rational coefficients and whose three complex roots are distinct, and let us consider the equation $E:Y^2=f(X)$, which defines a curve in the $(X,Y)$ plane. This curve (or strictly speaking its projectivisation) is a so-called elliptic curve (or an elliptic curve over $\\mathbb{Q}$ if we\u2026","title":"Galois representations and elliptic curves"},{"data_url":"scopes/flt/first-reductions-of-the-problem/the-frey-curve.json","id":"flt/first-reductions-of-the-problem/the-frey-curve","kind":"scope","parent":4236,"result":false,"route":"../roadmap/flt/first-reductions-of-the-problem/the-frey-curve/index.html","scope":"flt/first-reductions-of-the-problem/the-frey-curve","status":"fully_proved","summary":"Recall that a *Frey package* $(a,b,c,p)$ is simply a prime $p\\geq5$ and nonzero pairwise-coprime integers $a,b,c$ satisfying $a^p+b^p=c^p$ and satisfying the congruences $a\\equiv3\\pmod4$ and $b\\equiv0\\pmod2$. We have shown above that if Fermat's Last Theorem is false, then a Frey package exists.","title":"The Frey curve"},{"id":"flt/first-reductions-of-the-problem/the-frey-curve/freycurve","kind":"node","lean":"FreyPackage.freyCurve","parent":4247,"result":true,"route":"../roadmap/flt/first-reductions-of-the-problem/the-frey-curve/index.html#freycurve","scope":"flt/first-reductions-of-the-problem/the-frey-curve","status":"fully_proved","summary":"Given a Frey package $(a,b,c,p)$, the corresponding *Frey curve* (considered by Frey and, before him, Hellegouarch) is the elliptic curve over $\\mathbb{Q}$ defined by the equation $Y^2=X(X-a^p)(X+b^p).$","title":"Frey"},{"data_url":"scopes/flt/first-reductions-of-the-problem/reduction-to-two-big-theorems.json","id":"flt/first-reductions-of-the-problem/reduction-to-two-big-theorems","kind":"scope","parent":4236,"result":false,"route":"../roadmap/flt/first-reductions-of-the-problem/reduction-to-two-big-theorems/index.html","scope":"flt/first-reductions-of-the-problem/reduction-to-two-big-theorems","status":"stated","summary":"Recall that a representation of a group $G$ on a vector space $W$ is said to be *irreducible* if there are precisely two $G$-stable subspaces of $W$, namely $0$ and $W$. The representation is said to be *reducible* otherwise.","title":"Reduction to two big theorems."},{"id":"flt/first-reductions-of-the-problem/reduction-to-two-big-theorems/mazur-frey","kind":"node","lean":"FreyPackage.mazur","parent":4249,"result":true,"route":"../roadmap/flt/first-reductions-of-the-problem/reduction-to-two-big-theorems/index.html#mazur-frey","scope":"flt/first-reductions-of-the-problem/reduction-to-two-big-theorems","status":"can_prove","summary":"If $\\rho$ is the mod $p$ Galois representation associated to a Frey package $(a,b,c,p)$ then $\\rho$ is irreducible.","title":"Mazur"},{"id":"flt/first-reductions-of-the-problem/reduction-to-two-big-theorems/wiles-frey","kind":"node","lean":"FLT.Bosses.B4_proof","parent":4249,"result":true,"route":"../roadmap/flt/first-reductions-of-the-problem/reduction-to-two-big-theorems/index.html#wiles-frey","scope":"flt/first-reductions-of-the-problem/reduction-to-two-big-theorems","status":"can_prove","summary":"If $\\rho$ is the mod $p$ Galois representation associated to a Frey package $(a,b,c,p)$ then $\\rho$ is reducible.","title":"Wiles,Taylor\u2013Wiles, Ribet,\u2026"},{"id":"flt/first-reductions-of-the-problem/reduction-to-two-big-theorems/freypackage-false","kind":"node","lean":"FLT.Bosses.B3_proof","parent":4249,"result":true,"route":"../roadmap/flt/first-reductions-of-the-problem/reduction-to-two-big-theorems/index.html#freypackage-false","scope":"flt/first-reductions-of-the-problem/reduction-to-two-big-theorems","status":"can_prove","summary":"There is no Frey package.","title":"FreyPackage.false"},{"id":"flt/first-reductions-of-the-problem/reduction-to-two-big-theorems/flt","kind":"node","lean":"flt","parent":4249,"result":true,"route":"../roadmap/flt/first-reductions-of-the-problem/reduction-to-two-big-theorems/index.html#flt","scope":"flt/first-reductions-of-the-problem/reduction-to-two-big-theorems","status":"stated","summary":"Fermat's Last Theorem is true. In other words, there are no positive integers $a,b,c$ and natural $n\\geq3$ such that $a^n+b^n=c^n$.","title":"FLT"},{"data_url":"scopes/flt/reducibility-of-p-torsion-of-the-frey-curve.json","id":"flt/reducibility-of-p-torsion-of-the-frey-curve","kind":"scope","parent":4231,"result":false,"route":"../roadmap/flt/reducibility-of-p-torsion-of-the-frey-curve/index.html","scope":"flt/reducibility-of-p-torsion-of-the-frey-curve","status":"stated","title":"Reducibility of p-torsion of the Frey curve"},{"id":"flt/reducibility-of-p-torsion-of-the-frey-curve/overview","kind":"node","parent":4254,"result":false,"route":"../roadmap/flt/reducibility-of-p-torsion-of-the-frey-curve/overview/index.html","scope":"flt/reducibility-of-p-torsion-of-the-frey-curve","status":"can_state","summary":"In chapter First reductions of the problem we reduced FLT, modulo a hard theorem from the 1970s, to Theorem Wiles,Taylor\u2013Wiles, Ribet,\u2026, the assertion that $p$-torsion in the Frey curve is reducible. In this chapter we deduce this assertion from three more complex claims about \u201chardly ramified\u201d Galois representations. It is relatively straightforward to reduce one of these three claims to a result of Fontaine\u2026","title":"Overview"},{"data_url":"scopes/flt/reducibility-of-p-torsion-of-the-frey-curve/hardly-ramified-representations.json","id":"flt/reducibility-of-p-torsion-of-the-frey-curve/hardly-ramified-representations","kind":"scope","parent":4254,"result":false,"route":"../roadmap/flt/reducibility-of-p-torsion-of-the-frey-curve/hardly-ramified-representations/index.html","scope":"flt/reducibility-of-p-torsion-of-the-frey-curve/hardly-ramified-representations","status":"stated","summary":"Let $(a,b,c,p)$ be a Frey package (so in particular $p\\geq5$ is prime and $a^p+b^p=c^p$), let $E$ be the corresponding Frey curve over $\\mathbb{Q}$, and let $\\rho:\\operatorname{Gal}(\\overline{\\mathbb{Q}}/\\mathbb{Q})\\to\\operatorname{Aut}(E(\\overline{\\mathbb{Q}})[p])$ be the 2-dimensional Galois representation on the $p$-torsion of $E$. Recall that our goal is to prove that $\\rho$ is reducible.","title":"Hardly ramified representations"},{"id":"flt/reducibility-of-p-torsion-of-the-frey-curve/hardly-ramified-representations/hardly-ramified","kind":"node","lean":"GaloisRepresentation.IsHardlyRamified","parent":4256,"result":true,"route":"../roadmap/flt/reducibility-of-p-torsion-of-the-frey-curve/hardly-ramified-representations/index.html#hardly-ramified","scope":"flt/reducibility-of-p-torsion-of-the-frey-curve/hardly-ramified-representations","status":"fully_proved","summary":"Let $R$ be a coefficient ring with finite residue field of characteristic $\\ell\\geq3$. Let $V$ be a finite free $R$-module of rank 2, equipped with the product topology. A continuous representation $\\rho: \\operatorname{Gal}(\\overline{\\mathbb{Q}}/\\mathbb{Q})\\to \\operatorname{GL}_R(V)$ is said to be *hardly ramified* if it satisfies the following four conditions:","title":"hardly_ramified"},{"id":"flt/reducibility-of-p-torsion-of-the-frey-curve/hardly-ramified-representations/frey-curve-hardly-ramified","kind":"node","lean":"FreyCurve.torsion_isHardlyRamified","parent":4256,"result":true,"route":"../roadmap/flt/reducibility-of-p-torsion-of-the-frey-curve/hardly-ramified-representations/index.html#frey-curve-hardly-ramified","scope":"flt/reducibility-of-p-torsion-of-the-frey-curve/hardly-ramified-representations","status":"can_prove","summary":"The $\\ell$-torsion $\\rho:\\operatorname{Gal}(\\overline{\\mathbb{Q}}/\\mathbb{Q})\\to\\operatorname{GL}_2(\\mathbb{Z}/\\ell\\mathbb{Z})$ in the Frey curve associated to a Frey package $(a,b,c,\\ell)$ is hardly ramified.","title":"Frey_curve_hardly_ramified"},{"id":"flt/reducibility-of-p-torsion-of-the-frey-curve/hardly-ramified-representations/hardly-ramified-reducible","kind":"node","lean":"FreyCurve.torsion_not_isIrreducible","parent":4256,"result":true,"route":"../roadmap/flt/reducibility-of-p-torsion-of-the-frey-curve/hardly-ramified-representations/index.html#hardly-ramified-reducible","scope":"flt/reducibility-of-p-torsion-of-the-frey-curve/hardly-ramified-representations","status":"stated","summary":"If $\\ell\\geq 3$ is a prime and $\\rho:\\operatorname{Gal}(\\overline{\\mathbb{Q}}/\\mathbb{Q})\\to\\operatorname{GL}_2(\\mathbb{Z}/\\ell\\mathbb{Z})$ is hardly ramified, then $\\rho$ is reducible.","title":"hardly_ramified_reducible"},{"id":"flt/reducibility-of-p-torsion-of-the-frey-curve/hardly-ramified-representations/wiles-frey-again","kind":"node","lean":"FLT.Bosses.B4_proof","parent":4256,"result":true,"route":"../roadmap/flt/reducibility-of-p-torsion-of-the-frey-curve/hardly-ramified-representations/index.html#wiles-frey-again","scope":"flt/reducibility-of-p-torsion-of-the-frey-curve/hardly-ramified-representations","status":"stated","summary":"If $\\overline{\\rho}$ is the mod $p$ Galois representation associated to a Frey package $(a,b,c,p)$ then $\\overline{\\rho}$ is reducible.","title":"Wiles_Frey_again"},{"id":"flt/reducibility-of-p-torsion-of-the-frey-curve/hardly-ramified-representations/hardly-ramified-lifts","kind":"node","lean":"GaloisRepresentation.IsHardlyRamified.lifts","parent":4256,"result":true,"route":"../roadmap/flt/reducibility-of-p-torsion-of-the-frey-curve/hardly-ramified-representations/index.html#hardly-ramified-lifts","scope":"flt/reducibility-of-p-torsion-of-the-frey-curve/hardly-ramified-representations","status":"can_prove","summary":"If $\\ell\\geq3$ is prime and $\\overline{\\rho}:\\operatorname{Gal}(\\overline{\\mathbb{Q}}/\\mathbb{Q})\\to\\operatorname{GL}_2(\\mathbb{Z}/\\ell\\mathbb{Z})$ is hardly ramified and irreducible, then there exists a finite extension $K$ of $\\mathbb{Q}_\\ell$ with integer ring $\\mathcal{O}$ and maximal ideal $\\mathfrak{m}$ and a hardly ramified representation\u2026","title":"hardly_ramified_lifts"},{"id":"flt/reducibility-of-p-torsion-of-the-frey-curve/hardly-ramified-representations/hardly-ramified-spreads-out","kind":"node","lean":"GaloisRepresentation.IsHardlyRamified.mem_isCompatible","parent":4256,"result":true,"route":"../roadmap/flt/reducibility-of-p-torsion-of-the-frey-curve/hardly-ramified-representations/index.html#hardly-ramified-spreads-out","scope":"flt/reducibility-of-p-torsion-of-the-frey-curve/hardly-ramified-representations","status":"can_prove","summary":"If $\\ell\\geq3$ is prime, $K$ is a finite extension of $\\mathbb{Q}_\\ell$ with integers $\\mathcal{O}$ and if $\\rho:\\operatorname{Gal}(\\overline{\\mathbb{Q}}/\\mathbb{Q})\\to\\operatorname{GL}_2(\\mathcal{O})$ is a hardly ramified representation whose reduction is irreducible, then there exists a number field $M$ and, for each finite place $\\mu$ of $M$ of characteristic prime to 2, with completion $M_\\mu$ having integer\u2026","title":"hardly_ramified_spreads_out"},{"id":"flt/reducibility-of-p-torsion-of-the-frey-curve/hardly-ramified-representations/hardly-ramified-mod3-reducible","kind":"node","lean":"GaloisRepresentation.IsHardlyRamified.mod_three","parent":4256,"result":true,"route":"../roadmap/flt/reducibility-of-p-torsion-of-the-frey-curve/hardly-ramified-representations/index.html#hardly-ramified-mod3-reducible","scope":"flt/reducibility-of-p-torsion-of-the-frey-curve/hardly-ramified-representations","status":"can_prove","summary":"Suppose $k$ is a finite field of characteristic 3, and suppose $\\overline{rho}:\\operatorname{Gal}(\\overline{\\mathbb{Q}}/\\mathbb{Q})\\to\\operatorname{GL}_2(k)$ is hardly ramified. Then $\\overline{\\rho}$ is an extension of the cyclotomic character by the trivial representation.","title":"hardly_ramified_mod3_reducible"},{"id":"flt/reducibility-of-p-torsion-of-the-frey-curve/hardly-ramified-representations/hardly-ramified-3adic-reducible","kind":"node","lean":"GaloisRepresentation.IsHardlyRamified.three_adic","parent":4256,"result":true,"route":"../roadmap/flt/reducibility-of-p-torsion-of-the-frey-curve/hardly-ramified-representations/index.html#hardly-ramified-3adic-reducible","scope":"flt/reducibility-of-p-torsion-of-the-frey-curve/hardly-ramified-representations","status":"can_prove","summary":"Suppose $L/\\mathbb{Q}_3$ is a finite extension, with integer ring $\\mathcal{O}_L$, and suppose $\\rho_3:\\operatorname{Gal}(\\overline{\\mathbb{Q}}/\\mathbb{Q})\\to\\operatorname{GL}_2(\\mathcal{O}_L)$ is hardly ramified. Then (considered as a representation to $\\operatorname{GL}_2(L)$) $\\rho_3^{ss}=1\\oplus\\chi_3$ where $1$ is the trivial character and $\\chi_3$ is the 3-adic cyclotomic character.","title":"hardly_ramified_3adic_reducible"},{"data_url":"scopes/flt/an-example-of-an-automorphic-form.json","id":"flt/an-example-of-an-automorphic-form","kind":"scope","parent":4231,"result":false,"route":"../roadmap/flt/an-example-of-an-automorphic-form/index.html","scope":"flt/an-example-of-an-automorphic-form","status":"can_prove","title":"An example of an automorphic form"},{"id":"flt/an-example-of-an-automorphic-form/introduction","kind":"node","parent":4265,"result":false,"route":"../roadmap/flt/an-example-of-an-automorphic-form/introduction/index.html","scope":"flt/an-example-of-an-automorphic-form","status":"can_state","summary":"The key ingredient in Wiles' proof of Fermat's Last Theorem is a *modularity lifting theorem*, sometimes called an $R=T$ theorem. For Wiles, the $R$ came from elliptic curves, the $T$ came from classical modular forms, and the fact that they're equal is basically the Shimura\u2013Taniyama\u2013Weil conjecture, now known as the Breuil\u2013Conrad\u2013Diamond\u2013Taylor modularity theorem: any elliptic curve over the rationals is modular.","title":"Introduction"},{"id":"flt/an-example-of-an-automorphic-form/a-quaternion-algebra","kind":"node","parent":4265,"result":false,"route":"../roadmap/flt/an-example-of-an-automorphic-form/a-quaternion-algebra/index.html","scope":"flt/an-example-of-an-automorphic-form","status":"can_state","summary":"Let's define $D$ to be the quaternion algebra $\\mathbb{Q}\\oplus\\mathbb{Q} i\\oplus\\mathbb{Q} j\\oplus\\mathbb{Q} k$. As a vector space, $D$ is 4-dimensional over $\\mathbb{Q}$ with $[1,i,j,k]$ giving a basis. It has a (non-commutative)ring structure, with multiplication satisfying the usual quaternion algebra relations $i^2=j^2=k^2=ijk=-1$. You can think of $D$ as an analogue of $2\\times 2$ matrices with rational\u2026","title":"A quaternion algebra"},{"data_url":"scopes/flt/an-example-of-an-automorphic-form/widehat-z.json","id":"flt/an-example-of-an-automorphic-form/widehat-z","kind":"scope","parent":4265,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\widehat{\\mathbb{Z}}\\)"]],"source":"$\\widehat{\\mathbb{Z}}$"},"result":false,"route":"../roadmap/flt/an-example-of-an-automorphic-form/widehat-z/index.html","scope":"flt/an-example-of-an-automorphic-form/widehat-z","status":"fully_proved","summary":"Classically automorphic forms were defined as functions on symmetric spaces (like the upper half plane) which transformed well under the action of certain discrete groups (for example $\\operatorname{SL}_2(\\mathbb{Z})$). However such definitions became combinatorially problematic when generalised to number fields with nontrivial class group, because the classical theory needed a *number* $p$ to define the Hecke\u2026","title":"$\\widehat{\\mathbb{Z}}$"},{"id":"flt/an-example-of-an-automorphic-form/widehat-z/zhat","kind":"node","lean":"ZHat","parent":4268,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/widehat-z/index.html#zhat","scope":"flt/an-example-of-an-automorphic-form/widehat-z","status":"fully_proved","summary":"The profinite completion $\\widehat{\\mathbb{Z}}$ of $\\mathbb{Z}$ is the set of all compatible collections $c=(c_N)_N$ of elements of $\\mathbb{Z}/N\\mathbb{Z}$ indexed by $\\mathbb{N}^+:=\\{1,2,3,\\ldots\\}$. A collection is said to be *compatible* if for all positive integers $D\\mid N$, we have $c_N$ mod $D$ equals $c_D$.","title":"ZHat"},{"id":"flt/an-example-of-an-automorphic-form/widehat-z/zhat-commring","kind":"node","lean":"ZHat.commRing","parent":4268,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/widehat-z/index.html#zhat-commring","scope":"flt/an-example-of-an-automorphic-form/widehat-z","status":"fully_proved","summary":"$\\widehat{\\mathbb{Z}}$ is a subring of $\\prod_{N\\geq1}(Z/N\\mathbb{Z})$ and in particular is a ring.","title":"ZHat.commRing"},{"id":"flt/an-example-of-an-automorphic-form/widehat-z/zhat-nontrivial","kind":"node","lean":"ZHat.nontrivial","parent":4268,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/widehat-z/index.html#zhat-nontrivial","scope":"flt/an-example-of-an-automorphic-form/widehat-z","status":"fully_proved","summary":"$0\\not=1$ in $\\widehat{\\mathbb{Z}}$.","title":"ZHat.nontrivial"},{"id":"flt/an-example-of-an-automorphic-form/widehat-z/zhat-charzero","kind":"node","lean":"ZHat.charZero","parent":4268,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/widehat-z/index.html#zhat-charzero","scope":"flt/an-example-of-an-automorphic-form/widehat-z","status":"fully_proved","summary":"The map from the naturals into $\\widehat{\\mathbb{Z}}$ sending $n$ to $n$ is injective.","title":"ZHat.charZero"},{"id":"flt/an-example-of-an-automorphic-form/widehat-z/zhat-e","kind":"node","lean":"ZHat.e","parent":4268,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/widehat-z/index.html#zhat-e","scope":"flt/an-example-of-an-automorphic-form/widehat-z","status":"fully_proved","summary":"The infinite sum $0!+1!+2!+3!+4!+5!+\\cdots$ looks like it makes no sense at all; it is the sum of an infinite series of larger and larger positive numbers. However, the sum is *finite* modulo $N$ for every positive integer $N$, because all the terms from $N!$ onwards are multiples of $N$ and thus are zero in $\\mathbb{Z}/N\\mathbb{Z}$. Thus it makes sense to define $e_N$ to be the value of the finite sum modulo $N$.\u2026","title":"ZHat.e"},{"id":"flt/an-example-of-an-automorphic-form/widehat-z/zhat-e-def","kind":"node","lean":"ZHat.e_def","parent":4268,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/widehat-z/index.html#zhat-e-def","scope":"flt/an-example-of-an-automorphic-form/widehat-z","status":"fully_proved","summary":"The collection $(e_N)_N$ is an element of $\\widehat{\\mathbb{Z}}$.","title":"ZHat.e_def"},{"id":"flt/an-example-of-an-automorphic-form/widehat-z/zhat-e-not-in-int","kind":"node","lean":"ZHat.e_not_in_Int","parent":4268,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/widehat-z/index.html#zhat-e-not-in-int","scope":"flt/an-example-of-an-automorphic-form/widehat-z","status":"fully_proved","summary":"The element $(e_N)_N$ of $\\widehat{\\mathbb{Z}}$ is not in $\\mathbb{Z}$.","title":"ZHat.e_not_in_Int"},{"id":"flt/an-example-of-an-automorphic-form/widehat-z/zhat-torsionfree","kind":"node","lean":"ZHat.torsionfree","parent":4268,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/widehat-z/index.html#zhat-torsionfree","scope":"flt/an-example-of-an-automorphic-form/widehat-z","status":"fully_proved","summary":"If $0\\lt N$ is an integer then multiplication by $N$ is injective on $\\widehat{\\mathbb{Z}}$.","title":"ZHat.torsionfree"},{"id":"flt/an-example-of-an-automorphic-form/widehat-z/zhat-multiples","kind":"node","lean":"ZHat.multiples","parent":4268,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/widehat-z/index.html#zhat-multiples","scope":"flt/an-example-of-an-automorphic-form/widehat-z","status":"fully_proved","summary":"The multiples of $N$ in $\\widehat{\\mathbb{Z}}$ are precisely the compatible collections $(z_i)_i\\in\\widehat{\\mathbb{Z}}$ with $z_N=0$.","title":"ZHat.multiples"},{"id":"flt/an-example-of-an-automorphic-form/more-advanced-remarks-on-widehat-z-versus-q","kind":"node","parent":4265,"reading_title":{"format":"inline-parts-v1","parts":[["text","More advanced remarks on "],["math","\\(\\widehat{\\mathbb{Z}}\\)"],["text"," versus "],["math","\\(\\mathbb{Q}\\)"]],"source":"More advanced remarks on $\\widehat{\\mathbb{Z}}$ versus $\\mathbb{Q}$"},"result":false,"route":"../roadmap/flt/an-example-of-an-automorphic-form/more-advanced-remarks-on-widehat-z-versus-q/index.html","scope":"flt/an-example-of-an-automorphic-form","status":"can_state","summary":"This section can be skipped on first reading.","title":"More advanced remarks on $\\widehat{\\mathbb{Z}}$ versus $\\mathbb{Q}$"},{"data_url":"scopes/flt/an-example-of-an-automorphic-form/widehat-q-and-tensor-products.json","id":"flt/an-example-of-an-automorphic-form/widehat-q-and-tensor-products","kind":"scope","parent":4265,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\widehat{\\mathbb{Q}}\\)"],["text"," and tensor products."]],"source":"$\\widehat{\\mathbb{Q}}$ and tensor products."},"result":false,"route":"../roadmap/flt/an-example-of-an-automorphic-form/widehat-q-and-tensor-products/index.html","scope":"flt/an-example-of-an-automorphic-form/widehat-q-and-tensor-products","status":"fully_proved","summary":"The definition of $\\widehat{\\mathbb{Q}}$ is easy if you know about tensor products of additive abelian groups.","title":"$\\widehat{\\mathbb{Q}}$ and tensor products."},{"id":"flt/an-example-of-an-automorphic-form/widehat-q-and-tensor-products/qhat","kind":"node","lean":"QHat","parent":4279,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/widehat-q-and-tensor-products/index.html#qhat","scope":"flt/an-example-of-an-automorphic-form/widehat-q-and-tensor-products","status":"fully_proved","summary":"The profinite completion $\\widehat{\\mathbb{Q}}$ of $\\mathbb{Q}$ is the tensor product $\\mathbb{Q}\\otimes_{\\mathbb{Z}}\\widehat{\\mathbb{Z}}$, or $\\widehat{\\mathbb{Q}}=\\mathbb{Q}\\otimes\\widehat{\\mathbb{Z}}$ for short.","title":"QHat"},{"data_url":"scopes/flt/an-example-of-an-automorphic-form/a-crash-course-in-tensor-products.json","id":"flt/an-example-of-an-automorphic-form/a-crash-course-in-tensor-products","kind":"scope","parent":4265,"result":false,"route":"../roadmap/flt/an-example-of-an-automorphic-form/a-crash-course-in-tensor-products/index.html","scope":"flt/an-example-of-an-automorphic-form/a-crash-course-in-tensor-products","status":"can_prove","summary":"We've defined $\\widehat{\\mathbb{Q}}$ to be $\\mathbb{Q}\\otimes\\widehat{\\mathbb{Z}}$. Whatever does this mean? Well just to orient yourself, if $A$ and $B$ are additive abelian groups, then $A\\otimes B$ is also an abelian group. And if $A$ and $B$ are commutative rings (as they are in our case), then $A\\otimes B$ is also a commutative ring.","title":"A crash course in tensor products"},{"id":"flt/an-example-of-an-automorphic-form/a-crash-course-in-tensor-products/qhat-canonicalform","kind":"node","lean":"QHat.canonicalForm","parent":4281,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/a-crash-course-in-tensor-products/index.html#qhat-canonicalform","scope":"flt/an-example-of-an-automorphic-form/a-crash-course-in-tensor-products","status":"fully_proved","summary":"Every element of $\\widehat{\\mathbb{Q}}:=\\mathbb{Q}\\otimes\\widehat{\\mathbb{Z}}$ can be written as $q\\otimes_t z$ with $q\\in\\mathbb{Q}$ and $z\\in\\widehat{\\mathbb{Z}}$. Furthermore one can even assume that $q=\\frac{1}{N}$ for some positive integer $N$.","title":"QHat.canonicalForm"},{"id":"flt/an-example-of-an-automorphic-form/a-crash-course-in-tensor-products/qhat-iscoprime","kind":"node","lean":"QHat.IsCoprime","parent":4281,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/a-crash-course-in-tensor-products/index.html#qhat-iscoprime","scope":"flt/an-example-of-an-automorphic-form/a-crash-course-in-tensor-products","status":"fully_proved","summary":"If $N\\in\\mathbb{N}^+$ and $z\\in\\widehat{\\mathbb{Z}}$ then we say that $N$ and $z$ are *coprime* if $z_N\\in(\\mathbb{Z}/N\\mathbb{Z})^\\times$. We write $z/N$ as notation for the element $\\frac{1}{N}\\otimes_tz$.","title":"QHat.IsCoprime"},{"id":"flt/an-example-of-an-automorphic-form/a-crash-course-in-tensor-products/qhat-lowestterms","kind":"node","lean":"QHat.lowestTerms","parent":4281,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/a-crash-course-in-tensor-products/index.html#qhat-lowestterms","scope":"flt/an-example-of-an-automorphic-form/a-crash-course-in-tensor-products","status":"can_prove","summary":"Every element of $\\widehat{\\mathbb{Q}}$ can be uniquely written as $z/N$ with $z\\in\\widehat{\\mathbb{Z}}$, $N\\in\\mathbb{N}^+$, and with $N$ and $z$ coprime.","title":"QHat.lowestTerms"},{"id":"flt/an-example-of-an-automorphic-form/a-crash-course-in-tensor-products/qhat-injective-rat","kind":"node","lean":"QHat.injective_rat","parent":4281,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/a-crash-course-in-tensor-products/index.html#qhat-injective-rat","scope":"flt/an-example-of-an-automorphic-form/a-crash-course-in-tensor-products","status":"fully_proved","summary":"The ring homomorphism $\\mathbb{Q}\\to\\widehat{\\mathbb{Q}}$ sending $q$ to $q\\otimes_t 1$ is injective.","title":"QHat.injective_rat"},{"id":"flt/an-example-of-an-automorphic-form/a-crash-course-in-tensor-products/qhat-injective-zhat","kind":"node","lean":"QHat.injective_zHat","parent":4281,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/a-crash-course-in-tensor-products/index.html#qhat-injective-zhat","scope":"flt/an-example-of-an-automorphic-form/a-crash-course-in-tensor-products","status":"fully_proved","summary":"The ring homomorphism $\\widehat{\\mathbb{Z}}\\to\\widehat{\\mathbb{Q}}$ sending $z$ to $1\\otimes_t z$ is injective.","title":"QHat.injective_zHat"},{"data_url":"scopes/flt/an-example-of-an-automorphic-form/additive-structure-of-widehat-q.json","id":"flt/an-example-of-an-automorphic-form/additive-structure-of-widehat-q","kind":"scope","parent":4265,"reading_title":{"format":"inline-parts-v1","parts":[["text","Additive structure of "],["math","\\(\\widehat{\\mathbb{Q}}\\)"],["text","."]],"source":"Additive structure of $\\widehat{\\mathbb{Q}}$."},"result":false,"route":"../roadmap/flt/an-example-of-an-automorphic-form/additive-structure-of-widehat-q/index.html","scope":"flt/an-example-of-an-automorphic-form/additive-structure-of-widehat-q","status":"proved","summary":"Here we forget the ring structure on everything, and analyse $\\widehat{\\mathbb{Q}}$ as an additive abelian group, and in particular how the subgroups $\\mathbb{Z}$, $\\mathbb{Q}$ and $\\widehat{\\mathbb{Z}}$ sit within it.","title":"Additive structure of $\\widehat{\\mathbb{Q}}$."},{"id":"flt/an-example-of-an-automorphic-form/additive-structure-of-widehat-q/qhat-rat-meet-zhat","kind":"node","lean":"QHat.rat_meet_zHat","parent":4287,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/additive-structure-of-widehat-q/index.html#qhat-rat-meet-zhat","scope":"flt/an-example-of-an-automorphic-form/additive-structure-of-widehat-q","status":"proved","summary":"The intersection of $\\mathbb{Q}$ and $\\widehat{\\mathbb{Z}}$ in $\\widehat{\\mathbb{Q}}$ is $\\mathbb{Z}$.","title":"QHat.rat_meet_zHat"},{"id":"flt/an-example-of-an-automorphic-form/additive-structure-of-widehat-q/qhat-rat-join-zhat","kind":"node","lean":"QHat.rat_join_zHat","parent":4287,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/additive-structure-of-widehat-q/index.html#qhat-rat-join-zhat","scope":"flt/an-example-of-an-automorphic-form/additive-structure-of-widehat-q","status":"fully_proved","summary":"The sum of $\\mathbb{Q}$ and $\\widehat{\\mathbb{Z}}$ in $\\widehat{\\mathbb{Q}}$ is $\\widehat{\\mathbb{Q}}$. More precisely, every element of $\\widehat{\\mathbb{Q}}$ can be written as $q+z$ with $q\\in\\mathbb{Q}$ and $z\\in\\widehat{\\mathbb{Z}}$, or more precisely as $q\\otimes_t 1+1\\otimes_t z$.","title":"QHat.rat_join_zHat"},{"data_url":"scopes/flt/an-example-of-an-automorphic-form/multiplicative-structure-of-the-units-of-widehat.json","id":"flt/an-example-of-an-automorphic-form/multiplicative-structure-of-the-units-of-widehat","kind":"scope","parent":4265,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiplicative structure of the units of "],["math","\\(\\widehat{\\mathbb{Q}}\\)"],["text","."]],"source":"Multiplicative structure of the units of $\\widehat{\\mathbb{Q}}$."},"result":false,"route":"../roadmap/flt/an-example-of-an-automorphic-form/multiplicative-structure-of-the-units-of-widehat/index.html","scope":"flt/an-example-of-an-automorphic-form/multiplicative-structure-of-the-units-of-widehat","status":"proved","summary":"We now forget the additive structure on the commutative ring $\\widehat{\\mathbb{Q}}$ and consider the multiplicative structure of its group of units $\\widehat{\\mathbb{Q}}^\\times$ (which I couldn't get into the section title). We have the obvious subgroups $\\mathbb{Q}^\\times$, $\\mathbb{Z}^\\times$ and $\\widehat{\\mathbb{Z}}^\\times$.","title":"Multiplicative structure of the units of $\\widehat{\\mathbb{Q}}$."},{"id":"flt/an-example-of-an-automorphic-form/multiplicative-structure-of-the-units-of-widehat/qhat-unitsrat-meet-unitszhat","kind":"node","lean":"QHat.unitsrat_meet_unitszHat","parent":4290,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/multiplicative-structure-of-the-units-of-widehat/index.html#qhat-unitsrat-meet-unitszhat","scope":"flt/an-example-of-an-automorphic-form/multiplicative-structure-of-the-units-of-widehat","status":"proved","summary":"The intersection of $\\mathbb{Q}^\\times$ and $\\widehat{\\mathbb{Z}}^\\times$ in $\\widehat{\\mathbb{Q}}^\\times$ is $\\mathbb{Z}^\\times$.","title":"Qhat.unitsrat_meet_unitszHat"},{"id":"flt/an-example-of-an-automorphic-form/multiplicative-structure-of-the-units-of-widehat/qhat-unitsrat-join-unitszhat","kind":"node","lean":"QHat.unitsrat_join_unitszHat","parent":4290,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/multiplicative-structure-of-the-units-of-widehat/index.html#qhat-unitsrat-join-unitszhat","scope":"flt/an-example-of-an-automorphic-form/multiplicative-structure-of-the-units-of-widehat","status":"fully_proved","summary":"The product of $\\mathbb{Q}^\\times$ and $\\widehat{\\mathbb{Z}}^\\times$ in $\\widehat{\\mathbb{Q}}^\\times$ is all of $\\widehat{\\mathbb{Q}}^\\times$. More precisely, every element of $\\widehat{\\mathbb{Q}}^\\times$ can be written as $qz$ with $q\\in\\mathbb{Q}^\\times$ and $z\\in\\widehat{\\mathbb{Z}}^\\times$.","title":"QHat.unitsrat_join_unitszHat"},{"data_url":"scopes/flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions.json","id":"flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions","kind":"scope","parent":4265,"result":false,"route":"../roadmap/flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions/index.html","scope":"flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions","status":"fully_proved","summary":"This ring is isomorphic to $\\mathbb{Z}^4$ as an additive group, and $\\mathcal{O}\\otimes_{\\mathbb{Z}}\\mathbb{R}=\\mathbb{R}\\oplus \\mathbb{R} i\\oplus \\mathbb{R} j\\oplus\\mathbb{R} \\omega$ is the usual Hamilton quaternions.","title":"The Hurwitz quaternions"},{"id":"flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions/hurwitz","kind":"node","lean":"Hurwitz","parent":4293,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions/index.html#hurwitz","scope":"flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions","status":"fully_proved","summary":"The Hurwitz quaternions are the set $\\mathcal{O} := \\mathbb{Z}\\oplus\\mathbb{Z} \\omega\\oplus\\mathbb{Z} i\\oplus \\mathbb{Z} i\\omega$ (as an abstract abelian group or as a subgroup of the usual quaternions). Here $\\omega=\\frac{-1+(i+j+k)}{2}$ and note that $(i+j+k)^2=-3$. We have $\\overline{\\omega}=\\omega^2=-(\\omega+1)$. A general quaternion $a+bi+cj+dk$ is a Hurwitz quaternion if either $a,b,c,d\\in\\mathbb{Z}$ or\u2026","title":"Hurwitz"},{"id":"flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions/hurwitz-ring","kind":"node","lean":"Hurwitz.ring","parent":4293,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions/index.html#hurwitz-ring","scope":"flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions","status":"fully_proved","summary":"The Hurwitz quaternions form a ring.","title":"Hurwitz.ring"},{"id":"flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions/hurwitz-starring","kind":"node","lean":"Hurwitz.starRing","parent":4293,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions/index.html#hurwitz-starring","scope":"flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions","status":"fully_proved","summary":"There's a conjugation map (which we'll call \"star\") from the Hurwitz quaternions to themselves, sending integers to themselves and purely imaginary elements like $2\\omega+1$ to minus themselves. It satisfies $(x^*)^*=x$, $(xy)^*=y^*x^*$ and $(x+y)^*=x^*+y^*$. In particular, the Hurwitz quaternions are a \"star ring\" in the sense of mathlib.","title":"Hurwitz.starRing"},{"id":"flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions/hurwitz-norm","kind":"node","lean":"Hurwitz.norm","parent":4293,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions/index.html#hurwitz-norm","scope":"flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions","status":"fully_proved","summary":"The Hurwitz quaternions come equipped with an integer-valued norm, which is $a^2+b^2+c^2+d^2$ on $a+bi+cj+dk$ but needs to be modified a bit to deal with $\\omega$.","title":"Hurwitz.norm"},{"id":"flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions/hurwitz-norm-eq-mul-conj","kind":"node","lean":"Hurwitz.norm_eq_mul_conj","parent":4293,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions/index.html#hurwitz-norm-eq-mul-conj","scope":"flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions","status":"fully_proved","summary":"We have $N(x)=x\\overline{x}$.","title":"Hurwitz.norm_eq_mul_conj"},{"id":"flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions/hurwitz-norm-zero","kind":"node","lean":"Hurwitz.norm_zero","parent":4293,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions/index.html#hurwitz-norm-zero","scope":"flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions","status":"fully_proved","summary":"The norm of $0$ is $0$.","title":"Hurwitz.norm_zero"},{"id":"flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions/hurwitz-norm-one","kind":"node","lean":"Hurwitz.norm_one","parent":4293,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions/index.html#hurwitz-norm-one","scope":"flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions","status":"fully_proved","summary":"The norm of $1$ is $1$.","title":"Hurwitz.norm_one"},{"id":"flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions/hurwitz-norm-mul","kind":"node","lean":"Hurwitz.norm_mul","parent":4293,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions/index.html#hurwitz-norm-mul","scope":"flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions","status":"fully_proved","summary":"The norm of a product is the product of the norms.","title":"Hurwitz.norm_mul"},{"id":"flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions/hurwitz-norm-nonneg","kind":"node","lean":"Hurwitz.norm_nonneg","parent":4293,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions/index.html#hurwitz-norm-nonneg","scope":"flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions","status":"fully_proved","summary":"The norm of an element is nonnegative.","title":"Hurwitz.norm_nonneg"},{"id":"flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions/hurwitz-norm-eq-zero","kind":"node","lean":"Hurwitz.norm_eq_zero","parent":4293,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions/index.html#hurwitz-norm-eq-zero","scope":"flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions","status":"fully_proved","summary":"The norm of an element is zero if and only if the element is zero.","title":"Hurwitz.norm_eq_zero"},{"id":"flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions/hurwitz-exists-near","kind":"node","lean":"Hurwitz.exists_near","parent":4293,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions/index.html#hurwitz-exists-near","scope":"flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions","status":"fully_proved","summary":"Given a \u201cusual\u201d quaternion $a=x+yi+zj+wk$ with $x,y,z,w\\in\\mathbb{R}$, there exists a Hurwitz quaternion $q$ such that $N(a-q)<1$.","title":"Hurwitz.exists_near"},{"id":"flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions/hurwitz-quot-rem","kind":"node","lean":"Hurwitz.quot_rem","parent":4293,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions/index.html#hurwitz-quot-rem","scope":"flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions","status":"fully_proved","summary":"Given two Hurwitz quaternions $a$ and $b$ with $b$ nonzero, there exists $q$ and $r$ such that $a=qb+r$ and $N(r)\\lt N(b)$.","title":"Hurwitz.quot_rem"},{"id":"flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions/hurwitz-left-ideal-princ","kind":"node","lean":"Hurwitz.left_ideal_princ","parent":4293,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions/index.html#hurwitz-left-ideal-princ","scope":"flt/an-example-of-an-automorphic-form/the-hurwitz-quaternions","status":"fully_proved","summary":"All left ideals of $\\mathcal{O}$ are principal.","title":"Hurwitz.left_ideal_princ"},{"data_url":"scopes/flt/an-example-of-an-automorphic-form/profinite-completion-of-the-hurwitz-quaternions.json","id":"flt/an-example-of-an-automorphic-form/profinite-completion-of-the-hurwitz-quaternions","kind":"scope","parent":4265,"result":false,"route":"../roadmap/flt/an-example-of-an-automorphic-form/profinite-completion-of-the-hurwitz-quaternions/index.html","scope":"flt/an-example-of-an-automorphic-form/profinite-completion-of-the-hurwitz-quaternions","status":"can_prove","summary":"We define $\\widehat{\\mathcal{O}}$ to be $\\mathcal{O}\\otimes\\widehat{\\mathbb{Z}}$, so it's elements $a+bi+cj+d\\omega$ with $a,b,c,d\\in\\widehat{\\mathbb{Z}}$.","title":"Profinite completion of the Hurwitz quaternions"},{"id":"flt/an-example-of-an-automorphic-form/profinite-completion-of-the-hurwitz-quaternions/hurwitzrathat-canonicalform","kind":"node","lean":"HurwitzRatHat.canonicalForm","parent":4307,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/profinite-completion-of-the-hurwitz-quaternions/index.html#hurwitzrathat-canonicalform","scope":"flt/an-example-of-an-automorphic-form/profinite-completion-of-the-hurwitz-quaternions","status":"can_prove","summary":"Every element of $\\widehat{D}$ can be written as $z/N$ with $z\\in\\widehat{\\mathcal{O}}$ and $N\\in\\mathbb{N}^+$.","title":"HurwitzRatHat.canonicalForm"},{"id":"flt/an-example-of-an-automorphic-form/profinite-completion-of-the-hurwitz-quaternions/hurwitzrathat-completed-units","kind":"node","lean":"HurwitzRatHat.completed_units","parent":4307,"result":true,"route":"../roadmap/flt/an-example-of-an-automorphic-form/profinite-completion-of-the-hurwitz-quaternions/index.html#hurwitzrathat-completed-units","scope":"flt/an-example-of-an-automorphic-form/profinite-completion-of-the-hurwitz-quaternions","status":"can_prove","summary":"The group of units of $\\widehat{D}$ is $D^\\times\\widehat{\\mathcal{O}}^\\times$. More precisely, every element of $\\widehat{D}^\\times$ can be written as a product $\\delta u$ with $\\delta\\in D^\\times$ and $u\\in\\widehat{\\mathcal{O}}^\\times$.","title":"HurwitzRatHat.completed_units"},{"data_url":"scopes/flt/automorphic-forms-and-the-langlands-conjectures.json","id":"flt/automorphic-forms-and-the-langlands-conjectures","kind":"scope","parent":4231,"result":false,"route":"../roadmap/flt/automorphic-forms-and-the-langlands-conjectures/index.html","scope":"flt/automorphic-forms-and-the-langlands-conjectures","status":"fully_proved","summary":"This chapter came from discussions between Patrick, Mario and myself, all currently visiting the Hausdorff Research Institute for Mathematics in Bonn. The ultimate goal is to formally state some version of the global Langlands reciprocity conjectures for $\\operatorname{GL}_n$ over $\\mathbb{Q}$.","title":"Automorphic forms and the Langlands Conjectures"},{"id":"flt/automorphic-forms-and-the-langlands-conjectures/definition-of-an-automorphic-form-for-gl-n-over","kind":"node","parent":4310,"reading_title":{"format":"inline-parts-v1","parts":[["text","Definition of an automorphic form for "],["math","\\(\\operatorname{GL}_n\\)"],["text"," over "],["math","\\(\\mathbb{Q}\\)"],["text","."]],"source":"Definition of an automorphic form for $\\operatorname{GL}_n$ over $\\mathbb{Q}$."},"result":false,"route":"../roadmap/flt/automorphic-forms-and-the-langlands-conjectures/definition-of-an-automorphic-form-for-gl-n-over/index.html","scope":"flt/automorphic-forms-and-the-langlands-conjectures","status":"can_state","summary":"The global Langlands reciprocity conjectures relate automorphic forms to Galois representations. The statements for a general connected reductive group involve the construction of the Langlands dual group, and we do not have quite enough Lie algebra theory to push this definition through in general. However if we restrict the special case of the group $\\operatorname{GL}_n/\\mathbb{Q}$, the dual group is just\u2026","title":"Definition of an automorphic form for $\\operatorname{GL}_n$ over $\\mathbb{Q}$."},{"id":"flt/automorphic-forms-and-the-langlands-conjectures/the-finite-adeles-of-the-rationals","kind":"node","parent":4310,"result":false,"route":"../roadmap/flt/automorphic-forms-and-the-langlands-conjectures/the-finite-adeles-of-the-rationals/index.html","scope":"flt/automorphic-forms-and-the-langlands-conjectures","status":"can_state","summary":"Mathlib already has the definition of the finite adeles $\\mathbb{A}_{\\mathbb{Q}}^f$ of the rationals as a commutative $\\mathbb{Q}$-algebra, and the proof that it's a topological ring.","title":"The finite adeles of the rationals."},{"id":"flt/automorphic-forms-and-the-langlands-conjectures/the-group-gl-n-a-q","kind":"node","parent":4310,"reading_title":{"format":"inline-parts-v1","parts":[["text","The group "],["math","\\(\\operatorname{GL}_n(\\mathbb{A}_{\\mathbb{Q}})\\)"],["text","."]],"source":"The group $\\operatorname{GL}_n(\\mathbb{A}_{\\mathbb{Q}})$."},"result":false,"route":"../roadmap/flt/automorphic-forms-and-the-langlands-conjectures/the-group-gl-n-a-q/index.html","scope":"flt/automorphic-forms-and-the-langlands-conjectures","status":"can_state","summary":"The adeles $\\mathbb{A}_{\\mathbb{Q}}$ of $\\mathbb{Q}$ are the product $\\mathbb{A}_{\\mathbb{Q}}^f\\times\\mathbb{R}$, with the product topology. They are a topological ring. Hence $\\operatorname{GL}_n(\\mathbb{A}_{\\mathbb{Q}})=\\operatorname{GL}_n(\\mathbb{A}_{\\mathbb{Q}}^f)\\times\\operatorname{GL}_n(\\mathbb{R})$ is a topological group, where we are being a bit liberal with our use of the equality symbol.","title":"The group $\\operatorname{GL}_n(\\mathbb{A}_{\\mathbb{Q}})$."},{"data_url":"scopes/flt/automorphic-forms-and-the-langlands-conjectures/slowly-increasing-functions.json","id":"flt/automorphic-forms-and-the-langlands-conjectures/slowly-increasing-functions","kind":"scope","parent":4310,"result":false,"route":"../roadmap/flt/automorphic-forms-and-the-langlands-conjectures/slowly-increasing-functions/index.html","scope":"flt/automorphic-forms-and-the-langlands-conjectures/slowly-increasing-functions","status":"fully_proved","summary":"Automorphic representations satisfy a growth condition which we may as well factor out into a separate definition.","title":"Slowly-increasing functions"},{"id":"flt/automorphic-forms-and-the-langlands-conjectures/slowly-increasing-functions/automorphicform-gln-isslowlyincreasing","kind":"node","lean":"AutomorphicForm.GLn.IsSlowlyIncreasing","parent":4314,"result":true,"route":"../roadmap/flt/automorphic-forms-and-the-langlands-conjectures/slowly-increasing-functions/index.html#automorphicform-gln-isslowlyincreasing","scope":"flt/automorphic-forms-and-the-langlands-conjectures/slowly-increasing-functions","status":"fully_proved","summary":"We say that a function $f:\\operatorname{GL}_n(\\mathbb{R})\\to\\mathbb{C}$ is *slowly-increasing* if there's some real constant $C$ and positive integer $n$ such that $|f(M)|\\leq Cs(M)^n$ for all $M\\in\\operatorname{GL}_n(\\mathbb{R})$.","title":"AutomorphicForm.GLn.IsSlowlyIncreasing"},{"data_url":"scopes/flt/miniproject-frobenius-elements.json","id":"flt/miniproject-frobenius-elements","kind":"scope","parent":4231,"result":false,"route":"../roadmap/flt/miniproject-frobenius-elements/index.html","scope":"flt/miniproject-frobenius-elements","status":"mathlib","title":"Miniproject: Frobenius elements"},{"id":"flt/miniproject-frobenius-elements/status","kind":"node","parent":4316,"result":false,"route":"../roadmap/flt/miniproject-frobenius-elements/status/index.html","scope":"flt/miniproject-frobenius-elements","status":"can_state","summary":"This miniproject has been a success: the main results are sorry-free and merged into mathlib (see this PR). As a result there will be no more work on this miniproject in the FLT repo. Below is a fairly detailed sketch of the argument used.","title":"Status"},{"id":"flt/miniproject-frobenius-elements/introduction-and-goal","kind":"node","parent":4316,"result":false,"route":"../roadmap/flt/miniproject-frobenius-elements/introduction-and-goal/index.html","scope":"flt/miniproject-frobenius-elements","status":"can_state","summary":"When this project started, I had thought that the existence of Frobenius elements was specific to the theory of local and global fields, and a slightly more general result held for Dedekind domains. Then Joel Riou pointed out on the Lean Zulip an extremely general result from from Bourbaki's Commutative Algebra (Chapter V, Section 2, number 2, theorem 2, part (ii)). This beautiful result is surely what we want to\u2026","title":"Introduction and goal"},{"data_url":"scopes/flt/miniproject-frobenius-elements/statement-of-the-theorem.json","id":"flt/miniproject-frobenius-elements/statement-of-the-theorem","kind":"scope","parent":4316,"result":false,"route":"../roadmap/flt/miniproject-frobenius-elements/statement-of-the-theorem/index.html","scope":"flt/miniproject-frobenius-elements/statement-of-the-theorem","status":"mathlib","summary":"The set-up throughout this project: $G$ is a finite group acting (via ring isomorphisms) on a commutative ring $B$, and $A$ is the subring of $G$-invariants.","title":"Statement of the theorem"},{"id":"flt/miniproject-frobenius-elements/statement-of-the-theorem/isfractionring-stabilizerhom","kind":"node","parent":4319,"result":true,"route":"../roadmap/flt/miniproject-frobenius-elements/statement-of-the-theorem/index.html#isfractionring-stabilizerhom","scope":"flt/miniproject-frobenius-elements/statement-of-the-theorem","status":"mathlib","summary":"Choose $g\\in D_Q$. Then the action of $g$ on $B$ gives us an induced $A/P$-algebra automorphism of $B/Q$ which extends to a $K$-algebra automorphism $\\phi(g)$ of $L$. This construction $g\\mapsto \\phi(g)$ defines a group homomorphism from $D_Q$ to $\\operatorname{Aut}(L/K)$ (all the proofs implicit in the definition here are straightforward).","title":"IsFractionRing.stabilizerHom"},{"id":"flt/miniproject-frobenius-elements/statement-of-the-theorem/isfractionring-stabilizerhom-surjective","kind":"node","parent":4319,"result":true,"route":"../roadmap/flt/miniproject-frobenius-elements/statement-of-the-theorem/index.html#isfractionring-stabilizerhom-surjective","scope":"flt/miniproject-frobenius-elements/statement-of-the-theorem","status":"mathlib","summary":"The map $g\\mapsto \\phi_g$ from $D_Q$ to $\\operatorname{Aut}(L/K)$ defined above is surjective.","title":"IsFractionRing.stabilizerHom_surjective"},{"data_url":"scopes/flt/miniproject-frobenius-elements/the-extension-b-a.json","id":"flt/miniproject-frobenius-elements/the-extension-b-a","kind":"scope","parent":4316,"reading_title":{"format":"inline-parts-v1","parts":[["text","The extension "],["math","\\(B/A\\)"],["text","."]],"source":"The extension $B/A$."},"result":false,"route":"../roadmap/flt/miniproject-frobenius-elements/the-extension-b-a/index.html","scope":"flt/miniproject-frobenius-elements/the-extension-b-a","status":"mathlib","summary":"The precise set-up we'll work in is the following. We fix $G$ a finite group acting on $B$ a commutative ring, and we have another commutative ring $A$ such that $B$ is an $A$-algebra and the image of $A$ in $B$ is precisely the $G$-invariant elements of $B$. We don't ever need the map $A\\to B$ to be injective so we don't assume this.","title":"The extension $B/A$."},{"id":"flt/miniproject-frobenius-elements/the-extension-b-a/mulsemiringaction-charpoly","kind":"node","parent":4322,"result":true,"route":"../roadmap/flt/miniproject-frobenius-elements/the-extension-b-a/index.html#mulsemiringaction-charpoly","scope":"flt/miniproject-frobenius-elements/the-extension-b-a","status":"mathlib","summary":"If $b\\in B$ then define the *characteristic polynomial* $F_b(X) \\in B[X]$ of $b$ to be $\\prod_{g\\in G}(X-g\\cdot b)$.","title":"MulSemiringAction.charpoly"},{"id":"flt/miniproject-frobenius-elements/the-extension-b-a/mulsemiringaction-monic-charpoly","kind":"node","parent":4322,"result":true,"route":"../roadmap/flt/miniproject-frobenius-elements/the-extension-b-a/index.html#mulsemiringaction-monic-charpoly","scope":"flt/miniproject-frobenius-elements/the-extension-b-a","status":"mathlib","summary":"$F_b$ is monic.","title":"MulSemiringAction.monic_charpoly"},{"id":"flt/miniproject-frobenius-elements/the-extension-b-a/algebra-isinvariant-charpoly-mem-lifts","kind":"node","parent":4322,"result":true,"route":"../roadmap/flt/miniproject-frobenius-elements/the-extension-b-a/index.html#algebra-isinvariant-charpoly-mem-lifts","scope":"flt/miniproject-frobenius-elements/the-extension-b-a","status":"mathlib","summary":"$F_b$ is the lift of some monic polynomial $M_b$ in $A[X]$.","title":"Algebra.IsInvariant.charpoly_mem_lifts"},{"id":"flt/miniproject-frobenius-elements/the-extension-b-a/algebra-isinvariant-isintegral","kind":"node","parent":4322,"result":true,"route":"../roadmap/flt/miniproject-frobenius-elements/the-extension-b-a/index.html#algebra-isinvariant-isintegral","scope":"flt/miniproject-frobenius-elements/the-extension-b-a","status":"mathlib","summary":"$B/A$ is integral.","title":"Algebra.IsInvariant.isIntegral"},{"data_url":"scopes/flt/miniproject-frobenius-elements/the-extension-l-k.json","id":"flt/miniproject-frobenius-elements/the-extension-l-k","kind":"scope","parent":4316,"reading_title":{"format":"inline-parts-v1","parts":[["text","The extension "],["math","\\(L/K\\)"],["text","."]],"source":"The extension $L/K$."},"result":false,"route":"../roadmap/flt/miniproject-frobenius-elements/the-extension-l-k/index.html","scope":"flt/miniproject-frobenius-elements/the-extension-l-k","status":"mathlib","summary":"Let $L^{D_Q}$ denote the fixed field of the action $D_Q$ on $L$. Our strategy for proving surjectivity of $D_Q \\to \\operatorname{Aut}(L/K)$ will be to write this map as the composition $D_Q \\to \\operatorname{Aut}(L/L^{D_Q}) \\to \\operatorname{Aut}(L/K)$.","title":"The extension $L/K$."},{"id":"flt/miniproject-frobenius-elements/the-extension-l-k/fixedpoints-toalgaut-surjective","kind":"node","parent":4327,"result":true,"route":"../roadmap/flt/miniproject-frobenius-elements/the-extension-l-k/index.html#fixedpoints-toalgaut-surjective","scope":"flt/miniproject-frobenius-elements/the-extension-l-k","status":"mathlib","summary":"Let $H$ be a finite group acting on a field $F$ by field automorphisms. Then the map $H \\to \\operatorname{Aut}(F/F^H)$ is surjective.","title":"FixedPoints.toAlgAut_surjective"},{"id":"flt/miniproject-frobenius-elements/the-extension-l-k/isalgebraic-exists-smul-eq-mul","kind":"node","parent":4327,"result":true,"route":"../roadmap/flt/miniproject-frobenius-elements/the-extension-l-k/index.html#isalgebraic-exists-smul-eq-mul","scope":"flt/miniproject-frobenius-elements/the-extension-l-k","status":"mathlib","summary":"If $R/S$ is an algebraic extension of integral domains, then any fraction $a/b$ with $a,b\\in R$ can be written as $c/d$ with $c\\in R$ and $d\\in S$.","title":"IsAlgebraic.exists_smul_eq_mul"},{"data_url":"scopes/flt/miniproject-adeles.json","id":"flt/miniproject-adeles","kind":"scope","parent":4231,"result":false,"route":"../roadmap/flt/miniproject-adeles/index.html","scope":"flt/miniproject-adeles","status":"can_prove","title":"Miniproject: Adeles"},{"id":"flt/miniproject-adeles/status","kind":"node","parent":4330,"result":false,"route":"../roadmap/flt/miniproject-adeles/status/index.html","scope":"flt/miniproject-adeles","status":"can_state","summary":"This is an active miniproject.","title":"Status"},{"id":"flt/miniproject-adeles/the-goal","kind":"node","parent":4330,"result":false,"route":"../roadmap/flt/miniproject-adeles/the-goal/index.html","scope":"flt/miniproject-adeles","status":"can_state","summary":"There are several goals to this miniproject.","title":"The goal"},{"data_url":"scopes/flt/miniproject-adeles/local-compactness.json","id":"flt/miniproject-adeles/local-compactness","kind":"scope","parent":4330,"result":false,"route":"../roadmap/flt/miniproject-adeles/local-compactness/index.html","scope":"flt/miniproject-adeles/local-compactness","status":"can_prove","summary":"As mentioned above, Salvatore Mercuri was the first to give a complete formalisation of the proof that the adele ring is locally compact as a topological space. His work is in his own repo and proved the result using the \u201cad hoc\u201d topology on the adeles which we initially had. Since then, adeles have been refactored to have the direct limit topology and mathlib has\u2026","title":"Local compactness"},{"id":"flt/miniproject-adeles/local-compactness/numberfield-instcompactspaceadiccompletionintege","kind":"node","lean":"NumberField.instCompactSpaceAdicCompletionIntegers","parent":4333,"result":true,"route":"../roadmap/flt/miniproject-adeles/local-compactness/index.html#numberfield-instcompactspaceadiccompletionintege","scope":"flt/miniproject-adeles/local-compactness","status":"can_prove","summary":"If $K$ is a number field and $v$ is a nonzero prime ideal of the integers of $K$, then the integers of $K_v$ is a compact open subgroup.","title":"NumberField.instCompactSpaceAdicCompletionIntegers"},{"id":"flt/miniproject-adeles/local-compactness/numberfield-adelering-locallycompactspace","kind":"node","lean":"NumberField.AdeleRing.locallyCompactSpace","parent":4333,"result":true,"route":"../roadmap/flt/miniproject-adeles/local-compactness/index.html#numberfield-adelering-locallycompactspace","scope":"flt/miniproject-adeles/local-compactness","status":"can_prove","summary":"The adeles of a number field are locally compact.","title":"NumberField.AdeleRing.locallyCompactSpace"},{"data_url":"scopes/flt/miniproject-adeles/base-change.json","id":"flt/miniproject-adeles/base-change","kind":"scope","parent":4330,"result":false,"route":"../roadmap/flt/miniproject-adeles/base-change/index.html","scope":"flt/miniproject-adeles/base-change","status":"can_prove","summary":"The \u201ctheorem\u201d we want is that if $L/K$ is a finite extension of number fields, then $\\mathbb{A}_L=L\\otimes_K\\mathbb{A}_K$. This isn't a theorem though, this is actually a *definition* (the map between the two objects) and a theorem about the definition (that it's an isomorphism). In fact the full claim is that it is both a homeomorphism and an $L$-algebra isomorphism. Before we can prove the theorem, we need to\u2026","title":"Base change"},{"data_url":"scopes/flt/miniproject-adeles/base-change/part-01.json","id":"flt/miniproject-adeles/base-change/part-01","kind":"scope","navigation":"transparent","parent":4336,"result":false,"route":"../roadmap/flt/miniproject-adeles/base-change/part-01/index.html","scope":"flt/miniproject-adeles/base-change/part-01","status":"can_prove","summary":"Because of the commutative diagram","title":"IsDedekindDomain.HeightOneSpectr\u2026 to NumberField.InfinitePlace.Comple\u2026"},{"id":"flt/miniproject-adeles/base-change/part-01/isdedekinddomain-heightonespectrum-valuation-com","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.valuation_comap","parent":4337,"result":true,"route":"../roadmap/flt/miniproject-adeles/base-change/part-01/index.html#isdedekinddomain-heightonespectrum-valuation-com","scope":"flt/miniproject-adeles/base-change/part-01","status":"fully_proved","summary":"If $i:K\\to L$ denotes the inclusion then for $k\\in K$ we have $e\\times w(i(k))=v(k)$, where $e$ is the ramification index of $w/v$ (recall that valuations here are written additively, unlike in mathlib).","title":"IsDedekindDomain.HeightOneSpectrum.valuation_comap"},{"id":"flt/miniproject-adeles/base-change/part-01/isdedekinddomain-heightonespectrum-extension-adi","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.Extension.adicCompletionSemialgHom","parent":4337,"result":true,"route":"../roadmap/flt/miniproject-adeles/base-change/part-01/index.html#isdedekinddomain-heightonespectrum-extension-adi","scope":"flt/miniproject-adeles/base-change/part-01","status":"fully_proved","summary":"There's a natural ring map $K_v\\to L_w$ extending the map $K\\to L$. It is defined by completing the inclusion $K\\to L$ at the finite places $v$ and $w$ (which can be done because the previous lemma shows that the map is uniformly continuous for the $v$-adic and $w$-adic topologies).","title":"IsDedekindDomain.HeightOneSpectrum.Extension.adicCompletionSemialgHom"},{"id":"flt/miniproject-adeles/base-change/part-01/isdedekinddomain-heightonespectrum-extension-val","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.Extension.valued_adicCompletionSemialgHom","parent":4337,"result":true,"route":"../roadmap/flt/miniproject-adeles/base-change/part-01/index.html#isdedekinddomain-heightonespectrum-extension-val","scope":"flt/miniproject-adeles/base-change/part-01","status":"fully_proved","summary":"If $i_v:K_v\\to L_w$ denotes the map of the previous definition then for $x\\in K_v$ we have $e\\times w(i(k))=v(k)$, where $e$ is the ramification index of $w/v$.","title":"IsDedekindDomain.HeightOneSpectrum.Extension.valued_adicCompletionSemialgHom"},{"id":"flt/miniproject-adeles/base-change/part-01/isdedekinddomain-heightonespectrum-extension-adi-2","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.Extension.adicCompletionSemialgHom_image_adicCompletionIntegers","parent":4337,"result":true,"route":"../roadmap/flt/miniproject-adeles/base-change/part-01/index.html#isdedekinddomain-heightonespectrum-extension-adi-2","scope":"flt/miniproject-adeles/base-change/part-01","status":"fully_proved","summary":"The map $i_v:K_v\\to L_w$ sends the integer ring $A_v$ into $B_w$.","title":"IsDedekindDomain.HeightOneSpectrum.Extension.adicCompletionSemialgHom_image_adicCompletionIntegers"},{"id":"flt/miniproject-adeles/base-change/part-01/isdedekinddomain-heightonespectrum-adiccompletio","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.adicCompletion.instIsModuleTopology","parent":4337,"result":true,"route":"../roadmap/flt/miniproject-adeles/base-change/part-01/index.html#isdedekinddomain-heightonespectrum-adiccompletio","scope":"flt/miniproject-adeles/base-change/part-01","status":"can_prove","summary":"Giving $L_w$ the $K_v$-algebra structure coming from the natural map $K_v\\to L_w$, the $w$-adic topology on $L_w$ is the $K_v$-module topology.","title":"IsDedekindDomain.HeightOneSpectrum.adicCompletion.instIsModuleTopology"},{"id":"flt/miniproject-adeles/base-change/part-01/isdedekinddomain-heightonespectrum-extension-fin","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.Extension.finite","parent":4337,"result":true,"route":"../roadmap/flt/miniproject-adeles/base-change/part-01/index.html#isdedekinddomain-heightonespectrum-extension-fin","scope":"flt/miniproject-adeles/base-change/part-01","status":"fully_proved","summary":"There are only finitely many primes $w$ of $B$ lying above $v$.","title":"IsDedekindDomain.HeightOneSpectrum.Extension.finite"},{"id":"flt/miniproject-adeles/base-change/part-01/isdedekinddomain-heightonespectrum-adiccompletio-2","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.adicCompletion.semialgHomPi","parent":4337,"result":true,"route":"../roadmap/flt/miniproject-adeles/base-change/part-01/index.html#isdedekinddomain-heightonespectrum-adiccompletio-2","scope":"flt/miniproject-adeles/base-change/part-01","status":"fully_proved","summary":"The product of the maps $K_v\\to L_w$ for $w|v$ is a natural ring map $K_v\\to\\prod_{w|v}L_w$ lying over $K\\to L$.","title":"IsDedekindDomain.HeightOneSpectrum.adicCompletion.semialgHomPi"},{"id":"flt/miniproject-adeles/base-change/part-01/isdedekinddomain-heightonespectrum-adiccompletio-3","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.adicCompletion.baseChangeAlgEquiv","parent":4337,"result":true,"route":"../roadmap/flt/miniproject-adeles/base-change/part-01/index.html#isdedekinddomain-heightonespectrum-adiccompletio-3","scope":"flt/miniproject-adeles/base-change/part-01","status":"can_prove","summary":"The induced $L$-algebra homomorphism $L\\otimes_KK_v\\to\\prod_{w|v}L_w$ is an isomorphism of rings.","title":"IsDedekindDomain.HeightOneSpectrum.adicCompletion.baseChangeAlgEquiv"},{"id":"flt/miniproject-adeles/base-change/part-01/isdedekinddomain-heightonespectrum-adiccompletio-4","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.adicCompletion.instIsModuleTopologyPi","parent":4337,"result":true,"route":"../roadmap/flt/miniproject-adeles/base-change/part-01/index.html#isdedekinddomain-heightonespectrum-adiccompletio-4","scope":"flt/miniproject-adeles/base-change/part-01","status":"proved","summary":"For $v$ fixed, the product topology on $\\prod_{w|v}L_w$ is the $K_v$-module topology.","title":"IsDedekindDomain.HeightOneSpectrum.adicCompletion.instIsModuleTopologyPi"},{"id":"flt/miniproject-adeles/base-change/part-01/isdedekinddomain-heightonespectrum-range-basecha","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.range_baseChange_comp_tensorAdicCompletionTo_eq_pi","parent":4337,"result":true,"route":"../roadmap/flt/miniproject-adeles/base-change/part-01/index.html#isdedekinddomain-heightonespectrum-range-basecha","scope":"flt/miniproject-adeles/base-change/part-01","status":"proved","summary":"The isomorphism $L\\otimes_KK_v\\to\\prod_{w|v}L_w$ induces an isomorphism $B\\otimes_AA_v\\to \\prod_{w|v}B_w$ for all $v$ in the height one spectrum of $A$.","title":"IsDedekindDomain.HeightOneSpectrum.range_baseChange_comp_tensorAdicCompletionTo_eq_pi"},{"id":"flt/miniproject-adeles/base-change/part-01/isdedekinddomain-finiteadelering-mapsemialghom","kind":"node","lean":"IsDedekindDomain.FiniteAdeleRing.mapSemialgHom","parent":4337,"result":true,"route":"../roadmap/flt/miniproject-adeles/base-change/part-01/index.html#isdedekinddomain-finiteadelering-mapsemialghom","scope":"flt/miniproject-adeles/base-change/part-01","status":"fully_proved","summary":"There's a natural ring homomorphism $\\mathbb{A}_{A,K}^\\infty\\to\\mathbb{A}_{B,L}^\\infty$ lying over $K\\to L$.","title":"IsDedekindDomain.FiniteAdeleRing.mapSemialgHom"},{"id":"flt/miniproject-adeles/base-change/part-01/isdedekinddomain-finiteadelering-basechangealgeq","kind":"node","lean":"IsDedekindDomain.FiniteAdeleRing.baseChangeAlgEquiv","parent":4337,"result":true,"route":"../roadmap/flt/miniproject-adeles/base-change/part-01/index.html#isdedekinddomain-finiteadelering-basechangealgeq","scope":"flt/miniproject-adeles/base-change/part-01","status":"can_prove","summary":"This natural map $L\\otimes_K\\mathbb{A}_{A,K}^\\infty\\to\\mathbb{A}_{B,L}^\\infty$ is an isomorphism.","title":"IsDedekindDomain.FiniteAdeleRing.baseChangeAlgEquiv"},{"id":"flt/miniproject-adeles/base-change/part-01/isdedekinddomain-finiteadelering-basechangeconti","kind":"node","lean":"IsDedekindDomain.FiniteAdeleRing.baseChangeContinuousAlgEquiv","parent":4337,"result":true,"route":"../roadmap/flt/miniproject-adeles/base-change/part-01/index.html#isdedekinddomain-finiteadelering-basechangeconti","scope":"flt/miniproject-adeles/base-change/part-01","status":"can_prove","summary":"The induced $L$-algebra morphism $L\\otimes_K\\mathbb{A}_{A,K}^\\infty\\to\\mathbb{A}_{B,L}^\\infty$ is a topological isomorphism.","title":"IsDedekindDomain.FiniteAdeleRing.baseChangeContinuousAlgEquiv"},{"id":"flt/miniproject-adeles/base-change/part-01/isdedekinddomain-aklb-tensorproduct-module-algeq","kind":"node","lean":"IsDedekindDomain.linearEquivTensorProductModule","parent":4337,"result":true,"route":"../roadmap/flt/miniproject-adeles/base-change/part-01/index.html#isdedekinddomain-aklb-tensorproduct-module-algeq","scope":"flt/miniproject-adeles/base-change/part-01","status":"fully_proved","summary":"If $M$ is any $K$-module then the canonical map $B\\otimes_A M\\to L\\otimes_K M$ is an isomorphism.","title":"IsDedekindDomain.AKLB.tensorProduct_module_algEquiv"},{"id":"flt/miniproject-adeles/base-change/part-01/numberfield-infiniteplace-completion-denserange","kind":"node","lean":"NumberField.InfinitePlace.Completion.denseRange_algebraMap_subtype_pi","parent":4337,"result":true,"route":"../roadmap/flt/miniproject-adeles/base-change/part-01/index.html#numberfield-infiniteplace-completion-denserange","scope":"flt/miniproject-adeles/base-change/part-01","status":"can_prove","summary":"Let $S$ be a set of infinite places of $K$. The image of $K$ under the embedding $K\\hookrightarrow (K_v)_{v\\in S}; k \\mapsto (k)_v$ is dense in the product topology.","title":"NumberField.InfinitePlace.Completion.denseRange_algebraMap_subtype_pi"},{"id":"flt/miniproject-adeles/base-change/part-01/numberfield-infiniteplace-completion-piextension","kind":"node","lean":"NumberField.InfinitePlace.Completion.piExtension","parent":4337,"result":true,"route":"../roadmap/flt/miniproject-adeles/base-change/part-01/index.html#numberfield-infiniteplace-completion-piextension","scope":"flt/miniproject-adeles/base-change/part-01","status":"fully_proved","summary":"Let $v$ be an infinite place of $K$. There is a continuous $K$-algebra homomorphism $K_v \\to \\prod_{w\\mid v}L_w$, whose restriction to $K$ corresponds to the global embedding of $K$ into $(L_w)_w$.","title":"NumberField.InfinitePlace.Completion.piExtension"},{"id":"flt/miniproject-adeles/base-change/part-01/numberfield-infiniteplace-completion-basechange","kind":"node","lean":"NumberField.InfinitePlace.Completion.baseChange","parent":4337,"result":true,"route":"../roadmap/flt/miniproject-adeles/base-change/part-01/index.html#numberfield-infiniteplace-completion-basechange","scope":"flt/miniproject-adeles/base-change/part-01","status":"fully_proved","summary":"Let $v$ be an infinite place of $K$. There is a natural $L$-algebra homomorphism $L\\otimes_K K_v \\to \\prod_{w\\mid v}L_w$, whose restriction to $1\\otimes_K K_v$ corresponds to the map in NumberField.InfinitePlace.Completion.piExtension.","title":"NumberField.InfinitePlace.Completion.baseChange"},{"id":"flt/miniproject-adeles/base-change/part-01/numberfield-infiniteplace-completion-basechange-2","kind":"node","lean":"NumberField.InfinitePlace.Completion.baseChange_surjective","parent":4337,"result":true,"route":"../roadmap/flt/miniproject-adeles/base-change/part-01/index.html#numberfield-infiniteplace-completion-basechange-2","scope":"flt/miniproject-adeles/base-change/part-01","status":"can_prove","summary":"For a fixed infinite place $v$ of $K$, the map $L\\otimes_K K_v \\to\\prod_{w\\mid v}L_w$ is surjective.","title":"NumberField.InfinitePlace.Completion.baseChange_surjective"},{"id":"flt/miniproject-adeles/base-change/part-01/numberfield-infiniteplace-completion-basechange-3","kind":"node","lean":"NumberField.InfinitePlace.Completion.baseChange_injective","parent":4337,"result":true,"route":"../roadmap/flt/miniproject-adeles/base-change/part-01/index.html#numberfield-infiniteplace-completion-basechange-3","scope":"flt/miniproject-adeles/base-change/part-01","status":"can_prove","summary":"For a fixed infinite place $v$ of $K$, the map $L\\otimes_K K_v \\to\\prod_{w\\mid v}L_w$ is injective.","title":"NumberField.InfinitePlace.Completion.baseChange_injective"},{"id":"flt/miniproject-adeles/base-change/part-01/numberfield-infiniteplace-completion-instismodul","kind":"node","lean":"NumberField.InfinitePlace.Completion.instIsModuleTopologyValEqComapAlgebraMap_fLT","parent":4337,"result":true,"route":"../roadmap/flt/miniproject-adeles/base-change/part-01/index.html#numberfield-infiniteplace-completion-instismodul","scope":"flt/miniproject-adeles/base-change/part-01","status":"can_prove","summary":"If $w \\mid v$ is an infinite place of $L$ lying above the infinite place $v$ of $K$, then $L_w$ has the $K_v$-module topology.","title":"NumberField.InfinitePlace.Completion.instIsModuleTopologyValEqComapAlgebraMap_fLT"},{"data_url":"scopes/flt/miniproject-adeles/base-change/part-02.json","id":"flt/miniproject-adeles/base-change/part-02","kind":"scope","navigation":"transparent","parent":4336,"result":false,"route":"../roadmap/flt/miniproject-adeles/base-change/part-02/index.html","scope":"flt/miniproject-adeles/base-change/part-02","status":"can_prove","summary":"First, we induce a $K_{\\infty}$-algebra on $L_{\\infty}$ from the action of each $K_v$ on $\\prod_{w\\mid v}L_w$. Specifically, this means that for $x \\in K_{\\infty}$ and $y \\in L_{\\infty}$, we have $(x \\cdot y)_w = x_{v_w} \\cdot y_w$, where $v_w$ is the restriction of $w$ to $K$. We show that $L_{\\infty}$ has the $K_{\\infty}$-module topology.","title":"NumberField.InfinitePlace.Comple\u2026 to NumberField.AdeleRing.baseChange\u2026"},{"id":"flt/miniproject-adeles/base-change/part-02/numberfield-infiniteplace-completion-basechangee","kind":"node","lean":"NumberField.InfinitePlace.Completion.baseChangeEquiv","parent":4358,"result":true,"route":"../roadmap/flt/miniproject-adeles/base-change/part-02/index.html#numberfield-infiniteplace-completion-basechangee","scope":"flt/miniproject-adeles/base-change/part-02","status":"can_prove","summary":"Let $v$ be an infinite place of $K$. There is a natural $L$-algebra homeomorphism $L\\otimes_K K_v \\cong_L \\prod_{w\\mid v}L_w$, whose restriction to $1\\otimes_K K_v$ corresponds to the map in NumberField.InfinitePlace.Completion.piExtension.","title":"NumberField.InfinitePlace.Completion.baseChangeEquiv"},{"id":"flt/miniproject-adeles/base-change/part-02/numberfield-infiniteplace-completion-piequiv","kind":"node","lean":"NumberField.InfinitePlace.Completion.piEquiv","parent":4358,"result":true,"route":"../roadmap/flt/miniproject-adeles/base-change/part-02/index.html#numberfield-infiniteplace-completion-piequiv","scope":"flt/miniproject-adeles/base-change/part-02","status":"can_prove","summary":"Let $v$ be an infinite place of $K$. There is a natural $K_v$-linear homeomorphism $K_v^{[L:K]} \\cong_{K_v} \\prod_{w\\mid v}L_w$.","title":"NumberField.InfinitePlace.Completion.piEquiv"},{"id":"flt/miniproject-adeles/base-change/part-02/numberfield-infiniteadelering-piequiv","kind":"node","lean":"NumberField.InfiniteAdeleRing.piEquiv","parent":4358,"result":true,"route":"../roadmap/flt/miniproject-adeles/base-change/part-02/index.html#numberfield-infiniteadelering-piequiv","scope":"flt/miniproject-adeles/base-change/part-02","status":"can_prove","summary":"There is a natural $K_{\\infty}$-linear homeomorphism $K_{\\infty}^{[L:K]} \\cong_{K_{\\infty}} L_{\\infty}$.","title":"NumberField.InfiniteAdeleRing.piEquiv"},{"id":"flt/miniproject-adeles/base-change/part-02/numberfield-infiniteadelering-instismoduletopolo","kind":"node","lean":"NumberField.InfiniteAdeleRing.instIsModuleTopology_fLT","parent":4358,"result":true,"route":"../roadmap/flt/miniproject-adeles/base-change/part-02/index.html#numberfield-infiniteadelering-instismoduletopolo","scope":"flt/miniproject-adeles/base-change/part-02","status":"can_prove","summary":"$L_{\\infty}$ has the $K_{\\infty}$-module topology.","title":"NumberField.InfiniteAdeleRing.instIsModuleTopology_fLT"},{"id":"flt/miniproject-adeles/base-change/part-02/numberfield-infiniteadelering-basechangealgequiv","kind":"node","lean":"NumberField.InfiniteAdeleRing.baseChangeAlgEquiv","parent":4358,"result":true,"route":"../roadmap/flt/miniproject-adeles/base-change/part-02/index.html#numberfield-infiniteadelering-basechangealgequiv","scope":"flt/miniproject-adeles/base-change/part-02","status":"can_prove","summary":"There is a natural $L$-algebra isomorphism $L \\otimes_K K_{\\infty} \\cong_L L_{\\infty}$.","title":"NumberField.InfiniteAdeleRing.baseChangeAlgEquiv"},{"id":"flt/miniproject-adeles/base-change/part-02/numberfield-infiniteadelering-basechangeequiv","kind":"node","lean":"NumberField.InfiniteAdeleRing.baseChangeEquiv, NumberField.InfiniteAdeleRing.instIsModuleTopology_fLT","parent":4358,"result":true,"route":"../roadmap/flt/miniproject-adeles/base-change/part-02/index.html#numberfield-infiniteadelering-basechangeequiv","scope":"flt/miniproject-adeles/base-change/part-02","status":"can_prove","summary":"If $K\\to L$ is a ring homomorphism between two number fields then there is a natural isomorphism (both topological and algebraic) $L\\otimes_KK_\\infty\\cong L_\\infty$.","title":"NumberField.InfiniteAdeleRing.baseChangeEquiv"},{"id":"flt/miniproject-adeles/base-change/part-02/numberfield-adelering-basechangeequiv","kind":"node","lean":"NumberField.AdeleRing.baseChangeEquiv","parent":4358,"result":true,"route":"../roadmap/flt/miniproject-adeles/base-change/part-02/index.html#numberfield-adelering-basechangeequiv","scope":"flt/miniproject-adeles/base-change/part-02","status":"proved","summary":"If $K\\to L$ is a ring homomorphism between two number fields then there is a natural isomorphism (both topological and algebraic) $L\\otimes_K\\mathbb{A}_K\\cong\\mathbb{A}_L$.","title":"NumberField.AdeleRing.baseChangeEquiv"},{"id":"flt/miniproject-adeles/base-change/part-02/numberfield-adelering-basechange-moduletopology","kind":"node","lean":"NumberField.AdeleRing.instIsModuleTopology","parent":4358,"result":true,"route":"../roadmap/flt/miniproject-adeles/base-change/part-02/index.html#numberfield-adelering-basechange-moduletopology","scope":"flt/miniproject-adeles/base-change/part-02","status":"proved","summary":"If $K\\to L$ is a ring homomorphism between two number fields then the topology on $\\mathbb{A}_L$ is the $\\mathbb{A}_K$-module topology, where the module structure comes from the natural map $\\mathbb{A}_K\\to\\mathbb{A}_L$.","title":"NumberField.AdeleRing.baseChange_moduleTopology"},{"data_url":"scopes/flt/miniproject-adeles/discreteness-and-compactness.json","id":"flt/miniproject-adeles/discreteness-and-compactness","kind":"scope","parent":4330,"result":false,"route":"../roadmap/flt/miniproject-adeles/discreteness-and-compactness/index.html","scope":"flt/miniproject-adeles/discreteness-and-compactness","status":"can_prove","summary":"We need that if $K$ is a number field then $K\\subseteq\\mathbb{A}_K$ is discrete, and the quotient (with the quotient topology) is compact. Here is a proposed proof.","title":"Discreteness and compactness"},{"id":"flt/miniproject-adeles/discreteness-and-compactness/rat-adelering-zero-discrete","kind":"node","lean":"Rat.AdeleRing.zero_discrete","parent":4367,"result":true,"route":"../roadmap/flt/miniproject-adeles/discreteness-and-compactness/index.html#rat-adelering-zero-discrete","scope":"flt/miniproject-adeles/discreteness-and-compactness","status":"can_prove","summary":"There's an open subset of $\\mathbb{A}_{\\mathbb{Q}}$ whose intersection with $\\mathbb{Q}$ is $\\{0\\}$.","title":"Rat.AdeleRing.zero_discrete"},{"id":"flt/miniproject-adeles/discreteness-and-compactness/numberfield-adelering-zero-discrete","kind":"node","lean":"NumberField.AdeleRing.zero_discrete","parent":4367,"result":true,"route":"../roadmap/flt/miniproject-adeles/discreteness-and-compactness/index.html#numberfield-adelering-zero-discrete","scope":"flt/miniproject-adeles/discreteness-and-compactness","status":"can_prove","summary":"There's an open subset of $\\mathbb{A}_{K}$ whose intersection with $K$ is $\\{0\\}$.","title":"NumberField.AdeleRing.zero_discrete"},{"id":"flt/miniproject-adeles/discreteness-and-compactness/numberfield-adelering-discrete","kind":"node","lean":"NumberField.AdeleRing.discrete","parent":4367,"result":true,"route":"../roadmap/flt/miniproject-adeles/discreteness-and-compactness/index.html#numberfield-adelering-discrete","scope":"flt/miniproject-adeles/discreteness-and-compactness","status":"can_prove","summary":"The additive subgroup $K$ of $\\mathbb{A}_K$ is discrete.","title":"NumberField.AdeleRing.discrete"},{"id":"flt/miniproject-adeles/discreteness-and-compactness/rat-adelering-cocompact","kind":"node","lean":"Rat.AdeleRing.cocompact","parent":4367,"result":true,"route":"../roadmap/flt/miniproject-adeles/discreteness-and-compactness/index.html#rat-adelering-cocompact","scope":"flt/miniproject-adeles/discreteness-and-compactness","status":"can_prove","summary":"The quotient $\\mathbb{A}_{\\mathbb{Q}}/\\mathbb{Q}$ is compact.","title":"Rat.AdeleRing.cocompact"},{"id":"flt/miniproject-adeles/discreteness-and-compactness/numberfield-adelering-cocompact","kind":"node","lean":"NumberField.AdeleRing.cocompact","parent":4367,"result":true,"route":"../roadmap/flt/miniproject-adeles/discreteness-and-compactness/index.html#numberfield-adelering-cocompact","scope":"flt/miniproject-adeles/discreteness-and-compactness","status":"can_prove","summary":"The quotient $\\mathbb{A}_K/K$ is compact.","title":"NumberField.AdeleRing.cocompact"},{"data_url":"scopes/flt/miniproject-haar-characters.json","id":"flt/miniproject-haar-characters","kind":"scope","parent":4231,"result":false,"route":"../roadmap/flt/miniproject-haar-characters/index.html","scope":"flt/miniproject-haar-characters","status":"stated","title":"Miniproject: Haar Characters"},{"id":"flt/miniproject-haar-characters/the-goal","kind":"node","parent":4373,"result":false,"route":"../roadmap/flt/miniproject-haar-characters/the-goal/index.html","scope":"flt/miniproject-haar-characters","status":"can_state","summary":"The goal of this miniproject is to develop the theory (i.e., the basic API) of Haar characters. \u201cHaar character\u201d is a name I've made up to describe a certain character of the units of a locally compact topological ring. The main result we need here is that if $B$ is a finite-dimensional algebra over a number field $K$, then $B^\\times$ is in the kernel of the Haar character of $B\\otimes_K\\mathbb{A}_K$, where\u2026","title":"The goal"},{"data_url":"scopes/flt/miniproject-haar-characters/initial-definitions.json","id":"flt/miniproject-haar-characters/initial-definitions","kind":"scope","parent":4373,"result":false,"route":"../roadmap/flt/miniproject-haar-characters/initial-definitions/index.html","scope":"flt/miniproject-haar-characters/initial-definitions","status":"fully_proved","summary":"Let $A$ be a locally compact topological additive abelian group. There's then a regular additive Haar measure $\\mu$ on $A$, unique up to a positive scalar factor. If $\\phi:(A,+)\\cong(A,+)$ is a homeomorphism and an additive automorphism of $A$, then we can push forward $\\mu$ along $\\phi$ to get a second measure $\\phi_*\\mu$ on $A$, with the property that $(\\phi_*\\mu)(X)=\\mu(\\phi^{-1}X)$ for any Borel subset $X$ of\u2026","title":"Initial definitions"},{"id":"flt/miniproject-haar-characters/initial-definitions/measuretheory-addequivaddhaarchar","kind":"node","lean":"MeasureTheory.addEquivAddHaarChar","parent":4375,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/initial-definitions/index.html#measuretheory-addequivaddhaarchar","scope":"flt/miniproject-haar-characters/initial-definitions","status":"fully_proved","summary":"If $A$ is a locally compact topological additive abelian group, if $\\mu$ is a regular Haar measure on $A$, and if $\\phi:A\\to A$ is an additive homeomorphism, then we let $d_A(\\phi)$ denote the unique positive real number such that $\\mu(X)=d_A(\\phi)(\\phi_*\\mu)(X)$ for any Borel set $X$.","title":"MeasureTheory.addEquivAddHaarChar"},{"id":"flt/miniproject-haar-characters/initial-definitions/measuretheory-addequivaddhaarchar-eq","kind":"node","lean":"MeasureTheory.addEquivAddHaarChar_eq","parent":4375,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/initial-definitions/index.html#measuretheory-addequivaddhaarchar-eq","scope":"flt/miniproject-haar-characters/initial-definitions","status":"fully_proved","summary":"$d_A(\\phi)$ is independent of choice of regular Haar measure.","title":"MeasureTheory.addEquivAddHaarChar_eq"},{"id":"flt/miniproject-haar-characters/initial-definitions/measuretheory-addequivaddhaarchar-smul-map","kind":"node","lean":"MeasureTheory.addEquivAddHaarChar_smul_map","parent":4375,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/initial-definitions/index.html#measuretheory-addequivaddhaarchar-smul-map","scope":"flt/miniproject-haar-characters/initial-definitions","status":"fully_proved","summary":"If $\\mu$ is any regular Haar measure on $A$ then $d_A(\\phi)(\\phi_*\\mu) = \\mu.$","title":"MeasureTheory.addEquivAddHaarChar_smul_map"},{"id":"flt/miniproject-haar-characters/initial-definitions/measuretheory-addequivaddhaarchar-comap","kind":"node","lean":"MeasureTheory.addEquivAddHaarChar_smul_eq_comap","parent":4375,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/initial-definitions/index.html#measuretheory-addequivaddhaarchar-comap","scope":"flt/miniproject-haar-characters/initial-definitions","status":"fully_proved","summary":"If $\\mu$ is any regular Haar measure on $A$ then $d_A(\\phi)\\mu = \\phi^*\\mu.$","title":"MeasureTheory.addEquivAddHaarChar_comap"},{"id":"flt/miniproject-haar-characters/initial-definitions/measuretheory-addequivaddhaarchar-smul-preimage","kind":"node","lean":"MeasureTheory.addEquivAddHaarChar_smul_preimage","parent":4375,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/initial-definitions/index.html#measuretheory-addequivaddhaarchar-smul-preimage","scope":"flt/miniproject-haar-characters/initial-definitions","status":"fully_proved","summary":"If $X$ is a Borel set then $\\mu(X)=d_A(\\phi)\\mu(\\phi^{-1}X)$.","title":"MeasureTheory.addEquivAddHaarChar_smul_preimage"},{"id":"flt/miniproject-haar-characters/initial-definitions/measuretheory-addequivaddhaarchar-smul-integral","kind":"node","lean":"MeasureTheory.addEquivAddHaarChar_smul_integral_map","parent":4375,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/initial-definitions/index.html#measuretheory-addequivaddhaarchar-smul-integral","scope":"flt/miniproject-haar-characters/initial-definitions","status":"fully_proved","summary":"If $f:A\\to\\mathbb{R}$ is a Borel measurable function then $d_A(\\phi)\\int f(x)d\\phi_*\\mu(x)=\\int f(x)d\\mu(x)$.","title":"MeasureTheory.addEquivAddHaarChar_smul_integral_map"},{"id":"flt/miniproject-haar-characters/initial-definitions/measuretheory-addequivaddhaarchar-smul-integral-2","kind":"node","lean":"MeasureTheory.integral_comap_eq_addEquivAddHaarChar_smul","parent":4375,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/initial-definitions/index.html#measuretheory-addequivaddhaarchar-smul-integral-2","scope":"flt/miniproject-haar-characters/initial-definitions","status":"fully_proved","summary":"If $f:A\\to\\mathbb{R}$ is a Borel measurable function then $d_A(\\phi)\\int f(x)d\\mu(x)=\\int f(x)d\\phi^*\\mu(x)$.","title":"MeasureTheory.addEquivAddHaarChar_smul_integral_comap"},{"id":"flt/miniproject-haar-characters/initial-definitions/measuretheory-addequivaddhaarchar-refl","kind":"node","lean":"MeasureTheory.addEquivAddHaarChar_refl","parent":4375,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/initial-definitions/index.html#measuretheory-addequivaddhaarchar-refl","scope":"flt/miniproject-haar-characters/initial-definitions","status":"fully_proved","summary":"$d_A(id)=1.$","title":"MeasureTheory.addEquivAddHaarChar_refl"},{"id":"flt/miniproject-haar-characters/initial-definitions/measuretheory-addequivaddhaarchar-trans","kind":"node","lean":"MeasureTheory.addEquivAddHaarChar_trans","parent":4375,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/initial-definitions/index.html#measuretheory-addequivaddhaarchar-trans","scope":"flt/miniproject-haar-characters/initial-definitions","status":"fully_proved","summary":"$d_A(\\phi\\circ\\psi)=d_A(\\phi)d_A(\\psi).$","title":"MeasureTheory.addEquivAddHaarChar_trans"},{"id":"flt/miniproject-haar-characters/initial-definitions/measuretheory-ringhaarchar","kind":"node","lean":"MeasureTheory.ringHaarChar","parent":4375,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/initial-definitions/index.html#measuretheory-ringhaarchar","scope":"flt/miniproject-haar-characters/initial-definitions","status":"fully_proved","summary":"We define $\\delta_R(u)$ (or just $\\delta(u)$ when the ring $R$ is clear) to be $d_R(\\ell_u)$.","title":"MeasureTheory.ringHaarChar"},{"id":"flt/miniproject-haar-characters/initial-definitions/measuretheory-ringhaarchar-mul-integral","kind":"node","lean":"MeasureTheory.ringHaarChar_mul_integral","parent":4375,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/initial-definitions/index.html#measuretheory-ringhaarchar-mul-integral","scope":"flt/miniproject-haar-characters/initial-definitions","status":"fully_proved","summary":"If $f:R\\to\\mathbb{R}$ is a Borel measurable function and $u\\in R^\\times$ then $\\delta_R(u)\\int f(ux)d\\mu(x)=\\int f(x)d\\mu(x)$.","title":"MeasureTheory.ringHaarChar_mul_integral"},{"id":"flt/miniproject-haar-characters/initial-definitions/measuretheory-ringhaarchar-mul-volume","kind":"node","lean":"MeasureTheory.ringHaarChar_mul_volume","parent":4375,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/initial-definitions/index.html#measuretheory-ringhaarchar-mul-volume","scope":"flt/miniproject-haar-characters/initial-definitions","status":"fully_proved","summary":"If $X$ is a Borel subset of $R$ and $r\\in R^\\times$ then $\\mu(rX)=\\delta_R(r)\\mu(X)$.","title":"MeasureTheory.ringHaarChar_mul_volume"},{"id":"flt/miniproject-haar-characters/initial-definitions/measuretheory-ringhaarchar-continuous","kind":"node","lean":"MeasureTheory.ringHaarChar_continuous","parent":4375,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/initial-definitions/index.html#measuretheory-ringhaarchar-continuous","scope":"flt/miniproject-haar-characters/initial-definitions","status":"fully_proved","summary":"The function $\\delta_R:R^\\times\\to\\mathbb{R}_{>0}$ is continuous.","title":"MeasureTheory.ringHaarChar_continuous"},{"data_url":"scopes/flt/miniproject-haar-characters/examples.json","id":"flt/miniproject-haar-characters/examples","kind":"scope","parent":4373,"result":false,"route":"../roadmap/flt/miniproject-haar-characters/examples/index.html","scope":"flt/miniproject-haar-characters/examples","status":"fully_proved","summary":"We discuss some examples of Haar characters.","title":"Examples"},{"id":"flt/miniproject-haar-characters/examples/measuretheory-ringhaarchar-real","kind":"node","lean":"MeasureTheory.ringHaarChar_real","parent":4389,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/examples/index.html#measuretheory-ringhaarchar-real","scope":"flt/miniproject-haar-characters/examples","status":"fully_proved","summary":"If $R=\\mathbb{R}$ then $\\delta_R(u)=|u|$.","title":"MeasureTheory.ringHaarChar_real"},{"id":"flt/miniproject-haar-characters/examples/measuretheory-ringhaarchar-complex","kind":"node","lean":"MeasureTheory.ringHaarChar_complex","parent":4389,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/examples/index.html#measuretheory-ringhaarchar-complex","scope":"flt/miniproject-haar-characters/examples","status":"fully_proved","summary":"If $R=\\mathbb{C}$ then $\\delta_R(u)=|u|^2$.","title":"MeasureTheory.ringHaarChar_complex"},{"id":"flt/miniproject-haar-characters/examples/measuretheory-ringhaarchar-padic","kind":"node","lean":"MeasureTheory.ringHaarChar_padic","parent":4389,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/examples/index.html#measuretheory-ringhaarchar-padic","scope":"flt/miniproject-haar-characters/examples","status":"fully_proved","summary":"If $R=\\mathbb{Q}_p$ then $\\delta_R(u)=|u|_p$, the usual $p$-adic norm.","title":"MeasureTheory.ringHaarChar_padic"},{"data_url":"scopes/flt/miniproject-haar-characters/algebras.json","id":"flt/miniproject-haar-characters/algebras","kind":"scope","parent":4373,"result":false,"route":"../roadmap/flt/miniproject-haar-characters/algebras/index.html","scope":"flt/miniproject-haar-characters/algebras","status":"fully_proved","summary":"Say $F$ is a locally compact topological ring (for example $\\mathbb{R}$ or $\\mathbb{C}$ or $\\mathbb{Q}_p$, or the adeles of a number field), $V$ is a finite free $F$-module, and $\\phi:V\\to V$ is an invertible $F$-linear map. Then $V$ with its module topology (which is the product topology if one picks a basis) is a locally compact topological abelian group, and $\\phi$ is additive. One can check that linearity\u2026","title":"Algebras"},{"id":"flt/miniproject-haar-characters/algebras/measuretheory-addequivaddhaarchar-eq-ringhaarcha","kind":"node","lean":"MeasureTheory.addEquivAddHaarChar_eq_ringHaarChar_det, MeasureTheory.addEquivAddHaarChar_eq_ringHaarChar_det_of_existsListTransvecEtc","parent":4393,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/algebras/index.html#measuretheory-addequivaddhaarchar-eq-ringhaarcha","scope":"flt/miniproject-haar-characters/algebras","status":"fully_proved","summary":"Assume that there's an $F$-basis for $V$ such that $\\phi$ is a product of elementary and diagonal matrices (note that this is automatic if $F$ is a field and mathlib has this). Then $d_V(\\phi)=\\delta_F(\\det(\\phi))$, where $\\det(\\phi)\\in F$ is the determinant of $\\phi$ as an $F$-linear map.","title":"MeasureTheory.addEquivAddHaarChar_eq_ringHaarChar_det, MeasureTheory.addEquivAddHaarChar_eq_ringHaarChar_det_of_existsListTransvecEtc"},{"id":"flt/miniproject-haar-characters/algebras/measuretheory-algebra-ringhaarchar-eq-ringhaarch","kind":"node","lean":"MeasureTheory.algebra_ringHaarChar_eq_ringHaarChar_det","parent":4393,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/algebras/index.html#measuretheory-algebra-ringhaarchar-eq-ringhaarch","scope":"flt/miniproject-haar-characters/algebras","status":"fully_proved","summary":"If $u\\in R^\\times$ then $\\delta_R(u)=\\delta_F(\\det(\\ell_u))$.","title":"MeasureTheory.algebra_ringHaarChar_eq_ringHaarChar_det"},{"data_url":"scopes/flt/miniproject-haar-characters/left-and-right-multiplication.json","id":"flt/miniproject-haar-characters/left-and-right-multiplication","kind":"scope","parent":4373,"result":false,"route":"../roadmap/flt/miniproject-haar-characters/left-and-right-multiplication/index.html","scope":"flt/miniproject-haar-characters/left-and-right-multiplication","status":"fully_proved","summary":"If $R$ is a locally compact topological ring, and if multiplication on $R$ is not commutative, then left and right multiplication by an element of $R$ can scale Haar measure in different ways. For example if $R$ is the upper-triangular $2\\times 2$ matrices with real entries, then left multiplication by $\\begin{pmatrix}a&0\\\\0&1\\end{pmatrix}$ sends $\\begin{pmatrix}x&y\\\\0&z\\end{pmatrix}$ to\u2026","title":"Left and right multiplication"},{"id":"flt/miniproject-haar-characters/left-and-right-multiplication/issimplering-mulleft-det-eq-mulright-det","kind":"node","lean":"IsSimpleRing.mulLeft_det_eq_mulRight_det","parent":4396,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/left-and-right-multiplication/index.html#issimplering-mulleft-det-eq-mulright-det","scope":"flt/miniproject-haar-characters/left-and-right-multiplication","status":"fully_proved","summary":"Say $B$ is a finite-dimensional central simple algebra over a field $k$, and $u\\in B^\\times$. Let $\\ell_u:B\\to B$ be the $k$-linear mapping $x$ to $ux$ and let $r_u:B\\to B$ be the $k$-linear map sending $x$ to $xu$. Then $\\det(\\ell_u)=\\det(r_u)$.","title":"IsSimpleRing.mulLeft_det_eq_mulRight_det"},{"id":"flt/miniproject-haar-characters/left-and-right-multiplication/issimplering-ringhaarchar-eq-addequivaddhaarchar","kind":"node","lean":"IsSimpleRing.ringHaarChar_eq_addEquivAddHaarChar_mulRight","parent":4396,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/left-and-right-multiplication/index.html#issimplering-ringhaarchar-eq-addequivaddhaarchar","scope":"flt/miniproject-haar-characters/left-and-right-multiplication","status":"fully_proved","summary":"If $B$ is a central simple algebra over a locally compact field $F$, and if $u\\in B^\\times$, then $d_B(r_u)=\\delta_B(u)$ (recall that the latter is defined to mean $d_B(\\ell_u)$).","title":"IsSimpleRing.ringHaarChar_eq_addEquivAddHaarChar_mulRight"},{"data_url":"scopes/flt/miniproject-haar-characters/finite-products.json","id":"flt/miniproject-haar-characters/finite-products","kind":"scope","parent":4373,"result":false,"route":"../roadmap/flt/miniproject-haar-characters/finite-products/index.html","scope":"flt/miniproject-haar-characters/finite-products","status":"fully_proved","summary":"Here are two facts which we will need about products.","title":"Finite Products"},{"id":"flt/miniproject-haar-characters/finite-products/measuretheory-addequivaddhaarchar-prodcongr","kind":"node","lean":"MeasureTheory.addEquivAddHaarChar_prodCongr","parent":4399,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/finite-products/index.html#measuretheory-addequivaddhaarchar-prodcongr","scope":"flt/miniproject-haar-characters/finite-products","status":"fully_proved","summary":"If $(A,+)$ and $(B,+)$ are locally compact topological abelian groups, and if $\\phi:A\\to A$ and $\\psi:B\\to B$ are additive homeomorphisms, then $\\phi\\times\\psi:A\\times B\\to A\\times B$ is an additive homeomorphism (this is obvious), and $d_{A\\times B}(\\phi\\times\\psi)=d_A(\\phi)d_B(\\psi)$.","title":"MeasureTheory.addEquivAddHaarChar_prodCongr"},{"id":"flt/miniproject-haar-characters/finite-products/measuretheory-addequivaddhaarchar-picongrright","kind":"node","lean":"MeasureTheory.addEquivAddHaarChar_piCongrRight","parent":4399,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/finite-products/index.html#measuretheory-addequivaddhaarchar-picongrright","scope":"flt/miniproject-haar-characters/finite-products","status":"fully_proved","summary":"If $A_i$ are a finite collection of locally compact topological abelian groups, with $\\phi_i:A_i\\to A_i$ additive homeomorphisms, then $d_{\\prod_i A_i}(\\prod_i\\phi_i)=\\prod_i d_{A_i}(\\phi_i)$.","title":"MeasureTheory.addEquivAddHaarChar_piCongrRight"},{"id":"flt/miniproject-haar-characters/finite-products/measuretheory-ringhaarchar-prod","kind":"node","lean":"MeasureTheory.ringHaarChar_prod","parent":4399,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/finite-products/index.html#measuretheory-ringhaarchar-prod","scope":"flt/miniproject-haar-characters/finite-products","status":"fully_proved","summary":"If $R$ and $S$ are locally compact topological rings, then $\\delta_{R\\times S}(r,s)=\\delta_R(r)\\times\\delta_S(s)$.","title":"MeasureTheory.ringHaarChar_prod"},{"id":"flt/miniproject-haar-characters/finite-products/measuretheory-ringhaarchar-pi","kind":"node","lean":"MeasureTheory.ringHaarChar_pi","parent":4399,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/finite-products/index.html#measuretheory-ringhaarchar-pi","scope":"flt/miniproject-haar-characters/finite-products","status":"fully_proved","summary":"If $R_i$ are a finite collection of locally compact topological rings, and $u_i\\in R_i^\\times$ then $\\delta_{\\prod_i R_i}((u_i)_i)=\\prod_i\\delta_{R_i}(u_i)$.","title":"MeasureTheory.ringHaarChar_pi"},{"data_url":"scopes/flt/miniproject-haar-characters/some-measure-theoretic-preliminaries.json","id":"flt/miniproject-haar-characters/some-measure-theoretic-preliminaries","kind":"scope","parent":4373,"result":false,"route":"../roadmap/flt/miniproject-haar-characters/some-measure-theoretic-preliminaries/index.html","scope":"flt/miniproject-haar-characters/some-measure-theoretic-preliminaries","status":"fully_proved","title":"Some measure-theoretic preliminaries"},{"id":"flt/miniproject-haar-characters/some-measure-theoretic-preliminaries/topology-isopenembedding-ishaarmeasure-comap","kind":"node","lean":"Topology.IsOpenEmbedding.isHaarMeasure_comap","parent":4404,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/some-measure-theoretic-preliminaries/index.html#topology-isopenembedding-ishaarmeasure-comap","scope":"flt/miniproject-haar-characters/some-measure-theoretic-preliminaries","status":"fully_proved","summary":"Let $A$ and $B$ be locally compact topological groups and let $f:A\\to B$ be both a group homomorphism and open embedding. The pullback along $f$ of a Haar measure on $B$ is a Haar measure on $A$.","title":"Topology.IsOpenEmbedding.isHaarMeasure_comap"},{"id":"flt/miniproject-haar-characters/some-measure-theoretic-preliminaries/topology-isopenembedding-regular-comap","kind":"node","lean":"Topology.IsOpenEmbedding.regular_comap","parent":4404,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/some-measure-theoretic-preliminaries/index.html#topology-isopenembedding-regular-comap","scope":"flt/miniproject-haar-characters/some-measure-theoretic-preliminaries","status":"fully_proved","summary":"The pullback of a regular Borel measure along an open embedding is a regular Borel measure.","title":"Topology.IsOpenEmbedding.regular_comap"},{"id":"flt/miniproject-haar-characters/some-measure-theoretic-preliminaries/measuretheory-mulequivhaarchar-eq-one-of-compact","kind":"node","lean":"MeasureTheory.mulEquivHaarChar_eq_one_of_compactSpace","parent":4404,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/some-measure-theoretic-preliminaries/index.html#measuretheory-mulequivhaarchar-eq-one-of-compact","scope":"flt/miniproject-haar-characters/some-measure-theoretic-preliminaries","status":"fully_proved","summary":"Say $A$ is a compact topological additive group and $\\phi:A\\to A$ is an additive isomorphism. Then $d_A(\\phi)=1.$","title":"MeasureTheory.mulEquivHaarChar_eq_one_of_compactSpace"},{"id":"flt/miniproject-haar-characters/some-measure-theoretic-preliminaries/measuretheory-addequivaddhaarchar-eq-addequivadd","kind":"node","lean":"MeasureTheory.addEquivAddHaarChar_eq_addEquivAddHaarChar_of_isOpenEmbedding","parent":4404,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/some-measure-theoretic-preliminaries/index.html#measuretheory-addequivaddhaarchar-eq-addequivadd","scope":"flt/miniproject-haar-characters/some-measure-theoretic-preliminaries","status":"fully_proved","summary":"If $f:A\\to B$ is a group homomorphism and open embedding between locally compact topological additive groups and if $\\alpha:A\\to A$ and $\\beta:B\\to B$ are additive homeomorphisms such that the square commutes (i.e., $f\\circ\\alpha=\\beta\\circ f$) then $d_A(\\alpha)=d_B(\\beta)$.","title":"MeasureTheory.addEquivAddHaarChar_eq_addEquivAddHaarChar_of_isOpenEmbedding"},{"data_url":"scopes/flt/miniproject-haar-characters/restricted-products.json","id":"flt/miniproject-haar-characters/restricted-products","kind":"scope","parent":4373,"result":false,"route":"../roadmap/flt/miniproject-haar-characters/restricted-products/index.html","scope":"flt/miniproject-haar-characters/restricted-products","status":"can_prove","summary":"Now say $A=\\prod'_i A_i$ is the restricted product of a collection of types $A_i$ with respect to the subsets $C_i$. Recall that this is the subset of $\\prod_i A_i$ consisting of y Say $B=\\prod'_i B_i$ is the restricted product of types $B_i$ over the same index set, with respect to subsets $D_i$. Say $\\phi_i:A_i\\to B_i$ are functions with the property that $\\phi_i(C_i)\\subseteq D_i$ for all but finitely many $i$.\u2026","title":"Restricted products"},{"id":"flt/miniproject-haar-characters/restricted-products/continuous-restrictedproduct-congrright","kind":"node","lean":"Continuous.restrictedProduct_congrRight","parent":4409,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/restricted-products/index.html#continuous-restrictedproduct-congrright","scope":"flt/miniproject-haar-characters/restricted-products","status":"fully_proved","summary":"If the $A_i$ and $B_i$ are topological spaces and the $\\phi_i$ are continuous functions, then the restricted product $\\phi = \\prod'_i\\phi_i$ is a continuous function.","title":"Continuous.restrictedProduct_congrRight"},{"id":"flt/miniproject-haar-characters/restricted-products/measuretheory-addequivaddhaarchar-restrictedprod","kind":"node","lean":"MeasureTheory.addEquivAddHaarChar_restrictedProductCongrRight","parent":4409,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/restricted-products/index.html#measuretheory-addequivaddhaarchar-restrictedprod","scope":"flt/miniproject-haar-characters/restricted-products","status":"can_prove","summary":"With $A$, $A_i$, $C_i$, $\\phi_i$, $\\phi$ defined as above, we have $\\delta_A(\\phi)=\\prod_i\\delta_{A_i}(\\phi_i)$.","title":"MeasureTheory.addEquivAddHaarChar_restrictedProductCongrRight"},{"id":"flt/miniproject-haar-characters/restricted-products/measuretheory-ringhaarchar-restrictedproduct","kind":"node","lean":"MeasureTheory.ringHaarChar_restrictedProduct","parent":4409,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/restricted-products/index.html#measuretheory-ringhaarchar-restrictedproduct","scope":"flt/miniproject-haar-characters/restricted-products","status":"proved","summary":"If $u=(u_i)_i\\in R^\\times$ then $\\delta_R(u)=\\prod_i\\delta_{R_i}(u_i)$.","title":"MeasureTheory.ringHaarChar_restrictedProduct"},{"data_url":"scopes/flt/miniproject-haar-characters/adeles.json","id":"flt/miniproject-haar-characters/adeles","kind":"scope","parent":4373,"result":false,"route":"../roadmap/flt/miniproject-haar-characters/adeles/index.html","scope":"flt/miniproject-haar-characters/adeles","status":"stated","summary":"We finish this miniproject by proving some results about Haar characters for algebras over adele rings. So let $K$ be a number field and let $\\mathbb{A}_K$ be the adeles of $K$. We will prove some theorems about $\\mathbb{A}_K$-algebras $R$ which are finite and free as $\\mathbb{A}_K$-modules. Such algebras can be given the $\\mathbb{A}_K$-module topology and this makes them into locally compact topological rings. In\u2026","title":"Adeles"},{"id":"flt/miniproject-haar-characters/adeles/ismoduletopology-continuous-bilinear-of-finite-l","kind":"node","lean":"IsModuleTopology.continuous_bilinear_of_finite_left","parent":4413,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/adeles/index.html#ismoduletopology-continuous-bilinear-of-finite-l","scope":"flt/miniproject-haar-characters/adeles","status":"fully_proved","summary":"Say $R$ and $S$ are topological rings, and $S$ is an $R$-algebra, finite as an $R$-module. Assume that the topology on $S$ is the $R$-module topology. Now say $M$ is an $S$-module, and give it the induced $R$-module structure. Then the $R$-module topology and $S$-module topology on $M$ coincide.","title":"IsModuleTopology.continuous_bilinear_of_finite_left"},{"id":"flt/miniproject-haar-characters/adeles/numberfield-adelering-modulebasechangecontinuous","kind":"node","lean":"NumberField.AdeleRing.ModuleBaseChangeContinuousLinearEquiv","parent":4413,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/adeles/index.html#numberfield-adelering-modulebasechangecontinuous","scope":"flt/miniproject-haar-characters/adeles","status":"fully_proved","summary":"If $K$ is a number field and $V$ is an $K$-module, then the natural isomorphism $V\\otimes_K\\mathbb{A}_K=V\\otimes_{\\mathbb{Q}}\\mathbb{A}_{\\mathbb{Q}}$ induced by the natural isomorphism $\\mathbb{A}_K=K\\otimes_K\\mathbb{A}_{\\mathbb{Q}}$ is a homeomorphism if the left hand side has the $\\mathbb{A}_K$-module topology and the right hand side has the $\\mathbb{A}_{\\mathbb{Q}}$-module topology.","title":"NumberField.AdeleRing.ModuleBaseChangeContinuousLinearEquiv"},{"id":"flt/miniproject-haar-characters/adeles/numberfield-adelering-iscentralsimple-addhaarsca","kind":"node","lean":"NumberField.AdeleRing.isCentralSimple_addHaarScalarFactor_left_mul_eq_right_mul","parent":4413,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/adeles/index.html#numberfield-adelering-iscentralsimple-addhaarsca","scope":"flt/miniproject-haar-characters/adeles","status":"stated","summary":"Let $B$ be a finite-dimensional central simple $K$-algebra. Say $u\\in B_{\\mathbb{A}}^\\times$, and define $\\ell_u$ and $r_u:B_{\\mathbb{A}}\\to B_{\\mathbb{A}}$ by $\\ell_u(x)=ux$ and $r_u(x)=xu$. Then $d_{B_{\\mathbb{A}}}(\\ell_u)=d_{B_{\\mathbb{A}}}(r_u)$.","title":"NumberField.AdeleRing.isCentralSimple_addHaarScalarFactor_left_mul_eq_right_mul"},{"id":"flt/miniproject-haar-characters/adeles/measuretheory-ringhaarchar-adeles-rat","kind":"node","lean":"MeasureTheory.ringHaarChar_adeles_rat","parent":4413,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/adeles/index.html#measuretheory-ringhaarchar-adeles-rat","scope":"flt/miniproject-haar-characters/adeles","status":"proved","summary":"If $x\\in\\mathbb{A}_{\\mathbb{Q}}^\\times$ then $\\delta_{\\mathbb{A}_{\\mathbb{Q}}}(x)=\\prod_v|x_v|_v.$","title":"MeasureTheory.ringHaarChar_adeles_rat"},{"id":"flt/miniproject-haar-characters/adeles/measuretheory-ringhaarchar-adeles-units-rat-eq-o","kind":"node","lean":"MeasureTheory.ringHaarChar_adeles_units_rat_eq_one","parent":4413,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/adeles/index.html#measuretheory-ringhaarchar-adeles-units-rat-eq-o","scope":"flt/miniproject-haar-characters/adeles","status":"proved","summary":"If $x\\in\\mathbb{Q}^\\times\\subseteq\\mathbb{A}_{\\mathbb{Q}}^\\times$ then $\\delta_{\\mathbb{A}_{\\mathbb{Q}}}(x)=1.$","title":"MeasureTheory.ringHaarChar_adeles_units_rat_eq_one"},{"id":"flt/miniproject-haar-characters/adeles/measuretheory-addhaarscalarfactor-tensor-adeles","kind":"node","lean":"MeasureTheory.addHaarScalarFactor_tensor_adeles_eq_one","parent":4413,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/adeles/index.html#measuretheory-addhaarscalarfactor-tensor-adeles","scope":"flt/miniproject-haar-characters/adeles","status":"proved","summary":"In the above situation ($V$ a finite-dimensional $\\mathbb{Q}$-vector space, $\\phi:V\\cong V$ is $\\mathbb{Q}$-linear, $\\phi_{\\mathbb{A}}$ the base extension to $V_{\\mathbb{A}}:=V\\otimes_{\\mathbb{Q}}{\\mathbb{A}_{\\mathbb{Q}}}$, a continuous linear endomorphism of $V_{\\mathbb{A}}$ with the $\\mathbb{A}_{\\mathbb{Q}}$-module topology), we have $d_{V_{\\mathbb{A}}}(\\phi_{\\mathbb{A}})=1.$","title":"MeasureTheory.addHaarScalarFactor_tensor_adeles_eq_one"},{"id":"flt/miniproject-haar-characters/adeles/numberfield-adelering-units-mem-ringhaarcharacte","kind":"node","lean":"NumberField.AdeleRing.units_mem_ringHaarCharacter_ker","parent":4413,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/adeles/index.html#numberfield-adelering-units-mem-ringhaarcharacte","scope":"flt/miniproject-haar-characters/adeles","status":"proved","summary":"If $B$ is a finite-dimensional $\\mathbb{Q}$-algebra (for example a number field, or a quaternion algebra over a number field), if $B_{\\mathbb{A}}$ denotes the ring $B\\otimes_{\\mathbb{Q}}\\mathbb{A}_{\\mathbb{Q}}$, and if $b\\in B^\\times$, then $\\delta_{B_{\\mathbb{A}}}(b)=1$.","title":"NumberField.AdeleRing.units_mem_ringHaarCharacter_ker"},{"id":"flt/miniproject-haar-characters/adeles/numberfield-adelering-addequivaddhaarchar-mulrig","kind":"node","lean":"NumberField.AdeleRing.addEquivAddHaarChar_mulRight_unit_eq_one","parent":4413,"result":true,"route":"../roadmap/flt/miniproject-haar-characters/adeles/index.html#numberfield-adelering-addequivaddhaarchar-mulrig","scope":"flt/miniproject-haar-characters/adeles","status":"proved","summary":"If $B$ is a finite-dimensional $\\mathbb{Q}$-algebra and if $b\\in B^\\times$ then right multiplication by $b$ does not change Haar measure on $B_{\\mathbb{A}}$.","title":"NumberField.AdeleRing.addEquivAddHaarChar_mulRight_unit_eq_one"},{"data_url":"scopes/flt/miniproject-fujisakis-lemma.json","id":"flt/miniproject-fujisakis-lemma","kind":"scope","parent":4231,"result":false,"route":"../roadmap/flt/miniproject-fujisakis-lemma/index.html","scope":"flt/miniproject-fujisakis-lemma","status":"fully_proved","title":"Miniproject: Fujisaki's Lemma"},{"id":"flt/miniproject-fujisakis-lemma/the-goal","kind":"node","parent":4422,"result":false,"route":"../roadmap/flt/miniproject-fujisakis-lemma/the-goal/index.html","scope":"flt/miniproject-fujisakis-lemma","status":"can_state","summary":"There is an idelic compactness statement which encapsulates both finiteness of the class group of a number field and Dirichlet's units theorem about the rank of the unit group. In fact there is even a noncommutative version of this statement. In John Voight's book (Voight 2021) this is Main Theorem 27.6.14(a) and Voight calls it Fujisaki\u2019s lemma. I know nothing of the history but I'm happy to adopt this name. In\u2026","title":"The goal"},{"id":"flt/miniproject-fujisakis-lemma/initial-definitions","kind":"node","parent":4422,"result":false,"route":"../roadmap/flt/miniproject-fujisakis-lemma/initial-definitions/index.html","scope":"flt/miniproject-fujisakis-lemma","status":"can_state","summary":"Let $K$ be a field. A *central simple $K$-algebra* is a $K$-algebra $B$ (not necessarily commutative) with centre $K$ such that $B$ has exactly two two-sided ideals, namely ${0}$ and $B$ (or $\\bot$ and $\\top$, as Lean would call them). We will be concerned only with central simple $K$-algebras which are finite-dimensional as $K$-vector spaces, and when $K$ is clear we will just refer to them as central simple\u2026","title":"Initial definitions"},{"data_url":"scopes/flt/miniproject-fujisakis-lemma/enter-the-adeles.json","id":"flt/miniproject-fujisakis-lemma/enter-the-adeles","kind":"scope","parent":4422,"result":false,"route":"../roadmap/flt/miniproject-fujisakis-lemma/enter-the-adeles/index.html","scope":"flt/miniproject-fujisakis-lemma/enter-the-adeles","status":"fully_proved","summary":"The adeles of a number field are discussed in far more detail in the adele miniproject Miniproject: Adeles. We just recall here that if $K$ is a number field then there are two huge commutative topological $K$-algebras called the *finite adeles* $\\mathbb{A}_K^\\infty$ and the *adeles* $\\mathbb{A}_K$ of $K$, and that they're both locally compact as topological spaces. We also know from theorem\u2026","title":"Enter the adeles"},{"id":"flt/miniproject-fujisakis-lemma/enter-the-adeles/numberfield-adelering-divisionalgebra-compact-qu","kind":"node","lean":"NumberField.AdeleRing.DivisionAlgebra.compact_quotient","parent":4425,"result":true,"route":"../roadmap/flt/miniproject-fujisakis-lemma/enter-the-adeles/index.html#numberfield-adelering-divisionalgebra-compact-qu","scope":"flt/miniproject-fujisakis-lemma/enter-the-adeles","status":"fully_proved","summary":"If $D$ is a division algebra then the quotient $D^\\times\\backslash D_{\\mathbb{A}}^{(1)}$ with its quotient topology coming from $D_{\\mathbb{A}}^{(1)}$, is compact.","title":"NumberField.AdeleRing.DivisionAlgebra.compact_quotient"},{"data_url":"scopes/flt/miniproject-fujisakis-lemma/the-proof.json","id":"flt/miniproject-fujisakis-lemma/the-proof","kind":"scope","parent":4422,"result":false,"route":"../roadmap/flt/miniproject-fujisakis-lemma/the-proof/index.html","scope":"flt/miniproject-fujisakis-lemma/the-proof","status":"fully_proved","summary":"We prove the theorem via a series of lemmas.","title":"The proof"},{"id":"flt/miniproject-fujisakis-lemma/the-proof/numberfield-adelering-divisionalgebra-aux-exists","kind":"node","lean":"NumberField.AdeleRing.DivisionAlgebra.Aux.existsE","parent":4427,"result":true,"route":"../roadmap/flt/miniproject-fujisakis-lemma/the-proof/index.html#numberfield-adelering-divisionalgebra-aux-exists","scope":"flt/miniproject-fujisakis-lemma/the-proof","status":"fully_proved","summary":"There's a compact subset $E$ of $D_{\\mathbb{A}}$ with the property that for all $x\\in D_{\\mathbb{A}}^{(1)}$, the obvious map $xE\\to D\\backslash D_{\\mathbb{A}}$ is not injective.","title":"NumberField.AdeleRing.DivisionAlgebra.Aux.existsE"},{"id":"flt/miniproject-fujisakis-lemma/the-proof/numberfield-adelering-divisionalgebra-aux-e","kind":"node","lean":"NumberField.AdeleRing.DivisionAlgebra.Aux.E","parent":4427,"result":true,"route":"../roadmap/flt/miniproject-fujisakis-lemma/the-proof/index.html#numberfield-adelering-divisionalgebra-aux-e","scope":"flt/miniproject-fujisakis-lemma/the-proof","status":"fully_proved","summary":"We let $E$ denote any compact set satisfying the hypothesis of the previous lemma.","title":"NumberField.AdeleRing.DivisionAlgebra.Aux.E"},{"id":"flt/miniproject-fujisakis-lemma/the-proof/numberfield-adelering-divisionalgebra-aux-x","kind":"node","lean":"NumberField.AdeleRing.DivisionAlgebra.Aux.X","parent":4427,"result":true,"route":"../roadmap/flt/miniproject-fujisakis-lemma/the-proof/index.html#numberfield-adelering-divisionalgebra-aux-x","scope":"flt/miniproject-fujisakis-lemma/the-proof","status":"fully_proved","summary":"Define $X:=E-E:=\\{e-f:e,f\\in E\\}\\subseteq D_{\\mathbb{A}}$.","title":"NumberField.AdeleRing.DivisionAlgebra.Aux.X"},{"id":"flt/miniproject-fujisakis-lemma/the-proof/numberfield-adelering-divisionalgebra-aux-y","kind":"node","lean":"NumberField.AdeleRing.DivisionAlgebra.Aux.Y","parent":4427,"result":true,"route":"../roadmap/flt/miniproject-fujisakis-lemma/the-proof/index.html#numberfield-adelering-divisionalgebra-aux-y","scope":"flt/miniproject-fujisakis-lemma/the-proof","status":"fully_proved","summary":"Define $Y:=X.X:=\\{xy:x,y\\in X\\}\\subseteq D_{\\mathbb{A}}$.","title":"NumberField.AdeleRing.DivisionAlgebra.Aux.Y"},{"id":"flt/miniproject-fujisakis-lemma/the-proof/numberfield-adelering-divisionalgebra-aux-x-comp","kind":"node","lean":"NumberField.AdeleRing.DivisionAlgebra.Aux.X_compact","parent":4427,"result":true,"route":"../roadmap/flt/miniproject-fujisakis-lemma/the-proof/index.html#numberfield-adelering-divisionalgebra-aux-x-comp","scope":"flt/miniproject-fujisakis-lemma/the-proof","status":"fully_proved","summary":"$X$ is a compact subset of $D_{\\mathbb{A}}$.","title":"NumberField.AdeleRing.DivisionAlgebra.Aux.X_compact"},{"id":"flt/miniproject-fujisakis-lemma/the-proof/numberfield-adelering-divisionalgebra-aux-y-comp","kind":"node","lean":"NumberField.AdeleRing.DivisionAlgebra.Aux.Y_compact","parent":4427,"result":true,"route":"../roadmap/flt/miniproject-fujisakis-lemma/the-proof/index.html#numberfield-adelering-divisionalgebra-aux-y-comp","scope":"flt/miniproject-fujisakis-lemma/the-proof","status":"fully_proved","summary":"$Y$ is a compact subset of $D_{\\mathbb{A}}$.","title":"NumberField.AdeleRing.DivisionAlgebra.Aux.Y_compact"},{"id":"flt/miniproject-fujisakis-lemma/the-proof/numberfield-adelering-divisionalgebra-aux-x-meet","kind":"node","lean":"NumberField.AdeleRing.DivisionAlgebra.Aux.X_meets_kernel","parent":4427,"result":true,"route":"../roadmap/flt/miniproject-fujisakis-lemma/the-proof/index.html#numberfield-adelering-divisionalgebra-aux-x-meet","scope":"flt/miniproject-fujisakis-lemma/the-proof","status":"fully_proved","summary":"If $\\beta\\in D_{\\mathbb{A}}^{(1)}$ then $\\beta X\\cap D^\\times\\not=\\emptyset$.","title":"NumberField.AdeleRing.DivisionAlgebra.Aux.X_meets_kernel"},{"id":"flt/miniproject-fujisakis-lemma/the-proof/numberfield-adelering-divisionalgebra-aux-x-meet-2","kind":"node","lean":"NumberField.AdeleRing.DivisionAlgebra.Aux.X_meets_kernel'","parent":4427,"result":true,"route":"../roadmap/flt/miniproject-fujisakis-lemma/the-proof/index.html#numberfield-adelering-divisionalgebra-aux-x-meet-2","scope":"flt/miniproject-fujisakis-lemma/the-proof","status":"fully_proved","summary":"Similarly, if $\\beta\\in D_{\\mathbb{A}}^{(1)}$ then $X\\beta^{-1}\\cap D^\\times\\not=\\emptyset$.","title":"NumberField.AdeleRing.DivisionAlgebra.Aux.X_meets_kernel'"},{"id":"flt/miniproject-fujisakis-lemma/the-proof/numberfield-adelering-divisionalgebra-aux-t","kind":"node","lean":"NumberField.AdeleRing.DivisionAlgebra.Aux.T","parent":4427,"result":true,"route":"../roadmap/flt/miniproject-fujisakis-lemma/the-proof/index.html#numberfield-adelering-divisionalgebra-aux-t","scope":"flt/miniproject-fujisakis-lemma/the-proof","status":"fully_proved","summary":"Let $T:=Y\\cap D^\\times$.","title":"NumberField.AdeleRing.DivisionAlgebra.Aux.T"},{"id":"flt/miniproject-fujisakis-lemma/the-proof/numberfield-adelering-divisionalgebra-aux-t-fini","kind":"node","lean":"NumberField.AdeleRing.DivisionAlgebra.Aux.T_finite","parent":4427,"result":true,"route":"../roadmap/flt/miniproject-fujisakis-lemma/the-proof/index.html#numberfield-adelering-divisionalgebra-aux-t-fini","scope":"flt/miniproject-fujisakis-lemma/the-proof","status":"fully_proved","summary":"$T$ is finite.","title":"NumberField.AdeleRing.DivisionAlgebra.Aux.T_finite"},{"id":"flt/miniproject-fujisakis-lemma/the-proof/numberfield-adelering-divisionalgebra-aux-c","kind":"node","lean":"NumberField.AdeleRing.DivisionAlgebra.Aux.C","parent":4427,"result":true,"route":"../roadmap/flt/miniproject-fujisakis-lemma/the-proof/index.html#numberfield-adelering-divisionalgebra-aux-c","scope":"flt/miniproject-fujisakis-lemma/the-proof","status":"fully_proved","summary":"Define $C:= (T^{-1}.X) \\times X\\subset D_{\\mathbb{A}}\\times D_{\\mathbb{A}}$.","title":"NumberField.AdeleRing.DivisionAlgebra.Aux.C"},{"id":"flt/miniproject-fujisakis-lemma/the-proof/numberfield-adelering-divisionalgebra-aux-c-comp","kind":"node","lean":"NumberField.AdeleRing.DivisionAlgebra.Aux.C_compact","parent":4427,"result":true,"route":"../roadmap/flt/miniproject-fujisakis-lemma/the-proof/index.html#numberfield-adelering-divisionalgebra-aux-c-comp","scope":"flt/miniproject-fujisakis-lemma/the-proof","status":"fully_proved","summary":"$C$ is compact.","title":"NumberField.AdeleRing.DivisionAlgebra.Aux.C_compact"},{"id":"flt/miniproject-fujisakis-lemma/the-proof/numberfield-adelering-divisionalgebra-aux-antidi","kind":"node","lean":"NumberField.AdeleRing.DivisionAlgebra.Aux.antidiag_mem_C","parent":4427,"result":true,"route":"../roadmap/flt/miniproject-fujisakis-lemma/the-proof/index.html#numberfield-adelering-divisionalgebra-aux-antidi","scope":"flt/miniproject-fujisakis-lemma/the-proof","status":"fully_proved","summary":"For every $\\beta\\in D_{\\mathbb{A}}^{(1)}$, there exists $b\\in D^\\times$ and $\\nu\\in D_{\\mathbb{A}}^{(1)}$ such that $\\beta=b\\nu$ and $(\\nu,\\nu^{-1})\\in C.$","title":"NumberField.AdeleRing.DivisionAlgebra.Aux.antidiag_mem_C"},{"id":"flt/miniproject-fujisakis-lemma/the-proof/numberfield-finiteadelering-divisionalgebra-unit","kind":"node","lean":"NumberField.FiniteAdeleRing.DivisionAlgebra.units_cocompact","parent":4427,"result":true,"route":"../roadmap/flt/miniproject-fujisakis-lemma/the-proof/index.html#numberfield-finiteadelering-divisionalgebra-unit","scope":"flt/miniproject-fujisakis-lemma/the-proof","status":"fully_proved","summary":"$D^\\times\\backslash(D\\otimes_K\\mathbb{A}_K^\\infty)^\\times$ is compact.","title":"NumberField.FiniteAdeleRing.DivisionAlgebra.units_cocompact"},{"id":"flt/miniproject-fujisakis-lemma/the-proof/numberfield-finiteadelering-divisionalgebra-fini","kind":"node","lean":"NumberField.FiniteAdeleRing.DivisionAlgebra.finiteDoubleCoset","parent":4427,"result":true,"route":"../roadmap/flt/miniproject-fujisakis-lemma/the-proof/index.html#numberfield-finiteadelering-divisionalgebra-fini","scope":"flt/miniproject-fujisakis-lemma/the-proof","status":"fully_proved","summary":"If $U$ is an open subgroup of $(D\\otimes_K \\mathbb{A}_K^\\infty)^\\times$ then the double coset space $D^\\times\\backslash(D\\otimes_K \\mathbb{A}_K^\\infty)^\\times/U$ is finite.","title":"NumberField.FiniteAdeleRing.DivisionAlgebra.finiteDoubleCoset"},{"data_url":"scopes/flt/miniproject-quaternion-algebras.json","id":"flt/miniproject-quaternion-algebras","kind":"scope","parent":4231,"result":false,"route":"../roadmap/flt/miniproject-quaternion-algebras/index.html","scope":"flt/miniproject-quaternion-algebras","status":"fully_proved","title":"Miniproject: Quaternion algebras"},{"id":"flt/miniproject-quaternion-algebras/the-goal","kind":"node","parent":4443,"result":false,"route":"../roadmap/flt/miniproject-quaternion-algebras/the-goal/index.html","scope":"flt/miniproject-quaternion-algebras","status":"can_state","summary":"At the time of writing, mathlib still does not have a proof that the space of classical modular forms of a fixed weight, level and character is finite-dimensional. The main result in this miniproject is to prove that certain spaces of quaternionic modular forms are finite-dimensional. We need finiteness results like this in order to control the Hecke algebras which act on these spaces; these Hecke algebras are the\u2026","title":"The goal"},{"id":"flt/miniproject-quaternion-algebras/initial-definitions","kind":"node","parent":4443,"result":false,"route":"../roadmap/flt/miniproject-quaternion-algebras/initial-definitions/index.html","scope":"flt/miniproject-quaternion-algebras","status":"can_state","summary":"Our first goal in this miniproject is the definition of these spaces of quaternionic modular forms. We start with some preliminary work towards this.","title":"Initial definitions"},{"id":"flt/miniproject-quaternion-algebras/brief-introduction-to-automorphic-forms-in-this","kind":"node","parent":4443,"result":false,"route":"../roadmap/flt/miniproject-quaternion-algebras/brief-introduction-to-automorphic-forms-in-this/index.html","scope":"flt/miniproject-quaternion-algebras","status":"can_state","summary":"Having made assumptions on $D$ which makes the theory of automorphic forms over $D^\\times$ far less technical, we will now make it a little more technical by using the modern adelic approach to the theory. Note that many results about the adeles of a number field are proved in the adele miniproject (section Miniproject: Adeles). Our automorphic forms will be certain functions on the units of the ring\u2026","title":"Brief introduction to automorphic forms in this setting"},{"data_url":"scopes/flt/miniproject-quaternion-algebras/definition-of-spaces-of-automorphic-forms.json","id":"flt/miniproject-quaternion-algebras/definition-of-spaces-of-automorphic-forms","kind":"scope","parent":4443,"result":false,"route":"../roadmap/flt/miniproject-quaternion-algebras/definition-of-spaces-of-automorphic-forms/index.html","scope":"flt/miniproject-quaternion-algebras/definition-of-spaces-of-automorphic-forms","status":"fully_proved","summary":"Let us now give some precise definitions. Recall that by $\\mathbb{A}_F^\\infty$ we mean the finite adeles of the totally real number field $F$.","title":"Definition of spaces of automorphic forms"},{"id":"flt/miniproject-quaternion-algebras/definition-of-spaces-of-automorphic-forms/totallydefinitequaternionalgebra-weighttwoautomo","kind":"node","lean":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm","parent":4447,"result":true,"route":"../roadmap/flt/miniproject-quaternion-algebras/definition-of-spaces-of-automorphic-forms/index.html#totallydefinitequaternionalgebra-weighttwoautomo","scope":"flt/miniproject-quaternion-algebras/definition-of-spaces-of-automorphic-forms","status":"fully_proved","summary":"The space of $R$-valued *automorphic forms* for $D^\\times$ is the set of functions $f:D_{\\mathbb{A}^\\infty}^\\times\\to R$ satisfying the following axioms:","title":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm"},{"id":"flt/miniproject-quaternion-algebras/definition-of-spaces-of-automorphic-forms/totallydefinitequaternionalgebra-weighttwoautomo-2","kind":"node","lean":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.addCommGroup","parent":4447,"result":true,"route":"../roadmap/flt/miniproject-quaternion-algebras/definition-of-spaces-of-automorphic-forms/index.html#totallydefinitequaternionalgebra-weighttwoautomo-2","scope":"flt/miniproject-quaternion-algebras/definition-of-spaces-of-automorphic-forms","status":"fully_proved","summary":"Pointwise addition $(f_1+f_2)(g):=f_1(g)+f_2(g)$ makes $S^D(R)$ into an additive abelian group.","title":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.addCommGroup"},{"id":"flt/miniproject-quaternion-algebras/definition-of-spaces-of-automorphic-forms/totallydefinitequaternionalgebra-weighttwoautomo-3","kind":"node","lean":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.module","parent":4447,"result":true,"route":"../roadmap/flt/miniproject-quaternion-algebras/definition-of-spaces-of-automorphic-forms/index.html#totallydefinitequaternionalgebra-weighttwoautomo-3","scope":"flt/miniproject-quaternion-algebras/definition-of-spaces-of-automorphic-forms","status":"fully_proved","summary":"If $R$ is a commutative ring then pointwise scalar multiplication $(r\\cdot f)(g):= r\\cdot(f(g))$ makes $S^D(R)$ into an $R$-module.","title":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.module"},{"id":"flt/miniproject-quaternion-algebras/definition-of-spaces-of-automorphic-forms/totallydefinitequaternionalgebra-weighttwoautomo-4","kind":"node","lean":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.distribMulAction","parent":4447,"result":true,"route":"../roadmap/flt/miniproject-quaternion-algebras/definition-of-spaces-of-automorphic-forms/index.html#totallydefinitequaternionalgebra-weighttwoautomo-4","scope":"flt/miniproject-quaternion-algebras/definition-of-spaces-of-automorphic-forms","status":"fully_proved","summary":"The group $D_{\\mathbb{A}^f}^\\times$ acts on the additive abelian group $S^D(R)$ by $(g\\cdot f)(x)=f(xg)$.","title":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.distribMulAction"},{"id":"flt/miniproject-quaternion-algebras/definition-of-spaces-of-automorphic-forms/totallydefinitequaternionalgebra-weighttwoautomo-5","kind":"node","lean":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.LevelStruct.form","parent":4447,"result":true,"route":"../roadmap/flt/miniproject-quaternion-algebras/definition-of-spaces-of-automorphic-forms/index.html#totallydefinitequaternionalgebra-weighttwoautomo-5","scope":"flt/miniproject-quaternion-algebras/definition-of-spaces-of-automorphic-forms","status":"fully_proved","summary":"The quaternionic modular forms of level $U$, with notation $S^D(U;R)$, are the $U$-invariants for the $D_{\\mathbb{A}^\\infty}^\\times$-action on $S^D(R)$.","title":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicFormOfLevel"},{"data_url":"scopes/flt/miniproject-hecke-operators.json","id":"flt/miniproject-hecke-operators","kind":"scope","parent":4231,"result":false,"route":"../roadmap/flt/miniproject-hecke-operators/index.html","scope":"flt/miniproject-hecke-operators","status":"can_prove","title":"Miniproject: Hecke Operators"},{"id":"flt/miniproject-hecke-operators/status","kind":"node","parent":4453,"result":false,"route":"../roadmap/flt/miniproject-hecke-operators/status/index.html","scope":"flt/miniproject-hecke-operators","status":"can_state","summary":"This is an active miniproject. The abstract theory is completely formalized; at the time of writing the concrete theory has no sorried definitions but it does have some sorried proofs.","title":"Status"},{"id":"flt/miniproject-hecke-operators/the-goal","kind":"node","parent":4453,"result":false,"route":"../roadmap/flt/miniproject-hecke-operators/the-goal/index.html","scope":"flt/miniproject-hecke-operators","status":"can_state","summary":"The goal of this project is to get sorry-free definitions of Hecke operators acting on spaces of automorphic forms. These Hecke operators generate Hecke algebras, which are the rings called $T$ in the modularity lifting theorems, or $R=T$ theorems, crucially introduced by Wiles in order to prove FLT.","title":"The goal"},{"data_url":"scopes/flt/miniproject-hecke-operators/the-abstract-theory.json","id":"flt/miniproject-hecke-operators/the-abstract-theory","kind":"scope","parent":4453,"result":false,"route":"../roadmap/flt/miniproject-hecke-operators/the-abstract-theory/index.html","scope":"flt/miniproject-hecke-operators/the-abstract-theory","status":"fully_proved","summary":"The set-up: we have a commutative ring $R$, the coefficient ring, and all of our spaces which the operators act on will be $R$-modules.","title":"The abstract theory"},{"id":"flt/miniproject-hecke-operators/the-abstract-theory/abstractheckeoperator-heckeoperatortofun","kind":"node","lean":"AbstractHeckeOperator.heckeOperator.toFun","parent":4456,"result":true,"route":"../roadmap/flt/miniproject-hecke-operators/the-abstract-theory/index.html#abstractheckeoperator-heckeoperatortofun","scope":"flt/miniproject-hecke-operators/the-abstract-theory","status":"fully_proved","summary":"Assuming $UgV$ is a finite union of cosets $g_iV$, we define $[UgV]:A^V\\to A^U$ to be the map sending $a\\in A^V$ to $\\sum_i g_ia.$","title":"AbstractHeckeOperator.HeckeOperatorToFun"},{"id":"flt/miniproject-hecke-operators/the-abstract-theory/abstractheckeoperator-heckeoperator","kind":"node","lean":"AbstractHeckeOperator.heckeOperator","parent":4456,"result":true,"route":"../roadmap/flt/miniproject-hecke-operators/the-abstract-theory/index.html#abstractheckeoperator-heckeoperator","scope":"flt/miniproject-hecke-operators/the-abstract-theory","status":"fully_proved","summary":"This function is well-defined (that is, independent of the choice of $g_i$), has image in $A^U$ and is $R$-linear.","title":"AbstractHeckeOperator.HeckeOperator"},{"id":"flt/miniproject-hecke-operators/the-abstract-theory/abstractheckeoperator-comm","kind":"node","lean":"AbstractHeckeOperator.comm","parent":4456,"result":true,"route":"../roadmap/flt/miniproject-hecke-operators/the-abstract-theory/index.html#abstractheckeoperator-comm","scope":"flt/miniproject-hecke-operators/the-abstract-theory","status":"fully_proved","summary":"Say $g,h\\in G$ and we have $UgU=\\coprod_i g_iU$ and $UhU=\\coprod_j h_j$ and we have $g_ih_j=h_jg_i$ for all $i,j$. Then $[UgU][UhU]=[UhU][UgU]$, that is, the Hecke operators acting on $A^U$ commute.","title":"AbstractHeckeOperator.comm"},{"id":"flt/miniproject-hecke-operators/the-abstract-theory/quotientgroup-mk-image-finite-of-compact-of-open","kind":"node","lean":"QuotientGroup.mk_image_finite_of_compact_of_open","parent":4456,"result":true,"route":"../roadmap/flt/miniproject-hecke-operators/the-abstract-theory/index.html#quotientgroup-mk-image-finite-of-compact-of-open","scope":"flt/miniproject-hecke-operators/the-abstract-theory","status":"fully_proved","summary":"If $U$ and $V$ are compact subgroups of a topological group $G$, if $V$ is also open, and if $g\\in G$, then the double coset space $UgV$ is a finite union of left cosets $g_iV$.","title":"QuotientGroup.mk_image_finite_of_compact_of_open"},{"data_url":"scopes/flt/miniproject-hecke-operators/restricted-products.json","id":"flt/miniproject-hecke-operators/restricted-products","kind":"scope","parent":4453,"result":false,"route":"../roadmap/flt/miniproject-hecke-operators/restricted-products/index.html","scope":"flt/miniproject-hecke-operators/restricted-products","status":"can_prove","summary":"In the concrete example of Hecke operators which we care about, the invariants $A^G$ will be spaces of quaternionic automorphic forms (by definition). We do not need to worry about the definition of $A$ in this project at all. However we will need to do various computations with the specific groups $G$ which we are interested in, and they are restricted products. So we now develop some theory for restricted\u2026","title":"Restricted products"},{"id":"flt/miniproject-hecke-operators/restricted-products/homeomorph-restrictedproductprod","kind":"node","lean":"Homeomorph.restrictedProductProd","parent":4461,"result":true,"route":"../roadmap/flt/miniproject-hecke-operators/restricted-products/index.html#homeomorph-restrictedproductprod","scope":"flt/miniproject-hecke-operators/restricted-products","status":"fully_proved","summary":"If $A_i$ is a family of topological spaces equipped with open subsets $B_i$, and if $C_i$ is a family of topological spaces equipped with open subsets $D_i$, and if we equip $A_i\\times C_i$ with the open subset $B_i\\times D_i$, then the natural bijection $\\prod'_i(A_i\\times C_i)=\\left(\\prod'_iA_i\\right)\\times\\left(\\prod'_iB_i\\right)$ is a homeomorphism.","title":"Homeomorph.restrictedProductProd"},{"id":"flt/miniproject-hecke-operators/restricted-products/homeomorph-restrictedproductpi","kind":"node","lean":"Homeomorph.restrictedProductPi","parent":4461,"result":true,"route":"../roadmap/flt/miniproject-hecke-operators/restricted-products/index.html#homeomorph-restrictedproductpi","scope":"flt/miniproject-hecke-operators/restricted-products","status":"fully_proved","summary":"Restricted products (with respect to open subspaces) commute with finite products. In other words, if $j$ runs through a finite set $J$ and $i$ runs through an arbitrary set $I$, and if $X_{ji}$ are topological spaces equipped with open subspaces $Y_{ji}$, then the obvious bijection $\\prod'_i(\\prod_j X_{ji})=\\prod_j\\left(\\prod'_i X_{ji}\\right)$ is a homeomorphism.","title":"Homeomorph.restrictedProductPi"},{"id":"flt/miniproject-hecke-operators/restricted-products/homeomorph-restrictedproductmatrix","kind":"node","lean":"Homeomorph.restrictedProductMatrix","parent":4461,"result":true,"route":"../roadmap/flt/miniproject-hecke-operators/restricted-products/index.html#homeomorph-restrictedproductmatrix","scope":"flt/miniproject-hecke-operators/restricted-products","status":"fully_proved","summary":"If $X_i$ are topological spaces and the $Y_i$ are open subspaces, then the obvious map $M_n(\\prod'_iX_i)=\\prod'_iM_n(X_i)$ is a homeomorphism.","title":"Homeomorph.restrictedProductMatrix"},{"id":"flt/miniproject-hecke-operators/restricted-products/submonoid-units-isopen","kind":"node","lean":"Submonoid.units_isOpen","parent":4461,"result":true,"route":"../roadmap/flt/miniproject-hecke-operators/restricted-products/index.html#submonoid-units-isopen","scope":"flt/miniproject-hecke-operators/restricted-products","status":"fully_proved","summary":"If $M$ is a topological monoid and $U$ is an open submonoid, then the units $U^\\times$ of $U$ are naturally an open subgroup of $M^\\times$.","title":"Submonoid.units_isOpen"},{"id":"flt/miniproject-hecke-operators/restricted-products/submonoid-units-iscompact","kind":"node","lean":"Submonoid.units_isCompact","parent":4461,"result":true,"route":"../roadmap/flt/miniproject-hecke-operators/restricted-products/index.html#submonoid-units-iscompact","scope":"flt/miniproject-hecke-operators/restricted-products","status":"can_prove","summary":"If $M$ is a Hausdorff topological monoid and $U$ is a compact submonoid, then the units $U^\\times$ of $U$ are naturally a compact subgroup of $M^\\times$.","title":"Submonoid.units_isCompact"},{"id":"flt/miniproject-hecke-operators/restricted-products/continuousmulequiv-piunits","kind":"node","lean":"ContinuousMulEquiv.piUnits","parent":4461,"result":true,"route":"../roadmap/flt/miniproject-hecke-operators/restricted-products/index.html#continuousmulequiv-piunits","scope":"flt/miniproject-hecke-operators/restricted-products","status":"fully_proved","summary":"If $U_i$ are topological monoids then the canonical group isomorphism $(\\prod_i U_i)^\\times=\\prod_i(U_i^\\times)$ is a homeomorphism.","title":"ContinuousMulEquiv.piUnits"},{"id":"flt/miniproject-hecke-operators/restricted-products/continuousmulequiv-restrictedproductunits","kind":"node","lean":"ContinuousMulEquiv.restrictedProductUnits","parent":4461,"result":true,"route":"../roadmap/flt/miniproject-hecke-operators/restricted-products/index.html#continuousmulequiv-restrictedproductunits","scope":"flt/miniproject-hecke-operators/restricted-products","status":"fully_proved","summary":"If $M_i$ are a family of topological monoids equipped with open submonoids $U_i$, then the canonical map $(\\prod'_iM_i)^\\times\\to\\prod'_i(M_i^\\times)$ is a homeomorphism.","title":"ContinuousMulEquiv.restrictedProductUnits"},{"data_url":"scopes/flt/miniproject-hecke-operators/some-local-theory.json","id":"flt/miniproject-hecke-operators/some-local-theory","kind":"scope","parent":4453,"result":false,"route":"../roadmap/flt/miniproject-hecke-operators/some-local-theory/index.html","scope":"flt/miniproject-hecke-operators/some-local-theory","status":"proved","summary":"We could work over a general nonarchimedean normed field but we still do not have them in mathlib, so we stick to the case of interest which is the completion of a number field $K$ at a finite place $v$. Such a completion is a topological field $K_v$ equipped with a discrete valuation, a ring of integers $\\mathcal{O}_v$ having a principal maximal ideal $(\\varpi)$, and a finite residue field\u2026","title":"Some local theory"},{"id":"flt/miniproject-hecke-operators/some-local-theory/numberfield-isopenadiccompletionintegers","kind":"node","lean":"NumberField.isOpenAdicCompletionIntegers","parent":4469,"result":true,"route":"../roadmap/flt/miniproject-hecke-operators/some-local-theory/index.html#numberfield-isopenadiccompletionintegers","scope":"flt/miniproject-hecke-operators/some-local-theory","status":"fully_proved","summary":"$\\mathcal{O}_v$ is an open subring of $K_v$.","title":"NumberField.isOpenAdicCompletionIntegers"},{"id":"flt/miniproject-hecke-operators/some-local-theory/m2-localfulllevel-isopen","kind":"node","lean":"IsDedekindDomain.M2.localFullLevel.isOpen","parent":4469,"result":true,"route":"../roadmap/flt/miniproject-hecke-operators/some-local-theory/index.html#m2-localfulllevel-isopen","scope":"flt/miniproject-hecke-operators/some-local-theory","status":"fully_proved","summary":"$M_2(\\mathcal{O}_v)$ is an open subring of $M_2(K_v)$.","title":"M2.localFullLevel.isOpen"},{"id":"flt/miniproject-hecke-operators/some-local-theory/m2-localfulllevel-iscompact","kind":"node","lean":"IsDedekindDomain.M2.localFullLevel.isCompact","parent":4469,"result":true,"route":"../roadmap/flt/miniproject-hecke-operators/some-local-theory/index.html#m2-localfulllevel-iscompact","scope":"flt/miniproject-hecke-operators/some-local-theory","status":"proved","summary":"$M_2(\\mathcal{O}_v)$ is a compact subring of $M_2(K_v)$.","title":"M2.localFullLevel.isCompact"},{"id":"flt/miniproject-hecke-operators/some-local-theory/bijon-unipotent-mul-diagu1-u1diagu1","kind":"node","lean":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.Local.bijOn_unipotentMulDiagU1","parent":4469,"result":true,"route":"../roadmap/flt/miniproject-hecke-operators/some-local-theory/index.html#bijon-unipotent-mul-diagu1-u1diagu1","scope":"flt/miniproject-hecke-operators/some-local-theory","status":"fully_proved","summary":"The double coset space $UgU$ is the disjoint union of $g_tU$ as $t$ ranges through $\\mathcal{O}_v/\\alpha\\mathcal{O}_v$ and $g_t:=\\begin{pmatrix}\\alpha&\\tilde{t}\\\\0&1\\end{pmatrix}$, where $\\tilde{t}$ is any lift of $t$ to $\\mathcal{O}_v$.","title":"bijOn_unipotent_mul_diagU1_U1diagU1"},{"data_url":"scopes/flt/miniproject-hecke-operators/adelic-groups.json","id":"flt/miniproject-hecke-operators/adelic-groups","kind":"scope","parent":4453,"result":false,"route":"../roadmap/flt/miniproject-hecke-operators/adelic-groups/index.html","scope":"flt/miniproject-hecke-operators/adelic-groups","status":"fully_proved","summary":"We are finally ready to discuss the group $G$ and the subgroups $U$ which we will be using to define our Hecke operators. Let $K$ be a number field, let $D$ a quaternion algebra over $K$ and let $\\mathbb{A}_K^\\infty$ be the finite adeles of $K$; recall that this is a commutative topological ring, defined to be the restricted product of the commutative topological fields $K_v$ as $v$ runs through the finite places\u2026","title":"Adelic groups"},{"id":"flt/miniproject-hecke-operators/adelic-groups/gl2-restrictedproduct","kind":"node","lean":"IsDedekindDomain.FiniteAdeleRing.GL2.restrictedProduct","parent":4474,"result":true,"route":"../roadmap/flt/miniproject-hecke-operators/adelic-groups/index.html#gl2-restrictedproduct","scope":"flt/miniproject-hecke-operators/adelic-groups","status":"fully_proved","summary":"$G$ is isomorphic and homeomorphic to the restricted product of $GL_2(K_v)$ with respect to the compact open subgroups $GL_2(\\mathcal{O}_v)$.","title":"GL2.restrictedProduct"},{"id":"flt/miniproject-hecke-operators/automorphic-forms","kind":"node","parent":4453,"result":false,"route":"../roadmap/flt/miniproject-hecke-operators/automorphic-forms/index.html","scope":"flt/miniproject-hecke-operators","status":"can_state","summary":"We recall some of the definitions of spaces of automorphic forms, from the quaternion algebra project, section Miniproject: Quaternion algebras.","title":"Automorphic forms"},{"data_url":"chapters/pnt-plus.json","id":"pnt-plus","kind":"scope","parent":0,"result":false,"route":"../roadmap/pnt-plus/index.html","scope":"pnt-plus","status":"stated","summary":"The blueprint of PNT+, read at commit 127b50e3 (2026-10-19): one article for each of its 383 nodes whose Lean declarations it names and marks formalized, or in Mathlib. Its other 20 nodes have no article, and a section of only those is left out: the map records what the project contains, not what it plans. A statement for which the blueprint marks no proof `\\leanok` says so. Dependencies are the blueprint's\u2026","title":"PNT+"},{"id":"pnt-plus/the-project","kind":"node","parent":4477,"result":false,"route":"../roadmap/pnt-plus/the-project/index.html","scope":"pnt-plus","status":"can_state","summary":"The project github page is .","title":"The project"},{"data_url":"scopes/pnt-plus/basic-definitions-and-notation.json","id":"pnt-plus/basic-definitions-and-notation","kind":"scope","parent":4477,"result":false,"route":"../roadmap/pnt-plus/basic-definitions-and-notation/index.html","scope":"pnt-plus/basic-definitions-and-notation","status":"fully_proved","title":"Basic definitions and notation"},{"id":"pnt-plus/basic-definitions-and-notation/first-gap-def","kind":"node","lean":"first_gap","parent":4479,"result":true,"route":"../roadmap/pnt-plus/basic-definitions-and-notation/index.html#first-gap-def","scope":"pnt-plus/basic-definitions-and-notation","status":"fully_proved","summary":"$P(g)$ is the first prime $p_n$ for which the prime gap $p_{n+1}-p_n$ is equal to $g$, or $0$ if no such gap exists.","title":"First prime gap"},{"id":"pnt-plus/basic-definitions-and-notation/pi-def","kind":"node","lean":"pi","parent":4479,"result":true,"route":"../roadmap/pnt-plus/basic-definitions-and-notation/index.html#pi-def","scope":"pnt-plus/basic-definitions-and-notation","status":"fully_proved","summary":"$\\pi(x)$ is the number of primes less than or equal to $x$.","title":"pi"},{"id":"pnt-plus/basic-definitions-and-notation/pi-star-def","kind":"node","lean":"pi_star","parent":4479,"result":true,"route":"../roadmap/pnt-plus/basic-definitions-and-notation/index.html#pi-star-def","scope":"pnt-plus/basic-definitions-and-notation","status":"fully_proved","summary":"$\\pi^*(x) = \\sum_{k \\geq 1} \\pi(x^{1/k}) / k$.","title":"pi star"},{"id":"pnt-plus/basic-definitions-and-notation/li-def","kind":"node","lean":"li, Li","parent":4479,"result":true,"route":"../roadmap/pnt-plus/basic-definitions-and-notation/index.html#li-def","scope":"pnt-plus/basic-definitions-and-notation","status":"fully_proved","summary":"$\\mathrm{li}(x) = \\int_0^x \\frac{dt}{\\log t}$ (in the principal value sense) and $\\mathrm{Li}(x) = \\int_2^x \\frac{dt}{\\log t}$.","title":"li and Li"},{"id":"pnt-plus/basic-definitions-and-notation/epsi-def","kind":"node","lean":"E\u03c8","parent":4479,"result":true,"route":"../roadmap/pnt-plus/basic-definitions-and-notation/index.html#epsi-def","scope":"pnt-plus/basic-definitions-and-notation","status":"fully_proved","summary":"$E_\\psi(x) = |\\psi(x) - x| / x$","title":"Equation (2) of FKS2"},{"id":"pnt-plus/basic-definitions-and-notation/classical-bound-psi","kind":"node","lean":"E\u03c8.classicalBound","parent":4479,"result":true,"route":"../roadmap/pnt-plus/basic-definitions-and-notation/index.html#classical-bound-psi","scope":"pnt-plus/basic-definitions-and-notation","status":"fully_proved","summary":"We say that $E_\\psi$ satisfies a *classical bound* with parameters $A, B, C, R, x_0$ if for all $x \\geq x_0$ we have","title":"Definitions 1, 5, FKS2"},{"id":"pnt-plus/basic-definitions-and-notation/epi-def","kind":"node","lean":"E\u03c0","parent":4479,"result":true,"route":"../roadmap/pnt-plus/basic-definitions-and-notation/index.html#epi-def","scope":"pnt-plus/basic-definitions-and-notation","status":"fully_proved","summary":"$E_\\pi(x) = |\\pi(x) - \\mathrm{Li}(x)| / (x / \\log x)$.","title":"Equation (1) of FKS2"},{"id":"pnt-plus/basic-definitions-and-notation/etheta-def","kind":"node","lean":"E\u03b8","parent":4479,"result":true,"route":"../roadmap/pnt-plus/basic-definitions-and-notation/index.html#etheta-def","scope":"pnt-plus/basic-definitions-and-notation","status":"fully_proved","summary":"$E_\\theta(x) = |\\theta(x) - x| / x$","title":"Equation (2) of FKS2"},{"id":"pnt-plus/basic-definitions-and-notation/classical-bound-theta","kind":"node","lean":"E\u03b8.classicalBound","parent":4479,"result":true,"route":"../roadmap/pnt-plus/basic-definitions-and-notation/index.html#classical-bound-theta","scope":"pnt-plus/basic-definitions-and-notation","status":"fully_proved","summary":"We say that $E_\\theta$ satisfies a *classical bound* with parameters $A, B, C, R, x_0$ if for all $x \\geq x_0$ we have","title":"Definitions 1, 5, FKS2"},{"id":"pnt-plus/basic-definitions-and-notation/classical-bound-pi","kind":"node","lean":"E\u03c0.classicalBound","parent":4479,"result":true,"route":"../roadmap/pnt-plus/basic-definitions-and-notation/index.html#classical-bound-pi","scope":"pnt-plus/basic-definitions-and-notation","status":"fully_proved","summary":"We say that $E_\\pi$ satisfies a *classical bound* with parameters $A, B, C, R, x_0$ if for all $x \\geq x_0$ we have","title":"Definitions 1, 5, FKS2"},{"id":"pnt-plus/basic-definitions-and-notation/admissible-bound-monotone","kind":"node","lean":"admissible_bound.mono","parent":4479,"result":true,"route":"../roadmap/pnt-plus/basic-definitions-and-notation/index.html#admissible-bound-monotone","scope":"pnt-plus/basic-definitions-and-notation","status":"fully_proved","summary":"If $A,B,C,R > 0$ then the classical bound is monotone decreasing for $x \\geq \\exp( R (2B/C)^2 )$.","title":"Admissible bound decreasing for large x"},{"id":"pnt-plus/basic-definitions-and-notation/classical-to-numeric","kind":"node","lean":"E\u03c8.classicalBound.to_numericalBound, E\u03b8.classicalBound.to_numericalBound, E\u03c0.classicalBound.to_numericalBound","parent":4479,"result":true,"route":"../roadmap/pnt-plus/basic-definitions-and-notation/index.html#classical-to-numeric","scope":"pnt-plus/basic-definitions-and-notation","status":"fully_proved","summary":"A classical bound for $x \\geq x_0$ implies a numerical bound for $x \\geq \\max(x_0, \\exp( R (2B/C)^2 ))$.","title":"Classic bound implies numerical bound"},{"data_url":"scopes/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem.json","id":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem","kind":"scope","parent":4477,"result":false,"route":"../roadmap/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/index.html","scope":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem","status":"stated","title":"First approach: Wiener-Ikehara Tauberian theorem"},{"data_url":"scopes/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t.json","id":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t","kind":"scope","parent":4492,"result":false,"route":"../roadmap/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/index.html","scope":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t","status":"stated","summary":"The Fourier transform of an absolutely integrable function $\\psi: \\mathbb{R} \\to \\mathbb{C}$ is defined by the formula","title":"A Fourier-analytic proof of the Wiener-Ikehara theorem"},{"id":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/first-fourier","kind":"node","lean":"first_fourier","parent":4493,"result":true,"route":"../roadmap/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/index.html#first-fourier","scope":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t","status":"fully_proved","summary":"If $\\psi: \\mathbb{R} \\to \\mathbb{C}$ is integrable and $x > 0$, then for any $\\sigma>1$","title":"first-fourier"},{"id":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/second-fourier","kind":"node","lean":"second_fourier","parent":4493,"result":true,"route":"../roadmap/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/index.html#second-fourier","scope":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t","status":"fully_proved","summary":"If $\\psi: \\mathbb{R} \\to \\mathbb{C}$ is absolutely integrable and $x > 0$, then for any $\\sigma>1$","title":"second-fourier"},{"id":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/prelim-decay","kind":"node","lean":"prelim_decay","parent":4493,"result":true,"route":"../roadmap/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/index.html#prelim-decay","scope":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t","status":"fully_proved","summary":"If $\\psi:\\mathbb{R} \\to \\mathbb{C}$ is absolutely integrable then","title":"Preliminary decay bound I"},{"id":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/prelim-decay-2","kind":"node","lean":"prelim_decay_2","parent":4493,"result":true,"route":"../roadmap/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/index.html#prelim-decay-2","scope":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t","status":"can_prove","summary":"If $\\psi:\\mathbb{R} \\to \\mathbb{C}$ is absolutely integrable and of bounded variation, then","title":"Preliminary decay bound II"},{"id":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/prelim-decay-3","kind":"node","lean":"prelim_decay_3","parent":4493,"result":true,"route":"../roadmap/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/index.html#prelim-decay-3","scope":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t","status":"stated","summary":"If $\\psi:\\mathbb{R} \\to \\mathbb{C}$ is absolutely integrable, absolutely continuous, and $\\psi'$ is of bounded variation, then","title":"Preliminary decay bound III"},{"id":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/decay-alt","kind":"node","lean":"decay_alt","parent":4493,"result":true,"route":"../roadmap/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/index.html#decay-alt","scope":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t","status":"proved","summary":"If $\\psi:\\mathbb{R} \\to \\mathbb{C}$ is absolutely integrable, absolutely continuous, and $\\psi'$ is of bounded variation, then","title":"Decay bound, alternate form"},{"id":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/decay","kind":"node","lean":"decay_bounds","parent":4493,"result":true,"route":"../roadmap/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/index.html#decay","scope":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t","status":"fully_proved","summary":"If $\\psi:\\mathbb{R} \\to \\mathbb{C}$ is $C^2$ and obeys the bounds","title":"Decay bounds"},{"id":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/limiting","kind":"node","lean":"limiting_fourier","parent":4493,"result":true,"route":"../roadmap/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/index.html#limiting","scope":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t","status":"fully_proved","summary":"If $\\psi: \\mathbb{R} \\to \\mathbb{C}$ is $C^2$ and compactly supported and $x \\geq 1$, then","title":"Limiting Fourier identity"},{"id":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/limiting-cor","kind":"node","lean":"limiting_cor","parent":4493,"result":true,"route":"../roadmap/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/index.html#limiting-cor","scope":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t","status":"fully_proved","summary":"With the hypotheses as above, we have","title":"Corollary of limiting identity"},{"id":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/smooth-ury","kind":"node","lean":"smooth_urysohn","parent":4493,"result":true,"route":"../roadmap/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/index.html#smooth-ury","scope":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t","status":"fully_proved","summary":"If $I$ is a closed interval contained in an open interval $J$, then there exists a smooth function $\\Psi: \\mathbb{R} \\to \\mathbb{R}$ with $1_I \\leq \\Psi \\leq 1_J$.","title":"Smooth Urysohn lemma"},{"id":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/schwarz-id","kind":"node","lean":"limiting_cor_schwartz","parent":4493,"result":true,"route":"../roadmap/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/index.html#schwarz-id","scope":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t","status":"fully_proved","summary":"The previous corollary also holds for functions $\\psi$ that are assumed to be in the Schwartz class, as opposed to being $C^2$ and compactly supported.","title":"Limiting identity for Schwartz functions"},{"id":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/bij","kind":"node","lean":"fourier_surjection_on_schwartz","parent":4493,"result":true,"route":"../roadmap/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/index.html#bij","scope":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t","status":"fully_proved","summary":"The Fourier transform is a bijection on the Schwartz class. [Note: only surjectivity is actually used.]","title":"Bijectivity of Fourier transform"},{"id":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/wienerikeharasmooth","kind":"node","lean":"wiener_ikehara_smooth","parent":4493,"result":true,"route":"../roadmap/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/index.html#wienerikeharasmooth","scope":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t","status":"fully_proved","summary":"If $\\Psi: (0,\\infty) \\to \\mathbb{C}$ is smooth and compactly supported away from the origin, then,","title":"Smoothed Wiener-Ikehara"},{"id":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/wienerikeharainterval","kind":"node","lean":"WienerIkeharaInterval","parent":4493,"result":true,"route":"../roadmap/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/index.html#wienerikeharainterval","scope":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t","status":"fully_proved","summary":"For any closed interval $I \\subset (0,+\\infty)$, we have","title":"Wiener-Ikehara in an interval"},{"id":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/wienerikehara","kind":"node","lean":"WienerIkeharaTheorem'","parent":4493,"result":true,"route":"../roadmap/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t/index.html#wienerikehara","scope":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/a-fourier-analytic-proof-of-the-wiener-ikehara-t","status":"fully_proved","summary":"We have","title":"Wiener-Ikehara Theorem (1)"},{"data_url":"scopes/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/weak-pnt.json","id":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/weak-pnt","kind":"scope","parent":4492,"result":false,"route":"../roadmap/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/weak-pnt/index.html","scope":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/weak-pnt","status":"fully_proved","title":"Weak PNT"},{"id":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/weak-pnt/weakpnt","kind":"node","lean":"WeakPNT","parent":4509,"result":true,"route":"../roadmap/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/weak-pnt/index.html#weakpnt","scope":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/weak-pnt","status":"fully_proved","summary":"We have","title":"WeakPNT"},{"data_url":"scopes/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/removing-the-chebyshev-hypothesis.json","id":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/removing-the-chebyshev-hypothesis","kind":"scope","parent":4492,"result":false,"route":"../roadmap/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/removing-the-chebyshev-hypothesis/index.html","scope":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/removing-the-chebyshev-hypothesis","status":"fully_proved","summary":"In this section we do \\*not\\* assume the bound (1), but instead derive it from the other hypotheses.","title":"Removing the Chebyshev hypothesis"},{"id":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/removing-the-chebyshev-hypothesis/limiting-fourier-variant","kind":"node","lean":"limiting_fourier_variant","parent":4511,"result":true,"route":"../roadmap/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/removing-the-chebyshev-hypothesis/index.html#limiting-fourier-variant","scope":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/removing-the-chebyshev-hypothesis","status":"fully_proved","summary":"If $\\psi: \\mathbb{R} \\to \\mathbb{C}$ is $C^2$ and compactly supported with $f$ and $\\hat \\psi$ non-negative, and $0 < x$, then","title":"limiting-fourier-variant"},{"id":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/removing-the-chebyshev-hypothesis/crude-upper-bound","kind":"node","lean":"crude_upper_bound","parent":4511,"result":true,"route":"../roadmap/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/removing-the-chebyshev-hypothesis/index.html#crude-upper-bound","scope":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/removing-the-chebyshev-hypothesis","status":"fully_proved","summary":"If $\\psi: \\mathbb{R} \\to \\mathbb{C}$ is $C^2$ and compactly supported with $f$ and $\\hat \\psi$ non-negative, then there exists a constant $B$ such that","title":"crude-upper-bound"},{"id":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/removing-the-chebyshev-hypothesis/auto-cheby","kind":"node","lean":"auto_cheby","parent":4511,"result":true,"route":"../roadmap/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/removing-the-chebyshev-hypothesis/index.html#auto-cheby","scope":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/removing-the-chebyshev-hypothesis","status":"fully_proved","summary":"One has","title":"auto-cheby"},{"id":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/removing-the-chebyshev-hypothesis/wienerikehara2","kind":"node","lean":"WienerIkeharaTheorem''","parent":4511,"result":true,"route":"../roadmap/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/removing-the-chebyshev-hypothesis/index.html#wienerikehara2","scope":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/removing-the-chebyshev-hypothesis","status":"fully_proved","summary":"We have","title":"Wiener-Ikehara Theorem (2)"},{"data_url":"scopes/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/the-prime-number-theorem-in-arithmetic-progressi.json","id":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/the-prime-number-theorem-in-arithmetic-progressi","kind":"scope","parent":4492,"result":false,"route":"../roadmap/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/the-prime-number-theorem-in-arithmetic-progressi/index.html","scope":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/the-prime-number-theorem-in-arithmetic-progressi","status":"fully_proved","title":"The prime number theorem in arithmetic progressions"},{"id":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/the-prime-number-theorem-in-arithmetic-progressi/weakpnt-character","kind":"node","lean":"WeakPNT_character","parent":4516,"result":true,"route":"../roadmap/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/the-prime-number-theorem-in-arithmetic-progressi/index.html#weakpnt-character","scope":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/the-prime-number-theorem-in-arithmetic-progressi","status":"fully_proved","summary":"If $q \u2265 1$ and $a$ is coprime to $q$, and $\\mathrm{Re} s > 1$, we have","title":"WeakPNT-character"},{"id":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/the-prime-number-theorem-in-arithmetic-progressi/weakpnt-ap-prelim","kind":"node","lean":"WeakPNT_AP_prelim","parent":4516,"result":true,"route":"../roadmap/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/the-prime-number-theorem-in-arithmetic-progressi/index.html#weakpnt-ap-prelim","scope":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/the-prime-number-theorem-in-arithmetic-progressi","status":"fully_proved","summary":"If $q \u2265 1$ and $a$ is coprime to $q$, the Dirichlet series $\\sum_{n \\leq x: n = a\\ (q)} \\frac{\\Lambda(n)}{n^s}$ converges for $\\mathrm{Re}(s) > 1$ to $\\frac{1}{\\varphi(q)} \\frac{1}{s-1} + G(s)$ where $G$ has a continuous extension to $\\mathrm{Re}(s)=1$.","title":"WeakPNT-AP-prelim"},{"id":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/the-prime-number-theorem-in-arithmetic-progressi/weakpnt-ap","kind":"node","lean":"WeakPNT_AP","parent":4516,"result":true,"route":"../roadmap/pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/the-prime-number-theorem-in-arithmetic-progressi/index.html#weakpnt-ap","scope":"pnt-plus/first-approach-wiener-ikehara-tauberian-theorem/the-prime-number-theorem-in-arithmetic-progressi","status":"fully_proved","summary":"If $q \u2265 1$ and $a$ is coprime to $q$, we have","title":"WeakPNT-AP"},{"data_url":"scopes/pnt-plus/second-approach.json","id":"pnt-plus/second-approach","kind":"scope","parent":4477,"result":false,"route":"../roadmap/pnt-plus/second-approach/index.html","scope":"pnt-plus/second-approach","status":"fully_proved","title":"Second approach"},{"data_url":"scopes/pnt-plus/second-approach/residue-calculus-on-rectangles.json","id":"pnt-plus/second-approach/residue-calculus-on-rectangles","kind":"scope","parent":4520,"result":false,"route":"../roadmap/pnt-plus/second-approach/residue-calculus-on-rectangles/index.html","scope":"pnt-plus/second-approach/residue-calculus-on-rectangles","status":"fully_proved","summary":"This files gathers definitions and basic properties about rectangles.","title":"Residue calculus on rectangles"},{"id":"pnt-plus/second-approach/residue-calculus-on-rectangles/rectangleborder","kind":"node","lean":"RectangleBorder","parent":4521,"result":true,"route":"../roadmap/pnt-plus/second-approach/residue-calculus-on-rectangles/index.html#rectangleborder","scope":"pnt-plus/second-approach/residue-calculus-on-rectangles","status":"fully_proved","summary":"A Rectangle's border, given corners $z$ and $w$ is the union of the four sides.","title":"RectangleBorder"},{"id":"pnt-plus/second-approach/residue-calculus-on-rectangles/rectangleintegral","kind":"node","lean":"RectangleIntegral","parent":4521,"result":true,"route":"../roadmap/pnt-plus/second-approach/residue-calculus-on-rectangles/index.html#rectangleintegral","scope":"pnt-plus/second-approach/residue-calculus-on-rectangles","status":"fully_proved","summary":"A RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\mathbb{C}$. We will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)","title":"RectangleIntegral"},{"id":"pnt-plus/second-approach/residue-calculus-on-rectangles/upperuintegral","kind":"node","lean":"UpperUIntegral","parent":4521,"result":true,"route":"../roadmap/pnt-plus/second-approach/residue-calculus-on-rectangles/index.html#upperuintegral","scope":"pnt-plus/second-approach/residue-calculus-on-rectangles","status":"fully_proved","summary":"An UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.","title":"UpperUIntegral"},{"id":"pnt-plus/second-approach/residue-calculus-on-rectangles/loweruintegral","kind":"node","lean":"LowerUIntegral","parent":4521,"result":true,"route":"../roadmap/pnt-plus/second-approach/residue-calculus-on-rectangles/index.html#loweruintegral","scope":"pnt-plus/second-approach/residue-calculus-on-rectangles","status":"fully_proved","summary":"A LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.","title":"LowerUIntegral"},{"id":"pnt-plus/second-approach/residue-calculus-on-rectangles/verticalintegral","kind":"node","lean":"VerticalIntegral","parent":4521,"result":true,"route":"../roadmap/pnt-plus/second-approach/residue-calculus-on-rectangles/index.html#verticalintegral","scope":"pnt-plus/second-approach/residue-calculus-on-rectangles","status":"fully_proved","summary":"Let $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define","title":"VerticalIntegral"},{"id":"pnt-plus/second-approach/residue-calculus-on-rectangles/diffvertrect-eq-upperlowerus","kind":"node","lean":"DiffVertRect_eq_UpperLowerUs","parent":4521,"result":true,"route":"../roadmap/pnt-plus/second-approach/residue-calculus-on-rectangles/index.html#diffvertrect-eq-upperlowerus","scope":"pnt-plus/second-approach/residue-calculus-on-rectangles","status":"fully_proved","summary":"The difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.","title":"DiffVertRect-eq-UpperLowerUs"},{"id":"pnt-plus/second-approach/residue-calculus-on-rectangles/existsdifferentiableon-of-bddabove","kind":"node","lean":"existsDifferentiableOn_of_bddAbove","parent":4521,"result":true,"route":"../roadmap/pnt-plus/second-approach/residue-calculus-on-rectangles/index.html#existsdifferentiableon-of-bddabove","scope":"pnt-plus/second-approach/residue-calculus-on-rectangles","status":"fully_proved","summary":"If $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.","title":"existsDifferentiableOn-of-bddAbove"},{"id":"pnt-plus/second-approach/residue-calculus-on-rectangles/holomorphicon-vanishesonrectangle","kind":"node","lean":"HolomorphicOn.vanishesOnRectangle","parent":4521,"result":true,"route":"../roadmap/pnt-plus/second-approach/residue-calculus-on-rectangles/index.html#holomorphicon-vanishesonrectangle","scope":"pnt-plus/second-approach/residue-calculus-on-rectangles","status":"fully_proved","summary":"If $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.","title":"HolomorphicOn.vanishesOnRectangle"},{"id":"pnt-plus/second-approach/residue-calculus-on-rectangles/rectanglepulltonhdofpole","kind":"node","lean":"RectanglePullToNhdOfPole","parent":4521,"result":true,"route":"../roadmap/pnt-plus/second-approach/residue-calculus-on-rectangles/index.html#rectanglepulltonhdofpole","scope":"pnt-plus/second-approach/residue-calculus-on-rectangles","status":"fully_proved","summary":"If $f$ is holomorphic on a rectangle $z$ and $w$ except at a point $p$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is the same as the integral of $f$ over a small square centered at $p$.","title":"RectanglePullToNhdOfPole"},{"id":"pnt-plus/second-approach/residue-calculus-on-rectangles/residuetheorematorigin","kind":"node","lean":"ResidueTheoremAtOrigin","parent":4521,"result":true,"route":"../roadmap/pnt-plus/second-approach/residue-calculus-on-rectangles/index.html#residuetheorematorigin","scope":"pnt-plus/second-approach/residue-calculus-on-rectangles","status":"fully_proved","summary":"The rectangle (square) integral of $f(s) = 1/s$ with corners $-1-i$ and $1+i$ is equal to $2\\pi i$.","title":"ResidueTheoremAtOrigin"},{"id":"pnt-plus/second-approach/residue-calculus-on-rectangles/residuetheoremonrectanglewithsimplepole","kind":"node","lean":"ResidueTheoremOnRectangleWithSimplePole","parent":4521,"result":true,"route":"../roadmap/pnt-plus/second-approach/residue-calculus-on-rectangles/index.html#residuetheoremonrectanglewithsimplepole","scope":"pnt-plus/second-approach/residue-calculus-on-rectangles","status":"fully_proved","summary":"Suppose that $f$ is a holomorphic function on a rectangle, except for a simple pole at $p$. By the latter, we mean that there is a function $g$ holomorphic on the rectangle such that, $f = g + A/(s-p)$ for some $A\\in\\mathbb{C}$. Then the integral of $f$ over the rectangle is $A$.","title":"ResidueTheoremOnRectangleWithSimplePole"},{"data_url":"scopes/pnt-plus/second-approach/perron-formula.json","id":"pnt-plus/second-approach/perron-formula","kind":"scope","parent":4520,"result":false,"route":"../roadmap/pnt-plus/second-approach/perron-formula/index.html","scope":"pnt-plus/second-approach/perron-formula","status":"fully_proved","summary":"In this section, we prove the Perron formula, which plays a key role in our proof of Mellin inversion.","title":"Perron Formula"},{"data_url":"scopes/pnt-plus/second-approach/perron-formula/part-01.json","id":"pnt-plus/second-approach/perron-formula/part-01","kind":"scope","navigation":"transparent","parent":4533,"result":false,"route":"../roadmap/pnt-plus/second-approach/perron-formula/part-01/index.html","scope":"pnt-plus/second-approach/perron-formula/part-01","status":"fully_proved","summary":"TODO : Move to general section","title":"zeroTendstoDiff to residueAtZero"},{"id":"pnt-plus/second-approach/perron-formula/part-01/zerotendstodiff","kind":"node","lean":"zeroTendstoDiff","parent":4534,"result":true,"route":"../roadmap/pnt-plus/second-approach/perron-formula/part-01/index.html#zerotendstodiff","scope":"pnt-plus/second-approach/perron-formula/part-01","status":"fully_proved","summary":"If the limit of $0$ is $L_1 - L_2$, then $L_1 = L_2$.","title":"zeroTendstoDiff"},{"id":"pnt-plus/second-approach/perron-formula/part-01/rectangleintegral-tendsto-verticalintegral","kind":"node","lean":"RectangleIntegral_tendsTo_VerticalIntegral","parent":4534,"result":true,"route":"../roadmap/pnt-plus/second-approach/perron-formula/part-01/index.html#rectangleintegral-tendsto-verticalintegral","scope":"pnt-plus/second-approach/perron-formula/part-01","status":"fully_proved","summary":"Let $\\sigma,\\sigma' \\in \\mathbb{R}$, and $f : \\mathbb{C} \\to \\mathbb{C}$ such that the vertical integrals $\\int_{(\\sigma)}f(s)ds$ and $\\int_{(\\sigma')}f(s)ds$ exist and the horizontal integral $\\int_{(\\sigma)}^{\\sigma'}f(x + yi)dx$ vanishes as $y \\to \\pm \\infty$. Then the limit of rectangle integrals","title":"RectangleIntegral-tendsTo-VerticalIntegral"},{"id":"pnt-plus/second-approach/perron-formula/part-01/rectangleintegral-tendsto-upperu","kind":"node","lean":"RectangleIntegral_tendsTo_UpperU","parent":4534,"result":true,"route":"../roadmap/pnt-plus/second-approach/perron-formula/part-01/index.html#rectangleintegral-tendsto-upperu","scope":"pnt-plus/second-approach/perron-formula/part-01","status":"fully_proved","summary":"Let $\\sigma,\\sigma' \\in \\mathbb{R}$, and $f : \\mathbb{C} \\to \\mathbb{C}$ such that the vertical integrals $\\int_{(\\sigma)}f(s)ds$ and $\\int_{(\\sigma')}f(s)ds$ exist and the horizontal integral $\\int_{(\\sigma)}^{\\sigma'}f(x + yi)dx$ vanishes as $y \\to \\pm \\infty$. Then the limit of rectangle integrals","title":"RectangleIntegral-tendsTo-UpperU"},{"id":"pnt-plus/second-approach/perron-formula/part-01/rectangleintegral-tendsto-loweru","kind":"node","lean":"RectangleIntegral_tendsTo_LowerU","parent":4534,"result":true,"route":"../roadmap/pnt-plus/second-approach/perron-formula/part-01/index.html#rectangleintegral-tendsto-loweru","scope":"pnt-plus/second-approach/perron-formula/part-01","status":"fully_proved","summary":"Let $\\sigma,\\sigma' \\in \\mathbb{R}$, and $f : \\mathbb{C} \\to \\mathbb{C}$ such that the vertical integrals $\\int_{(\\sigma)}f(s)ds$ and $\\int_{(\\sigma')}f(s)ds$ exist and the horizontal integral $\\int_{(\\sigma)}^{\\sigma'}f(x + yi)dx$ vanishes as $y \\to -\\infty$. Then the limit of rectangle integrals","title":"RectangleIntegral-tendsTo-LowerU"},{"id":"pnt-plus/second-approach/perron-formula/part-01/limitofconstant","kind":"node","lean":"limitOfConstant","parent":4534,"result":true,"route":"../roadmap/pnt-plus/second-approach/perron-formula/part-01/index.html#limitofconstant","scope":"pnt-plus/second-approach/perron-formula/part-01","status":"fully_proved","summary":"Let $a:\\mathbb{R}\\to\\mathbb{C}$ be a function, and let $\\sigma>0$ be a real number. Suppose that, for all $\\sigma, \\sigma'>0$, we have $a(\\sigma')=a(\\sigma)$, and that $\\lim_{\\sigma\\to\\infty}a(\\sigma)=0$. Then $a(\\sigma)=0$.","title":"limitOfConstant"},{"id":"pnt-plus/second-approach/perron-formula/part-01/limitofconstantleft","kind":"node","lean":"limitOfConstantLeft","parent":4534,"result":true,"route":"../roadmap/pnt-plus/second-approach/perron-formula/part-01/index.html#limitofconstantleft","scope":"pnt-plus/second-approach/perron-formula/part-01","status":"fully_proved","summary":"Let $a:\\mathbb{R}\\to\\mathbb{C}$ be a function, and let $\\sigma<-3/2$ be a real number. Suppose that, for all $\\sigma, \\sigma'>0$, we have $a(\\sigma')=a(\\sigma)$, and that $\\lim_{\\sigma\\to-\\infty}a(\\sigma)=0$. Then $a(\\sigma)=0$.","title":"limitOfConstantLeft"},{"id":"pnt-plus/second-approach/perron-formula/part-01/tendsto-rpow-attop-nhds-zero-of-norm-lt-one","kind":"node","lean":"tendsto_rpow_atTop_nhds_zero_of_norm_lt_one","parent":4534,"result":true,"route":"../roadmap/pnt-plus/second-approach/perron-formula/part-01/index.html#tendsto-rpow-attop-nhds-zero-of-norm-lt-one","scope":"pnt-plus/second-approach/perron-formula/part-01","status":"fully_proved","summary":"Let $x>0$ and $x<1$. Then","title":"tendsto-rpow-atTop-nhds-zero-of-norm-lt-one"},{"id":"pnt-plus/second-approach/perron-formula/part-01/tendsto-rpow-attop-nhds-zero-of-norm-gt-one","kind":"node","lean":"tendsto_rpow_atTop_nhds_zero_of_norm_gt_one","parent":4534,"result":true,"route":"../roadmap/pnt-plus/second-approach/perron-formula/part-01/index.html#tendsto-rpow-attop-nhds-zero-of-norm-gt-one","scope":"pnt-plus/second-approach/perron-formula/part-01","status":"fully_proved","summary":"Let $x>1$. Then","title":"tendsto-rpow-atTop-nhds-zero-of-norm-gt-one"},{"id":"pnt-plus/second-approach/perron-formula/part-01/isholomorphicon","kind":"node","lean":"Perron.isHolomorphicOn","parent":4534,"result":true,"route":"../roadmap/pnt-plus/second-approach/perron-formula/part-01/index.html#isholomorphicon","scope":"pnt-plus/second-approach/perron-formula/part-01","status":"fully_proved","summary":"Let $x>0$. Then the function $f(s) = x^s/(s(s+1))$ is holomorphic on the half-plane $\\{s\\in\\mathbb{C}:\\Re(s)>0\\}$.","title":"isHolomorphicOn"},{"id":"pnt-plus/second-approach/perron-formula/part-01/integralposaux","kind":"node","lean":"Perron.integralPosAux","parent":4534,"result":true,"route":"../roadmap/pnt-plus/second-approach/perron-formula/part-01/index.html#integralposaux","scope":"pnt-plus/second-approach/perron-formula/part-01","status":"fully_proved","summary":"The integral","title":"integralPosAux"},{"id":"pnt-plus/second-approach/perron-formula/part-01/vertintbound","kind":"node","lean":"Perron.vertIntBound","parent":4534,"result":true,"route":"../roadmap/pnt-plus/second-approach/perron-formula/part-01/index.html#vertintbound","scope":"pnt-plus/second-approach/perron-formula/part-01","status":"fully_proved","summary":"Let $x>0$ and $\\sigma>1$. Then","title":"vertIntBound"},{"id":"pnt-plus/second-approach/perron-formula/part-01/vertintboundleft","kind":"node","lean":"Perron.vertIntBoundLeft","parent":4534,"result":true,"route":"../roadmap/pnt-plus/second-approach/perron-formula/part-01/index.html#vertintboundleft","scope":"pnt-plus/second-approach/perron-formula/part-01","status":"fully_proved","summary":"Let $x>1$ and $\\sigma<-3/2$. Then","title":"vertIntBoundLeft"},{"id":"pnt-plus/second-approach/perron-formula/part-01/isintegrable","kind":"node","lean":"Perron.isIntegrable","parent":4534,"result":true,"route":"../roadmap/pnt-plus/second-approach/perron-formula/part-01/index.html#isintegrable","scope":"pnt-plus/second-approach/perron-formula/part-01","status":"fully_proved","summary":"Let $x>0$ and $\\sigma\\in\\mathbb{R}$. Then","title":"isIntegrable"},{"id":"pnt-plus/second-approach/perron-formula/part-01/tendsto-zero-lower","kind":"node","lean":"Perron.tendsto_zero_Lower","parent":4534,"result":true,"route":"../roadmap/pnt-plus/second-approach/perron-formula/part-01/index.html#tendsto-zero-lower","scope":"pnt-plus/second-approach/perron-formula/part-01","status":"fully_proved","summary":"Let $x>0$ and $\\sigma',\\sigma''\\in\\mathbb{R}$. Then","title":"tendsto-zero-Lower"},{"id":"pnt-plus/second-approach/perron-formula/part-01/perron-tendsto-zero-upper","kind":"node","lean":"Perron.tendsto_zero_Upper","parent":4534,"result":true,"route":"../roadmap/pnt-plus/second-approach/perron-formula/part-01/index.html#perron-tendsto-zero-upper","scope":"pnt-plus/second-approach/perron-formula/part-01","status":"fully_proved","summary":"Let $x>0$ and $\\sigma',\\sigma''\\in\\mathbb{R}$. Then","title":"tendsto-zero-Upper"},{"id":"pnt-plus/second-approach/perron-formula/part-01/formulaltone","kind":"node","lean":"Perron.formulaLtOne","parent":4534,"result":true,"route":"../roadmap/pnt-plus/second-approach/perron-formula/part-01/index.html#formulaltone","scope":"pnt-plus/second-approach/perron-formula/part-01","status":"fully_proved","summary":"For $x>0$, $\\sigma>0$, and $x<1$, we have","title":"formulaLtOne"},{"id":"pnt-plus/second-approach/perron-formula/part-01/keyidentity","kind":"node","lean":"Perron.keyIdentity","parent":4534,"result":true,"route":"../roadmap/pnt-plus/second-approach/perron-formula/part-01/index.html#keyidentity","scope":"pnt-plus/second-approach/perron-formula/part-01","status":"fully_proved","summary":"Let $x\\in \\mathbb{R}$ and $s \\ne 0, -1$. Then","title":"keyIdentity"},{"id":"pnt-plus/second-approach/perron-formula/part-01/diffbddatzero","kind":"node","lean":"Perron.diffBddAtZero","parent":4534,"result":true,"route":"../roadmap/pnt-plus/second-approach/perron-formula/part-01/index.html#diffbddatzero","scope":"pnt-plus/second-approach/perron-formula/part-01","status":"fully_proved","summary":"Let $x>0$. Then for $0 < c < 1 /2$, we have that the function","title":"diffBddAtZero"},{"id":"pnt-plus/second-approach/perron-formula/part-01/diffbddatnegone","kind":"node","lean":"Perron.diffBddAtNegOne","parent":4534,"result":true,"route":"../roadmap/pnt-plus/second-approach/perron-formula/part-01/index.html#diffbddatnegone","scope":"pnt-plus/second-approach/perron-formula/part-01","status":"fully_proved","summary":"Let $x>0$. Then for $0 < c < 1 /2$, we have that the function","title":"diffBddAtNegOne"},{"id":"pnt-plus/second-approach/perron-formula/part-01/residueatzero","kind":"node","lean":"Perron.residueAtZero","parent":4534,"result":true,"route":"../roadmap/pnt-plus/second-approach/perron-formula/part-01/index.html#residueatzero","scope":"pnt-plus/second-approach/perron-formula/part-01","status":"fully_proved","summary":"Let $x>0$. Then for all sufficiently small $c>0$, we have that","title":"residueAtZero"},{"data_url":"scopes/pnt-plus/second-approach/perron-formula/part-02.json","id":"pnt-plus/second-approach/perron-formula/part-02","kind":"scope","navigation":"transparent","parent":4533,"result":false,"route":"../roadmap/pnt-plus/second-approach/perron-formula/part-02/index.html","scope":"pnt-plus/second-approach/perron-formula/part-02","status":"fully_proved","summary":"The two together give the Perron formula. (Which doesn't need to be a separate lemma.)","title":"residueAtNegOne to formulaGtOne"},{"id":"pnt-plus/second-approach/perron-formula/part-02/residueatnegone","kind":"node","lean":"Perron.residueAtNegOne","parent":4555,"result":true,"route":"../roadmap/pnt-plus/second-approach/perron-formula/part-02/index.html#residueatnegone","scope":"pnt-plus/second-approach/perron-formula/part-02","status":"fully_proved","summary":"Let $x>0$. Then for all sufficiently small $c>0$, we have that","title":"residueAtNegOne"},{"id":"pnt-plus/second-approach/perron-formula/part-02/residuepull1","kind":"node","lean":"Perron.residuePull1","parent":4555,"result":true,"route":"../roadmap/pnt-plus/second-approach/perron-formula/part-02/index.html#residuepull1","scope":"pnt-plus/second-approach/perron-formula/part-02","status":"fully_proved","summary":"For $x>1$ (of course $x>0$ would suffice) and $\\sigma>0$, we have","title":"residuePull1"},{"id":"pnt-plus/second-approach/perron-formula/part-02/residuepull2","kind":"node","lean":"Perron.residuePull2","parent":4555,"result":true,"route":"../roadmap/pnt-plus/second-approach/perron-formula/part-02/index.html#residuepull2","scope":"pnt-plus/second-approach/perron-formula/part-02","status":"fully_proved","summary":"For $x>1$, we have","title":"residuePull2"},{"id":"pnt-plus/second-approach/perron-formula/part-02/contourpull3","kind":"node","lean":"Perron.contourPull3","parent":4555,"result":true,"route":"../roadmap/pnt-plus/second-approach/perron-formula/part-02/index.html#contourpull3","scope":"pnt-plus/second-approach/perron-formula/part-02","status":"fully_proved","summary":"For $x>1$ and $\\sigma<-3/2$, we have","title":"contourPull3"},{"id":"pnt-plus/second-approach/perron-formula/part-02/formulagtone","kind":"node","lean":"Perron.formulaGtOne","parent":4555,"result":true,"route":"../roadmap/pnt-plus/second-approach/perron-formula/part-02/index.html#formulagtone","scope":"pnt-plus/second-approach/perron-formula/part-02","status":"fully_proved","summary":"For $x>1$ and $\\sigma>0$, we have","title":"formulaGtOne"},{"data_url":"scopes/pnt-plus/second-approach/mellin-transforms.json","id":"pnt-plus/second-approach/mellin-transforms","kind":"scope","parent":4520,"result":false,"route":"../roadmap/pnt-plus/second-approach/mellin-transforms/index.html","scope":"pnt-plus/second-approach/mellin-transforms","status":"fully_proved","title":"Mellin transforms"},{"data_url":"scopes/pnt-plus/second-approach/mellin-transforms/part-01.json","id":"pnt-plus/second-approach/mellin-transforms/part-01","kind":"scope","navigation":"transparent","parent":4561,"result":false,"route":"../roadmap/pnt-plus/second-approach/mellin-transforms/part-01/index.html","scope":"pnt-plus/second-approach/mellin-transforms/part-01","status":"fully_proved","summary":"In this section, we define the Mellin transform (already in Mathlib, thanks to David Loeffler), prove its inversion formula, and derive a number of important properties of some special functions and bumpfunctions.","title":"PartialIntegration to Smooth1ContinuousAt"},{"id":"pnt-plus/second-approach/mellin-transforms/part-01/partialintegration","kind":"node","lean":"PartialIntegration","parent":4562,"result":true,"route":"../roadmap/pnt-plus/second-approach/mellin-transforms/part-01/index.html#partialintegration","scope":"pnt-plus/second-approach/mellin-transforms/part-01","status":"fully_proved","summary":"Let $f, g$ be once differentiable functions from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$ so that $fg'$ and $f'g$ are both integrable, and $f\\cdot g (x)\\to 0$ as $x\\to 0^+,\\infty$. Then","title":"PartialIntegration"},{"id":"pnt-plus/second-approach/mellin-transforms/part-01/mellinconvolution","kind":"node","lean":"MellinConvolution","parent":4562,"result":true,"route":"../roadmap/pnt-plus/second-approach/mellin-transforms/part-01/index.html#mellinconvolution","scope":"pnt-plus/second-approach/mellin-transforms/part-01","status":"fully_proved","summary":"Let $f$ and $g$ be functions from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$. Then we define the Mellin convolution of $f$ and $g$ to be the function $f\\ast g$ from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$ defined by","title":"MellinConvolution"},{"id":"pnt-plus/second-approach/mellin-transforms/part-01/mellinconvolutionsymmetric","kind":"node","lean":"MellinConvolutionSymmetric","parent":4562,"result":true,"route":"../roadmap/pnt-plus/second-approach/mellin-transforms/part-01/index.html#mellinconvolutionsymmetric","scope":"pnt-plus/second-approach/mellin-transforms/part-01","status":"fully_proved","summary":"Let $f$ and $g$ be functions from $\\mathbb{R}_{>0}$ to $\\mathbb{R}$ or $\\mathbb{C}$, for $x\\neq0$,","title":"MellinConvolutionSymmetric"},{"id":"pnt-plus/second-approach/mellin-transforms/part-01/mellinconvolutiontransform","kind":"node","lean":"MellinConvolutionTransform","parent":4562,"result":true,"route":"../roadmap/pnt-plus/second-approach/mellin-transforms/part-01/index.html#mellinconvolutiontransform","scope":"pnt-plus/second-approach/mellin-transforms/part-01","status":"fully_proved","summary":"Let $f$ and $g$ be functions from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$ such that","title":"MellinConvolutionTransform"},{"id":"pnt-plus/second-approach/mellin-transforms/part-01/mellinofpsi","kind":"node","lean":"MellinOfPsi","parent":4562,"result":true,"route":"../roadmap/pnt-plus/second-approach/mellin-transforms/part-01/index.html#mellinofpsi","scope":"pnt-plus/second-approach/mellin-transforms/part-01","status":"fully_proved","summary":"The Mellin transform of $\\nu$ is","title":"MellinOfPsi"},{"id":"pnt-plus/second-approach/mellin-transforms/part-01/deltaspike","kind":"node","lean":"DeltaSpike","parent":4562,"result":true,"route":"../roadmap/pnt-plus/second-approach/mellin-transforms/part-01/index.html#deltaspike","scope":"pnt-plus/second-approach/mellin-transforms/part-01","status":"fully_proved","summary":"Let $\\nu$ be a bumpfunction supported in $[1/2,2]$. Then for any $\\epsilon>0$, we define the delta spike $\\nu_\\epsilon$ to be the function from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$ defined by","title":"DeltaSpike"},{"id":"pnt-plus/second-approach/mellin-transforms/part-01/deltaspikemass","kind":"node","lean":"DeltaSpikeMass","parent":4562,"result":true,"route":"../roadmap/pnt-plus/second-approach/mellin-transforms/part-01/index.html#deltaspikemass","scope":"pnt-plus/second-approach/mellin-transforms/part-01","status":"fully_proved","summary":"For any $\\epsilon>0$, we have","title":"DeltaSpikeMass"},{"id":"pnt-plus/second-approach/mellin-transforms/part-01/mellinofdeltaspike","kind":"node","lean":"MellinOfDeltaSpike","parent":4562,"result":true,"route":"../roadmap/pnt-plus/second-approach/mellin-transforms/part-01/index.html#mellinofdeltaspike","scope":"pnt-plus/second-approach/mellin-transforms/part-01","status":"fully_proved","summary":"For any $\\epsilon>0$, the Mellin transform of $\\nu_\\epsilon$ is","title":"MellinOfDeltaSpike"},{"id":"pnt-plus/second-approach/mellin-transforms/part-01/mellinofdeltaspikeat1","kind":"node","lean":"MellinOfDeltaSpikeAt1","parent":4562,"result":true,"route":"../roadmap/pnt-plus/second-approach/mellin-transforms/part-01/index.html#mellinofdeltaspikeat1","scope":"pnt-plus/second-approach/mellin-transforms/part-01","status":"fully_proved","summary":"For any $\\epsilon>0$, we have","title":"MellinOfDeltaSpikeAt1"},{"id":"pnt-plus/second-approach/mellin-transforms/part-01/mellinofdeltaspikeat1-asymp","kind":"node","lean":"MellinOfDeltaSpikeAt1_asymp","parent":4562,"result":true,"route":"../roadmap/pnt-plus/second-approach/mellin-transforms/part-01/index.html#mellinofdeltaspikeat1-asymp","scope":"pnt-plus/second-approach/mellin-transforms/part-01","status":"fully_proved","summary":"As $\\epsilon\\to 0$, we have","title":"MellinOfDeltaSpikeAt1-asymp"},{"id":"pnt-plus/second-approach/mellin-transforms/part-01/mellinof1","kind":"node","lean":"MellinOf1","parent":4562,"result":true,"route":"../roadmap/pnt-plus/second-approach/mellin-transforms/part-01/index.html#mellinof1","scope":"pnt-plus/second-approach/mellin-transforms/part-01","status":"fully_proved","summary":"The Mellin transform of $1_{(0,1]}$ is","title":"MellinOf1"},{"id":"pnt-plus/second-approach/mellin-transforms/part-01/smooth1","kind":"node","lean":"Smooth1","parent":4562,"result":true,"route":"../roadmap/pnt-plus/second-approach/mellin-transforms/part-01/index.html#smooth1","scope":"pnt-plus/second-approach/mellin-transforms/part-01","status":"fully_proved","summary":"Let $\\epsilon>0$. Then we define the smooth function $\\widetilde{1_{\\epsilon}}$ from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$ by","title":"Smooth1"},{"id":"pnt-plus/second-approach/mellin-transforms/part-01/smooth1properties-below","kind":"node","lean":"Smooth1Properties_below","parent":4562,"result":true,"route":"../roadmap/pnt-plus/second-approach/mellin-transforms/part-01/index.html#smooth1properties-below","scope":"pnt-plus/second-approach/mellin-transforms/part-01","status":"fully_proved","summary":"Fix $\\epsilon>0$. There is an absolute constant $c>0$ so that: If $0 < x \\leq (1-c\\epsilon)$, then","title":"Smooth1Properties-below"},{"id":"pnt-plus/second-approach/mellin-transforms/part-01/smooth1properties-above","kind":"node","lean":"Smooth1Properties_above","parent":4562,"result":true,"route":"../roadmap/pnt-plus/second-approach/mellin-transforms/part-01/index.html#smooth1properties-above","scope":"pnt-plus/second-approach/mellin-transforms/part-01","status":"fully_proved","summary":"Fix $0\\lt \\epsilon<1$. There is an absolute constant $c>0$ so that: if $x\\geq (1+c\\epsilon)$, then","title":"Smooth1Properties-above"},{"id":"pnt-plus/second-approach/mellin-transforms/part-01/smooth1nonneg","kind":"node","lean":"Smooth1Nonneg","parent":4562,"result":true,"route":"../roadmap/pnt-plus/second-approach/mellin-transforms/part-01/index.html#smooth1nonneg","scope":"pnt-plus/second-approach/mellin-transforms/part-01","status":"fully_proved","summary":"If $\\nu$ is nonnegative, then $\\widetilde{1_{\\epsilon}}(x)$ is nonnegative.","title":"Smooth1Nonneg"},{"id":"pnt-plus/second-approach/mellin-transforms/part-01/smooth1leone","kind":"node","lean":"Smooth1LeOne","parent":4562,"result":true,"route":"../roadmap/pnt-plus/second-approach/mellin-transforms/part-01/index.html#smooth1leone","scope":"pnt-plus/second-approach/mellin-transforms/part-01","status":"fully_proved","summary":"If $\\nu$ is nonnegative and has mass one, then $\\widetilde{1_{\\epsilon}}(x)\\le 1$, $\\forall x>0$.","title":"Smooth1LeOne"},{"id":"pnt-plus/second-approach/mellin-transforms/part-01/mellinofsmooth1a","kind":"node","lean":"MellinOfSmooth1a","parent":4562,"result":true,"route":"../roadmap/pnt-plus/second-approach/mellin-transforms/part-01/index.html#mellinofsmooth1a","scope":"pnt-plus/second-approach/mellin-transforms/part-01","status":"fully_proved","summary":"Fix $\\epsilon>0$. Then the Mellin transform of $\\widetilde{1_{\\epsilon}}$ is","title":"MellinOfSmooth1a"},{"id":"pnt-plus/second-approach/mellin-transforms/part-01/mellinofsmooth1b","kind":"node","lean":"MellinOfSmooth1b","parent":4562,"result":true,"route":"../roadmap/pnt-plus/second-approach/mellin-transforms/part-01/index.html#mellinofsmooth1b","scope":"pnt-plus/second-approach/mellin-transforms/part-01","status":"fully_proved","summary":"Given $0\\lt \\sigma_1\\le\\sigma_2$, for any $s$ such that $\\sigma_1\\le\\mathcal Re(s)\\le\\sigma_2$, we have","title":"MellinOfSmooth1b"},{"id":"pnt-plus/second-approach/mellin-transforms/part-01/mellinofsmooth1c","kind":"node","lean":"MellinOfSmooth1c","parent":4562,"result":true,"route":"../roadmap/pnt-plus/second-approach/mellin-transforms/part-01/index.html#mellinofsmooth1c","scope":"pnt-plus/second-approach/mellin-transforms/part-01","status":"fully_proved","summary":"At $s=1$, we have","title":"MellinOfSmooth1c"},{"id":"pnt-plus/second-approach/mellin-transforms/part-01/smooth1continuousat","kind":"node","lean":"Smooth1ContinuousAt","parent":4562,"result":true,"route":"../roadmap/pnt-plus/second-approach/mellin-transforms/part-01/index.html#smooth1continuousat","scope":"pnt-plus/second-approach/mellin-transforms/part-01","status":"fully_proved","summary":"Fix a nonnegative, continuously differentiable function $F$ on $\\mathbb{R}$ with support in $[1/2,2]$. Then for any $\\epsilon>0$, the function $x \\mapsto \\int_{(0,\\infty)} x^{1+it} \\widetilde{1_{\\epsilon}}(x) dx$ is continuous at any $y>0$.","title":"Smooth1ContinuousAt"},{"data_url":"scopes/pnt-plus/second-approach/mellin-transforms/part-02.json","id":"pnt-plus/second-approach/mellin-transforms/part-02","kind":"scope","navigation":"transparent","parent":4561,"result":false,"route":"../roadmap/pnt-plus/second-approach/mellin-transforms/part-02/index.html","scope":"pnt-plus/second-approach/mellin-transforms/part-02","status":"fully_proved","summary":"Let $\\nu$ be a bumpfunction.","title":"SmoothExistence"},{"id":"pnt-plus/second-approach/mellin-transforms/part-02/smoothexistence","kind":"node","lean":"SmoothExistence","parent":4583,"result":true,"route":"../roadmap/pnt-plus/second-approach/mellin-transforms/part-02/index.html#smoothexistence","scope":"pnt-plus/second-approach/mellin-transforms/part-02","status":"fully_proved","summary":"There exists a smooth (once differentiable would be enough), nonnegative \u201cbumpfunction\u201d $\\nu$, supported in $[1/2,2]$ with total mass one:","title":"SmoothExistence"},{"data_url":"scopes/pnt-plus/second-approach/zeta-bounds.json","id":"pnt-plus/second-approach/zeta-bounds","kind":"scope","parent":4520,"result":false,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/index.html","scope":"pnt-plus/second-approach/zeta-bounds","status":"fully_proved","summary":"Now in Mathlib:","title":"Zeta Bounds"},{"data_url":"scopes/pnt-plus/second-approach/zeta-bounds/part-01.json","id":"pnt-plus/second-approach/zeta-bounds/part-01","kind":"scope","navigation":"transparent","parent":4585,"result":false,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-01/index.html","scope":"pnt-plus/second-approach/zeta-bounds/part-01","status":"fully_proved","summary":"We record here some prelimiaries about the zeta function and general holomorphic functions.","title":"deriv-conj-conj to isPathConnected-aux"},{"id":"pnt-plus/second-approach/zeta-bounds/part-01/deriv-conj-conj","kind":"node","lean":"deriv_conj_conj'","parent":4586,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-01/index.html#deriv-conj-conj","scope":"pnt-plus/second-approach/zeta-bounds/part-01","status":"fully_proved","summary":"Let $f : \\mathbb{C} \\to \\mathbb{C}$ be a function at $p \\in \\mathbb{C}$ with derivative $a$. Then the derivative of the function $g(z) = \\overline{f(\\overline{z})}$ at $\\overline{p}$ is $\\overline{a}$.","title":"deriv-conj-conj"},{"id":"pnt-plus/second-approach/zeta-bounds/part-01/deriv-riemannzeta-conj","kind":"node","lean":"deriv_riemannZeta_conj","parent":4586,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-01/index.html#deriv-riemannzeta-conj","scope":"pnt-plus/second-approach/zeta-bounds/part-01","status":"fully_proved","summary":"Conjugation symmetry of the derivative of the Riemann zeta function. Let $s \\in \\mathbb{C}$. Then","title":"deriv-riemannZeta-conj"},{"id":"pnt-plus/second-approach/zeta-bounds/part-01/intervalintegral-conj","kind":"node","lean":"intervalIntegral_conj","parent":4586,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-01/index.html#intervalintegral-conj","scope":"pnt-plus/second-approach/zeta-bounds/part-01","status":"fully_proved","summary":"The conjugation symmetry of the interval integral. Let $f : \\mathbb{R} \\to \\mathbb{C}$ be a measurable function, and let $a, b \\in \\mathbb{R}$. Then","title":"intervalIntegral-conj"},{"id":"pnt-plus/second-approach/zeta-bounds/part-01/residueoftendsto","kind":"node","lean":"ResidueOfTendsTo","parent":4586,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-01/index.html#residueoftendsto","scope":"pnt-plus/second-approach/zeta-bounds/part-01","status":"fully_proved","summary":"If a function $f$ is holomorphic in a neighborhood of $p$ and $\\lim_{s\\to p} (s-p)f(s) = A$, then $f(s) = \\frac{A}{s-p} + O(1)$ near $p$.","title":"ResidueOfTendsTo"},{"id":"pnt-plus/second-approach/zeta-bounds/part-01/riemannzetaresidue","kind":"node","lean":"riemannZetaResidue","parent":4586,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-01/index.html#riemannzetaresidue","scope":"pnt-plus/second-approach/zeta-bounds/part-01","status":"fully_proved","summary":"The Riemann zeta function $\\zeta(s)$ has a simple pole at $s=1$ with residue $1$. In particular, the function $\\zeta(s) - \\frac{1}{s-1}$ is bounded in a neighborhood of $s=1$.","title":"riemannZetaResidue"},{"id":"pnt-plus/second-approach/zeta-bounds/part-01/nonzeroofbddabove","kind":"node","lean":"nonZeroOfBddAbove","parent":4586,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-01/index.html#nonzeroofbddabove","scope":"pnt-plus/second-approach/zeta-bounds/part-01","status":"fully_proved","summary":"If a function $f$ has a simple pole at a point $p$ with residue $A \\neq 0$, then $f$ is nonzero in a punctured neighborhood of $p$.","title":"nonZeroOfBddAbove"},{"id":"pnt-plus/second-approach/zeta-bounds/part-01/logderivresidue","kind":"node","lean":"logDerivResidue","parent":4586,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-01/index.html#logderivresidue","scope":"pnt-plus/second-approach/zeta-bounds/part-01","status":"fully_proved","summary":"If $f$ is holomorphic in a neighborhood of $p$, and there is a simple pole at $p$, then $f'/ f$ has a simple pole at $p$ with residue $-1$:","title":"logDerivResidue"},{"id":"pnt-plus/second-approach/zeta-bounds/part-01/bddabove-to-isbigo","kind":"node","lean":"BddAbove_to_IsBigO","parent":4586,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-01/index.html#bddabove-to-isbigo","scope":"pnt-plus/second-approach/zeta-bounds/part-01","status":"fully_proved","summary":"If $f$ is bounded above in a punctured neighborhood of $p$, then $f$ is $O(1)$ in that neighborhood.","title":"BddAbove-to-IsBigO"},{"id":"pnt-plus/second-approach/zeta-bounds/part-01/residuemult","kind":"node","lean":"ResidueMult","parent":4586,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-01/index.html#residuemult","scope":"pnt-plus/second-approach/zeta-bounds/part-01","status":"fully_proved","summary":"If $f$ has a simple pole at $p$ with residue $A$, and $g$ is holomorphic near $p$, then the residue of $f \\cdot g$ at $p$ is $A \\cdot g(p)$. That is, we assume that","title":"ResidueMult"},{"id":"pnt-plus/second-approach/zeta-bounds/part-01/riemannzetalogderivresidue","kind":"node","lean":"riemannZetaLogDerivResidue","parent":4586,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-01/index.html#riemannzetalogderivresidue","scope":"pnt-plus/second-approach/zeta-bounds/part-01","status":"fully_proved","summary":"The log derivative of the Riemann zeta function $\\zeta(s)$ has a simple pole at $s=1$ with residue $-1$: $-\\frac{\\zeta'(s)}{\\zeta(s)} - \\frac{1}{s-1} = O(1)$.","title":"riemannZetaLogDerivResidue"},{"id":"pnt-plus/second-approach/zeta-bounds/part-01/riemannzeta0","kind":"node","lean":"riemannZeta0","parent":4586,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-01/index.html#riemannzeta0","scope":"pnt-plus/second-approach/zeta-bounds/part-01","status":"fully_proved","summary":"For any natural $N\\ge1$, we define","title":"riemannZeta0"},{"id":"pnt-plus/second-approach/zeta-bounds/part-01/sum-eq-int-deriv","kind":"node","lean":"sum_eq_int_deriv","parent":4586,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-01/index.html#sum-eq-int-deriv","scope":"pnt-plus/second-approach/zeta-bounds/part-01","status":"fully_proved","summary":"Let $a < b$, and let $\\phi$ be continuously differentiable on $[a, b]$. Then","title":"sum-eq-int-deriv"},{"id":"pnt-plus/second-approach/zeta-bounds/part-01/zetasum-aux1","kind":"node","lean":"ZetaSum_aux1","parent":4586,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-01/index.html#zetasum-aux1","scope":"pnt-plus/second-approach/zeta-bounds/part-01","status":"fully_proved","summary":"Let $0 < a < b$ be natural numbers and $s\\in \\mathbb{C}$ with $s \\ne 1$ and $s \\ne 0$. Then","title":"ZetaSum-aux1"},{"id":"pnt-plus/second-approach/zeta-bounds/part-01/zetabnd-aux1a","kind":"node","lean":"ZetaBnd_aux1a","parent":4586,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-01/index.html#zetabnd-aux1a","scope":"pnt-plus/second-approach/zeta-bounds/part-01","status":"fully_proved","summary":"For any $0 < a < b$ and $s \\in \\mathbb{C}$ with $\\sigma=\\Re(s)>0$,","title":"ZetaBnd-aux1a"},{"id":"pnt-plus/second-approach/zeta-bounds/part-01/zetasum-aux2","kind":"node","lean":"ZetaSum_aux2","parent":4586,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-01/index.html#zetasum-aux2","scope":"pnt-plus/second-approach/zeta-bounds/part-01","status":"fully_proved","summary":"Let $N$ be a natural number and $s\\in \\mathbb{C}$, $\\Re(s)>1$. Then","title":"ZetaSum-aux2"},{"id":"pnt-plus/second-approach/zeta-bounds/part-01/zetabnd-aux1b","kind":"node","lean":"ZetaBnd_aux1b","parent":4586,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-01/index.html#zetabnd-aux1b","scope":"pnt-plus/second-approach/zeta-bounds/part-01","status":"fully_proved","summary":"For any $N\\ge1$ and $s = \\sigma + tI \\in \\mathbb{C}$, $\\sigma > 0$,","title":"ZetaBnd-aux1b"},{"id":"pnt-plus/second-approach/zeta-bounds/part-01/zetabnd-aux1","kind":"node","lean":"ZetaBnd_aux1","parent":4586,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-01/index.html#zetabnd-aux1","scope":"pnt-plus/second-approach/zeta-bounds/part-01","status":"fully_proved","summary":"For any $N\\ge1$ and $s = \\sigma + tI \\in \\mathbb{C}$, $\\sigma=\\in(0,2], 2 < |t|$,","title":"ZetaBnd-aux1"},{"id":"pnt-plus/second-approach/zeta-bounds/part-01/zetabnd-aux1p","kind":"node","lean":"ZetaBnd_aux1p","parent":4586,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-01/index.html#zetabnd-aux1p","scope":"pnt-plus/second-approach/zeta-bounds/part-01","status":"fully_proved","summary":"For any $N\\ge1$ and $s = \\sigma + tI \\in \\mathbb{C}$, $\\sigma=\\in(0,2], 2 < |t|$,","title":"ZetaBnd-aux1p"},{"id":"pnt-plus/second-approach/zeta-bounds/part-01/holomorphicon-riemannzeta0","kind":"node","lean":"HolomorphicOn_riemannZeta0","parent":4586,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-01/index.html#holomorphicon-riemannzeta0","scope":"pnt-plus/second-approach/zeta-bounds/part-01","status":"fully_proved","summary":"For any $N\\ge1$, the function $\\zeta_0(N,s)$ is holomorphic on $\\{s\\in \\mathbb{C}\\mid \\Re(s)>0 \u2227 s \\ne 1\\}$.","title":"HolomorphicOn-riemannZeta0"},{"id":"pnt-plus/second-approach/zeta-bounds/part-01/ispathconnected-aux","kind":"node","lean":"isPathConnected_aux","parent":4586,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-01/index.html#ispathconnected-aux","scope":"pnt-plus/second-approach/zeta-bounds/part-01","status":"fully_proved","summary":"The set $\\{s\\in \\mathbb{C}\\mid \\Re(s)>0 \u2227 s \\ne 1\\}$ is path-connected.","title":"isPathConnected-aux"},{"data_url":"scopes/pnt-plus/second-approach/zeta-bounds/part-02.json","id":"pnt-plus/second-approach/zeta-bounds/part-02","kind":"scope","navigation":"transparent","parent":4585,"result":false,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-02/index.html","scope":"pnt-plus/second-approach/zeta-bounds/part-02","status":"fully_proved","summary":"Annoyingly, it is not immediate from this that $\\zeta$ doesn't vanish there! That's because $1/0 = 0$ in Lean. So we give a second proof of the same fact (refactor this later), with a lower bound on $\\zeta$ instead of upper bound on $1 / \\zeta$.","title":"Zeta0EqZeta to LogDerivZetaHolcLargeT9"},{"id":"pnt-plus/second-approach/zeta-bounds/part-02/zeta0eqzeta","kind":"node","lean":"Zeta0EqZeta","parent":4607,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-02/index.html#zeta0eqzeta","scope":"pnt-plus/second-approach/zeta-bounds/part-02","status":"fully_proved","summary":"For $\\Re(s)>0$, $s\\ne1$, and for any $N$,","title":"Zeta0EqZeta"},{"id":"pnt-plus/second-approach/zeta-bounds/part-02/zetabnd-aux2","kind":"node","lean":"ZetaBnd_aux2","parent":4607,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-02/index.html#zetabnd-aux2","scope":"pnt-plus/second-approach/zeta-bounds/part-02","status":"fully_proved","summary":"Given $n \u2264 t$ and $\\sigma$ with $1-A/\\log t \\le \\sigma$, we have that","title":"ZetaBnd-aux2"},{"id":"pnt-plus/second-approach/zeta-bounds/part-02/zetaupperbnd","kind":"node","lean":"ZetaUpperBnd","parent":4607,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-02/index.html#zetaupperbnd","scope":"pnt-plus/second-approach/zeta-bounds/part-02","status":"fully_proved","summary":"For any $s = \\sigma + tI \\in \\mathbb{C}$, $1/2 \\le \\sigma\\le 2, 3 < |t|$ and any $0 < A < 1$ sufficiently small, and $1-A/\\log |t| \\le \\sigma$, we have","title":"ZetaUpperBnd"},{"id":"pnt-plus/second-approach/zeta-bounds/part-02/derivupperbnd-aux7","kind":"node","lean":"DerivUpperBnd_aux7","parent":4607,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-02/index.html#derivupperbnd-aux7","scope":"pnt-plus/second-approach/zeta-bounds/part-02","status":"fully_proved","summary":"For any $s = \\sigma + tI \\in \\mathbb{C}$, $1/2 \\le \\sigma\\le 2, 3 < |t|$, and any $0 < A < 1$ sufficiently small, and $1-A/\\log |t| \\le \\sigma$, we have","title":"DerivUpperBnd-aux7"},{"id":"pnt-plus/second-approach/zeta-bounds/part-02/zetaderivupperbnd","kind":"node","lean":"ZetaDerivUpperBnd","parent":4607,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-02/index.html#zetaderivupperbnd","scope":"pnt-plus/second-approach/zeta-bounds/part-02","status":"fully_proved","summary":"For any $s = \\sigma + tI \\in \\mathbb{C}$, $1/2 \\le \\sigma\\le 2, 3 < |t|$, there is an $A>0$ so that for $1-A/\\log t \\le \\sigma$, we have","title":"ZetaDerivUpperBnd"},{"id":"pnt-plus/second-approach/zeta-bounds/part-02/zetanear1bndfilter","kind":"node","lean":"ZetaNear1BndFilter","parent":4607,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-02/index.html#zetanear1bndfilter","scope":"pnt-plus/second-approach/zeta-bounds/part-02","status":"fully_proved","summary":"As $\\sigma\\to1^+$,","title":"ZetaNear1BndFilter"},{"id":"pnt-plus/second-approach/zeta-bounds/part-02/zetanear1bndexact","kind":"node","lean":"ZetaNear1BndExact","parent":4607,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-02/index.html#zetanear1bndexact","scope":"pnt-plus/second-approach/zeta-bounds/part-02","status":"fully_proved","summary":"There exists a $c>0$ such that for all $1 < \\sigma \u2264 2$,","title":"ZetaNear1BndExact"},{"id":"pnt-plus/second-approach/zeta-bounds/part-02/zetalowerbound3","kind":"node","lean":"ZetaLowerBound3","parent":4607,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-02/index.html#zetalowerbound3","scope":"pnt-plus/second-approach/zeta-bounds/part-02","status":"fully_proved","summary":"There exists a $c>0$ such that for all $1 < \\sigma <= 2$ and $3 < |t|$,","title":"ZetaLowerBound3"},{"id":"pnt-plus/second-approach/zeta-bounds/part-02/zetainvbound1","kind":"node","lean":"ZetaInvBound1","parent":4607,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-02/index.html#zetainvbound1","scope":"pnt-plus/second-approach/zeta-bounds/part-02","status":"fully_proved","summary":"For all $\\sigma>1$,","title":"ZetaInvBound1"},{"id":"pnt-plus/second-approach/zeta-bounds/part-02/zetainvbound2","kind":"node","lean":"ZetaInvBound2","parent":4607,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-02/index.html#zetainvbound2","scope":"pnt-plus/second-approach/zeta-bounds/part-02","status":"fully_proved","summary":"For $\\sigma>1$ (and $\\sigma \\le 2$),","title":"ZetaInvBound2"},{"id":"pnt-plus/second-approach/zeta-bounds/part-02/zeta-eq-int-derivzeta","kind":"node","lean":"Zeta_eq_int_derivZeta","parent":4607,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-02/index.html#zeta-eq-int-derivzeta","scope":"pnt-plus/second-approach/zeta-bounds/part-02","status":"fully_proved","summary":"For any $t\\ne0$ (so we don't pass through the pole), and $\\sigma_1 < \\sigma_2$,","title":"Zeta-eq-int-derivZeta"},{"id":"pnt-plus/second-approach/zeta-bounds/part-02/zeta-diff-bnd","kind":"node","lean":"Zeta_diff_Bnd","parent":4607,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-02/index.html#zeta-diff-bnd","scope":"pnt-plus/second-approach/zeta-bounds/part-02","status":"fully_proved","summary":"For any $A>0$ sufficiently small, there is a constant $C>0$ so that whenever $1- A / \\log t \\le \\sigma_1 < \\sigma_2\\le 2$ and $3 < |t|$, we have that:","title":"Zeta-diff-Bnd"},{"id":"pnt-plus/second-approach/zeta-bounds/part-02/zetainvbnd","kind":"node","lean":"ZetaInvBnd","parent":4607,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-02/index.html#zetainvbnd","scope":"pnt-plus/second-approach/zeta-bounds/part-02","status":"fully_proved","summary":"For any $A>0$ sufficiently small, there is a constant $C>0$ so that whenever $1- A / \\log^9 |t| \\le \\sigma < 1+A/\\log^9 |t|$ and $3 < |t|$, we have that:","title":"ZetaInvBnd"},{"id":"pnt-plus/second-approach/zeta-bounds/part-02/zetalowerbnd","kind":"node","lean":"ZetaLowerBnd","parent":4607,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-02/index.html#zetalowerbnd","scope":"pnt-plus/second-approach/zeta-bounds/part-02","status":"fully_proved","summary":"For any $A>0$ sufficiently small, there is a constant $C>0$ so that whenever $1- A / \\log^9 |t| \\le \\sigma < 1$ and $3 < |t|$, we have that:","title":"ZetaLowerBnd"},{"id":"pnt-plus/second-approach/zeta-bounds/part-02/zetazerofree9","kind":"node","lean":"ZetaZeroFree9","parent":4607,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-02/index.html#zetazerofree9","scope":"pnt-plus/second-approach/zeta-bounds/part-02","status":"fully_proved","summary":"There is an $A>0$ so that for $1-A/\\log^9 |t| \\le \\sigma < 1$ and $3 < |t|$,","title":"ZetaZeroFree9"},{"id":"pnt-plus/second-approach/zeta-bounds/part-02/logderivzetabnd","kind":"node","lean":"LogDerivZetaBnd","parent":4607,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-02/index.html#logderivzetabnd","scope":"pnt-plus/second-approach/zeta-bounds/part-02","status":"fully_proved","summary":"There is an $A>0$ so that for $1-A/\\log^9 |t| \\le \\sigma < 1+A/\\log^9 |t|$ and $3 < |t|$,","title":"LogDerivZetaBnd"},{"id":"pnt-plus/second-approach/zeta-bounds/part-02/zetanozeroson1line","kind":"node","lean":"ZetaNoZerosOn1Line","parent":4607,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-02/index.html#zetanozeroson1line","scope":"pnt-plus/second-approach/zeta-bounds/part-02","status":"fully_proved","summary":"The zeta function does not vanish on the 1-line.","title":"ZetaNoZerosOn1Line"},{"id":"pnt-plus/second-approach/zeta-bounds/part-02/zetanozerosinbox","kind":"node","lean":"ZetaNoZerosInBox","parent":4607,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-02/index.html#zetanozerosinbox","scope":"pnt-plus/second-approach/zeta-bounds/part-02","status":"fully_proved","summary":"For any $T>0$, there is a constant $\\sigma<1$ so that","title":"ZetaNoZerosInBox"},{"id":"pnt-plus/second-approach/zeta-bounds/part-02/logderivzetaholcsmallt","kind":"node","lean":"LogDerivZetaHolcSmallT","parent":4607,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-02/index.html#logderivzetaholcsmallt","scope":"pnt-plus/second-approach/zeta-bounds/part-02","status":"fully_proved","summary":"There is a $\\sigma_2 < 1$ so that the function","title":"LogDerivZetaHolcSmallT"},{"id":"pnt-plus/second-approach/zeta-bounds/part-02/logderivzetaholclarget9","kind":"node","lean":"LogDerivZetaHolcLargeT9","parent":4607,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-02/index.html#logderivzetaholclarget9","scope":"pnt-plus/second-approach/zeta-bounds/part-02","status":"fully_proved","summary":"There is an $A>0$ so that for all $T>3$, the function $\\frac {\\zeta'}{\\zeta}(s)$ is holomorphic on $\\{1-A/\\log^9 T \\le \\Re s \\le 2, |\\Im s|\\le T \\}\\setminus\\{1\\}$.","title":"LogDerivZetaHolcLargeT9"},{"data_url":"scopes/pnt-plus/second-approach/zeta-bounds/part-03.json","id":"pnt-plus/second-approach/zeta-bounds/part-03","kind":"scope","navigation":"transparent","parent":4585,"result":false,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-03/index.html","scope":"pnt-plus/second-approach/zeta-bounds/part-03","status":"fully_proved","title":"LogDerivZetaBndUnif99"},{"id":"pnt-plus/second-approach/zeta-bounds/part-03/logderivzetabndunif99","kind":"node","lean":"LogDerivZetaBndUnif99","parent":4628,"result":true,"route":"../roadmap/pnt-plus/second-approach/zeta-bounds/part-03/index.html#logderivzetabndunif99","scope":"pnt-plus/second-approach/zeta-bounds/part-03","status":"fully_proved","summary":"There exist $A, C > 0$ such that","title":"LogDerivZetaBndUnif99"},{"data_url":"scopes/pnt-plus/second-approach/proof-of-medium-pnt.json","id":"pnt-plus/second-approach/proof-of-medium-pnt","kind":"scope","parent":4520,"result":false,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/index.html","scope":"pnt-plus/second-approach/proof-of-medium-pnt","status":"fully_proved","summary":"The approach here is completely standard. We follow the use of $\\mathcal{M}(\\widetilde{1_{\\epsilon}})$ as in [Kontorovich 2015].","title":"Proof of Medium PNT"},{"data_url":"scopes/pnt-plus/second-approach/proof-of-medium-pnt/part-01.json","id":"pnt-plus/second-approach/proof-of-medium-pnt/part-01","kind":"scope","navigation":"transparent","parent":4630,"result":false,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-01/index.html","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-01","status":"fully_proved","summary":"It has already been established that zeta doesn't vanish on the 1 line, and has a pole at $s=1$ of order 1. We also have the following.","title":"ChebyshevPsi to SmoothedChebyshevPull1-aux-integ\u2026"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-01/chebyshevpsi","kind":"node","lean":"ChebyshevPsi","parent":4631,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-01/index.html#chebyshevpsi","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-01","status":"fully_proved","summary":"The (second) Chebyshev Psi function is defined as","title":"ChebyshevPsi"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-01/logderivativedirichlet","kind":"node","lean":"LogDerivativeDirichlet","parent":4631,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-01/index.html#logderivativedirichlet","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-01","status":"fully_proved","summary":"We have that, for $\\Re(s)>1$,","title":"LogDerivativeDirichlet"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-01/smoothedchebyshev","kind":"node","lean":"SmoothedChebyshev","parent":4631,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-01/index.html#smoothedchebyshev","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-01","status":"fully_proved","summary":"Fix $\\epsilon>0$, and a bumpfunction supported in $[1/2,2]$. Then we define the smoothed Chebyshev function $\\psi_{\\epsilon}$ from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$ by","title":"SmoothedChebyshev"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-01/smoothedchebyshevdirichlet-aux-integrable","kind":"node","lean":"SmoothedChebyshevDirichlet_aux_integrable","parent":4631,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-01/index.html#smoothedchebyshevdirichlet-aux-integrable","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-01","status":"fully_proved","summary":"Fix a nonnegative, continuously differentiable function $F$ on $\\mathbb{R}$ with support in $[1/2,2]$, and total mass one, $\\int_{(0,\\infty)} F(x)/x dx = 1$. Then for any $\\epsilon>0$, and $\\sigma\\in (1, 2]$, the function","title":"SmoothedChebyshevDirichlet-aux-integrable"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-01/smoothedchebyshevdirichlet-aux-tsum-integral","kind":"node","lean":"SmoothedChebyshevDirichlet_aux_tsum_integral","parent":4631,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-01/index.html#smoothedchebyshevdirichlet-aux-tsum-integral","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-01","status":"fully_proved","summary":"Fix a nonnegative, continuously differentiable function $F$ on $\\mathbb{R}$ with support in $[1/2,2]$, and total mass one, $\\int_{(0,\\infty)} F(x)/x dx = 1$. Then for any $\\epsilon>0$ and $\\sigma\\in(1,2]$, the function $x \\mapsto \\sum_{n=1}^\\infty \\frac{\\Lambda(n)}{n^{\\sigma+it}} \\mathcal{M}(\\widetilde{1_{\\epsilon}})(\\sigma+it) x^{\\sigma+it}$ is equal to\u2026","title":"SmoothedChebyshevDirichlet-aux-tsum-integral"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-01/smoothedchebyshevdirichlet","kind":"node","lean":"SmoothedChebyshevDirichlet","parent":4631,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-01/index.html#smoothedchebyshevdirichlet","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-01","status":"fully_proved","summary":"We have that","title":"SmoothedChebyshevDirichlet"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-01/smoothedchebyshevclose","kind":"node","lean":"SmoothedChebyshevClose","parent":4631,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-01/index.html#smoothedchebyshevclose","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-01","status":"fully_proved","summary":"We have that","title":"SmoothedChebyshevClose"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-01/logderivzetahasbound","kind":"node","lean":"LogDerivZetaHasBound","parent":4631,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-01/index.html#logderivzetahasbound","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-01","status":"fully_proved","summary":"For constants $A$, $C$, $n_1$, and $n_2$ this is true if for all $3<|t|$ one has","title":"LogDerivZetaHasBound"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-01/i1","kind":"node","lean":"I\u2081","parent":4631,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-01/index.html#i1","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-01","status":"fully_proved","summary":"I_1(\\nu, \\epsilon, X, T) := \\frac{1}{2\\pi i} \\int_{-\\infty}^{-T} \\left( \\frac{-\\zeta'}\\zeta(\\sigma_0 + t i) \\right) \\mathcal M(\\widetilde 1_\\epsilon)(\\sigma_0 + t i) X^{\\sigma_0 + t i} \\ i \\ dt","title":"I\u2081"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-01/i2","kind":"node","lean":"I\u2082","parent":4631,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-01/index.html#i2","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-01","status":"fully_proved","summary":"I_2(\\nu, \\epsilon, X, T, \\sigma_1) := \\frac{1}{2\\pi i} \\int_{\\sigma_1}^{\\sigma_0} \\left( \\frac{-\\zeta'}\\zeta(\\sigma - i T) \\right) \\mathcal M(\\widetilde 1_\\epsilon)(\\sigma - i T) X^{\\sigma - i T} \\ d\\sigma","title":"I\u2082"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-01/i37","kind":"node","lean":"I\u2083\u2087","parent":4631,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-01/index.html#i37","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-01","status":"fully_proved","summary":"I_{37}(\\nu, \\epsilon, X, T, \\sigma_1) := \\frac{1}{2\\pi i} \\int_{-T}^{T} \\left( \\frac{-\\zeta'}\\zeta(\\sigma_1 + t i) \\right) \\mathcal M(\\widetilde 1_\\epsilon)(\\sigma_1 + t i) X^{\\sigma_1 + t i} \\ i \\ dt","title":"I\u2083\u2087"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-01/i8","kind":"node","lean":"I\u2088","parent":4631,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-01/index.html#i8","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-01","status":"fully_proved","summary":"I_8(\\nu, \\epsilon, X, T, \\sigma_1) := \\frac{1}{2\\pi i} \\int_{\\sigma_1}^{\\sigma_0} \\left( \\frac{-\\zeta'}\\zeta(\\sigma + T i) \\right) \\mathcal M(\\widetilde 1_\\epsilon)(\\sigma + T i) X^{\\sigma + T i} \\ d\\sigma","title":"I\u2088"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-01/i9","kind":"node","lean":"I\u2089","parent":4631,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-01/index.html#i9","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-01","status":"fully_proved","summary":"I_9(\\nu, \\epsilon, X, T) := \\frac{1}{2\\pi i} \\int_{T}^{\\infty} \\left( \\frac{-\\zeta'}\\zeta(\\sigma_0 + t i) \\right) \\mathcal M(\\widetilde 1_\\epsilon)(\\sigma_0 + t i) X^{\\sigma_0 + t i} \\ i \\ dt","title":"I\u2089"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-01/i3","kind":"node","lean":"I\u2083","parent":4631,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-01/index.html#i3","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-01","status":"fully_proved","summary":"I_3(\\nu, \\epsilon, X, T, \\sigma_1) := \\frac{1}{2\\pi i} \\int_{-T}^{-3} \\left( \\frac{-\\zeta'}\\zeta(\\sigma_1 + t i) \\right) \\mathcal M(\\widetilde 1_\\epsilon)(\\sigma_1 + t i) X^{\\sigma_1 + t i} \\ i \\ dt","title":"I\u2083"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-01/i7","kind":"node","lean":"I\u2087","parent":4631,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-01/index.html#i7","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-01","status":"fully_proved","summary":"I_7(\\nu, \\epsilon, X, T, \\sigma_1) := \\frac{1}{2\\pi i} \\int_{3}^{T} \\left( \\frac{-\\zeta'}\\zeta(\\sigma_1 + t i) \\right) \\mathcal M(\\widetilde 1_\\epsilon)(\\sigma_1 + t i) X^{\\sigma_1 + t i} \\ i \\ dt","title":"I\u2087"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-01/i4","kind":"node","lean":"I\u2084","parent":4631,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-01/index.html#i4","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-01","status":"fully_proved","summary":"I_4(\\nu, \\epsilon, X, \\sigma_1, \\sigma_2) := \\frac{1}{2\\pi i} \\int_{\\sigma_2}^{\\sigma_1} \\left( \\frac{-\\zeta'}\\zeta(\\sigma - 3 i) \\right) \\mathcal M(\\widetilde 1_\\epsilon)(\\sigma - 3 i) X^{\\sigma - 3 i} \\ d\\sigma","title":"I\u2084"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-01/i6","kind":"node","lean":"I\u2086","parent":4631,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-01/index.html#i6","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-01","status":"fully_proved","summary":"I_6(\\nu, \\epsilon, X, \\sigma_1, \\sigma_2) := \\frac{1}{2\\pi i} \\int_{\\sigma_2}^{\\sigma_1} \\left( \\frac{-\\zeta'}\\zeta(\\sigma + 3 i) \\right) \\mathcal M(\\widetilde 1_\\epsilon)(\\sigma + 3 i) X^{\\sigma + 3 i} \\ d\\sigma","title":"I\u2086"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-01/i5","kind":"node","lean":"I\u2085","parent":4631,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-01/index.html#i5","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-01","status":"fully_proved","summary":"I_5(\\nu, \\epsilon, X, \\sigma_2) := \\frac{1}{2\\pi i} \\int_{-3}^{3} \\left( \\frac{-\\zeta'}\\zeta(\\sigma_2 + t i) \\right) \\mathcal M(\\widetilde 1_\\epsilon)(\\sigma_2 + t i) X^{\\sigma_2 + t i} \\ i \\ dt","title":"I\u2085"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-01/dlog-riemannzeta-bdd-on-vertical-lines","kind":"node","lean":"dlog_riemannZeta_bdd_on_vertical_lines","parent":4631,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-01/index.html#dlog-riemannzeta-bdd-on-vertical-lines","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-01","status":"fully_proved","summary":"For $\\sigma_0 > 1$, there exists a constant $C > 0$ such that","title":"dlog-riemannZeta-bdd-on-vertical-lines"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-01/smoothedchebyshevpull1-aux-integrable","kind":"node","lean":"SmoothedChebyshevPull1_aux_integrable","parent":4631,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-01/index.html#smoothedchebyshevpull1-aux-integrable","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-01","status":"fully_proved","summary":"The integrand","title":"SmoothedChebyshevPull1-aux-integrable"},{"data_url":"scopes/pnt-plus/second-approach/proof-of-medium-pnt/part-02.json","id":"pnt-plus/second-approach/proof-of-medium-pnt/part-02","kind":"scope","navigation":"transparent","parent":4630,"result":false,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-02/index.html","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-02","status":"fully_proved","summary":"Next pull contours to another box.","title":"BddAboveOnRect to I5Bound"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-02/bddaboveonrect","kind":"node","lean":"BddAboveOnRect","parent":4652,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-02/index.html#bddaboveonrect","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-02","status":"fully_proved","summary":"Let $g : \\mathbb{C} \\to \\mathbb{C}$ be a holomorphic function on a rectangle, then $g$ is bounded above on the rectangle.","title":"BddAboveOnRect"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-02/smoothedchebyshevpull1","kind":"node","lean":"SmoothedChebyshevPull1","parent":4652,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-02/index.html#smoothedchebyshevpull1","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-02","status":"fully_proved","summary":"We have that","title":"SmoothedChebyshevPull1"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-02/smoothedchebyshevpull2","kind":"node","lean":"SmoothedChebyshevPull2","parent":4652,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-02/index.html#smoothedchebyshevpull2","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-02","status":"fully_proved","summary":"We have that","title":"SmoothedChebyshevPull2"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-02/zetaboxeval","kind":"node","lean":"ZetaBoxEval","parent":4652,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-02/index.html#zetaboxeval","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-02","status":"fully_proved","summary":"For all $\\epsilon > 0$ sufficiently close to $0$, the rectangle integral over $[1-\\delta,2] \\times_{\u2102} [-T,T]$ of the integrand in $\\psi_{\\epsilon}$ is","title":"ZetaBoxEval"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-02/ibound-aux1","kind":"node","lean":"IBound_aux1","parent":4652,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-02/index.html#ibound-aux1","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-02","status":"fully_proved","summary":"Given a natural number $k$ and a real number $X_0 > 0$, there exists $C \\geq 1$ so that for all $X \\geq X_0$,","title":"IBound-aux1"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-02/i1bound","kind":"node","lean":"I1Bound","parent":4652,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-02/index.html#i1bound","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-02","status":"fully_proved","summary":"We have that","title":"I1Bound"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-02/i9bound","kind":"node","lean":"I9Bound","parent":4652,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-02/index.html#i9bound","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-02","status":"fully_proved","summary":"We have that","title":"I9Bound"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-02/i2genbound","kind":"node","lean":"I2GenBound","parent":4652,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-02/index.html#i2genbound","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-02","status":"fully_proved","summary":"Assuming a bound of the form of Definition LogDerivZetaHasBound we have that","title":"I2GenBound"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-02/i8i2","kind":"node","lean":"I8I2","parent":4652,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-02/index.html#i8i2","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-02","status":"fully_proved","summary":"Symmetry between $I_2$ and $I_8$:","title":"I8I2"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-02/i8genbound","kind":"node","lean":"I8GenBound","parent":4652,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-02/index.html#i8genbound","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-02","status":"fully_proved","summary":"Assuming a bound of the form of Definition LogDerivZetaHasBound, we have that","title":"I8GenBound"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-02/real-log-le-const-mul-rpow","kind":"node","lean":"Real.log_le_const_mul_rpow","parent":4652,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-02/index.html#real-log-le-const-mul-rpow","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-02","status":"fully_proved","summary":"For all $0\\lt c$ and $0\\lt x$ we have that","title":"Real.log-le-const-mul-rpow"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-02/log-pow-over-xsq-integral-bounded","kind":"node","lean":"log_pow_over_xsq_integral_bounded","parent":4652,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-02/index.html#log-pow-over-xsq-integral-bounded","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-02","status":"fully_proved","summary":"For every $0\\lt n$ there exists a constant $C>0$ (depending on $n$) such that","title":"log-pow-over-xsq-integral-bounded"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-02/i3genbound","kind":"node","lean":"I3GenBound","parent":4652,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-02/index.html#i3genbound","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-02","status":"fully_proved","summary":"Assuming a bound of the form of Definition LogDerivZetaHasBound, we have that","title":"I3GenBound"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-02/i7genbound","kind":"node","lean":"I7GenBound","parent":4652,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-02/index.html#i7genbound","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-02","status":"fully_proved","summary":"Assuming a bound of the form of Definition LogDerivZetaHasBound, we have that","title":"I7GenBound"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-02/i4genbound","kind":"node","lean":"I4GenBound","parent":4652,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-02/index.html#i4genbound","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-02","status":"fully_proved","summary":"Assuming a bound of the form of Definition LogDerivZetaHasBound, we have that","title":"I4GenBound"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-02/i6genbound","kind":"node","lean":"I6GenBound","parent":4652,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-02/index.html#i6genbound","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-02","status":"fully_proved","summary":"Assuming a bound of the form of Definition LogDerivZetaHasBound, we have that","title":"I6GenBound"},{"id":"pnt-plus/second-approach/proof-of-medium-pnt/part-02/i5bound","kind":"node","lean":"I5Bound","parent":4652,"result":true,"route":"../roadmap/pnt-plus/second-approach/proof-of-medium-pnt/part-02/index.html#i5bound","scope":"pnt-plus/second-approach/proof-of-medium-pnt/part-02","status":"fully_proved","summary":"We have that","title":"I5Bound"},{"data_url":"scopes/pnt-plus/second-approach/mediumpnt.json","id":"pnt-plus/second-approach/mediumpnt","kind":"scope","parent":4520,"result":false,"route":"../roadmap/pnt-plus/second-approach/mediumpnt/index.html","scope":"pnt-plus/second-approach/mediumpnt","status":"fully_proved","title":"MediumPNT"},{"id":"pnt-plus/second-approach/mediumpnt/genstrengthpnt","kind":"node","lean":"GenStrengthPNT","parent":4670,"result":true,"route":"../roadmap/pnt-plus/second-approach/mediumpnt/index.html#genstrengthpnt","scope":"pnt-plus/second-approach/mediumpnt","status":"fully_proved","summary":"Assuming a bound of the form of Definition LogDerivZetaHasBound, we have that","title":"GenStrengthPNT"},{"id":"pnt-plus/second-approach/mediumpnt/mediumpnt","kind":"node","lean":"MediumPNT","parent":4670,"result":true,"route":"../roadmap/pnt-plus/second-approach/mediumpnt/index.html#mediumpnt","scope":"pnt-plus/second-approach/mediumpnt","status":"fully_proved","summary":"We have","title":"MediumPNT"},{"data_url":"scopes/pnt-plus/third-approach.json","id":"pnt-plus/third-approach","kind":"scope","parent":4477,"result":false,"route":"../roadmap/pnt-plus/third-approach/index.html","scope":"pnt-plus/third-approach","status":"fully_proved","title":"Third Approach"},{"id":"pnt-plus/third-approach/hadamard-factorization","kind":"node","parent":4673,"result":false,"route":"../roadmap/pnt-plus/third-approach/hadamard-factorization/index.html","scope":"pnt-plus/third-approach","status":"can_state","summary":"In this file, we prove the Hadamard Factorization theorem for functions of finite order, and prove that the zeta function is such.","title":"Hadamard factorization"},{"data_url":"scopes/pnt-plus/third-approach/strong-pnt.json","id":"pnt-plus/third-approach/strong-pnt","kind":"scope","parent":4673,"result":false,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/index.html","scope":"pnt-plus/third-approach/strong-pnt","status":"fully_proved","title":"Strong PNT"},{"data_url":"scopes/pnt-plus/third-approach/strong-pnt/part-01.json","id":"pnt-plus/third-approach/strong-pnt/part-01","kind":"scope","navigation":"transparent","parent":4675,"result":false,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-01/index.html","scope":"pnt-plus/third-approach/strong-pnt/part-01","status":"fully_proved","summary":"This upstreamed from https://github.com/math-inc/strongpnt/tree/main","title":"AnalyticOn.norm-le-of-norm-le-on\u2026 to JBlaschke"},{"id":"pnt-plus/third-approach/strong-pnt/part-01/analyticon-norm-le-of-norm-le-on-sphere","kind":"node","lean":"AnalyticOn.norm_le_of_norm_le_on_sphere","parent":4676,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-01/index.html#analyticon-norm-le-of-norm-le-on-sphere","scope":"pnt-plus/third-approach/strong-pnt/part-01","status":"fully_proved","summary":"An application of the Maximum modulus principle.","title":"AnalyticOn.norm-le-of-norm-le-on-sphere"},{"id":"pnt-plus/third-approach/strong-pnt/part-01/borelcaratheodory","kind":"node","lean":"borelCaratheodory'","parent":4676,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-01/index.html#borelcaratheodory","scope":"pnt-plus/third-approach/strong-pnt/part-01","status":"fully_proved","summary":"An application of","title":"borelCaratheodory'"},{"id":"pnt-plus/third-approach/strong-pnt/part-01/cauchy-formula-deriv","kind":"node","lean":"cauchy_formula_deriv","parent":4676,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-01/index.html#cauchy-formula-deriv","scope":"pnt-plus/third-approach/strong-pnt/part-01","status":"fully_proved","summary":"Let $f$ be analytic on $|z|\\leq R$. For any $z$ with $|z|\\leq r$ and any $r'$ with $0 < r < r' < R$ we have","title":"cauchy-formula-deriv"},{"id":"pnt-plus/third-approach/strong-pnt/part-01/derivativebound","kind":"node","lean":"DerivativeBound","parent":4676,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-01/index.html#derivativebound","scope":"pnt-plus/third-approach/strong-pnt/part-01","status":"fully_proved","summary":"Let $R,\\,M>0$ and $0 < r < r' < R$. Let $f$ be analytic on $|z|\\leq R$ such that $f(0)=0$ and suppose $\\Re f(z)\\leq M$ for all $|z|\\leq R$. Then we have that","title":"DerivativeBound"},{"id":"pnt-plus/third-approach/strong-pnt/part-01/borelcaratheodoryderiv","kind":"node","lean":"BorelCaratheodoryDeriv","parent":4676,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-01/index.html#borelcaratheodoryderiv","scope":"pnt-plus/third-approach/strong-pnt/part-01","status":"fully_proved","summary":"Let $R,\\,M>0$. Let $f$ be analytic on $|z|\\leq R$ such that $f(0)=0$ and suppose $\\Re f(z)\\leq M$ for all $|z|\\leq R$. Then for any $0 < r < R$,","title":"BorelCaratheodoryDeriv"},{"id":"pnt-plus/third-approach/strong-pnt/part-01/logofanalyticfunction","kind":"node","lean":"LogOfAnalyticFunction","parent":4676,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-01/index.html#logofanalyticfunction","scope":"pnt-plus/third-approach/strong-pnt/part-01","status":"fully_proved","summary":"Let $0\\lt r\\lt R$. Let $B:\\overline{\\mathbb{D}_{R}}\\to\\mathbb{C}$ be analytic on neighborhoods of points in $\\overline{\\mathbb{D}_{R}}$ with $B(z)\\neq 0$ for all $z\\in\\overline{\\mathbb{D}_{R}}$.Then there exists $J_B:\\mathbb{D}_R\\to\\mathbb{C}$ that is analytic on neighborhoods of points in $\\mathbb{D}_R$ such that","title":"LogOfAnalyticFunction"},{"id":"pnt-plus/third-approach/strong-pnt/part-01/logofanalyticfunction-2","kind":"node","lean":"LogOfAnalyticFunction'","parent":4676,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-01/index.html#logofanalyticfunction-2","scope":"pnt-plus/third-approach/strong-pnt/part-01","status":"fully_proved","summary":"A wrapper of the above theorem that will be useful later on.","title":"LogOfAnalyticFunction'"},{"id":"pnt-plus/third-approach/strong-pnt/part-01/setofzeros","kind":"node","lean":"SetOfZeros","parent":4676,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-01/index.html#setofzeros","scope":"pnt-plus/third-approach/strong-pnt/part-01","status":"fully_proved","summary":"Let $R>0$ and $f:\\mathbb{C}\\to\\mathbb{C}$. Define the set of zeros $\\mathcal{K}_f(R)=\\{\\rho\\in\\mathbb{C}:|\\rho|\\leq R,\\,f(\\rho)=0\\}$.","title":"SetOfZeros"},{"id":"pnt-plus/third-approach/strong-pnt/part-01/zerofactor","kind":"node","lean":"ZeroFactor","parent":4676,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-01/index.html#zerofactor","scope":"pnt-plus/third-approach/strong-pnt/part-01","status":"fully_proved","summary":"Let $f:\\mathbb{C}\\to\\mathbb{C}$ and $\\rho\\in\\mathbb{C}$. Then there exists $h_\\rho$ such that","title":"ZeroFactor"},{"id":"pnt-plus/third-approach/strong-pnt/part-01/zerofactorization","kind":"node","lean":"ZeroFactorization","parent":4676,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-01/index.html#zerofactorization","scope":"pnt-plus/third-approach/strong-pnt/part-01","status":"fully_proved","summary":"Let $f:\\mathbb{C}\\to\\mathbb{C}$ be analytic on $\\overline{\\mathbb{D}_1}$ with $f(0)\\neq 0$. For all $\\rho\\in\\mathcal{K}_f(R)$ with $R<1$ there exists $h_\\rho(z)$ such that $h_\\rho(z)$ is analytic at $\\rho$, $h_\\rho(\\rho)\\neq 0$, and $f(z)=(z-\\rho)^{m_f(\\rho)}\\,h_\\rho(z)$.","title":"ZeroFactorization"},{"id":"pnt-plus/third-approach/strong-pnt/part-01/cfunction","kind":"node","lean":"Cf","parent":4676,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-01/index.html#cfunction","scope":"pnt-plus/third-approach/strong-pnt/part-01","status":"fully_proved","summary":"Let $0 < r < 1$, and $f:\\mathbb{C}\\to\\mathbb{C}$ be analytic on $\\overline{\\mathbb{D}_1}$ with $f(0)\\neq 0$. We define a function $C_f:\\mathbb{C}\\to\\mathbb{C}$ as follows. This function is constructed by dividing $f(z)$ by a polynomial whose roots are the zeros of $f$ inside $\\overline{\\mathbb{D}_r}$.","title":"CFunction"},{"id":"pnt-plus/third-approach/strong-pnt/part-01/cfanalytic","kind":"node","lean":"CfAnalytic","parent":4676,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-01/index.html#cfanalytic","scope":"pnt-plus/third-approach/strong-pnt/part-01","status":"fully_proved","summary":"If $f:\\mathbb{C}\\to\\mathbb{C}$ is analytic on $\\overline{\\mathbb{D}_1}$ then so too is $C_f$.","title":"CfAnalytic"},{"id":"pnt-plus/third-approach/strong-pnt/part-01/blaschkeb","kind":"node","lean":"BlaschkeB","parent":4676,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-01/index.html#blaschkeb","scope":"pnt-plus/third-approach/strong-pnt/part-01","status":"fully_proved","summary":"Let $0 < r < R < 1$, and $f:\\mathbb{C}\\to\\mathbb{C}$ be analytic $\\overline{\\mathbb{D}_1}$ with $f(0)\\neq 0$. We define a function $B_f:\\mathbb{C}\\to\\mathbb{C}$ as follows.","title":"BlaschkeB"},{"id":"pnt-plus/third-approach/strong-pnt/part-01/blaschkeanalytic","kind":"node","lean":"BlaschkeAnalytic","parent":4676,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-01/index.html#blaschkeanalytic","scope":"pnt-plus/third-approach/strong-pnt/part-01","status":"fully_proved","summary":"If $f:\\mathbb{C}\\to\\mathbb{C}$ is analytic on $\\overline{\\mathbb{D}_R}$ then so too is $B_f$.","title":"BlaschkeAnalytic"},{"id":"pnt-plus/third-approach/strong-pnt/part-01/blaschkeofzero","kind":"node","lean":"BlaschkeOfZero","parent":4676,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-01/index.html#blaschkeofzero","scope":"pnt-plus/third-approach/strong-pnt/part-01","status":"fully_proved","summary":"Let $0 < r < R<1$, and $f:\\mathbb{C}\\to\\mathbb{C}$ be analytic on $\\overline{\\mathbb{D}_1}$ with $f(0)\\neq 0$. Then","title":"BlaschkeOfZero"},{"id":"pnt-plus/third-approach/strong-pnt/part-01/norm-fofzero-le-norm-blaschkeofzero","kind":"node","lean":"norm_fOfZero_le_norm_BlaschkeOfZero","parent":4676,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-01/index.html#norm-fofzero-le-norm-blaschkeofzero","scope":"pnt-plus/third-approach/strong-pnt/part-01","status":"fully_proved","summary":"Let $0 < r < R<1$, and $f:\\mathbb{C}\\to\\mathbb{C}$ be analytic on $\\overline{\\mathbb{D}_1}$ with $f(0)\\neq 0$. Then","title":"norm-fOfZero-le-norm-BlaschkeOfZero"},{"id":"pnt-plus/third-approach/strong-pnt/part-01/diskbound","kind":"node","lean":"DiskBound","parent":4676,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-01/index.html#diskbound","scope":"pnt-plus/third-approach/strong-pnt/part-01","status":"fully_proved","summary":"Let $0 < r < R<1$. If $f:\\mathbb{C}\\to\\mathbb{C}$ is a function analytic on $\\overline{\\mathbb{D}_1}$ with $f(0)\\neq0$ such that $|f(z)|\\leq B$ for $|z|\\leq R$, then $|B_f(z)|\\leq B$ for $|z|\\leq R$ also.","title":"DiskBound"},{"id":"pnt-plus/third-approach/strong-pnt/part-01/blaschkenonzero","kind":"node","lean":"BlaschkeNonzero","parent":4676,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-01/index.html#blaschkenonzero","scope":"pnt-plus/third-approach/strong-pnt/part-01","status":"fully_proved","summary":"Let $0 < r < R<1$ and $f:\\overline{\\mathbb{D}_1}\\to\\mathbb{C}$ be analytic on neighborhoods of points in $\\overline{\\mathbb{D}_1}$ with $f(0)\\neq 0$. Then $B_f(z)\\neq 0$ for all $z\\in\\overline{\\mathbb{D}_r}$.","title":"BlaschkeNonZero"},{"id":"pnt-plus/third-approach/strong-pnt/part-01/zerosbound","kind":"node","lean":"ZerosBound","parent":4676,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-01/index.html#zerosbound","scope":"pnt-plus/third-approach/strong-pnt/part-01","status":"fully_proved","summary":"Let $0< r < R<1$. If $f:\\mathbb{C}\\to\\mathbb{C}$ is a function analytic on neighborhoods of points in $\\overline{\\mathbb{D}_1}$ with $f(0)=1$ and $|f(z)|\\leq B$ for $|z|\\leq R$, then","title":"ZerosBound"},{"id":"pnt-plus/third-approach/strong-pnt/part-01/jblaschke","kind":"node","lean":"JBlaschke","parent":4676,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-01/index.html#jblaschke","scope":"pnt-plus/third-approach/strong-pnt/part-01","status":"fully_proved","summary":"Let $0 < r < R<1$. If $f:\\mathbb{C}\\to\\mathbb{C}$ is a function analytic on neighborhoods of points in $\\overline{\\mathbb{D}_1}$ with $f(0)=1$, define $L_f(z)=J_{B_f}(z)$ where $J$ is from Theorem LogOfAnalyticFunction and $B_f$ is from Definition BlaschkeB.","title":"JBlaschke"},{"data_url":"scopes/pnt-plus/third-approach/strong-pnt/part-02.json","id":"pnt-plus/third-approach/strong-pnt/part-02","kind":"scope","navigation":"transparent","parent":4675,"result":false,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-02/index.html","scope":"pnt-plus/third-approach/strong-pnt/part-02","status":"fully_proved","title":"JBlaschkeDerivBound to DeltaRange"},{"id":"pnt-plus/third-approach/strong-pnt/part-02/jblaschkederivbound","kind":"node","lean":"JBlaschkeDerivBound","parent":4697,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-02/index.html#jblaschkederivbound","scope":"pnt-plus/third-approach/strong-pnt/part-02","status":"fully_proved","summary":"Let $B>1$ and $0 < r' < r < R<1$. If $f:\\mathbb{C}\\to\\mathbb{C}$ is a function analytic on neighborhoods of points in $\\overline{\\mathbb{D}_1}$ with $f(0)=1$ and $|f(z)|\\leq B$ for all $|z|\\leq R$, then for all $|z|\\leq r'$","title":"JBlaschkeDerivBound"},{"id":"pnt-plus/third-approach/strong-pnt/part-02/finalbound","kind":"node","lean":"FinalBound","parent":4697,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-02/index.html#finalbound","scope":"pnt-plus/third-approach/strong-pnt/part-02","status":"fully_proved","summary":"Let $B>1$ and $0 < r' < r < R' < R<1$. If $f:\\mathbb{C}\\to\\mathbb{C}$ is a function analytic on neighborhoods of points in $\\overline{\\mathbb{D}_1}$ with $f(0)=1$ and $|f(z)|\\leq B$ for all $|z|\\leq R$, then for all $z\\in\\overline{\\mathbb{D}_{r'}}\\setminus\\mathcal{K}_f(R')$ we have","title":"FinalBound"},{"id":"pnt-plus/third-approach/strong-pnt/part-02/zetafixedlowerbound","kind":"node","lean":"ZetaFixedLowerBound","parent":4697,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-02/index.html#zetafixedlowerbound","scope":"pnt-plus/third-approach/strong-pnt/part-02","status":"fully_proved","summary":"For all $t\\in\\mathbb{R}$ one has","title":"ZetaFixedLowerBound"},{"id":"pnt-plus/third-approach/strong-pnt/part-02/riemannzeta1","kind":"node","lean":"riemannZeta1","parent":4697,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-02/index.html#riemannzeta1","scope":"pnt-plus/third-approach/strong-pnt/part-02","status":"fully_proved","summary":"Let","title":"riemannZeta1"},{"id":"pnt-plus/third-approach/strong-pnt/part-02/zeta1altformula","kind":"node","lean":"Zeta1AltFormula","parent":4697,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-02/index.html#zeta1altformula","scope":"pnt-plus/third-approach/strong-pnt/part-02","status":"fully_proved","summary":"We have that","title":"Zeta1AltFormula"},{"id":"pnt-plus/third-approach/strong-pnt/part-02/zetaaltformula","kind":"node","lean":"ZetaAltFormula","parent":4697,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-02/index.html#zetaaltformula","scope":"pnt-plus/third-approach/strong-pnt/part-02","status":"fully_proved","summary":"We have that","title":"ZetaAltFormula"},{"id":"pnt-plus/third-approach/strong-pnt/part-02/globalbound","kind":"node","lean":"GlobalBound","parent":4697,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-02/index.html#globalbound","scope":"pnt-plus/third-approach/strong-pnt/part-02","status":"fully_proved","summary":"For all $s\\in\\mathbb{C}$ with $|s|\\leq 1$ and $t\\in\\mathbb{R}$ with $|t|\\geq 2$, we have that","title":"GlobalBound"},{"id":"pnt-plus/third-approach/strong-pnt/part-02/norm-zeta-strict-mono-ofreal","kind":"node","lean":"norm_zeta_strict_mono_ofReal","parent":4697,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-02/index.html#norm-zeta-strict-mono-ofreal","scope":"pnt-plus/third-approach/strong-pnt/part-02","status":"fully_proved","summary":"If $a,b\\in\\mathbb{R}$ with $1\\lt b\\lt a$, then","title":"norm-zeta-strict-mono-ofReal"},{"id":"pnt-plus/third-approach/strong-pnt/part-02/logderivzetafinalbound","kind":"node","lean":"LogDerivZetaFinalBound","parent":4697,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-02/index.html#logderivzetafinalbound","scope":"pnt-plus/third-approach/strong-pnt/part-02","status":"fully_proved","summary":"Let $t\\in\\mathbb{R}$ with $|t|\\geq 2$ and $0 < r' < r < R' < R<1$. If $f(z)=\\zeta(z+3/2+it)$, then for all $z\\in\\overline{\\mathbb{D}_{r'}}\\setminus\\mathcal{K}_f(R')$ we have that","title":"LogDerivZetaFinalBound"},{"id":"pnt-plus/third-approach/strong-pnt/part-02/zetashiftfinitezeros","kind":"node","lean":"ZetaShiftFiniteZeros","parent":4697,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-02/index.html#zetashiftfinitezeros","scope":"pnt-plus/third-approach/strong-pnt/part-02","status":"fully_proved","summary":"For all $|t|\\geq 2$, if $f(z)=\\zeta(z+3/2+it)$, then $f(z)$ has a finite number of zeros in $\\overline{\\mathbb{D}_1}$.","title":"ZetaShiftFiniteZeros"},{"id":"pnt-plus/third-approach/strong-pnt/part-02/zerowindow","kind":"node","lean":"ZeroWindow","parent":4697,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-02/index.html#zerowindow","scope":"pnt-plus/third-approach/strong-pnt/part-02","status":"fully_proved","summary":"Let $\\mathcal{Z}_t=\\{\\rho\\in\\mathbb{C}:\\zeta(\\rho)=0,\\,|\\rho-(3/2+it)|\\leq 3/4\\}$.","title":"ZeroWindow"},{"id":"pnt-plus/third-approach/strong-pnt/part-02/sumboundi","kind":"node","lean":"SumBoundI","parent":4697,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-02/index.html#sumboundi","scope":"pnt-plus/third-approach/strong-pnt/part-02","status":"fully_proved","summary":"For all $\\delta\\in (0,1)$ and $t\\in\\mathbb{R}$ with $|t|\\geq 2$ we have","title":"SumBoundI"},{"id":"pnt-plus/third-approach/strong-pnt/part-02/shifttwobound","kind":"node","lean":"ShiftTwoBound","parent":4697,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-02/index.html#shifttwobound","scope":"pnt-plus/third-approach/strong-pnt/part-02","status":"fully_proved","summary":"For all $\\delta\\in (0,1)$ and $t\\in\\mathbb{R}$ with $|t|\\geq 2$ we have","title":"ShiftTwoBound"},{"id":"pnt-plus/third-approach/strong-pnt/part-02/shiftonebound","kind":"node","lean":"ShiftOneBound","parent":4697,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-02/index.html#shiftonebound","scope":"pnt-plus/third-approach/strong-pnt/part-02","status":"fully_proved","summary":"There exists $C>0$ such that for all $\\delta\\in(0,1)$ and $t\\in\\mathbb{R}$ with $|t|\\geq 2$; if $\\zeta(\\rho)=0$ with $\\rho=\\sigma+it$, then","title":"ShiftOneBound"},{"id":"pnt-plus/third-approach/strong-pnt/part-02/shiftzerobound","kind":"node","lean":"ShiftZeroBound","parent":4697,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-02/index.html#shiftzerobound","scope":"pnt-plus/third-approach/strong-pnt/part-02","status":"fully_proved","summary":"For all $\\delta\\in(0,1)$ we have","title":"ShiftZeroBound"},{"id":"pnt-plus/third-approach/strong-pnt/part-02/vonmangoldtlseriessummable","kind":"node","lean":"vonMangoldtLSeriesSummable","parent":4697,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-02/index.html#vonmangoldtlseriessummable","scope":"pnt-plus/third-approach/strong-pnt/part-02","status":"fully_proved","summary":"The series","title":"vonMangoldtLSeriesSummable"},{"id":"pnt-plus/third-approach/strong-pnt/part-02/zerowindowfinite","kind":"node","lean":"ZeroWindowFinite","parent":4697,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-02/index.html#zerowindowfinite","scope":"pnt-plus/third-approach/strong-pnt/part-02","status":"fully_proved","summary":"For all $|t|\\geq 2$, $\\mathcal{Z}(t)$ contains a finite number of points.","title":"ZeroWindowFinite"},{"id":"pnt-plus/third-approach/strong-pnt/part-02/zeroinequality","kind":"node","lean":"ZeroInequality","parent":4697,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-02/index.html#zeroinequality","scope":"pnt-plus/third-approach/strong-pnt/part-02","status":"fully_proved","summary":"There exists a constant $0 < E<1/14$ such that for all $\\rho=\\sigma+it$ with $\\zeta(\\rho)=0$ and $|t|\\geq 2$, one has","title":"ZeroInequality"},{"id":"pnt-plus/third-approach/strong-pnt/part-02/deltat","kind":"node","lean":"DeltaT","parent":4697,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-02/index.html#deltat","scope":"pnt-plus/third-approach/strong-pnt/part-02","status":"fully_proved","summary":"Let $\\delta_t=E/\\log|t|$ where $E$ is the constant coming from Theorem ZeroInequality.","title":"DeltaT"},{"id":"pnt-plus/third-approach/strong-pnt/part-02/deltarange","kind":"node","lean":"DeltaRange","parent":4697,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-02/index.html#deltarange","scope":"pnt-plus/third-approach/strong-pnt/part-02","status":"fully_proved","summary":"For all $t\\in\\mathbb{R}$ with $|t|\\geq 2$ we have that","title":"DeltaRange"},{"data_url":"scopes/pnt-plus/third-approach/strong-pnt/part-03.json","id":"pnt-plus/third-approach/strong-pnt/part-03","kind":"scope","navigation":"transparent","parent":4675,"result":false,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-03/index.html","scope":"pnt-plus/third-approach/strong-pnt/part-03","status":"fully_proved","title":"SumBoundII to StrongPNT"},{"id":"pnt-plus/third-approach/strong-pnt/part-03/sumboundii","kind":"node","lean":"SumBoundII","parent":4718,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-03/index.html#sumboundii","scope":"pnt-plus/third-approach/strong-pnt/part-03","status":"fully_proved","summary":"For all $t\\in\\mathbb{R}$ with $|t|\\geq 2$ and $z=\\sigma+it$ where $1-\\delta_t/3\\leq\\sigma\\leq 3/2$, we have that","title":"SumBoundII"},{"id":"pnt-plus/third-approach/strong-pnt/part-03/gapsize","kind":"node","lean":"GapSize","parent":4718,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-03/index.html#gapsize","scope":"pnt-plus/third-approach/strong-pnt/part-03","status":"fully_proved","summary":"Let $t\\in\\mathbb{R}$ with $|t|\\geq 3$ and $z=\\sigma+it$ where $1-\\delta_t/3\\leq\\sigma\\leq 3/2$. Additionally, let $\\rho\\in\\mathcal{Z}_t$. Then we have that","title":"GapSize"},{"id":"pnt-plus/third-approach/strong-pnt/part-03/logderivzetauniformlogsquaredboundstrip","kind":"node","lean":"LogDerivZetaUniformLogSquaredBoundStrip","parent":4718,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-03/index.html#logderivzetauniformlogsquaredboundstrip","scope":"pnt-plus/third-approach/strong-pnt/part-03","status":"fully_proved","summary":"There exists a constant $F\\in(0,1/42)$ such that for all $t\\in\\mathbb{R}$ with $|t|\\geq 3$ one has","title":"LogDerivZetaUniformLogSquaredBoundStrip"},{"id":"pnt-plus/third-approach/strong-pnt/part-03/logderivzetabdd-of-re-ge-three-halves","kind":"node","lean":"LogDerivZetaBdd_of_Re_ge_three_halves","parent":4718,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-03/index.html#logderivzetabdd-of-re-ge-three-halves","scope":"pnt-plus/third-approach/strong-pnt/part-03","status":"fully_proved","summary":"There exists a uniform constant $C$ such that for $z=\\sigma+it$ one has","title":"LogDerivZetaBdd-of-Re-ge-three-halves"},{"id":"pnt-plus/third-approach/strong-pnt/part-03/logderivzetauniformlogsquaredbound","kind":"node","lean":"LogDerivZetaUniformLogSquaredBound","parent":4718,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-03/index.html#logderivzetauniformlogsquaredbound","scope":"pnt-plus/third-approach/strong-pnt/part-03","status":"fully_proved","summary":"There exists a constant $F\\in(0,1/42)$ such that for all $t\\in\\mathbb{R}$ with $|t|\\geq 3$ one has","title":"LogDerivZetaUniformLogSquaredBound"},{"id":"pnt-plus/third-approach/strong-pnt/part-03/strongpnt","kind":"node","lean":"StrongPNT","parent":4718,"result":true,"route":"../roadmap/pnt-plus/third-approach/strong-pnt/part-03/index.html#strongpnt","scope":"pnt-plus/third-approach/strong-pnt/part-03","status":"fully_proved","summary":"We have","title":"StrongPNT"},{"data_url":"scopes/pnt-plus/elementary-corollaries.json","id":"pnt-plus/elementary-corollaries","kind":"scope","parent":4477,"result":false,"route":"../roadmap/pnt-plus/elementary-corollaries/index.html","scope":"pnt-plus/elementary-corollaries","status":"fully_proved","title":"Elementary Corollaries"},{"id":"pnt-plus/elementary-corollaries/finsum-range-eq-sum-range","kind":"node","lean":"finsum_range_eq_sum_range","parent":4725,"result":true,"route":"../roadmap/pnt-plus/elementary-corollaries/index.html#finsum-range-eq-sum-range","scope":"pnt-plus/elementary-corollaries","status":"fully_proved","summary":"For any arithmetic function $f$ and real number $x$, one has","title":"finsum-range-eq-sum-range"},{"id":"pnt-plus/elementary-corollaries/chebyshev-asymptotic","kind":"node","lean":"chebyshev_asymptotic","parent":4725,"result":true,"route":"../roadmap/pnt-plus/elementary-corollaries/index.html#chebyshev-asymptotic","scope":"pnt-plus/elementary-corollaries","status":"fully_proved","summary":"One has","title":"chebyshev-asymptotic"},{"id":"pnt-plus/elementary-corollaries/primorial-bounds","kind":"node","lean":"primorial_bounds","parent":4725,"result":true,"route":"../roadmap/pnt-plus/elementary-corollaries/index.html#primorial-bounds","scope":"pnt-plus/elementary-corollaries","status":"fully_proved","summary":"We have","title":"primorial-bounds"},{"id":"pnt-plus/elementary-corollaries/pi-asymp","kind":"node","lean":"pi_asymp","parent":4725,"result":true,"route":"../roadmap/pnt-plus/elementary-corollaries/index.html#pi-asymp","scope":"pnt-plus/elementary-corollaries","status":"fully_proved","summary":"There exists a function $c(x)$ such that $c(x) = o(1)$ as $x \\to \\infty$ and","title":"pi-asymp"},{"id":"pnt-plus/elementary-corollaries/pi-alt","kind":"node","lean":"pi_alt","parent":4725,"result":true,"route":"../roadmap/pnt-plus/elementary-corollaries/index.html#pi-alt","scope":"pnt-plus/elementary-corollaries","status":"fully_proved","summary":"One has","title":"pi-alt"},{"id":"pnt-plus/elementary-corollaries/pn-asymptotic","kind":"node","lean":"pn_asymptotic","parent":4725,"result":true,"route":"../roadmap/pnt-plus/elementary-corollaries/index.html#pn-asymptotic","scope":"pnt-plus/elementary-corollaries","status":"fully_proved","summary":"One has","title":"pn-asymptotic"},{"id":"pnt-plus/elementary-corollaries/pn-pn-plus-one","kind":"node","lean":"pn_pn_plus_one","parent":4725,"result":true,"route":"../roadmap/pnt-plus/elementary-corollaries/index.html#pn-pn-plus-one","scope":"pnt-plus/elementary-corollaries","status":"fully_proved","summary":"We have $p_{n+1} - p_n = o(p_n)$ as $n \\to \\infty$.","title":"pn-pn-plus-one"},{"id":"pnt-plus/elementary-corollaries/prime-between","kind":"node","lean":"prime_between","parent":4725,"result":true,"route":"../roadmap/pnt-plus/elementary-corollaries/index.html#prime-between","scope":"pnt-plus/elementary-corollaries","status":"fully_proved","summary":"For every $\\varepsilon>0$, there is a prime between $x$ and $(1+\\varepsilon)x$ for all sufficiently large $x$.","title":"prime-between"},{"id":"pnt-plus/elementary-corollaries/mun","kind":"node","lean":"sum_mobius_div_self_le","parent":4725,"result":true,"route":"../roadmap/pnt-plus/elementary-corollaries/index.html#mun","scope":"pnt-plus/elementary-corollaries","status":"fully_proved","summary":"We have $|\\sum_{n \\leq x} \\frac{\\mu(n)}{n}| \\leq 1$.","title":"mun"},{"id":"pnt-plus/elementary-corollaries/mu-pnt","kind":"node","lean":"mu_pnt","parent":4725,"result":true,"route":"../roadmap/pnt-plus/elementary-corollaries/index.html#mu-pnt","scope":"pnt-plus/elementary-corollaries","status":"fully_proved","summary":"We have $\\sum_{n \\leq x} \\mu(n) = o(x)$.","title":"M\u00f6bius form of prime number theorem"},{"id":"pnt-plus/elementary-corollaries/lambda-pnt","kind":"node","lean":"lambda_pnt","parent":4725,"result":true,"route":"../roadmap/pnt-plus/elementary-corollaries/index.html#lambda-pnt","scope":"pnt-plus/elementary-corollaries","status":"fully_proved","summary":"We have $\\sum_{n \\leq x} \\lambda(n) = o(x)$.","title":"lambda-pnt"},{"id":"pnt-plus/elementary-corollaries/mu-pnt-alt","kind":"node","lean":"mu_pnt_alt","parent":4725,"result":true,"route":"../roadmap/pnt-plus/elementary-corollaries/index.html#mu-pnt-alt","scope":"pnt-plus/elementary-corollaries","status":"fully_proved","summary":"We have $\\sum_{n \\leq x} \\mu(n)/n = o(1)$.","title":"Alternate M\u00f6bius form of prime number theorem"},{"data_url":"scopes/pnt-plus/elementary-corollaries/consequences-of-the-pnt-in-arithmetic-progressio.json","id":"pnt-plus/elementary-corollaries/consequences-of-the-pnt-in-arithmetic-progressio","kind":"scope","parent":4725,"result":false,"route":"../roadmap/pnt-plus/elementary-corollaries/consequences-of-the-pnt-in-arithmetic-progressio/index.html","scope":"pnt-plus/elementary-corollaries/consequences-of-the-pnt-in-arithmetic-progressio","status":"fully_proved","title":"Consequences of the PNT in arithmetic progressions"},{"id":"pnt-plus/elementary-corollaries/consequences-of-the-pnt-in-arithmetic-progressio/chebyshev-asymptotic-pnt","kind":"node","lean":"chebyshev_asymptotic_pnt","parent":4738,"result":true,"route":"../roadmap/pnt-plus/elementary-corollaries/consequences-of-the-pnt-in-arithmetic-progressio/index.html#chebyshev-asymptotic-pnt","scope":"pnt-plus/elementary-corollaries/consequences-of-the-pnt-in-arithmetic-progressio","status":"fully_proved","summary":"If $a\\ (q)$ is a primitive residue class, then one has","title":"Prime number theorem in AP"},{"id":"pnt-plus/elementary-corollaries/consequences-of-the-pnt-in-arithmetic-progressio/dirichlet-thm","kind":"node","lean":"dirichlet_thm","parent":4738,"result":true,"route":"../roadmap/pnt-plus/elementary-corollaries/consequences-of-the-pnt-in-arithmetic-progressio/index.html#dirichlet-thm","scope":"pnt-plus/elementary-corollaries/consequences-of-the-pnt-in-arithmetic-progressio","status":"fully_proved","summary":"Any primitive residue class contains an infinite number of primes.","title":"Dirichlet's theorem"},{"data_url":"scopes/pnt-plus/explicit-estimates.json","id":"pnt-plus/explicit-estimates","kind":"scope","parent":4477,"result":false,"route":"../roadmap/pnt-plus/explicit-estimates/index.html","scope":"pnt-plus/explicit-estimates","status":"fully_proved","summary":"We will try to systematically collect explicit estimates related to the prime number theorem from the literature, and formalize them in a modular fashion. We divide such estimates into four classes:","title":"Explicit estimates"},{"data_url":"scopes/pnt-plus/explicit-estimates/mertens-theorems.json","id":"pnt-plus/explicit-estimates/mertens-theorems","kind":"scope","parent":4741,"result":false,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/index.html","scope":"pnt-plus/explicit-estimates/mertens-theorems","status":"fully_proved","summary":"In this section we give explicit versions of Mertens' theorems:","title":"Mertens' theorems"},{"data_url":"scopes/pnt-plus/explicit-estimates/mertens-theorems/part-01.json","id":"pnt-plus/explicit-estimates/mertens-theorems/part-01","kind":"scope","navigation":"transparent","parent":4742,"result":false,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-01/index.html","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-01","status":"fully_proved","title":"Partial sum of logarithm\u2026 to Integral form for second error\u2026"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-01/mertens-sum-log","kind":"node","lean":"Mertens.sum_log_eq","parent":4743,"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-01/index.html#mertens-sum-log","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-01","status":"fully_proved","summary":"For any $x \\geq 1$, one has","title":"Partial sum of logarithm identity"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-01/mertens-sum-log-le","kind":"node","lean":"Mertens.sum_log_le","parent":4743,"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-01/index.html#mertens-sum-log-le","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-01","status":"fully_proved","summary":"For any $x \\geq 1$, one has","title":"Partial sum of logarithm upper bound"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-01/mertens-sum-log-ge","kind":"node","lean":"Mertens.sum_log_ge","parent":4743,"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-01/index.html#mertens-sum-log-ge","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-01","status":"fully_proved","summary":"For any $x \\geq 1$, one has","title":"Partial sum of logarithm lower bound"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-01/mertens-sum-log-eq-log-factorial","kind":"node","lean":"Mertens.sum_log_eq_log_factorial","parent":4743,"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-01/index.html#mertens-sum-log-eq-log-factorial","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-01","status":"fully_proved","summary":"For any $x \\geq 1$, one has","title":"Partial sum of logarithm as logarithm of factorial"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-01/mertens-sum-log-eq-sum-mangoldt","kind":"node","lean":"Mertens.sum_log_eq_sum_mangoldt","parent":4743,"reading_title":{"format":"inline-parts-v1","parts":[["text","Partial sum of logarithm as sum of "],["math","\\(\\Lambda(d)/d\\)"]],"source":"Partial sum of logarithm as sum of $\\Lambda(d)/d$"},"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-01/index.html#mertens-sum-log-eq-sum-mangoldt","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-01","status":"fully_proved","summary":"For any real $x$, one has","title":"Partial sum of logarithm as sum of $\\Lambda(d)/d$"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-01/mertens-first-error-mangoldt","kind":"node","lean":"Mertens.E\u2081\u039b, Mertens.E\u2081\u039b.bounded, Mertens.sum_mangoldt_div_eq_log'","parent":4743,"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-01/index.html#mertens-first-error-mangoldt","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-01","status":"fully_proved","summary":"We define $E_{1,\\Lambda}(x) := \\sum_{d \\leq x} \\frac{\\Lambda(d)}{d} - \\log x$.","title":"The remainder term in Mertens first theorem (von Mangoldt form)"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-01/mertens-first-error-mangoldt-ge","kind":"node","lean":"Mertens.E\u2081\u039b.ge","parent":4743,"reading_title":{"format":"inline-parts-v1","parts":[["text","Partial sum of "],["math","\\(\\Lambda(d)/d\\)"],["text"," lower bound"]],"source":"Partial sum of $\\Lambda(d)/d$ lower bound"},"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-01/index.html#mertens-first-error-mangoldt-ge","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-01","status":"fully_proved","summary":"For any $x \\geq 1$, one has","title":"Partial sum of $\\Lambda(d)/d$ lower bound"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-01/mertens-first-error-mangoldt-le","kind":"node","lean":"Mertens.E\u2081\u039b.le","parent":4743,"reading_title":{"format":"inline-parts-v1","parts":[["text","Partial sum of "],["math","\\(\\Lambda(d)/d\\)"],["text"," upper bound"]],"source":"Partial sum of $\\Lambda(d)/d$ upper bound"},"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-01/index.html#mertens-first-error-mangoldt-le","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-01","status":"fully_proved","summary":"For any $x \\geq 1$, one has","title":"Partial sum of $\\Lambda(d)/d$ upper bound"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-01/mertens-first-theorem-mangoldt","kind":"node","lean":"Mertens.sum_mangoldt_div_eq_log","parent":4743,"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-01/index.html#mertens-first-theorem-mangoldt","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-01","status":"fully_proved","summary":"For any $x \\geq 1$, one has","title":"Mertens' first theorem (von Mangoldt form)"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-01/mertens-first-error-prime","kind":"node","lean":"Mertens.E\u2081p","parent":4743,"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-01/index.html#mertens-first-error-prime","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-01","status":"fully_proved","summary":"We define $E_{1,p}(x) := \\sum_{p \\leq x} \\frac{\\log p}{p} - \\log x$.","title":"The remainder term in Mertens first theorem (prime form)"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-01/mertens-first-error-prime-le-mangoldt","kind":"node","lean":"Mertens.E\u2081p.le_E\u2081\u039b","parent":4743,"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-01/index.html#mertens-first-error-prime-le-mangoldt","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-01","status":"fully_proved","summary":"For any $x \\geq 1$, one has","title":"Prime error for Mertens first theorem bounded by von Mangoldt error"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-01/mertens-first-error-prime-le","kind":"node","lean":"Mertens.E\u2081p.le","parent":4743,"reading_title":{"format":"inline-parts-v1","parts":[["text","Partial sum of "],["math","\\(\\frac{\\log p}{p}\\)"],["text"," upper bound"]],"source":"Partial sum of $\\frac{\\log p}{p}$ upper bound"},"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-01/index.html#mertens-first-error-prime-le","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-01","status":"fully_proved","summary":"For any $x \\geq 1$, one has","title":"Partial sum of $\\frac{\\log p}{p}$ upper bound"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-01/e1-summable","kind":"node","lean":"Mertens.E\u2081.summable","parent":4743,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(E_1\\)"],["text"," summable"]],"source":"$E_1$ summable"},"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-01/index.html#e1-summable","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-01","status":"fully_proved","summary":"The series $E_1 := \\sum_p \\frac{\\log p}{p(p-1)}$ converges.","title":"$E_1$ summable"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-01/e1-bound","kind":"node","lean":"Mertens.E\u2081.le","parent":4743,"reading_title":{"format":"inline-parts-v1","parts":[["text","Upper bound on "],["math","\\(E_1\\)"]],"source":"Upper bound on $E_1$"},"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-01/index.html#e1-bound","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-01","status":"fully_proved","summary":"One has $E_1 \\leq \\frac{5 \\log 2 + 3}{4}$","title":"Upper bound on $E_1$"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-01/mertens-first-error-prime-ge","kind":"node","lean":"Mertens.E\u2081\u039b.le_E\u2081p_add_E\u2081","parent":4743,"reading_title":{"format":"inline-parts-v1","parts":[["text","Partial sum of "],["math","\\(\\frac{\\log p}{p}\\)"],["text"," lower bound"]],"source":"Partial sum of $\\frac{\\log p}{p}$ lower bound"},"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-01/index.html#mertens-first-error-prime-ge","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-01","status":"fully_proved","summary":"For any $x \\geq 1$, one has","title":"Partial sum of $\\frac{\\log p}{p}$ lower bound"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-01/mertens-first-theorem-prime-bounded","kind":"node","lean":"Mertens.sum_log_prime_div_eq_log, Mertens.sum_log_prime_div_eq_log', Mertens.sum_log_prime_div_eq_log''","parent":4743,"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-01/index.html#mertens-first-theorem-prime-bounded","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-01","status":"fully_proved","summary":"For any $x \\geq 1$, one has","title":"Error term in Mertens' first theorem bounded (prime form)"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-01/euler-mascheroni-const-alt","kind":"node","lean":"Mertens.\u03b3","parent":4743,"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-01/index.html#euler-mascheroni-const-alt","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-01","status":"fully_proved","summary":"We set $\\gamma := \\int_2^\\infty \\frac{E_{1,\\Lambda}(t)}{t \\log^2 t} \\, dt + 1 - \\log \\log 2$.","title":"Alternate Formula for Euler-Mascheroni constant"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-01/mertens-second-error-mangoldt","kind":"node","lean":"Mertens.E\u2082\u039b, Mertens.E\u2082p","parent":4743,"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-01/index.html#mertens-second-error-mangoldt","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-01","status":"fully_proved","summary":"We define $E_{2,\\Lambda}(x) := \\sum_{d \\leq x} \\frac{\\Lambda(d)}{d \\log d} - \\log \\log x - \\gamma$.","title":"The remainder term in Mertens second theorem (von Mangoldt form)"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-01/mertens-integral-ident","kind":"node","lean":"_private.PrimeNumberTheoremAnd.IEANTN.Mertens.0.Mertens.sum_div_log_eq","parent":4743,"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-01/index.html#mertens-integral-ident","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-01","status":"fully_proved","summary":"For any $x \\geq 2$ and any $f : {\\mathbb N} \\mapsto {\\mathbb R}$, one has","title":"Integral identity involving inverse log weight"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-01/mertens-second-error-mangoldt-eq","kind":"node","lean":"Mertens.E\u2082\u039b.eq","parent":4743,"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-01/index.html#mertens-second-error-mangoldt-eq","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-01","status":"fully_proved","summary":"For any $x \\geq 2$, one has","title":"Integral form for second error (von Mangoldt form)"},{"data_url":"scopes/pnt-plus/explicit-estimates/mertens-theorems/part-02.json","id":"pnt-plus/explicit-estimates/mertens-theorems/part-02","kind":"scope","navigation":"transparent","parent":4742,"result":false,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-02/index.html","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-02","status":"fully_proved","title":"Bound for second Mertens error\u2026 to Mertens' third theorem error\u2026"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-02/mertens-second-error-mangoldt-bound","kind":"node","lean":"Mertens.E\u2082\u039b.abs_le, Mertens.E\u2082\u039b.bound, Mertens.E\u2082\u039b.bound'","parent":4764,"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-02/index.html#mertens-second-error-mangoldt-bound","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-02","status":"fully_proved","summary":"For any $x \\geq 2$, one has","title":"Bound for second Mertens error (von Mangoldt form)"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-02/log-zeta-eq-1","kind":"node","lean":"Mertens.log_zeta_eq_sum","parent":4764,"reading_title":{"format":"inline-parts-v1","parts":[["text","Dirichlet series for "],["math","\\(\\log \\zeta(s)\\)"]],"source":"Dirichlet series for $\\log \\zeta(s)$"},"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-02/index.html#log-zeta-eq-1","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-02","status":"fully_proved","summary":"If $s > 1$ then $\\log\\zeta(s) = - \\log (s-1) + \\Gamma'(1) + \\gamma + (s-1) \\int_1^\\infty E_{2,\\Lambda}(x) x^{-s}\\ ds$.","title":"Dirichlet series for $\\log \\zeta(s)$"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-02/log-zeta-eq-2","kind":"node","lean":"_private.PrimeNumberTheoremAnd.IEANTN.Mertens.0.Mertens.log_zeta_eq_integ","parent":4764,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integration by parts identity for "],["math","\\(\\log \\zeta(s)\\)"]],"source":"Integration by parts identity for $\\log \\zeta(s)$"},"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-02/index.html#log-zeta-eq-2","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-02","status":"fully_proved","summary":"If $s > 1$ then $\\log\\zeta(s) = (s-1) \\int_1^\\infty (\\log \\log x + \\gamma + E_{2,\\Lambda}(x)) x^{-s})\\ dx$.","title":"Integration by parts identity for $\\log \\zeta(s)$"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-02/log-zeta-eq-3","kind":"node","lean":"_private.PrimeNumberTheoremAnd.IEANTN.Mertens.0.Mertens.mul_integ_log_log_eq","parent":4764,"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-02/index.html#log-zeta-eq-3","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-02","status":"fully_proved","summary":"If $s > 1$ then $(s-1) \\int_1^\\infty \\log \\log x \\cdot x^{-s}\\ dx = -\\log (s-1) + \\Gamma'(1)$.","title":"First integral identity"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-02/log-zeta-eq-4","kind":"node","lean":"_private.PrimeNumberTheoremAnd.IEANTN.Mertens.0.Mertens.mul_integ_gamma_eq","parent":4764,"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-02/index.html#log-zeta-eq-4","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-02","status":"fully_proved","summary":"If $s > 1$ then $(s-1) \\int_1^\\infty \\gamma \\cdot x^{-s}\\ dx = \\gamma$.","title":"Second integral identity"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-02/log-zeta-eq","kind":"node","lean":"_private.PrimeNumberTheoremAnd.IEANTN.Mertens.0.Mertens.log_zeta_eq","parent":4764,"reading_title":{"format":"inline-parts-v1","parts":[["text","An identity for "],["math","\\(\\log \\zeta(s)\\)"]],"source":"An identity for $\\log \\zeta(s)$"},"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-02/index.html#log-zeta-eq","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-02","status":"fully_proved","summary":"If $s > 1$ then $\\log\\zeta(s) = - \\log (s-1) + \\Gamma'(1) + \\gamma + (s-1) \\int_1^\\infty E_{2,\\Lambda}(x) x^{-s}\\ ds$.","title":"An identity for $\\log \\zeta(s)$"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-02/log-zeta-limit","kind":"node","lean":"_private.PrimeNumberTheoremAnd.IEANTN.Mertens.0.Mertens.log_zeta_limit","parent":4764,"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-02/index.html#log-zeta-limit","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-02","status":"fully_proved","summary":"One has $\\log \\zeta(s) = - \\log(s-1) + o(1)$ as $s \\to 1^+$.","title":"limiting behavior of log zeta"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-02/euler-mascheroni-eq","kind":"node","lean":"Mertens.deriv_gamma_add_\u03b3_eq_zero","parent":4764,"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-02/index.html#euler-mascheroni-eq","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-02","status":"fully_proved","summary":"$\\gamma$ is the Euler\u2013Mascheroni constant.","title":"Compatibility with Mathlib Euler-Mascheroni constant"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-02/mertens-second-theorem-mangoldt-weak","kind":"node","lean":"Mertens.sum_mangoldt_div_log_eq_log_log, Mertens.sum_mangoldt_div_log_eq_log_log', Mertens.sum_mangoldt_div_log_eq_log_log''","parent":4764,"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-02/index.html#mertens-second-theorem-mangoldt-weak","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-02","status":"fully_proved","summary":"For any $x \\geq 2$, one has","title":"Mertens' second theorem (weak von Mangoldt form)"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-02/meissel-mertens-constant","kind":"node","lean":"Mertens.M","parent":4764,"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-02/index.html#meissel-mertens-constant","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-02","status":"fully_proved","summary":"We define $M := \\int_2^\\infty \\frac{E_{1,p}(t)}{t \\log^2 t} \\, dt + 1 - \\log \\log 2$.","title":"The Meissel-Mertens constant"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-02/mertens-second-constant-prime-le","kind":"node","lean":"Mertens.M.le","parent":4764,"reading_title":{"format":"inline-parts-v1","parts":[["text","Upper bound for "],["math","\\(M\\)"]],"source":"Upper bound for $M$"},"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-02/index.html#mertens-second-constant-prime-le","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-02","status":"fully_proved","summary":"One has $M \\leq \\frac{\\log 4 + 4}{\\log 2} + 1 - \\log \\log 2$.","title":"Upper bound for $M$"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-02/mertens-second-constant-prime-ge","kind":"node","lean":"Mertens.M.ge","parent":4764,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lower bound for "],["math","\\(M\\)"]],"source":"Lower bound for $M$"},"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-02/index.html#mertens-second-constant-prime-ge","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-02","status":"fully_proved","summary":"One has $M \\geq -\\frac{2 + E_1}{\\log 2} + 1 - \\log \\log 2$.","title":"Lower bound for $M$"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-02/mertens-second-error-prime-eq","kind":"node","lean":"Mertens.E\u2082p.eq","parent":4764,"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-02/index.html#mertens-second-error-prime-eq","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-02","status":"fully_proved","summary":"For any $x \\geq 2$, one has","title":"Integral form for second error (prime form)"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-02/mertens-second-error-prime-abs-le","kind":"node","lean":"Mertens.E\u2082p.abs_le, Mertens.E\u2082p.bound, Mertens.E\u2082p.bound'","parent":4764,"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-02/index.html#mertens-second-error-prime-abs-le","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-02","status":"fully_proved","summary":"For any $x \\geq 2$, one has","title":"Bound for second error (prime form)"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-02/mertens-second-theorem-prime-weak","kind":"node","lean":"Mertens.sum_prime_div_eq_log_log, Mertens.sum_prime_div_eq_log_log', Mertens.sum_prime_div_eq_log_log''","parent":4764,"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-02/index.html#mertens-second-theorem-prime-weak","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-02","status":"fully_proved","summary":"For any $x \\geq 2$, one has","title":"Mertens' second theorem (weak prime form)"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-02/meissel-mertens-eq","kind":"node","lean":"Mertens.M.eq","parent":4764,"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-02/index.html#meissel-mertens-eq","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-02","status":"fully_proved","summary":"One has $M = \\gamma + \\sum_p \\log(1-\\frac{1}{p}) + \\frac{1}{p}$.","title":"Formula for Meissel-Mertens constant"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-02/mertens-third-error","kind":"node","lean":"Mertens.E\u2083","parent":4764,"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-02/index.html#mertens-third-error","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-02","status":"fully_proved","summary":"We define $E_3(x) := \\sum_{p \\leq x} \\log (1 - \\frac{1}{p}) + \\log\\log x + \\gamma$.","title":"The remainder term in Mertens third theorem"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-02/mertens-third-theorem-error","kind":"node","lean":"Mertens.prod_one_minus_div_prime_eq","parent":4764,"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-02/index.html#mertens-third-theorem-error","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-02","status":"fully_proved","summary":"For any $x \\geq 2$, one has","title":"Mertens' third theorem error term"},{"id":"pnt-plus/explicit-estimates/mertens-theorems/part-02/mertens-third-theorem-error-le","kind":"node","lean":"Mertens.E\u2083.abs_le, Mertens.E\u2083.bound, Mertens.E\u2083.bound', Mertens.E\u2083.bound'', Mertens.E\u2083.bound'''","parent":4764,"result":true,"route":"../roadmap/pnt-plus/explicit-estimates/mertens-theorems/part-02/index.html#mertens-third-theorem-error-le","scope":"pnt-plus/explicit-estimates/mertens-theorems/part-02","status":"fully_proved","summary":"For any $x \\geq 2$, one has","title":"Mertens' third theorem error bound"},{"data_url":"scopes/pnt-plus/tertiary-explicit-estimates.json","id":"pnt-plus/tertiary-explicit-estimates","kind":"scope","parent":4477,"result":false,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/index.html","scope":"pnt-plus/tertiary-explicit-estimates","status":"fully_proved","title":"Tertiary explicit estimates"},{"data_url":"scopes/pnt-plus/tertiary-explicit-estimates/erdos-problem-392.json","id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392","kind":"scope","parent":4784,"result":false,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/index.html","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392","status":"fully_proved","summary":"The proof here is adapted from which in turn is inspired by the arguments in .","title":"Erdos problem 392"},{"data_url":"scopes/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01.json","id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01","kind":"scope","navigation":"transparent","parent":4785,"result":false,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/index.html","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01","status":"fully_proved","title":"factorization-def to initial-factorization-medium-pri\u2026"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/factorization-def","kind":"node","lean":"Erdos392.Factorization","parent":4786,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/index.html#factorization-def","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01","status":"fully_proved","summary":"We work with (approximate) factorizations $a_1 \\dots a_t$ of a factorial $n!$.","title":"factorization-def"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/waste-def","kind":"node","lean":"Erdos392.Factorization.waste","parent":4786,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/index.html#waste-def","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01","status":"fully_proved","summary":"The waste of a factorizations $a_1 \\dots a_t$ is defined as $\\sum_i \\log (n / a_i)$.","title":"waste-def"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/balance-def","kind":"node","lean":"Erdos392.Factorization.balance, Erdos392.Factorization.total_imbalance","parent":4786,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/index.html#balance-def","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01","status":"fully_proved","summary":"The balance of a factorization $a_1 \\dots a_t$ at a prime $p$ is defined as the number of times $p$ divides $a_1 \\dots a_t$, minus the number of times $p$ divides $n!$.","title":"balance-def"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/balance-zero","kind":"node","lean":"Erdos392.Factorization.zero_total_imbalance","parent":4786,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/index.html#balance-zero","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01","status":"fully_proved","summary":"If a factorization has zero total imbalance, then it exactly factors $n!$.","title":"balance-zero"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/waste-eq","kind":"node","lean":"Erdos392.Factorization.waste_eq","parent":4786,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/index.html#waste-eq","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01","status":"fully_proved","summary":"The waste of a factorization is equal to $t \\log n - \\log n!$, where $t$ is the number of elements.","title":"waste-eq"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/score-def","kind":"node","lean":"Erdos392.Factorization.score","parent":4786,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/index.html#score-def","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01","status":"fully_proved","summary":"The score of a factorization (relative to a cutoff parameter $L$) is equal to its waste, plus $\\log p$ for every surplus prime $p$, $\\log (n/p)$ for every deficit prime above $L$, $\\log L$ for every deficit prime below $L$ and an additional $\\log n$ if one is not in total balance.","title":"score-def"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/score-eq","kind":"node","lean":"Erdos392.Factorization.score_eq","parent":4786,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/index.html#score-eq","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01","status":"fully_proved","summary":"If one is in total balance, then the score is equal to the waste.","title":"score-eq"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/score-lower-1","kind":"node","lean":"Erdos392.Factorization.lower_score_1","parent":4786,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/index.html#score-lower-1","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01","status":"fully_proved","summary":"If there is a prime $p$ in surplus, one can remove it without increasing the score.","title":"score-lower-1"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/score-lower-2","kind":"node","lean":"Erdos392.Factorization.lower_score_2","parent":4786,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/index.html#score-lower-2","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01","status":"fully_proved","summary":"If there is a prime $p$ in deficit larger than $L$, one can remove it without increasing the score.","title":"score-lower-2"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/score-lower-3","kind":"node","lean":"Erdos392.Factorization.lower_score_3","parent":4786,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/index.html#score-lower-3","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01","status":"fully_proved","summary":"If there is a prime $p$ in deficit less than $L$, one can remove it without increasing the score.","title":"score-lower-3"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/score-lowest","kind":"node","lean":"Erdos392.Factorization.lowest_score","parent":4786,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/index.html#score-lowest","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01","status":"fully_proved","summary":"One can bring any factorization into balance without increasing the score.","title":"score-lowest"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/card-bound","kind":"node","lean":"Erdos392.Factorization.card_bound","parent":4786,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/index.html#card-bound","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01","status":"fully_proved","summary":"Starting from any factorization $f$, one can find a factorization $f'$ in balance whose cardinality is at most $\\log n!$ plus the score of $f$, divided by $\\log n$.","title":"card-bound"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/params-set","kind":"node","lean":"Erdos392.Params","parent":4786,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/index.html#params-set","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01","status":"fully_proved","summary":"Now let $M,L$ be additional parameters with $n > L^2$; we also need the minor variant $\\lfloor n/L \\rfloor > \\sqrt{n}$.","title":"params-set"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/initial-factorization-def","kind":"node","lean":"Erdos392.Params.initial","parent":4786,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/index.html#initial-factorization-def","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01","status":"fully_proved","summary":"We perform an initial factorization by taking the natural numbers between $n-n/M$ (inclusive) and $n$ (exclusive) repeated $M$ times, deleting those elements that are not $n/L$-smooth (i.e., have a prime factor greater than or equal to $n/L$).","title":"initial-factorization-def"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/initial-factorization-card","kind":"node","lean":"Erdos392.Params.initial.card","parent":4786,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/index.html#initial-factorization-card","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01","status":"fully_proved","summary":"The number of elements in this initial factorization is at most $n$.","title":"initial-factorization-card"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/initial-factorization-waste","kind":"node","lean":"Erdos392.Params.initial.waste","parent":4786,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/index.html#initial-factorization-waste","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01","status":"fully_proved","summary":"The total waste in this initial factorization is at most $n \\log \\frac{1}{1-1/M}$.","title":"initial-factorization-waste"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/initial-factorization-large-prime-le","kind":"node","lean":"Erdos392.Params.initial.balance_large_prime_le","parent":4786,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/index.html#initial-factorization-large-prime-le","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01","status":"fully_proved","summary":"A large prime $p \\geq n/L$ cannot be in surplus.","title":"initial-factorization-large-prime-le"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/initial-factorization-large-prime-ge","kind":"node","lean":"Erdos392.Params.initial.balance_large_prime_ge","parent":4786,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/index.html#initial-factorization-large-prime-ge","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01","status":"fully_proved","summary":"A large prime $p \\geq n/L$ can be in deficit by at most $n/p$.","title":"initial-factorization-large-prime-ge"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/initial-factorization-medium-prime-le","kind":"node","lean":"Erdos392.Params.initial.balance_medium_prime_le","parent":4786,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/index.html#initial-factorization-medium-prime-le","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01","status":"fully_proved","summary":"A medium prime $\\sqrt{n} < p \u2264 n/L$ can be in surplus by at most $M$.","title":"initial-factorization-medium-prime-le"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/initial-factorization-medium-prime-ge","kind":"node","lean":"Erdos392.Params.initial.balance_medium_prime_ge","parent":4786,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01/index.html#initial-factorization-medium-prime-ge","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-01","status":"fully_proved","summary":"A medium prime $\\sqrt{n} < p \u2264 n/L$ can be in deficit by at most $M$.","title":"initial-factorization-medium-prime-ge"},{"data_url":"scopes/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02.json","id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02","kind":"scope","navigation":"transparent","parent":4785,"result":false,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02/index.html","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02","status":"fully_proved","title":"initial-factorization-small-prim\u2026 to erdos-sol-2"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02/initial-factorization-small-prime-le","kind":"node","lean":"Erdos392.Params.initial.balance_small_prime_le","parent":4807,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02/index.html#initial-factorization-small-prime-le","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02","status":"fully_proved","summary":"A small prime $p \\leq \\sqrt{n}$ can be in surplus by at most $M\\log n$.","title":"initial-factorization-small-prime-le"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02/initial-factorization-small-prime-ge","kind":"node","lean":"Erdos392.Params.initial.balance_small_prime_ge","parent":4807,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02/index.html#initial-factorization-small-prime-ge","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02","status":"fully_proved","summary":"A small prime $L < p \\leq \\sqrt{n}$ can be in deficit by at most $M\\log n$.","title":"initial-factorization-small-prime-ge"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02/initial-factorization-tiny-prime-ge","kind":"node","lean":"Erdos392.Params.initial.balance_tiny_prime_ge","parent":4807,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02/index.html#initial-factorization-tiny-prime-ge","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02","status":"fully_proved","summary":"A tiny prime $p \\leq L$ can be in deficit by at most $M\\log n + ML\\pi(n)$.","title":"initial-factorization-tiny-prime-ge"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02/initial-score-bound","kind":"node","lean":"Erdos392.Params.initial.score_bound","parent":4807,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02/index.html#initial-score-bound","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02","status":"fully_proved","summary":"The initial score is bounded by","title":"initial-score-bound"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02/bound-score-1","kind":"node","lean":"Erdos392.Params.initial.bound_score_1","parent":4807,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02/index.html#bound-score-1","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02","status":"fully_proved","summary":"If $M$ is sufficiently large depending on $\\varepsilon$, then $n \\log(1-1/M)^{-1} \\leq \\varepsilon n$.","title":"bound-score-1"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02/bound-score-2","kind":"node","lean":"Erdos392.Params.initial.bound_score_2","parent":4807,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02/index.html#bound-score-2","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02","status":"fully_proved","summary":"If $L$ is sufficiently large depending on $M, \\varepsilon$, and $n$ sufficiently large depending on $L$, then $\\sum_{p \\leq n/L} M \\log n \\leq \\varepsilon n$.","title":"bound-score-2"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02/bound-score-3","kind":"node","lean":"Erdos392.Params.initial.bound_score_3","parent":4807,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02/index.html#bound-score-3","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02","status":"fully_proved","summary":"If $n$ sufficiently large depending on $M, \\varepsilon$, then $\\sum_{p \\leq \\sqrt{n}} M \\log^2 n / \\log 2 \\leq \\varepsilon n$.","title":"bound-score-3"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02/primecounting-is-o-id","kind":"node","lean":"Erdos392.primeCounting_is_o_id","parent":4807,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02/index.html#primecounting-is-o-id","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02","status":"fully_proved","summary":"\\pi(n) = o(n) \\quad \\text{as } n \\to \\infty.","title":"primeCounting-is-o-id"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02/bound-score-4","kind":"node","lean":"Erdos392.Params.initial.bound_score_4","parent":4807,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02/index.html#bound-score-4","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02","status":"fully_proved","summary":"If $n$ sufficiently large depending on $L, \\varepsilon$, then $\\sum_{n/L < p \\leq n} \\frac{n}{p} \\log \\frac{n}{p} \\leq \\varepsilon n$.","title":"bound-score-4"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02/primecounting-le-bound","kind":"node","lean":"Erdos392.primeCounting_le_bound","parent":4807,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02/index.html#primecounting-le-bound","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02","status":"fully_proved","summary":"For all $n \\geq 2$, one has","title":"primeCounting-le-bound"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02/bound-score-5","kind":"node","lean":"Erdos392.Params.initial.bound_score_5","parent":4807,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02/index.html#bound-score-5","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02","status":"fully_proved","summary":"If $n$ sufficiently large depending on $M, L, \\varepsilon$, then $\\sum_{p \\leq L} (M \\log n + M L \\pi(n)) \\log L \\leq \\varepsilon n$.","title":"bound-score-5"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02/initial-score","kind":"node","lean":"Erdos392.Params.initial.score","parent":4807,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02/index.html#initial-score","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02","status":"fully_proved","summary":"The score of the initial factorization can be taken to be $o(n)$.","title":"initial-score"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02/erdos-sol-1","kind":"node","lean":"Erdos392.Solution_1","parent":4807,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02/index.html#erdos-sol-1","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02","status":"fully_proved","summary":"One can find a balanced factorization of $n!$ with cardinality at most $n - n / \\log n + o(n / \\log n)$.-","title":"erdos-sol-1"},{"id":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02/erdos-sol-2","kind":"node","lean":"Erdos392.Solution_2","parent":4807,"result":true,"route":"../roadmap/pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02/index.html#erdos-sol-2","scope":"pnt-plus/tertiary-explicit-estimates/erdos-problem-392/part-02","status":"fully_proved","summary":"One can factorize $n!$ into at most $n/2 - n / 2\\log n + o(n / \\log n)$ numbers of size at most $n^2$.-","title":"erdos-sol-2"},{"data_url":"scopes/pnt-plus/iwaniec-kowalski.json","id":"pnt-plus/iwaniec-kowalski","kind":"scope","parent":4477,"result":false,"route":"../roadmap/pnt-plus/iwaniec-kowalski/index.html","scope":"pnt-plus/iwaniec-kowalski","status":"can_prove","title":"Iwaniec-Kowalski"},{"data_url":"scopes/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1.json","id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1","kind":"scope","parent":4822,"result":false,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/index.html","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1","status":"can_prove","summary":"Here we collect facts from Chapter 1 that are not already in Mathlib. We will try to upstream as much as possible.","title":"Blueprint for Iwaniec-Kowalski Chapter 1"},{"data_url":"scopes/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01.json","id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01","kind":"scope","navigation":"transparent","parent":4823,"result":false,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/index.html","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01","status":"can_prove","title":"IsAdditive to sigmaR-natCast"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/isadditive","kind":"node","lean":"ArithmeticFunction.IsAdditive","parent":4824,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/index.html#isadditive","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01","status":"fully_proved","summary":"Additive arithmetic function: satisfies $f(mn) = f(m) + f(n)$ for coprime $m$, $n$.","title":"IsAdditive"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/iscompletelyadditive","kind":"node","lean":"ArithmeticFunction.IsCompletelyAdditive","parent":4824,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/index.html#iscompletelyadditive","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01","status":"fully_proved","summary":"Completely additive arithmetic function: satisfies $f(mn) = f(m) + f(n)$ for all $m, n \\ne 0$.","title":"IsCompletelyAdditive"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/iscompletelyadditive-isadditive","kind":"node","lean":"ArithmeticFunction.IsCompletelyAdditive.isAdditive","parent":4824,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/index.html#iscompletelyadditive-isadditive","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01","status":"fully_proved","summary":"A completely additive function is additive.","title":"IsCompletelyAdditive.isAdditive"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/unique-divisor-decomposition","kind":"node","lean":"ArithmeticFunction.unique_divisor_decomposition","parent":4824,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/index.html#unique-divisor-decomposition","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01","status":"can_prove","summary":"If $a$ and $b$ are coprime, then any divisor $d$ of $ab$ can be uniquely expressed as a product of a divisor of $a$ and a divisor of $b$.","title":"unique divisor decomposition"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/sum-divisors-mul-of-coprime","kind":"node","lean":"ArithmeticFunction.sum_divisors_mul_of_coprime","parent":4824,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/index.html#sum-divisors-mul-of-coprime","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01","status":"can_prove","summary":"If $f$ is a multiplicative arithmetic function, then for coprime, nonzero $a$ and $b$, we have that $\\sum_{d | ab} f(d) = (\\sum_{d | a} f(d)) \\cdot (\\sum_{d | b} f(d))$.","title":"sum divisors mul of coprime"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/sum-moebius-pmul-eq-prod-one-sub","kind":"node","lean":"ArithmeticFunction.sum_moebius_pmul_eq_prod_one_sub","parent":4824,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/index.html#sum-moebius-pmul-eq-prod-one-sub","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01","status":"proved","summary":"If $g$ is a multiplicative arithmetic function, then for any $n \\neq 0$, $\\sum_{d | n} \\mu(d) \\cdot g(d) = \\prod_{p | n} (1 - g(p))$.","title":"sum moebius pmul eq prod one sub"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/zeta-mul-zeta","kind":"node","lean":"ArithmeticFunction.zeta_mul_zeta","parent":4824,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/index.html#zeta-mul-zeta","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01","status":"fully_proved","summary":"The Dirichlet convolution of $\\zeta$ with itself is $\\tau$ (the divisor count function).","title":"zeta mul zeta"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/lseries-tau-eq-riemannzeta-sq","kind":"node","lean":"ArithmeticFunction.LSeries_tau_eq_riemannZeta_sq","parent":4824,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/index.html#lseries-tau-eq-riemannzeta-sq","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01","status":"fully_proved","summary":"The L-series of $\\tau$ equals the square of the Riemann zeta function for $\\Re(s) > 1$.","title":"LSeries tau eq riemannZeta sq"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/d","kind":"node","lean":"ArithmeticFunction.d","parent":4824,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/index.html#d","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01","status":"fully_proved","summary":"$d_k$ is the $k$-fold divisor function: the number of ways to write $n$ as an ordered product of $k$ natural numbers. Equivalently, the Dirichlet convolution of $\\zeta$ with itself $k$ times.","title":"d"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/d-zero","kind":"node","lean":"ArithmeticFunction.d_zero","parent":4824,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/index.html#d-zero","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01","status":"fully_proved","summary":"$d_0$ is the multiplicative identity (indicator at 1).","title":"d zero"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/d-one","kind":"node","lean":"ArithmeticFunction.d_one","parent":4824,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/index.html#d-one","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01","status":"fully_proved","summary":"$d_1$ is $\\zeta$.","title":"d one"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/d-two","kind":"node","lean":"ArithmeticFunction.d_two","parent":4824,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/index.html#d-two","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01","status":"fully_proved","summary":"$d_2$ is the classical divisor count function $\\tau$.","title":"d two"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/d-succ","kind":"node","lean":"ArithmeticFunction.d_succ","parent":4824,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/index.html#d-succ","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01","status":"fully_proved","summary":"Recurrence: $d_{k+1} = d_k * \\zeta$.","title":"d succ"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/lseries-d-summable","kind":"node","lean":"ArithmeticFunction.LSeries_d_summable","parent":4824,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/index.html#lseries-d-summable","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01","status":"fully_proved","summary":"The L-series for $d_k$ is summable for $\\Re(s) > 1$.","title":"LSeries d summable"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/lseries-d-eq-riemannzeta-pow","kind":"node","lean":"ArithmeticFunction.LSeries_d_eq_riemannZeta_pow","parent":4824,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/index.html#lseries-d-eq-riemannzeta-pow","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01","status":"fully_proved","summary":"The $L$-series of $d_k$ equals $\\zeta(s)^k$ for $\\Re(s) > 1$.","title":"LSeries d eq riemannZeta pow"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/d-ismultiplicative","kind":"node","lean":"ArithmeticFunction.d_isMultiplicative","parent":4824,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/index.html#d-ismultiplicative","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01","status":"fully_proved","summary":"$d_k$ is multiplicative for all $k$.","title":"d isMultiplicative"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/d-apply-prime-pow","kind":"node","lean":"ArithmeticFunction.d_apply_prime_pow","parent":4824,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/index.html#d-apply-prime-pow","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01","status":"fully_proved","summary":"Explicit formula: $d_k (p^a) = (a + k - 1).choose (k - 1)$ for prime $p$ and $k \\geq 1$.","title":"d apply prime pow"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/d-apply","kind":"node","lean":"ArithmeticFunction.d_apply","parent":4824,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/index.html#d-apply","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01","status":"fully_proved","summary":"(1.25) in Iwaniec-Kowalski: a formula for $d_k$ for all $n$.","title":"d apply"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/sigmar","kind":"node","lean":"ArithmeticFunction.sigmaR","parent":4824,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/index.html#sigmar","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01","status":"fully_proved","summary":"Divisor power sum with complex exponent.","title":"sigmaR"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/sigmar-natcast","kind":"node","lean":"ArithmeticFunction.sigmaR_natCast","parent":4824,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01/index.html#sigmar-natcast","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-01","status":"fully_proved","summary":"For natural exponents, $\\sigma^R$ agrees with $\\sigma$.","title":"sigmaR-natCast"},{"data_url":"scopes/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02.json","id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02","kind":"scope","navigation":"transparent","parent":4823,"result":false,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/index.html","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02","status":"can_prove","title":"sigmaR-apply to zeta mul tau square eq"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/sigmar-apply","kind":"node","lean":"ArithmeticFunction.sigmaR_apply","parent":4845,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/index.html#sigmar-apply","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02","status":"fully_proved","summary":"We have that $\\sigma^R_s(n)=\\sum_{d\\mid n}d^s.$","title":"sigmaR-apply"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/sigmar-natcast","kind":"node","lean":"ArithmeticFunction.sigmaR_natCast'","parent":4845,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/index.html#sigmar-natcast","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02","status":"fully_proved","summary":"A casting lemma for $\\sigma^R$.","title":"sigmaR-natCast'"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/sigmar-apply-prime-pow","kind":"node","lean":"ArithmeticFunction.sigmaR_apply_prime_pow","parent":4845,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/index.html#sigmar-apply-prime-pow","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02","status":"fully_proved","summary":"For a prime power, we have that $\\sigma^R_s(p^i)=\\sum_{j=0}^ip^{js}$.","title":"sigmaR-apply-prime-pow"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/sigmar-one-apply","kind":"node","lean":"ArithmeticFunction.sigmaR_one_apply","parent":4845,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/index.html#sigmar-one-apply","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02","status":"fully_proved","summary":"Same as the previous lemma, but with a different casting structure.","title":"sigmaR-one-apply"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/sigmar-one-apply-prime-pow","kind":"node","lean":"ArithmeticFunction.sigmaR_one_apply_prime_pow","parent":4845,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/index.html#sigmar-one-apply-prime-pow","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02","status":"fully_proved","summary":"Same as the previous lemma, but with a different casting structure.","title":"sigmaR-one-apply-prime-pow"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/sigmar-eq-sum-div","kind":"node","lean":"ArithmeticFunction.sigmaR_eq_sum_div","parent":4845,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/index.html#sigmar-eq-sum-div","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02","status":"fully_proved","summary":"We have that $\\sigma^R_s(n)=\\sum_{d\\mid n}(n/d)^s$.","title":"sigmaR-eq-sum-div"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/sigmar-zero-apply","kind":"node","lean":"ArithmeticFunction.sigmaR_zero_apply","parent":4845,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/index.html#sigmar-zero-apply","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02","status":"fully_proved","summary":"Same as the previous lemma, but with a different casting structure.","title":"sigmaR-zero-apply"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/sigmar-zero-apply-prime-pow","kind":"node","lean":"ArithmeticFunction.sigmaR_zero_apply_prime_pow","parent":4845,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/index.html#sigmar-zero-apply-prime-pow","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02","status":"fully_proved","summary":"Same as the previous lemma, but with a different casting structure.","title":"sigmaR-zero-apply-prime-pow"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/sigmar-one","kind":"node","lean":"ArithmeticFunction.sigmaR_one","parent":4845,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/index.html#sigmar-one","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02","status":"fully_proved","summary":"We have that $\\sigma^R_s(1)=1$.","title":"sigmaR-one"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/powr","kind":"node","lean":"ArithmeticFunction.powR","parent":4845,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/index.html#powr","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02","status":"fully_proved","summary":"Arithmetic function with complex parameter $\\nu$. Evaluates as $n\\mapsto n^{\\nu}$ for $n\\neq 0$ and $0$ at $n=0$.","title":"powR"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/ismultiplicative-powr","kind":"node","lean":"ArithmeticFunction.isMultiplicative_powR","parent":4845,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/index.html#ismultiplicative-powr","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02","status":"fully_proved","summary":"For fixed $\\nu$ the function $n\\mapsto n^\\nu$ is multiplicative.","title":"isMultiplicative-powR"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/sigmar-eq-zeta-mul-powr","kind":"node","lean":"ArithmeticFunction.sigmaR_eq_zeta_mul_powR","parent":4845,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/index.html#sigmar-eq-zeta-mul-powr","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02","status":"fully_proved","summary":"$\\sigma^R(\\nu) = \\zeta * \\text{pow}^R(\\nu)$, where $\\zeta$ is the constant function $1$.","title":"sigmaR-eq-zeta-mul-powR"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/ismultiplicative-sigmar","kind":"node","lean":"ArithmeticFunction.isMultiplicative_sigmaR","parent":4845,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/index.html#ismultiplicative-sigmar","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02","status":"fully_proved","summary":"For fixed $s$ function $n\\mapsto\\sigma^R_s(n)$ is multiplicative.","title":"isMultiplicative-sigmaR"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/sigmar-eq-prod-primefactors-sum-range-factorizat","kind":"node","lean":"ArithmeticFunction.sigmaR_eq_prod_primeFactors_sum_range_factorization_pow_mul","parent":4845,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/index.html#sigmar-eq-prod-primefactors-sum-range-factorizat","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02","status":"fully_proved","summary":"We have that","title":"sigmaR-eq-prod-primeFactors-sum-range-factorization-pow-mul"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/lseries-powr-eq","kind":"node","lean":"ArithmeticFunction.LSeries_powR_eq","parent":4845,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/index.html#lseries-powr-eq","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02","status":"fully_proved","summary":"$L(\\text{pow}^R(\\nu), s) = \\zeta(s - \\nu)$ for $\\Re(s - \\nu) > 1$.","title":"LSeries powR eq"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/abscissa-powr-le","kind":"node","lean":"ArithmeticFunction.abscissa_powR_le","parent":4845,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/index.html#abscissa-powr-le","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02","status":"fully_proved","summary":"The abscissa of absolute convergence of $L(\\text{pow}^R(\\nu), s)$ is at most $\\Re(\\nu) + 1$.","title":"abscissa powR le"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/lseries-sigma-eq-riemannzeta-mul","kind":"node","lean":"ArithmeticFunction.LSeries_sigma_eq_riemannZeta_mul","parent":4845,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/index.html#lseries-sigma-eq-riemannzeta-mul","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02","status":"fully_proved","summary":"$\\zeta(s)\\zeta(s - \\nu) = \\sum_{n=1}^{\\infty} \\sigma_\\nu(n) n^{-s}$ for $\\Re(s) > 1$ and $\\Re(s - \\nu) > 1$.","title":"LSeries sigma eq riemannZeta mul"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/zeta-mul-zeta-mul-zeta-mul-zeta-eq","kind":"node","lean":"ArithmeticFunction.zeta_mul_zeta_mul_zeta_mul_zeta_eq","parent":4845,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/index.html#zeta-mul-zeta-mul-zeta-mul-zeta-eq","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02","status":"can_prove","summary":"Ramanujan formula: $\\zeta(s)\\zeta(s-\\alpha)\\zeta(s-\\beta)\\zeta(s-\\alpha-\\beta)=\\zeta(2s-\\alpha-\\beta) \\sum_{n=1}^{\\infty} \\sigma_\\alpha(n)\\sigma_\\beta(n)n^{-s}$ for $\\Re(s) > 1$, $\\Re(s-\\alpha) > 1$, $\\Re(s-\\beta) > 1$, and $\\Re(s-\\alpha-\\beta) > 1$.","title":"zeta mul zeta mul zeta mul zeta eq"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/zeta-pow-four-eq","kind":"node","lean":"ArithmeticFunction.zeta_pow_four_eq","parent":4845,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/index.html#zeta-pow-four-eq","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02","status":"proved","summary":"Corollary: $\\zeta(s)^4 = \\zeta(2s) \\sum_{n=1}^{\\infty} \\tau(n)^2 n^{-s}$ for $\\Re(s) > 1$.","title":"zeta pow four eq"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/zeta-mul-tau-square-eq","kind":"node","lean":"ArithmeticFunction.zeta_mul_tau_square_eq","parent":4845,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02/index.html#zeta-mul-tau-square-eq","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-02","status":"fully_proved","summary":"Baby Rankin-Selberg: $\\zeta(s)\\sum_{n=1}^{\\infty}\\tau(n^2)n^{-s} = \\sum_{n=1}^{\\infty}\\tau(n)^2 n^{-s}$ for $\\Re(s) > 1$.","title":"zeta mul tau square eq"},{"data_url":"scopes/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03.json","id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03","kind":"scope","navigation":"transparent","parent":4823,"result":false,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/index.html","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03","status":"can_prove","title":"zeta pow three eq to moebius-sq-tsum-prime-pow"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/zeta-pow-three-eq","kind":"node","lean":"ArithmeticFunction.zeta_pow_three_eq'","parent":4866,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/index.html#zeta-pow-three-eq","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03","status":"proved","summary":"Zeta cubed: $\\zeta(s)^3 = \\zeta(2s) \\sum_{n=1}^{\\infty}\\tau(n^2) n^{-s}$.","title":"zeta pow three eq"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/zeta-pow-three-eq-alt","kind":"node","lean":"ArithmeticFunction.zeta_pow_three_eq_alt","parent":4866,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/index.html#zeta-pow-three-eq-alt","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03","status":"can_prove","summary":"symmetric square $L$-function for $\\zeta^2$:","title":"zeta pow three eq alt"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/two-pow-omega-le-sigma-zero","kind":"node","lean":"ArithmeticFunction.two_pow_omega_le_sigma_zero","parent":4866,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/index.html#two-pow-omega-le-sigma-zero","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03","status":"fully_proved","summary":"We have the inequality $2^{\\omega(n)}\\leq\\sigma_0(n)$ when $n\\neq 0$.","title":"two-pow-omega-le-sigma-zero"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/lseriessummable-two-pow-omega","kind":"node","lean":"ArithmeticFunction.LSeriesSummable.of_norm_le_norm, ArithmeticFunction.LSeriesSummable_two_pow_omega","parent":4866,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/index.html#lseriessummable-two-pow-omega","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03","status":"fully_proved","summary":"An L-series is convergent if the absolute value of each term is term wise less than a summable series.","title":"LSeriesSummable-two-pow-omega"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/lseries-term-ismultiplicative-if-fun-ismultiplic","kind":"node","lean":"ArithmeticFunction.LSeries.term_isMultiplicative_if_fun_isMultiplicative","parent":4866,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/index.html#lseries-term-ismultiplicative-if-fun-ismultiplic","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03","status":"fully_proved","summary":"If $f$ is a multiplicative function, then so to is $n\\mapsto f(n)/n^s$.","title":"LSeries.term-isMultiplicative-if-fun-isMultiplicative"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/powofadditive-ismultiplicative","kind":"node","lean":"ArithmeticFunction.powOfAdditive_isMultiplicative","parent":4866,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/index.html#powofadditive-ismultiplicative","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03","status":"fully_proved","summary":"If $f$ is an additive function, then so to is $n\\mapsto k^{f(n)}$ for any fixed $k$.","title":"powOfAdditive-isMultiplicative"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/two-pow-omega-ismultiplicative","kind":"node","lean":"ArithmeticFunction.two_pow_omega_isMultiplicative","parent":4866,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/index.html#two-pow-omega-ismultiplicative","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03","status":"fully_proved","summary":"The function $n\\mapsto 2^{\\omega(n)}$ is multiplicative.","title":"two-pow-omega-isMultiplicative"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/two-pow-omega-lseries-term-ismultiplicative","kind":"node","lean":"ArithmeticFunction.two_pow_omega_LSeries.term_isMultiplicative","parent":4866,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/index.html#two-pow-omega-lseries-term-ismultiplicative","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03","status":"fully_proved","summary":"We have that $n\\mapsto 2^{\\omega(n)}/n^{s}$ is a multiplicative function.","title":"two-pow-omega-LSeries.term-isMultiplicative"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/sumonprimepows","kind":"node","lean":"ArithmeticFunction.sumOnPrimePows","parent":4866,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/index.html#sumonprimepows","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03","status":"fully_proved","summary":"Shorthand for the sum of a function $f:\\mathbb{N}\\to\\mathbb{C}$ at prime powers for a prime $p$.","title":"sumOnPrimePows"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/sumonprimepows-apply","kind":"node","lean":"ArithmeticFunction.sumOnPrimePows_apply","parent":4866,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/index.html#sumonprimepows-apply","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03","status":"fully_proved","summary":"Helper lemma for sumOnPrimePows that rewrites the shorthand as its sum.","title":"sumOnPrimePows-apply"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/two-pow-omega-tsum-prime-pow","kind":"node","lean":"ArithmeticFunction.two_pow_omega_tsum_prime_pow","parent":4866,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/index.html#two-pow-omega-tsum-prime-pow","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03","status":"fully_proved","summary":"For $1\\lt \\Re(s)$ and $p$ prime, we have that","title":"two-pow-omega-tsum-prime-pow"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/complex-one-add-prime-cpow-ne-zero","kind":"node","lean":"ArithmeticFunction.Complex.one_add_prime_cpow_ne_zero","parent":4866,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/index.html#complex-one-add-prime-cpow-ne-zero","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03","status":"fully_proved","summary":"For $1\\lt \\Re(s)$ and $p$ prime, we have that $1+p^{-s}\\neq 0$. The naming convention is to mimic","title":"Complex.one-add-prime-cpow-ne-zero"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/two-pow-omega-lseries-eulerproduct-tprod","kind":"node","lean":"ArithmeticFunction.two_pow_omega_LSeries_eulerProduct_tprod","parent":4866,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/index.html#two-pow-omega-lseries-eulerproduct-tprod","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03","status":"fully_proved","summary":"For $1\\lt \\Re(s)$ we have that","title":"two-pow-omega-LSeries-eulerProduct-tprod"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/two-pow-omega-lseries-eulerproduct-hasprod","kind":"node","lean":"ArithmeticFunction.two_pow_omega_LSeries_eulerProduct_hasProd","parent":4866,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/index.html#two-pow-omega-lseries-eulerproduct-hasprod","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03","status":"fully_proved","summary":"For $1\\lt \\Re(s)$ we have that","title":"two-pow-omega-LSeries-eulerProduct-hasProd"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/zeta-pow-two","kind":"node","lean":"ArithmeticFunction.zeta_pow_two","parent":4866,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/index.html#zeta-pow-two","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03","status":"fully_proved","summary":"\\zeta(s)^2 =\\zeta(2s) \\sum_{n=1}^{\\infty} 2^{\\omega(n)} n^{-s}","title":"zeta pow two"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/zeta-pow-three-eq-2","kind":"node","lean":"ArithmeticFunction.zeta_pow_three_eq","parent":4866,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/index.html#zeta-pow-three-eq-2","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03","status":"fully_proved","summary":"Zeta cubed: $\\zeta(s)^3 = \\zeta(2s) \\sum_{n=1}^{\\infty}\\tau(n^2) n^{-s}$ for $\\Re(s) > 1$.","title":"zeta pow three eq"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/lseriessummable-moebius-sq","kind":"node","lean":"ArithmeticFunction.LSeriesSummable_moebius_sq","parent":4866,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/index.html#lseriessummable-moebius-sq","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03","status":"fully_proved","summary":"The $L$-series with coefficients given by $\\mu^2(n)$ converges on the region $1\\lt \\Re(s)$.","title":"LSeriesSummable-moebius-sq"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/powofmultiplicative-ismultiplicative","kind":"node","lean":"ArithmeticFunction.powOfMultiplicative_isMultiplicative","parent":4866,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/index.html#powofmultiplicative-ismultiplicative","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03","status":"fully_proved","summary":"If $f$ is a multiplicative function, then so to is $n\\mapsto f(n)^k$ for any $k\\in\\mathbb{N}$.","title":"powOfMultiplicative-isMultiplicative"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/moebius-sq-lseries-term-ismultiplicative","kind":"node","lean":"ArithmeticFunction.moebius_sq_LSeries.term_isMultiplicative","parent":4866,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/index.html#moebius-sq-lseries-term-ismultiplicative","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03","status":"fully_proved","summary":"We have that $n\\mapsto \\mu^2(n)/n^{s}$ is a multiplicative function.","title":"moebius-sq-LSeries.term-isMultiplicative"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/moebius-sq-tsum-prime-pow","kind":"node","lean":"ArithmeticFunction.moebius_sq_tsum_prime_pow","parent":4866,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03/index.html#moebius-sq-tsum-prime-pow","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-03","status":"fully_proved","summary":"For $p$ prime, we have that","title":"moebius-sq-tsum-prime-pow"},{"data_url":"scopes/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04.json","id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04","kind":"scope","navigation":"transparent","parent":4823,"result":false,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/index.html","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04","status":"can_prove","title":"moebius-sq-LSeries-eulerProduct-\u2026 to LSeries totient eq"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/moebius-sq-lseries-eulerproduct-tprod","kind":"node","lean":"ArithmeticFunction.moebius_sq_LSeries_eulerProduct_tprod","parent":4887,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/index.html#moebius-sq-lseries-eulerproduct-tprod","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04","status":"fully_proved","summary":"For $1\\lt \\Re(s)$ we have that","title":"moebius-sq-LSeries-eulerProduct-tprod"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/moebius-sq-lseries-eulerproduct-hasprod","kind":"node","lean":"ArithmeticFunction.moebius_sq_LSeries_eulerProduct_hasProd","parent":4887,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/index.html#moebius-sq-lseries-eulerproduct-hasprod","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04","status":"fully_proved","summary":"For $1\\lt \\Re(s)$ we have that","title":"moebius-sq-LSeries-eulerProduct-hasProd"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/zeta-alt","kind":"node","lean":"ArithmeticFunction.zeta_alt","parent":4887,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/index.html#zeta-alt","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04","status":"fully_proved","summary":"\\zeta(s) =\\zeta(2s) \\sum_{n=1}^{\\infty}\\mu^2(n)n^{-s}.","title":"zeta alt"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/pow-divisors-mul","kind":"node","lean":"ArithmeticFunction.pow_divisors_mul","parent":4887,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/index.html#pow-divisors-mul","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04","status":"fully_proved","summary":"Let $m$ and $n$ be coprime natural numbers and let $k$ be fixed. The divisors $x$ of $mn$ with $x^k \\mid mn$ are exactly the products $ab$ where $a \\mid m$, $a^k \\mid m$, $b \\mid n$, and $b^k \\mid n$.","title":"pow-divisors-mul"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/divisors-mul-injective","kind":"node","lean":"ArithmeticFunction.divisors_mul_injective","parent":4887,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/index.html#divisors-mul-injective","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04","status":"can_prove","summary":"Let $m$ and $n$ be coprime natural numbers. The function $(a,b) \\mapsto ab$ is injective on the product of the divisors of $m$ and $n$.","title":"divisors-mul-injective"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/pow-divisors-mul-injective","kind":"node","lean":"ArithmeticFunction.pow_divisors_mul_injective","parent":4887,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/index.html#pow-divisors-mul-injective","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04","status":"proved","summary":"Let $m$ and $n$ be coprime natural numbers and let $k$ be fixed. The map $(a,b) \\mapsto ab$ is injective on the product of $\\{a \\mid m : a^k \\mid m\\}$ and $\\{b \\mid n : b^k \\mid n\\}$.","title":"pow-divisors-mul-injective"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/sum-moebius-sq-divisors","kind":"node","lean":"ArithmeticFunction.sum_moebius_sq_divisors","parent":4887,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/index.html#sum-moebius-sq-divisors","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04","status":"fully_proved","summary":"The function $n \\mapsto \\sum_{d^2|n} \\mu(d)$.","title":"sum-moebius-sq-divisors"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/sum-moebius-sq-divisors-apply","kind":"node","lean":"ArithmeticFunction.sum_moebius_sq_divisors_apply","parent":4887,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/index.html#sum-moebius-sq-divisors-apply","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04","status":"fully_proved","summary":"A simple helper lemma for the above definition.","title":"sum-moebius-sq-divisors-apply"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/sum-moebius-sq-divisors-ismultiplicative","kind":"node","lean":"ArithmeticFunction.sum_moebius_sq_divisors_IsMultiplicative","parent":4887,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/index.html#sum-moebius-sq-divisors-ismultiplicative","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04","status":"proved","summary":"The function $n \\mapsto \\sum_{d^2|n} \\mu(d)$ is multiplicative.","title":"sum-moebius-sq-divisors-is-multiplicative"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/sum-moebius-sq-divisors-apply-prime-pow","kind":"node","lean":"ArithmeticFunction.sum_moebius_sq_divisors_apply_prime_pow","parent":4887,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/index.html#sum-moebius-sq-divisors-apply-prime-pow","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04","status":"fully_proved","summary":"Applied at prime powers, sum-moebius-sq-divisors coincides with $\\mu^2$.","title":"sum-moebius-sq-divisors-apply-prime-pow"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/moebius-sq-eq","kind":"node","lean":"ArithmeticFunction.moebius_sq_eq","parent":4887,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/index.html#moebius-sq-eq","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04","status":"proved","summary":"I-K (1.33): $\\mu^2(n) = \\sum_{d^2|n} \\mu(d)$.","title":"moebius-sq-eq"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/liouville","kind":"node","lean":"ArithmeticFunction.liouville","parent":4887,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/index.html#liouville","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04","status":"fully_proved","summary":"Liouville function: $\\lambda(n) = (-1)^{\\Omega(n)}$.","title":"liouville"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/iscompletelymultiplicative","kind":"node","lean":"ArithmeticFunction.IsCompletelyMultiplicative","parent":4887,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/index.html#iscompletelymultiplicative","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04","status":"fully_proved","summary":"Define Complete Multiplicativity for an arithmetic function.","title":"IsCompletelyMultiplicative"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/iscompletelymultiplicative-ismultiplicative","kind":"node","lean":"ArithmeticFunction.IsCompletelyMultiplicative.isMultiplicative","parent":4887,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/index.html#iscompletelymultiplicative-ismultiplicative","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04","status":"fully_proved","summary":"A function that is completely multiplicative is also multiplicative.","title":"IsCompletelyMultiplicative.isMultiplicative"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/iscompletelymultiplicative-liouville","kind":"node","lean":"ArithmeticFunction.isCompletelyMultiplicative_liouville","parent":4887,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/index.html#iscompletelymultiplicative-liouville","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04","status":"can_prove","summary":"The Liouville function is completely multiplicative.","title":"isCompletelyMultiplicative liouville"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/lseries-liouville-eq","kind":"node","lean":"ArithmeticFunction.LSeries_liouville_eq","parent":4887,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/index.html#lseries-liouville-eq","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04","status":"can_prove","summary":"The Dirichlet series of the Liouville function is $\\zeta(2s)/\\zeta(s)$.","title":"LSeries liouville eq"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/liouville-eq-moebius-on-squarefree","kind":"node","lean":"ArithmeticFunction.liouville_eq_moebius_on_squarefree","parent":4887,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/index.html#liouville-eq-moebius-on-squarefree","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04","status":"fully_proved","summary":"The Liouville function agrees with the M\u00f6bius function on square-free numbers.","title":"liouville eq moebius on squarefree"},{"id":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/lseries-totient-eq","kind":"node","lean":"ArithmeticFunction.LSeries_totient_eq","parent":4887,"result":true,"route":"../roadmap/pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04/index.html#lseries-totient-eq","scope":"pnt-plus/iwaniec-kowalski/blueprint-for-iwaniec-kowalski-chapter-1/part-04","status":"fully_proved","summary":"Euler totient series: $\\sum_{n=1}^{\\infty} \\varphi(n) n^{-s} = \\zeta(s-1)/\\zeta(s)$ for $\\Re(s) > 2$.","title":"LSeries totient eq"},{"id":"pnt-plus/bombieri-vinogradov","kind":"node","parent":4477,"result":false,"route":"../roadmap/pnt-plus/bombieri-vinogradov/index.html","scope":"pnt-plus","status":"can_state","summary":"A formalization of Bombieri-Vinogradov (modulo Siegel-Walfisz) is under way at","title":"Bombieri-Vinogradov"},{"data_url":"chapters/pfr.json","id":"pfr","kind":"scope","parent":0,"result":false,"route":"../roadmap/pfr/index.html","scope":"pfr","status":"fully_proved","summary":"The blueprint of Polynomial Freiman-Ruzsa, read at commit 1ef23c3c (2026-11-05): one article for each of its 213 nodes whose Lean declarations it names and marks formalized, or in Mathlib. Its other 6 nodes have no article, and a section of only those is left out: the map records what the project contains, not what it plans. A statement for which the blueprint marks no proof `\\leanok` says so. Dependencies are the\u2026","title":"Polynomial Freiman-Ruzsa"},{"data_url":"scopes/pfr/applications-of-jensens-inequality.json","id":"pfr/applications-of-jensens-inequality","kind":"scope","parent":4907,"result":false,"route":"../roadmap/pfr/applications-of-jensens-inequality/index.html","scope":"pfr/applications-of-jensens-inequality","status":"fully_proved","summary":"In this chapter, $h$ denotes the function $h(x) := x \\log \\frac{1}{x}$ for $x \\in [0,1]$.","title":"Applications of Jensen's inequality"},{"id":"pfr/applications-of-jensens-inequality/concave","kind":"node","lean":"Real.strictConcaveOn_negMulLog","parent":4908,"result":true,"route":"../roadmap/pfr/applications-of-jensens-inequality/index.html#concave","scope":"pfr/applications-of-jensens-inequality","status":"fully_proved","summary":"$h$ is strictly concave on $[0,\\infty)$.","title":"Concavity"},{"id":"pfr/applications-of-jensens-inequality/log-sum","kind":"node","lean":"Real.sum_mul_log_div_leq","parent":4908,"result":true,"route":"../roadmap/pfr/applications-of-jensens-inequality/index.html#log-sum","scope":"pfr/applications-of-jensens-inequality","status":"fully_proved","summary":"If $S$ is a finite set, and $a_s,b_s$ are non-negative for $s\\in S$, then","title":"log sum inequality"},{"id":"pfr/applications-of-jensens-inequality/converse-log-sum","kind":"node","lean":"Real.sum_mul_log_div_eq_iff","parent":4908,"result":true,"route":"../roadmap/pfr/applications-of-jensens-inequality/index.html#converse-log-sum","scope":"pfr/applications-of-jensens-inequality","status":"fully_proved","summary":"If equality holds in log sum inequality, then $a_s=r\\cdot b_s$ for every $s\\in S$, for some constant $r\\in \\mathbb{R}$.","title":"converse log sum"},{"data_url":"scopes/pfr/shannon-entropy-inequalities.json","id":"pfr/shannon-entropy-inequalities","kind":"scope","parent":4907,"result":false,"route":"../roadmap/pfr/shannon-entropy-inequalities/index.html","scope":"pfr/shannon-entropy-inequalities","status":"fully_proved","summary":"Random variables in this paper are measurable maps $X : \\Omega \\to S$ from a probability space $\\Omega$ to a measurable space $S$, and called $S$-valued random variables. In many cases we will assume that singletons in $S$ are measurable. Often we will restrict further to the case when $S$ is finite with the discrete $\\sigma$-algebra, which of course implies that $S$ has measurable singletons.","title":"Shannon entropy inequalities"},{"data_url":"scopes/pfr/shannon-entropy-inequalities/part-01.json","id":"pfr/shannon-entropy-inequalities/part-01","kind":"scope","navigation":"transparent","parent":4912,"result":false,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-01/index.html","scope":"pfr/shannon-entropy-inequalities/part-01","status":"fully_proved","summary":"We use $X,Y$ to denote the pair $\\omega \\mapsto (X(\\omega),Y(\\omega)$).","title":"Entropy to Submodularity"},{"id":"pfr/shannon-entropy-inequalities/part-01/entropy-def","kind":"node","lean":"ProbabilityTheory.entropy","parent":4913,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-01/index.html#entropy-def","scope":"pfr/shannon-entropy-inequalities/part-01","status":"fully_proved","summary":"If $X$ is an $S$-valued random variable, the entropy $\\mathbb{H}[X]$ of $X$ is defined","title":"Entropy"},{"id":"pfr/shannon-entropy-inequalities/part-01/relabeled-entropy","kind":"node","lean":"ProbabilityTheory.entropy_comp_of_injective, ProbabilityTheory.entropy_of_comp_eq_of_comp","parent":4913,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-01/index.html#relabeled-entropy","scope":"pfr/shannon-entropy-inequalities/part-01","status":"fully_proved","summary":"Expand out both entropies and rearrange.","title":"Entropy and relabeling"},{"id":"pfr/shannon-entropy-inequalities/part-01/jensen-bound","kind":"node","lean":"ProbabilityTheory.entropy_le_log_card, ProbabilityTheory.entropy_le_log_card_of_mem","parent":4913,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-01/index.html#jensen-bound","scope":"pfr/shannon-entropy-inequalities/part-01","status":"fully_proved","summary":"If $X$ is an $S$-valued random variable, then $\\mathbb{H}[X] \\leq \\log |S|$.","title":"Jensen bound"},{"id":"pfr/shannon-entropy-inequalities/part-01/uniform-def","kind":"node","lean":"ProbabilityTheory.IsUniform","parent":4913,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-01/index.html#uniform-def","scope":"pfr/shannon-entropy-inequalities/part-01","status":"fully_proved","summary":"If $H$ is a subset of $S$, an $S$-random variable $X$ is said to be uniformly distributed on $H$ if $\\mathbb{P}[X = s] = 1/|H|$ for $s \\in X$ and $\\mathbb{P}[X=s] = 0$ otherwise.","title":"Uniform distribution"},{"id":"pfr/shannon-entropy-inequalities/part-01/unif-exist","kind":"node","lean":"ProbabilityTheory.exists_isUniform, ProbabilityTheory.exists_isUniform_measureSpace","parent":4913,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-01/index.html#unif-exist","scope":"pfr/shannon-entropy-inequalities/part-01","status":"fully_proved","summary":"Given a finite non-empty subset $H$ of a set $S$, there exists a random variable $X$ (on some probability space) that is uniformly distributed on $H$.","title":"Uniform distributions exist"},{"id":"pfr/shannon-entropy-inequalities/part-01/uniform-entropy","kind":"node","lean":"ProbabilityTheory.IsUniform.entropy_eq","parent":4913,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-01/index.html#uniform-entropy","scope":"pfr/shannon-entropy-inequalities/part-01","status":"fully_proved","summary":"If $X$ is $S$-valued random variable, then $\\mathbb{H}[X] = \\log |S|$ if and only if $X$ is uniformly distributed on $S$.","title":"Entropy of uniform random variable"},{"id":"pfr/shannon-entropy-inequalities/part-01/uniform-entropy-ii","kind":"node","lean":"ProbabilityTheory.IsUniform.entropy_eq","parent":4913,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-01/index.html#uniform-entropy-ii","scope":"pfr/shannon-entropy-inequalities/part-01","status":"fully_proved","summary":"If $X$ is uniformly distributed on $H$, then, then $\\mathbb{H}[X] = \\log |H|$.","title":"Entropy of uniform random variable, II"},{"id":"pfr/shannon-entropy-inequalities/part-01/bound-conc","kind":"node","lean":"ProbabilityTheory.prob_ge_exp_neg_entropy","parent":4913,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-01/index.html#bound-conc","scope":"pfr/shannon-entropy-inequalities/part-01","status":"fully_proved","summary":"If $X$ is an $S$-valued random variable, then there exists $s \\in S$ such that $\\mathbb{P}[X=s] \\geq \\exp(-\\mathbb{H}[X])$.","title":"Bounded entropy implies concentration"},{"id":"pfr/shannon-entropy-inequalities/part-01/entropy-comm","kind":"node","lean":"ProbabilityTheory.entropy_comm, ProbabilityTheory.entropy_assoc","parent":4913,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-01/index.html#entropy-comm","scope":"pfr/shannon-entropy-inequalities/part-01","status":"fully_proved","summary":"If $X: \\Omega \\to S$, $Y: \\Omega \\to T$, and $Z: \\Omega \\to U$ are random variables, then $\\mathbb{H}[X, Y] = \\mathbb{H}[Y, X]$ and $\\mathbb{H}[X, (Y,Z)] = \\mathbb{H}[(X,Y), Z]$.","title":"Commutativity and associativity of joint entropy"},{"id":"pfr/shannon-entropy-inequalities/part-01/condition-event-def","kind":"node","lean":"ProbabilityTheory.cond","parent":4913,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-01/index.html#condition-event-def","scope":"pfr/shannon-entropy-inequalities/part-01","status":"fully_proved","summary":"If $X: \\Omega \\to S$ is an $S$-valued random variable and $E$ is an event in $\\Omega$, then the conditioned event $(X|E)$ is defined to be the same random variable as $X$, but now the ambient probability measure has been conditioned to $E$.","title":"Conditioned event"},{"id":"pfr/shannon-entropy-inequalities/part-01/conditional-entropy-def","kind":"node","lean":"ProbabilityTheory.condEntropy","parent":4913,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-01/index.html#conditional-entropy-def","scope":"pfr/shannon-entropy-inequalities/part-01","status":"fully_proved","summary":"If $X: \\Omega \\to S$ and $Y: \\Omega \\to T$ are random variables, the conditional entropy $\\mathbb{H}[X|Y]$ is defined as","title":"Conditional entropy"},{"id":"pfr/shannon-entropy-inequalities/part-01/relabeled-entropy-cond","kind":"node","lean":"ProbabilityTheory.condEntropy_comp_of_injective, ProbabilityTheory.condEntropy_of_injective'","parent":4913,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-01/index.html#relabeled-entropy-cond","scope":"pfr/shannon-entropy-inequalities/part-01","status":"fully_proved","summary":"If $X: \\Omega \\to S$, $Y: \\Omega \\to T$, and $Z: \\Omega \\to U$ are random variables, and $Y = f(X,Z)$ almost surely for some map $f: S \\times U \\to T$ that is injective for each fixed $U$, then $\\mathbb{H}[X|Z] = \\mathbb{H}[Y|Z]$.","title":"Conditional entropy and relabeling"},{"id":"pfr/shannon-entropy-inequalities/part-01/chain-rule","kind":"node","lean":"ProbabilityTheory.chain_rule, ProbabilityTheory.chain_rule'","parent":4913,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-01/index.html#chain-rule","scope":"pfr/shannon-entropy-inequalities/part-01","status":"fully_proved","summary":"If $X: \\Omega \\to S$ and $Y: \\Omega \\to T$ are random variables, then","title":"Chain rule"},{"id":"pfr/shannon-entropy-inequalities/part-01/conditional-chain-rule","kind":"node","lean":"ProbabilityTheory.cond_chain_rule, ProbabilityTheory.cond_chain_rule'","parent":4913,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-01/index.html#conditional-chain-rule","scope":"pfr/shannon-entropy-inequalities/part-01","status":"fully_proved","summary":"If $X: \\Omega \\to S$, $Y: \\Omega \\to T$, $Z: \\Omega \\to U$ are random variables, then","title":"Conditional chain rule"},{"id":"pfr/shannon-entropy-inequalities/part-01/information-def","kind":"node","lean":"ProbabilityTheory.mutualInfo, ProbabilityTheory.mutualInfo_def","parent":4913,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-01/index.html#information-def","scope":"pfr/shannon-entropy-inequalities/part-01","status":"fully_proved","summary":"If $X: \\Omega \\to S$, $Y: \\Omega \\to T$ are random variables, then","title":"Mutual information"},{"id":"pfr/shannon-entropy-inequalities/part-01/alternative-mutual","kind":"node","lean":"ProbabilityTheory.mutualInfo_comm, ProbabilityTheory.mutualInfo_eq_entropy_sub_condEntropy","parent":4913,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-01/index.html#alternative-mutual","scope":"pfr/shannon-entropy-inequalities/part-01","status":"fully_proved","summary":"With notation as above, we have","title":"Alternative formulae for mutual information"},{"id":"pfr/shannon-entropy-inequalities/part-01/mutual-nonneg","kind":"node","lean":"ProbabilityTheory.mutualInfo_nonneg","parent":4913,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-01/index.html#mutual-nonneg","scope":"pfr/shannon-entropy-inequalities/part-01","status":"fully_proved","summary":"We have $\\mathbb{I}[X:Y] \\geq 0$.","title":"Nonnegativity of mutual information"},{"id":"pfr/shannon-entropy-inequalities/part-01/subadditive","kind":"node","lean":"ProbabilityTheory.entropy_pair_le_add","parent":4913,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-01/index.html#subadditive","scope":"pfr/shannon-entropy-inequalities/part-01","status":"fully_proved","summary":"With notation as above, we have $\\mathbb{H}[X,Y] \\leq \\mathbb{H}[X] + \\mathbb{H}[Y]$.","title":"Subadditivity"},{"id":"pfr/shannon-entropy-inequalities/part-01/cond-reduce","kind":"node","lean":"ProbabilityTheory.condEntropy_le_entropy","parent":4913,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-01/index.html#cond-reduce","scope":"pfr/shannon-entropy-inequalities/part-01","status":"fully_proved","summary":"With notation as above, we have $\\mathbb{H}[X|Y] \\leq \\mathbb{H}[X]$.","title":"Conditioning reduces entropy"},{"id":"pfr/shannon-entropy-inequalities/part-01/submodularity","kind":"node","lean":"ProbabilityTheory.entropy_submodular, ProbabilityTheory.condEntropy_comp_ge","parent":4913,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-01/index.html#submodularity","scope":"pfr/shannon-entropy-inequalities/part-01","status":"fully_proved","summary":"With three random variables $X,Y,Z$, one has $\\mathbb{H}[X|Y,Z] \\leq \\mathbb{H}[X|Z]$.","title":"Submodularity"},{"data_url":"scopes/pfr/shannon-entropy-inequalities/part-02.json","id":"pfr/shannon-entropy-inequalities/part-02","kind":"scope","navigation":"transparent","parent":4912,"result":false,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-02/index.html","scope":"pfr/shannon-entropy-inequalities/part-02","status":"fully_proved","title":"Alternate form of submodularity to Entropy of conditionally\u2026"},{"id":"pfr/shannon-entropy-inequalities/part-02/alt-submodularity","kind":"node","lean":"ProbabilityTheory.entropy_triple_add_entropy_le","parent":4934,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-02/index.html#alt-submodularity","scope":"pfr/shannon-entropy-inequalities/part-02","status":"fully_proved","summary":"With three random variables $X,Y,Z$, one has","title":"Alternate form of submodularity"},{"id":"pfr/shannon-entropy-inequalities/part-02/independent-def","kind":"node","lean":"ProbabilityTheory.IndepFun","parent":4934,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-02/index.html#independent-def","scope":"pfr/shannon-entropy-inequalities/part-02","status":"fully_proved","summary":"Two random variables $X: \\Omega \\to S$ and $Y: \\Omega \\to T$ are independent if the law of $(X,Y)$ is the product of the law of $X$ and the law of $Y$. Similarly for more than two variables.","title":"Independent random variables"},{"id":"pfr/shannon-entropy-inequalities/part-02/vanish-entropy","kind":"node","lean":"ProbabilityTheory.mutualInfo_eq_zero","parent":4934,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-02/index.html#vanish-entropy","scope":"pfr/shannon-entropy-inequalities/part-02","status":"fully_proved","summary":"If $X,Y$ are random variables, then $\\mathbb{I}[X:Y] = 0$ if and only if $X,Y$ are independent.","title":"Vanishing of mutual information"},{"id":"pfr/shannon-entropy-inequalities/part-02/add-entropy","kind":"node","lean":"ProbabilityTheory.entropy_pair_eq_add","parent":4934,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-02/index.html#add-entropy","scope":"pfr/shannon-entropy-inequalities/part-02","status":"fully_proved","summary":"If $X,Y$ are random variables, then $\\mathbb{H}[X,Y] = \\mathbb{H}[X] + \\mathbb{H}[Y]$ if and only if $X,Y$ are independent.","title":"Additivity of entropy"},{"id":"pfr/shannon-entropy-inequalities/part-02/conditional-mutual-def","kind":"node","lean":"ProbabilityTheory.condMutualInfo","parent":4934,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-02/index.html#conditional-mutual-def","scope":"pfr/shannon-entropy-inequalities/part-02","status":"fully_proved","summary":"If $X,Y,Z$ are random variables, with $Z$ $U$-valued, then","title":"Conditional mutual information"},{"id":"pfr/shannon-entropy-inequalities/part-02/conditional-mutual-alt","kind":"node","lean":"ProbabilityTheory.condMutualInfo_eq, ProbabilityTheory.condMutualInfo_eq'","parent":4934,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-02/index.html#conditional-mutual-alt","scope":"pfr/shannon-entropy-inequalities/part-02","status":"fully_proved","summary":"We have","title":"Alternate formula for conditional mutual information"},{"id":"pfr/shannon-entropy-inequalities/part-02/conditional-nonneg","kind":"node","lean":"ProbabilityTheory.condMutualInfo_nonneg","parent":4934,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-02/index.html#conditional-nonneg","scope":"pfr/shannon-entropy-inequalities/part-02","status":"fully_proved","summary":"If $X,Y,Z$ are random variables, then $\\mathbb{I}[X:Y|Z] \\ge 0$.","title":"Nonnegativity of conditional mutual information"},{"id":"pfr/shannon-entropy-inequalities/part-02/conditional-independent-def","kind":"node","lean":"ProbabilityTheory.CondIndepFun","parent":4934,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-02/index.html#conditional-independent-def","scope":"pfr/shannon-entropy-inequalities/part-02","status":"fully_proved","summary":"Two random variables $X: \\Omega \\to S$ and $Y: \\Omega \\to T$ are conditionally independent relative to another random variable $Z: \\Omega \\to U$ if $P[X = s \\wedge Y = t| Z=u] = P[X=s|Z=u] P[Y=t|Z=u]$ for all $s \\in S, t \\in T, u \\in U$. (We won't need conditional independence for more variables than this.)","title":"Conditionally independent random variables"},{"id":"pfr/shannon-entropy-inequalities/part-02/conditional-vanish","kind":"node","lean":"ProbabilityTheory.condMutualInfo_eq_zero","parent":4934,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-02/index.html#conditional-vanish","scope":"pfr/shannon-entropy-inequalities/part-02","status":"fully_proved","summary":"If $X,Y,Z$ are random variables, then $\\mathbb{I}[X:Y|Z] = 0$ iff $X,Y$ are conditionally independent over $Z$.","title":"Vanishing conditional mutual information"},{"id":"pfr/shannon-entropy-inequalities/part-02/cond-trial-ent","kind":"node","lean":"ProbabilityTheory.ent_of_cond_indep","parent":4934,"result":true,"route":"../roadmap/pfr/shannon-entropy-inequalities/part-02/index.html#cond-trial-ent","scope":"pfr/shannon-entropy-inequalities/part-02","status":"fully_proved","summary":"If $X, Y$ are conditionally independent over $Z$, then","title":"Entropy of conditionally independent variables"},{"data_url":"scopes/pfr/entropic-ruzsa-calculus.json","id":"pfr/entropic-ruzsa-calculus","kind":"scope","parent":4907,"result":false,"route":"../roadmap/pfr/entropic-ruzsa-calculus/index.html","scope":"pfr/entropic-ruzsa-calculus","status":"fully_proved","summary":"In this section $G$ will be a finite additive group. (May eventually want to generalize to infinite $G$.)","title":"Entropic Ruzsa calculus"},{"data_url":"scopes/pfr/entropic-ruzsa-calculus/part-01.json","id":"pfr/entropic-ruzsa-calculus/part-01","kind":"scope","navigation":"transparent","parent":4945,"result":false,"route":"../roadmap/pfr/entropic-ruzsa-calculus/part-01/index.html","scope":"pfr/entropic-ruzsa-calculus/part-01","status":"fully_proved","summary":"One random variable is said to be a copy of another if they have the same distribution.","title":"Negation preserves entropy to Alternate form of distance"},{"id":"pfr/entropic-ruzsa-calculus/part-01/neg-ent","kind":"node","lean":"ProbabilityTheory.entropy_neg","parent":4946,"result":true,"route":"../roadmap/pfr/entropic-ruzsa-calculus/part-01/index.html#neg-ent","scope":"pfr/entropic-ruzsa-calculus/part-01","status":"fully_proved","summary":"If $X$ is $G$-valued, then $\\mathbb{H}[-X]=\\mathbb{H}[X]$.","title":"Negation preserves entropy"},{"id":"pfr/entropic-ruzsa-calculus/part-01/shear-ent","kind":"node","lean":"ProbabilityTheory.condEntropy_add_right, ProbabilityTheory.condEntropy_add_left, ProbabilityTheory.condEntropy_sub_right, ProbabilityTheory.condEntropy_sub_left","parent":4946,"result":true,"route":"../roadmap/pfr/entropic-ruzsa-calculus/part-01/index.html#shear-ent","scope":"pfr/entropic-ruzsa-calculus/part-01","status":"fully_proved","summary":"If $X,Y$ are $G$-valued, then $\\mathbb{H}[X \\pm Y | Y]=\\mathbb{H}[X|Y]$ and $\\mathbb{H}[X \\pm Y, Y] = \\mathbb{H}[X, Y]$.","title":"Shearing preserves entropy"},{"id":"pfr/entropic-ruzsa-calculus/part-01/sumset-lower-gen","kind":"node","lean":"ProbabilityTheory.max_entropy_sub_mutualInfo_le_entropy_add, ProbabilityTheory.max_entropy_sub_mutualInfo_le_entropy_sub","parent":4946,"result":true,"route":"../roadmap/pfr/entropic-ruzsa-calculus/part-01/index.html#sumset-lower-gen","scope":"pfr/entropic-ruzsa-calculus/part-01","status":"fully_proved","summary":"If $X,Y$ are $G$-valued random variables on $\\Omega$, we have","title":"Lower bound of sumset"},{"id":"pfr/entropic-ruzsa-calculus/part-01/sumset-lower-gen-cond","kind":"node","lean":"ProbabilityTheory.max_condEntropy_sub_condMutualInfo_le_condEntropy_mul, ProbabilityTheory.max_condEntropy_sub_condMutualInfo_le_condEntropy_div","parent":4946,"result":true,"route":"../roadmap/pfr/entropic-ruzsa-calculus/part-01/index.html#sumset-lower-gen-cond","scope":"pfr/entropic-ruzsa-calculus/part-01","status":"fully_proved","summary":"If $X,Y$ are $G$-valued random variables on $\\Omega$ and $Z$ is another random variable on $\\Omega$ then","title":"Conditional lower bound on sumset"},{"id":"pfr/entropic-ruzsa-calculus/part-01/sumset-lower","kind":"node","lean":"ProbabilityTheory.max_entropy_le_entropy_add, ProbabilityTheory.max_entropy_le_entropy_sub","parent":4946,"result":true,"route":"../roadmap/pfr/entropic-ruzsa-calculus/part-01/index.html#sumset-lower","scope":"pfr/entropic-ruzsa-calculus/part-01","status":"fully_proved","summary":"If $X,Y$ are independent $G$-valued random variables, then","title":"Independent lower bound on sumset"},{"id":"pfr/entropic-ruzsa-calculus/part-01/copy-ent","kind":"node","lean":"ProbabilityTheory.IdentDistrib.entropy_congr","parent":4946,"result":true,"route":"../roadmap/pfr/entropic-ruzsa-calculus/part-01/index.html#copy-ent","scope":"pfr/entropic-ruzsa-calculus/part-01","status":"fully_proved","summary":"If $X'$ is a copy of $X$ then $\\mathbb{H}[X'] = \\mathbb{H}[X]$.","title":"Copy preserves entropy"},{"id":"pfr/entropic-ruzsa-calculus/part-01/independent-exist","kind":"node","lean":"ProbabilityTheory.independent_copies, ProbabilityTheory.independent_copies', ProbabilityTheory.independent_copies_two, ProbabilityTheory.independent_copies3_nondep","parent":4946,"result":true,"route":"../roadmap/pfr/entropic-ruzsa-calculus/part-01/index.html#independent-exist","scope":"pfr/entropic-ruzsa-calculus/part-01","status":"fully_proved","summary":"Let $X_i : \\Omega_i \\to S_i$ be random variables for $i=1,\\dots,k$. Then if one gives $\\prod_{i=1}^k S_i$ the product measure of the laws of $X_i$, the coordinate functions $(x_j)_{j=1}^k \\mapsto x_i$ are jointly independent random variables which are copies of the $X_1,\\dots,X_k$.","title":"Existence of independent copies"},{"id":"pfr/entropic-ruzsa-calculus/part-01/ruz-dist-def","kind":"node","lean":"rdist_def","parent":4946,"result":true,"route":"../roadmap/pfr/entropic-ruzsa-calculus/part-01/index.html#ruz-dist-def","scope":"pfr/entropic-ruzsa-calculus/part-01","status":"fully_proved","summary":"Let $X,Y$ be $G$-valued random variables (not necessarily on the same sample space). The *Ruzsa distance* $d[X ;Y]$ between $X$ and $Y$ is defined to be","title":"Ruzsa distance"},{"id":"pfr/entropic-ruzsa-calculus/part-01/dist-zero","kind":"node","lean":"rdist_zero_eq_half_ent","parent":4946,"result":true,"route":"../roadmap/pfr/entropic-ruzsa-calculus/part-01/index.html#dist-zero","scope":"pfr/entropic-ruzsa-calculus/part-01","status":"fully_proved","summary":"If $X$ is a $G$-valued random variable and $0$ is the random variable taking the value $0$ everywhere then","title":"Distance from zero"},{"id":"pfr/entropic-ruzsa-calculus/part-01/ruz-copy","kind":"node","lean":"ProbabilityTheory.IdentDistrib.rdist_congr","parent":4946,"result":true,"route":"../roadmap/pfr/entropic-ruzsa-calculus/part-01/index.html#ruz-copy","scope":"pfr/entropic-ruzsa-calculus/part-01","status":"fully_proved","summary":"If $X',Y'$ are copies of $X,Y$ respectively then $d[X';Y']=d[X ;Y]$.","title":"Copy preserves Ruzsa distance"},{"id":"pfr/entropic-ruzsa-calculus/part-01/ruz-indep","kind":"node","lean":"ProbabilityTheory.IndepFun.rdist_eq","parent":4946,"result":true,"route":"../roadmap/pfr/entropic-ruzsa-calculus/part-01/index.html#ruz-indep","scope":"pfr/entropic-ruzsa-calculus/part-01","status":"fully_proved","summary":"If $X,Y$ are independent $G$-random variables then","title":"Ruzsa distance in independent case"},{"id":"pfr/entropic-ruzsa-calculus/part-01/ruzsa-symm","kind":"node","lean":"rdist_symm","parent":4946,"result":true,"route":"../roadmap/pfr/entropic-ruzsa-calculus/part-01/index.html#ruzsa-symm","scope":"pfr/entropic-ruzsa-calculus/part-01","status":"fully_proved","summary":"If $X,Y$ are $G$-valued random variables, then","title":"Distance symmetric"},{"id":"pfr/entropic-ruzsa-calculus/part-01/ruzsa-diff","kind":"node","lean":"diff_ent_le_rdist","parent":4946,"result":true,"route":"../roadmap/pfr/entropic-ruzsa-calculus/part-01/index.html#ruzsa-diff","scope":"pfr/entropic-ruzsa-calculus/part-01","status":"fully_proved","summary":"If $X,Y$ are $G$-valued random variables, then","title":"Distance controls entropy difference"},{"id":"pfr/entropic-ruzsa-calculus/part-01/ruzsa-growth","kind":"node","lean":"diff_ent_le_rdist', diff_ent_le_rdist''","parent":4946,"result":true,"route":"../roadmap/pfr/entropic-ruzsa-calculus/part-01/index.html#ruzsa-growth","scope":"pfr/entropic-ruzsa-calculus/part-01","status":"fully_proved","summary":"If $X,Y$ are independent $G$-valued random variables, then","title":"Distance controls entropy growth"},{"id":"pfr/entropic-ruzsa-calculus/part-01/ruzsa-nonneg","kind":"node","lean":"rdist_nonneg","parent":4946,"result":true,"route":"../roadmap/pfr/entropic-ruzsa-calculus/part-01/index.html#ruzsa-nonneg","scope":"pfr/entropic-ruzsa-calculus/part-01","status":"fully_proved","summary":"If $X,Y$ are $G$-valued random variables, then","title":"Distance nonnegative"},{"id":"pfr/entropic-ruzsa-calculus/part-01/dist-projection","kind":"node","lean":"ent_of_proj_le","parent":4946,"result":true,"route":"../roadmap/pfr/entropic-ruzsa-calculus/part-01/index.html#dist-projection","scope":"pfr/entropic-ruzsa-calculus/part-01","status":"fully_proved","summary":"If $G$ is an additive group and $X$ is a $G$-valued random variable and $H\\leq G$ is a finite subgroup then, with $\\pi:G\\to G/H$ the natural homomorphism we have (where $U_H$ is uniform on $H$)","title":"Projection entropy and distance"},{"id":"pfr/entropic-ruzsa-calculus/part-01/ruzsa-triangle-improved","kind":"node","lean":"ent_of_diff_le","parent":4946,"result":true,"route":"../roadmap/pfr/entropic-ruzsa-calculus/part-01/index.html#ruzsa-triangle-improved","scope":"pfr/entropic-ruzsa-calculus/part-01","status":"fully_proved","summary":"If $X,Y,Z$ are $G$-valued random variables on $\\Omega$ with $(X,Y)$ independent of $Z$, then","title":"Improved Ruzsa triangle inequality"},{"id":"pfr/entropic-ruzsa-calculus/part-01/ruzsa-triangle","kind":"node","lean":"rdist_triangle","parent":4946,"result":true,"route":"../roadmap/pfr/entropic-ruzsa-calculus/part-01/index.html#ruzsa-triangle","scope":"pfr/entropic-ruzsa-calculus/part-01","status":"fully_proved","summary":"If $X,Y,Z$ are $G$-valued random variables, then","title":"Ruzsa triangle inequality"},{"id":"pfr/entropic-ruzsa-calculus/part-01/cond-dist-def","kind":"node","lean":"condRuzsaDist","parent":4946,"result":true,"route":"../roadmap/pfr/entropic-ruzsa-calculus/part-01/index.html#cond-dist-def","scope":"pfr/entropic-ruzsa-calculus/part-01","status":"fully_proved","summary":"If $(X, Z)$ and $(Y, W)$ are random variables (where $X$ and $Y$ are $G$-valued) we define","title":"Conditioned Ruzsa distance"},{"id":"pfr/entropic-ruzsa-calculus/part-01/cond-dist-alt","kind":"node","lean":"condRuzsaDist_of_copy, condRuzsaDist'_of_copy, condRuzsaDist_of_indep, condRuzsaDist'_of_indep","parent":4946,"result":true,"route":"../roadmap/pfr/entropic-ruzsa-calculus/part-01/index.html#cond-dist-alt","scope":"pfr/entropic-ruzsa-calculus/part-01","status":"fully_proved","summary":"The expression $d[X | Z;Y | W]$ is unchanged if $(X,Z)$ or $(Y,W)$ is replaced by a copy. Furthermore, if $(X,Z)$ and $(Y,W)$ are independent, then","title":"Alternate form of distance"},{"data_url":"scopes/pfr/entropic-ruzsa-calculus/part-02.json","id":"pfr/entropic-ruzsa-calculus/part-02","kind":"scope","navigation":"transparent","parent":4945,"result":false,"route":"../roadmap/pfr/entropic-ruzsa-calculus/part-02/index.html","scope":"pfr/entropic-ruzsa-calculus/part-02","status":"fully_proved","title":"Kaimanovich-Vershik-Madiman\u2026 to Comparison of Ruzsa distances\u2026"},{"id":"pfr/entropic-ruzsa-calculus/part-02/kv","kind":"node","lean":"kaimanovich_vershik","parent":4967,"result":true,"route":"../roadmap/pfr/entropic-ruzsa-calculus/part-02/index.html#kv","scope":"pfr/entropic-ruzsa-calculus/part-02","status":"fully_proved","summary":"Suppose that $X, Y, Z$ are independent $G$-valued random variables. Then","title":"Kaimanovich-Vershik-Madiman inequality"},{"id":"pfr/entropic-ruzsa-calculus/part-02/cond-indep-exist","kind":"node","lean":"ProbabilityTheory.condIndep_copies","parent":4967,"result":true,"route":"../roadmap/pfr/entropic-ruzsa-calculus/part-02/index.html#cond-indep-exist","scope":"pfr/entropic-ruzsa-calculus/part-02","status":"fully_proved","summary":"For $X,Y$ random variables, there exist random variables $X_1,X_2,Y'$ on a common probability space with $(X_1, Y'), (X_2, Y')$ both having the distribution of $(X,Y)$, and $X_1, X_2$ conditionally independent over $Y'$ in the sense of Conditionally independent random variables.","title":"Existence of conditional independent trials"},{"id":"pfr/entropic-ruzsa-calculus/part-02/entropic-bsg","kind":"node","lean":"ent_bsg","parent":4967,"result":true,"route":"../roadmap/pfr/entropic-ruzsa-calculus/part-02/index.html#entropic-bsg","scope":"pfr/entropic-ruzsa-calculus/part-02","status":"fully_proved","summary":"Let $A,B$ be $G$-valued random variables on $\\Omega$, and set $Z := A+B$. Then","title":"Balog-Szemer\u00e9di-Gowers"},{"id":"pfr/entropic-ruzsa-calculus/part-02/cond-dist-fact","kind":"node","lean":"condRuzsaDist_le, condRuzsaDist_le'","parent":4967,"result":true,"route":"../roadmap/pfr/entropic-ruzsa-calculus/part-02/index.html#cond-dist-fact","scope":"pfr/entropic-ruzsa-calculus/part-02","status":"fully_proved","summary":"Suppose that $(X, Z)$ and $(Y, W)$ are random variables, where $X, Y$ take values in an abelian group. Then","title":"Upper bound on conditioned Ruzsa distance"},{"id":"pfr/entropic-ruzsa-calculus/part-02/first-useful","kind":"node","lean":"condRuzsaDist_diff_le, condRuzsaDist_diff_le', condRuzsaDist_diff_le'', condRuzsaDist_diff_le'''","parent":4967,"result":true,"route":"../roadmap/pfr/entropic-ruzsa-calculus/part-02/index.html#first-useful","scope":"pfr/entropic-ruzsa-calculus/part-02","status":"fully_proved","summary":"Let $X, Y, Z$ be random variables taking values in some abelian group of characteristic $2$, and with $Y, Z$ independent. Then we have","title":"Comparison of Ruzsa distances, I"},{"id":"pfr/entropic-ruzsa-calculus/part-02/second-useful","kind":"node","lean":"condRuzsaDist_diff_ofsum_le","parent":4967,"result":true,"route":"../roadmap/pfr/entropic-ruzsa-calculus/part-02/index.html#second-useful","scope":"pfr/entropic-ruzsa-calculus/part-02","status":"fully_proved","summary":"Let $X, Y, Z, Z'$ be random variables taking values in some abelian group, and with $Y, Z, Z'$ independent. Then we have","title":"Comparison of Ruzsa distances, II"},{"data_url":"scopes/pfr/the-100-version-of-pfr.json","id":"pfr/the-100-version-of-pfr","kind":"scope","parent":4907,"result":false,"route":"../roadmap/pfr/the-100-version-of-pfr/index.html","scope":"pfr/the-100-version-of-pfr","status":"fully_proved","title":"The 100% version of PFR"},{"id":"pfr/the-100-version-of-pfr/sym-group-def","kind":"node","lean":"symmGroup","parent":4974,"result":true,"route":"../roadmap/pfr/the-100-version-of-pfr/index.html#sym-group-def","scope":"pfr/the-100-version-of-pfr","status":"fully_proved","summary":"If $X$ is a $G$-valued random variable, then the symmetry group $\\mathrm{Sym}[X]$ is the set of all $h \\in G$ such that $X +h$ has the same distribution as $X$.","title":"Symmetry group"},{"id":"pfr/the-100-version-of-pfr/sym-group","kind":"node","lean":"symmGroup","parent":4974,"result":true,"route":"../roadmap/pfr/the-100-version-of-pfr/index.html#sym-group","scope":"pfr/the-100-version-of-pfr","status":"fully_proved","summary":"If $X$ is a $G$-valued random variable, then $\\mathrm{Sym}[X]$ is a subgroup of $G$.","title":"Symmetry group is a group"},{"id":"pfr/the-100-version-of-pfr/zero-large","kind":"node","lean":"sub_mem_symmGroup","parent":4974,"result":true,"route":"../roadmap/pfr/the-100-version-of-pfr/index.html#zero-large","scope":"pfr/the-100-version-of-pfr","status":"fully_proved","summary":"If $X$ is a $G$-valued random variable such that $d[X ;X]=0$, and $x,y \\in G$ are such that $P[X=x], P[X=y]>0$, then $x-y \\in \\mathrm{Sym}[X]$.","title":"Zero Ruzsa distance implies large symmetry group"},{"id":"pfr/the-100-version-of-pfr/sym-zero","kind":"node","lean":"isUniform_sub_const_of_rdist_eq_zero","parent":4974,"result":true,"route":"../roadmap/pfr/the-100-version-of-pfr/index.html#sym-zero","scope":"pfr/the-100-version-of-pfr","status":"fully_proved","summary":"If $X$ is a $G$-valued random variable with $d[X ;X]=0$, and $x_0$ is a point with $P[X=x_0] > 0$, then $X-x_0$ is uniformly distributed on $\\mathrm{Sym}[X]$.","title":"Translate is uniform on symmetry group"},{"id":"pfr/the-100-version-of-pfr/lem-100pc-self","kind":"node","lean":"exists_isUniform_of_rdist_self_eq_zero","parent":4974,"result":true,"route":"../roadmap/pfr/the-100-version-of-pfr/index.html#lem-100pc-self","scope":"pfr/the-100-version-of-pfr","status":"fully_proved","summary":"Suppose that $X$ is a $G$-valued random variable such that $d[X ;X]=0$. Then there exists a subgroup $H \\leq G$ such that $d[X ;U_H] = 0$.","title":"Symmetric 100% inverse theorem"},{"id":"pfr/the-100-version-of-pfr/lem-100pc","kind":"node","lean":"exists_isUniform_of_rdist_eq_zero","parent":4974,"result":true,"route":"../roadmap/pfr/the-100-version-of-pfr/index.html#lem-100pc","scope":"pfr/the-100-version-of-pfr","status":"fully_proved","summary":"Suppose that $X_1,X_2$ are $G$-valued random variables such that $d[X_1;X_2]=0$. Then there exists a subgroup $H \\leq G$ such that $d[X_1;U_H] = d[X_2;U_H] = 0$.","title":"General 100% inverse theorem"},{"data_url":"scopes/pfr/the-fibring-lemma.json","id":"pfr/the-fibring-lemma","kind":"scope","parent":4907,"result":false,"route":"../roadmap/pfr/the-fibring-lemma/index.html","scope":"pfr/the-fibring-lemma","status":"fully_proved","title":"The Fibring lemma"},{"id":"pfr/the-fibring-lemma/fibring-ident","kind":"node","lean":"rdist_of_indep_eq_sum_fibre, rdist_le_sum_fibre","parent":4981,"result":true,"route":"../roadmap/pfr/the-fibring-lemma/index.html#fibring-ident","scope":"pfr/the-fibring-lemma","status":"fully_proved","summary":"Let $\\pi : H \\to H'$ be a homomorphism additive groups, and let $Z_1,Z_2$ be $H$-valued random variables. Then we have","title":"General fibring identity"},{"id":"pfr/the-fibring-lemma/fibring-ineq","kind":"node","lean":"rdist_of_hom_le","parent":4981,"result":true,"route":"../roadmap/pfr/the-fibring-lemma/index.html#fibring-ineq","scope":"pfr/the-fibring-lemma","status":"fully_proved","summary":"If $\\pi:G\\to H$ is a homomorphism of additive groups and $X,Y$ are $G$-valued random variables then","title":"fibring-ineq"},{"id":"pfr/the-fibring-lemma/cor-fibre","kind":"node","lean":"sum_of_rdist_eq","parent":4981,"result":true,"route":"../roadmap/pfr/the-fibring-lemma/index.html#cor-fibre","scope":"pfr/the-fibring-lemma","status":"fully_proved","summary":"Let $Y_1,Y_2,Y_3$ and $Y_4$ be independent $G$-valued random variables. Then","title":"Specific fibring identity"},{"data_url":"scopes/pfr/entropy-version-of-pfr.json","id":"pfr/entropy-version-of-pfr","kind":"scope","parent":4907,"result":false,"route":"../roadmap/pfr/entropy-version-of-pfr/index.html","scope":"pfr/entropy-version-of-pfr","status":"fully_proved","summary":"Throughout this chapter, $G = \\mathbb{F}_2^n$, and $X^0_1, X^0_2$ are $G$-valued random variables.","title":"Entropy version of PFR"},{"id":"pfr/entropy-version-of-pfr/tau-def","kind":"node","lean":"tau","parent":4985,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\tau\\)"],["text"," functional"]],"source":"$\\tau$ functional"},"result":true,"route":"../roadmap/pfr/entropy-version-of-pfr/index.html#tau-def","scope":"pfr/entropy-version-of-pfr","status":"fully_proved","summary":"If $X_1,X_2$ are two $G$-valued random variables, then","title":"$\\tau$ functional"},{"id":"pfr/entropy-version-of-pfr/tau-copy","kind":"node","lean":"ProbabilityTheory.IdentDistrib.tau_eq","parent":4985,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\tau\\)"],["text"," depends only on distribution"]],"source":"$\\tau$ depends only on distribution"},"result":true,"route":"../roadmap/pfr/entropy-version-of-pfr/index.html#tau-copy","scope":"pfr/entropy-version-of-pfr","status":"fully_proved","summary":"If $X'_1, X'_2$ are copies of $X_1,X_2$, then $\\tau[X'_1;X'_2] = \\tau[X_1;X_2]$.","title":"$\\tau$ depends only on distribution"},{"id":"pfr/entropy-version-of-pfr/tau-min-def","kind":"node","lean":"TauMinimizes","parent":4985,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\tau\\)"],["text","-minimizer"]],"source":"$\\tau$-minimizer"},"result":true,"route":"../roadmap/pfr/entropy-version-of-pfr/index.html#tau-min-def","scope":"pfr/entropy-version-of-pfr","status":"fully_proved","summary":"A pair of $G$-valued random variables $X_1, X_2$ are said to be a $\\tau$-minimizer if one has","title":"$\\tau$-minimizer"},{"id":"pfr/entropy-version-of-pfr/tau-min","kind":"node","lean":"tau_minimizer_exists","parent":4985,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\tau\\)"],["text"," has minimum"]],"source":"$\\tau$ has minimum"},"result":true,"route":"../roadmap/pfr/entropy-version-of-pfr/index.html#tau-min","scope":"pfr/entropy-version-of-pfr","status":"fully_proved","summary":"A pair $X_1, X_2$ of $\\tau$-minimizers exist.","title":"$\\tau$ has minimum"},{"data_url":"scopes/pfr/entropy-version-of-pfr/basic-facts-about-minimizers.json","id":"pfr/entropy-version-of-pfr/basic-facts-about-minimizers","kind":"scope","parent":4985,"result":false,"route":"../roadmap/pfr/entropy-version-of-pfr/basic-facts-about-minimizers/index.html","scope":"pfr/entropy-version-of-pfr/basic-facts-about-minimizers","status":"fully_proved","summary":"In this section we assume that $X_1,X_2$ are $\\tau$-minimizers. We also write $k := d[X_1;X_2]$.","title":"Basic facts about minimizers"},{"id":"pfr/entropy-version-of-pfr/basic-facts-about-minimizers/distance-lower","kind":"node","lean":"distance_ge_of_min","parent":4990,"result":true,"route":"../roadmap/pfr/entropy-version-of-pfr/basic-facts-about-minimizers/index.html#distance-lower","scope":"pfr/entropy-version-of-pfr/basic-facts-about-minimizers","status":"fully_proved","summary":"For any $G$-valued random variables $X'_1,X'_2$, one has","title":"Distance lower bound"},{"id":"pfr/entropy-version-of-pfr/basic-facts-about-minimizers/cond-distance-lower","kind":"node","lean":"condRuzsaDistance_ge_of_min","parent":4990,"result":true,"route":"../roadmap/pfr/entropy-version-of-pfr/basic-facts-about-minimizers/index.html#cond-distance-lower","scope":"pfr/entropy-version-of-pfr/basic-facts-about-minimizers","status":"fully_proved","summary":"For any $G$-valued random variables $X'_1,X'_2$ and random variables $Z,W$, one has","title":"Conditional distance lower bound"},{"data_url":"scopes/pfr/entropy-version-of-pfr/first-estimate.json","id":"pfr/entropy-version-of-pfr/first-estimate","kind":"scope","parent":4985,"result":false,"route":"../roadmap/pfr/entropy-version-of-pfr/first-estimate/index.html","scope":"pfr/entropy-version-of-pfr/first-estimate","status":"fully_proved","summary":"We continue the assumptions from the preceding section.","title":"First estimate"},{"id":"pfr/entropy-version-of-pfr/first-estimate/first-fibre","kind":"node","lean":"rdist_add_rdist_add_condMutual_eq","parent":4993,"result":true,"route":"../roadmap/pfr/entropy-version-of-pfr/first-estimate/index.html#first-fibre","scope":"pfr/entropy-version-of-pfr/first-estimate","status":"fully_proved","summary":"We have","title":"Fibring identity for first estimate"},{"id":"pfr/entropy-version-of-pfr/first-estimate/first-dist-sum","kind":"node","lean":"rdist_of_sums_ge","parent":4993,"result":true,"route":"../roadmap/pfr/entropy-version-of-pfr/first-estimate/index.html#first-dist-sum","scope":"pfr/entropy-version-of-pfr/first-estimate","status":"fully_proved","summary":"We have","title":"Lower bound on distances"},{"id":"pfr/entropy-version-of-pfr/first-estimate/first-cond","kind":"node","lean":"condRuzsaDist_of_sums_ge","parent":4993,"result":true,"route":"../roadmap/pfr/entropy-version-of-pfr/first-estimate/index.html#first-cond","scope":"pfr/entropy-version-of-pfr/first-estimate","status":"fully_proved","summary":"We have","title":"Lower bound on conditional distances"},{"id":"pfr/entropy-version-of-pfr/first-estimate/first-upper","kind":"node","lean":"diff_rdist_le_1, diff_rdist_le_2, diff_rdist_le_3, diff_rdist_le_4","parent":4993,"result":true,"route":"../roadmap/pfr/entropy-version-of-pfr/first-estimate/index.html#first-upper","scope":"pfr/entropy-version-of-pfr/first-estimate","status":"fully_proved","summary":"We have","title":"Upper bound on distance differences"},{"id":"pfr/entropy-version-of-pfr/first-estimate/first-estimate","kind":"node","lean":"first_estimate","parent":4993,"result":true,"route":"../roadmap/pfr/entropy-version-of-pfr/first-estimate/index.html#first-estimate","scope":"pfr/entropy-version-of-pfr/first-estimate","status":"fully_proved","summary":"We have $I_1 \\leq 2 \\eta k$.","title":"First estimate"},{"id":"pfr/entropy-version-of-pfr/first-estimate/foursum-bound","kind":"node","lean":"ent_ofsum_le","parent":4993,"result":true,"route":"../roadmap/pfr/entropy-version-of-pfr/first-estimate/index.html#foursum-bound","scope":"pfr/entropy-version-of-pfr/first-estimate","status":"fully_proved","summary":"With the same notation, we have","title":"Entropy bound on quadruple sum"},{"data_url":"scopes/pfr/entropy-version-of-pfr/second-estimate.json","id":"pfr/entropy-version-of-pfr/second-estimate","kind":"scope","parent":4985,"result":false,"route":"../roadmap/pfr/entropy-version-of-pfr/second-estimate/index.html","scope":"pfr/entropy-version-of-pfr/second-estimate","status":"fully_proved","summary":"We continue the assumptions from the preceding section. We introduce the quantity","title":"Second estimate"},{"id":"pfr/entropy-version-of-pfr/second-estimate/dist-sums","kind":"node","lean":"rdist_of_sums_ge'","parent":5000,"result":true,"route":"../roadmap/pfr/entropy-version-of-pfr/second-estimate/index.html#dist-sums","scope":"pfr/entropy-version-of-pfr/second-estimate","status":"fully_proved","summary":"We have","title":"Distance between sums"},{"id":"pfr/entropy-version-of-pfr/second-estimate/second-estimate-aux","kind":"node","lean":"second_estimate_aux","parent":5000,"result":true,"route":"../roadmap/pfr/entropy-version-of-pfr/second-estimate/index.html#second-estimate-aux","scope":"pfr/entropy-version-of-pfr/second-estimate","status":"fully_proved","summary":"We have","title":"second-estimate-aux"},{"id":"pfr/entropy-version-of-pfr/second-estimate/second-estimate","kind":"node","lean":"second_estimate","parent":5000,"result":true,"route":"../roadmap/pfr/entropy-version-of-pfr/second-estimate/index.html#second-estimate","scope":"pfr/entropy-version-of-pfr/second-estimate","status":"fully_proved","summary":"We have","title":"Second estimate"},{"data_url":"scopes/pfr/entropy-version-of-pfr/endgame.json","id":"pfr/entropy-version-of-pfr/endgame","kind":"scope","parent":4985,"result":false,"route":"../roadmap/pfr/entropy-version-of-pfr/endgame/index.html","scope":"pfr/entropy-version-of-pfr/endgame","status":"fully_proved","summary":"Let $X_1,X_2,\\tilde X_1,\\tilde X_2$ be as before, and introduce the random variables","title":"Endgame"},{"id":"pfr/entropy-version-of-pfr/endgame/symm-lemma","kind":"node","lean":"I\u2083_eq","parent":5004,"result":true,"route":"../roadmap/pfr/entropy-version-of-pfr/endgame/index.html#symm-lemma","scope":"pfr/entropy-version-of-pfr/endgame","status":"fully_proved","summary":"We have","title":"Symmetry identity"},{"id":"pfr/entropy-version-of-pfr/endgame/uvw-s","kind":"node","lean":"I\u2083_eq, sum_condMutual_le","parent":5004,"result":true,"route":"../roadmap/pfr/entropy-version-of-pfr/endgame/index.html#uvw-s","scope":"pfr/entropy-version-of-pfr/endgame","status":"fully_proved","summary":"We have","title":"Bound on conditional mutual informations"},{"id":"pfr/entropy-version-of-pfr/endgame/total-dist","kind":"node","lean":"sum_dist_diff_le","parent":5004,"result":true,"route":"../roadmap/pfr/entropy-version-of-pfr/endgame/index.html#total-dist","scope":"pfr/entropy-version-of-pfr/endgame","status":"fully_proved","summary":"We have","title":"Bound on distance increments"},{"id":"pfr/entropy-version-of-pfr/endgame/key-ident","kind":"node","lean":"sum_uvw_eq_zero","parent":5004,"result":true,"route":"../roadmap/pfr/entropy-version-of-pfr/endgame/index.html#key-ident","scope":"pfr/entropy-version-of-pfr/endgame","status":"fully_proved","summary":"We have $U+V+W=0$.","title":"Key identity"},{"id":"pfr/entropy-version-of-pfr/endgame/construct-good-prelim","kind":"node","lean":"construct_good_prelim","parent":5004,"result":true,"route":"../roadmap/pfr/entropy-version-of-pfr/endgame/index.html#construct-good-prelim","scope":"pfr/entropy-version-of-pfr/endgame","status":"fully_proved","summary":"One has","title":"Constructing good variables, I"},{"id":"pfr/entropy-version-of-pfr/endgame/construct-good","kind":"node","lean":"construct_good","parent":5004,"result":true,"route":"../roadmap/pfr/entropy-version-of-pfr/endgame/index.html#construct-good","scope":"pfr/entropy-version-of-pfr/endgame","status":"fully_proved","summary":"One has","title":"Constructing good variables, II"},{"id":"pfr/entropy-version-of-pfr/endgame/de-prop","kind":"node","lean":"tau_strictly_decreases","parent":5004,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\tau\\)"],["text","-decrement"]],"source":"$\\tau$-decrement"},"result":true,"route":"../roadmap/pfr/entropy-version-of-pfr/endgame/index.html#de-prop","scope":"pfr/entropy-version-of-pfr/endgame","status":"fully_proved","summary":"Let $X_1, X_2$ be tau-minimizers. Then $d[X_1;X_2] = 0$.","title":"$\\tau$-decrement"},{"data_url":"scopes/pfr/entropy-version-of-pfr/conclusion.json","id":"pfr/entropy-version-of-pfr/conclusion","kind":"scope","parent":4985,"result":false,"route":"../roadmap/pfr/entropy-version-of-pfr/conclusion/index.html","scope":"pfr/entropy-version-of-pfr/conclusion","status":"fully_proved","summary":"Note: a \u201cstretch goal\u201d for this project would be to obtain a \u2018decidable\u2018 analogue of this result (see the remark at the end of Section 2 for some related discussion).","title":"Conclusion"},{"id":"pfr/entropy-version-of-pfr/conclusion/entropy-pfr","kind":"node","lean":"entropic_PFR_conjecture","parent":5012,"result":true,"route":"../roadmap/pfr/entropy-version-of-pfr/conclusion/index.html#entropy-pfr","scope":"pfr/entropy-version-of-pfr/conclusion","status":"fully_proved","summary":"Let $G = \\mathbb{F}_2^n$, and suppose that $X^0_1, X^0_2$ are $G$-valued random variables. Then there is some subgroup $H \\leq G$ such that","title":"Entropy version of PFR"},{"data_url":"scopes/pfr/proof-of-pfr.json","id":"pfr/proof-of-pfr","kind":"scope","parent":4907,"result":false,"route":"../roadmap/pfr/proof-of-pfr/index.html","scope":"pfr/proof-of-pfr","status":"fully_proved","title":"Proof of PFR"},{"id":"pfr/proof-of-pfr/ruz-cov","kind":"node","lean":"Finset.ruzsa_covering_add","parent":5014,"result":true,"route":"../roadmap/pfr/proof-of-pfr/index.html#ruz-cov","scope":"pfr/proof-of-pfr","status":"fully_proved","summary":"If $A,B$ are finite non-empty subsets of a group $G$, then $A$ can be covered by at most $|A+B|/|B|$ translates of $B-B$.","title":"Ruzsa covering lemma"},{"id":"pfr/proof-of-pfr/pfr-aux","kind":"node","lean":"PFR_conjecture_aux","parent":5014,"result":true,"route":"../roadmap/pfr/proof-of-pfr/index.html#pfr-aux","scope":"pfr/proof-of-pfr","status":"fully_proved","summary":"If $A \\subset {\\bf F}_2^n$ is non-empty and $|A+A| \\leq K|A|$, then $A$ can be covered by at most $K ^ {13/2}|A|^{1/2}/|H|^{1/2}$ translates of a subspace $H$ of ${\\bf F}_2^n$ with","title":"pfr_aux"},{"id":"pfr/proof-of-pfr/pfr","kind":"node","lean":"PFR_conjecture","parent":5014,"result":true,"route":"../roadmap/pfr/proof-of-pfr/index.html#pfr","scope":"pfr/proof-of-pfr","status":"fully_proved","summary":"If $A \\subset {\\bf F}_2^n$ is non-empty and $|A+A| \\leq K|A|$, then $A$ can be covered by most $2K^{12}$ translates of a subspace $H$ of ${\\bf F}_2^n$ with $|H| \\leq |A|$.","title":"PFR"},{"id":"pfr/proof-of-pfr/pfr-cor","kind":"node","lean":"PFR_conjecture","parent":5014,"result":true,"route":"../roadmap/pfr/proof-of-pfr/index.html#pfr-cor","scope":"pfr/proof-of-pfr","status":"fully_proved","summary":"If $G$ is an abelian $2$-torsion group, $A \\subset G$ is non-empty finite, and $|A+A| \\leq K|A|$, then $A$ can be covered by most $2K^{12}$ translates of a finite group $H$ of $G$ with $|H| \\leq |A|$.","title":"PFR in infinite groups"},{"data_url":"scopes/pfr/improving-the-exponents.json","id":"pfr/improving-the-exponents","kind":"scope","parent":4907,"result":false,"route":"../roadmap/pfr/improving-the-exponents/index.html","scope":"pfr/improving-the-exponents","status":"fully_proved","summary":"The arguments here are due to Jyun-Jie Liao.","title":"Improving the exponents"},{"id":"pfr/improving-the-exponents/construct-good-prelim-improv","kind":"node","lean":"construct_good_prelim'","parent":5019,"result":true,"route":"../roadmap/pfr/improving-the-exponents/index.html#construct-good-prelim-improv","scope":"pfr/improving-the-exponents","status":"fully_proved","summary":"One has","title":"Constructing good variables, I'"},{"id":"pfr/improving-the-exponents/construct-good-improv","kind":"node","lean":"construct_good_improved'","parent":5019,"result":true,"route":"../roadmap/pfr/improving-the-exponents/index.html#construct-good-improv","scope":"pfr/improving-the-exponents","status":"fully_proved","summary":"One has","title":"Constructing good variables, II'"},{"id":"pfr/improving-the-exponents/averaged-construct-good","kind":"node","lean":"averaged_construct_good","parent":5019,"result":true,"route":"../roadmap/pfr/improving-the-exponents/index.html#averaged-construct-good","scope":"pfr/improving-the-exponents","status":"fully_proved","summary":"One has","title":"Constructing good variables, III'"},{"id":"pfr/improving-the-exponents/gen-ineq","kind":"node","lean":"gen_ineq_00, gen_ineq_01, gen_ineq_10","parent":5019,"result":true,"route":"../roadmap/pfr/improving-the-exponents/index.html#gen-ineq","scope":"pfr/improving-the-exponents","status":"fully_proved","summary":"Let $X_1, X_2, X_3, X_4$ be independent $G$-valued random variables, and let $Y$ be another $G$-valued random variable. Set $S := X_1+X_2+X_3+X_4$. Then","title":"General inequality"},{"id":"pfr/improving-the-exponents/dist-diff-bound","kind":"node","lean":"dist_diff_bound_1, dist_diff_bound_2","parent":5019,"result":true,"route":"../roadmap/pfr/improving-the-exponents/index.html#dist-diff-bound","scope":"pfr/improving-the-exponents","status":"fully_proved","summary":"We have","title":"Bound on distance differences"},{"id":"pfr/improving-the-exponents/de-prop-improv","kind":"node","lean":"tau_strictly_decreases'","parent":5019,"reading_title":{"format":"inline-parts-v1","parts":[["text","Improved "],["math","\\(\\tau\\)"],["text","-decrement"]],"source":"Improved $\\tau$-decrement"},"result":true,"route":"../roadmap/pfr/improving-the-exponents/index.html#de-prop-improv","scope":"pfr/improving-the-exponents","status":"fully_proved","summary":"Suppose $0 < \\eta < 1/8$. Let $X_1, X_2$ be tau-minimizers. Then $d[X_1;X_2] = 0$.","title":"Improved $\\tau$-decrement"},{"id":"pfr/improving-the-exponents/de-prop-lim-improv","kind":"node","lean":"tau_minimizer_exists_rdist_eq_zero","parent":5019,"reading_title":{"format":"inline-parts-v1","parts":[["text","Limiting improved "],["math","\\(\\tau\\)"],["text","-decrement"]],"source":"Limiting improved $\\tau$-decrement"},"result":true,"route":"../roadmap/pfr/improving-the-exponents/index.html#de-prop-lim-improv","scope":"pfr/improving-the-exponents","status":"fully_proved","summary":"For $\\eta = 1/8$, there exist tau-minimizers $X_1, X_2$ satisfying $d[X_1;X_2] = 0$.","title":"Limiting improved $\\tau$-decrement"},{"id":"pfr/improving-the-exponents/entropy-pfr-improv","kind":"node","lean":"entropic_PFR_conjecture_improv","parent":5019,"result":true,"route":"../roadmap/pfr/improving-the-exponents/index.html#entropy-pfr-improv","scope":"pfr/improving-the-exponents","status":"fully_proved","summary":"Let $G = \\mathbb{F}_2^n$, and suppose that $X^0_1, X^0_2$ are $G$-valued random variables. Then there is some subgroup $H \\leq G$ such that","title":"Improved entropy version of PFR"},{"id":"pfr/improving-the-exponents/pfr-aux-improv","kind":"node","lean":"PFR_conjecture_improv_aux","parent":5019,"result":true,"route":"../roadmap/pfr/improving-the-exponents/index.html#pfr-aux-improv","scope":"pfr/improving-the-exponents","status":"fully_proved","summary":"If $A \\subset {\\bf F}_2^n$ is non-empty and $|A+A| \\leq K|A|$, then $A$ can be covered by at most $K^6 |A|^{1/2}/|H|^{1/2}$ translates of a subspace $H$ of ${\\bf F}_2^n$ with","title":"pfr_aux-improv"},{"id":"pfr/improving-the-exponents/pfr-improv","kind":"node","lean":"PFR_conjecture_improv","parent":5019,"result":true,"route":"../roadmap/pfr/improving-the-exponents/index.html#pfr-improv","scope":"pfr/improving-the-exponents","status":"fully_proved","summary":"If $A \\subset {\\bf F}_2^n$ is non-empty and $|A+A| \\leq K|A|$, then $A$ can be covered by most $2K^{11}$ translates of a subspace $H$ of ${\\bf F}_2^n$ with $|H| \\leq |A|$.","title":"Improved PFR"},{"data_url":"scopes/pfr/homomorphism-version-of-pfr.json","id":"pfr/homomorphism-version-of-pfr","kind":"scope","parent":4907,"result":false,"route":"../roadmap/pfr/homomorphism-version-of-pfr/index.html","scope":"pfr/homomorphism-version-of-pfr","status":"fully_proved","summary":"In this section, $G, G'$ are finite abelian $2$-groups.","title":"Homomorphism version of PFR"},{"id":"pfr/homomorphism-version-of-pfr/hb-thm","kind":"node","lean":"hahn_banach","parent":5030,"result":true,"route":"../roadmap/pfr/homomorphism-version-of-pfr/index.html#hb-thm","scope":"pfr/homomorphism-version-of-pfr","status":"fully_proved","summary":"Let $H_0$ be a subgroup of $G$. Then every homomorphism $\\phi: H_0 \\to G'$ can be extended to a homomorphism $\\tilde \\phi: G \\to G'$.","title":"Hahn-Banach type theorem"},{"id":"pfr/homomorphism-version-of-pfr/goursat","kind":"node","lean":"goursat","parent":5030,"result":true,"route":"../roadmap/pfr/homomorphism-version-of-pfr/index.html#goursat","scope":"pfr/homomorphism-version-of-pfr","status":"fully_proved","summary":"Let $H$ be a subgroup of $G \\times G'$. Then there exists a subgroup $H_0$ of $G$, a subgroup $H_1$ of $G'$, and a homomorphism $\\phi: G \\to G'$ such that","title":"Goursat type theorem"},{"id":"pfr/homomorphism-version-of-pfr/hom-pfr","kind":"node","lean":"homomorphism_pfr","parent":5030,"result":true,"route":"../roadmap/pfr/homomorphism-version-of-pfr/index.html#hom-pfr","scope":"pfr/homomorphism-version-of-pfr","status":"fully_proved","summary":"Let $f: G \\to G'$ be a function, and let $S$ denote the set","title":"Homomorphism form of PFR"},{"data_url":"scopes/pfr/approximate-homomorphism-version-of-pfr.json","id":"pfr/approximate-homomorphism-version-of-pfr","kind":"scope","parent":4907,"result":false,"route":"../roadmap/pfr/approximate-homomorphism-version-of-pfr/index.html","scope":"pfr/approximate-homomorphism-version-of-pfr","status":"fully_proved","summary":"With a bit more effort, we can remove the constant term $c$, at the cost of reducing the set of agreement slightly. We need some preliminary lemmas.","title":"Approximate homomorphism version of PFR"},{"id":"pfr/approximate-homomorphism-version-of-pfr/energy-def","kind":"node","lean":"Finset.addEnergy'","parent":5034,"result":true,"route":"../roadmap/pfr/approximate-homomorphism-version-of-pfr/index.html#energy-def","scope":"pfr/approximate-homomorphism-version-of-pfr","status":"fully_proved","summary":"If $G$ is a finite additive group and $A$ is a subset of $G$, the *additive energy* $E(A)$ of $A$ is the normalized count","title":"Additive energy"},{"id":"pfr/approximate-homomorphism-version-of-pfr/cs-bound","kind":"node","lean":"Finset.card_sq_le_card_mul_addEnergy'","parent":5034,"result":true,"route":"../roadmap/pfr/approximate-homomorphism-version-of-pfr/index.html#cs-bound","scope":"pfr/approximate-homomorphism-version-of-pfr","status":"fully_proved","summary":"If $G$ is a finite additive group and $A,B$ are subsets of $G$, then","title":"Cauchy\u2013Schwarz bound"},{"id":"pfr/approximate-homomorphism-version-of-pfr/bsg","kind":"node","lean":"BSG_self","parent":5034,"result":true,"route":"../roadmap/pfr/approximate-homomorphism-version-of-pfr/index.html#bsg","scope":"pfr/approximate-homomorphism-version-of-pfr","status":"fully_proved","summary":"Let $G$ be a finite abelian group, and let $A$ be a non-empty subset of $G$ with $E(A) \\geq \\frac{1}{K} \\left( \\frac{|A|}{|G|} \\right)^3$ for some $K \\geq 1$. Then there is a subset $A'$ of $A$ with $|A'| \\geq |A| / (C_1 K^{C_2})$ and $|A'-A'| \\leq C_3 K^{C_4} |A|$, where","title":"Balog\u2013Szemer\u00e9di\u2013Gowers lemma"},{"id":"pfr/approximate-homomorphism-version-of-pfr/approx-hom-pfr","kind":"node","lean":"approx_hom_pfr","parent":5034,"result":true,"route":"../roadmap/pfr/approximate-homomorphism-version-of-pfr/index.html#approx-hom-pfr","scope":"pfr/approximate-homomorphism-version-of-pfr","status":"fully_proved","summary":"Let $G,G'$ be finite abelian $2$-groups. Let $f: G \\to G'$ be a function, and suppose that there are at least $|G|^2 / K$ pairs $(x,y) \\in G^2$ such that","title":"Approximate homomorphism form of PFR"},{"id":"pfr/approximate-homomorphism-version-of-pfr/gdual","kind":"node","lean":"card_of_dual","parent":5034,"result":true,"route":"../roadmap/pfr/approximate-homomorphism-version-of-pfr/index.html#gdual","scope":"pfr/approximate-homomorphism-version-of-pfr","status":"fully_proved","summary":"Let $G$ be a finite abelian $2$-group. Then the finite abelian $2$-group $\\mathrm{Hom}(G,\\mathbb{Z}/2\\mathbb{Z})$ of homomorphisms from $G$ to $\\mathbb{Z}/2\\mathbb{Z}$ has the same order as $G$.","title":"Duality"},{"id":"pfr/approximate-homomorphism-version-of-pfr/gcount","kind":"node","lean":"card_of_dual_constrained","parent":5034,"result":true,"route":"../roadmap/pfr/approximate-homomorphism-version-of-pfr/index.html#gcount","scope":"pfr/approximate-homomorphism-version-of-pfr","status":"fully_proved","summary":"Let $G$ be a finite abelian $2$-group, and let $x \\in G$ be non-zero. Then there are $|G|/2$ homomorphisms $\\phi: G \\to \\mathbb{Z}/2\\mathbb{Z}$ such that $\\phi(x) = 1$.","title":"Counting"},{"id":"pfr/approximate-homomorphism-version-of-pfr/gslice","kind":"node","lean":"card_of_slice","parent":5034,"result":true,"route":"../roadmap/pfr/approximate-homomorphism-version-of-pfr/index.html#gslice","scope":"pfr/approximate-homomorphism-version-of-pfr","status":"fully_proved","summary":"Let $G$ be a finite abelian $2$-group, and let $A$ be a subset of $G$. Then there exists a homomorphism $\\phi: G \\to \\mathbb{Z}/2\\mathbb{Z}$ such that $|A \\cap \\phi^{-1}(1)| \\geq (|A|-1)/2$.","title":"Slicing"},{"id":"pfr/approximate-homomorphism-version-of-pfr/approx-hom-pfr-no-const","kind":"node","lean":"approx_hom_pfr'","parent":5034,"result":true,"route":"../roadmap/pfr/approximate-homomorphism-version-of-pfr/index.html#approx-hom-pfr-no-const","scope":"pfr/approximate-homomorphism-version-of-pfr","status":"fully_proved","summary":"Let $G,G'$ be finite abelian $2$-groups. Let $f: G \\to G'$ be a function, and suppose that there are at least $|G|^2 / K$ pairs $(x,y) \\in G^2$ such that","title":"Approximate homomorphism form of PFR, no constant term"},{"data_url":"scopes/pfr/weak-pfr-over-the-integers.json","id":"pfr/weak-pfr-over-the-integers","kind":"scope","parent":4907,"result":false,"route":"../roadmap/pfr/weak-pfr-over-the-integers/index.html","scope":"pfr/weak-pfr-over-the-integers","status":"fully_proved","summary":"We could use the previous lemma for any value of $\\alpha \\in (0,1)$, which would give a whole range of estimates in Theorem weak-pfr-int. For definiteness, we specialize only to $\\alpha=3/5$, which gives a constant $2$ in the first bound below.","title":"Weak PFR over the integers"},{"id":"pfr/weak-pfr-over-the-integers/torsion-free-doubling","kind":"node","lean":"torsion_free_doubling","parent":5043,"result":true,"route":"../roadmap/pfr/weak-pfr-over-the-integers/index.html#torsion-free-doubling","scope":"pfr/weak-pfr-over-the-integers","status":"fully_proved","summary":"If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$.","title":"torsion-free-doubling"},{"id":"pfr/weak-pfr-over-the-integers/torsion-dist-shrinking","kind":"node","lean":"torsion_dist_shrinking","parent":5043,"result":true,"route":"../roadmap/pfr/weak-pfr-over-the-integers/index.html#torsion-dist-shrinking","scope":"pfr/weak-pfr-over-the-integers","status":"fully_proved","summary":"If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and $\\phi:G\\to \\mathbb{F}_2^d$ is a homomorphism then","title":"torsion-dist-shrinking"},{"id":"pfr/weak-pfr-over-the-integers/app-ent-pfr","kind":"node","lean":"app_ent_PFR","parent":5043,"result":true,"route":"../roadmap/pfr/weak-pfr-over-the-integers/index.html#app-ent-pfr","scope":"pfr/weak-pfr-over-the-integers","status":"fully_proved","summary":"Let $G=\\mathbb{F}_2^n$ and $\\alpha\\in (0,1)$ and let $X,Y$ be $G$-valued random variables such that","title":"app-ent-pfr"},{"id":"pfr/weak-pfr-over-the-integers/pfr-projection","kind":"node","lean":"PFR_projection'","parent":5043,"result":true,"route":"../roadmap/pfr/weak-pfr-over-the-integers/index.html#pfr-projection","scope":"pfr/weak-pfr-over-the-integers","status":"fully_proved","summary":"If $G=\\mathbb{F}_2^d$ and $\\alpha\\in (0,1)$ and $X,Y$ are $G$-valued random variables then there is a subgroup $H\\leq \\mathbb{F}_2^d$ such that","title":"pfr-projection'"},{"id":"pfr/weak-pfr-over-the-integers/pfr-projection-2","kind":"node","lean":"PFR_projection","parent":5043,"result":true,"route":"../roadmap/pfr/weak-pfr-over-the-integers/index.html#pfr-projection-2","scope":"pfr/weak-pfr-over-the-integers","status":"fully_proved","summary":"If $G=\\mathbb{F}_2^d$ and $\\alpha\\in (0,1)$ and $X,Y$ are $G$-valued random variables then there is a subgroup $H\\leq \\mathbb{F}_2^d$ such that","title":"pfr-projection"},{"id":"pfr/weak-pfr-over-the-integers/single-fibres","kind":"node","lean":"single_fibres","parent":5043,"result":true,"route":"../roadmap/pfr/weak-pfr-over-the-integers/index.html#single-fibres","scope":"pfr/weak-pfr-over-the-integers","status":"fully_proved","summary":"Let $\\phi:G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and","title":"single-fibres"},{"id":"pfr/weak-pfr-over-the-integers/dimension-def","kind":"node","lean":"AffineSpace.finrank","parent":5043,"result":true,"route":"../roadmap/pfr/weak-pfr-over-the-integers/index.html#dimension-def","scope":"pfr/weak-pfr-over-the-integers","status":"fully_proved","summary":"If $A\\subseteq \\mathbb{Z}^{d}$ then by $\\dim(A)$ we mean the dimension of the span of $A-A$ over the reals \u2013 equivalently, the smallest $d'$ such that $A$ lies in a coset of a subgroup isomorphic to $\\mathbb{Z}^{d'}$.","title":"dimension-def"},{"id":"pfr/weak-pfr-over-the-integers/weak-pfr-asymm","kind":"node","lean":"weak_PFR_asymm","parent":5043,"result":true,"route":"../roadmap/pfr/weak-pfr-over-the-integers/index.html#weak-pfr-asymm","scope":"pfr/weak-pfr-over-the-integers","status":"fully_proved","summary":"If $A,B\\subseteq \\mathbb{Z}^d$ are finite non-empty sets then there exist non-empty $A'\\subseteq A$ and $B'\\subseteq B$ such that","title":"weak-pfr-asymm"},{"id":"pfr/weak-pfr-over-the-integers/weak-pfr-symm","kind":"node","lean":"weak_PFR","parent":5043,"result":true,"route":"../roadmap/pfr/weak-pfr-over-the-integers/index.html#weak-pfr-symm","scope":"pfr/weak-pfr-over-the-integers","status":"fully_proved","summary":"If $A\\subseteq \\mathbb{Z}^d$ is a finite non-empty set with $d[U_A;U_A]\\leq \\log K$ then there exists a non-empty $A'\\subseteq A$ such that","title":"weak-pfr-symm"},{"id":"pfr/weak-pfr-over-the-integers/weak-pfr-int","kind":"node","lean":"weak_PFR_int","parent":5043,"result":true,"route":"../roadmap/pfr/weak-pfr-over-the-integers/index.html#weak-pfr-int","scope":"pfr/weak-pfr-over-the-integers","status":"fully_proved","summary":"Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A-A\\rvert\\leq K\\lvert A\\rvert$. There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and $\\dim A' \\leq \\frac{40}{\\log 2}\\log K$.","title":"weak-pfr-int"},{"id":"pfr/weak-pfr-over-the-integers/weak-pfr-int-sumset","kind":"node","lean":"weak_PFR_int_sumset","parent":5043,"result":true,"route":"../roadmap/pfr/weak-pfr-over-the-integers/index.html#weak-pfr-int-sumset","scope":"pfr/weak-pfr-over-the-integers","status":"fully_proved","summary":"Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$. There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-34}\\lvert A\\rvert$ and $\\dim A' \\leq \\frac{80}{\\log 2}\\log K$.","title":"weak-pfr-int-sumset"},{"data_url":"scopes/pfr/the-m-torsion-case.json","id":"pfr/the-m-torsion-case","kind":"scope","parent":4907,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(m\\)"],["text","-torsion case"]],"source":"The $m$-torsion case"},"result":false,"route":"../roadmap/pfr/the-m-torsion-case/index.html","scope":"pfr/the-m-torsion-case","status":"fully_proved","title":"The $m$-torsion case"},{"data_url":"scopes/pfr/the-m-torsion-case/data-processing-inequality.json","id":"pfr/the-m-torsion-case/data-processing-inequality","kind":"scope","parent":5055,"result":false,"route":"../roadmap/pfr/the-m-torsion-case/data-processing-inequality/index.html","scope":"pfr/the-m-torsion-case/data-processing-inequality","status":"fully_proved","title":"Data processing inequality"},{"id":"pfr/the-m-torsion-case/data-processing-inequality/data-process-single","kind":"node","lean":"ProbabilityTheory.entropy_comp_le","parent":5056,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/data-processing-inequality/index.html#data-process-single","scope":"pfr/the-m-torsion-case/data-processing-inequality","status":"fully_proved","summary":"Let $X$ be a random variable. Then for any function $f$ on the range of $X$, one has $\\mathbb{H}[f(X)] \\leq \\mathbb{H}[X]$.","title":"Data processing for a single variable"},{"id":"pfr/the-m-torsion-case/data-processing-inequality/data-process-unc-one","kind":"node","lean":"ProbabilityTheory.mutual_comp_le","parent":5056,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/data-processing-inequality/index.html#data-process-unc-one","scope":"pfr/the-m-torsion-case/data-processing-inequality","status":"fully_proved","summary":"Let $X,Y$ be random variables. For any function $f, g$ on the range of $X$, we have $\\mathbb{I}[f(X) : Y] \\leq \\mathbb{I}[X:Y]$.","title":"One-sided unconditional data processing inequality"},{"id":"pfr/the-m-torsion-case/data-processing-inequality/data-process-unc","kind":"node","lean":"ProbabilityTheory.mutual_comp_comp_le","parent":5056,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/data-processing-inequality/index.html#data-process-unc","scope":"pfr/the-m-torsion-case/data-processing-inequality","status":"fully_proved","summary":"Let $X,Y$ be random variables. For any functions $f, g$ on the ranges of $X, Y$ respectively, we have $\\mathbb{I}[f(X) : g(Y )] \\leq \\mathbb{I}[X : Y]$.","title":"Unconditional data processing inequality"},{"id":"pfr/the-m-torsion-case/data-processing-inequality/data-process","kind":"node","lean":"ProbabilityTheory.condMutual_comp_comp_le","parent":5056,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/data-processing-inequality/index.html#data-process","scope":"pfr/the-m-torsion-case/data-processing-inequality","status":"fully_proved","summary":"Let $X,Y,Z$. For any functions $f, g$ on the ranges of $X, Y$ respectively, we have $\\mathbb{I}[f(X) : g(Y )|Z] \\leq \\mathbb{I}[X :Y |Z]$.","title":"Data processing inequality"},{"data_url":"scopes/pfr/the-m-torsion-case/more-ruzsa-distance-estimates.json","id":"pfr/the-m-torsion-case/more-ruzsa-distance-estimates","kind":"scope","parent":5055,"result":false,"route":"../roadmap/pfr/the-m-torsion-case/more-ruzsa-distance-estimates/index.html","scope":"pfr/the-m-torsion-case/more-ruzsa-distance-estimates","status":"fully_proved","summary":"Let $G$ be an additive group.","title":"More Ruzsa distance estimates"},{"id":"pfr/the-m-torsion-case/more-ruzsa-distance-estimates/sign-flip","kind":"node","lean":"rdist_of_neg_le","parent":5061,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/more-ruzsa-distance-estimates/index.html#sign-flip","scope":"pfr/the-m-torsion-case/more-ruzsa-distance-estimates","status":"fully_proved","summary":"If $X,Y$ are $G$-valued, then","title":"Flipping a sign"},{"id":"pfr/the-m-torsion-case/more-ruzsa-distance-estimates/klm-1","kind":"node","lean":"kvm_ineq_I","parent":5061,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/more-ruzsa-distance-estimates/index.html#klm-1","scope":"pfr/the-m-torsion-case/more-ruzsa-distance-estimates","status":"fully_proved","summary":"If $n \\geq 0$ and $X, Y_1, \\dots, Y_n$ are jointly independent $G$-valued random variables, then","title":"Kaimonovich\u2013Vershik\u2013Madiman inequality"},{"id":"pfr/the-m-torsion-case/more-ruzsa-distance-estimates/klm-2","kind":"node","lean":"kvm_ineq_II","parent":5061,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/more-ruzsa-distance-estimates/index.html#klm-2","scope":"pfr/the-m-torsion-case/more-ruzsa-distance-estimates","status":"fully_proved","summary":"If $n \\geq 1$ and $X, Y_1, \\dots, Y_n$ are jointly independent $G$-valued random variables, then","title":"Kaimonovich\u2013Vershik\u2013Madiman inequality, II"},{"id":"pfr/the-m-torsion-case/more-ruzsa-distance-estimates/klm-3","kind":"node","lean":"kvm_ineq_III","parent":5061,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/more-ruzsa-distance-estimates/index.html#klm-3","scope":"pfr/the-m-torsion-case/more-ruzsa-distance-estimates","status":"fully_proved","summary":"If $n \\geq 1$ and $X, Y_1, \\dots, Y_n$ are jointly independent $G$-valued random variables, then","title":"Kaimonovich\u2013Vershik\u2013Madiman inequality, III"},{"id":"pfr/the-m-torsion-case/more-ruzsa-distance-estimates/compare-sums","kind":"node","lean":"ent_of_sum_le_ent_of_sum","parent":5061,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/more-ruzsa-distance-estimates/index.html#compare-sums","scope":"pfr/the-m-torsion-case/more-ruzsa-distance-estimates","status":"fully_proved","summary":"Let $(X_i)_{1 \\leq i \\leq m}$ and $(Y_j)_{1 \\leq j \\leq l}$ be tuples of jointly independent random variables (so the $X$'s and $Y$'s are also independent of each other), and let $f: \\{1,\\dots,l\\} \\to \\{1,\\dots,m\\}$ be a function, then","title":"Comparing sums"},{"id":"pfr/the-m-torsion-case/more-ruzsa-distance-estimates/sum-dilate-i","kind":"node","lean":"ent_sub_nsmul_le, ent_of_sub_smul'","parent":5061,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/more-ruzsa-distance-estimates/index.html#sum-dilate-i","scope":"pfr/the-m-torsion-case/more-ruzsa-distance-estimates","status":"fully_proved","summary":"Let $X,Y,X'$ be independent $G$-valued random variables, with $X'$ a copy of $X$, and let $a$ be an integer. Then","title":"Sums of dilates I"},{"id":"pfr/the-m-torsion-case/more-ruzsa-distance-estimates/sum-dilate-ii","kind":"node","lean":"ent_sub_zsmul_sub_ent_le","parent":5061,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/more-ruzsa-distance-estimates/index.html#sum-dilate-ii","scope":"pfr/the-m-torsion-case/more-ruzsa-distance-estimates","status":"fully_proved","summary":"Let $X,Y$ be independent $G$-valued random variables, and let $a$ be an integer. Then","title":"Sums of dilates II"},{"data_url":"scopes/pfr/the-m-torsion-case/multidistance.json","id":"pfr/the-m-torsion-case/multidistance","kind":"scope","parent":5055,"result":false,"route":"../roadmap/pfr/the-m-torsion-case/multidistance/index.html","scope":"pfr/the-m-torsion-case/multidistance","status":"fully_proved","summary":"We continue to let $G$ be an abelian group.","title":"Multidistance"},{"id":"pfr/the-m-torsion-case/multidistance/multidist-def","kind":"node","lean":"multiDist","parent":5069,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/multidistance/index.html#multidist-def","scope":"pfr/the-m-torsion-case/multidistance","status":"fully_proved","summary":"Let $m$ be a positive integer, and let $X_{[m]} = (X_i)_{1 \\leq i \\leq m}$ be an $m$-tuple of $G$-valued random variables $X_i$. Then we define","title":"Multidistance"},{"id":"pfr/the-m-torsion-case/multidistance/multidist-copy","kind":"node","lean":"multiDist_copy","parent":5069,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/multidistance/index.html#multidist-copy","scope":"pfr/the-m-torsion-case/multidistance","status":"fully_proved","summary":"If $X_{[m]} = (X_i)_{1 \\leq i \\leq m}$ and $Y_{[m]} = (Y_i)_{1 \\leq i \\leq m}$ are such that $X_i$ and $Y_i$ have the same distribution for each $i$, then $D[X_{[m]}] = D[Y_{[m]}]$.","title":"Multidistance of copy"},{"id":"pfr/the-m-torsion-case/multidistance/multidist-indep","kind":"node","lean":"multiDist_indep","parent":5069,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/multidistance/index.html#multidist-indep","scope":"pfr/the-m-torsion-case/multidistance","status":"fully_proved","summary":"If $X_{[m]} = (X_i)_{1 \\leq i \\leq m}$ are jointly independent, then $D[X_{[m]}] = \\mathbb{H}[\\sum_{i=1}^m X_i] - \\frac{1}{m} \\sum_{i=1}^m \\mathbb{H}[X_i]$.","title":"Multidistance of independent variables"},{"id":"pfr/the-m-torsion-case/multidistance/multidist-nonneg","kind":"node","lean":"multiDist_nonneg","parent":5069,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/multidistance/index.html#multidist-nonneg","scope":"pfr/the-m-torsion-case/multidistance","status":"fully_proved","summary":"For any such tuple, we have $D[X_{[m]}] \\geq 0$.","title":"Nonnegativity"},{"id":"pfr/the-m-torsion-case/multidistance/multidist-perm","kind":"node","lean":"multiDist_of_perm","parent":5069,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/multidistance/index.html#multidist-perm","scope":"pfr/the-m-torsion-case/multidistance","status":"fully_proved","summary":"If $\\phi: \\{1,\\dots,m\\} \\to \\{1,\\dots,m\\}$ is a bijection, then $D[X_{[m]}] = D[(X_{\\phi(j)})_{1 \\leq j \\leq m}]$.","title":"Relabeling"},{"id":"pfr/the-m-torsion-case/multidistance/multidist-ruzsa-i","kind":"node","lean":"multidist_ruzsa_I","parent":5069,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/multidistance/index.html#multidist-ruzsa-i","scope":"pfr/the-m-torsion-case/multidistance","status":"fully_proved","summary":"Let $m \\ge 2$, and let $X_{[m]}$ be a tuple of $G$-valued random variables. Then","title":"Multidistance and Ruzsa distance, I"},{"id":"pfr/the-m-torsion-case/multidistance/multidist-ruzsa-ii","kind":"node","lean":"multidist_ruzsa_II","parent":5069,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/multidistance/index.html#multidist-ruzsa-ii","scope":"pfr/the-m-torsion-case/multidistance","status":"fully_proved","summary":"Let $m \\ge 2$, and let $X_{[m]}$ be a tuple of $G$-valued random variables. Then","title":"Multidistance and Ruzsa distance, II"},{"id":"pfr/the-m-torsion-case/multidistance/multidist-ruzsa-iii","kind":"node","lean":"multidist_ruzsa_III","parent":5069,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/multidistance/index.html#multidist-ruzsa-iii","scope":"pfr/the-m-torsion-case/multidistance","status":"fully_proved","summary":"Let $m \\ge 2$, and let $X_{[m]}$ be a tuple of $G$-valued random variables. If the $X_i$ all have the same distribution, then $D[X_{[m]}] \\leq m d[X_i;X_i]$ for any $1 \\leq i \\leq m$.","title":"Multidistance and Ruzsa distance, III"},{"id":"pfr/the-m-torsion-case/multidistance/multidist-ruzsa-iv","kind":"node","lean":"multidist_ruzsa_IV","parent":5069,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/multidistance/index.html#multidist-ruzsa-iv","scope":"pfr/the-m-torsion-case/multidistance","status":"fully_proved","summary":"Let $m \\ge 2$, and let $X_{[m]}$ be a tuple of independent $G$-valued random variables. Let $W := \\sum_{i=1}^m X_i$. Then","title":"Multidistance and Ruzsa distance, IV"},{"id":"pfr/the-m-torsion-case/multidistance/multi-zero","kind":"node","lean":"multidist_eq_zero","parent":5069,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/multidistance/index.html#multi-zero","scope":"pfr/the-m-torsion-case/multidistance","status":"fully_proved","summary":"If $D[X_{[m]}]=0$, then for each $1 \\leq i \\leq m$ there is a finite subgroup $H_i \\leq G$ such that $d[X_i; U_{H_i}] = 0$.","title":"Vanishing"},{"data_url":"scopes/pfr/the-m-torsion-case/the-tau-functional.json","id":"pfr/the-m-torsion-case/the-tau-functional","kind":"scope","parent":5055,"result":false,"route":"../roadmap/pfr/the-m-torsion-case/the-tau-functional/index.html","scope":"pfr/the-m-torsion-case/the-tau-functional","status":"fully_proved","summary":"Fix $m \\geq 2$, and a reference variable $X^0$ in $G$.","title":"The tau functional"},{"id":"pfr/the-m-torsion-case/the-tau-functional/eta-def-multi","kind":"node","lean":"multiRefPackage","parent":5080,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\eta\\)"]],"source":"$\\eta$"},"result":true,"route":"../roadmap/pfr/the-m-torsion-case/the-tau-functional/index.html#eta-def-multi","scope":"pfr/the-m-torsion-case/the-tau-functional","status":"fully_proved","summary":"We set $\\eta := \\frac{1}{32m^3}$.","title":"$\\eta$"},{"id":"pfr/the-m-torsion-case/the-tau-functional/tau-def-multi","kind":"node","lean":"multiTau","parent":5080,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\tau\\)"],["text","-functional"]],"source":"$\\tau$-functional"},"result":true,"route":"../roadmap/pfr/the-m-torsion-case/the-tau-functional/index.html#tau-def-multi","scope":"pfr/the-m-torsion-case/the-tau-functional","status":"fully_proved","summary":"If $(X_i)_{1 \\leq i \\leq m}$ is a tuple, we define its $\\tau$-functional","title":"$\\tau$-functional"},{"id":"pfr/the-m-torsion-case/the-tau-functional/tau-min-multi","kind":"node","lean":"multiTauMinimizes","parent":5080,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\tau\\)"],["text","-minimizer"]],"source":"$\\tau$-minimizer"},"result":true,"route":"../roadmap/pfr/the-m-torsion-case/the-tau-functional/index.html#tau-min-multi","scope":"pfr/the-m-torsion-case/the-tau-functional","status":"fully_proved","summary":"A $\\tau$-minimizer is a tuple $(X_i)_{1 \\leq i \\leq m}$ that minimizes the $\\tau$-functional among all tuples of $G$-valued random variables.","title":"$\\tau$-minimizer"},{"id":"pfr/the-m-torsion-case/the-tau-functional/tau-min-exist-multi","kind":"node","lean":"multiTau_continuous, multiTau_min_exists","parent":5080,"reading_title":{"format":"inline-parts-v1","parts":[["text","Existence of "],["math","\\(\\tau\\)"],["text","-minimizer"]],"source":"Existence of $\\tau$-minimizer"},"result":true,"route":"../roadmap/pfr/the-m-torsion-case/the-tau-functional/index.html#tau-min-exist-multi","scope":"pfr/the-m-torsion-case/the-tau-functional","status":"fully_proved","summary":"If $G$ is finite, then a $\\tau$-minimizer exists.","title":"Existence of $\\tau$-minimizer"},{"id":"pfr/the-m-torsion-case/the-tau-functional/tau-ref","kind":"node","lean":"multiTau_min_sum_le","parent":5080,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/the-tau-functional/index.html#tau-ref","scope":"pfr/the-m-torsion-case/the-tau-functional","status":"fully_proved","summary":"If $(X_i)_{1 \\leq i \\leq m}$ is a $\\tau$-minimizer, then $\\sum_{i=1}^m d[X_i; X^0] \\leq \\frac{2m}{\\eta} d[X^0; X^0]$.","title":"Minimizer close to reference variables"},{"id":"pfr/the-m-torsion-case/the-tau-functional/multidist-lower","kind":"node","lean":"sub_multiDistance_le","parent":5080,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/the-tau-functional/index.html#multidist-lower","scope":"pfr/the-m-torsion-case/the-tau-functional","status":"fully_proved","summary":"If $(X_i)_{1 \\leq i \\leq m}$ is a $\\tau$-minimizer, and $k := D[(X_i)_{1 \\leq i \\leq m}]$, then for any other tuple $(X'_i)_{1 \\leq i \\leq m}$, one has","title":"Lower bound on multidistance"},{"id":"pfr/the-m-torsion-case/the-tau-functional/cond-multidist-def","kind":"node","lean":"condMultiDist","parent":5080,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/the-tau-functional/index.html#cond-multidist-def","scope":"pfr/the-m-torsion-case/the-tau-functional","status":"fully_proved","summary":"If $X_{[m]} = (X_i)_{1 \\leq i \\leq m}$ and $Y_{[m]} = (Y_i)_{1 \\leq i \\leq m}$ are tuples of random variables, with the $X_i$ being $G$-valued (but the $Y_i$ need not be), then we define","title":"Conditional multidistance"},{"id":"pfr/the-m-torsion-case/the-tau-functional/cond-multidist-alt","kind":"node","lean":"condMultiDist_eq","parent":5080,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/the-tau-functional/index.html#cond-multidist-alt","scope":"pfr/the-m-torsion-case/the-tau-functional","status":"fully_proved","summary":"If the $(X_i,Y_i)$ are independent,","title":"Alternate form of conditional multidistance"},{"id":"pfr/the-m-torsion-case/the-tau-functional/cond-multidist-nonneg","kind":"node","lean":"condMultiDist_nonneg","parent":5080,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/the-tau-functional/index.html#cond-multidist-nonneg","scope":"pfr/the-m-torsion-case/the-tau-functional","status":"fully_proved","summary":"If $X_{[m]} = (X_i)_{1 \\leq i \\leq m}$ and $Y_{[m]} = (Y_i)_{1 \\leq i \\leq m}$ are tuples of random variables, then $D[ X_{[m]} | Y_{[m]} ] \\geq 0$.","title":"Conditional multidistance nonnegative"},{"id":"pfr/the-m-torsion-case/the-tau-functional/cond-multidist-lower","kind":"node","lean":"sub_condMultiDistance_le","parent":5080,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/the-tau-functional/index.html#cond-multidist-lower","scope":"pfr/the-m-torsion-case/the-tau-functional","status":"fully_proved","summary":"If $(X_i)_{1 \\leq i \\leq m}$ is a $\\tau$-minimizer, and $k := D[(X_i)_{1 \\leq i \\leq m}]$, then for any other tuples $(X'_i)_{1 \\leq i \\leq m}$ and $(Y_i)_{1 \\leq i \\leq m}$ with the $X'_i$ $G$-valued, one has","title":"Lower bound on conditional multidistance"},{"id":"pfr/the-m-torsion-case/the-tau-functional/cond-multidist-lower-ii","kind":"node","lean":"sub_condMultiDistance_le'","parent":5080,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/the-tau-functional/index.html#cond-multidist-lower-ii","scope":"pfr/the-m-torsion-case/the-tau-functional","status":"fully_proved","summary":"With the notation of the previous lemma, we have","title":"Lower bound on conditional multidistance, II"},{"data_url":"scopes/pfr/the-m-torsion-case/the-multidistance-chain-rule.json","id":"pfr/the-m-torsion-case/the-multidistance-chain-rule","kind":"scope","parent":5055,"result":false,"route":"../roadmap/pfr/the-m-torsion-case/the-multidistance-chain-rule/index.html","scope":"pfr/the-m-torsion-case/the-multidistance-chain-rule","status":"fully_proved","summary":"We will need to iterate the multidistance chain rule, so it is convenient to observe a conditional version of this rule, as follows.","title":"The multidistance chain rule"},{"id":"pfr/the-m-torsion-case/the-multidistance-chain-rule/multidist-chain-rule","kind":"node","lean":"multiDist_chainRule","parent":5092,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/the-multidistance-chain-rule/index.html#multidist-chain-rule","scope":"pfr/the-m-torsion-case/the-multidistance-chain-rule","status":"fully_proved","summary":"Let $\\pi \\colon G \\to H$ be a homomorphism of abelian groups and let $X_{[m]}$ be a tuple of jointly independent $G$-valued random variables. Then $D[X_{[m]}]$ is equal to","title":"Multidistance chain rule"},{"id":"pfr/the-m-torsion-case/the-multidistance-chain-rule/multidist-chain-rule-cond","kind":"node","lean":"cond_multiDist_chainRule","parent":5092,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/the-multidistance-chain-rule/index.html#multidist-chain-rule-cond","scope":"pfr/the-m-torsion-case/the-multidistance-chain-rule","status":"fully_proved","summary":"Let $\\pi \\colon G \\to H$ be a homomorphism of abelian groups. Let $I$ be a finite index set and let $X_{[m]}$ be a tuple of $G$-valued random variables. Let $Y_{[m]}$ be another tuple of random variables (not necessarily $G$-valued). Suppose that the pairs $(X_i, Y_i)$ are jointly independent of one another (but $X_i$ need not be independent of $Y_i$). Then","title":"Conditional multidistance chain rule"},{"id":"pfr/the-m-torsion-case/the-multidistance-chain-rule/multidist-chain-rule-iter","kind":"node","lean":"iter_multiDist_chainRule, iter_multiDist_chainRule'","parent":5092,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/the-multidistance-chain-rule/index.html#multidist-chain-rule-iter","scope":"pfr/the-m-torsion-case/the-multidistance-chain-rule","status":"fully_proved","summary":"Let $m$ be a positive integer. Suppose one has a sequence","title":"multidist-chain-rule-iter"},{"id":"pfr/the-m-torsion-case/the-multidistance-chain-rule/cor-multid","kind":"node","lean":"cor_multiDist_chainRule","parent":5092,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/the-multidistance-chain-rule/index.html#cor-multid","scope":"pfr/the-m-torsion-case/the-multidistance-chain-rule","status":"fully_proved","summary":"Let $G$ be an abelian group and let $m \\geq 2$. Suppose that $X_{i,j}$, $1 \\leq i, j \\leq m$, are independent $G$-valued random variables. Then","title":"cor-multid"},{"data_url":"scopes/pfr/the-m-torsion-case/bounding-the-mutual-information.json","id":"pfr/the-m-torsion-case/bounding-the-mutual-information","kind":"scope","parent":5055,"result":false,"route":"../roadmap/pfr/the-m-torsion-case/bounding-the-mutual-information/index.html","scope":"pfr/the-m-torsion-case/bounding-the-mutual-information","status":"fully_proved","summary":"As before, $G$ is an abelian group, and $m\\geq 2$. We let $X_{[m]} = (X_i)_{i =1}^m$ be a $\\tau$-minimizer.","title":"Bounding the mutual information"},{"id":"pfr/the-m-torsion-case/bounding-the-mutual-information/key","kind":"node","lean":"mutual_information_le","parent":5097,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/bounding-the-mutual-information/index.html#key","scope":"pfr/the-m-torsion-case/bounding-the-mutual-information","status":"fully_proved","summary":"Suppose that $X_{i,j}$, $1 \\leq i,j \\leq m$, are jointly independent $G$-valued random variables, such that for each $j = 1,\\dots,m$, the random variables $(X_{i,j})_{i = 1}^m$ coincide in distribution with some permutation of $X_{[m]}$. Write","title":"Bounding mutual information"},{"data_url":"scopes/pfr/the-m-torsion-case/endgame.json","id":"pfr/the-m-torsion-case/endgame","kind":"scope","parent":5055,"result":false,"route":"../roadmap/pfr/the-m-torsion-case/endgame/index.html","scope":"pfr/the-m-torsion-case/endgame","status":"fully_proved","summary":"Now let $m \\geq 2$, let $G$ be an $m$-torsion abelian group, and let $(X_i)_{1 \\leq i \\leq m}$ be a $\\tau$-minimizer.","title":"Endgame"},{"id":"pfr/the-m-torsion-case/endgame/zero-sum","kind":"node","lean":"sum_of_z_eq_zero","parent":5099,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/endgame/index.html#zero-sum","scope":"pfr/the-m-torsion-case/endgame","status":"fully_proved","summary":"We have","title":"Zero-sum"},{"id":"pfr/the-m-torsion-case/endgame/prop-52","kind":"node","lean":"mutual_information_le_t_12, mutual_information_le_t_13, mutual_information_le_t_23","parent":5099,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/endgame/index.html#prop-52","scope":"pfr/the-m-torsion-case/endgame","status":"fully_proved","summary":"We have","title":"Mutual information bound"},{"id":"pfr/the-m-torsion-case/endgame/ent-w","kind":"node","lean":"entropy_of_W_le","parent":5099,"reading_title":{"format":"inline-parts-v1","parts":[["text","Entropy of "],["math","\\(W\\)"]],"source":"Entropy of $W$"},"result":true,"route":"../roadmap/pfr/the-m-torsion-case/endgame/index.html#ent-w","scope":"pfr/the-m-torsion-case/endgame","status":"fully_proved","summary":"We have $\\mathbb{H}[W] \\leq (2m-1)k + \\frac1m \\sum_{i=1}^m \\mathbb{H}[X_i]$.","title":"Entropy of $W$"},{"id":"pfr/the-m-torsion-case/endgame/ent-z2","kind":"node","lean":"entropy_of_Z_two_le","parent":5099,"reading_title":{"format":"inline-parts-v1","parts":[["text","Entropy of "],["math","\\(Z_2\\)"]],"source":"Entropy of $Z_2$"},"result":true,"route":"../roadmap/pfr/the-m-torsion-case/endgame/index.html#ent-z2","scope":"pfr/the-m-torsion-case/endgame","status":"fully_proved","summary":"We have $\\mathbb{H}[Z_2] \\leq (8m^2-16m+1) k + \\frac{1}{m} \\sum_{i=1}^m \\mathbb{H}[X_i]$.","title":"Entropy of $Z_2$"},{"id":"pfr/the-m-torsion-case/endgame/mutual-w-z2","kind":"node","lean":"mutual_of_W_Z_two_le","parent":5099,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/endgame/index.html#mutual-w-z2","scope":"pfr/the-m-torsion-case/endgame","status":"fully_proved","summary":"We have $\\mathbb{I}[W : Z_2] \\leq 2 (m-1) k$.","title":"Mutual information bound"},{"id":"pfr/the-m-torsion-case/endgame/xi-z2-w-dist","kind":"node","lean":"sum_of_conditional_distance_le","parent":5099,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/endgame/index.html#xi-z2-w-dist","scope":"pfr/the-m-torsion-case/endgame","status":"fully_proved","summary":"We have $\\sum_{i=1}^m d[X_i;Z_2|W] \\leq 4(m^3-m^2) k$.","title":"Distance bound"},{"id":"pfr/the-m-torsion-case/endgame/lem-get-better","kind":"node","lean":"dist_of_U_add_le","parent":5099,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/endgame/index.html#lem-get-better","scope":"pfr/the-m-torsion-case/endgame","status":"fully_proved","summary":"Let $G$ be an abelian group, let $(T_1,T_2,T_3)$ be a $G^3$-valued random variable such that $T_1+T_2+T_3=0$ holds identically, and write","title":"Application of BSG"},{"id":"pfr/the-m-torsion-case/endgame/k-vanish","kind":"node","lean":"k_eq_zero","parent":5099,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/endgame/index.html#k-vanish","scope":"pfr/the-m-torsion-case/endgame","status":"fully_proved","summary":"We have $k = 0$.","title":"Vanishing entropy"},{"data_url":"scopes/pfr/the-m-torsion-case/wrapping-up.json","id":"pfr/the-m-torsion-case/wrapping-up","kind":"scope","parent":5055,"result":false,"route":"../roadmap/pfr/the-m-torsion-case/wrapping-up/index.html","scope":"pfr/the-m-torsion-case/wrapping-up","status":"fully_proved","title":"Wrapping up"},{"id":"pfr/the-m-torsion-case/wrapping-up/main-entropy","kind":"node","lean":"dist_of_X_U_H_le","parent":5108,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/wrapping-up/index.html#main-entropy","scope":"pfr/the-m-torsion-case/wrapping-up","status":"fully_proved","summary":"Suppose that $G$ is a finite abelian group of torsion $m$. Suppose that $X$ is a $G$-valued random variable. Then there exists a subgroup $H \\leq G$ such that","title":"Entropy form of PFR"},{"id":"pfr/the-m-torsion-case/wrapping-up/pfr-aux-torsion","kind":"node","lean":"torsion_PFR_conjecture_aux","parent":5108,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/wrapping-up/index.html#pfr-aux-torsion","scope":"pfr/the-m-torsion-case/wrapping-up","status":"fully_proved","summary":"Suppose that $G$ is a finite abelian group of torsion $m$. If $A \\subset G$ is non-empty and $|A+A| \\leq K|A|$, then $A$ can be covered by at most $K ^ {128m^3+1}|A|^{1/2}/|H|^{1/2}$ translates of a subspace $H$ of $G$ with","title":"pfr_aux_torsion"},{"id":"pfr/the-m-torsion-case/wrapping-up/pfr-torsion","kind":"node","lean":"torsion_PFR","parent":5108,"result":true,"route":"../roadmap/pfr/the-m-torsion-case/wrapping-up/index.html#pfr-torsion","scope":"pfr/the-m-torsion-case/wrapping-up","status":"fully_proved","summary":"Suppose that $G$ is an abelian group of torsion $m$ (not necessarily finite). If $A \\subset G$ is a finite non-empty set with $|A+A| \\leq K|A|$, then $A$ can be covered by at most $mK^{256m^3+1}$ translates of a finite subspace $H$ of $G$ with $|H| \\leq |A|$.","title":"PFR"},{"data_url":"scopes/pfr/further-improvement-to-exponent.json","id":"pfr/further-improvement-to-exponent","kind":"scope","parent":4907,"result":false,"route":"../roadmap/pfr/further-improvement-to-exponent/index.html","scope":"pfr/further-improvement-to-exponent","status":"fully_proved","title":"Further improvement to exponent"},{"data_url":"scopes/pfr/further-improvement-to-exponent/kullback-leibler-divergence.json","id":"pfr/further-improvement-to-exponent/kullback-leibler-divergence","kind":"scope","parent":5112,"result":false,"route":"../roadmap/pfr/further-improvement-to-exponent/kullback-leibler-divergence/index.html","scope":"pfr/further-improvement-to-exponent/kullback-leibler-divergence","status":"fully_proved","summary":"In the definitions below, $G$ is a set.","title":"Kullback\u2013Leibler divergence"},{"id":"pfr/further-improvement-to-exponent/kullback-leibler-divergence/kl-div","kind":"node","lean":"KLDiv","parent":5113,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/kullback-leibler-divergence/index.html#kl-div","scope":"pfr/further-improvement-to-exponent/kullback-leibler-divergence","status":"fully_proved","summary":"If $X,Y$ are two $G$-valued random variables, the Kullback\u2013Leibler divergence is defined as","title":"Kullback\u2013Leibler divergence"},{"id":"pfr/further-improvement-to-exponent/kullback-leibler-divergence/kl-div-copy","kind":"node","lean":"ProbabilityTheory.IdentDistrib.KLDiv_eq","parent":5113,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/kullback-leibler-divergence/index.html#kl-div-copy","scope":"pfr/further-improvement-to-exponent/kullback-leibler-divergence","status":"fully_proved","summary":"If $X'$ is a copy of $X$, and $Y'$ is a copy of $Y$, then $D_{KL}(X'\\Vert Y') = D_{KL}(X\\Vert Y)$.","title":"Kullback\u2013Leibler divergence of copy"},{"id":"pfr/further-improvement-to-exponent/kullback-leibler-divergence/gibbs","kind":"node","lean":"KLDiv_nonneg","parent":5113,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/kullback-leibler-divergence/index.html#gibbs","scope":"pfr/further-improvement-to-exponent/kullback-leibler-divergence","status":"fully_proved","summary":"$D_{KL}(X\\Vert Y) \\geq 0$.","title":"Gibbs inequality"},{"id":"pfr/further-improvement-to-exponent/kullback-leibler-divergence/gibbs-converse","kind":"node","lean":"KLDiv_eq_zero_iff_identDistrib","parent":5113,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/kullback-leibler-divergence/index.html#gibbs-converse","scope":"pfr/further-improvement-to-exponent/kullback-leibler-divergence","status":"fully_proved","summary":"If $D_{KL}(X\\Vert Y) = 0$, then $Y$ is a copy of $X$.","title":"Converse Gibbs inequality"},{"id":"pfr/further-improvement-to-exponent/kullback-leibler-divergence/kl-div-convex","kind":"node","lean":"KLDiv_of_convex","parent":5113,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/kullback-leibler-divergence/index.html#kl-div-convex","scope":"pfr/further-improvement-to-exponent/kullback-leibler-divergence","status":"fully_proved","summary":"If $S$ is a finite set, $\\sum_{s \\in S} w_s = 1$ for some non-negative $w_s$, and ${\\bf P}(X=x) = \\sum_{s\\in S} w_s {\\bf P}(X_s=x)$, ${\\bf P}(Y=x) = \\sum_{s\\in S} w_s {\\bf P}(Y_s=x)$ for all $x$, then","title":"Convexity of Kullback\u2013Leibler"},{"id":"pfr/further-improvement-to-exponent/kullback-leibler-divergence/kl-div-inj","kind":"node","lean":"KLDiv_of_comp_inj","parent":5113,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/kullback-leibler-divergence/index.html#kl-div-inj","scope":"pfr/further-improvement-to-exponent/kullback-leibler-divergence","status":"fully_proved","summary":"If $f:G \\to H$ is an injection, then $D_{KL}(f(X)\\Vert f(Y)) = D_{KL}(X\\Vert Y)$.","title":"Kullback\u2013Leibler and injections"},{"id":"pfr/further-improvement-to-exponent/kullback-leibler-divergence/kl-sums","kind":"node","lean":"KLDiv_add_le_KLDiv_of_indep","parent":5113,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/kullback-leibler-divergence/index.html#kl-sums","scope":"pfr/further-improvement-to-exponent/kullback-leibler-divergence","status":"fully_proved","summary":"If $X, Y, Z$ are independent $G$-valued random variables, then","title":"Kullback\u2013Leibler and sums"},{"id":"pfr/further-improvement-to-exponent/kullback-leibler-divergence/kl-cond","kind":"node","lean":"condKLDiv_eq","parent":5113,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/kullback-leibler-divergence/index.html#kl-cond","scope":"pfr/further-improvement-to-exponent/kullback-leibler-divergence","status":"fully_proved","summary":"If $X, Y$ are independent $G$-valued random variables, and $Z$ is another random variable defined on the same sample space as $X$, then","title":"Kullback\u2013Leibler and conditioning"},{"id":"pfr/further-improvement-to-exponent/kullback-leibler-divergence/conditional-gibbs","kind":"node","lean":"condKLDiv_nonneg","parent":5113,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/kullback-leibler-divergence/index.html#conditional-gibbs","scope":"pfr/further-improvement-to-exponent/kullback-leibler-divergence","status":"fully_proved","summary":"$D_{KL}((X|W)\\Vert Y) \\geq 0$.","title":"Conditional Gibbs inequality"},{"data_url":"scopes/pfr/further-improvement-to-exponent/rho-functionals.json","id":"pfr/further-improvement-to-exponent/rho-functionals","kind":"scope","parent":5112,"result":false,"route":"../roadmap/pfr/further-improvement-to-exponent/rho-functionals/index.html","scope":"pfr/further-improvement-to-exponent/rho-functionals","status":"fully_proved","summary":"Let $G$ be an additive group, and let $A$ be a non-empty subset of $G$.","title":"Rho functionals"},{"id":"pfr/further-improvement-to-exponent/rho-functionals/rhominus-def","kind":"node","lean":"rhoMinus","parent":5123,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/rho-functionals/index.html#rhominus-def","scope":"pfr/further-improvement-to-exponent/rho-functionals","status":"fully_proved","summary":"For any $G$-valued random variable $X$, we define $\\rho^-(X)$ to be the infimum of $D_{KL}(X \\Vert U_A + T)$, where $U_A$ is uniform on $A$ and $T$ ranges over $G$-valued random variables independent of $U_A$.","title":"Rho minus"},{"id":"pfr/further-improvement-to-exponent/rho-functionals/rhoplus-def","kind":"node","lean":"rhoPlus","parent":5123,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/rho-functionals/index.html#rhoplus-def","scope":"pfr/further-improvement-to-exponent/rho-functionals","status":"fully_proved","summary":"For any $G$-valued random variable $X$, we define $\\rho^+(X) := \\rho^-(X) + \\mathbb{H}(X) - \\mathbb{H}(U_A)$.","title":"Rho plus"},{"id":"pfr/further-improvement-to-exponent/rho-functionals/rhominus-subgroup","kind":"node","lean":"rhoMinus_of_subgroup","parent":5123,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/rho-functionals/index.html#rhominus-subgroup","scope":"pfr/further-improvement-to-exponent/rho-functionals","status":"fully_proved","summary":"If $H$ is a finite subgroup of $G$, then $\\rho^-(U_H) = \\log |A| - \\log \\max_t |A \\cap (H+t)|$.","title":"Rho minus of subgroup"},{"id":"pfr/further-improvement-to-exponent/rho-functionals/rhoplus-subgroup","kind":"node","lean":"rhoPlus_of_subgroup","parent":5123,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/rho-functionals/index.html#rhoplus-subgroup","scope":"pfr/further-improvement-to-exponent/rho-functionals","status":"fully_proved","summary":"If $H$ is a finite subgroup of $G$, then $\\rho^+(U_H) = \\log |H| - \\log \\max_t |A \\cap (H+t)|$.","title":"Rho plus of subgroup"},{"id":"pfr/further-improvement-to-exponent/rho-functionals/rho-def","kind":"node","lean":"rho","parent":5123,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/rho-functionals/index.html#rho-def","scope":"pfr/further-improvement-to-exponent/rho-functionals","status":"fully_proved","summary":"We define $\\rho(X) := (\\rho^+(X) + \\rho^-(X))/2$.","title":"Rho functional"},{"id":"pfr/further-improvement-to-exponent/rho-functionals/rho-subgroup","kind":"node","lean":"rho_of_subgroup","parent":5123,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/rho-functionals/index.html#rho-subgroup","scope":"pfr/further-improvement-to-exponent/rho-functionals","status":"fully_proved","summary":"If $H$ is a finite subgroup of $G$, and $\\rho(U_H) \\leq r$, then there exists $t$ such that $|A \\cap (H+t)| \\geq e^{-r} \\sqrt{|A||H|}$, and $|H|/|A|\\in[e^{-2r},e^{2r}]$.","title":"Rho of subgroup"},{"id":"pfr/further-improvement-to-exponent/rho-functionals/rho-invariant","kind":"node","lean":"rho_of_translate","parent":5123,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/rho-functionals/index.html#rho-invariant","scope":"pfr/further-improvement-to-exponent/rho-functionals","status":"fully_proved","summary":"For any $s \\in G$, $\\rho(X+s) = \\rho(X)$.","title":"Rho invariant"},{"id":"pfr/further-improvement-to-exponent/rho-functionals/rho-cts","kind":"node","lean":"rho_continuous","parent":5123,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/rho-functionals/index.html#rho-cts","scope":"pfr/further-improvement-to-exponent/rho-functionals","status":"fully_proved","summary":"$\\rho(X)$ depends continuously on the distribution of $X$.","title":"Rho continuous"},{"id":"pfr/further-improvement-to-exponent/rho-functionals/rho-sums","kind":"node","lean":"rhoMinus_of_sum, rhoPlus_of_sum, rho_of_sum","parent":5123,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/rho-functionals/index.html#rho-sums","scope":"pfr/further-improvement-to-exponent/rho-functionals","status":"fully_proved","summary":"If $X,Y$ are independent, one has","title":"Rho and sums"},{"id":"pfr/further-improvement-to-exponent/rho-functionals/rho-cond-def","kind":"node","lean":"condRho","parent":5123,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/rho-functionals/index.html#rho-cond-def","scope":"pfr/further-improvement-to-exponent/rho-functionals","status":"fully_proved","summary":"We define $\\rho(X|Y) := \\sum_y {\\bf P}(Y=y) \\rho(X|Y=y)$.","title":"Conditional Rho functional"},{"id":"pfr/further-improvement-to-exponent/rho-functionals/rho-cond-invariant","kind":"node","lean":"condRho_of_translate","parent":5123,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/rho-functionals/index.html#rho-cond-invariant","scope":"pfr/further-improvement-to-exponent/rho-functionals","status":"fully_proved","summary":"For any $s\\in G$, $\\rho(X+s|Y)=\\rho(X|Y)$.","title":"Conditional rho and translation"},{"id":"pfr/further-improvement-to-exponent/rho-functionals/rho-cond-relabeled","kind":"node","lean":"condRho_of_injective","parent":5123,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/rho-functionals/index.html#rho-cond-relabeled","scope":"pfr/further-improvement-to-exponent/rho-functionals","status":"fully_proved","summary":"If $f$ is injective, then $\\rho(X|f(Y))=\\rho(X|Y)$.","title":"Conditional rho and relabeling"},{"id":"pfr/further-improvement-to-exponent/rho-functionals/rho-cond","kind":"node","lean":"condRhoMinus_le, condRhoPlus_le, condRho_le","parent":5123,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/rho-functionals/index.html#rho-cond","scope":"pfr/further-improvement-to-exponent/rho-functionals","status":"fully_proved","summary":"If $X,Z$ are defined on the same space, one has","title":"Rho and conditioning"},{"id":"pfr/further-improvement-to-exponent/rho-functionals/rho-sums-sym","kind":"node","lean":"rho_of_sum_le","parent":5123,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/rho-functionals/index.html#rho-sums-sym","scope":"pfr/further-improvement-to-exponent/rho-functionals","status":"fully_proved","summary":"If $X,Y$ are independent, then","title":"Rho and sums, symmetrized"},{"id":"pfr/further-improvement-to-exponent/rho-functionals/rho-cond-sym","kind":"node","lean":"condRho_of_sum_le","parent":5123,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/rho-functionals/index.html#rho-cond-sym","scope":"pfr/further-improvement-to-exponent/rho-functionals","status":"fully_proved","summary":"If $X,Y$ are independent, then","title":"Rho and conditioning, symmetrized"},{"data_url":"scopes/pfr/further-improvement-to-exponent/studying-a-minimizer.json","id":"pfr/further-improvement-to-exponent/studying-a-minimizer","kind":"scope","parent":5112,"result":false,"route":"../roadmap/pfr/further-improvement-to-exponent/studying-a-minimizer/index.html","scope":"pfr/further-improvement-to-exponent/studying-a-minimizer","status":"fully_proved","summary":"Set $\\eta < 1/8$. In this section, consider $G=\\mathbb{F}_2^n$.","title":"Studying a minimizer"},{"id":"pfr/further-improvement-to-exponent/studying-a-minimizer/phi-min-def","kind":"node","lean":"phiMinimizes","parent":5139,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/studying-a-minimizer/index.html#phi-min-def","scope":"pfr/further-improvement-to-exponent/studying-a-minimizer","status":"fully_proved","summary":"Given $G$-valued random variables $X,Y$, define","title":"phi-min-def"},{"id":"pfr/further-improvement-to-exponent/studying-a-minimizer/phi-min-exist","kind":"node","lean":"phi_min_exists","parent":5139,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\phi\\)"],["text","-minimizers exist"]],"source":"$\\phi$-minimizers exist"},"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/studying-a-minimizer/index.html#phi-min-exist","scope":"pfr/further-improvement-to-exponent/studying-a-minimizer","status":"fully_proved","summary":"There exists a $\\phi$-minimizer.","title":"$\\phi$-minimizers exist"},{"id":"pfr/further-improvement-to-exponent/studying-a-minimizer/phi-first-estimate","kind":"node","lean":"I_one_le","parent":5139,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/studying-a-minimizer/index.html#phi-first-estimate","scope":"pfr/further-improvement-to-exponent/studying-a-minimizer","status":"fully_proved","summary":"$I_1\\le 2\\eta d[X_1;X_2]$","title":"phi-first-estimate"},{"id":"pfr/further-improvement-to-exponent/studying-a-minimizer/i1-i2-diff","kind":"node","lean":"rdist_add_rdist_eq","parent":5139,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/studying-a-minimizer/index.html#i1-i2-diff","scope":"pfr/further-improvement-to-exponent/studying-a-minimizer","status":"fully_proved","summary":"$d[X_1;X_1]+d[X_2;X_2]= 2d[X_1;X_2]+(I_2-I_1)$.","title":"I1-I2-diff"},{"id":"pfr/further-improvement-to-exponent/studying-a-minimizer/phi-second-estimate","kind":"node","lean":"I_two_le","parent":5139,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/studying-a-minimizer/index.html#phi-second-estimate","scope":"pfr/further-improvement-to-exponent/studying-a-minimizer","status":"fully_proved","summary":"$I_2\\le 2\\eta d[X_1;X_2] + \\frac{\\eta}{1-\\eta}(2\\eta d[X_1;X_2]-I_1)$.","title":"phi-second-estimate"},{"id":"pfr/further-improvement-to-exponent/studying-a-minimizer/rho-bsg-triplet","kind":"node","lean":"dist_le_of_sum_zero","parent":5139,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/studying-a-minimizer/index.html#rho-bsg-triplet","scope":"pfr/further-improvement-to-exponent/studying-a-minimizer","status":"fully_proved","summary":"If $G$-valued random variables $T_1,T_2,T_3$ satisfy $T_1+T_2+T_3=0$, then","title":"rho-BSG-triplet"},{"id":"pfr/further-improvement-to-exponent/studying-a-minimizer/rho-bsg-triplet-symmetrized","kind":"node","lean":"dist_le_of_sum_zero'","parent":5139,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/studying-a-minimizer/index.html#rho-bsg-triplet-symmetrized","scope":"pfr/further-improvement-to-exponent/studying-a-minimizer","status":"fully_proved","summary":"If $G$-valued random variables $T_1,T_2,T_3$ satisfy $T_1+T_2+T_3=0$, then","title":"rho-BSG-triplet-symmetrized"},{"id":"pfr/further-improvement-to-exponent/studying-a-minimizer/rho-increase","kind":"node","lean":"condRho_sum_le","parent":5139,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/studying-a-minimizer/index.html#rho-increase","scope":"pfr/further-improvement-to-exponent/studying-a-minimizer","status":"fully_proved","summary":"For independent random variables $Y_1,Y_2,Y_3,Y_4$ over $G$, define $S:=Y_1+Y_2+Y_3+Y_4$, $T_1:=Y_1+Y_2$, $T_2:=Y_1+Y_3$. Then","title":"rho-increase"},{"id":"pfr/further-improvement-to-exponent/studying-a-minimizer/rho-increase-symmetrized","kind":"node","lean":"condRho_sum_le'","parent":5139,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/studying-a-minimizer/index.html#rho-increase-symmetrized","scope":"pfr/further-improvement-to-exponent/studying-a-minimizer","status":"fully_proved","summary":"For independent random variables $Y_1,Y_2,Y_3,Y_4$ over $G$, define $T_1:=Y_1+Y_2,T_2:=Y_1+Y_3,T_3:=Y_2+Y_3$ and $S:=Y_1+Y_2+Y_3+Y_4$. Then","title":"rho-increase-symmetrized"},{"id":"pfr/further-improvement-to-exponent/studying-a-minimizer/phi-minimizer-zero-distance","kind":"node","lean":"dist_of_min_eq_zero","parent":5139,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/studying-a-minimizer/index.html#phi-minimizer-zero-distance","scope":"pfr/further-improvement-to-exponent/studying-a-minimizer","status":"fully_proved","summary":"If $X_1,X_2$ is a $\\phi$-minimizer, then $d[X_1;X_2] = 0$.","title":"phi-minimizer-zero-distance"},{"id":"pfr/further-improvement-to-exponent/studying-a-minimizer/pfr-rho","kind":"node","lean":"rho_PFR_conjecture","parent":5139,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/studying-a-minimizer/index.html#pfr-rho","scope":"pfr/further-improvement-to-exponent/studying-a-minimizer","status":"fully_proved","summary":"For any random variables $Y_1,Y_2$, there exist a subgroup $H$ such that","title":"pfr-rho"},{"id":"pfr/further-improvement-to-exponent/studying-a-minimizer/pfr-9-aux","kind":"node","lean":"better_PFR_conjecture_aux0","parent":5139,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/studying-a-minimizer/index.html#pfr-9-aux","scope":"pfr/further-improvement-to-exponent/studying-a-minimizer","status":"fully_proved","summary":"If $|A+A| \\leq K|A|$, then there exists a subgroup $H$ and $t\\in G$ such that $|A \\cap (H+t)| \\geq K^{-4} \\sqrt{|A||H|}$, and $|H|/|A|\\in[K^{-8},K^8]$.","title":"pfr-9-aux"},{"id":"pfr/further-improvement-to-exponent/studying-a-minimizer/pfr-9-aux-2","kind":"node","lean":"better_PFR_conjecture_aux","parent":5139,"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/studying-a-minimizer/index.html#pfr-9-aux-2","scope":"pfr/further-improvement-to-exponent/studying-a-minimizer","status":"fully_proved","summary":"If $|A+A| \\leq K|A|$, then there exist a subgroup $H$ and a subset $c$ of $G$ with $A \\subseteq c + H$, such that $|c| \\leq K^{5} |A|^{1/2}/|H|^{1/2}$ and $|H|/|A|\\in[K^{-8},K^8]$.","title":"pfr-9-aux'"},{"id":"pfr/further-improvement-to-exponent/studying-a-minimizer/pfr-9","kind":"node","lean":"better_PFR_conjecture","parent":5139,"reading_title":{"format":"inline-parts-v1","parts":[["text","PFR with "],["math","\\(C=9\\)"]],"source":"PFR with $C=9$"},"result":true,"route":"../roadmap/pfr/further-improvement-to-exponent/studying-a-minimizer/index.html#pfr-9","scope":"pfr/further-improvement-to-exponent/studying-a-minimizer","status":"fully_proved","summary":"If $A \\subset {\\bf F}_2^n$ is finite non-empty with $|A+A| \\leq K|A|$, then there exists a subgroup $H$ of ${\\bf F}_2^n$ with $|H| \\leq |A|$ such that $A$ can be covered by at most $2K^9$ translates of $H$.","title":"PFR with $C=9$"},{"data_url":"chapters/equational-theories.json","id":"equational-theories","kind":"scope","parent":0,"result":false,"route":"../roadmap/equational-theories/index.html","scope":"equational-theories","status":"stated","summary":"The blueprint of Equational Theories, read at commit 8b87204d (2026-11-08): one article for each of its 137 nodes whose Lean declarations it names and marks formalized, or in Mathlib. Its other 77 nodes have no article, and a section of only those is left out: the map records what the project contains, not what it plans. A statement for which the blueprint marks no proof `\\leanok` says so. Dependencies are the\u2026","title":"Equational Theories"},{"data_url":"scopes/equational-theories/part-01.json","id":"equational-theories/part-01","kind":"scope","navigation":"transparent","parent":5154,"result":false,"route":"../roadmap/equational-theories/part-01/index.html","scope":"equational-theories/part-01","status":"stated","title":"Basic theory of magmas to Enumerating Small Finite Magmas"},{"data_url":"scopes/equational-theories/part-01/basic-theory-of-magmas.json","id":"equational-theories/part-01/basic-theory-of-magmas","kind":"scope","parent":5155,"result":false,"route":"../roadmap/equational-theories/part-01/basic-theory-of-magmas/index.html","scope":"equational-theories/part-01/basic-theory-of-magmas","status":"fully_proved","summary":"Groups, semi-groups, and monoids are familiar examples of magmas. However, in general we do not expect magmas to have any associative properties. In some literature, magmas are also known as groupoids, although this term is also used for a slightly different object (a category with inverses).","title":"Basic theory of magmas"},{"id":"equational-theories/part-01/basic-theory-of-magmas/magma-def","kind":"node","lean":"Magma","parent":5156,"result":true,"route":"../roadmap/equational-theories/part-01/basic-theory-of-magmas/index.html#magma-def","scope":"equational-theories/part-01/basic-theory-of-magmas","status":"fully_proved","summary":"A *magma* is a set $G$ equipped with a binary operation $\\diamond: G \\times G \\to G$. A *homomorphism* $\\varphi : G \\to H$ between two magmas is a map such that $\\varphi(x \\diamond y) = \\varphi(x) \\diamond \\varphi(y)$ for all $x,y \\in G$. An *isomorphism* is an invertible homomorphism.","title":"Magma"},{"id":"equational-theories/part-01/basic-theory-of-magmas/free-magma-def","kind":"node","lean":"FreeMagma","parent":5156,"result":true,"route":"../roadmap/equational-theories/part-01/basic-theory-of-magmas/index.html#free-magma-def","scope":"equational-theories/part-01/basic-theory-of-magmas","status":"fully_proved","summary":"The *free magma* $M_X$ generated by a set $X$ (which we call an *alphabet*) is the set of all finite formal expressions built from elements of $X$ and the operation $\\diamond$. An element of $M_X$ will be called a *word* with alphabet $X$. The *order* of a word is the number of $\\diamond$ symbols needed to generate the word. Thus for instance $X$ is precisely the set of words of order $0$ in $M_X$.","title":"Free Magma"},{"id":"equational-theories/part-01/basic-theory-of-magmas/lemma-intro-1","kind":"node","lean":"FreeMagma.elementsOfNumNodesEq_card_eq_catalan_mul_pow","parent":5156,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lemma: For a finite alphabet "],["math","\\(X\\)"],["text",", the number of\u2026"]],"source":"Lemma: For a finite alphabet $X$, the number of\u2026"},"result":true,"route":"../roadmap/equational-theories/part-01/basic-theory-of-magmas/index.html#lemma-intro-1","scope":"equational-theories/part-01/basic-theory-of-magmas","status":"fully_proved","summary":"For a finite alphabet $X$, the number of words of order $n$ is $C_n |X|^{n+1}$, where $C_n$ is the $n^{\\mathrm{th}}$ Catalan number and $X$ is the cardinality of $X$.","title":"Lemma: For a finite alphabet $X$, the number of\u2026"},{"id":"equational-theories/part-01/basic-theory-of-magmas/law-def","kind":"node","lean":"Law.MagmaLaw","parent":5156,"result":true,"route":"../roadmap/equational-theories/part-01/basic-theory-of-magmas/index.html#law-def","scope":"equational-theories/part-01/basic-theory-of-magmas","status":"fully_proved","summary":"Let $X$ be a set. A *law* with alphabet $X$ is a formal expression of the form $w \\simeq w'$, where $w, w' \\in M_X$ are words with alphabet $X$ (thus one can identify laws with alphabet $X$ with elements of $M_X \\times M_X$). A magma $G$ *satisfies* the law $w \\simeq w'$ if we have $\\varphi_f( w ) = \\varphi_f ( w' )$ for all $f: X \\to G$, in which case we write $G \\models w \\simeq w'$.","title":"Law"},{"id":"equational-theories/part-01/basic-theory-of-magmas/models-def","kind":"node","lean":"models","parent":5156,"result":true,"route":"../roadmap/equational-theories/part-01/basic-theory-of-magmas/index.html#models-def","scope":"equational-theories/part-01/basic-theory-of-magmas","status":"fully_proved","summary":"A *theory* is a set $\\Gamma$ of laws. Given a theory $\\Gamma$, a magma $G$ is a *model* of $\\Gamma$ with the (overloaded) notation $G\\models\\Gamma$ if $G\\models w\\simeq w'$ for every $w\\simeq w'$ in $\\Gamma$; we also say that $G$ *satisfies* $\\Gamma$. Given a law $E$, we write $\\Gamma \\models E$ if every magma $G$ that models $\\Gamma$, also models $E$.","title":"Models"},{"id":"equational-theories/part-01/basic-theory-of-magmas/derivation-def","kind":"node","lean":"derive","parent":5156,"result":true,"route":"../roadmap/equational-theories/part-01/basic-theory-of-magmas/index.html#derivation-def","scope":"equational-theories/part-01/basic-theory-of-magmas","status":"fully_proved","summary":"Given a theory $\\Gamma$ and a law $w\\simeq w'$ over a fixed alphabet $X$, we say that $\\Gamma$ *derives* $w\\simeq w'$, and write $\\Gamma\\vdash w\\simeq w'$, if the law can be obtained using a finite number of applications of the following rules:","title":"Derivation"},{"id":"equational-theories/part-01/basic-theory-of-magmas/sound-complete","kind":"node","lean":"Completeness","parent":5156,"result":true,"route":"../roadmap/equational-theories/part-01/basic-theory-of-magmas/index.html#sound-complete","scope":"equational-theories/part-01/basic-theory-of-magmas","status":"fully_proved","summary":"For any theory $\\Gamma$ and words $w, w'$ over a fixed alphabet","title":"Birkhoff's completeness theorem"},{"id":"equational-theories/part-01/basic-theory-of-magmas/alternate-formula","kind":"node","parent":5156,"result":false,"route":"../roadmap/equational-theories/part-01/basic-theory-of-magmas/alternate-formula/index.html","scope":"equational-theories/part-01/basic-theory-of-magmas","status":"can_state","summary":"For integers $0 \\le a \\le b$ let $E(a,b)$ be the number of magma equations $t_1 = t_2$ with $t_1$ of order $a$, $t_2$ of order $b$, counting up to relabelling, up to switching terms, and only allowing the equation $x=x$ of the form $t=t$. Then one has","title":"Alternate formula"},{"data_url":"scopes/equational-theories/part-01/selected-laws.json","id":"equational-theories/part-01/selected-laws","kind":"scope","parent":5155,"result":false,"route":"../roadmap/equational-theories/part-01/selected-laws/index.html","scope":"equational-theories/part-01/selected-laws","status":"fully_proved","summary":"In this project we study the 4694 laws (up to symmetry and relabeling) of total order at most $4$.","title":"Selected laws"},{"data_url":"scopes/equational-theories/part-01/selected-laws/part-01.json","id":"equational-theories/part-01/selected-laws/part-01","kind":"scope","navigation":"transparent","parent":5165,"result":false,"route":"../roadmap/equational-theories/part-01/selected-laws/part-01/index.html","scope":"equational-theories/part-01/selected-laws/part-01","status":"fully_proved","summary":"This is the trivial law, satisfied by all magmas. It is self-dual.","title":"Equation 1 to Equation 46"},{"id":"equational-theories/part-01/selected-laws/part-01/eq1","kind":"node","lean":"Equation1","parent":5166,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-01/index.html#eq1","scope":"equational-theories/part-01/selected-laws/part-01","status":"fully_proved","summary":"Equation 1 is the law $0 \\simeq 0$ (or the equation $x=x$).","title":"Equation 1"},{"id":"equational-theories/part-01/selected-laws/part-01/eq2","kind":"node","lean":"Equation2","parent":5166,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-01/index.html#eq2","scope":"equational-theories/part-01/selected-laws/part-01","status":"fully_proved","summary":"Equation 2 is the law $0 \\simeq 1$ (or the equation $x=y$).","title":"Equation 2"},{"id":"equational-theories/part-01/selected-laws/part-01/eq3","kind":"node","lean":"Equation3","parent":5166,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-01/index.html#eq3","scope":"equational-theories/part-01/selected-laws/part-01","status":"fully_proved","summary":"Equation 3 is the law $0 \\simeq 0 \\diamond 0$ (or the equation $x = x \\diamond x$).","title":"Equation 3"},{"id":"equational-theories/part-01/selected-laws/part-01/eq4","kind":"node","lean":"Equation4","parent":5166,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-01/index.html#eq4","scope":"equational-theories/part-01/selected-laws/part-01","status":"fully_proved","summary":"Equation 4 is the law $0 \\simeq 0 \\diamond 1$ (or the equation $x = x \\diamond y$).","title":"Equation 4"},{"id":"equational-theories/part-01/selected-laws/part-01/eq5","kind":"node","lean":"Equation5","parent":5166,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-01/index.html#eq5","scope":"equational-theories/part-01/selected-laws/part-01","status":"fully_proved","summary":"Equation 5 is the law $0 \\simeq 1 \\diamond 0$ (or the equation $x = y \\diamond x$).","title":"Equation 5"},{"id":"equational-theories/part-01/selected-laws/part-01/eq6","kind":"node","lean":"Equation6","parent":5166,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-01/index.html#eq6","scope":"equational-theories/part-01/selected-laws/part-01","status":"fully_proved","summary":"Equation 6 is the law $0 \\simeq 1 \\diamond 1$ (or the equation $x = y \\diamond y$).","title":"Equation 6"},{"id":"equational-theories/part-01/selected-laws/part-01/eq7","kind":"node","lean":"Equation7","parent":5166,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-01/index.html#eq7","scope":"equational-theories/part-01/selected-laws/part-01","status":"fully_proved","summary":"Equation 7 is the law $0 \\simeq 1 \\diamond 2$ (or the equation $x = y \\diamond z$).","title":"Equation 7"},{"id":"equational-theories/part-01/selected-laws/part-01/eq8","kind":"node","lean":"Equation8","parent":5166,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-01/index.html#eq8","scope":"equational-theories/part-01/selected-laws/part-01","status":"fully_proved","summary":"Equation 8 is the law $0 \\simeq 0 \\diamond (0 \\diamond 0)$ (or the equation $x = x \\diamond (x \\diamond x)$).","title":"Equation 8"},{"id":"equational-theories/part-01/selected-laws/part-01/eq14","kind":"node","lean":"Equation14","parent":5166,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-01/index.html#eq14","scope":"equational-theories/part-01/selected-laws/part-01","status":"fully_proved","summary":"Equation 14 is the law $0 \\simeq 1 \\diamond (0 \\diamond 1)$ (or the equation $x = y \\diamond (x \\diamond y))$.","title":"Equation 14"},{"id":"equational-theories/part-01/selected-laws/part-01/eq16","kind":"node","lean":"Equation16","parent":5166,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-01/index.html#eq16","scope":"equational-theories/part-01/selected-laws/part-01","status":"fully_proved","summary":"Equation 16 is the law $0 \\simeq 1 \\diamond (1 \\diamond 0)$ (or the equation $x = y \\diamond (y \\diamond x))$.","title":"Equation 16"},{"id":"equational-theories/part-01/selected-laws/part-01/eq23","kind":"node","lean":"Equation23","parent":5166,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-01/index.html#eq23","scope":"equational-theories/part-01/selected-laws/part-01","status":"fully_proved","summary":"Equation 23 is the law $0 \\simeq (0 \\diamond 0) \\diamond 0$ (or the equation $x = (x \\diamond x) \\diamond x$).","title":"Equation 23"},{"id":"equational-theories/part-01/selected-laws/part-01/eq29","kind":"node","lean":"Equation29","parent":5166,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-01/index.html#eq29","scope":"equational-theories/part-01/selected-laws/part-01","status":"fully_proved","summary":"Equation 29 is the law $0 \\simeq (1 \\diamond 0) \\diamond 1$ (or the equation $x = (y \\diamond x) \\diamond y)$.","title":"Equation 29"},{"id":"equational-theories/part-01/selected-laws/part-01/eq38","kind":"node","lean":"Equation38","parent":5166,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-01/index.html#eq38","scope":"equational-theories/part-01/selected-laws/part-01","status":"fully_proved","summary":"Equation 38 is the law $0 \\diamond 0 \\simeq 0 \\diamond 1$ (or the equation $x \\diamond x = x \\diamond y$).","title":"Equation 38"},{"id":"equational-theories/part-01/selected-laws/part-01/eq39","kind":"node","lean":"Equation39","parent":5166,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-01/index.html#eq39","scope":"equational-theories/part-01/selected-laws/part-01","status":"fully_proved","summary":"Equation 39 is the law $0 \\diamond 0 \\simeq 1 \\diamond 0$ (or the equation $x \\diamond x = y \\diamond x$).","title":"Equation 39"},{"id":"equational-theories/part-01/selected-laws/part-01/eq40","kind":"node","lean":"Equation40","parent":5166,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-01/index.html#eq40","scope":"equational-theories/part-01/selected-laws/part-01","status":"fully_proved","summary":"Equation 40 is the law $0 \\diamond 0 \\simeq 1 \\diamond 1$ (or the equation $x \\diamond x = y \\diamond y$).","title":"Equation 40"},{"id":"equational-theories/part-01/selected-laws/part-01/eq41","kind":"node","lean":"Equation41","parent":5166,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-01/index.html#eq41","scope":"equational-theories/part-01/selected-laws/part-01","status":"fully_proved","summary":"Equation 41 is the law $0 \\diamond 0 \\simeq 1 \\diamond 2$ (or the equation $x \\diamond x = y \\diamond z$).","title":"Equation 41"},{"id":"equational-theories/part-01/selected-laws/part-01/eq42","kind":"node","lean":"Equation42","parent":5166,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-01/index.html#eq42","scope":"equational-theories/part-01/selected-laws/part-01","status":"fully_proved","summary":"Equation 42 is the law $0 \\diamond 1 \\simeq 0 \\diamond 2$ (or the equation $x \\diamond y = x \\diamond z$).","title":"Equation 42"},{"id":"equational-theories/part-01/selected-laws/part-01/eq43","kind":"node","lean":"Equation43","parent":5166,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-01/index.html#eq43","scope":"equational-theories/part-01/selected-laws/part-01","status":"fully_proved","summary":"Equation 43 is the law $0 \\diamond 1 \\simeq 1 \\diamond 0$ (or the equation $x \\diamond y = y \\diamond x$).","title":"Equation 43"},{"id":"equational-theories/part-01/selected-laws/part-01/eq45","kind":"node","lean":"Equation45","parent":5166,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-01/index.html#eq45","scope":"equational-theories/part-01/selected-laws/part-01","status":"fully_proved","summary":"Equation 45 is the law $0 \\diamond 1 \\simeq 2 \\diamond 1$ (or the equation $x \\diamond y = z \\diamond y$).","title":"Equation 45"},{"id":"equational-theories/part-01/selected-laws/part-01/eq46","kind":"node","lean":"Equation46","parent":5166,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-01/index.html#eq46","scope":"equational-theories/part-01/selected-laws/part-01","status":"fully_proved","summary":"Equation 46 is the law $0 \\diamond 1 \\simeq 2 \\diamond 3$ (or the equation $x \\diamond y = z \\diamond w$).","title":"Equation 46"},{"data_url":"scopes/equational-theories/part-01/selected-laws/part-02.json","id":"equational-theories/part-01/selected-laws/part-02","kind":"scope","navigation":"transparent","parent":5165,"result":false,"route":"../roadmap/equational-theories/part-01/selected-laws/part-02/index.html","scope":"equational-theories/part-01/selected-laws/part-02","status":"fully_proved","summary":"The constant law: all products are constant. It is self-dual.","title":"Equation 63 to Equation 3167"},{"id":"equational-theories/part-01/selected-laws/part-02/eq63","kind":"node","lean":"Equation63","parent":5187,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-02/index.html#eq63","scope":"equational-theories/part-01/selected-laws/part-02","status":"fully_proved","summary":"Equation 63 is the law $0 \\simeq 1 \\diamond (0 \\diamond (0 \\diamond 1))$ (or the equation $x = y \\diamond (x \\diamond (x \\diamond y))$).","title":"Equation 63"},{"id":"equational-theories/part-01/selected-laws/part-02/eq65","kind":"node","lean":"Equation65","parent":5187,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-02/index.html#eq65","scope":"equational-theories/part-01/selected-laws/part-02","status":"fully_proved","summary":"Equation 65 is the law $0 \\simeq 1 \\diamond (0 \\diamond (1 \\diamond 0))$ (or the equation $x = y \\diamond (x \\diamond (y \\diamond x))$).","title":"Equation 65"},{"id":"equational-theories/part-01/selected-laws/part-02/eq168","kind":"node","lean":"Equation168","parent":5187,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-02/index.html#eq168","scope":"equational-theories/part-01/selected-laws/part-02","status":"fully_proved","summary":"Equation 168 is the law $0 \\simeq (1 \\diamond 0) \\diamond (0 \\diamond 2)$ (or the equation $x = (y \\diamond x) \\diamond (x \\diamond z)$).","title":"Equation 168"},{"id":"equational-theories/part-01/selected-laws/part-02/eq206","kind":"node","lean":"Equation206","parent":5187,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-02/index.html#eq206","scope":"equational-theories/part-01/selected-laws/part-02","status":"fully_proved","summary":"Equation 206 is the law $0 \\simeq (0 \\diamond (0 \\diamond 1)) \\diamond 1$ (or the equation $x = (x \\diamond (x \\diamond y)) \\diamond y$).","title":"Equation 206"},{"id":"equational-theories/part-01/selected-laws/part-02/eq381","kind":"node","lean":"Equation381","parent":5187,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-02/index.html#eq381","scope":"equational-theories/part-01/selected-laws/part-02","status":"fully_proved","summary":"Equation 381 is the law $0 \\diamond 1 \\simeq (0 \\diamond 2) \\diamond 1$ (or the equation $x \\diamond y = (x \\diamond z) \\diamond y$).","title":"Equation 381"},{"id":"equational-theories/part-01/selected-laws/part-02/eq387","kind":"node","lean":"Equation387","parent":5187,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-02/index.html#eq387","scope":"equational-theories/part-01/selected-laws/part-02","status":"fully_proved","summary":"Equation 387 is the law $0 \\diamond 1 \\simeq (1 \\diamond 1) \\diamond 0$ (or the equation $x \\diamond y = (y \\diamond y) \\diamond x$).","title":"Equation 387"},{"id":"equational-theories/part-01/selected-laws/part-02/eq477","kind":"node","lean":"Equation477","parent":5187,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-02/index.html#eq477","scope":"equational-theories/part-01/selected-laws/part-02","status":"fully_proved","summary":"Equation 477 is the law $0 \\simeq 1 \\diamond (0 \\diamond (1 \\diamond (1 \\diamond 1)))$ (or the equation $x = y \\diamond (x \\diamond (y \\diamond (y \\diamond y)))$).","title":"Equation 477"},{"id":"equational-theories/part-01/selected-laws/part-02/eq854","kind":"node","lean":"Equation953","parent":5187,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-02/index.html#eq854","scope":"equational-theories/part-01/selected-laws/part-02","status":"fully_proved","summary":"Equation 854 is the law $0 = 0 \\diamond ((1 \\diamond 2) \\diamond (0 \\diamond 2))$ (or the equation $x = x \\diamond ((y \\diamond z) \\diamond (x \\diamond z))$).","title":"Equation 854"},{"id":"equational-theories/part-01/selected-laws/part-02/eq953","kind":"node","lean":"Equation953","parent":5187,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-02/index.html#eq953","scope":"equational-theories/part-01/selected-laws/part-02","status":"fully_proved","summary":"Equation 953 is the law $0 = 1 \\diamond ((2 \\diamond 0) \\diamond (2 \\diamond 2))$ (or the equation $x = y \\diamond ((z \\diamond x) \\diamond (z \\diamond z))$).","title":"Equation 953"},{"id":"equational-theories/part-01/selected-laws/part-02/eq1485","kind":"node","lean":"Equation1491","parent":5187,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-02/index.html#eq1485","scope":"equational-theories/part-01/selected-laws/part-02","status":"fully_proved","summary":"Equation 1485 is the law $0 \\simeq (1 \\diamond 0) \\diamond (0 \\diamond (2 \\diamond 1))$ (or the equation $x = (y \\diamond x) \\diamond (x \\diamond (z \\diamond y))$).","title":"Equation 1485"},{"id":"equational-theories/part-01/selected-laws/part-02/eq1491","kind":"node","lean":"Equation1491","parent":5187,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-02/index.html#eq1491","scope":"equational-theories/part-01/selected-laws/part-02","status":"fully_proved","summary":"Equation 1491 is the law $0 \\simeq (1 \\diamond 0) \\diamond (1 \\diamond (1 \\diamond 0))$ (or the equation $x = (y \\diamond x) \\diamond (y \\diamond (y \\diamond x))$).","title":"Equation 1491"},{"id":"equational-theories/part-01/selected-laws/part-02/eq1571","kind":"node","lean":"Equation1571","parent":5187,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-02/index.html#eq1571","scope":"equational-theories/part-01/selected-laws/part-02","status":"fully_proved","summary":"Equation 1571 is the law $0 \\simeq (1 \\diamond 2) \\diamond (1 \\diamond (0 \\diamond 2))$ (or the equation $x = (y \\diamond z) \\diamond (y \\diamond (x \\diamond z))$).","title":"Equation 1571"},{"id":"equational-theories/part-01/selected-laws/part-02/eq1648","kind":"node","lean":"Equation1648","parent":5187,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-02/index.html#eq1648","scope":"equational-theories/part-01/selected-laws/part-02","status":"fully_proved","summary":"Equation 1648 is the law $0 \\simeq (0 \\diamond 1) \\diamond ((0 \\diamond 1) \\diamond 1)$ (or the equation $x = (x \\diamond y) \\diamond ((x \\diamond y) \\diamond y)$).","title":"Equation 1648"},{"id":"equational-theories/part-01/selected-laws/part-02/eq1657","kind":"node","lean":"Equation1657","parent":5187,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-02/index.html#eq1657","scope":"equational-theories/part-01/selected-laws/part-02","status":"fully_proved","summary":"Equation 1657 is the law $0 \\simeq (0 \\diamond 1) \\diamond ((1 \\diamond 1) \\diamond 0)$ (or the equation $x = (x \\diamond y) \\diamond ((y \\diamond y) \\diamond x)$).","title":"Equation 1657"},{"id":"equational-theories/part-01/selected-laws/part-02/eq1659","kind":"node","lean":"Equation1659","parent":5187,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-02/index.html#eq1659","scope":"equational-theories/part-01/selected-laws/part-02","status":"fully_proved","summary":"Equation 1659 is the law $0 \\simeq (0 \\diamond 1) \\diamond ((1 \\diamond 1) \\diamond 2)$ (or the equation $x = (x \\diamond y) \\diamond ((y \\diamond y) \\diamond z)$).","title":"Equation 1659"},{"id":"equational-theories/part-01/selected-laws/part-02/eq1661","kind":"node","lean":"Equation1661","parent":5187,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-02/index.html#eq1661","scope":"equational-theories/part-01/selected-laws/part-02","status":"fully_proved","summary":"Equation 1661 is the law $0 \\simeq (0 \\diamond 1) \\diamond ((1 \\diamond 2) \\diamond 1)$ (or the equation $x = (x \\diamond y) \\diamond ((y \\diamond z) \\diamond y)$).","title":"Equation 1661"},{"id":"equational-theories/part-01/selected-laws/part-02/eq1689","kind":"node","lean":"Equation1689","parent":5187,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-02/index.html#eq1689","scope":"equational-theories/part-01/selected-laws/part-02","status":"fully_proved","summary":"Equation 1689 is the law $0 \\simeq (1 \\diamond 0) \\diamond ((0 \\diamond 2) \\diamond 2)$ (or the equation $x = (y \\diamond x) \\diamond ((x \\diamond z) \\diamond z)$).","title":"Equation 1689"},{"id":"equational-theories/part-01/selected-laws/part-02/eq1701","kind":"node","lean":"Equation1701","parent":5187,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-02/index.html#eq1701","scope":"equational-theories/part-01/selected-laws/part-02","status":"fully_proved","summary":"Equation 1701 is the law $0 \\simeq (1 \\diamond x) \\diamond ((2 \\diamond 0) \\diamond 0)$ (or the equation $x = (y \\diamond x) \\diamond ((z \\diamond x) \\diamond x)$).","title":"Equation 1701"},{"id":"equational-theories/part-01/selected-laws/part-02/eq2662","kind":"node","lean":"Equation2662","parent":5187,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-02/index.html#eq2662","scope":"equational-theories/part-01/selected-laws/part-02","status":"fully_proved","summary":"Equation 2662 is the law $0 \\simeq ((0 \\diamond 1) \\diamond (0 \\diamond 1)) \\diamond 0$ (or the equation $x = ((x \\diamond y) \\diamond (x \\diamond y)) \\diamond x$).","title":"Equation 2662"},{"id":"equational-theories/part-01/selected-laws/part-02/eq3167","kind":"node","lean":"Equation3167","parent":5187,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-02/index.html#eq3167","scope":"equational-theories/part-01/selected-laws/part-02","status":"fully_proved","summary":"Equation 3167 is the law $0 \\simeq (((1 \\diamond 1) \\diamond 2) \\diamond 2) \\diamond 0$ (or the equation $x = (((y \\diamond y) \\diamond z) \\diamond z) \\diamond x$).","title":"Equation 3167"},{"data_url":"scopes/equational-theories/part-01/selected-laws/part-03.json","id":"equational-theories/part-01/selected-laws/part-03","kind":"scope","navigation":"transparent","parent":5165,"result":false,"route":"../roadmap/equational-theories/part-01/selected-laws/part-03/index.html","scope":"equational-theories/part-01/selected-laws/part-03","status":"fully_proved","summary":"Our project located this law as one member of an \u201cAustin pair\u201d; see Finite implications and infinite models.","title":"Equation 3588 to Equations of order greater than\u2026"},{"id":"equational-theories/part-01/selected-laws/part-03/eq3588","kind":"node","lean":"Equation3588","parent":5208,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-03/index.html#eq3588","scope":"equational-theories/part-01/selected-laws/part-03","status":"fully_proved","summary":"Equation 3588 is the law $0 \\diamond 1 \\simeq 2 \\diamond ((0 \\diamond 1) \\diamond 2)$ (or the equation $x \\diamond y = z \\diamond ((x \\diamond y) \\diamond z)$).","title":"Equation 3588"},{"id":"equational-theories/part-01/selected-laws/part-03/eq3722","kind":"node","lean":"Equation3722","parent":5208,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-03/index.html#eq3722","scope":"equational-theories/part-01/selected-laws/part-03","status":"fully_proved","summary":"Equation 3722 is the law $0 \\diamond 1 \\simeq (0 \\diamond 1) \\diamond (0 \\diamond 1)$ (or the equation $x \\diamond y = (x \\diamond y) \\diamond (x \\diamond y)$).","title":"Equation 3722"},{"id":"equational-theories/part-01/selected-laws/part-03/eq3744","kind":"node","lean":"Equation3744","parent":5208,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-03/index.html#eq3744","scope":"equational-theories/part-01/selected-laws/part-03","status":"fully_proved","summary":"Equation 3744 is the law $0 \\diamond 1 \\simeq (0 \\diamond 2) \\diamond (3 \\diamond 1)$ (or the equation $x \\diamond y = (x \\diamond z) \\diamond (w \\diamond y)$).","title":"Equation 3744"},{"id":"equational-theories/part-01/selected-laws/part-03/eq4315","kind":"node","lean":"Equation4315","parent":5208,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-03/index.html#eq4315","scope":"equational-theories/part-01/selected-laws/part-03","status":"fully_proved","summary":"Equation 4315 is the law $0 \\diamond (1 \\diamond 0) \\simeq 0 \\diamond (1 \\diamond 2)$ (or the equation $x \\diamond (y \\diamond x) = x \\diamond (y \\diamond z)$).","title":"Equation 4315"},{"id":"equational-theories/part-01/selected-laws/part-03/eq4512","kind":"node","lean":"Equation4512","parent":5208,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-03/index.html#eq4512","scope":"equational-theories/part-01/selected-laws/part-03","status":"fully_proved","summary":"Equation 4512 is the law $0 \\diamond (1 \\diamond 2) \\simeq (0 \\diamond 1) \\diamond 2$ (or the equation $x \\diamond (y \\diamond z) = (x \\diamond y) \\diamond z$).","title":"Equation 4512"},{"id":"equational-theories/part-01/selected-laws/part-03/eq4513","kind":"node","lean":"Equation4513","parent":5208,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-03/index.html#eq4513","scope":"equational-theories/part-01/selected-laws/part-03","status":"fully_proved","summary":"Equation 4513 is the law $0 \\diamond (1 \\diamond 2) \\simeq (0 \\diamond 1) \\diamond 3$ (or the equation $x \\diamond (y \\diamond z) = (x \\diamond y) \\diamond w$).","title":"Equation 4513"},{"id":"equational-theories/part-01/selected-laws/part-03/eq4522","kind":"node","lean":"Equation4522","parent":5208,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-03/index.html#eq4522","scope":"equational-theories/part-01/selected-laws/part-03","status":"fully_proved","summary":"Equation 4522 is the law $0 \\diamond (1 \\diamond 2) \\simeq (0 \\diamond 3) \\diamond 4$ (or the equation $x \\diamond (y \\diamond z) = (x \\diamond w) \\diamond u$).","title":"Equation 4522"},{"id":"equational-theories/part-01/selected-laws/part-03/eq4564","kind":"node","lean":"Equation4564","parent":5208,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-03/index.html#eq4564","scope":"equational-theories/part-01/selected-laws/part-03","status":"fully_proved","summary":"Equation 4564 is the law $0 \\diamond (1 \\diamond 2) \\simeq (3 \\diamond 1) \\diamond 2$ (or the equation $x \\diamond (y \\diamond z) = (w \\diamond y) \\diamond z$).","title":"Equation 4564"},{"id":"equational-theories/part-01/selected-laws/part-03/eq4579","kind":"node","lean":"Equation4579","parent":5208,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-03/index.html#eq4579","scope":"equational-theories/part-01/selected-laws/part-03","status":"fully_proved","summary":"Equation 4579 is the law $0 \\diamond (1 \\diamond 2) \\simeq (3 \\diamond 4) \\diamond 2$ (or the equation $x \\diamond (y \\diamond z) = (w \\diamond u) \\diamond z$).","title":"Equation 4579"},{"id":"equational-theories/part-01/selected-laws/part-03/eq4582","kind":"node","lean":"Equation4582","parent":5208,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-03/index.html#eq4582","scope":"equational-theories/part-01/selected-laws/part-03","status":"fully_proved","summary":"Equation 4582 is the law $0 \\diamond (1 \\diamond 2) \\simeq (3 \\diamond 4) \\diamond 5$ (or the equation $x \\diamond (y \\diamond z) = (w \\diamond u) \\diamond v$).","title":"Equation 4582"},{"data_url":"scopes/equational-theories/part-01/selected-laws/part-03/equations-of-order-greater-than-4.json","id":"equational-theories/part-01/selected-laws/part-03/equations-of-order-greater-than-4","kind":"scope","parent":5208,"reading_title":{"format":"inline-parts-v1","parts":[["text","Equations of order greater than "],["math","\\(4\\)"]],"source":"Equations of order greater than $4$"},"result":false,"route":"../roadmap/equational-theories/part-01/selected-laws/part-03/equations-of-order-greater-than-4/index.html","scope":"equational-theories/part-01/selected-laws/part-03/equations-of-order-greater-than-4","status":"fully_proved","summary":"We note some selected laws of order more than $5$, which are used in some later chapters of the blueprint.","title":"Equations of order greater than $4$"},{"id":"equational-theories/part-01/selected-laws/part-03/equations-of-order-greater-than-4/eq5093","kind":"node","lean":"Equation5093","parent":5219,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-03/equations-of-order-greater-than-4/index.html#eq5093","scope":"equational-theories/part-01/selected-laws/part-03/equations-of-order-greater-than-4","status":"fully_proved","summary":"Equation 5093 is the law $0 \\simeq 1 \\diamond (1 \\diamond (1 \\diamond (0 \\diamond (2 \\diamond 1))))$ (or the equation $x = y \\diamond (y \\diamond (y \\diamond (x \\diamond (z \\diamond y))))$).","title":"Equation 5093"},{"id":"equational-theories/part-01/selected-laws/part-03/equations-of-order-greater-than-4/eq28770","kind":"node","lean":"Equation28770","parent":5219,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-03/equations-of-order-greater-than-4/index.html#eq28770","scope":"equational-theories/part-01/selected-laws/part-03/equations-of-order-greater-than-4","status":"fully_proved","summary":"Equation 28770 is the law $0 \\simeq (((1 \\diamond 1) \\diamond 1) \\diamond 0) \\diamond (1 \\diamond 2)$ (or the equation $x = (((y \\diamond y) \\diamond y) \\diamond x) \\diamond (y \\diamond z)$).","title":"Equation 28770"},{"id":"equational-theories/part-01/selected-laws/part-03/equations-of-order-greater-than-4/eq374794","kind":"node","lean":"Equation374794","parent":5219,"result":true,"route":"../roadmap/equational-theories/part-01/selected-laws/part-03/equations-of-order-greater-than-4/index.html#eq374794","scope":"equational-theories/part-01/selected-laws/part-03/equations-of-order-greater-than-4","status":"fully_proved","summary":"Equation 374794 is the law $0 \\simeq (((1 \\diamond 1) \\diamond 1) \\diamond 0) \\diamond ((1 \\diamond 1) \\diamond 2)$ (or the equation $x = (((y \\diamond y) \\diamond y) \\diamond x) \\diamond ((y \\diamond y) \\diamond z)$).","title":"Equation 374794"},{"data_url":"scopes/equational-theories/part-01/general-implications.json","id":"equational-theories/part-01/general-implications","kind":"scope","parent":5155,"result":false,"route":"../roadmap/equational-theories/part-01/general-implications/index.html","scope":"equational-theories/part-01/general-implications","status":"can_prove","summary":"We will be interested in seeing which laws imply which other laws, in the sense that magmas satisfying the former law automatically satisfy the latter. We will also be interested in *anti-implications* showing that one law does *not* imply another, by producing examples of magmas that satisfy the former law but not the latter. Here is a formal definition.","title":"General implications"},{"id":"equational-theories/part-01/general-implications/maximal","kind":"node","lean":"Law.MagmaLaw.Equation1_maximal","parent":5223,"result":true,"route":"../roadmap/equational-theories/part-01/general-implications/index.html#maximal","scope":"equational-theories/part-01/general-implications","status":"can_prove","summary":"The law $0 \\simeq 0$ is the maximal element in this pre-order.","title":"Maximal element"},{"id":"equational-theories/part-01/general-implications/minimal","kind":"node","lean":"Law.MagmaLaw.Equation2_minimal","parent":5223,"result":true,"route":"../roadmap/equational-theories/part-01/general-implications/index.html#minimal","scope":"equational-theories/part-01/general-implications","status":"can_prove","summary":"The law $0 \\simeq 1$ is the minimal element in this pre-order.","title":"Minimal element"},{"id":"equational-theories/part-01/general-implications/duality","kind":"node","lean":"Law.MagmaLaw.implies_iff_dual","parent":5223,"result":true,"route":"../roadmap/equational-theories/part-01/general-implications/index.html#duality","scope":"equational-theories/part-01/general-implications","status":"can_prove","summary":"The law $w \\simeq w'$ implies $w''\\simeq w'''$, if and only if $w^{\\mathrm{op}} \\simeq (w')^{\\mathrm{op}}$ implies $w''^{\\mathrm{op}} \\simeq (w''')^{\\mathrm{op}}$.","title":"Duality of laws"},{"id":"equational-theories/part-01/general-implications/variable-impl","kind":"node","lean":"Law.MagmaLaw.SameCount.derive","parent":5223,"result":true,"route":"../roadmap/equational-theories/part-01/general-implications/index.html#variable-impl","scope":"equational-theories/part-01/general-implications","status":"can_prove","summary":"If $w \\simeq w'$ is such that every variable appears the same number of times in both $w$ and $w'$, and $w \\simeq w'$ implies another law $w'' \\simeq w'''$, then every variable appears the same number of times in both $w''$ and $w'''$.","title":"Criterion for implication"},{"data_url":"scopes/equational-theories/part-01/implications-between-selected-laws.json","id":"equational-theories/part-01/implications-between-selected-laws","kind":"scope","parent":5155,"result":false,"route":"../roadmap/equational-theories/part-01/implications-between-selected-laws/index.html","scope":"equational-theories/part-01/implications-between-selected-laws","status":"proved","summary":"We collect here some notable implications between the selected laws in Selected laws. By Birkhoff's completeness theorem, every implication can basically be established by a finite number of rewrites. In most cases, the sequence of rewrites is quite straightforward, and the implication is very easy, but we record some less obvious examples.","title":"Implications between selected laws"},{"id":"equational-theories/part-01/implications-between-selected-laws/387-implies-43","kind":"node","lean":"Subgraph.Equation387_implies_Equation43","parent":5228,"result":true,"route":"../roadmap/equational-theories/part-01/implications-between-selected-laws/index.html#387-implies-43","scope":"equational-theories/part-01/implications-between-selected-laws","status":"fully_proved","summary":"E387 $({x \\diamond y = (y \\diamond y) \\diamond x})$ implies E43 $({x \\diamond y = y \\diamond x})$.","title":"387 implies 43"},{"id":"equational-theories/part-01/implications-between-selected-laws/29-equiv-14","kind":"node","lean":"Subgraph.Equation29_implies_Equation14","parent":5228,"result":true,"route":"../roadmap/equational-theories/part-01/implications-between-selected-laws/index.html#29-equiv-14","scope":"equational-theories/part-01/implications-between-selected-laws","status":"proved","summary":"E29 $({x = (y \\diamond x) \\diamond y)})$ is equivalent to E14 $({x = y \\diamond (x \\diamond y))})$.","title":"29 equivalent to 14"},{"id":"equational-theories/part-01/implications-between-selected-laws/14-implies-29","kind":"node","lean":"Subgraph.Equation14_implies_Equation29","parent":5228,"result":true,"route":"../roadmap/equational-theories/part-01/implications-between-selected-laws/index.html#14-implies-29","scope":"equational-theories/part-01/implications-between-selected-laws","status":"fully_proved","summary":"E14 implies E29.","title":"14 implies 29"},{"id":"equational-theories/part-01/implications-between-selected-laws/3744-implies-3722-381","kind":"node","lean":"Subgraph.Equation3744_implies_Equation3722, Subgraph.Equation3744_implies_Equation381","parent":5228,"result":true,"route":"../roadmap/equational-theories/part-01/implications-between-selected-laws/index.html#3744-implies-3722-381","scope":"equational-theories/part-01/implications-between-selected-laws","status":"fully_proved","summary":"E3744 $({x \\diamond y = (x \\diamond z) \\diamond (w \\diamond y)})$ implies E3722 $({x \\diamond y = (x \\diamond y) \\diamond (x \\diamond y)})$ and E381 $({x \\diamond y = (x \\diamond z) \\diamond y})$.","title":"3744 implies 3722, 381"},{"id":"equational-theories/part-01/implications-between-selected-laws/1689-equiv-2","kind":"node","lean":"Subgraph.Equation1689_implies_Equation2, Subgraph.Equation2_implies_Equation1689","parent":5228,"result":true,"route":"../roadmap/equational-theories/part-01/implications-between-selected-laws/index.html#1689-equiv-2","scope":"equational-theories/part-01/implications-between-selected-laws","status":"fully_proved","summary":"E1689 $({x = (y \\diamond x) \\diamond ((x \\diamond z) \\diamond z)})$ is equivalent to E2.","title":"1689 is equivalent to 2"},{"id":"equational-theories/part-01/implications-between-selected-laws/1571-impl","kind":"node","lean":"Subgraph.Equation1571_implies_Equation2662, Subgraph.Equation1571_implies_Equation40, Subgraph.Equation1571_implies_Equation23, Subgraph.Equation1571_implies_Equation8, Subgraph.Equation1571_implies_Equation16, Subgraph.Equation1571_implies_Equation43, Subgraph.Equation1571_implies_Equation4512","parent":5228,"result":true,"route":"../roadmap/equational-theories/part-01/implications-between-selected-laws/index.html#1571-impl","scope":"equational-theories/part-01/implications-between-selected-laws","status":"fully_proved","summary":"Magmas satisfying E1571 $({x = (y \\diamond z) \\diamond (y \\diamond (x \\diamond z))})$ also satisfy E2662, E40, E23, E8, E16, E14, E43, and E4512, and are in fact abelian groups of exponent two. Conversely, all abelian groups of exponent two satisfy E1571.","title":"Consequences of 1571"},{"id":"equational-theories/part-01/implications-between-selected-laws/953-equiv-2","kind":"node","lean":"Subgraph.Equation953_implies_Equation2","parent":5228,"result":true,"route":"../roadmap/equational-theories/part-01/implications-between-selected-laws/index.html#953-equiv-2","scope":"equational-theories/part-01/implications-between-selected-laws","status":"fully_proved","summary":"E953 $({x = y \\diamond ((z \\diamond x) \\diamond (z \\diamond z))})$ is equivalent to E2.","title":"953 is equivalent to 2"},{"id":"equational-theories/part-01/implications-between-selected-laws/sheffer","kind":"node","lean":"Sheffer.Equation345169_is_Boolean","parent":5228,"result":true,"route":"../roadmap/equational-theories/part-01/implications-between-selected-laws/index.html#sheffer","scope":"equational-theories/part-01/implications-between-selected-laws","status":"fully_proved","summary":"Definition E345169 $({x = (y \\diamond ((x \\diamond y) \\diamond y)) \\diamond (x \\diamond (z \\diamond y))})$ axiomatizes the Sheffer stroke operation $x \\diamond y = \\overline{xy}$ in a Boolean algebra.","title":"Sheffer stroke axiom"},{"data_url":"scopes/equational-theories/part-01/finite-implications-and-infinite-models.json","id":"equational-theories/part-01/finite-implications-and-infinite-models","kind":"scope","parent":5155,"result":false,"route":"../roadmap/equational-theories/part-01/finite-implications-and-infinite-models/index.html","scope":"equational-theories/part-01/finite-implications-and-infinite-models","status":"fully_proved","summary":"In this chapter we discuss implications that hold for finite magmas but not infinite ones. The $820$ such implications between laws of order up to $4$ are listed at . The implications are proven using consequences of finiteness such as the equivalence of injectivity and surjectivity. The non-implications for infinite magmas are obtained by constructing infinite models, which are more challenging to construct\u2014they\u2026","title":"Finite implications and infinite models"},{"data_url":"scopes/equational-theories/part-01/finite-implications-and-infinite-models/austin-laws.json","id":"equational-theories/part-01/finite-implications-and-infinite-models/austin-laws","kind":"scope","parent":5237,"result":false,"route":"../roadmap/equational-theories/part-01/finite-implications-and-infinite-models/austin-laws/index.html","scope":"equational-theories/part-01/finite-implications-and-infinite-models/austin-laws","status":"fully_proved","summary":"The singleton or empty magma satisfies all equational laws. It is natural to ask whether an equational law admits nontrivial finite or infinite models. If no model exists, then the law is equivalent to E2 $({x = y})$. An *Austin law* is a law which admits infinite models, but no nontrivial finite models: the law is equivalent to E2 for finite magmas. Austin (Austin 1965) established the first such law, namely the\u2026","title":"Austin laws"},{"id":"equational-theories/part-01/finite-implications-and-infinite-models/austin-laws/kis-thm","kind":"node","lean":"InfModel.Finite.Equation374794_implies_Equation2, InfModel.Equation374794_not_implies_Equation2","parent":5238,"result":true,"route":"../roadmap/equational-theories/part-01/finite-implications-and-infinite-models/austin-laws/index.html#kis-thm","scope":"equational-theories/part-01/finite-implications-and-infinite-models/austin-laws","status":"fully_proved","summary":"E374794 $({x = (((y \\diamond y) \\diamond y) \\diamond x) \\diamond ((y \\diamond y) \\diamond z)})$ is an Austin law.","title":"Kisielewicz's first Austin law"},{"id":"equational-theories/part-01/finite-implications-and-infinite-models/austin-laws/kis-thm2","kind":"node","lean":"InfModel.Equation28770_not_implies_Equation2","parent":5238,"result":true,"route":"../roadmap/equational-theories/part-01/finite-implications-and-infinite-models/austin-laws/index.html#kis-thm2","scope":"equational-theories/part-01/finite-implications-and-infinite-models/austin-laws","status":"fully_proved","summary":"E28770 $({x = (((y \\diamond y) \\diamond y) \\diamond x) \\diamond (y \\diamond z)})$ is an Austin law.","title":"Kisielewicz's second Austin law"},{"id":"equational-theories/part-01/finite-implications-and-infinite-models/austin-laws/5093-nontrivial","kind":"node","lean":"InfModel.Finite.Equation5093_implies_Equation2","parent":5238,"result":true,"route":"../roadmap/equational-theories/part-01/finite-implications-and-infinite-models/austin-laws/index.html#5093-nontrivial","scope":"equational-theories/part-01/finite-implications-and-infinite-models/austin-laws","status":"fully_proved","summary":"E5093 has no non-trivial finite models.","title":"Equation 5093 has no non-trivial finite models"},{"id":"equational-theories/part-01/finite-implications-and-infinite-models/austin-laws/austin-two","kind":"node","lean":"InfModel.Finite.two_variable_laws","parent":5238,"result":true,"route":"../roadmap/equational-theories/part-01/finite-implications-and-infinite-models/austin-laws/index.html#austin-two","scope":"equational-theories/part-01/finite-implications-and-infinite-models/austin-laws","status":"fully_proved","summary":"Any law with at most two variables has a non-trivial finite model.","title":"Austin's finite model theorem"},{"data_url":"scopes/equational-theories/part-01/finite-implications-and-infinite-models/austin-implications.json","id":"equational-theories/part-01/finite-implications-and-infinite-models/austin-implications","kind":"scope","parent":5237,"result":false,"route":"../roadmap/equational-theories/part-01/finite-implications-and-infinite-models/austin-implications/index.html","scope":"equational-theories/part-01/finite-implications-and-infinite-models/austin-implications","status":"fully_proved","summary":"We now broaden the discussion to consider not only the implications to E2 $({x = y})$, but also between any pair of equational laws. Many implications for finite magmas rely on the fact that if $f, g:X \\to X$ are functions on a finite set $X$, then $f \\circ g = I$ if and only if $g \\circ f = I$. Two more complicated variants of this are as follows.","title":"Austin implications"},{"id":"equational-theories/part-01/finite-implications-and-infinite-models/austin-implications/ffg","kind":"node","lean":"FiniteModel.Finite.f_ffg_implies_f_fgf","parent":5243,"result":true,"route":"../roadmap/equational-theories/part-01/finite-implications-and-infinite-models/austin-implications/index.html#ffg","scope":"equational-theories/part-01/finite-implications-and-infinite-models/austin-implications","status":"fully_proved","summary":"Let $X$ be finite, and let $f, g: X \\to X$ be such that $f = f \\circ f \\circ g$. Then $f = f \\circ g \\circ f$.","title":"ffg"},{"id":"equational-theories/part-01/finite-implications-and-infinite-models/austin-implications/gff","kind":"node","lean":"FiniteModel.Finite.f_gff_implies_f_fgf","parent":5243,"result":true,"route":"../roadmap/equational-theories/part-01/finite-implications-and-infinite-models/austin-implications/index.html#gff","scope":"equational-theories/part-01/finite-implications-and-infinite-models/austin-implications","status":"fully_proved","summary":"Let $X$ be finite, and let $f, g: X \\to X$ be such that $f = g \\circ f \\circ f$. Then $f = f \\circ g \\circ f$.","title":"gff"},{"id":"equational-theories/part-01/finite-implications-and-infinite-models/austin-implications/period","kind":"node","lean":"FiniteModel.Finite.fn_eventually_periodic'","parent":5243,"result":true,"route":"../roadmap/equational-theories/part-01/finite-implications-and-infinite-models/austin-implications/index.html#period","scope":"equational-theories/part-01/finite-implications-and-infinite-models/austin-implications","status":"fully_proved","summary":"Let $X$ be finite and $f: X \\to X$. Then there exists $n \\geq 1$ such that $f^{2n} = f^n$.","title":"Eventual period"},{"id":"equational-theories/part-01/finite-implications-and-infinite-models/austin-implications/finite-imp-3994-3588","kind":"node","lean":"InfModel.Finite.Equation3994_implies_Equation3588","parent":5243,"result":true,"route":"../roadmap/equational-theories/part-01/finite-implications-and-infinite-models/austin-implications/index.html#finite-imp-3994-3588","scope":"equational-theories/part-01/finite-implications-and-infinite-models/austin-implications","status":"fully_proved","summary":"All finite magmas which satisfy E3994 $({x \\diamond y = (z \\diamond (x \\diamond y)) \\diamond z})$ also satisfy E3588 $({x \\diamond y = z \\diamond ((x \\diamond y) \\diamond z)})$.","title":"3994 implies 3588 for finite models"},{"id":"equational-theories/part-01/finite-implications-and-infinite-models/austin-implications/non-imp-3994-3588-thm","kind":"node","lean":"InfModel.Equation3994_not_implies_Equation3588","parent":5243,"result":true,"route":"../roadmap/equational-theories/part-01/finite-implications-and-infinite-models/austin-implications/index.html#non-imp-3994-3588-thm","scope":"equational-theories/part-01/finite-implications-and-infinite-models/austin-implications","status":"fully_proved","summary":"There exists a magma which satisfies E3994 and not E3588.","title":"3994 does not imply 3588 for infinite models"},{"id":"equational-theories/part-01/finite-implications-and-infinite-models/austin-implications/3342","kind":"node","lean":"Eq3342.Finite.Equation3342_implies_Equation3522, Eq3342.Finite.Equation3342_implies_Equation4118","parent":5243,"result":true,"route":"../roadmap/equational-theories/part-01/finite-implications-and-infinite-models/austin-implications/index.html#3342","scope":"equational-theories/part-01/finite-implications-and-infinite-models/austin-implications","status":"fully_proved","summary":"On a finite magma $M$, equation 3342, $x \\diamond y= y \\diamond (x \\diamond (x \\diamond x))$, implies equation 3522, $x \\diamond y = x \\diamond ((y \\diamond y) \\diamond y)$, as well as equation 4118, $x \\diamond y = ((x \\diamond x) \\diamond x) \\diamond y$.","title":"3342"},{"id":"equational-theories/part-01/finite-implications-and-infinite-models/austin-implications/1167-1096","kind":"node","lean":"Eq1133.Finite.Equation1167_implies_Equation1096","parent":5243,"result":true,"route":"../roadmap/equational-theories/part-01/finite-implications-and-infinite-models/austin-implications/index.html#1167-1096","scope":"equational-theories/part-01/finite-implications-and-infinite-models/austin-implications","status":"fully_proved","summary":"For finite magmas, Equation 1167,","title":"1167 implies 1096"},{"id":"equational-theories/part-01/finite-implications-and-infinite-models/austin-implications/1133-1167","kind":"node","lean":"Eq1133.Finite.Equation1133_implies_Equation1167","parent":5243,"result":true,"route":"../roadmap/equational-theories/part-01/finite-implications-and-infinite-models/austin-implications/index.html#1133-1167","scope":"equational-theories/part-01/finite-implications-and-infinite-models/austin-implications","status":"fully_proved","summary":"For finite magmas, Equation 1133,","title":"1133 implies 1167"},{"id":"equational-theories/part-01/finite-implications-and-infinite-models/austin-implications/1441-4067-1443-3055","kind":"node","lean":"Eq1441.Finite.Equation1441_implies_Equation4067, Eq1441.Finite.Equation1443_implies_Equation3055","parent":5243,"result":true,"route":"../roadmap/equational-theories/part-01/finite-implications-and-infinite-models/austin-implications/index.html#1441-4067-1443-3055","scope":"equational-theories/part-01/finite-implications-and-infinite-models/austin-implications","status":"fully_proved","summary":"For finite magmas, Equation 1441,","title":"1441 implies 4067, 1443 implies 3055"},{"id":"equational-theories/part-01/finite-implications-and-infinite-models/austin-implications/1681-3877-1701-1035","kind":"node","lean":"Eq1441.Finite.Equation1681_implies_Equation3877, Eq1441.Finite.Equation1701_implies_Equation1035","parent":5243,"result":true,"route":"../roadmap/equational-theories/part-01/finite-implications-and-infinite-models/austin-implications/index.html#1681-3877-1701-1035","scope":"equational-theories/part-01/finite-implications-and-infinite-models/austin-implications","status":"fully_proved","summary":"For finite magmas, Equation 1681,","title":"1681 implies 3877, 1701 implies 1035"},{"id":"equational-theories/part-01/selected-magmas","kind":"node","parent":5155,"result":false,"route":"../roadmap/equational-theories/part-01/selected-magmas/index.html","scope":"equational-theories/part-01","status":"can_state","summary":"Each magma can be used to establish anti-implications: if $\\Gamma$ is the set of all laws satisfied by a magma $G$, then we have $\\neg E \\leq E'$ whenever $E \\in \\Gamma$ and $E' \\not \\in \\Gamma$. Large numbers of implications can already be obtained from","title":"Selected magmas"},{"data_url":"scopes/equational-theories/part-01/infinite-magma-constructions.json","id":"equational-theories/part-01/infinite-magma-constructions","kind":"scope","parent":5155,"result":false,"route":"../roadmap/equational-theories/part-01/infinite-magma-constructions/index.html","scope":"equational-theories/part-01/infinite-magma-constructions","status":"can_prove","summary":"The need to construct infinite magmas primarily arises in the context of *Austin laws* and *Austin pairs*. An Austin law admits infinite models but no nontrivial finite ones, while an Austin pair consists of laws $P$ and $Q$ such that every finite model satisfying the law $P$ also satisfies $Q$, but some infinite magma satisfies $P$ without also satisfying $Q$. Examples are given in Finite implications and infinite\u2026","title":"Infinite magma constructions"},{"id":"equational-theories/part-01/infinite-magma-constructions/translation-invariant-magmas","kind":"node","parent":5255,"result":false,"route":"../roadmap/equational-theories/part-01/infinite-magma-constructions/translation-invariant-magmas/index.html","scope":"equational-theories/part-01/infinite-magma-constructions","status":"can_state","summary":"A *translation-invariant magma* is a magma whose carrier $G$ is an abelian group $(G,+,-,0)$, and whose magma operation takes the form","title":"Translation-invariant magmas"},{"data_url":"scopes/equational-theories/part-01/infinite-magma-constructions/greedy-algorithm-constructions.json","id":"equational-theories/part-01/infinite-magma-constructions/greedy-algorithm-constructions","kind":"scope","parent":5255,"result":false,"route":"../roadmap/equational-theories/part-01/infinite-magma-constructions/greedy-algorithm-constructions/index.html","scope":"equational-theories/part-01/infinite-magma-constructions/greedy-algorithm-constructions","status":"can_prove","summary":"In The Asterix equation a magma satisfying one law and refuting another was obtained by using the fact that over translation-invariant magmas, equational laws simplify to univariate functional equations, and solving the resulting equation using a greedy algorithm.","title":"Greedy algorithm constructions"},{"id":"equational-theories/part-01/infinite-magma-constructions/greedy-algorithm-constructions/1722-extension","kind":"node","lean":"Eq1722.Greedy.lift","parent":5257,"result":true,"route":"../roadmap/equational-theories/part-01/infinite-magma-constructions/greedy-algorithm-constructions/index.html#1722-extension","scope":"equational-theories/part-01/infinite-magma-constructions/greedy-algorithm-constructions","status":"can_prove","summary":"Suppose that $\\diamond$ is a partial solution, and $a \\diamond b$ is currently undefined. Then there exists an extension $\\diamond'$ of $\\diamond$ for which $a \\diamond' b$ is defined.","title":"1722 extension"},{"id":"equational-theories/part-01/infinite-magma-constructions/greedy-algorithm-constructions/713-extension","kind":"node","lean":"Eq1722.Greedy.lift","parent":5257,"result":true,"route":"../roadmap/equational-theories/part-01/infinite-magma-constructions/greedy-algorithm-constructions/index.html#713-extension","scope":"equational-theories/part-01/infinite-magma-constructions/greedy-algorithm-constructions","status":"can_prove","summary":"Suppose that $\\diamond$ is a partial solution, and $a \\diamond b$ is currently undefined. Then there exists an extension $\\diamond'$ of $\\diamond$ for which $a \\diamond' b$ is defined.","title":"713 extension"},{"id":"equational-theories/part-01/infinite-magma-constructions/greedy-algorithm-constructions/1289-extension","kind":"node","lean":"Eq1289.Greedy.lift","parent":5257,"result":true,"route":"../roadmap/equational-theories/part-01/infinite-magma-constructions/greedy-algorithm-constructions/index.html#1289-extension","scope":"equational-theories/part-01/infinite-magma-constructions/greedy-algorithm-constructions","status":"can_prove","summary":"Suppose that $\\diamond$ is a partial solution, and $a \\diamond b$ is currently undefined. Then there exists an extension $\\diamond'$ of $\\diamond$ for which $a \\diamond' b$ is defined.","title":"1289 extension"},{"id":"equational-theories/part-01/infinite-magma-constructions/greedy-algorithm-constructions/1076-extension","kind":"node","lean":"Eq1076.Greedy.lift","parent":5257,"result":true,"route":"../roadmap/equational-theories/part-01/infinite-magma-constructions/greedy-algorithm-constructions/index.html#1076-extension","scope":"equational-theories/part-01/infinite-magma-constructions/greedy-algorithm-constructions","status":"can_prove","summary":"Suppose that $\\diamond$ is a partial solution, and $a \\diamond b$ is currently undefined. Then there exists an extension $\\diamond'$ of $\\diamond$ for which $a \\diamond' b$ is defined.","title":"1076 extension"},{"data_url":"scopes/equational-theories/part-01/infinite-magma-constructions/a-survey-of-examples.json","id":"equational-theories/part-01/infinite-magma-constructions/a-survey-of-examples","kind":"scope","parent":5255,"result":false,"route":"../roadmap/equational-theories/part-01/infinite-magma-constructions/a-survey-of-examples/index.html","scope":"equational-theories/part-01/infinite-magma-constructions/a-survey-of-examples","status":"can_prove","summary":"The implication between the laws E1648 $({x = (x \\diamond y) \\diamond ((x \\diamond y) \\diamond y)})$ and E206 $({x = (x \\diamond (x \\diamond y)) \\diamond y})$ can be refuted by a translation-invariant magma on the integers. Define the function $f: \\mathbb{Z} \\to \\mathbb{Z}$ by","title":"A survey of examples"},{"id":"equational-theories/part-01/infinite-magma-constructions/a-survey-of-examples/non-imp-1648-206-thm","kind":"node","lean":"Equation1648_not_implies_Equation206","parent":5262,"result":true,"route":"../roadmap/equational-theories/part-01/infinite-magma-constructions/a-survey-of-examples/index.html#non-imp-1648-206-thm","scope":"equational-theories/part-01/infinite-magma-constructions/a-survey-of-examples","status":"can_prove","summary":"There exists a magma which satisfies E1648 and not E206.","title":"1648 does not imply 206"},{"id":"equational-theories/part-01/infinite-magma-constructions/a-survey-of-examples/non-imp-1659-4315-thm","kind":"node","lean":"Equation1659_facts","parent":5262,"result":true,"route":"../roadmap/equational-theories/part-01/infinite-magma-constructions/a-survey-of-examples/index.html#non-imp-1659-4315-thm","scope":"equational-theories/part-01/infinite-magma-constructions/a-survey-of-examples","status":"can_prove","summary":"There exists a magma which satisfies E1659 and not E4315.","title":"1659 does not imply 4315"},{"data_url":"scopes/equational-theories/part-01/infinite-magma-constructions/the-dupont-equation.json","id":"equational-theories/part-01/infinite-magma-constructions/the-dupont-equation","kind":"scope","parent":5255,"result":false,"route":"../roadmap/equational-theories/part-01/infinite-magma-constructions/the-dupont-equation/index.html","scope":"equational-theories/part-01/infinite-magma-constructions/the-dupont-equation","status":"can_prove","summary":"Now we consider the Dupont equation, E63 $({x = y \\diamond (x \\diamond (x \\diamond y))})$, which can be treated by a greedy translation invariant construction that is more complicated than the one considered for the Asterix law in The Asterix equation.","title":"The Dupont equation"},{"id":"equational-theories/part-01/infinite-magma-constructions/the-dupont-equation/extension-lemma","kind":"node","lean":"Eq63.Greedy.Extension.next","parent":5265,"result":true,"route":"../roadmap/equational-theories/part-01/infinite-magma-constructions/the-dupont-equation/index.html#extension-lemma","scope":"equational-theories/part-01/infinite-magma-constructions/the-dupont-equation","status":"can_prove","summary":"If $f: E \\to \\mathbb{Z}$ is a partial solution and $h_0 \\in \\mathbb{Z}$, then there exists an extension $f': E' \\to \\mathbb{Z}$ for which axiom (ii) applies, i.e., $h_0 \\in E'$, $f'(h_0) \\in E'$, $(f')^2(h_0)-h_0 \\in E'$, and $f'((f')^2(h_0)-h_0) = -h_0$.","title":"Extension"},{"id":"equational-theories/part-01/infinite-magma-constructions/the-dupont-equation/dupont-iter","kind":"node","lean":"Eq63.Greedy.exists_extension","parent":5265,"result":true,"route":"../roadmap/equational-theories/part-01/infinite-magma-constructions/the-dupont-equation/index.html#dupont-iter","scope":"equational-theories/part-01/infinite-magma-constructions/the-dupont-equation","status":"can_prove","summary":"Every partial solution $f: E \\to \\mathbb{Z}$ can be extended to a global solution $f': \\mathbb{Z} \\to\\mathbb{Z}$ of (1).","title":"Greedy completion of Dupont"},{"id":"equational-theories/part-01/infinite-magma-constructions/the-dupont-equation/non-inject","kind":"node","lean":"Eq63.Equation63_not_implies_Equation1692","parent":5265,"result":true,"route":"../roadmap/equational-theories/part-01/infinite-magma-constructions/the-dupont-equation/index.html#non-inject","scope":"equational-theories/part-01/infinite-magma-constructions/the-dupont-equation","status":"can_prove","summary":"The Dupont equation admits non-injective solutions, and hence can violate Equation 1692.","title":"Non-injective Dupont solution"},{"id":"equational-theories/part-01/equivalence-with-the-constant-and-singleton-laws","kind":"node","parent":5155,"result":false,"route":"../roadmap/equational-theories/part-01/equivalence-with-the-constant-and-singleton-laws/index.html","scope":"equational-theories/part-01","status":"can_state","summary":"85 laws have been shown to be equivalent to the constant law (E46 $({x \\diamond y = z \\diamond w})$), and 815 laws have been shown to be equivalent to the singleton law (E2 $({x = y})$).","title":"Equivalence with the constant and singleton laws"},{"data_url":"scopes/equational-theories/part-01/metatheorems-from-invariants.json","id":"equational-theories/part-01/metatheorems-from-invariants","kind":"scope","parent":5155,"result":false,"route":"../roadmap/equational-theories/part-01/metatheorems-from-invariants/index.html","scope":"equational-theories/part-01/metatheorems-from-invariants","status":"can_prove","summary":"For the purposes of this chapter, a *theorem* is a (true) statement about particular equations, for example \u2018(387 implies 43)\u2019 is a theorem. A *metatheorem* is a general statement about implications; one can usually get many theorems from a single metatheorem. This chapter is all about generating many interesting metatheorems using a *meta-metatheorem*, called the fundamental property of invariants. If all this is\u2026","title":"Metatheorems from Invariants"},{"id":"equational-theories/part-01/metatheorems-from-invariants/invariants","kind":"node","parent":5270,"result":false,"route":"../roadmap/equational-theories/part-01/metatheorems-from-invariants/invariants/index.html","scope":"equational-theories/part-01/metatheorems-from-invariants","status":"can_state","summary":"Let $E, E_1$, and $E_2$ be equations. If $E \\Rightarrow E_1$ and $E_1 \\Rightarrow E_2$, then $E \\Rightarrow E_2$. Very trivial. Rephrasing this, we see that if $E \\Rightarrow E_1$ and $E \\nRightarrow E_2$, then $E_1\\nRightarrow E_2$.","title":"Invariants"},{"id":"equational-theories/part-01/metatheorems-from-invariants/expanding-the-language","kind":"node","parent":5270,"result":false,"route":"../roadmap/equational-theories/part-01/metatheorems-from-invariants/expanding-the-language/index.html","scope":"equational-theories/part-01/metatheorems-from-invariants","status":"can_state","summary":"The method of invariants really shines when we expand our formal language. Right now our language consists of variables, parentheses, the equal sign, and $\\diamond$ (there is also an implicit use of $\\forall$ but let's ignore that for now). Let $\\Pi$ denote the preorder of equations (built from the language described) ordered by implication.","title":"Expanding the language"},{"data_url":"scopes/equational-theories/part-01/metatheorems-from-invariants/proving-metatheorems-from-invariants-in-lean.json","id":"equational-theories/part-01/metatheorems-from-invariants/proving-metatheorems-from-invariants-in-lean","kind":"scope","parent":5270,"result":false,"route":"../roadmap/equational-theories/part-01/metatheorems-from-invariants/proving-metatheorems-from-invariants-in-lean/index.html","scope":"equational-theories/part-01/metatheorems-from-invariants/proving-metatheorems-from-invariants-in-lean","status":"can_prove","summary":"For the rest of this chapter we readopt the convention of calling \u2018theorem\u2019 an important result, not necessarily pertaining to specific equations.","title":"Proving metatheorems from invariants in Lean"},{"id":"equational-theories/part-01/metatheorems-from-invariants/proving-metatheorems-from-invariants-in-lean/lifting-magma-family","kind":"node","lean":"LiftingMagmaFamily","parent":5273,"result":true,"route":"../roadmap/equational-theories/part-01/metatheorems-from-invariants/proving-metatheorems-from-invariants-in-lean/index.html#lifting-magma-family","scope":"equational-theories/part-01/metatheorems-from-invariants/proving-metatheorems-from-invariants-in-lean","status":"fully_proved","summary":"A *lifting magma family* is a family of magmas $\\{G_\\alpha\\}$, one for each type $\\alpha$, satisfying the following properties:","title":"Lifting Magma Family"},{"id":"equational-theories/part-01/metatheorems-from-invariants/proving-metatheorems-from-invariants-in-lean/lifting-magma-basis-evaluation","kind":"node","lean":"MagmaLaw.models_iff_satisfies_\u03b9","parent":5273,"result":true,"route":"../roadmap/equational-theories/part-01/metatheorems-from-invariants/proving-metatheorems-from-invariants-in-lean/index.html#lifting-magma-basis-evaluation","scope":"equational-theories/part-01/metatheorems-from-invariants/proving-metatheorems-from-invariants-in-lean","status":"can_prove","summary":"Suppose $E$ is an equation involving a set of variables $X$, and let $G$ be a lifting magma family.","title":"Evaluation theorem for lifting magma families"},{"data_url":"scopes/equational-theories/part-01/metatheorems-from-invariants/generating-laws-from-equations.json","id":"equational-theories/part-01/metatheorems-from-invariants/generating-laws-from-equations","kind":"scope","parent":5270,"result":false,"route":"../roadmap/equational-theories/part-01/metatheorems-from-invariants/generating-laws-from-equations/index.html","scope":"equational-theories/part-01/metatheorems-from-invariants/generating-laws-from-equations","status":"can_prove","summary":"The invariants defined in this chapter are properties of the *syntax* of the equations being considered. In other words, they are properties of the laws associated with the equations, rather than of the equations themselves. Proving non-implications using invariants requires a way to operate on the syntax of the equations and then translate the reasoning back to results about the original equations.","title":"Generating laws from equations"},{"id":"equational-theories/part-01/metatheorems-from-invariants/generating-laws-from-equations/compatibility-between-magma-laws","kind":"node","lean":"Law.satisfies_fin_satisfies_nat","parent":5276,"result":true,"route":"../roadmap/equational-theories/part-01/metatheorems-from-invariants/generating-laws-from-equations/index.html#compatibility-between-magma-laws","scope":"equational-theories/part-01/metatheorems-from-invariants/generating-laws-from-equations","status":"can_prove","summary":"[Compatibility between magma laws over finite sets and the natural numbers] Let $E$ be a magma law defined over $n$ variables and let $\\tilde{E}$ be the same equation with variables ranging over the natural numbers (formally, $\\tilde{E}$ is the image of $E$ under the canonical map from the finite set with $n$ elements to the natural numbers). Then any magma $M$ satisfies $E$ if and only if it satisfies $\\tilde{E}$.","title":"compatibility-between-magma-laws"},{"id":"equational-theories/part-01/metatheorems-from-invariants/conclusion-beyond-invariants","kind":"node","parent":5270,"result":false,"route":"../roadmap/equational-theories/part-01/metatheorems-from-invariants/conclusion-beyond-invariants/index.html","scope":"equational-theories/part-01/metatheorems-from-invariants","status":"can_state","summary":"We are still lacking:","title":"Conclusion: beyond invariants"},{"data_url":"scopes/equational-theories/part-01/some-abstract-nonsense.json","id":"equational-theories/part-01/some-abstract-nonsense","kind":"scope","parent":5155,"result":false,"route":"../roadmap/equational-theories/part-01/some-abstract-nonsense/index.html","scope":"equational-theories/part-01/some-abstract-nonsense","status":"fully_proved","summary":"This is an alternate presentation of the material of the previous section, where we use the \u201cabstract nonsense\u201d of free magmas in the presence of a theory as the conceptual foundation.","title":"Some abstract nonsense"},{"id":"equational-theories/part-01/some-abstract-nonsense/free-theory","kind":"node","lean":"FreeMagmaWithLaws","parent":5279,"result":true,"route":"../roadmap/equational-theories/part-01/some-abstract-nonsense/index.html#free-theory","scope":"equational-theories/part-01/some-abstract-nonsense","status":"fully_proved","summary":"Let $\\Gamma$ be a theory with an alphabet $X$. A *free magma* with alphabet $X$ subject to the theory $\\Gamma$ is a magma $M_{X,\\Gamma}$ together with a function $\\iota_{X,\\Gamma} : X \\to M_{X,\\Gamma}$, with the following properties:","title":"Free magma relative to a theory"},{"id":"equational-theories/part-01/some-abstract-nonsense/freemag-exist","kind":"node","lean":"FreeMagma.EvalFreeMagmaWithLawsUniversalProperty","parent":5279,"result":true,"route":"../roadmap/equational-theories/part-01/some-abstract-nonsense/index.html#freemag-exist","scope":"equational-theories/part-01/some-abstract-nonsense","status":"fully_proved","summary":"Let $\\Gamma$ be a theory with alphabet $X$.","title":"Existence and uniqueness of free magmas"},{"id":"equational-theories/part-01/magma-cohomology","kind":"node","parent":5155,"result":false,"route":"../roadmap/equational-theories/part-01/magma-cohomology/index.html","scope":"equational-theories/part-01","status":"can_state","summary":"Group cohomology is a theory that constructs certain abelian groups $H^n(G,M)$ to a group $G$ acting as a module on an abelian group $M$, via a chain complex","title":"Magma cohomology"},{"data_url":"scopes/equational-theories/part-01/weak-central-groupoids.json","id":"equational-theories/part-01/weak-central-groupoids","kind":"scope","parent":5155,"result":false,"route":"../roadmap/equational-theories/part-01/weak-central-groupoids/index.html","scope":"equational-theories/part-01/weak-central-groupoids","status":"stated","summary":"In this chapter we study weak central groupoids E1485,","title":"Weak central groupoids"},{"id":"equational-theories/part-01/weak-central-groupoids/1485-dual","kind":"node","lean":"WeakCentralGroupoid.dual_eqn","parent":5283,"result":true,"route":"../roadmap/equational-theories/part-01/weak-central-groupoids/index.html#1485-dual","scope":"equational-theories/part-01/weak-central-groupoids","status":"can_prove","summary":"E1485 is equivalent to the dual law","title":"1485 equivalent to 2162"},{"id":"equational-theories/part-01/weak-central-groupoids/graph-dual","kind":"node","lean":"WeakCentralGroupoid.Path.def'","parent":5283,"result":true,"route":"../roadmap/equational-theories/part-01/weak-central-groupoids/index.html#graph-dual","scope":"equational-theories/part-01/weak-central-groupoids","status":"stated","summary":"One has $x \\to y$ if and only if $x = w \\diamond y$ for some $w$.","title":"Equivalent characterization of graph"},{"id":"equational-theories/part-01/weak-central-groupoids/claim-4","kind":"node","lean":"WeakCentralGroupoid.isGood_five","parent":5283,"result":true,"route":"../roadmap/equational-theories/part-01/weak-central-groupoids/index.html#claim-4","scope":"equational-theories/part-01/weak-central-groupoids","status":"stated","summary":"If $a \\to b \\to c \\to d \\to e \\to a$ is a 5-cycle in the directed graph, and $a \\to b \\to c$ and $c \\to d \\to e$ are good paths, then $b \\to c \\to d$ is also good.","title":"Claim 4"},{"id":"equational-theories/part-01/weak-central-groupoids/rev-claim","kind":"node","lean":"RelaxedWeakCentralGroupoid.strictify","parent":5283,"result":true,"route":"../roadmap/equational-theories/part-01/weak-central-groupoids/index.html#rev-claim","scope":"equational-theories/part-01/weak-central-groupoids","status":"can_prove","summary":"Let $G$ be a directed graph, with some paths of length two in the graph designated as \u201cgood\u201d, in such a way that Claims 1-4 hold. Then there is a weak central groupoid structure on the vertices of $G$ such that the good paths are precisely the paths of the form $x \\to x \\diamond y \\to y$.","title":"Reversing the claims"},{"id":"equational-theories/part-01/weak-central-groupoids/greedy-prop","kind":"node","lean":"RelaxedVeryWeakCentralGroupoid.Greedy.exists_extension","parent":5283,"result":true,"route":"../roadmap/equational-theories/part-01/weak-central-groupoids/index.html#greedy-prop","scope":"equational-theories/part-01/weak-central-groupoids","status":"can_prove","summary":"Let $G_0$ be a directed graph satisfying claims 1', 2, 3, 4. Then any finite partial extension of $G_0$ with carrier $G_0 \\times \\mathbb{N}$ (and projection map $\\pi(a,n) = a$) can be completed to a complete extension.","title":"Completion property"},{"id":"equational-theories/part-01/weak-central-groupoids/1485-refutes","kind":"node","lean":"Refutation_1485.not_3457, Refutation_1485.not_3511, Refutation_1485.not_2087_2124","parent":5283,"result":true,"route":"../roadmap/equational-theories/part-01/weak-central-groupoids/index.html#1485-refutes","scope":"equational-theories/part-01/weak-central-groupoids","status":"stated","summary":"E1485 does not imply any of the following laws:","title":"1485 does not imply 1483"},{"id":"equational-theories/part-01/weak-central-groupoids/twisting-a-weak-central-groupoid","kind":"node","parent":5283,"result":false,"route":"../roadmap/equational-theories/part-01/weak-central-groupoids/twisting-a-weak-central-groupoid/index.html","scope":"equational-theories/part-01/weak-central-groupoids","status":"can_state","summary":"Occasionally, an equational law is preserved under a \u201ctwist\u201d operation in which one replaces the magma operation $x \\diamond y$ by $x \\diamond' y := Tx \\diamond Uy$ for some automorphisms $T,U$ of the magma $G$ that satisfy additional relations. In the case of the weak central groupoid law (1), we see that","title":"Twisting a weak central groupoid"},{"data_url":"scopes/equational-theories/part-01/equation-854.json","id":"equational-theories/part-01/equation-854","kind":"scope","parent":5155,"result":false,"route":"../roadmap/equational-theories/part-01/equation-854/index.html","scope":"equational-theories/part-01/equation-854","status":"can_prove","summary":"In this chapter we study magmas that satisfy E854, thus","title":"Equation 854"},{"id":"equational-theories/part-01/equation-854/unique-factorization","kind":"node","lean":"Refutation_854.unique_factorization","parent":5291,"result":true,"route":"../roadmap/equational-theories/part-01/equation-854/index.html#unique-factorization","scope":"equational-theories/part-01/equation-854","status":"fully_proved","summary":"Two irreducible words $w, w'$ are equivalent if and only if they are either the same generator of $X$, or are of the form $w = w_1 \\diamond w_2$, $w' = w'_1 \\diamond w'_2$ with $w_1 \\sim w'_1$ and $w_2 \\sim w'_2$.","title":"Unique factorization"},{"id":"equational-theories/part-01/equation-854/854-anti","kind":"node","lean":"Refutation_854.not_3316_3925","parent":5291,"result":true,"route":"../roadmap/equational-theories/part-01/equation-854/index.html#854-anti","scope":"equational-theories/part-01/equation-854","status":"can_prove","summary":"The laws","title":"854 does not imply 3316, 3925"},{"data_url":"scopes/equational-theories/part-01/equation-854/running-a-greedy-algorithm.json","id":"equational-theories/part-01/equation-854/running-a-greedy-algorithm","kind":"scope","parent":5291,"result":false,"route":"../roadmap/equational-theories/part-01/equation-854/running-a-greedy-algorithm/index.html","scope":"equational-theories/part-01/equation-854/running-a-greedy-algorithm","status":"can_prove","summary":"Define a *partial 854 magma* to be a partial function $\\diamond: \\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{N}$ satisfying the following axioms:","title":"Running a greedy algorithm"},{"id":"equational-theories/part-01/equation-854/running-a-greedy-algorithm/854-extend","kind":"node","lean":"Refutation_854.Greedy.Extension1.next_ok","parent":5294,"result":true,"route":"../roadmap/equational-theories/part-01/equation-854/running-a-greedy-algorithm/index.html#854-extend","scope":"equational-theories/part-01/equation-854/running-a-greedy-algorithm","status":"can_prove","summary":"Suppose one has a partial 854 magma on $\\mathbb{N}$ that is only finitely defined, and let $a,b \\in G$ be such that $a \\diamond b$ is currently undefined. Then it is possible to extend the magma to a larger partial 854 magma, such that $a \\diamond b$ is now defined.","title":"Greedy construction"},{"id":"equational-theories/part-01/equation-854/running-a-greedy-algorithm/extend-854","kind":"node","lean":"Refutation_854.Greedy.exists_extension","parent":5294,"result":true,"route":"../roadmap/equational-theories/part-01/equation-854/running-a-greedy-algorithm/index.html#extend-854","scope":"equational-theories/part-01/equation-854/running-a-greedy-algorithm","status":"can_prove","summary":"Suppose one has a partial 854 magma on $\\mathbb{N}$ that is only finitely defined. Then it can be extended to a complete 854 magma that additionally satisfies the no idempotence law, the monotonicity law, the auxiliary law, and the unique factorization law.","title":"854 extension"},{"id":"equational-theories/part-01/equation-854/running-a-greedy-algorithm/854-413","kind":"node","lean":"Refutation_854.not_413_1045","parent":5294,"result":true,"route":"../roadmap/equational-theories/part-01/equation-854/running-a-greedy-algorithm/index.html#854-413","scope":"equational-theories/part-01/equation-854/running-a-greedy-algorithm","status":"can_prove","summary":"There is an 854 magma which does not satisfy the 413 law $x = x \\diamond (x \\diamond (x \\diamond (y \\diamond x)))$.","title":"854 does not not imply 413"},{"id":"equational-theories/part-01/equation-854/running-a-greedy-algorithm/854-1045","kind":"node","lean":"Refutation_854.not_413_1045","parent":5294,"result":true,"route":"../roadmap/equational-theories/part-01/equation-854/running-a-greedy-algorithm/index.html#854-1045","scope":"equational-theories/part-01/equation-854/running-a-greedy-algorithm","status":"can_prove","summary":"There is an 854 magma which does not satisfy the 1045 law $x = x \\diamond ((y \\diamond (y \\diamond x)) \\diamond x)$.","title":"854 does not not imply 1045"},{"data_url":"scopes/equational-theories/part-01/equation-906.json","id":"equational-theories/part-01/equation-906","kind":"scope","parent":5155,"result":false,"route":"../roadmap/equational-theories/part-01/equation-906/index.html","scope":"equational-theories/part-01/equation-906","status":"fully_proved","summary":"In this chapter we study finite magmas that satisfy equation 906,","title":"Equation 906"},{"id":"equational-theories/part-01/equation-906/906-3862","kind":"node","lean":"Eq906.Finite.Equation906_implies_Equation3862","parent":5299,"result":true,"route":"../roadmap/equational-theories/part-01/equation-906/index.html#906-3862","scope":"equational-theories/part-01/equation-906","status":"fully_proved","summary":"For finite magmas, equation 906 implies equation 3862,","title":"906-3862"},{"data_url":"scopes/equational-theories/part-01/equation-1323.json","id":"equational-theories/part-01/equation-1323","kind":"scope","parent":5155,"result":false,"route":"../roadmap/equational-theories/part-01/equation-1323/index.html","scope":"equational-theories/part-01/equation-1323","status":"stated","summary":"In this chapter we study magmas that satisfy equation 1323,","title":"Equation 1323"},{"id":"equational-theories/part-01/equation-1323/1323-construct","kind":"node","lean":"Eq1323.eq1323_if_conditions","parent":5301,"result":true,"route":"../roadmap/equational-theories/part-01/equation-1323/index.html#1323-construct","scope":"equational-theories/part-01/equation-1323","status":"can_prove","summary":"Suppose that $M$ is a magma such that","title":"Construction of 1323 magmas"},{"id":"equational-theories/part-01/equation-1323/bij","kind":"node","lean":"Eq1323.\u03d5","parent":5301,"result":true,"route":"../roadmap/equational-theories/part-01/equation-1323/index.html#bij","scope":"equational-theories/part-01/equation-1323","status":"can_prove","summary":"Let $G$ be a countably infinite abelian torsion group of exponent $2$. Then there exists a bijection $\\phi_a: G \\to \\mathbb{Q}^\\times$ for each $a \\in G \\backslash \\{0\\}$ such that $\\phi_a(0) = 1$ and $\\phi_a(a+b) = -\\phi_a(b)$ for all $b \\in G$, so in particular $\\phi_a(a)=-1$.","title":"Bijections"},{"id":"equational-theories/part-01/equation-1323/build-magma","kind":"node","lean":"Eq1323.op_Ly_Ry_eq_LSy","parent":5301,"result":true,"route":"../roadmap/equational-theories/part-01/equation-1323/index.html#build-magma","scope":"equational-theories/part-01/equation-1323","status":"stated","summary":"Let $G$ be a countably infinite abelian torsion group of exponent $2$, and let $\\phi_a$ be as in the previous lemma. Let $N$ be the set of pairs $(x,a)$ with $x \\in \\mathbb{Q}^\\times$ and $a \\in G \\backslash \\{0\\}$, and let $M = G \\uplus N$ be the disjoint union of $G$ and $N$. Suppose that we have an operation $\\diamond : M \\times M \\to M$ satisfying the following axioms:","title":"Building a magma"},{"id":"equational-theories/part-01/equation-1323/greedy-1323","kind":"node","lean":"Eq1323.extend","parent":5301,"result":true,"route":"../roadmap/equational-theories/part-01/equation-1323/index.html#greedy-1323","scope":"equational-theories/part-01/equation-1323","status":"can_prove","summary":"If $\\diamond$ is defined by a partial solution, and $(x,a) \\diamond (y,b)$ is undefined for some $x,y \\in \\mathbb{Q}^\\times$ and distinct $a,b \\in G \\backslash \\{0\\}$, then it is possible to extend the partial solution so that $(x,a) \\diamond (y,b)$ is now defined.","title":"Greedy extension"},{"id":"equational-theories/part-01/equation-1323/greedy-iterate","kind":"node","lean":"Eq1323.exists_complete_function","parent":5301,"result":true,"route":"../roadmap/equational-theories/part-01/equation-1323/index.html#greedy-iterate","scope":"equational-theories/part-01/equation-1323","status":"stated","summary":"Every partial solution can be extended to a complete solution that satisfies (2) and (3), and hence 1323.","title":"Iterated greedy extension"},{"id":"equational-theories/part-01/equation-1323/1323-refute-2744","kind":"node","lean":"Eq1323.Equation1323_not_implies_Equation2744","parent":5301,"result":true,"route":"../roadmap/equational-theories/part-01/equation-1323/index.html#1323-refute-2744","scope":"equational-theories/part-01/equation-1323","status":"stated","summary":"There exists a 1323 magma which does not satisfy the 2744 equation $R_y L_{Sy} L_y x = x$.","title":"1323 does not imply 2744"},{"data_url":"scopes/equational-theories/part-01/equation-1729.json","id":"equational-theories/part-01/equation-1729","kind":"scope","parent":5155,"result":false,"route":"../roadmap/equational-theories/part-01/equation-1729/index.html","scope":"equational-theories/part-01/equation-1729","status":"fully_proved","summary":"In this chapter we study magmas that satisfy equation 1729,","title":"Equation 1729"},{"id":"equational-theories/part-01/equation-1729/sm-def","kind":"node","lean":"Eq1729.SM","parent":5308,"reading_title":{"format":"inline-parts-v1","parts":[["text","Definition of "],["math","\\(SM\\)"]],"source":"Definition of $SM$"},"result":true,"route":"../roadmap/equational-theories/part-01/equation-1729/index.html#sm-def","scope":"equational-theories/part-01/equation-1729","status":"fully_proved","summary":"Take $SM$ to be a countably infinite abelian group of exponent $4$, generated by generators $E_n$ for $n \\in \\mathbb{N}$ subject to the relations $4E_n=0$.","title":"Definition of $SM$"},{"id":"equational-theories/part-01/equation-1729/sm-1729","kind":"node","lean":"Eq1729.SM_square_eq_double, Eq1729.SM_square_square_eq_zero, Eq1729.SM_satisfies_1729","parent":5308,"reading_title":{"format":"inline-parts-v1","parts":[["text","Basic properties of "],["math","\\(SM\\)"]],"source":"Basic properties of $SM$"},"result":true,"route":"../roadmap/equational-theories/part-01/equation-1729/index.html#sm-1729","scope":"equational-theories/part-01/equation-1729","status":"fully_proved","summary":"$SM$ is a 1729 magma, the squaring operation $S: SM \\to SM$ is just the doubling map $Sa = 2a$, and the double squaring map $S^2: SM \\to SM$ is constant: $S^2 a = 0$ for all $a \\in SM$.","title":"Basic properties of $SM$"},{"id":"equational-theories/part-01/equation-1729/n-def","kind":"node","lean":"Eq1729.N, Eq1729.N_order, Eq1729.parent","parent":5308,"reading_title":{"format":"inline-parts-v1","parts":[["text","Definition of "],["math","\\(N\\)"]],"source":"Definition of $N$"},"result":true,"route":"../roadmap/equational-theories/part-01/equation-1729/index.html#n-def","scope":"equational-theories/part-01/equation-1729","status":"fully_proved","summary":"Take $N$ to be the free non-abelian group with a generator $e_a$ for each $a \\in SM$, thus $N$ is the set of reduced words using the alphabet $e_a$, $e_a^{-1}$. Two elements $x,y \\in SM$ are said to be *adjacent* if $x = e_a y$ or $y = e_a x$ for some $a \\in SM$; this defines a left Cayley graph on $N$. We make a partial ordering $\\leq$ on $N$ by declaring $y \\leq x$ if $y$ is a right subword of $x$ (or\u2026","title":"Definition of $N$"},{"id":"equational-theories/part-01/equation-1729/n-prop","kind":"node","lean":"Eq1729.N_countable, Eq1729.N_order","parent":5308,"reading_title":{"format":"inline-parts-v1","parts":[["text","Basic properties of "],["math","\\(N\\)"]],"source":"Basic properties of $N$"},"result":true,"route":"../roadmap/equational-theories/part-01/equation-1729/index.html#n-prop","scope":"equational-theories/part-01/equation-1729","status":"fully_proved","summary":"$N$ is countable, and $\\leq$ is a partial ordering.","title":"Basic properties of $N$"},{"id":"equational-theories/part-01/equation-1729/ra-defn","kind":"node","lean":"Eq1729.R'","parent":5308,"reading_title":{"format":"inline-parts-v1","parts":[["text","Definition of "],["math","\\(R'_a\\)"]],"source":"Definition of $R'_a$"},"result":true,"route":"../roadmap/equational-theories/part-01/equation-1729/index.html#ra-defn","scope":"equational-theories/part-01/equation-1729","status":"fully_proved","summary":"We set","title":"Definition of $R'_a$"},{"id":"equational-theories/part-01/equation-1729/ra-prop","kind":"node","lean":"Eq1729.R'_axiom_iia, Eq1729.R'_axiom_iib","parent":5308,"reading_title":{"format":"inline-parts-v1","parts":[["text","Basic properties of "],["math","\\(R'_a\\)"]],"source":"Basic properties of $R'_a$"},"result":true,"route":"../roadmap/equational-theories/part-01/equation-1729/index.html#ra-prop","scope":"equational-theories/part-01/equation-1729","status":"fully_proved","summary":"The operators $R'_a$ are bijective and satisfy axiom (ii).","title":"Basic properties of $R'_a$"},{"id":"equational-theories/part-01/equation-1729/l0-la","kind":"node","lean":"Eq1729.L', Eq1729.L'_0_eq_L\u2080'","parent":5308,"reading_title":{"format":"inline-parts-v1","parts":[["text","Using "],["math","\\(L'_0\\)"],["text"," to construct "],["math","\\(L'_a\\)"]],"source":"Using $L'_0$ to construct $L'_a$"},"result":true,"route":"../roadmap/equational-theories/part-01/equation-1729/index.html#l0-la","scope":"equational-theories/part-01/equation-1729","status":"fully_proved","summary":"Suppose we have a bijection $L'_0: N \\to N$ that satisfies axiom (i'), and then define $L'_a$ for all $a \\in SM$ by the formula (4). Then this recovers $L'_0$ when $a=0$ (to formalize this it may be convenient to give $L'_0$ and $L'_a$ distinct names), and the $L'_a$ are all bijections and satisfy axiom (i). Furthermore, we have","title":"Using $L'_0$ to construct $L'_a$"},{"id":"equational-theories/part-01/equation-1729/part-sol","kind":"node","lean":"Eq1729.PartialSolution","parent":5308,"result":true,"route":"../roadmap/equational-theories/part-01/equation-1729/index.html#part-sol","scope":"equational-theories/part-01/equation-1729","status":"fully_proved","summary":"A *partial solution* $(L'_0, \\diamond', S', {\\mathcal I})$ is a collection of the following data:","title":"Partial solution"},{"id":"equational-theories/part-01/equation-1729/partial-exist","kind":"node","lean":"Eq1729.TrivialPartialSolution","parent":5308,"result":true,"route":"../roadmap/equational-theories/part-01/equation-1729/index.html#partial-exist","scope":"equational-theories/part-01/equation-1729","status":"fully_proved","summary":"There exists a partial solution.","title":"Existence of partial solution"},{"id":"equational-theories/part-01/equation-1729/chain","kind":"node","lean":"Eq1729.use_chain","parent":5308,"result":true,"route":"../roadmap/equational-theories/part-01/equation-1729/index.html#chain","scope":"equational-theories/part-01/equation-1729","status":"fully_proved","summary":"Suppose that one has a sequence $(L'_{0,n}, \\diamond'_{n}, S'_{n}, {\\mathcal I}_n)$ of partial solutions, each one an extension of the previous, such that for any $x, y \\in N$, $L'_{0,n} x$, $x \\diamond'_{n} y$, and $S'_n x$ are defined for some $n$. Then there exists a 1729 magma that does not satisfy 817.","title":"Chain of partial solutions"},{"id":"equational-theories/part-01/equation-1729/enlarge-l0","kind":"node","lean":"Eq1729.enlarge_L\u2080'","parent":5308,"reading_title":{"format":"inline-parts-v1","parts":[["text","Enlarging "],["math","\\(L'_0\\)"]],"source":"Enlarging $L'_0$"},"result":true,"route":"../roadmap/equational-theories/part-01/equation-1729/index.html#enlarge-l0","scope":"equational-theories/part-01/equation-1729","status":"fully_proved","summary":"Suppose one has a partial solution in which $L'_0 x$ is undefined for some $x \\in N$. Then one can extend the partial solution so that $L'_0 x$ is now defined.","title":"Enlarging $L'_0$"},{"id":"equational-theories/part-01/equation-1729/enlarge-l0-many","kind":"node","lean":"Eq1729.enlarge_L\u2080'_multiple","parent":5308,"reading_title":{"format":"inline-parts-v1","parts":[["text","Enlarging "],["math","\\(L'_0\\)"],["text"," many times"]],"source":"Enlarging $L'_0$ many times"},"result":true,"route":"../roadmap/equational-theories/part-01/equation-1729/index.html#enlarge-l0-many","scope":"equational-theories/part-01/equation-1729","status":"fully_proved","summary":"Suppose one has a partial solution. Let $A$ be a finite subset of $N$. Then one can extend the partial solution so that $L'_0 x$ is now defined for all $x \\in A$.","title":"Enlarging $L'_0$ many times"},{"id":"equational-theories/part-01/equation-1729/enlarge-s-induct-axioms","kind":"node","lean":"Eq1729.enlarge_S'_induction_with_axioms","parent":5308,"reading_title":{"format":"inline-parts-v1","parts":[["text","Enlarging "],["math","\\(S'\\)"],["text"," with induction hypothesis and axioms"]],"source":"Enlarging $S'$ with induction hypothesis and axioms"},"result":true,"route":"../roadmap/equational-theories/part-01/equation-1729/index.html#enlarge-s-induct-axioms","scope":"equational-theories/part-01/equation-1729","status":"fully_proved","summary":"Suppose one has a partial solution in which $S'x$ is undefined for some $x \\in N$, but $S'y$ is defined for all $y < x$. (This hypothesis is vacuous for $x=1$.) Let $y_0$ be the parent of $x$ (if $x \\neq 1$), and assume the following additional axioms:","title":"Enlarging $S'$ with induction hypothesis and axioms"},{"id":"equational-theories/part-01/equation-1729/enlarge-s-induct","kind":"node","lean":"Eq1729.enlarge_S'_induction","parent":5308,"reading_title":{"format":"inline-parts-v1","parts":[["text","Enlarging "],["math","\\(S'\\)"],["text"," with induction hypothesis"]],"source":"Enlarging $S'$ with induction hypothesis"},"result":true,"route":"../roadmap/equational-theories/part-01/equation-1729/index.html#enlarge-s-induct","scope":"equational-theories/part-01/equation-1729","status":"fully_proved","summary":"Suppose one has a partial solution in which $S'x$ is undefined for some $x \\in N$, but $S'y$ is defined for all $y < x$. (This hypothesis is vacuous for $x=1$.) Then one can extend the partial solution so that $S'x$ is now defined.","title":"Enlarging $S'$ with induction hypothesis"},{"id":"equational-theories/part-01/equation-1729/enlarge-s","kind":"node","lean":"Eq1729.enlarge_S'","parent":5308,"reading_title":{"format":"inline-parts-v1","parts":[["text","Enlarging "],["math","\\(S'\\)"]],"source":"Enlarging $S'$"},"result":true,"route":"../roadmap/equational-theories/part-01/equation-1729/index.html#enlarge-s","scope":"equational-theories/part-01/equation-1729","status":"fully_proved","summary":"Suppose one has a partial solution in which $S'x$ is undefined for some $x \\in N$. Then one can extend the partial solution so that $S'x$ is now defined.","title":"Enlarging $S'$"},{"id":"equational-theories/part-01/equation-1729/enlarge-op","kind":"node","lean":"Eq1729.enlarge_op","parent":5308,"reading_title":{"format":"inline-parts-v1","parts":[["text","Enlarging "],["math","\\(\\diamond\\)"]],"source":"Enlarging $\\diamond$"},"result":true,"route":"../roadmap/equational-theories/part-01/equation-1729/index.html#enlarge-op","scope":"equational-theories/part-01/equation-1729","status":"fully_proved","summary":"Suppose one has a partial solution in which $x \\diamond' y$ is undefined for some $x,y \\in N$. Then one can extend the partial solution so that $x \\diamond' y$ is now defined.","title":"Enlarging $\\diamond$"},{"id":"equational-theories/part-01/equation-1729/1729-refute-817","kind":"node","lean":"Eq1729.not_817","parent":5308,"result":true,"route":"../roadmap/equational-theories/part-01/equation-1729/index.html#1729-refute-817","scope":"equational-theories/part-01/equation-1729","status":"fully_proved","summary":"There exists a magma that satisfies equation 1729 but not equation 817.","title":"1729 does not imply 817"},{"id":"equational-theories/part-01/order-5-austin-laws","kind":"node","parent":5155,"result":false,"route":"../roadmap/equational-theories/part-01/order-5-austin-laws/index.html","scope":"equational-theories/part-01","status":"can_state","summary":"Finite implications and infinite models has useful background context for this chapter. As noted in Finite implications and infinite models: an *Austin law* is a law which admits infinite models, but no nontrivial finite models. Austin laws have order 5 or greater (Kisielewicz 1997). In this chapter we report partial results of a classification of laws of order 5 on whether they do or do not allow non-trivial\u2026","title":"Order 5 Austin laws"},{"id":"equational-theories/part-01/simple-rewrites","kind":"node","parent":5155,"result":false,"route":"../roadmap/equational-theories/part-01/simple-rewrites/index.html","scope":"equational-theories/part-01","status":"can_state","summary":"53,905 implications were automatically generated by simple rewrites.","title":"Simple rewrites"},{"id":"equational-theories/part-01/trivial-auto-generated-theorems","kind":"node","parent":5155,"result":false,"route":"../roadmap/equational-theories/part-01/trivial-auto-generated-theorems/index.html","scope":"equational-theories/part-01","status":"can_state","summary":"Approximately 4.5m transitive implications were proven by a transitive reduction of about 15k theorems. Most of these implications were derived from being the first automated run to connect the largest equivalence classes, hence creating a large set of transitively closed implications.","title":"Trivial auto-generated theorems"},{"id":"equational-theories/part-01/enumerating-small-finite-magmas","kind":"node","parent":5155,"result":false,"route":"../roadmap/equational-theories/part-01/enumerating-small-finite-magmas/index.html","scope":"equational-theories/part-01","status":"can_state","summary":"describe the process of automatically generating these implications here.","title":"Enumerating Small Finite Magmas"},{"data_url":"scopes/equational-theories/part-02.json","id":"equational-theories/part-02","kind":"scope","navigation":"transparent","parent":5154,"result":false,"route":"../roadmap/equational-theories/part-02/index.html","scope":"equational-theories/part-02","status":"fully_proved","title":"Equation Search to Hard implications"},{"id":"equational-theories/part-02/equation-search","kind":"node","parent":5330,"result":false,"route":"../roadmap/equational-theories/part-02/equation-search/index.html","scope":"equational-theories/part-02","status":"can_state","summary":"Approximately 650k transitive implications were proven by a custom tool leveraging the implication graph. After previous brute force had derived many implications expressible as a small number of rewrites, this search tool uses substitutions implied by the implication graph to search further.","title":"Equation Search"},{"data_url":"scopes/equational-theories/part-02/e-graphs.json","id":"equational-theories/part-02/e-graphs","kind":"scope","parent":5330,"result":false,"route":"../roadmap/equational-theories/part-02/e-graphs/index.html","scope":"equational-theories/part-02/e-graphs","status":"fully_proved","summary":"For proving implications, we used another technique called equality saturation (Willsey et al. 2021) with the `lean-egg` tactic, to automatically construct proofs.","title":"E-Graphs"},{"data_url":"scopes/equational-theories/part-02/e-graphs/lean-egg.json","id":"equational-theories/part-02/e-graphs/lean-egg","kind":"scope","parent":5332,"result":false,"route":"../roadmap/equational-theories/part-02/e-graphs/lean-egg/index.html","scope":"equational-theories/part-02/e-graphs/lean-egg","status":"fully_proved","summary":"The basic methodology of equality saturation is based on E-Graphs, a data structure that can store equivalence classes of terms efficiently. We used the `lean-egg` tactic (), based on equality saturation as a tactic, which (re)constructs a proof from the E-graph (Koehler et al. 2024) in Lean. This means that we do not have to trust either the egg tool nor the tactic: if something goes wrong, Lean will not accept\u2026","title":"lean-egg"},{"id":"equational-theories/part-02/e-graphs/lean-egg/14-implies-23","kind":"node","lean":"Subgraph.Equation14_implies_Equation23","parent":5333,"result":true,"route":"../roadmap/equational-theories/part-02/e-graphs/lean-egg/index.html#14-implies-23","scope":"equational-theories/part-02/e-graphs/lean-egg","status":"fully_proved","summary":"E14 $({x = y \\diamond (x \\diamond y))})$ is equivalent to E23 $({x = (x \\diamond x) \\diamond x})$.","title":"14 implies 23"},{"id":"equational-theories/part-02/e-graphs/magmaegg","kind":"node","parent":5332,"result":false,"route":"../roadmap/equational-theories/part-02/e-graphs/magmaegg/index.html","scope":"equational-theories/part-02/e-graphs","status":"can_state","summary":"This is a simple but apparently at least somewhat effective Rust theorem prover based on egg `e-graph` library written for this project.","title":"MagmaEgg"},{"id":"equational-theories/part-02/using-the-vampire-theorem-prover","kind":"node","parent":5330,"result":false,"route":"../roadmap/equational-theories/part-02/using-the-vampire-theorem-prover/index.html","scope":"equational-theories/part-02","status":"can_state","summary":"1,775 implications were proven using the Vampire theorem prover.","title":"Using the Vampire theorem prover"},{"data_url":"scopes/equational-theories/part-02/hard-implications.json","id":"equational-theories/part-02/hard-implications","kind":"scope","parent":5330,"result":false,"route":"../roadmap/equational-theories/part-02/hard-implications/index.html","scope":"equational-theories/part-02/hard-implications","status":"planned","title":"Hard implications"},{"id":"equational-theories/part-02/hard-implications/infinite-magmas","kind":"node","parent":5337,"result":false,"route":"../roadmap/equational-theories/part-02/hard-implications/infinite-magmas/index.html","scope":"equational-theories/part-02/hard-implications","status":"can_state","summary":"The following forward implications can be solved by ATPs, but take unusually long:","title":"Infinite magmas"},{"id":"equational-theories/part-02/hard-implications/finite-magmas","kind":"node","parent":5337,"result":false,"route":"../roadmap/equational-theories/part-02/hard-implications/finite-magmas/index.html","scope":"equational-theories/part-02/hard-implications","status":"can_state","summary":"In the case of finite magmas, there are some non-trivial positive implications:","title":"Finite magmas"},{"data_url":"chapters/strongpnt.json","id":"strongpnt","kind":"scope","parent":0,"result":false,"route":"../roadmap/strongpnt/index.html","scope":"strongpnt","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc), read at commit 2f5835c3 (2025-09-10): one article for each of its 615 nodes whose Lean declarations it names and marks formalized, or in Mathlib. Its other 6 nodes have no article, and a section of only those is left out: the map records what the project contains, not what it plans. A statement for which the blueprint marks no proof `\\leanok` says so. Dependencies are the\u2026","title":"Strong PNT (Math Inc)"},{"data_url":"scopes/strongpnt/complex-analysis.json","id":"strongpnt/complex-analysis","kind":"scope","parent":5340,"result":false,"route":"../roadmap/strongpnt/complex-analysis/index.html","scope":"strongpnt/complex-analysis","status":"fully_proved","title":"Complex Analysis"},{"data_url":"scopes/strongpnt/complex-analysis/part-01.json","id":"strongpnt/complex-analysis/part-01","kind":"scope","navigation":"transparent","parent":5341,"result":false,"route":"../roadmap/strongpnt/complex-analysis/part-01/index.html","scope":"strongpnt/complex-analysis/part-01","status":"fully_proved","title":"Log growth to Trig identity"},{"id":"strongpnt/complex-analysis/part-01/lem-2logolog","kind":"node","lean":"lem_2logOlog","parent":5342,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-01/index.html#lem-2logolog","scope":"strongpnt/complex-analysis/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 4. The project gives no license, so trough does not reproduce its text.","title":"Log growth"},{"id":"strongpnt/complex-analysis/part-01/lem-logt22logt","kind":"node","lean":"lem_logt22logt","parent":5342,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-01/index.html#lem-logt22logt","scope":"strongpnt/complex-analysis/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 12. The project gives no license, so trough does not reproduce its text.","title":"Square log"},{"id":"strongpnt/complex-analysis/part-01/lem-log2tlogt2","kind":"node","lean":"lem_log2tlogt2","parent":5342,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-01/index.html#lem-log2tlogt2","scope":"strongpnt/complex-analysis/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 20. The project gives no license, so trough does not reproduce its text.","title":"Double log"},{"id":"strongpnt/complex-analysis/part-01/lem-log22log","kind":"node","lean":"lem_log22log","parent":5342,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-01/index.html#lem-log22log","scope":"strongpnt/complex-analysis/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 28. The project gives no license, so trough does not reproduce its text.","title":"Log compare"},{"id":"strongpnt/complex-analysis/part-01/lem-exprule","kind":"node","lean":"lem_exprule","parent":5342,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-01/index.html#lem-exprule","scope":"strongpnt/complex-analysis/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 37. The project gives no license, so trough does not reproduce its text.","title":"Exp rule"},{"id":"strongpnt/complex-analysis/part-01/lem-realbw","kind":"node","lean":"lem_realbw","parent":5342,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-01/index.html#lem-realbw","scope":"strongpnt/complex-analysis/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 45. The project gives no license, so trough does not reproduce its text.","title":"Real scale"},{"id":"strongpnt/complex-analysis/part-01/lem-sumreal","kind":"node","lean":"lem_sumReal","parent":5342,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-01/index.html#lem-sumreal","scope":"strongpnt/complex-analysis/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 52. The project gives no license, so trough does not reproduce its text.","title":"Real series"},{"id":"strongpnt/complex-analysis/part-01/lem-euler","kind":"node","lean":"lem_Euler","parent":5342,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-01/index.html#lem-euler","scope":"strongpnt/complex-analysis/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 61. The project gives no license, so trough does not reproduce its text.","title":"Euler's formula"},{"id":"strongpnt/complex-analysis/part-01/lem-reecos","kind":"node","lean":"lem_Reecos","parent":5342,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-01/index.html#lem-reecos","scope":"strongpnt/complex-analysis/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 68. The project gives no license, so trough does not reproduce its text.","title":"Real cosine"},{"id":"strongpnt/complex-analysis/part-01/lem-explog","kind":"node","lean":"lem_explog","parent":5342,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-01/index.html#lem-explog","scope":"strongpnt/complex-analysis/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 76. The project gives no license, so trough does not reproduce its text.","title":"Log inverse"},{"id":"strongpnt/complex-analysis/part-01/lem-coseven","kind":"node","lean":"lem_coseven","parent":5342,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-01/index.html#lem-coseven","scope":"strongpnt/complex-analysis/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 83. The project gives no license, so trough does not reproduce its text.","title":"Cos even"},{"id":"strongpnt/complex-analysis/part-01/lem-coseveny","kind":"node","lean":"lem_coseveny","parent":5342,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-01/index.html#lem-coseveny","scope":"strongpnt/complex-analysis/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 91. The project gives no license, so trough does not reproduce its text.","title":"Cos even"},{"id":"strongpnt/complex-analysis/part-01/lem-niyelog","kind":"node","lean":"lem_niyelog","parent":5342,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-01/index.html#lem-niyelog","scope":"strongpnt/complex-analysis/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 101. The project gives no license, so trough does not reproduce its text.","title":"Exp form"},{"id":"strongpnt/complex-analysis/part-01/lem-eacosalog","kind":"node","lean":"lem_eacosalog","parent":5342,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-01/index.html#lem-eacosalog","scope":"strongpnt/complex-analysis/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 109. The project gives no license, so trough does not reproduce its text.","title":"Real cosine"},{"id":"strongpnt/complex-analysis/part-01/lem-eacosalog2","kind":"node","lean":"lem_eacosalog2","parent":5342,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-01/index.html#lem-eacosalog2","scope":"strongpnt/complex-analysis/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 117. The project gives no license, so trough does not reproduce its text.","title":"Real cosine"},{"id":"strongpnt/complex-analysis/part-01/lem-eacosalog3","kind":"node","lean":"lem_eacosalog3","parent":5342,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-01/index.html#lem-eacosalog3","scope":"strongpnt/complex-analysis/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 125. The project gives no license, so trough does not reproduce its text.","title":"Real cosine"},{"id":"strongpnt/complex-analysis/part-01/lem-cos2t","kind":"node","lean":"lem_cos2t","parent":5342,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-01/index.html#lem-cos2t","scope":"strongpnt/complex-analysis/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 135. The project gives no license, so trough does not reproduce its text.","title":"Double angle"},{"id":"strongpnt/complex-analysis/part-01/lem-cos2t2","kind":"node","lean":"lem_cos2t2","parent":5342,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-01/index.html#lem-cos2t2","scope":"strongpnt/complex-analysis/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 142. The project gives no license, so trough does not reproduce its text.","title":"Cos square"},{"id":"strongpnt/complex-analysis/part-01/lem-cossquare","kind":"node","lean":"lem_cosSquare","parent":5342,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-01/index.html#lem-cossquare","scope":"strongpnt/complex-analysis/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 150. The project gives no license, so trough does not reproduce its text.","title":"Square expand"},{"id":"strongpnt/complex-analysis/part-01/lem-cos2cos341","kind":"node","lean":"lem_cos2cos341","parent":5342,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-01/index.html#lem-cos2cos341","scope":"strongpnt/complex-analysis/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 158. The project gives no license, so trough does not reproduce its text.","title":"Trig identity"},{"data_url":"scopes/strongpnt/complex-analysis/part-02.json","id":"strongpnt/complex-analysis/part-02","kind":"scope","navigation":"transparent","parent":5341,"result":false,"route":"../roadmap/strongpnt/complex-analysis/part-02/index.html","scope":"strongpnt/complex-analysis/part-02","status":"fully_proved","title":"Square nonneg to Mod diff"},{"id":"strongpnt/complex-analysis/part-02/lem-squarepos","kind":"node","lean":"lem_SquarePos","parent":5363,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-02/index.html#lem-squarepos","scope":"strongpnt/complex-analysis/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 166. The project gives no license, so trough does not reproduce its text.","title":"Square nonneg"},{"id":"strongpnt/complex-analysis/part-02/lem-squarepos2","kind":"node","lean":"lem_SquarePos2","parent":5363,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-02/index.html#lem-squarepos2","scope":"strongpnt/complex-analysis/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 173. The project gives no license, so trough does not reproduce its text.","title":"Double square"},{"id":"strongpnt/complex-analysis/part-02/lem-squareposcos","kind":"node","lean":"lem_SquarePoscos","parent":5363,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-02/index.html#lem-squareposcos","scope":"strongpnt/complex-analysis/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 181. The project gives no license, so trough does not reproduce its text.","title":"Cos square"},{"id":"strongpnt/complex-analysis/part-02/lem-postrig","kind":"node","lean":"lem_postrig","parent":5363,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-02/index.html#lem-postrig","scope":"strongpnt/complex-analysis/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 189. The project gives no license, so trough does not reproduce its text.","title":"Trig positive"},{"id":"strongpnt/complex-analysis/part-02/lem-postriglogn","kind":"node","lean":"lem_postriglogn","parent":5363,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-02/index.html#lem-postriglogn","scope":"strongpnt/complex-analysis/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 198. The project gives no license, so trough does not reproduce its text.","title":"Trig positive"},{"id":"strongpnt/complex-analysis/part-02/lem-seriespos","kind":"node","lean":"lem_seriesPos","parent":5363,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-02/index.html#lem-seriespos","scope":"strongpnt/complex-analysis/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 207. The project gives no license, so trough does not reproduce its text.","title":"Series positive"},{"id":"strongpnt/complex-analysis/part-02/lem-real-part-of-diff","kind":"node","lean":"real_part_of_diff","parent":5363,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-02/index.html#lem-real-part-of-diff","scope":"strongpnt/complex-analysis/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 215. The project gives no license, so trough does not reproduce its text.","title":"Real part diff"},{"id":"strongpnt/complex-analysis/part-02/lem-real-part-of-diffz","kind":"node","lean":"real_part_of_diffz","parent":5363,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-02/index.html#lem-real-part-of-diffz","scope":"strongpnt/complex-analysis/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 221. The project gives no license, so trough does not reproduce its text.","title":"Real part 2M"},{"id":"strongpnt/complex-analysis/part-02/lem-inequality-reversal","kind":"node","lean":"inequality_reversal","parent":5363,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-02/index.html#lem-inequality-reversal","scope":"strongpnt/complex-analysis/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 228. The project gives no license, so trough does not reproduce its text.","title":"Inequality reversal"},{"id":"strongpnt/complex-analysis/part-02/lem-real-part-lower-bound","kind":"node","lean":"real_part_lower_bound","parent":5363,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-02/index.html#lem-real-part-lower-bound","scope":"strongpnt/complex-analysis/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 234. The project gives no license, so trough does not reproduce its text.","title":"Real part lower bound"},{"id":"strongpnt/complex-analysis/part-02/lem-real-part-lower-bound2","kind":"node","lean":"real_part_lower_bound2","parent":5363,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-02/index.html#lem-real-part-lower-bound2","scope":"strongpnt/complex-analysis/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 241. The project gives no license, so trough does not reproduce its text.","title":"Real part bound"},{"id":"strongpnt/complex-analysis/part-02/lem-real-part-lower-bound3","kind":"node","lean":"real_part_lower_bound3","parent":5363,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-02/index.html#lem-real-part-lower-bound3","scope":"strongpnt/complex-analysis/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 248. The project gives no license, so trough does not reproduce its text.","title":"Real part >0"},{"id":"strongpnt/complex-analysis/part-02/lem-nonzero-if-real-part-positive","kind":"node","lean":"nonzero_if_real_part_positive","parent":5363,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-02/index.html#lem-nonzero-if-real-part-positive","scope":"strongpnt/complex-analysis/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 255. The project gives no license, so trough does not reproduce its text.","title":"Pos real nonzero"},{"id":"strongpnt/complex-analysis/part-02/lem-real-part-lower-bound4","kind":"node","lean":"lem_real_part_lower_bound4","parent":5363,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-02/index.html#lem-real-part-lower-bound4","scope":"strongpnt/complex-analysis/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 261. The project gives no license, so trough does not reproduce its text.","title":"2M minus nonzero"},{"id":"strongpnt/complex-analysis/part-02/lem-abspos","kind":"node","lean":"lem_abspos","parent":5363,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-02/index.html#lem-abspos","scope":"strongpnt/complex-analysis/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 268. The project gives no license, so trough does not reproduce its text.","title":"Absolute value positive"},{"id":"strongpnt/complex-analysis/part-02/lem-real-part-lower-bound5","kind":"node","lean":"lem_real_part_lower_bound5","parent":5363,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-02/index.html#lem-real-part-lower-bound5","scope":"strongpnt/complex-analysis/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 274. The project gives no license, so trough does not reproduce its text.","title":"2M minus mod"},{"id":"strongpnt/complex-analysis/part-02/lem-wreim","kind":"node","lean":"lem_wReIm","parent":5363,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-02/index.html#lem-wreim","scope":"strongpnt/complex-analysis/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 282. The project gives no license, so trough does not reproduce its text.","title":"Real imaginary"},{"id":"strongpnt/complex-analysis/part-02/lem-modaib","kind":"node","lean":"lem_modaib","parent":5363,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-02/index.html#lem-modaib","scope":"strongpnt/complex-analysis/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 289. The project gives no license, so trough does not reproduce its text.","title":"Mod square"},{"id":"strongpnt/complex-analysis/part-02/lem-modcaib","kind":"node","lean":"lem_modcaib","parent":5363,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-02/index.html#lem-modcaib","scope":"strongpnt/complex-analysis/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 295. The project gives no license, so trough does not reproduce its text.","title":"Shifted mod"},{"id":"strongpnt/complex-analysis/part-02/lem-diffmods","kind":"node","lean":"lem_diffmods","parent":5363,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-02/index.html#lem-diffmods","scope":"strongpnt/complex-analysis/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 301. The project gives no license, so trough does not reproduce its text.","title":"Mod diff"},{"data_url":"scopes/strongpnt/complex-analysis/part-03.json","id":"strongpnt/complex-analysis/part-03","kind":"scope","navigation":"transparent","parent":5341,"result":false,"route":"../roadmap/strongpnt/complex-analysis/part-03/index.html","scope":"strongpnt/complex-analysis/part-03","status":"fully_proved","title":"Square expand to Scaled triangle"},{"id":"strongpnt/complex-analysis/part-03/lem-casq","kind":"node","lean":"lem_casq","parent":5384,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-03/index.html#lem-casq","scope":"strongpnt/complex-analysis/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 308. The project gives no license, so trough does not reproduce its text.","title":"Square expand"},{"id":"strongpnt/complex-analysis/part-03/lem-casq2","kind":"node","lean":"lem_casq2","parent":5384,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-03/index.html#lem-casq2","scope":"strongpnt/complex-analysis/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 314. The project gives no license, so trough does not reproduce its text.","title":"Square diff"},{"id":"strongpnt/complex-analysis/part-03/lem-diffmods2","kind":"node","lean":"lem_diffmods2","parent":5384,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-03/index.html#lem-diffmods2","scope":"strongpnt/complex-analysis/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 321. The project gives no license, so trough does not reproduce its text.","title":"Mod diff"},{"id":"strongpnt/complex-analysis/part-03/lem-modulus-sq-reimw","kind":"node","lean":"lem_modulus_sq_ReImw","parent":5384,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-03/index.html#lem-modulus-sq-reimw","scope":"strongpnt/complex-analysis/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 328. The project gives no license, so trough does not reproduce its text.","title":"Modulus diff"},{"id":"strongpnt/complex-analysis/part-03/lem-modulus-sq-identity","kind":"node","lean":"lem_modulus_sq_identity","parent":5384,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-03/index.html#lem-modulus-sq-identity","scope":"strongpnt/complex-analysis/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 335. The project gives no license, so trough does not reproduce its text.","title":"Modulus identity"},{"id":"strongpnt/complex-analysis/part-03/lem-nonnegative-product","kind":"node","lean":"lem_nonnegative_product","parent":5384,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-03/index.html#lem-nonnegative-product","scope":"strongpnt/complex-analysis/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 342. The project gives no license, so trough does not reproduce its text.","title":"Nonneg product"},{"id":"strongpnt/complex-analysis/part-03/lem-nonnegative-product2","kind":"node","lean":"lem_nonnegative_product2","parent":5384,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-03/index.html#lem-nonnegative-product2","scope":"strongpnt/complex-analysis/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 348. The project gives no license, so trough does not reproduce its text.","title":"Nonneg product"},{"id":"strongpnt/complex-analysis/part-03/lem-nonnegative-product3","kind":"node","lean":"lem_nonnegative_product3","parent":5384,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-03/index.html#lem-nonnegative-product3","scope":"strongpnt/complex-analysis/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 356. The project gives no license, so trough does not reproduce its text.","title":"Modulus compare"},{"id":"strongpnt/complex-analysis/part-03/lem-nonnegative-product4","kind":"node","lean":"lem_nonnegative_product4","parent":5384,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-03/index.html#lem-nonnegative-product4","scope":"strongpnt/complex-analysis/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 363. The project gives no license, so trough does not reproduce its text.","title":"Modulus bound"},{"id":"strongpnt/complex-analysis/part-03/lem-nonnegative-product5","kind":"node","lean":"lem_nonnegative_product5","parent":5384,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-03/index.html#lem-nonnegative-product5","scope":"strongpnt/complex-analysis/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 370. The project gives no license, so trough does not reproduce its text.","title":"Modulus bound"},{"id":"strongpnt/complex-analysis/part-03/lem-nonnegative-product6","kind":"node","lean":"lem_nonnegative_product6","parent":5384,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-03/index.html#lem-nonnegative-product6","scope":"strongpnt/complex-analysis/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 377. The project gives no license, so trough does not reproduce its text.","title":"Modulus order"},{"id":"strongpnt/complex-analysis/part-03/lem-ineqmultr","kind":"node","lean":"lem_ineqmultr","parent":5384,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-03/index.html#lem-ineqmultr","scope":"strongpnt/complex-analysis/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 384. The project gives no license, so trough does not reproduce its text.","title":"Divide inequality"},{"id":"strongpnt/complex-analysis/part-03/lem-ineqmultrbb","kind":"node","lean":"lem_ineqmultrbb","parent":5384,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-03/index.html#lem-ineqmultrbb","scope":"strongpnt/complex-analysis/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 391. The project gives no license, so trough does not reproduce its text.","title":"Ratio bound"},{"id":"strongpnt/complex-analysis/part-03/lem-nonnegative-product7","kind":"node","lean":"lem_nonnegative_product7","parent":5384,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-03/index.html#lem-nonnegative-product7","scope":"strongpnt/complex-analysis/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 398. The project gives no license, so trough does not reproduce its text.","title":"Ratio bound"},{"id":"strongpnt/complex-analysis/part-03/lem-nonnegative-product8","kind":"node","lean":"lem_nonnegative_product8","parent":5384,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-03/index.html#lem-nonnegative-product8","scope":"strongpnt/complex-analysis/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 405. The project gives no license, so trough does not reproduce its text.","title":"Ratio bound"},{"id":"strongpnt/complex-analysis/part-03/lem-nonnegative-product9","kind":"node","lean":"lem_nonnegative_product9","parent":5384,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-03/index.html#lem-nonnegative-product9","scope":"strongpnt/complex-analysis/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 412. The project gives no license, so trough does not reproduce its text.","title":"Ratio bound"},{"id":"strongpnt/complex-analysis/part-03/lem-triangleineq","kind":"node","lean":"lem_triangle_ineq","parent":5384,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-03/index.html#lem-triangleineq","scope":"strongpnt/complex-analysis/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 420. The project gives no license, so trough does not reproduce its text.","title":"Triangle inequality"},{"id":"strongpnt/complex-analysis/part-03/lem-triangleineqminus","kind":"node","lean":"lem_triangleineqminus","parent":5384,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-03/index.html#lem-triangleineqminus","scope":"strongpnt/complex-analysis/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 426. The project gives no license, so trough does not reproduce its text.","title":"Triangle minus"},{"id":"strongpnt/complex-analysis/part-03/lem-rtriangle","kind":"node","lean":"lem_rtriangle","parent":5384,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-03/index.html#lem-rtriangle","scope":"strongpnt/complex-analysis/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 434. The project gives no license, so trough does not reproduce its text.","title":"Scaled triangle"},{"id":"strongpnt/complex-analysis/part-03/lem-rtriangle2","kind":"node","lean":"rtriangle2","parent":5384,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-03/index.html#lem-rtriangle2","scope":"strongpnt/complex-analysis/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 441. The project gives no license, so trough does not reproduce its text.","title":"Scaled triangle"},{"data_url":"scopes/strongpnt/complex-analysis/part-04.json","id":"strongpnt/complex-analysis/part-04","kind":"scope","navigation":"transparent","parent":5341,"result":false,"route":"../roadmap/strongpnt/complex-analysis/part-04/index.html","scope":"strongpnt/complex-analysis/part-04","status":"fully_proved","title":"Ineq step to Product analytic"},{"id":"strongpnt/complex-analysis/part-04/lem-rtriangle3","kind":"node","lean":"lem_rtriangle3","parent":5405,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-04/index.html#lem-rtriangle3","scope":"strongpnt/complex-analysis/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 448. The project gives no license, so trough does not reproduce its text.","title":"Ineq step"},{"id":"strongpnt/complex-analysis/part-04/lem-rtriangle4","kind":"node","lean":"lem_rtriangle4","parent":5405,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-04/index.html#lem-rtriangle4","scope":"strongpnt/complex-analysis/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 455. The project gives no license, so trough does not reproduce its text.","title":"Rearranged bound"},{"id":"strongpnt/complex-analysis/part-04/lem-absposeq","kind":"node","lean":"lem_absposeq","parent":5405,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-04/index.html#lem-absposeq","scope":"strongpnt/complex-analysis/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 462. The project gives no license, so trough does not reproduce its text.","title":"Abs positive"},{"id":"strongpnt/complex-analysis/part-04/lem-a2a","kind":"node","lean":"lem_a2a","parent":5405,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-04/index.html#lem-a2a","scope":"strongpnt/complex-analysis/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 468. The project gives no license, so trough does not reproduce its text.","title":"Double positive"},{"id":"strongpnt/complex-analysis/part-04/lem-absposeq2","kind":"node","lean":"lem_absposeq2","parent":5405,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-04/index.html#lem-absposeq2","scope":"strongpnt/complex-analysis/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 474. The project gives no license, so trough does not reproduce its text.","title":"Scaled abs"},{"id":"strongpnt/complex-analysis/part-04/lem-rtriangle5","kind":"node","lean":"lem_rtriangle5","parent":5405,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-04/index.html#lem-rtriangle5","scope":"strongpnt/complex-analysis/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 481. The project gives no license, so trough does not reproduce its text.","title":"Key bound"},{"id":"strongpnt/complex-analysis/part-04/lemrrfpos","kind":"node","lean":"lem_RrFpos","parent":5405,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-04/index.html#lemrrfpos","scope":"strongpnt/complex-analysis/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 488. The project gives no license, so trough does not reproduce its text.","title":"Nonneg factor"},{"id":"strongpnt/complex-analysis/part-04/lem-rtriangle6","kind":"node","lean":"lem_rtriangle6","parent":5405,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-04/index.html#lem-rtriangle6","scope":"strongpnt/complex-analysis/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 494. The project gives no license, so trough does not reproduce its text.","title":"Divide bound"},{"id":"strongpnt/complex-analysis/part-04/lem-rtriangle7","kind":"node","lean":"lem_rtriangle7","parent":5405,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-04/index.html#lem-rtriangle7","scope":"strongpnt/complex-analysis/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 501. The project gives no license, so trough does not reproduce its text.","title":"Final bound"},{"id":"strongpnt/complex-analysis/part-04/lem-orderne0","kind":"node","lean":"lem_orderne0","parent":5405,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-04/index.html#lem-orderne0","scope":"strongpnt/complex-analysis/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 509. The project gives no license, so trough does not reproduce its text.","title":"Order nonzero"},{"id":"strongpnt/complex-analysis/part-04/lem-ordernetop","kind":"node","lean":"lem_ordernetop","parent":5405,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-04/index.html#lem-ordernetop","scope":"strongpnt/complex-analysis/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 519. The project gives no license, so trough does not reproduce its text.","title":"Order natural"},{"id":"strongpnt/complex-analysis/part-04/lem-ordernatcast","kind":"node","lean":"lem_ordernatcast","parent":5405,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-04/index.html#lem-ordernatcast","scope":"strongpnt/complex-analysis/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 530. The project gives no license, so trough does not reproduce its text.","title":"Factor power"},{"id":"strongpnt/complex-analysis/part-04/lem-ordernatcast1","kind":"node","lean":"lem_ordernatcast1","parent":5405,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-04/index.html#lem-ordernatcast1","scope":"strongpnt/complex-analysis/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 541. The project gives no license, so trough does not reproduce its text.","title":"Factor linear"},{"id":"strongpnt/complex-analysis/part-04/lem-ordernatcast2","kind":"node","lean":"lem_ordernatcast2","parent":5405,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-04/index.html#lem-ordernatcast2","scope":"strongpnt/complex-analysis/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 554. The project gives no license, so trough does not reproduce its text.","title":"Divide zero"},{"id":"strongpnt/complex-analysis/part-04/lem-analmono","kind":"node","lean":"lem_analmono","parent":5405,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-04/index.html#lem-analmono","scope":"strongpnt/complex-analysis/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 577. The project gives no license, so trough does not reproduce its text.","title":"Analytic mono"},{"id":"strongpnt/complex-analysis/part-04/lem-not0mono","kind":"node","lean":"lem_not0mono","parent":5405,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-04/index.html#lem-not0mono","scope":"strongpnt/complex-analysis/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 587. The project gives no license, so trough does not reproduce its text.","title":"Nonzero subset"},{"id":"strongpnt/complex-analysis/part-04/lem-1zanaldr","kind":"node","lean":"lem_1zanalDR","parent":5405,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-04/index.html#lem-1zanaldr","scope":"strongpnt/complex-analysis/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 597. The project gives no license, so trough does not reproduce its text.","title":"Inverse analytic"},{"id":"strongpnt/complex-analysis/part-04/lem-analprod","kind":"node","lean":"lem_analprod","parent":5405,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-04/index.html#lem-analprod","scope":"strongpnt/complex-analysis/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 609. The project gives no license, so trough does not reproduce its text.","title":"Product analytic"},{"id":"strongpnt/complex-analysis/part-04/lem-analprodst","kind":"node","lean":"lem_analprodST","parent":5405,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-04/index.html#lem-analprodst","scope":"strongpnt/complex-analysis/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 619. The project gives no license, so trough does not reproduce its text.","title":"Product analytic"},{"id":"strongpnt/complex-analysis/part-04/lem-analprodtdr","kind":"node","lean":"lem_analprodTDR","parent":5405,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-04/index.html#lem-analprodtdr","scope":"strongpnt/complex-analysis/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 632. The project gives no license, so trough does not reproduce its text.","title":"Product analytic"},{"data_url":"scopes/strongpnt/complex-analysis/part-05.json","id":"strongpnt/complex-analysis/part-05","kind":"scope","navigation":"transparent","parent":5341,"result":false,"route":"../roadmap/strongpnt/complex-analysis/part-05/index.html","scope":"strongpnt/complex-analysis/part-05","status":"fully_proved","title":"Quotient analytic to Complex logarithm"},{"id":"strongpnt/complex-analysis/part-05/lem-fzztanal","kind":"node","lean":"lem_fzzTanal","parent":5426,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/index.html#lem-fzztanal","scope":"strongpnt/complex-analysis/part-05","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 644. The project gives no license, so trough does not reproduce its text.","title":"Quotient analytic"},{"id":"strongpnt/complex-analysis/part-05/lem-analontowithin","kind":"node","lean":"lem_AnalOntoWithin","parent":5426,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/index.html#lem-analontowithin","scope":"strongpnt/complex-analysis/part-05","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 657. The project gives no license, so trough does not reproduce its text.","title":"On implies within"},{"id":"strongpnt/complex-analysis/part-05/lem-analwithintoon","kind":"node","lean":"lem_AnalWithintoOn","parent":5426,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/index.html#lem-analwithintoon","scope":"strongpnt/complex-analysis/part-05","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 667. The project gives no license, so trough does not reproduce its text.","title":"Within implies on"},{"id":"strongpnt/complex-analysis/part-05/lem-dr0t","kind":"node","lean":"lem_DR0T","parent":5426,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/index.html#lem-dr0t","scope":"strongpnt/complex-analysis/part-05","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 678. The project gives no license, so trough does not reproduce its text.","title":"Disk split"},{"id":"strongpnt/complex-analysis/part-05/lem-analwwwithin","kind":"node","lean":"lem_analWWWithin","parent":5426,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/index.html#lem-analwwwithin","scope":"strongpnt/complex-analysis/part-05","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 689. The project gives no license, so trough does not reproduce its text.","title":"Within union"},{"id":"strongpnt/complex-analysis/part-05/lem-analwwithinaton","kind":"node","lean":"lem_analWWithinAtOn","parent":5426,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/index.html#lem-analwwithinaton","scope":"strongpnt/complex-analysis/part-05","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 700. The project gives no license, so trough does not reproduce its text.","title":"Within gives on"},{"id":"strongpnt/complex-analysis/part-05/lem-analattowithin","kind":"node","lean":"lem_AnalAttoWithin","parent":5426,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/index.html#lem-analattowithin","scope":"strongpnt/complex-analysis/part-05","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 713. The project gives no license, so trough does not reproduce its text.","title":"At to within"},{"id":"strongpnt/complex-analysis/part-05/lem-analatonon","kind":"node","lean":"lem_analAtOnOn","parent":5426,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/index.html#lem-analatonon","scope":"strongpnt/complex-analysis/part-05","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 723. The project gives no license, so trough does not reproduce its text.","title":"Local to global"},{"data_url":"scopes/strongpnt/complex-analysis/part-05/borel-caratheodory-i.json","id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i","kind":"scope","parent":5426,"result":false,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/index.html","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i","status":"fully_proved","title":"Borel-Carath\u00e9odory I"},{"data_url":"scopes/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01.json","id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01","kind":"scope","navigation":"transparent","parent":5435,"result":false,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/index.html","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01","status":"fully_proved","title":"Open disk to Boundary max"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/def-balldr","kind":"node","lean":"ballDR","parent":5436,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/index.html#def-balldr","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this definition at PNT/PNT1_ComplexAnalysis.tex, line 737. The project gives no license, so trough does not reproduce its text.","title":"Open disk"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/lem-balldr","kind":"node","lean":"lem_ballDR","parent":5436,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/index.html#lem-balldr","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 742. The project gives no license, so trough does not reproduce its text.","title":"Disk closure"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/lem-indr","kind":"node","lean":"lem_inDR","parent":5436,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/index.html#lem-indr","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 749. The project gives no license, so trough does not reproduce its text.","title":"In disk bound"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/lem-notindr","kind":"node","lean":"lem_notinDR","parent":5436,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/index.html#lem-notindr","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 758. The project gives no license, so trough does not reproduce its text.","title":"Outside disk bound"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/lem-leger","kind":"node","lean":"lem_legeR","parent":5436,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/index.html#lem-leger","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 767. The project gives no license, so trough does not reproduce its text.","title":"Modulus equal"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/lem-circledr","kind":"node","lean":"lem_circleDR","parent":5436,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/index.html#lem-circledr","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 775. The project gives no license, so trough does not reproduce its text.","title":"Boundary modulus"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/lem-rself","kind":"node","lean":"lem_Rself","parent":5436,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/index.html#lem-rself","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 784. The project gives no license, so trough does not reproduce its text.","title":"Positive modulus"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/lem-rself2","kind":"node","lean":"lem_Rself2","parent":5436,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/index.html#lem-rself2","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 791. The project gives no license, so trough does not reproduce its text.","title":"Modulus bound"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/lem-rself3","kind":"node","lean":"lem_Rself3","parent":5436,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/index.html#lem-rself3","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 800. The project gives no license, so trough does not reproduce its text.","title":"Positive radius belongs to its closed disk"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/lem-drcompact","kind":"node","lean":"lem_DRcompact","parent":5436,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/index.html#lem-drcompact","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 809. The project gives no license, so trough does not reproduce its text.","title":"Compactness"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/lem-extrvalthm","kind":"node","lean":"lem_ExtrValThm","parent":5436,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/index.html#lem-extrvalthm","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 816. The project gives no license, so trough does not reproduce its text.","title":"ExtrValThm"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/lem-extrvalthmdr","kind":"node","lean":"lem_ExtrValThmDR","parent":5436,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/index.html#lem-extrvalthmdr","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 823. The project gives no license, so trough does not reproduce its text.","title":"Disk Boundary"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/lem-analcont","kind":"node","lean":"lem_AnalCont","parent":5436,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/index.html#lem-analcont","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 832. The project gives no license, so trough does not reproduce its text.","title":"Analytic Continuation"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/lem-extrvalthmh","kind":"node","lean":"lem_ExtrValThmh","parent":5436,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/index.html#lem-extrvalthmh","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 839. The project gives no license, so trough does not reproduce its text.","title":"Max modulus"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/lem-maxmodp","kind":"node","lean":"lem_MaxModP","parent":5436,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/index.html#lem-maxmodp","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 849. The project gives no license, so trough does not reproduce its text.","title":"Interior max"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/lem-maxmodr","kind":"node","lean":"lem_MaxModR","parent":5436,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/index.html#lem-maxmodr","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 858. The project gives no license, so trough does not reproduce its text.","title":"Boundary value"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/lem-maxmodrr","kind":"node","lean":"lem_MaxModRR","parent":5436,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/index.html#lem-maxmodrr","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 867. The project gives no license, so trough does not reproduce its text.","title":"Boundary bound"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/lem-maxmodv2","kind":"node","lean":"lem_MaxModv2","parent":5436,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/index.html#lem-maxmodv2","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 877. The project gives no license, so trough does not reproduce its text.","title":"Boundary point"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/lem-maxmodv3","kind":"node","lean":"lem_MaxModv3","parent":5436,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/index.html#lem-maxmodv3","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 888. The project gives no license, so trough does not reproduce its text.","title":"Boundary point"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/lem-maxmodv4","kind":"node","lean":"lem_MaxModv4","parent":5436,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01/index.html#lem-maxmodv4","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 898. The project gives no license, so trough does not reproduce its text.","title":"Boundary max"},{"data_url":"scopes/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02.json","id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02","kind":"scope","navigation":"transparent","parent":5435,"result":false,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/index.html","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02","status":"fully_proved","title":"Max principle to Rearranged bound"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/lem-hardmmp","kind":"node","lean":"lem_HardMMP","parent":5457,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/index.html#lem-hardmmp","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 907. The project gives no license, so trough does not reproduce its text.","title":"Max principle"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/lem-easymmp","kind":"node","lean":"lem_EasyMMP","parent":5457,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/index.html#lem-easymmp","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 916. The project gives no license, so trough does not reproduce its text.","title":"Easy maximum principle"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/lem-mmp","kind":"node","lean":"lem_MMP","parent":5457,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/index.html#lem-mmp","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 924. The project gives no license, so trough does not reproduce its text.","title":"Maximum modulus principle"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/lem-denominator-nonzero","kind":"node","lean":"lem_denominator_nonzero","parent":5457,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/index.html#lem-denominator-nonzero","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 934. The project gives no license, so trough does not reproduce its text.","title":"Denom nonzero"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/lem-f-vs-2m-minus-f","kind":"node","lean":"lem_f_vs_2M_minus_f","parent":5457,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/index.html#lem-f-vs-2m-minus-f","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 944. The project gives no license, so trough does not reproduce its text.","title":"Ratio bound"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/lem-removable-singularity","kind":"node","lean":"lem_removable_singularity","parent":5457,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/index.html#lem-removable-singularity","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 954. The project gives no license, so trough does not reproduce its text.","title":"Removable zero"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/lem-quotient-analytic","kind":"node","lean":"lem_quotient_analytic","parent":5457,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/index.html#lem-quotient-analytic","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 963. The project gives no license, so trough does not reproduce its text.","title":"Quotient analytic"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/def-g","kind":"node","lean":"f_M","parent":5457,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/index.html#def-g","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this definition at PNT/PNT1_ComplexAnalysis.tex, line 970. The project gives no license, so trough does not reproduce its text.","title":"Modified function"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/lem-g-analytic","kind":"node","lean":"lem_g_analytic","parent":5457,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/index.html#lem-g-analytic","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 976. The project gives no license, so trough does not reproduce its text.","title":"g analytic"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/lem-absab","kind":"node","lean":"lem_absab","parent":5457,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/index.html#lem-absab","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 986. The project gives no license, so trough does not reproduce its text.","title":"Quotient modulus"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/lem-g-on-boundaryz","kind":"node","lean":"lem_g_on_boundaryz","parent":5457,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/index.html#lem-g-on-boundaryz","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 993. The project gives no license, so trough does not reproduce its text.","title":"g modulus"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/lem-fzzr","kind":"node","lean":"lem_fzzR","parent":5457,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/index.html#lem-fzzr","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1002. The project gives no license, so trough does not reproduce its text.","title":"Quotient radius"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/lem-g-on-boundary","kind":"node","lean":"lem_g_on_boundary","parent":5457,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/index.html#lem-g-on-boundary","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1010. The project gives no license, so trough does not reproduce its text.","title":"Boundary g"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/lem-f-vs-2m-minus-fr","kind":"node","lean":"lem_f_vs_2M_minus_fR","parent":5457,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/index.html#lem-f-vs-2m-minus-fr","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1019. The project gives no license, so trough does not reproduce its text.","title":"Scaled ratio"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/lem-g-boundary-bound0","kind":"node","lean":"lem_g_boundary_bound0","parent":5457,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/index.html#lem-g-boundary-bound0","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1028. The project gives no license, so trough does not reproduce its text.","title":"Boundary bound"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/lem-g-interior-bound","kind":"node","lean":"lem_g_interior_bound","parent":5457,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/index.html#lem-g-interior-bound","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1038. The project gives no license, so trough does not reproduce its text.","title":"Interior bound"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/lem-g-at-r","kind":"node","lean":"lem_g_at_r","parent":5457,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/index.html#lem-g-at-r","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1047. The project gives no license, so trough does not reproduce its text.","title":"g at r"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/lem-g-at-rr","kind":"node","lean":"lem_g_at_rR","parent":5457,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/index.html#lem-g-at-rr","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1056. The project gives no license, so trough does not reproduce its text.","title":"g bound r"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/lem-fracs","kind":"node","lean":"lem_fracs","parent":5457,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/index.html#lem-fracs","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1065. The project gives no license, so trough does not reproduce its text.","title":"Fraction swap"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/lem-f-bound-rearranged","kind":"node","lean":"lem_f_bound_rearranged","parent":5457,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02/index.html#lem-f-bound-rearranged","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1072. The project gives no license, so trough does not reproduce its text.","title":"Rearranged bound"},{"data_url":"scopes/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-03.json","id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-03","kind":"scope","navigation":"transparent","parent":5435,"result":false,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-03/index.html","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-03","status":"fully_proved","title":"Circle bound to Borel-Carath\u00e9odory I"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-03/lem-final-bound-on-circle0","kind":"node","lean":"lem_final_bound_on_circle0","parent":5478,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-03/index.html#lem-final-bound-on-circle0","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1082. The project gives no license, so trough does not reproduce its text.","title":"Circle bound"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-03/lem-final-bound-on-circle","kind":"node","lean":"lem_final_bound_on_circle","parent":5478,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-03/index.html#lem-final-bound-on-circle","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1092. The project gives no license, so trough does not reproduce its text.","title":"Circle bound"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-03/lem-bci","kind":"node","lean":"lem_BCI","parent":5478,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-03/index.html#lem-bci","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1102. The project gives no license, so trough does not reproduce its text.","title":"BC bound"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-03/thm-borelcaratheodoryi","kind":"node","lean":"thm_BorelCaratheodoryI","parent":5478,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-03/index.html#thm-borelcaratheodoryi","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-i/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this theorem at PNT/PNT1_ComplexAnalysis.tex, line 1114. The project gives no license, so trough does not reproduce its text.","title":"Borel-Carath\u00e9odory I"},{"data_url":"scopes/strongpnt/complex-analysis/part-05/borel-caratheodory-ii.json","id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii","kind":"scope","parent":5426,"result":false,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/index.html","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii","status":"fully_proved","title":"Borel-Carath\u00e9odory II"},{"data_url":"scopes/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01.json","id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01","kind":"scope","navigation":"transparent","parent":5483,"result":false,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/index.html","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01","status":"fully_proved","title":"Cauchy's Integral Formula for\u2026 to Reverse triangle"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/lem-cauchy-formula-deriv","kind":"node","lean":"cauchy_formula_deriv","parent":5484,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cauchy's Integral Formula for "],["math","\\(f'\\)"]],"source":"Cauchy's Integral Formula for $f'$"},"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/index.html#lem-cauchy-formula-deriv","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1127. The project gives no license, so trough does not reproduce its text.","title":"Cauchy's Integral Formula for $f'$"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/lem-dw-dt","kind":"node","lean":"lem_dw_dt","parent":5484,"reading_title":{"format":"inline-parts-v1","parts":[["text","Differential of "],["math","\\(w(t)\\)"]],"source":"Differential of $w(t)$"},"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/index.html#lem-dw-dt","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1137. The project gives no license, so trough does not reproduce its text.","title":"Differential of $w(t)$"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/lem-cif-deriv-param","kind":"node","lean":"lem_CIF_deriv_param","parent":5484,"reading_title":{"format":"inline-parts-v1","parts":[["text","CIF for "],["math","\\(f'\\)"],["text",", Parameterized"]],"source":"CIF for $f'$, Parameterized"},"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/index.html#lem-cif-deriv-param","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1146. The project gives no license, so trough does not reproduce its text.","title":"CIF for $f'$, Parameterized"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/lem-cif-deriv-simplified","kind":"node","lean":"lem_CIF_deriv_simplified","parent":5484,"reading_title":{"format":"inline-parts-v1","parts":[["text","CIF for "],["math","\\(f'\\)"],["text",", Simplified"]],"source":"CIF for $f'$, Simplified"},"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/index.html#lem-cif-deriv-simplified","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1157. The project gives no license, so trough does not reproduce its text.","title":"CIF for $f'$, Simplified"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/lem-modulus-of-f-prime0","kind":"node","lean":"lem_modulus_of_f_prime0","parent":5484,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/index.html#lem-modulus-of-f-prime0","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1170. The project gives no license, so trough does not reproduce its text.","title":"Derivative modulus"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/lem-integral-modulus-inequality","kind":"node","lean":"lem_integral_modulus_inequality","parent":5484,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/index.html#lem-integral-modulus-inequality","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1181. The project gives no license, so trough does not reproduce its text.","title":"Integral bound"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/lem-modulus-of-f-prime","kind":"node","lean":"lem_modulus_of_f_prime","parent":5484,"reading_title":{"format":"inline-parts-v1","parts":[["text","Modulus of "],["math","\\(f'\\)"]],"source":"Modulus of $f'$"},"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/index.html#lem-modulus-of-f-prime","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1192. The project gives no license, so trough does not reproduce its text.","title":"Modulus of $f'$"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/lem-modulus-of-integrand-product2","kind":"node","lean":"lem_modulus_of_integrand_product2","parent":5484,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/index.html#lem-modulus-of-integrand-product2","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1204. The project gives no license, so trough does not reproduce its text.","title":"Integrand modulus"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/lem-modeit","kind":"node","lean":"lem_modeit","parent":5484,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/index.html#lem-modeit","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1216. The project gives no license, so trough does not reproduce its text.","title":"Modulus one"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/lem-reit0","kind":"node","lean":"lem_Reit0","parent":5484,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/index.html#lem-reit0","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1225. The project gives no license, so trough does not reproduce its text.","title":"Cosine part"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/lem-ereite0","kind":"node","lean":"lem_eReite0","parent":5484,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/index.html#lem-ereite0","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1234. The project gives no license, so trough does not reproduce its text.","title":"Euler part"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/lem-e01","kind":"node","lean":"lem_e01","parent":5484,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/index.html#lem-e01","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1244. The project gives no license, so trough does not reproduce its text.","title":"Exp zero one"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/lem-ereit1","kind":"node","lean":"lem_eReit1","parent":5484,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/index.html#lem-ereit1","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1252. The project gives no license, so trough does not reproduce its text.","title":"Cosine relation"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/lem-modulus-of-e-it-is-one","kind":"node","lean":"lem_modulus_of_e_it_is_one","parent":5484,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/index.html#lem-modulus-of-e-it-is-one","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1262. The project gives no license, so trough does not reproduce its text.","title":"Unit modulus"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/lem-modulus-of-ae-it","kind":"node","lean":"lem_modulus_of_ae_it","parent":5484,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/index.html#lem-modulus-of-ae-it","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1272. The project gives no license, so trough does not reproduce its text.","title":"Scaled modulus"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/lem-modulus-of-integrand-product3","kind":"node","lean":"lem_modulus_of_integrand_product3","parent":5484,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/index.html#lem-modulus-of-integrand-product3","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1283. The project gives no license, so trough does not reproduce its text.","title":"Integrand modulus"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/lem-modulus-of-square","kind":"node","lean":"lem_modulus_of_square","parent":5484,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/index.html#lem-modulus-of-square","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1294. The project gives no license, so trough does not reproduce its text.","title":"Square modulus"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/lem-modulus-wz","kind":"node","lean":"lem_modulus_wz","parent":5484,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/index.html#lem-modulus-wz","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1302. The project gives no license, so trough does not reproduce its text.","title":"Shifted modulus"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/lem-reverse-triangle","kind":"node","lean":"lem_reverse_triangle","parent":5484,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/index.html#lem-reverse-triangle","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1312. The project gives no license, so trough does not reproduce its text.","title":"Reverse triangle"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/lem-reverse-triangle2","kind":"node","lean":"lem_reverse_triangle2","parent":5484,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01/index.html#lem-reverse-triangle2","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1320. The project gives no license, so trough does not reproduce its text.","title":"Reverse triangle"},{"data_url":"scopes/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02.json","id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02","kind":"scope","navigation":"transparent","parent":5483,"result":false,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/index.html","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02","status":"fully_proved","title":"Reverse triangle to Integral inequality"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/lem-reverse-triangle3","kind":"node","lean":"lem_reverse_triangle3","parent":5505,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/index.html#lem-reverse-triangle3","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1330. The project gives no license, so trough does not reproduce its text.","title":"Reverse triangle"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/lem-zrr1","kind":"node","lean":"lem_zrr1","parent":5505,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/index.html#lem-zrr1","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1340. The project gives no license, so trough does not reproduce its text.","title":"Radius relation"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/lem-zrr2","kind":"node","lean":"lem_zrr2","parent":5505,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/index.html#lem-zrr2","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1349. The project gives no license, so trough does not reproduce its text.","title":"Radius relation"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/lem-rr11","kind":"node","lean":"lem_rr11","parent":5505,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/index.html#lem-rr11","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1359. The project gives no license, so trough does not reproduce its text.","title":"Radius relation"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/lem-rr12","kind":"node","lean":"lem_rr12","parent":5505,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/index.html#lem-rr12","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1368. The project gives no license, so trough does not reproduce its text.","title":"Radius relation"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/lem-zrr3","kind":"node","lean":"lem_zrr3","parent":5505,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/index.html#lem-zrr3","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1378. The project gives no license, so trough does not reproduce its text.","title":"Radius relation"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/lem-zrr4","kind":"node","lean":"lem_zrr4","parent":5505,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/index.html#lem-zrr4","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1388. The project gives no license, so trough does not reproduce its text.","title":"Radius relation"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/lem-reverse-triangle4","kind":"node","lean":"lem_reverse_triangle4","parent":5505,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/index.html#lem-reverse-triangle4","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1398. The project gives no license, so trough does not reproduce its text.","title":"Reverse triangle"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/lem-wposneq0","kind":"node","lean":"lem_wposneq0","parent":5505,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/index.html#lem-wposneq0","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1408. The project gives no license, so trough does not reproduce its text.","title":"Positive nonzero"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/lem-reverse-triangle5","kind":"node","lean":"lem_reverse_triangle5","parent":5505,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/index.html#lem-reverse-triangle5","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1416. The project gives no license, so trough does not reproduce its text.","title":"Reverse triangle"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/lem-reverse-triangle6","kind":"node","lean":"lem_reverse_triangle6","parent":5505,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/index.html#lem-reverse-triangle6","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1426. The project gives no license, so trough does not reproduce its text.","title":"Reverse triangle"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/lem-absdiv","kind":"node","lean":"lem_absdiv","parent":5505,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/index.html#lem-absdiv","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1436. The project gives no license, so trough does not reproduce its text.","title":"Division bound"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/lem-modulus-of-integrand-product","kind":"node","lean":"lem_modulus_of_integrand_product","parent":5505,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/index.html#lem-modulus-of-integrand-product","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1444. The project gives no license, so trough does not reproduce its text.","title":"Integrand modulus"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/lem-modulus-of-product","kind":"node","lean":"lem_modulus_of_product","parent":5505,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/index.html#lem-modulus-of-product","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1456. The project gives no license, so trough does not reproduce its text.","title":"Product modulus"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/lem-modulus-of-product2","kind":"node","lean":"lem_modulus_of_product2","parent":5505,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/index.html#lem-modulus-of-product2","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1467. The project gives no license, so trough does not reproduce its text.","title":"Product modulus"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/lem-modulus-of-product3","kind":"node","lean":"lem_modulus_of_product3","parent":5505,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/index.html#lem-modulus-of-product3","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1478. The project gives no license, so trough does not reproduce its text.","title":"Product modulus"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/lem-modulus-of-product4","kind":"node","lean":"lem_modulus_of_product4","parent":5505,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/index.html#lem-modulus-of-product4","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1490. The project gives no license, so trough does not reproduce its text.","title":"Product modulus"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/lem-bound-on-f-at-r-prime","kind":"node","lean":"lem_bound_on_f_at_r_prime","parent":5505,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/index.html#lem-bound-on-f-at-r-prime","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1501. The project gives no license, so trough does not reproduce its text.","title":"Point bound"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/lem-bound-on-integrand-modulus","kind":"node","lean":"lem_bound_on_integrand_modulus","parent":5505,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/index.html#lem-bound-on-integrand-modulus","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1513. The project gives no license, so trough does not reproduce its text.","title":"Integrand bound"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/lem-inequality-under-integral","kind":"node","lean":"lem_integral_inequality","parent":5505,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02/index.html#lem-inequality-under-integral","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1523. The project gives no license, so trough does not reproduce its text.","title":"Integral inequality"},{"data_url":"scopes/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03.json","id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03","kind":"scope","navigation":"transparent","parent":5483,"result":false,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/index.html","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03","status":"fully_proved","title":"Derivative bound to Square identity"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/lem-f-prime-bound-by-integral-of-constant","kind":"node","lean":"lem_f_prime_bound_by_integral_of_constant","parent":5526,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/index.html#lem-f-prime-bound-by-integral-of-constant","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1532. The project gives no license, so trough does not reproduce its text.","title":"Derivative bound"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/lem-integral-of-1","kind":"node","lean":"lem_integral_of_1","parent":5526,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/index.html#lem-integral-of-1","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1543. The project gives no license, so trough does not reproduce its text.","title":"Integrate one"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/lem-integral-2","kind":"node","lean":"lem_integral_2","parent":5526,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/index.html#lem-integral-2","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1551. The project gives no license, so trough does not reproduce its text.","title":"Exponential integral"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/lem-f-prime-bound","kind":"node","lean":"lem_f_prime_bound","parent":5526,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/index.html#lem-f-prime-bound","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1561. The project gives no license, so trough does not reproduce its text.","title":"Derivative bound"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/lem-r-prime-gt-r","kind":"node","lean":"lem_r_prime_gt_r","parent":5526,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/index.html#lem-r-prime-gt-r","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1573. The project gives no license, so trough does not reproduce its text.","title":"Radius compare"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/lem-r-prime-lt-r","kind":"node","lean":"lem_r_prime_lt_R","parent":5526,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/index.html#lem-r-prime-lt-r","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1582. The project gives no license, so trough does not reproduce its text.","title":"Radius compare"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/lem-r-prime-is-intermediate","kind":"node","lean":"lem_r_prime_is_intermediate","parent":5526,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/index.html#lem-r-prime-is-intermediate","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1591. The project gives no license, so trough does not reproduce its text.","title":"Intermediate radius"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/lem-calc-r-minus-r-prime","kind":"node","lean":"lem_calc_R_minus_r_prime","parent":5526,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/index.html#lem-calc-r-minus-r-prime","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1602. The project gives no license, so trough does not reproduce its text.","title":"Radius formula"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/lem-calc-r-prime-minus-r","kind":"node","lean":"lem_calc_r_prime_minus_r","parent":5526,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/index.html#lem-calc-r-prime-minus-r","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1611. The project gives no license, so trough does not reproduce its text.","title":"Radius formula"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/lem-calc-denominator-specific","kind":"node","lean":"lem_calc_denominator_specific","parent":5526,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/index.html#lem-calc-denominator-specific","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1620. The project gives no license, so trough does not reproduce its text.","title":"Denominator form"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/lem-calc-numerator-specific","kind":"node","lean":"lem_calc_numerator_specific","parent":5526,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/index.html#lem-calc-numerator-specific","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1631. The project gives no license, so trough does not reproduce its text.","title":"Numerator form"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/lem-frac-simplify","kind":"node","lean":"lem_frac_simplify","parent":5526,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/index.html#lem-frac-simplify","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1640. The project gives no license, so trough does not reproduce its text.","title":"Fraction simplify"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/lem-frac-simplify2","kind":"node","lean":"lem_frac_simplify2","parent":5526,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/index.html#lem-frac-simplify2","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1650. The project gives no license, so trough does not reproduce its text.","title":"Fraction simplify"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/lem-frac-simplify3","kind":"node","lean":"lem_frac_simplify3","parent":5526,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/index.html#lem-frac-simplify3","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1659. The project gives no license, so trough does not reproduce its text.","title":"Fraction simplify"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/lem-ineq-r-plus-r-lt-2r","kind":"node","lean":"lem_ineq_R_plus_r_lt_2R","parent":5526,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/index.html#lem-ineq-r-plus-r-lt-2r","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1670. The project gives no license, so trough does not reproduce its text.","title":"Inequality fact"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/lem-r-plus-r-is-positive","kind":"node","lean":"lem_R_plus_r_is_positive","parent":5526,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/index.html#lem-r-plus-r-is-positive","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1680. The project gives no license, so trough does not reproduce its text.","title":"Sum positive"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/lem-2r-is-positive","kind":"node","lean":"lem_2R_is_positive","parent":5526,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/index.html#lem-2r-is-positive","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1688. The project gives no license, so trough does not reproduce its text.","title":"Double positive"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/lem-square-inequality-strict","kind":"node","lean":"lem_square_inequality_strict","parent":5526,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/index.html#lem-square-inequality-strict","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1696. The project gives no license, so trough does not reproduce its text.","title":"Square fact"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/lem-ineq-r-plus-r-sq-lt-2r-sq","kind":"node","lean":"lem_ineq_R_plus_r_sq_lt_2R_sq","parent":5526,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/index.html#lem-ineq-r-plus-r-sq-lt-2r-sq","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1704. The project gives no license, so trough does not reproduce its text.","title":"Square bound"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/lem-2r-sq-is-4r-sq","kind":"node","lean":"lem_2R_sq_is_4R_sq","parent":5526,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03/index.html#lem-2r-sq-is-4r-sq","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1714. The project gives no license, so trough does not reproduce its text.","title":"Square identity"},{"data_url":"scopes/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-04.json","id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-04","kind":"scope","navigation":"transparent","parent":5483,"result":false,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-04/index.html","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-04","status":"fully_proved","title":"Square bound to Borel-Carath\u00e9odory II"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-04/lem-ineq-r-plus-r-sq","kind":"node","lean":"lem_ineq_R_plus_r_sq","parent":5547,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-04/index.html#lem-ineq-r-plus-r-sq","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1723. The project gives no license, so trough does not reproduce its text.","title":"Square bound"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-04/lem-ineq-r-plus-r-sqm","kind":"node","lean":"lem_ineq_R_plus_r_sqM","parent":5547,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-04/index.html#lem-ineq-r-plus-r-sqm","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1733. The project gives no license, so trough does not reproduce its text.","title":"Square bound"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-04/lem-simplify-final-bound","kind":"node","lean":"lem_simplify_final_bound","parent":5547,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-04/index.html#lem-simplify-final-bound","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1743. The project gives no license, so trough does not reproduce its text.","title":"Bound simplify"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-04/lem-frac-simplify4","kind":"node","lean":"lem_bound_after_substitution","parent":5547,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-04/index.html#lem-frac-simplify4","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1753. The project gives no license, so trough does not reproduce its text.","title":"Fraction simplify"},{"id":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-04/thm-bcii","kind":"node","lean":"borel_caratheodory_II","parent":5547,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-04/index.html#thm-bcii","scope":"strongpnt/complex-analysis/part-05/borel-caratheodory-ii/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this theorem at PNT/PNT1_ComplexAnalysis.tex, line 1764. The project gives no license, so trough does not reproduce its text.","title":"Borel-Carath\u00e9odory II"},{"data_url":"scopes/strongpnt/complex-analysis/part-05/integral-antiderivative.json","id":"strongpnt/complex-analysis/part-05/integral-antiderivative","kind":"scope","parent":5426,"result":false,"route":"../roadmap/strongpnt/complex-analysis/part-05/integral-antiderivative/index.html","scope":"strongpnt/complex-analysis/part-05/integral-antiderivative","status":"fully_proved","title":"Integral Antiderivative"},{"id":"strongpnt/complex-analysis/part-05/integral-antiderivative/lem-cauchy-for-rectangles","kind":"node","lean":"cauchy_for_rectangles","parent":5553,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/integral-antiderivative/index.html#lem-cauchy-for-rectangles","scope":"strongpnt/complex-analysis/part-05/integral-antiderivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1777. The project gives no license, so trough does not reproduce its text.","title":"Cauchy rectangles"},{"id":"strongpnt/complex-analysis/part-05/integral-antiderivative/def-if-taxicab","kind":"node","lean":"If_taxicab","parent":5553,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/integral-antiderivative/index.html#def-if-taxicab","scope":"strongpnt/complex-analysis/part-05/integral-antiderivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this definition at PNT/PNT1_ComplexAnalysis.tex, line 1810. The project gives no license, so trough does not reproduce its text.","title":"Integral along the taxicab path"},{"id":"strongpnt/complex-analysis/part-05/integral-antiderivative/lem-def-if-z-plus-h","kind":"node","lean":"def_If_z_plus_h","parent":5553,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/integral-antiderivative/index.html#lem-def-if-z-plus-h","scope":"strongpnt/complex-analysis/part-05/integral-antiderivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1819. The project gives no license, so trough does not reproduce its text.","title":"Integral form"},{"id":"strongpnt/complex-analysis/part-05/integral-antiderivative/lem-def-if-z","kind":"node","lean":"def_If_z","parent":5553,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/integral-antiderivative/index.html#lem-def-if-z","scope":"strongpnt/complex-analysis/part-05/integral-antiderivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1830. The project gives no license, so trough does not reproduce its text.","title":"Integral form"},{"id":"strongpnt/complex-analysis/part-05/integral-antiderivative/lem-def-if-w","kind":"node","lean":"def_If_w","parent":5553,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/integral-antiderivative/index.html#lem-def-if-w","scope":"strongpnt/complex-analysis/part-05/integral-antiderivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1841. The project gives no license, so trough does not reproduce its text.","title":"Integral form"},{"id":"strongpnt/complex-analysis/part-05/integral-antiderivative/lem-diff-if-zh-w","kind":"node","lean":"diff_If_zh_w","parent":5553,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/integral-antiderivative/index.html#lem-diff-if-zh-w","scope":"strongpnt/complex-analysis/part-05/integral-antiderivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1853. The project gives no license, so trough does not reproduce its text.","title":"Difference form"},{"id":"strongpnt/complex-analysis/part-05/integral-antiderivative/lem-diff-if-w-z-initial-form","kind":"node","lean":"diff_If_w_z_initial_form","parent":5553,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/integral-antiderivative/index.html#lem-diff-if-w-z-initial-form","scope":"strongpnt/complex-analysis/part-05/integral-antiderivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1866. The project gives no license, so trough does not reproduce its text.","title":"Initial form"},{"id":"strongpnt/complex-analysis/part-05/integral-antiderivative/lem-cauchy-for-horizontal-strip","kind":"node","lean":"cauchy_for_horizontal_strip","parent":5553,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/integral-antiderivative/index.html#lem-cauchy-for-horizontal-strip","scope":"strongpnt/complex-analysis/part-05/integral-antiderivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1878. The project gives no license, so trough does not reproduce its text.","title":"Horizontal strip"},{"id":"strongpnt/complex-analysis/part-05/integral-antiderivative/lem-cauchy-rearrangement-step1","kind":"node","lean":"cauchy_rearrangement_step1","parent":5553,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/integral-antiderivative/index.html#lem-cauchy-rearrangement-step1","scope":"strongpnt/complex-analysis/part-05/integral-antiderivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1890. The project gives no license, so trough does not reproduce its text.","title":"Rearrangement step"},{"id":"strongpnt/complex-analysis/part-05/integral-antiderivative/lem-diff-if-w-z","kind":"node","lean":"diff_If_w_z","parent":5553,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/integral-antiderivative/index.html#lem-diff-if-w-z","scope":"strongpnt/complex-analysis/part-05/integral-antiderivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1909. The project gives no license, so trough does not reproduce its text.","title":"Shift integral"},{"id":"strongpnt/complex-analysis/part-05/integral-antiderivative/lem-if-difference-is-l-path-integral","kind":"node","lean":"If_difference_is_L_path_integral","parent":5553,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/integral-antiderivative/index.html#lem-if-difference-is-l-path-integral","scope":"strongpnt/complex-analysis/part-05/integral-antiderivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1931. The project gives no license, so trough does not reproduce its text.","title":"L path"},{"id":"strongpnt/complex-analysis/part-05/integral-antiderivative/lem-if-diff-add-sub-identity","kind":"node","lean":"If_diff_add_sub_identity","parent":5553,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/integral-antiderivative/index.html#lem-if-diff-add-sub-identity","scope":"strongpnt/complex-analysis/part-05/integral-antiderivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1943. The project gives no license, so trough does not reproduce its text.","title":"Add-sub step"},{"id":"strongpnt/complex-analysis/part-05/integral-antiderivative/lem-if-diff-linearity","kind":"node","lean":"If_diff_linearity","parent":5553,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/integral-antiderivative/index.html#lem-if-diff-linearity","scope":"strongpnt/complex-analysis/part-05/integral-antiderivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1955. The project gives no license, so trough does not reproduce its text.","title":"Linearity split"},{"id":"strongpnt/complex-analysis/part-05/integral-antiderivative/lem-integral-of-constant-over-l-path","kind":"node","lean":"integral_of_constant_over_L_path","parent":5553,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/integral-antiderivative/index.html#lem-integral-of-constant-over-l-path","scope":"strongpnt/complex-analysis/part-05/integral-antiderivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1981. The project gives no license, so trough does not reproduce its text.","title":"Integral of constant over L-path"},{"id":"strongpnt/complex-analysis/part-05/integral-antiderivative/lem-if-diff-decomposition-final","kind":"node","lean":"If_diff_decomposition_final","parent":5553,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/integral-antiderivative/index.html#lem-if-diff-decomposition-final","scope":"strongpnt/complex-analysis/part-05/integral-antiderivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 1996. The project gives no license, so trough does not reproduce its text.","title":"Difference decomposition"},{"id":"strongpnt/complex-analysis/part-05/integral-antiderivative/lem-bound-on-err","kind":"node","lean":"bound_on_Err","parent":5553,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/integral-antiderivative/index.html#lem-bound-on-err","scope":"strongpnt/complex-analysis/part-05/integral-antiderivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 2025. The project gives no license, so trough does not reproduce its text.","title":"Bound on error term"},{"id":"strongpnt/complex-analysis/part-05/integral-antiderivative/lem-bound-on-err-ratio","kind":"node","lean":"bound_on_Err_ratio","parent":5553,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/integral-antiderivative/index.html#lem-bound-on-err-ratio","scope":"strongpnt/complex-analysis/part-05/integral-antiderivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 2056. The project gives no license, so trough does not reproduce its text.","title":"Bound on error term ratio"},{"id":"strongpnt/complex-analysis/part-05/integral-antiderivative/lem-limit-of-s-is-zero","kind":"node","lean":"limit_of_S_is_zero","parent":5553,"reading_title":{"format":"inline-parts-v1","parts":[["text","Limit of "],["math","\\(S(z,h)\\)"],["text"," is zero"]],"source":"Limit of $S(z,h)$ is zero"},"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/integral-antiderivative/index.html#lem-limit-of-s-is-zero","scope":"strongpnt/complex-analysis/part-05/integral-antiderivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 2080. The project gives no license, so trough does not reproduce its text.","title":"Limit of $S(z,h)$ is zero"},{"id":"strongpnt/complex-analysis/part-05/integral-antiderivative/lem-limit-of-err-ratio-is-zero","kind":"node","lean":"limit_of_Err_ratio_is_zero","parent":5553,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/integral-antiderivative/index.html#lem-limit-of-err-ratio-is-zero","scope":"strongpnt/complex-analysis/part-05/integral-antiderivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 2116. The project gives no license, so trough does not reproduce its text.","title":"Limit of error term ratio is zero"},{"id":"strongpnt/complex-analysis/part-05/integral-antiderivative/lem-if-is-differentiable","kind":"node","lean":"If_is_differentiable_on","parent":5553,"reading_title":{"format":"inline-parts-v1","parts":[["text","Differentiability of "],["math","\\(I_f(z)\\)"]],"source":"Differentiability of $I_f(z)$"},"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/integral-antiderivative/index.html#lem-if-is-differentiable","scope":"strongpnt/complex-analysis/part-05/integral-antiderivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 2140. The project gives no license, so trough does not reproduce its text.","title":"Differentiability of $I_f(z)$"},{"data_url":"scopes/strongpnt/complex-analysis/part-05/complex-logarithm.json","id":"strongpnt/complex-analysis/part-05/complex-logarithm","kind":"scope","parent":5426,"result":false,"route":"../roadmap/strongpnt/complex-analysis/part-05/complex-logarithm/index.html","scope":"strongpnt/complex-analysis/part-05/complex-logarithm","status":"fully_proved","title":"Complex logarithm"},{"id":"strongpnt/complex-analysis/part-05/complex-logarithm/lem-log-deriv-is-analytic","kind":"node","lean":"log_deriv_is_analytic","parent":5574,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/complex-logarithm/index.html#lem-log-deriv-is-analytic","scope":"strongpnt/complex-analysis/part-05/complex-logarithm","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 2169. The project gives no license, so trough does not reproduce its text.","title":"Logarithmic derivative is analytic"},{"id":"strongpnt/complex-analysis/part-05/complex-logarithm/lem-i-is-antiderivative","kind":"node","lean":"I_is_antiderivative","parent":5574,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/complex-logarithm/index.html#lem-i-is-antiderivative","scope":"strongpnt/complex-analysis/part-05/complex-logarithm","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 2179. The project gives no license, so trough does not reproduce its text.","title":"Antiderivative of logarithmic derivative"},{"id":"strongpnt/complex-analysis/part-05/complex-logarithm/def-h-auxiliary","kind":"node","lean":"H_auxiliary","parent":5574,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/complex-logarithm/index.html#def-h-auxiliary","scope":"strongpnt/complex-analysis/part-05/complex-logarithm","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this definition at PNT/PNT1_ComplexAnalysis.tex, line 2191. The project gives no license, so trough does not reproduce its text.","title":"Auxiliary function"},{"id":"strongpnt/complex-analysis/part-05/complex-logarithm/lem-exp-i-at-zero","kind":"node","lean":"exp_I_at_zero","parent":5574,"reading_title":{"format":"inline-parts-v1","parts":[["text","Exponential of "],["math","\\(I_f\\)"],["text"," at zero"]],"source":"Exponential of $I_f$ at zero"},"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/complex-logarithm/index.html#lem-exp-i-at-zero","scope":"strongpnt/complex-analysis/part-05/complex-logarithm","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 2197. The project gives no license, so trough does not reproduce its text.","title":"Exponential of $I_f$ at zero"},{"id":"strongpnt/complex-analysis/part-05/complex-logarithm/lem-h-at-zero","kind":"node","lean":"H_at_zero","parent":5574,"reading_title":{"format":"inline-parts-v1","parts":[["text","Value of "],["math","\\(H\\)"],["text"," at zero"]],"source":"Value of $H$ at zero"},"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/complex-logarithm/index.html#lem-h-at-zero","scope":"strongpnt/complex-analysis/part-05/complex-logarithm","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 2208. The project gives no license, so trough does not reproduce its text.","title":"Value of $H$ at zero"},{"id":"strongpnt/complex-analysis/part-05/complex-logarithm/lem-log-deriv-id","kind":"node","lean":"log_deriv_id","parent":5574,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/complex-logarithm/index.html#lem-log-deriv-id","scope":"strongpnt/complex-analysis/part-05/complex-logarithm","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 2219. The project gives no license, so trough does not reproduce its text.","title":"Logarithmic derivative identity"},{"id":"strongpnt/complex-analysis/part-05/complex-logarithm/lem-log-deriv-identity","kind":"node","lean":"log_deriv_identity","parent":5574,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/complex-logarithm/index.html#lem-log-deriv-identity","scope":"strongpnt/complex-analysis/part-05/complex-logarithm","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 2231. The project gives no license, so trough does not reproduce its text.","title":"Logarithmic derivative identity"},{"id":"strongpnt/complex-analysis/part-05/complex-logarithm/lem-h-derivative-quotient-rule","kind":"node","lean":"H_derivative_quotient_rule","parent":5574,"reading_title":{"format":"inline-parts-v1","parts":[["text","Derivative of "],["math","\\(H(z)\\)"]],"source":"Derivative of $H(z)$"},"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/complex-logarithm/index.html#lem-h-derivative-quotient-rule","scope":"strongpnt/complex-analysis/part-05/complex-logarithm","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 2243. The project gives no license, so trough does not reproduce its text.","title":"Derivative of $H(z)$"},{"id":"strongpnt/complex-analysis/part-05/complex-logarithm/lem-exp-i-derivative-chain-rule","kind":"node","lean":"exp_I_derivative_chain_rule","parent":5574,"reading_title":{"format":"inline-parts-v1","parts":[["text","Derivative of "],["math","\\(\\exp(J(z))\\)"]],"source":"Derivative of $\\exp(J(z))$"},"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/complex-logarithm/index.html#lem-exp-i-derivative-chain-rule","scope":"strongpnt/complex-analysis/part-05/complex-logarithm","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 2255. The project gives no license, so trough does not reproduce its text.","title":"Derivative of $\\exp(J(z))$"},{"id":"strongpnt/complex-analysis/part-05/complex-logarithm/lem-h-derivative-calc","kind":"node","lean":"H_derivative_calc","parent":5574,"reading_title":{"format":"inline-parts-v1","parts":[["text","Derivative of "],["math","\\(H(z)\\)"]],"source":"Derivative of $H(z)$"},"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/complex-logarithm/index.html#lem-h-derivative-calc","scope":"strongpnt/complex-analysis/part-05/complex-logarithm","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 2267. The project gives no license, so trough does not reproduce its text.","title":"Derivative of $H(z)$"},{"id":"strongpnt/complex-analysis/part-05/complex-logarithm/lem-h-derivative-is-zero","kind":"node","lean":"H_derivative_is_zero","parent":5574,"reading_title":{"format":"inline-parts-v1","parts":[["text","Derivative of "],["math","\\(H(z)\\)"],["text"," is 0"]],"source":"Derivative of $H(z)$ is 0"},"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/complex-logarithm/index.html#lem-h-derivative-is-zero","scope":"strongpnt/complex-analysis/part-05/complex-logarithm","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 2281. The project gives no license, so trough does not reproduce its text.","title":"Derivative of $H(z)$ is 0"},{"id":"strongpnt/complex-analysis/part-05/complex-logarithm/lem-h-is-constant","kind":"node","lean":"H_is_constant","parent":5574,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(H(z)\\)"],["text"," is constant"]],"source":"$H(z)$ is constant"},"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/complex-logarithm/index.html#lem-h-is-constant","scope":"strongpnt/complex-analysis/part-05/complex-logarithm","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 2292. The project gives no license, so trough does not reproduce its text.","title":"$H(z)$ is constant"},{"id":"strongpnt/complex-analysis/part-05/complex-logarithm/lem-h-is-one","kind":"node","lean":"H_is_one","parent":5574,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(H=1\\)"]],"source":"$H=1$"},"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/complex-logarithm/index.html#lem-h-is-one","scope":"strongpnt/complex-analysis/part-05/complex-logarithm","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 2305. The project gives no license, so trough does not reproduce its text.","title":"$H=1$"},{"id":"strongpnt/complex-analysis/part-05/complex-logarithm/lem-analytic-log-exists","kind":"node","lean":"analytic_log_exists","parent":5574,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/complex-logarithm/index.html#lem-analytic-log-exists","scope":"strongpnt/complex-analysis/part-05/complex-logarithm","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 2316. The project gives no license, so trough does not reproduce its text.","title":"Existence of analytic logarithm"},{"id":"strongpnt/complex-analysis/part-05/complex-logarithm/lem-modulus-of-exp-i","kind":"node","lean":"modulus_of_exp_I","parent":5574,"reading_title":{"format":"inline-parts-v1","parts":[["text","Modulus of "],["math","\\(\\exp(J(z))\\)"]],"source":"Modulus of $\\exp(J(z))$"},"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/complex-logarithm/index.html#lem-modulus-of-exp-i","scope":"strongpnt/complex-analysis/part-05/complex-logarithm","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 2328. The project gives no license, so trough does not reproduce its text.","title":"Modulus of $\\exp(J(z))$"},{"id":"strongpnt/complex-analysis/part-05/complex-logarithm/lem-modulus-of-b-product-form","kind":"node","lean":"modulus_of_B_product_form","parent":5574,"reading_title":{"format":"inline-parts-v1","parts":[["text","Modulus of "],["math","\\(B(z)\\)"],["text"," in product form"]],"source":"Modulus of $B(z)$ in product form"},"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/complex-logarithm/index.html#lem-modulus-of-b-product-form","scope":"strongpnt/complex-analysis/part-05/complex-logarithm","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 2340. The project gives no license, so trough does not reproduce its text.","title":"Modulus of $B(z)$ in product form"},{"id":"strongpnt/complex-analysis/part-05/complex-logarithm/lem-modulus-of-exp-log","kind":"node","lean":"modulus_of_exp_log","parent":5574,"reading_title":{"format":"inline-parts-v1","parts":[["text","Modulus of "],["math","\\(\\exp(J(z))\\)"]],"source":"Modulus of $\\exp(J(z))$"},"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/complex-logarithm/index.html#lem-modulus-of-exp-log","scope":"strongpnt/complex-analysis/part-05/complex-logarithm","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 2351. The project gives no license, so trough does not reproduce its text.","title":"Modulus of $\\exp(J(z))$"},{"id":"strongpnt/complex-analysis/part-05/complex-logarithm/lem-log-modulus-as-sum","kind":"node","lean":"log_modulus_as_sum","parent":5574,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/complex-logarithm/index.html#lem-log-modulus-as-sum","scope":"strongpnt/complex-analysis/part-05/complex-logarithm","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 2363. The project gives no license, so trough does not reproduce its text.","title":"Logarithm of modulus as sum"},{"id":"strongpnt/complex-analysis/part-05/complex-logarithm/lem-real-log-of-modulus-difference","kind":"node","lean":"real_log_of_modulus_difference","parent":5574,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/complex-logarithm/index.html#lem-real-log-of-modulus-difference","scope":"strongpnt/complex-analysis/part-05/complex-logarithm","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 2374. The project gives no license, so trough does not reproduce its text.","title":"Real logarithm of modulus difference"},{"id":"strongpnt/complex-analysis/part-05/complex-logarithm/lem-log-of-analytic","kind":"node","lean":"log_of_analytic","parent":5574,"result":true,"route":"../roadmap/strongpnt/complex-analysis/part-05/complex-logarithm/index.html#lem-log-of-analytic","scope":"strongpnt/complex-analysis/part-05/complex-logarithm","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT1_ComplexAnalysis.tex, line 2385. The project gives no license, so trough does not reproduce its text.","title":"Logarithm of an analytic function"},{"data_url":"scopes/strongpnt/log-derivative.json","id":"strongpnt/log-derivative","kind":"scope","parent":5340,"result":false,"route":"../roadmap/strongpnt/log-derivative/index.html","scope":"strongpnt/log-derivative","status":"fully_proved","title":"Log Derivative"},{"id":"strongpnt/log-derivative/lem-drind1","kind":"node","lean":"DRinD1","parent":5595,"result":true,"route":"../roadmap/strongpnt/log-derivative/index.html#lem-drind1","scope":"strongpnt/log-derivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 4. The project gives no license, so trough does not reproduce its text.","title":"Disk inclusion"},{"id":"strongpnt/log-derivative/def-zerosetkfr","kind":"node","lean":"zerosetKfR","parent":5595,"result":true,"route":"../roadmap/strongpnt/log-derivative/index.html#def-zerosetkfr","scope":"strongpnt/log-derivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this definition at PNT/PNT2_LogDerivative.tex, line 14. The project gives no license, so trough does not reproduce its text.","title":"Zero set"},{"id":"strongpnt/log-derivative/lem-kindr","kind":"node","lean":"lemKinDR","parent":5595,"result":true,"route":"../roadmap/strongpnt/log-derivative/index.html#lem-kindr","scope":"strongpnt/log-derivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 20. The project gives no license, so trough does not reproduce its text.","title":"Zero containment"},{"id":"strongpnt/log-derivative/lem-krink1","kind":"node","lean":"lemKRinK1","parent":5595,"result":true,"route":"../roadmap/strongpnt/log-derivative/index.html#lem-krink1","scope":"strongpnt/log-derivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 31. The project gives no license, so trough does not reproduce its text.","title":"Zero in disk"},{"id":"strongpnt/log-derivative/lem-bolzano-weierstrass","kind":"node","lean":"lem_bolzano_weierstrass","parent":5595,"result":true,"route":"../roadmap/strongpnt/log-derivative/index.html#lem-bolzano-weierstrass","scope":"strongpnt/log-derivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 43. The project gives no license, so trough does not reproduce its text.","title":"Accumulation point"},{"id":"strongpnt/log-derivative/lem-zeros-have-limit-point","kind":"node","lean":"lem_zeros_have_limit_point","parent":5595,"result":true,"route":"../roadmap/strongpnt/log-derivative/index.html#lem-zeros-have-limit-point","scope":"strongpnt/log-derivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 53. The project gives no license, so trough does not reproduce its text.","title":"Zeros accumulate"},{"id":"strongpnt/log-derivative/lem-identity-theorem","kind":"node","lean":"lem_identity_theorem","parent":5595,"result":true,"route":"../roadmap/strongpnt/log-derivative/index.html#lem-identity-theorem","scope":"strongpnt/log-derivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 66. The project gives no license, so trough does not reproduce its text.","title":"Identity theorem"},{"id":"strongpnt/log-derivative/lem-identity-theoremr","kind":"node","lean":"lem_identity_theoremR","parent":5595,"result":true,"route":"../roadmap/strongpnt/log-derivative/index.html#lem-identity-theoremr","scope":"strongpnt/log-derivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 75. The project gives no license, so trough does not reproduce its text.","title":"Identity theorem R"},{"id":"strongpnt/log-derivative/lem-identity-theoremkr","kind":"node","lean":"lem_identity_theoremKR","parent":5595,"result":true,"route":"../roadmap/strongpnt/log-derivative/index.html#lem-identity-theoremkr","scope":"strongpnt/log-derivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 87. The project gives no license, so trough does not reproduce its text.","title":"Identity on K"},{"id":"strongpnt/log-derivative/lem-identity-infinitekr","kind":"node","lean":"lem_identity_infiniteKR","parent":5595,"result":true,"route":"../roadmap/strongpnt/log-derivative/index.html#lem-identity-infinitekr","scope":"strongpnt/log-derivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 98. The project gives no license, so trough does not reproduce its text.","title":"Infinite zeros imply"},{"id":"strongpnt/log-derivative/lem-contra-finitekr","kind":"node","lean":"lem_Contra_finiteKR","parent":5595,"result":true,"route":"../roadmap/strongpnt/log-derivative/index.html#lem-contra-finitekr","scope":"strongpnt/log-derivative","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 109. The project gives no license, so trough does not reproduce its text.","title":"Finite zeros"},{"data_url":"scopes/strongpnt/log-derivative/b-f-analytic-and-never-zero.json","id":"strongpnt/log-derivative/b-f-analytic-and-never-zero","kind":"scope","parent":5595,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(B_f\\)"],["text"," analytic and never zero"]],"source":"$B_f$ analytic and never zero"},"result":false,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/index.html","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero","status":"fully_proved","title":"$B_f$ analytic and never zero"},{"data_url":"scopes/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01.json","id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01","kind":"scope","navigation":"transparent","parent":5607,"result":false,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/index.html","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01","status":"fully_proved","title":"Order is integer to C nonzero on K"},{"id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/lem-m-rho-is-nat","kind":"node","lean":"lem_m_rho_is_nat","parent":5608,"result":true,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/index.html#lem-m-rho-is-nat","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 128. The project gives no license, so trough does not reproduce its text.","title":"Order is integer"},{"id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/lem-m-rho-ge-1","kind":"node","lean":"lem_m_rho_ge_1","parent":5608,"result":true,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/index.html#lem-m-rho-ge-1","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 143. The project gives no license, so trough does not reproduce its text.","title":"Order at least one"},{"id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/lem-analdiv","kind":"node","lean":"lem_analDiv","parent":5608,"result":true,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/index.html#lem-analdiv","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 158. The project gives no license, so trough does not reproduce its text.","title":"Analytic division"},{"id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/lem-denomanalat","kind":"node","lean":"lem_denomAnalAt","parent":5608,"result":true,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/index.html#lem-denomanalat","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 173. The project gives no license, so trough does not reproduce its text.","title":"Denominator analytic"},{"id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/lem-ratioanalat","kind":"node","lean":"lem_ratioAnalAt","parent":5608,"result":true,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/index.html#lem-ratioanalat","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 194. The project gives no license, so trough does not reproduce its text.","title":"Ratio analytic"},{"id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/lem-analytic-zero-factor","kind":"node","lean":"lem_analytic_zero_factor","parent":5608,"result":true,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/index.html#lem-analytic-zero-factor","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 217. The project gives no license, so trough does not reproduce its text.","title":"Zero factorization"},{"id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/lem-c-analytic-off-k","kind":"node","lean":"lem_Cf_analytic_off_K","parent":5608,"result":true,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/index.html#lem-c-analytic-off-k","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 249. The project gives no license, so trough does not reproduce its text.","title":"C analytic off K"},{"id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/lem-c-at-sigma-onk","kind":"node","lean":"lem_Cf_at_sigma_onK","parent":5608,"result":true,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/index.html#lem-c-at-sigma-onk","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 270. The project gives no license, so trough does not reproduce its text.","title":"C at zero"},{"id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/lem-k-isolated","kind":"node","lean":"lem_K_isolated","parent":5608,"result":true,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/index.html#lem-k-isolated","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 286. The project gives no license, so trough does not reproduce its text.","title":"Zeros isolated"},{"id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/lem-c-at-sigma-offk0","kind":"node","lean":"lem_Cf_at_sigma_offK0","parent":5608,"result":true,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/index.html#lem-c-at-sigma-offk0","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 302. The project gives no license, so trough does not reproduce its text.","title":"C near zero"},{"id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/lem-prod-no-sigma1","kind":"node","lean":"lem_prod_no_sigma1","parent":5608,"result":true,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/index.html#lem-prod-no-sigma1","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 321. The project gives no license, so trough does not reproduce its text.","title":"Product split"},{"id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/lem-prod-no-sigma2","kind":"node","lean":"lem_prod_no_sigma2","parent":5608,"result":true,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/index.html#lem-prod-no-sigma2","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 339. The project gives no license, so trough does not reproduce its text.","title":"Product quotient"},{"id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/lem-c-at-sigma-offk","kind":"node","lean":"lem_Cf_at_sigma_offK","parent":5608,"result":true,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/index.html#lem-c-at-sigma-offk","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 359. The project gives no license, so trough does not reproduce its text.","title":"C off K"},{"id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/lem-c-at-sigma","kind":"node","lean":"lem_Cf_at_sigma","parent":5608,"result":true,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/index.html#lem-c-at-sigma","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 378. The project gives no license, so trough does not reproduce its text.","title":"C local form"},{"id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/lem-h-ratio-anal","kind":"node","lean":"lem_h_ratio_anal","parent":5608,"result":true,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/index.html#lem-h-ratio-anal","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 393. The project gives no license, so trough does not reproduce its text.","title":"h ratio analytic"},{"id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/lem-c-analytic-at-k","kind":"node","lean":"lem_Cf_analytic_at_K","parent":5608,"result":true,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/index.html#lem-c-analytic-at-k","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 416. The project gives no license, so trough does not reproduce its text.","title":"C analytic at K"},{"id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/lem-c-is-analytic","kind":"node","lean":"lem_Cf_is_analytic","parent":5608,"result":true,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/index.html#lem-c-is-analytic","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 430. The project gives no license, so trough does not reproduce its text.","title":"C is analytic"},{"id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/lem-f-nonzero-off-k","kind":"node","lean":"lem_f_nonzero_off_K","parent":5608,"result":true,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/index.html#lem-f-nonzero-off-k","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 448. The project gives no license, so trough does not reproduce its text.","title":"f nonzero off K"},{"id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/lem-c-nonzero-off-k","kind":"node","lean":"lem_Cf_nonzero_off_K","parent":5608,"result":true,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/index.html#lem-c-nonzero-off-k","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 462. The project gives no license, so trough does not reproduce its text.","title":"C nonzero off K"},{"id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/lem-c-nonzero-on-k","kind":"node","lean":"lem_Cf_nonzero_on_K","parent":5608,"result":true,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01/index.html#lem-c-nonzero-on-k","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 477. The project gives no license, so trough does not reproduce its text.","title":"C nonzero on K"},{"data_url":"scopes/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-02.json","id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-02","kind":"scope","navigation":"transparent","parent":5607,"result":false,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-02/index.html","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-02","status":"fully_proved","title":"C never zero to B off K"},{"id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-02/lem-c-never-zero","kind":"node","lean":"lem_Cf_never_zero","parent":5629,"result":true,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-02/index.html#lem-c-never-zero","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 492. The project gives no license, so trough does not reproduce its text.","title":"C never zero"},{"id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-02/lem-bl-num-diff","kind":"node","lean":"bl_num_diff","parent":5629,"result":true,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-02/index.html#lem-bl-num-diff","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 507. The project gives no license, so trough does not reproduce its text.","title":"Blaschke diff"},{"id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-02/lem-bl-num-nonzero","kind":"node","lean":"lem_bl_num_nonzero","parent":5629,"result":true,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-02/index.html#lem-bl-num-nonzero","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 516. The project gives no license, so trough does not reproduce its text.","title":"Blaschke nonzero"},{"id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-02/def-bf","kind":"node","lean":"Bf","parent":5629,"result":true,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-02/index.html#def-bf","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this definition at PNT/PNT2_LogDerivative.tex, line 525. The project gives no license, so trough does not reproduce its text.","title":"Blaschke B"},{"id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-02/lem-bfcf","kind":"node","lean":"lem_BfCf","parent":5629,"result":true,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-02/index.html#lem-bfcf","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 529. The project gives no license, so trough does not reproduce its text.","title":"B and C relation"},{"id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-02/lem-bf-div","kind":"node","lean":"lem_Bf_div","parent":5629,"result":true,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-02/index.html#lem-bf-div","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 541. The project gives no license, so trough does not reproduce its text.","title":"B division"},{"id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-02/lem-bf-prodpow","kind":"node","lean":"lem_Bf_prodpow","parent":5629,"result":true,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-02/index.html#lem-bf-prodpow","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 554. The project gives no license, so trough does not reproduce its text.","title":"B product pow"},{"id":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-02/lem-bf-off-k","kind":"node","lean":"lem_Bf_off_K","parent":5629,"result":true,"route":"../roadmap/strongpnt/log-derivative/b-f-analytic-and-never-zero/part-02/index.html#lem-bf-off-k","scope":"strongpnt/log-derivative/b-f-analytic-and-never-zero/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 569. The project gives no license, so trough does not reproduce its text.","title":"B off K"},{"data_url":"scopes/strongpnt/log-derivative/bounding-k-le-3-log-b.json","id":"strongpnt/log-derivative/bounding-k-le-3-log-b","kind":"scope","parent":5595,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bounding "],["math","\\(K\\le 3\\log B\\)"]],"source":"Bounding $K\\le 3\\log B$"},"result":false,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/index.html","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b","status":"fully_proved","title":"Bounding $K\\le 3\\log B$"},{"data_url":"scopes/strongpnt/log-derivative/bounding-k-le-3-log-b/part-01.json","id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01","kind":"scope","navigation":"transparent","parent":5638,"result":false,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/index.html","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01","status":"fully_proved","title":"Zero value to Abs division"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/lem-frho-zero","kind":"node","lean":"lem_frho_zero","parent":5639,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/index.html#lem-frho-zero","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 584. The project gives no license, so trough does not reproduce its text.","title":"Zero value"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/lem-frho-zero-contra","kind":"node","lean":"lem_frho_zero_contra","parent":5639,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/index.html#lem-frho-zero-contra","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 592. The project gives no license, so trough does not reproduce its text.","title":"Zero contrapositive"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/lem-f-is-nonzero","kind":"node","lean":"lem_f_is_nonzero","parent":5639,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/index.html#lem-f-is-nonzero","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 600. The project gives no license, so trough does not reproduce its text.","title":"Not zero"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/lem-rho-in-disk-r1","kind":"node","lean":"lem_rho_in_disk_R1","parent":5639,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/index.html#lem-rho-in-disk-r1","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 608. The project gives no license, so trough does not reproduce its text.","title":"Disk bound"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/lem-zero-not-in-kf","kind":"node","lean":"lem_zero_not_in_Kf","parent":5639,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/index.html#lem-zero-not-in-kf","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 616. The project gives no license, so trough does not reproduce its text.","title":"Zero excluded"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/lem-rho-ne-zero","kind":"node","lean":"lem_rho_ne_zero","parent":5639,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/index.html#lem-rho-ne-zero","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 624. The project gives no license, so trough does not reproduce its text.","title":"Nonzero rho"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/lem-mod-pos-iff-ne-zero","kind":"node","lean":"lem_mod_pos_iff_ne_zero","parent":5639,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/index.html#lem-mod-pos-iff-ne-zero","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 632. The project gives no license, so trough does not reproduce its text.","title":"Mod positive"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/lem-mod-rho-pos","kind":"node","lean":"lem_mod_rho_pos","parent":5639,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/index.html#lem-mod-rho-pos","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 640. The project gives no license, so trough does not reproduce its text.","title":"Rho positive"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/lem-rho-in-disk-r1-repeat","kind":"node","lean":"lem_rho_in_disk_R1_repeat","parent":5639,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/index.html#lem-rho-in-disk-r1-repeat","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 648. The project gives no license, so trough does not reproduce its text.","title":"Disk bound"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/lem-inv-mono-decr","kind":"node","lean":"lem_inv_mono_decr","parent":5639,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/index.html#lem-inv-mono-decr","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 656. The project gives no license, so trough does not reproduce its text.","title":"Inverse mono"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/lem-inv-mod-rho-ge-inv-r1","kind":"node","lean":"lem_inv_mod_rho_ge_inv_R1","parent":5639,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/index.html#lem-inv-mod-rho-ge-inv-r1","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 662. The project gives no license, so trough does not reproduce its text.","title":"Inverse bound"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/lem-mul-pos-preserves-ineq","kind":"node","lean":"lem_mul_pos_preserves_ineq","parent":5639,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/index.html#lem-mul-pos-preserves-ineq","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 670. The project gives no license, so trough does not reproduce its text.","title":"Mult inequality"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/lem-r-div-mod-rho-ge-r-div-r1","kind":"node","lean":"lem_R_div_mod_rho_ge_R_div_R1","parent":5639,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/index.html#lem-r-div-mod-rho-ge-r-div-r1","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 676. The project gives no license, so trough does not reproduce its text.","title":"Ratio bound"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/lem-mod-of-prod2","kind":"node","lean":"lem_mod_of_prod2","parent":5639,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/index.html#lem-mod-of-prod2","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 685. The project gives no license, so trough does not reproduce its text.","title":"Mod product"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/lem-mod-bf-is-prod-mod","kind":"node","lean":"lem_mod_Bf_is_prod_mod","parent":5639,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/index.html#lem-mod-bf-is-prod-mod","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 691. The project gives no license, so trough does not reproduce its text.","title":"B modulus"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/lem-abs-pow","kind":"node","lean":"lem_abs_pow","parent":5639,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/index.html#lem-abs-pow","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 701. The project gives no license, so trough does not reproduce its text.","title":"Abs power"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/lem-bmod-pow","kind":"node","lean":"lem_Bmod_pow","parent":5639,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/index.html#lem-bmod-pow","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 707. The project gives no license, so trough does not reproduce its text.","title":"Power mod"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/lem-mod-bf-prod-mod","kind":"node","lean":"lem_mod_Bf_prod_mod","parent":5639,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/index.html#lem-mod-bf-prod-mod","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 716. The project gives no license, so trough does not reproduce its text.","title":"B modulus"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/lem-mod-bf-at-0","kind":"node","lean":"lem_mod_Bf_at_0","parent":5639,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/index.html#lem-mod-bf-at-0","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 725. The project gives no license, so trough does not reproduce its text.","title":"B at zero"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/lem-mod-div","kind":"node","lean":"lem_mod_div_","parent":5639,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-01/index.html#lem-mod-div","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 734. The project gives no license, so trough does not reproduce its text.","title":"Abs division"},{"data_url":"scopes/strongpnt/log-derivative/bounding-k-le-3-log-b/part-02.json","id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02","kind":"scope","navigation":"transparent","parent":5638,"result":false,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/index.html","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02","status":"fully_proved","title":"Abs neg to Combine bounds"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/lem-mod-neg","kind":"node","lean":"lem_mod_neg","parent":5660,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/index.html#lem-mod-neg","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 740. The project gives no license, so trough does not reproduce its text.","title":"Abs neg"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/lem-mod-div-and-neg","kind":"node","lean":"lem_mod_div_and_neg","parent":5660,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/index.html#lem-mod-div-and-neg","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 746. The project gives no license, so trough does not reproduce its text.","title":"Abs ratio"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/lem-mod-bf-at-0-eval","kind":"node","lean":"lem_mod_Bf_at_0_eval","parent":5660,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/index.html#lem-mod-bf-at-0-eval","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 755. The project gives no license, so trough does not reproduce its text.","title":"B zero form"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/lem-mod-of-pos-real","kind":"node","lean":"lem_mod_of_pos_real","parent":5660,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/index.html#lem-mod-of-pos-real","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 763. The project gives no license, so trough does not reproduce its text.","title":"Abs positive"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/lem-mod-bf-at-0-as-ratio","kind":"node","lean":"lem_mod_Bf_at_0_as_ratio","parent":5660,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/index.html#lem-mod-bf-at-0-as-ratio","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 770. The project gives no license, so trough does not reproduce its text.","title":"B zero form"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/lem-prod-ineq","kind":"node","lean":"lem_prod_ineq","parent":5660,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/index.html#lem-prod-ineq","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 779. The project gives no license, so trough does not reproduce its text.","title":"Prod inequality"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/lem-power-ineq","kind":"node","lean":"lem_power_ineq","parent":5660,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/index.html#lem-power-ineq","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 785. The project gives no license, so trough does not reproduce its text.","title":"Power bound"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/lem-power-ineq-1","kind":"node","lean":"lem_power_ineq_1","parent":5660,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/index.html#lem-power-ineq-1","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 791. The project gives no license, so trough does not reproduce its text.","title":"Power one"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/lem-prod-power-ineq","kind":"node","lean":"lem_prod_power_ineq","parent":5660,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/index.html#lem-prod-power-ineq","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 799. The project gives no license, so trough does not reproduce its text.","title":"Product power"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/lem-prod-1","kind":"node","lean":"lem_prod_1","parent":5660,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/index.html#lem-prod-1","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 808. The project gives no license, so trough does not reproduce its text.","title":"Product one"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/lem-prod-power-ineq1","kind":"node","lean":"lem_prod_power_ineq1","parent":5660,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/index.html#lem-prod-power-ineq1","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 814. The project gives no license, so trough does not reproduce its text.","title":"Power bound"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/lem-mod-lower-bound-1","kind":"node","lean":"lem_mod_lower_bound_1","parent":5660,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/index.html#lem-mod-lower-bound-1","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 823. The project gives no license, so trough does not reproduce its text.","title":"Modulus bound"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/lem-bf-is-analytic","kind":"node","lean":"lem_Bf_is_analytic","parent":5660,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/index.html#lem-bf-is-analytic","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 831. The project gives no license, so trough does not reproduce its text.","title":"B analytic"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/lem-mod-bf-eq-mod-f-on-boundary","kind":"node","lean":"lem_mod_Bf_eq_mod_f_on_boundary","parent":5660,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/index.html#lem-mod-bf-eq-mod-f-on-boundary","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 840. The project gives no license, so trough does not reproduce its text.","title":"Boundary mod"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/lem-bf-bounded-on-boundary","kind":"node","lean":"lem_Bf_bounded_on_boundary","parent":5660,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/index.html#lem-bf-bounded-on-boundary","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 847. The project gives no license, so trough does not reproduce its text.","title":"Boundary bound"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/lem-max-mod-principle-for-bf","kind":"node","lean":"lem_max_mod_principle_for_Bf","parent":5660,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/index.html#lem-max-mod-principle-for-bf","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 856. The project gives no license, so trough does not reproduce its text.","title":"Max modulus"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/lem-bf-bounded-in-disk-from-boundary","kind":"node","lean":"lem_Bf_bounded_in_disk_from_boundary","parent":5660,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/index.html#lem-bf-bounded-in-disk-from-boundary","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 864. The project gives no license, so trough does not reproduce its text.","title":"Disk bound"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/lem-bf-bounded-in-disk-from-f","kind":"node","lean":"lem_Bf_bounded_in_disk_from_f","parent":5660,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/index.html#lem-bf-bounded-in-disk-from-f","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 872. The project gives no license, so trough does not reproduce its text.","title":"Disk bound"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/lem-bf-at-0-le-m","kind":"node","lean":"lem_Bf_at_0_le_M","parent":5660,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/index.html#lem-bf-at-0-le-m","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 880. The project gives no license, so trough does not reproduce its text.","title":"Zero bound"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/lem-combine-bounds-on-bf0","kind":"node","lean":"lem_combine_bounds_on_Bf0","parent":5660,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-02/index.html#lem-combine-bounds-on-bf0","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 888. The project gives no license, so trough does not reproduce its text.","title":"Combine bounds"},{"data_url":"scopes/strongpnt/log-derivative/bounding-k-le-3-log-b/part-03.json","id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-03","kind":"scope","navigation":"transparent","parent":5638,"result":false,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-03/index.html","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-03","status":"fully_proved","title":"Jensen form to Zeros bound"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-03/lem-jensen-inequality-form","kind":"node","lean":"lem_jensen_inequality_form","parent":5681,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-03/index.html#lem-jensen-inequality-form","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 895. The project gives no license, so trough does not reproduce its text.","title":"Jensen form"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-03/lem-log-mono-inc","kind":"node","lean":"lem_log_mono_inc","parent":5681,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-03/index.html#lem-log-mono-inc","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 903. The project gives no license, so trough does not reproduce its text.","title":"Log monotone"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-03/lem-jensen-log-form","kind":"node","lean":"lem_jensen_log_form","parent":5681,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-03/index.html#lem-jensen-log-form","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 909. The project gives no license, so trough does not reproduce its text.","title":"Jensen log"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-03/lem-3-gt-e","kind":"node","lean":"lem_three_gt_e","parent":5681,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-03/index.html#lem-3-gt-e","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 917. The project gives no license, so trough does not reproduce its text.","title":"Three exceeds e"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-03/lem-sum-m-rho-bound","kind":"node","lean":"lem_sum_m_rho_bound","parent":5681,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-03/index.html#lem-sum-m-rho-bound","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 924. The project gives no license, so trough does not reproduce its text.","title":"Multiplicity bound"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-03/lem-sum-ineq","kind":"node","lean":"lem_sum_ineq","parent":5681,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-03/index.html#lem-sum-ineq","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 935. The project gives no license, so trough does not reproduce its text.","title":"Sum inequality"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-03/lem-sum-m-rho-1","kind":"node","lean":"lem_sum_m_rho_1","parent":5681,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-03/index.html#lem-sum-m-rho-1","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 941. The project gives no license, so trough does not reproduce its text.","title":"Multiplicity one"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-03/lem-sum-1-bound","kind":"node","lean":"lem_sum_1_bound","parent":5681,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-03/index.html#lem-sum-1-bound","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 951. The project gives no license, so trough does not reproduce its text.","title":"Ones bound"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-03/lem-sum-1-is-card","kind":"node","lean":"lem_sum_1_is_card","parent":5681,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-03/index.html#lem-sum-1-is-card","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 960. The project gives no license, so trough does not reproduce its text.","title":"Count identity"},{"id":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-03/lem-num-zeros-bound","kind":"node","lean":"lem_num_zeros_bound","parent":5681,"result":true,"route":"../roadmap/strongpnt/log-derivative/bounding-k-le-3-log-b/part-03/index.html#lem-num-zeros-bound","scope":"strongpnt/log-derivative/bounding-k-le-3-log-b/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 966. The project gives no license, so trough does not reproduce its text.","title":"Zeros bound"},{"data_url":"scopes/strongpnt/log-derivative/log-l-f.json","id":"strongpnt/log-derivative/log-l-f","kind":"scope","parent":5595,"reading_title":{"format":"inline-parts-v1","parts":[["text","Log "],["math","\\(L_f\\)"]],"source":"Log $L_f$"},"result":false,"route":"../roadmap/strongpnt/log-derivative/log-l-f/index.html","scope":"strongpnt/log-derivative/log-l-f","status":"fully_proved","title":"Log $L_f$"},{"id":"strongpnt/log-derivative/log-l-f/lem-bf-is-analytic-on-disk","kind":"node","lean":"Bf_is_analytic_on_disk","parent":5692,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-l-f/index.html#lem-bf-is-analytic-on-disk","scope":"strongpnt/log-derivative/log-l-f","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 987. The project gives no license, so trough does not reproduce its text.","title":"Disk analytic"},{"id":"strongpnt/log-derivative/log-l-f/lem-bf-never-zero","kind":"node","lean":"Bf_never_zero","parent":5692,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-l-f/index.html#lem-bf-never-zero","scope":"strongpnt/log-derivative/log-l-f","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 996. The project gives no license, so trough does not reproduce its text.","title":"Never zero"},{"id":"strongpnt/log-derivative/log-l-f/lem-bf0-not-zero","kind":"node","lean":"Bf0_not_zero","parent":5692,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-l-f/index.html#lem-bf0-not-zero","scope":"strongpnt/log-derivative/log-l-f","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1006. The project gives no license, so trough does not reproduce its text.","title":"B zero"},{"id":"strongpnt/log-derivative/log-l-f/lem-lf-is-analytic","kind":"node","lean":"Lf_is_analytic","parent":5692,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-l-f/index.html#lem-lf-is-analytic","scope":"strongpnt/log-derivative/log-l-f","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1015. The project gives no license, so trough does not reproduce its text.","title":"Lf analytic"},{"id":"strongpnt/log-derivative/log-l-f/lem-lf-at-0-is-0","kind":"node","lean":"Lf_at_0_is_0","parent":5692,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-l-f/index.html#lem-lf-at-0-is-0","scope":"strongpnt/log-derivative/log-l-f","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1024. The project gives no license, so trough does not reproduce its text.","title":"Lf at zero"},{"id":"strongpnt/log-derivative/log-l-f/lem-re-lf-as-diff-of-log-mods","kind":"node","lean":"re_Lf_as_diff_of_log_mods","parent":5692,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-l-f/index.html#lem-re-lf-as-diff-of-log-mods","scope":"strongpnt/log-derivative/log-l-f","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1034. The project gives no license, so trough does not reproduce its text.","title":"Real part diff"},{"id":"strongpnt/log-derivative/log-l-f/lem-log-bf-le-log-b","kind":"node","lean":"log_Bf_le_log_B","parent":5692,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-l-f/index.html#lem-log-bf-le-log-b","scope":"strongpnt/log-derivative/log-l-f","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1044. The project gives no license, so trough does not reproduce its text.","title":"Log bound"},{"id":"strongpnt/log-derivative/log-l-f/lem-log-bf-le-log-b2","kind":"node","lean":"log_Bf_le_log_B2","parent":5692,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-l-f/index.html#lem-log-bf-le-log-b2","scope":"strongpnt/log-derivative/log-l-f","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1053. The project gives no license, so trough does not reproduce its text.","title":"Log bound"},{"id":"strongpnt/log-derivative/log-l-f/lem-log-bf-le-log-b3","kind":"node","lean":"log_Bf_le_log_B3","parent":5692,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-l-f/index.html#lem-log-bf-le-log-b3","scope":"strongpnt/log-derivative/log-l-f","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1061. The project gives no license, so trough does not reproduce its text.","title":"Log bound"},{"id":"strongpnt/log-derivative/log-l-f/lem-log-bf0-ge-0","kind":"node","lean":"log_Bf0_ge_0","parent":5692,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-l-f/index.html#lem-log-bf0-ge-0","scope":"strongpnt/log-derivative/log-l-f","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1069. The project gives no license, so trough does not reproduce its text.","title":"Log nonnegative"},{"id":"strongpnt/log-derivative/log-l-f/lem-re-lf-le-log-b","kind":"node","lean":"re_Lf_le_log_B","parent":5692,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-l-f/index.html#lem-re-lf-le-log-b","scope":"strongpnt/log-derivative/log-l-f","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1077. The project gives no license, so trough does not reproduce its text.","title":"Real part bound"},{"id":"strongpnt/log-derivative/log-l-f/lem-bcii","kind":"node","lean":"lem_BCII","parent":5692,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-l-f/index.html#lem-bcii","scope":"strongpnt/log-derivative/log-l-f","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1086. The project gives no license, so trough does not reproduce its text.","title":"BC inequality"},{"id":"strongpnt/log-derivative/log-l-f/lem-apply-bc-to-lf","kind":"node","lean":"apply_BC_to_Lf","parent":5692,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-l-f/index.html#lem-apply-bc-to-lf","scope":"strongpnt/log-derivative/log-l-f","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1097. The project gives no license, so trough does not reproduce its text.","title":"Apply BC"},{"data_url":"scopes/strongpnt/log-derivative/log-derivative-l-f-expansion.json","id":"strongpnt/log-derivative/log-derivative-l-f-expansion","kind":"scope","parent":5595,"reading_title":{"format":"inline-parts-v1","parts":[["text","Log derivative "],["math","\\(L_f'\\)"],["text"," expansion"]],"source":"Log derivative $L_f'$ expansion"},"result":false,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/index.html","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion","status":"fully_proved","title":"Log derivative $L_f'$ expansion"},{"data_url":"scopes/strongpnt/log-derivative/log-derivative-l-f-expansion/part-01.json","id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01","kind":"scope","navigation":"transparent","parent":5706,"result":false,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/index.html","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01","status":"fully_proved","title":"Constant rule to Power multiple"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/lem-logderivconst","kind":"node","lean":"logDerivconst","parent":5707,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/index.html#lem-logderivconst","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1112. The project gives no license, so trough does not reproduce its text.","title":"Constant rule"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/lem-1bneq0","kind":"node","lean":"oneBneq0","parent":5707,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/index.html#lem-1bneq0","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1121. The project gives no license, so trough does not reproduce its text.","title":"One minus B"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/lem-lf-deriv-is-logbf-deriv","kind":"node","lean":"Lf_deriv_is_logBf_deriv","parent":5707,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/index.html#lem-lf-deriv-is-logbf-deriv","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1132. The project gives no license, so trough does not reproduce its text.","title":"Derivative form"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/lem-logderivmul","kind":"node","lean":"logDerivmul","parent":5707,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/index.html#lem-logderivmul","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1142. The project gives no license, so trough does not reproduce its text.","title":"Product rule"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/lem-logderivprod","kind":"node","lean":"logDerivprod","parent":5707,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/index.html#lem-logderivprod","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1152. The project gives no license, so trough does not reproduce its text.","title":"Product sum"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/lem-logderivdiv","kind":"node","lean":"logDerivdiv","parent":5707,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/index.html#lem-logderivdiv","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1161. The project gives no license, so trough does not reproduce its text.","title":"Quotient rule"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/lem-logderivfunpow","kind":"node","lean":"logDerivfunpow","parent":5707,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/index.html#lem-logderivfunpow","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1170. The project gives no license, so trough does not reproduce its text.","title":"Power rule"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/lem-z-minus-rho-diff-nonzero","kind":"node","lean":"z_minus_rho_diff_nonzero","parent":5707,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/index.html#lem-z-minus-rho-diff-nonzero","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1180. The project gives no license, so trough does not reproduce its text.","title":"Difference nonzero"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/lem-blaschke-num-diff-nonzero","kind":"node","lean":"blaschke_num_diff_nonzero","parent":5707,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/index.html#lem-blaschke-num-diff-nonzero","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1192. The project gives no license, so trough does not reproduce its text.","title":"Numerator nonzero"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/lem-blaschke-frac-diff-nonzero","kind":"node","lean":"blaschke_frac_diff_nonzero","parent":5707,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/index.html#lem-blaschke-frac-diff-nonzero","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1203. The project gives no license, so trough does not reproduce its text.","title":"Fraction nonzero"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/lem-blaschke-pow-diff-nonzero","kind":"node","lean":"blaschke_pow_diff_nonzero","parent":5707,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/index.html#lem-blaschke-pow-diff-nonzero","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1213. The project gives no license, so trough does not reproduce its text.","title":"Power nonzero"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/lem-blaschke-prod-diff-nonzero","kind":"node","lean":"blaschke_prod_diff_nonzero","parent":5707,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/index.html#lem-blaschke-prod-diff-nonzero","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1223. The project gives no license, so trough does not reproduce its text.","title":"Product nonzero"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/lem-f-diff-nonzero-outside-kf","kind":"node","lean":"f_diff_nonzero_outside_Kf","parent":5707,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/index.html#lem-f-diff-nonzero-outside-kf","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1233. The project gives no license, so trough does not reproduce its text.","title":"Outside zeros"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/lem-bf-diff-nonzero-outside-kf","kind":"node","lean":"Bf_diff_nonzero_outside_Kf","parent":5707,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/index.html#lem-bf-diff-nonzero-outside-kf","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1243. The project gives no license, so trough does not reproduce its text.","title":"Outside zeros"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/lem-logderiv-fprod-is-sum","kind":"node","lean":"logDeriv_fprod_is_sum","parent":5707,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/index.html#lem-logderiv-fprod-is-sum","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1254. The project gives no license, so trough does not reproduce its text.","title":"Log sum"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/lem-logderiv-bf-is-sum","kind":"node","lean":"logDeriv_Bf_is_sum","parent":5707,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/index.html#lem-logderiv-bf-is-sum","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1267. The project gives no license, so trough does not reproduce its text.","title":"Log sum"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/lem-logderiv-def-as-frac","kind":"node","lean":"logDeriv_def_as_frac","parent":5707,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/index.html#lem-logderiv-def-as-frac","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1278. The project gives no license, so trough does not reproduce its text.","title":"Fraction form"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/lem-lf-deriv-step1","kind":"node","lean":"Lf_deriv_step1","parent":5707,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/index.html#lem-lf-deriv-step1","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1287. The project gives no license, so trough does not reproduce its text.","title":"Step one"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/lem-logderiv-prod-is-sum","kind":"node","lean":"logDeriv_prod_is_sum","parent":5707,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/index.html#lem-logderiv-prod-is-sum","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1299. The project gives no license, so trough does not reproduce its text.","title":"Product sum"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/lem-logderiv-power-is-mul","kind":"node","lean":"logDeriv_power_is_mul","parent":5707,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-01/index.html#lem-logderiv-power-is-mul","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1314. The project gives no license, so trough does not reproduce its text.","title":"Power multiple"},{"data_url":"scopes/strongpnt/log-derivative/log-derivative-l-f-expansion/part-02.json","id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02","kind":"scope","navigation":"transparent","parent":5706,"result":false,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/index.html","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02","status":"fully_proved","title":"Sum multiple to Final bound"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/lem-logderiv-prod-is-sum-mul","kind":"node","lean":"logDeriv_prod_is_sum_mul","parent":5728,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/index.html#lem-logderiv-prod-is-sum-mul","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1327. The project gives no license, so trough does not reproduce its text.","title":"Sum multiple"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/lem-lf-deriv-step2","kind":"node","lean":"Lf_deriv_step2","parent":5728,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/index.html#lem-lf-deriv-step2","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1338. The project gives no license, so trough does not reproduce its text.","title":"Step two"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/lem-logderiv-blaschke-is-diff","kind":"node","lean":"logDeriv_Blaschke_is_diff","parent":5728,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/index.html#lem-logderiv-blaschke-is-diff","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1350. The project gives no license, so trough does not reproduce its text.","title":"Difference form"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/lem-logderiv-linear","kind":"node","lean":"logDeriv_linear","parent":5728,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/index.html#lem-logderiv-linear","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1363. The project gives no license, so trough does not reproduce its text.","title":"Linear rule"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/lem-logderiv-denominator","kind":"node","lean":"logDeriv_denominator","parent":5728,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/index.html#lem-logderiv-denominator","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1374. The project gives no license, so trough does not reproduce its text.","title":"Denominator rule"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/lem-logderiv-numerator-pre","kind":"node","lean":"logDeriv_numerator_pre","parent":5728,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/index.html#lem-logderiv-numerator-pre","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1384. The project gives no license, so trough does not reproduce its text.","title":"Numerator rule"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/lem-logderiv-numerator-rearranged","kind":"node","lean":"logDeriv_numerator_rearranged","parent":5728,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/index.html#lem-logderiv-numerator-rearranged","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1394. The project gives no license, so trough does not reproduce its text.","title":"Rearranged form"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/lem-logderiv-numerator","kind":"node","lean":"logDeriv_numerator","parent":5728,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/index.html#lem-logderiv-numerator","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1429. The project gives no license, so trough does not reproduce its text.","title":"Numerator form"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/lem-logderiv-blaschke-is-diff-frac","kind":"node","lean":"logDeriv_Blaschke_is_diff_frac","parent":5728,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/index.html#lem-logderiv-blaschke-is-diff-frac","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1441. The project gives no license, so trough does not reproduce its text.","title":"Diff fraction"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/lem-lf-deriv-step3","kind":"node","lean":"Lf_deriv_step3","parent":5728,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/index.html#lem-lf-deriv-step3","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1477. The project gives no license, so trough does not reproduce its text.","title":"Step three"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/lem-sum-of-diff","kind":"node","lean":"sum_of_diff","parent":5728,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/index.html#lem-sum-of-diff","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1504. The project gives no license, so trough does not reproduce its text.","title":"Sum difference"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/lem-sum-rearranged","kind":"node","lean":"sum_rearranged","parent":5728,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/index.html#lem-sum-rearranged","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1513. The project gives no license, so trough does not reproduce its text.","title":"Sum rearranged"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/lem-lf-deriv-final-formula","kind":"node","lean":"Lf_deriv_final_formula","parent":5728,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/index.html#lem-lf-deriv-final-formula","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1525. The project gives no license, so trough does not reproduce its text.","title":"Final formula"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/lem-rearrange-lf-deriv","kind":"node","lean":"rearrange_Lf_deriv","parent":5728,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/index.html#lem-rearrange-lf-deriv","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1536. The project gives no license, so trough does not reproduce its text.","title":"Rearranged deriv"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/lem-triangle-ineq-sum","kind":"node","lean":"triangle_ineq_sum","parent":5728,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/index.html#lem-triangle-ineq-sum","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1547. The project gives no license, so trough does not reproduce its text.","title":"Triangle sum"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/lem-target-inequality-setup","kind":"node","lean":"target_inequality_setup","parent":5728,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/index.html#lem-target-inequality-setup","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1556. The project gives no license, so trough does not reproduce its text.","title":"Setup inequality"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/lem-sum-bound-step2","kind":"node","lean":"lem_sum_bound_step2","parent":5728,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/index.html#lem-sum-bound-step2","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1568. The project gives no license, so trough does not reproduce its text.","title":"Step two"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/lem-final-sum-bound","kind":"node","lean":"final_sum_bound","parent":5728,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/index.html#lem-final-sum-bound","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1579. The project gives no license, so trough does not reproduce its text.","title":"Final sum"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/lem-final-inequality","kind":"node","lean":"final_inequality","parent":5728,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/index.html#lem-final-inequality","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1590. The project gives no license, so trough does not reproduce its text.","title":"Final bound"},{"id":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/lem-final-ineq1","kind":"node","lean":"final_ineq1","parent":5728,"result":true,"route":"../roadmap/strongpnt/log-derivative/log-derivative-l-f-expansion/part-02/index.html#lem-final-ineq1","scope":"strongpnt/log-derivative/log-derivative-l-f-expansion/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT2_LogDerivative.tex, line 1601. The project gives no license, so trough does not reproduce its text.","title":"Final bound"},{"data_url":"scopes/strongpnt/riemann-zeta-function.json","id":"strongpnt/riemann-zeta-function","kind":"scope","parent":5340,"result":false,"route":"../roadmap/strongpnt/riemann-zeta-function/index.html","scope":"strongpnt/riemann-zeta-function","status":"fully_proved","title":"Riemann Zeta Function"},{"data_url":"scopes/strongpnt/riemann-zeta-function/zeta-lower-bound.json","id":"strongpnt/riemann-zeta-function/zeta-lower-bound","kind":"scope","parent":5749,"result":false,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/index.html","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound","status":"fully_proved","title":"Zeta lower bound"},{"data_url":"scopes/strongpnt/riemann-zeta-function/zeta-lower-bound/part-01.json","id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01","kind":"scope","navigation":"transparent","parent":5750,"result":false,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/index.html","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01","status":"fully_proved","title":"Prime decay to Prime power"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/lem-p-s-abs-1","kind":"node","lean":"p_s_abs_1","parent":5751,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/index.html#lem-p-s-abs-1","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 9. The project gives no license, so trough does not reproduce its text.","title":"Prime decay"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/lem-zetaeulerprod","kind":"node","lean":"zetaEulerprod","parent":5751,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/index.html#lem-zetaeulerprod","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 23. The project gives no license, so trough does not reproduce its text.","title":"Euler product"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/lem-abs-of-tprod","kind":"node","lean":"abs_of_tprod","parent":5751,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/index.html#lem-abs-of-tprod","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 34. The project gives no license, so trough does not reproduce its text.","title":"Abs product"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/lem-abs-p-prod","kind":"node","lean":"abs_P_prod","parent":5751,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/index.html#lem-abs-p-prod","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 43. The project gives no license, so trough does not reproduce its text.","title":"Abs primes"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/lem-abs-zeta-prod","kind":"node","lean":"abs_zeta_prod","parent":5751,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/index.html#lem-abs-zeta-prod","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 55. The project gives no license, so trough does not reproduce its text.","title":"Abs zeta"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/lem-abs-of-inv","kind":"node","lean":"abs_of_inv","parent":5751,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/index.html#lem-abs-of-inv","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 66. The project gives no license, so trough does not reproduce its text.","title":"Abs inverse"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/lem-one-minus-p-s-neq-0","kind":"node","lean":"one_minus_p_s_neq_0","parent":5751,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/index.html#lem-one-minus-p-s-neq-0","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 76. The project gives no license, so trough does not reproduce its text.","title":"Nonzero factor"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/lem-abs-zeta-prod-prime","kind":"node","lean":"abs_zeta_prod_prime","parent":5751,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/index.html#lem-abs-zeta-prod-prime","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 87. The project gives no license, so trough does not reproduce its text.","title":"Abs product"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/lem-re2s","kind":"node","lean":"Re2s","parent":5751,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/index.html#lem-re2s","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 98. The project gives no license, so trough does not reproduce its text.","title":"Real double"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/lem-re2sge1","kind":"node","lean":"Re2sge1","parent":5751,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/index.html#lem-re2sge1","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 106. The project gives no license, so trough does not reproduce its text.","title":"Real bound"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/lem-zeta-ratio-prod","kind":"node","lean":"zeta_ratio_prod","parent":5751,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/index.html#lem-zeta-ratio-prod","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 116. The project gives no license, so trough does not reproduce its text.","title":"Zeta ratio"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/lem-prod-of-ratios","kind":"node","lean":"prod_of_ratios","parent":5751,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/index.html#lem-prod-of-ratios","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 127. The project gives no license, so trough does not reproduce its text.","title":"Ratio product"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/lem-simplify-prod-ratio","kind":"node","lean":"simplify_prod_ratio","parent":5751,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/index.html#lem-simplify-prod-ratio","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 186. The project gives no license, so trough does not reproduce its text.","title":"Ratio split"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/lem-zeta-ratios","kind":"node","lean":"zeta_ratios","parent":5751,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/index.html#lem-zeta-ratios","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 196. The project gives no license, so trough does not reproduce its text.","title":"Ratio form"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/lem-diff-of-squares","kind":"node","lean":"diff_of_squares","parent":5751,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/index.html#lem-diff-of-squares","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 207. The project gives no license, so trough does not reproduce its text.","title":"Diff squares"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/lem-ratio-invs","kind":"node","lean":"ratio_invs","parent":5751,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/index.html#lem-ratio-invs","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 216. The project gives no license, so trough does not reproduce its text.","title":"Inverse ratio"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/thm-zeta-ratio-identity","kind":"node","lean":"zeta_ratio_identity","parent":5751,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/index.html#thm-zeta-ratio-identity","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this theorem at PNT/PNT3_RiemannZeta.tex, line 227. The project gives no license, so trough does not reproduce its text.","title":"Ratio identity"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/lem-zeta-ratio-at-3-2","kind":"node","lean":"zeta_ratio_at_3_2","parent":5751,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/index.html#lem-zeta-ratio-at-3-2","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 238. The project gives no license, so trough does not reproduce its text.","title":"Three halves"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/lem-triangle-inequality-specific","kind":"node","lean":"triangle_inequality_specific","parent":5751,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/index.html#lem-triangle-inequality-specific","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 249. The project gives no license, so trough does not reproduce its text.","title":"Triangle abs"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/lem-abs-p-pow-s","kind":"node","lean":"abs_p_pow_s","parent":5751,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-01/index.html#lem-abs-p-pow-s","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 258. The project gives no license, so trough does not reproduce its text.","title":"Prime power"},{"data_url":"scopes/strongpnt/riemann-zeta-function/zeta-lower-bound/part-02.json","id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-02","kind":"scope","navigation":"transparent","parent":5750,"result":false,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-02/index.html","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-02","status":"fully_proved","title":"Term bound to Fixed lower"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-02/lem-abs-term-bound","kind":"node","lean":"abs_term_bound","parent":5772,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-02/index.html#lem-abs-term-bound","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 268. The project gives no license, so trough does not reproduce its text.","title":"Term bound"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-02/lem-inv-inequality","kind":"node","lean":"inv_inequality","parent":5772,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-02/index.html#lem-inv-inequality","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 278. The project gives no license, so trough does not reproduce its text.","title":"Inv order"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-02/lem-condp32","kind":"node","lean":"condp32","parent":5772,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-02/index.html#lem-condp32","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 287. The project gives no license, so trough does not reproduce its text.","title":"Nonzero term"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-02/lem-abs-term-inv-bound","kind":"node","lean":"abs_term_inv_bound","parent":5772,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-02/index.html#lem-abs-term-inv-bound","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 298. The project gives no license, so trough does not reproduce its text.","title":"Inverse bound"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-02/lem-prod-inequality","kind":"node","lean":"prod_inequality","parent":5772,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-02/index.html#lem-prod-inequality","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 309. The project gives no license, so trough does not reproduce its text.","title":"Prod order"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-02/lem-abs-zeta-inequality","kind":"node","lean":"abs_zeta_inequality","parent":5772,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-02/index.html#lem-abs-zeta-inequality","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 318. The project gives no license, so trough does not reproduce its text.","title":"Zeta compare"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-02/thm-zeta-lower-bound","kind":"node","lean":"zeta_lower_bound","parent":5772,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-02/index.html#thm-zeta-lower-bound","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this theorem at PNT/PNT3_RiemannZeta.tex, line 329. The project gives no license, so trough does not reproduce its text.","title":"Zeta lower"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-02/lem-zetapos","kind":"node","lean":"zetapos","parent":5772,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-02/index.html#lem-zetapos","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 340. The project gives no license, so trough does not reproduce its text.","title":"Zeta positive"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-02/lem-zeta332pos","kind":"node","lean":"zeta332pos","parent":5772,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-02/index.html#lem-zeta332pos","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 350. The project gives no license, so trough does not reproduce its text.","title":"Ratio positive"},{"id":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-02/lem-zeta-low-332","kind":"node","lean":"zeta_low_332","parent":5772,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-lower-bound/part-02/index.html#lem-zeta-low-332","scope":"strongpnt/riemann-zeta-function/zeta-lower-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 360. The project gives no license, so trough does not reproduce its text.","title":"Fixed lower"},{"data_url":"scopes/strongpnt/riemann-zeta-function/zeta-bound.json","id":"strongpnt/riemann-zeta-function/zeta-bound","kind":"scope","parent":5749,"result":false,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/index.html","scope":"strongpnt/riemann-zeta-function/zeta-bound","status":"fully_proved","title":"Zeta bound"},{"data_url":"scopes/strongpnt/riemann-zeta-function/zeta-bound/part-01.json","id":"strongpnt/riemann-zeta-function/zeta-bound/part-01","kind":"scope","navigation":"transparent","parent":5783,"result":false,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-01/index.html","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-01","status":"fully_proved","title":"Series form to Integral conv"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-01/lem-zetalimit","kind":"node","lean":"lem_zetaLimit","parent":5784,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-01/index.html#lem-zetalimit","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 373. The project gives no license, so trough does not reproduce its text.","title":"Series form"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-01/def-zetapartialsum","kind":"node","lean":"zetaPartialSum","parent":5784,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-01/index.html#def-zetapartialsum","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this definition at PNT/PNT3_RiemannZeta.tex, line 382. The project gives no license, so trough does not reproduce its text.","title":"Partial sum"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-01/lem-abelsummation","kind":"node","lean":"lem_abelSummation","parent":5784,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-01/index.html#lem-abelsummation","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 388. The project gives no license, so trough does not reproduce its text.","title":"Abel sum"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-01/lem-partialsumiszetan","kind":"node","lean":"lem_partialSumIsZetaN","parent":5784,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-01/index.html#lem-partialsumiszetan","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 399. The project gives no license, so trough does not reproduce its text.","title":"Sum identity"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-01/lem-sumofan","kind":"node","lean":"lem_sumOfAn","parent":5784,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-01/index.html#lem-sumofan","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 409. The project gives no license, so trough does not reproduce its text.","title":"Count sum"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-01/lem-fderiv","kind":"node","lean":"lem_fDeriv","parent":5784,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-01/index.html#lem-fderiv","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 418. The project gives no license, so trough does not reproduce its text.","title":"Power deriv"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-01/lem-applyabel","kind":"node","lean":"lem_applyAbel","parent":5784,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-01/index.html#lem-applyabel","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 427. The project gives no license, so trough does not reproduce its text.","title":"Apply Abel"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-01/lem-floornisn","kind":"node","lean":"lem_floorNisN","parent":5784,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-01/index.html#lem-floornisn","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 439. The project gives no license, so trough does not reproduce its text.","title":"Floor int"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-01/lem-zetansimplified1","kind":"node","lean":"lem_zetaNsimplified1","parent":5784,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-01/index.html#lem-zetansimplified1","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 448. The project gives no license, so trough does not reproduce its text.","title":"First form"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-01/lem-floorudecomp","kind":"node","lean":"lem_floorUdecomp","parent":5784,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-01/index.html#lem-floorudecomp","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 460. The project gives no license, so trough does not reproduce its text.","title":"Floor split"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-01/lem-integralsplit","kind":"node","lean":"lem_integralSplit","parent":5784,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-01/index.html#lem-integralsplit","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 469. The project gives no license, so trough does not reproduce its text.","title":"Integral split"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-01/lem-zetansimplified2","kind":"node","lean":"lem_zetaNsimplified2","parent":5784,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-01/index.html#lem-zetansimplified2","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 480. The project gives no license, so trough does not reproduce its text.","title":"Second form"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-01/lem-evalmainintegral","kind":"node","lean":"lem_evalMainIntegral","parent":5784,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-01/index.html#lem-evalmainintegral","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 492. The project gives no license, so trough does not reproduce its text.","title":"Main integral"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-01/lem-zetanfinal","kind":"node","lean":"lem_zetaNfinal","parent":5784,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-01/index.html#lem-zetanfinal","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 501. The project gives no license, so trough does not reproduce its text.","title":"Final form"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-01/lem-limitterm1","kind":"node","lean":"lem_limitTerm1","parent":5784,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-01/index.html#lem-limitterm1","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 515. The project gives no license, so trough does not reproduce its text.","title":"Limit term"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-01/lem-fracpartbound","kind":"node","lean":"lem_fracPartBound","parent":5784,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-01/index.html#lem-fracpartbound","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 524. The project gives no license, so trough does not reproduce its text.","title":"Frac bound"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-01/lem-integrandbound","kind":"node","lean":"lem_integrandBound","parent":5784,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-01/index.html#lem-integrandbound","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 534. The project gives no license, so trough does not reproduce its text.","title":"Term bound"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-01/lem-integrandboundeps","kind":"node","lean":"lem_integrandBoundeps","parent":5784,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-01/index.html#lem-integrandboundeps","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 544. The project gives no license, so trough does not reproduce its text.","title":"Eps bound"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-01/lem-triangleinequality","kind":"node","lean":"lem_triangleInequality_add","parent":5784,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-01/index.html#lem-triangleinequality","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 554. The project gives no license, so trough does not reproduce its text.","title":"Triangle int"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-01/lem-integralconvergence","kind":"node","lean":"lem_integralConvergence","parent":5784,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-01/index.html#lem-integralconvergence","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 563. The project gives no license, so trough does not reproduce its text.","title":"Integral conv"},{"data_url":"scopes/strongpnt/riemann-zeta-function/zeta-bound/part-02.json","id":"strongpnt/riemann-zeta-function/zeta-bound/part-02","kind":"scope","navigation":"transparent","parent":5783,"result":false,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-02/index.html","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-02","status":"fully_proved","title":"Zeta formula to Upper bound"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-02/lem-zetaformula","kind":"node","lean":"lem_zetaFormula","parent":5805,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-02/index.html#lem-zetaformula","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 574. The project gives no license, so trough does not reproduce its text.","title":"Zeta formula"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-02/lem-zetaanals","kind":"node","lean":"lem_zetaanalS","parent":5805,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-02/index.html#lem-zetaanals","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 584. The project gives no license, so trough does not reproduce its text.","title":"Analytic off"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-02/lem-s-isopen","kind":"node","lean":"lem_S_isOpen","parent":5805,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-02/index.html#lem-s-isopen","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 594. The project gives no license, so trough does not reproduce its text.","title":"S open"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-02/lem-t-isopen","kind":"node","lean":"lem_T_isOpen","parent":5805,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-02/index.html#lem-t-isopen","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 603. The project gives no license, so trough does not reproduce its text.","title":"T open"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-02/lem-t-ispreconnected","kind":"node","lean":"lem_T_isPreconnected","parent":5805,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-02/index.html#lem-t-ispreconnected","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 613. The project gives no license, so trough does not reproduce its text.","title":"T connected"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-02/lem-integralanalytic","kind":"node","lean":"lem_integralAnalytic","parent":5805,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-02/index.html#lem-integralanalytic","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 622. The project gives no license, so trough does not reproduce its text.","title":"Integral analytic"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-02/lem-zetaformulaac","kind":"node","lean":"lem_zetaFormulaAC","parent":5805,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-02/index.html#lem-zetaformulaac","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 629. The project gives no license, so trough does not reproduce its text.","title":"Analytic ext"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-02/lem-div-eq-one-plus-one-div","kind":"node","lean":"lem_div_eq_one_plus_one_div","parent":5805,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-02/index.html#lem-div-eq-one-plus-one-div","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 639. The project gives no license, so trough does not reproduce its text.","title":"Divide split"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-02/lem-zetaanalyticcontinuation","kind":"node","lean":"lem_zetaAnalyticContinuation","parent":5805,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-02/index.html#lem-zetaanalyticcontinuation","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 648. The project gives no license, so trough does not reproduce its text.","title":"Zeta extend"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-02/lem-zetabound1","kind":"node","lean":"lem_zetaBound1","parent":5805,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-02/index.html#lem-zetabound1","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 661. The project gives no license, so trough does not reproduce its text.","title":"First bound"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-02/lem-integralboundvalue","kind":"node","lean":"lem_integralBoundValue","parent":5805,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-02/index.html#lem-integralboundvalue","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 671. The project gives no license, so trough does not reproduce its text.","title":"Integral value"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-02/lem-zetabound2","kind":"node","lean":"lem_zetaBound2","parent":5805,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-02/index.html#lem-zetabound2","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 680. The project gives no license, so trough does not reproduce its text.","title":"Second bound"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-02/lem-soversminus1bound","kind":"node","lean":"lem_sOverSminus1Bound","parent":5805,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-02/index.html#lem-soversminus1bound","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 691. The project gives no license, so trough does not reproduce its text.","title":"Inverse mod"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-02/lem-zetabound3","kind":"node","lean":"lem_zetaBound3","parent":5805,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-02/index.html#lem-zetabound3","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 701. The project gives no license, so trough does not reproduce its text.","title":"Third bound"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-02/lem-sbound","kind":"node","lean":"lem_sBound","parent":5805,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-02/index.html#lem-sbound","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 712. The project gives no license, so trough does not reproduce its text.","title":"s bound"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-02/lem-invresbound","kind":"node","lean":"lem_invReSbound","parent":5805,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-02/index.html#lem-invresbound","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 721. The project gives no license, so trough does not reproduce its text.","title":"Real inv"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-02/lem-invsminus1bound","kind":"node","lean":"lem_invSminus1bound","parent":5805,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-02/index.html#lem-invsminus1bound","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 730. The project gives no license, so trough does not reproduce its text.","title":"Shift bound"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-02/lem-finalboundcombination","kind":"node","lean":"lem_finalBoundCombination","parent":5805,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-02/index.html#lem-finalboundcombination","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 739. The project gives no license, so trough does not reproduce its text.","title":"Combine bounds"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-02/lem-finalalgebra","kind":"node","lean":"lem_finalAlgebra","parent":5805,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-02/index.html#lem-finalalgebra","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 750. The project gives no license, so trough does not reproduce its text.","title":"Algebra step"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-02/lem-zetauppbd","kind":"node","lean":"lem_zetaUppBd","parent":5805,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-02/index.html#lem-zetauppbd","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 759. The project gives no license, so trough does not reproduce its text.","title":"Upper bound"},{"data_url":"scopes/strongpnt/riemann-zeta-function/zeta-bound/part-03.json","id":"strongpnt/riemann-zeta-function/zeta-bound/part-03","kind":"scope","navigation":"transparent","parent":5783,"result":false,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-03/index.html","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-03","status":"fully_proved","title":"Shift calc to Global bound"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-03/lem-zfroms","kind":"node","lean":"lem_zfroms_calc","parent":5826,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-03/index.html#lem-zfroms","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 769. The project gives no license, so trough does not reproduce its text.","title":"Shift calc"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-03/lem-zfroms-conditions","kind":"node","lean":"lem_zfroms_conditions","parent":5826,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-03/index.html#lem-zfroms-conditions","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 778. The project gives no license, so trough does not reproduce its text.","title":"Shift cond"},{"id":"strongpnt/riemann-zeta-function/zeta-bound/part-03/lem-zetauppbound","kind":"node","lean":"lem_zetaUppBound","parent":5826,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-bound/part-03/index.html#lem-zetauppbound","scope":"strongpnt/riemann-zeta-function/zeta-bound/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 788. The project gives no license, so trough does not reproduce its text.","title":"Global bound"},{"data_url":"scopes/strongpnt/riemann-zeta-function/zeta-derivatives.json","id":"strongpnt/riemann-zeta-function/zeta-derivatives","kind":"scope","parent":5749,"result":false,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-derivatives/index.html","scope":"strongpnt/riemann-zeta-function/zeta-derivatives","status":"fully_proved","title":"Zeta derivatives"},{"data_url":"scopes/strongpnt/riemann-zeta-function/zeta-derivatives/part-01.json","id":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01","kind":"scope","navigation":"transparent","parent":5830,"result":false,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-derivatives/part-01/index.html","scope":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01","status":"fully_proved","title":"Diff off pole to Log expansion"},{"id":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01/lem-zetadiffatnot1","kind":"node","lean":"zetadiffAtnot1","parent":5831,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-derivatives/part-01/index.html#lem-zetadiffatnot1","scope":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 802. The project gives no license, so trough does not reproduce its text.","title":"Diff off pole"},{"id":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01/lem-diffatwithinat","kind":"node","lean":"DiffAtWithinAt","parent":5831,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-derivatives/part-01/index.html#lem-diffatwithinat","scope":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 812. The project gives no license, so trough does not reproduce its text.","title":"At to within"},{"id":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01/lem-diffwithinatallon","kind":"node","lean":"DiffWithinAtallOn","parent":5831,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-derivatives/part-01/index.html#lem-diffwithinatallon","scope":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 822. The project gives no license, so trough does not reproduce its text.","title":"Within to on"},{"id":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01/lem-diffaton","kind":"node","lean":"DiffAtOn","parent":5831,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-derivatives/part-01/index.html#lem-diffaton","scope":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 832. The project gives no license, so trough does not reproduce its text.","title":"At to on"},{"id":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01/lem-diffonanalonnhd","kind":"node","lean":"DiffOnanalOnNhd","parent":5831,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-derivatives/part-01/index.html#lem-diffonanalonnhd","scope":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 845. The project gives no license, so trough does not reproduce its text.","title":"Diff to anal"},{"id":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01/lem-diffatallanalonnhd","kind":"node","lean":"DiffAtallanalOnNhd","parent":5831,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-derivatives/part-01/index.html#lem-diffatallanalonnhd","scope":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 855. The project gives no license, so trough does not reproduce its text.","title":"At gives anal"},{"id":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01/lem-zetaanalonnot1","kind":"node","lean":"zetaanalOnnot1","parent":5831,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-derivatives/part-01/index.html#lem-zetaanalonnot1","scope":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 867. The project gives no license, so trough does not reproduce its text.","title":"Analytic off"},{"id":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01/lem-d1cintt-pre","kind":"node","lean":"D1cinTt_pre","parent":5831,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-derivatives/part-01/index.html#lem-d1cintt-pre","scope":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 879. The project gives no license, so trough does not reproduce its text.","title":"Disk avoid"},{"id":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01/lem-d1cintt","kind":"node","lean":"D1cinTt","parent":5831,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-derivatives/part-01/index.html#lem-d1cintt","scope":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 890. The project gives no license, so trough does not reproduce its text.","title":"Disk subset"},{"id":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01/lem-zetaanalond1c","kind":"node","lean":"zetaanalOnD1c_general","parent":5831,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-derivatives/part-01/index.html#lem-zetaanalond1c","scope":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 901. The project gives no license, so trough does not reproduce its text.","title":"Disk analytic"},{"id":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01/lem-sigmageq1","kind":"node","lean":"sigmageq1","parent":5831,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-derivatives/part-01/index.html#lem-sigmageq1","scope":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 913. The project gives no license, so trough does not reproduce its text.","title":"Zero free"},{"id":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01/lem-zetacnot0","kind":"node","lean":"zetacnot0","parent":5831,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-derivatives/part-01/index.html#lem-zetacnot0","scope":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 922. The project gives no license, so trough does not reproduce its text.","title":"Point nonzero"},{"id":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01/lem-fc-analytic-normalized","kind":"node","lean":"fc_analytic_normalized","parent":5831,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-derivatives/part-01/index.html#lem-fc-analytic-normalized","scope":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 934. The project gives no license, so trough does not reproduce its text.","title":"Normalize analytic"},{"id":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01/lem-fc-log-deriv","kind":"node","lean":"fc_log_deriv","parent":5831,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-derivatives/part-01/index.html#lem-fc-log-deriv","scope":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 944. The project gives no license, so trough does not reproduce its text.","title":"Log derivative"},{"id":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01/lem-fc-bound","kind":"node","lean":"fc_bound","parent":5831,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-derivatives/part-01/index.html#lem-fc-bound","scope":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 953. The project gives no license, so trough does not reproduce its text.","title":"Shift bound"},{"id":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01/lem-fc-zeros","kind":"node","lean":"fc_zeros","parent":5831,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-derivatives/part-01/index.html#lem-fc-zeros","scope":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 962. The project gives no license, so trough does not reproduce its text.","title":"Zero shift"},{"id":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01/lem-fc-m-order","kind":"node","lean":"fc_m_order","parent":5831,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-derivatives/part-01/index.html#lem-fc-m-order","scope":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 972. The project gives no license, so trough does not reproduce its text.","title":"Order shift"},{"id":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01/lem-dminusk","kind":"node","lean":"DminusK","parent":5831,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-derivatives/part-01/index.html#lem-dminusk","scope":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 982. The project gives no license, so trough does not reproduce its text.","title":"Disk minus K"},{"id":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01/lem-final-ineq2","kind":"node","lean":"final_ineq2","parent":5831,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-derivatives/part-01/index.html#lem-final-ineq2","scope":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 993. The project gives no license, so trough does not reproduce its text.","title":"Final bound"},{"id":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01/lem-log-deriv-expansion-zeta","kind":"node","lean":"log_Deriv_Expansion_Zeta","parent":5831,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-derivatives/part-01/index.html#lem-log-deriv-expansion-zeta","scope":"strongpnt/riemann-zeta-function/zeta-derivatives/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 1004. The project gives no license, so trough does not reproduce its text.","title":"Log expansion"},{"data_url":"scopes/strongpnt/riemann-zeta-function/zeta-derivatives/part-02.json","id":"strongpnt/riemann-zeta-function/zeta-derivatives/part-02","kind":"scope","navigation":"transparent","parent":5830,"result":false,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-derivatives/part-02/index.html","scope":"strongpnt/riemann-zeta-function/zeta-derivatives/part-02","status":"fully_proved","title":"Lower shift to Final expansion"},{"id":"strongpnt/riemann-zeta-function/zeta-derivatives/part-02/lem-zeta32lower","kind":"node","lean":"zeta32lower","parent":5852,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-derivatives/part-02/index.html#lem-zeta32lower","scope":"strongpnt/riemann-zeta-function/zeta-derivatives/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 1018. The project gives no license, so trough does not reproduce its text.","title":"Lower shift"},{"id":"strongpnt/riemann-zeta-function/zeta-derivatives/part-02/lem-zeta32lower-log","kind":"node","lean":"zeta32lower_log","parent":5852,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-derivatives/part-02/index.html#lem-zeta32lower-log","scope":"strongpnt/riemann-zeta-function/zeta-derivatives/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 1069. The project gives no license, so trough does not reproduce its text.","title":"Log lower"},{"id":"strongpnt/riemann-zeta-function/zeta-derivatives/part-02/lem-zeta32upper-pre","kind":"node","lean":"zeta32upper_pre","parent":5852,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-derivatives/part-02/index.html#lem-zeta32upper-pre","scope":"strongpnt/riemann-zeta-function/zeta-derivatives/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 1085. The project gives no license, so trough does not reproduce its text.","title":"Upper pre"},{"id":"strongpnt/riemann-zeta-function/zeta-derivatives/part-02/lem-zeta32upper","kind":"node","lean":"zeta32upper","parent":5852,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-derivatives/part-02/index.html#lem-zeta32upper","scope":"strongpnt/riemann-zeta-function/zeta-derivatives/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 1096. The project gives no license, so trough does not reproduce its text.","title":"Upper disk"},{"id":"strongpnt/riemann-zeta-function/zeta-derivatives/part-02/lem-zeta1-zeta-expand","kind":"node","lean":"Zeta1_Zeta_Expand","parent":5852,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-derivatives/part-02/index.html#lem-zeta1-zeta-expand","scope":"strongpnt/riemann-zeta-function/zeta-derivatives/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 1154. The project gives no license, so trough does not reproduce its text.","title":"Expand bound"},{"id":"strongpnt/riemann-zeta-function/zeta-derivatives/part-02/lem-zeta1-zeta-expansion","kind":"node","lean":"Zeta1_Zeta_Expansion","parent":5852,"result":true,"route":"../roadmap/strongpnt/riemann-zeta-function/zeta-derivatives/part-02/index.html#lem-zeta1-zeta-expansion","scope":"strongpnt/riemann-zeta-function/zeta-derivatives/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT3_RiemannZeta.tex, line 1167. The project gives no license, so trough does not reproduce its text.","title":"Final expansion"},{"data_url":"scopes/strongpnt/zero-free-region.json","id":"strongpnt/zero-free-region","kind":"scope","parent":5340,"result":false,"route":"../roadmap/strongpnt/zero-free-region/index.html","scope":"strongpnt/zero-free-region","status":"fully_proved","title":"Zero Free Region"},{"data_url":"scopes/strongpnt/zero-free-region/part-01.json","id":"strongpnt/zero-free-region/part-01","kind":"scope","navigation":"transparent","parent":5859,"result":false,"route":"../roadmap/strongpnt/zero-free-region/part-01/index.html","scope":"strongpnt/zero-free-region/part-01","status":"fully_proved","title":"Log derivative to Real delta"},{"id":"strongpnt/zero-free-region/part-01/def-logderivzeta","kind":"node","lean":"logDerivZeta","parent":5860,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-01/index.html#def-logderivzeta","scope":"strongpnt/zero-free-region/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this definition at PNT/PNT4_ZeroFreeRegion.tex, line 3. The project gives no license, so trough does not reproduce its text.","title":"Log derivative"},{"id":"strongpnt/zero-free-region/part-01/def-zeroz","kind":"node","lean":"zeroZ","parent":5860,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-01/index.html#def-zeroz","scope":"strongpnt/zero-free-region/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this definition at PNT/PNT4_ZeroFreeRegion.tex, line 9. The project gives no license, so trough does not reproduce its text.","title":"Zero set"},{"id":"strongpnt/zero-free-region/part-01/def-z-t","kind":"node","lean":"ZetaZerosNearPoint","parent":5860,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-01/index.html#def-z-t","scope":"strongpnt/zero-free-region/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this definition at PNT/PNT4_ZeroFreeRegion.tex, line 15. The project gives no license, so trough does not reproduce its text.","title":"Window zeros"},{"id":"strongpnt/zero-free-region/part-01/lem-ztfinite","kind":"node","lean":"ZetaZerosNearPoint_finite","parent":5860,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-01/index.html#lem-ztfinite","scope":"strongpnt/zero-free-region/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 20. The project gives no license, so trough does not reproduce its text.","title":"Finite set"},{"id":"strongpnt/zero-free-region/part-01/lem-re1zge0","kind":"node","lean":"lem_Re1zge0","parent":5860,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-01/index.html#lem-re1zge0","scope":"strongpnt/zero-free-region/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 32. The project gives no license, so trough does not reproduce its text.","title":"Reciprocal real"},{"id":"strongpnt/zero-free-region/part-01/lem-sigmage1","kind":"node","lean":"lem_sigmage1","parent":5860,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-01/index.html#lem-sigmage1","scope":"strongpnt/zero-free-region/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 40. The project gives no license, so trough does not reproduce its text.","title":"Zero free"},{"id":"strongpnt/zero-free-region/part-01/lem-sigmale1","kind":"node","lean":"lem_sigmale1","parent":5860,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-01/index.html#lem-sigmale1","scope":"strongpnt/zero-free-region/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 50. The project gives no license, so trough does not reproduce its text.","title":"Zero bound"},{"id":"strongpnt/zero-free-region/part-01/lem-sigmale1zt","kind":"node","lean":"lem_sigmale1Zt","parent":5860,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-01/index.html#lem-sigmale1zt","scope":"strongpnt/zero-free-region/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 59. The project gives no license, so trough does not reproduce its text.","title":"Zero bound"},{"id":"strongpnt/zero-free-region/part-01/lem-s-notin-zt","kind":"node","lean":"lem_s_notin_Zt","parent":5860,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-01/index.html#lem-s-notin-zt","scope":"strongpnt/zero-free-region/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 68. The project gives no license, so trough does not reproduce its text.","title":"Outside zeros"},{"id":"strongpnt/zero-free-region/part-01/lem-s-in-d12","kind":"node","lean":"s_in_D12","parent":5860,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-01/index.html#lem-s-in-d12","scope":"strongpnt/zero-free-region/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 76. The project gives no license, so trough does not reproduce its text.","title":"Half disk"},{"id":"strongpnt/zero-free-region/part-01/lem-explicit1deltat","kind":"node","lean":"lem_explicit1deltat","parent":5860,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-01/index.html#lem-explicit1deltat","scope":"strongpnt/zero-free-region/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 84. The project gives no license, so trough does not reproduce its text.","title":"Sum bound"},{"id":"strongpnt/zero-free-region/part-01/lem-explicit1realreal","kind":"node","lean":"lem_explicit1RealReal","parent":5860,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-01/index.html#lem-explicit1realreal","scope":"strongpnt/zero-free-region/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 106. The project gives no license, so trough does not reproduce its text.","title":"Split real"},{"id":"strongpnt/zero-free-region/part-01/lem-explicit2real","kind":"node","lean":"lem_explicit2Real","parent":5860,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-01/index.html#lem-explicit2real","scope":"strongpnt/zero-free-region/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 113. The project gives no license, so trough does not reproduce its text.","title":"Double real"},{"id":"strongpnt/zero-free-region/part-01/lem-realsum","kind":"node","lean":"lem_Realsum","parent":5860,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-01/index.html#lem-realsum","scope":"strongpnt/zero-free-region/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 124. The project gives no license, so trough does not reproduce its text.","title":"Real sum"},{"id":"strongpnt/zero-free-region/part-01/lem-sumrho1","kind":"node","lean":"lem_sumrho1","parent":5860,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-01/index.html#lem-sumrho1","scope":"strongpnt/zero-free-region/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 131. The project gives no license, so trough does not reproduce its text.","title":"Sum split"},{"id":"strongpnt/zero-free-region/part-01/lem-sumrho2","kind":"node","lean":"lem_sumrho2","parent":5860,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-01/index.html#lem-sumrho2","scope":"strongpnt/zero-free-region/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 142. The project gives no license, so trough does not reproduce its text.","title":"Sum split"},{"id":"strongpnt/zero-free-region/part-01/lem-1deltatrho1","kind":"node","lean":"lem_1deltatrho1","parent":5860,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-01/index.html#lem-1deltatrho1","scope":"strongpnt/zero-free-region/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 153. The project gives no license, so trough does not reproduce its text.","title":"Difference form"},{"id":"strongpnt/zero-free-region/part-01/lem-re1deltatrho1","kind":"node","lean":"lem_Re1deltatrho1","parent":5860,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-01/index.html#lem-re1deltatrho1","scope":"strongpnt/zero-free-region/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 164. The project gives no license, so trough does not reproduce its text.","title":"Real part"},{"id":"strongpnt/zero-free-region/part-01/lem-re1delta1","kind":"node","lean":"lem_Re1delta1","parent":5860,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-01/index.html#lem-re1delta1","scope":"strongpnt/zero-free-region/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 175. The project gives no license, so trough does not reproduce its text.","title":"Delta bound"},{"id":"strongpnt/zero-free-region/part-01/lem-re1deltatge","kind":"node","lean":"lem_Re1deltatge","parent":5860,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-01/index.html#lem-re1deltatge","scope":"strongpnt/zero-free-region/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 187. The project gives no license, so trough does not reproduce its text.","title":"Real delta"},{"data_url":"scopes/strongpnt/zero-free-region/part-02.json","id":"strongpnt/zero-free-region/part-02","kind":"scope","navigation":"transparent","parent":5859,"result":false,"route":"../roadmap/strongpnt/zero-free-region/part-02/index.html","scope":"strongpnt/zero-free-region/part-02","status":"fully_proved","title":"Positive real to Real inverse"},{"id":"strongpnt/zero-free-region/part-02/lem-re1deltatneq0","kind":"node","lean":"lem_Re1deltatneq0","parent":5881,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-02/index.html#lem-re1deltatneq0","scope":"strongpnt/zero-free-region/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 198. The project gives no license, so trough does not reproduce its text.","title":"Positive real"},{"id":"strongpnt/zero-free-region/part-02/lem-re1deltatge0","kind":"node","lean":"lem_Re1deltatge0","parent":5881,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-02/index.html#lem-re1deltatge0","scope":"strongpnt/zero-free-region/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 209. The project gives no license, so trough does not reproduce its text.","title":"Inverse real"},{"id":"strongpnt/zero-free-region/part-02/lem-re1deltatge0m","kind":"node","lean":"lem_Re1deltatge0m","parent":5881,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-02/index.html#lem-re1deltatge0m","scope":"strongpnt/zero-free-region/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 220. The project gives no license, so trough does not reproduce its text.","title":"Scaled real"},{"id":"strongpnt/zero-free-region/part-02/lem-re1delta2tge0","kind":"node","lean":"lem_Re1delta2tge0","parent":5881,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-02/index.html#lem-re1delta2tge0","scope":"strongpnt/zero-free-region/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 230. The project gives no license, so trough does not reproduce its text.","title":"Double real"},{"id":"strongpnt/zero-free-region/part-02/lem-sumrho2ge","kind":"node","lean":"lem_sumrho2ge","parent":5881,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-02/index.html#lem-sumrho2ge","scope":"strongpnt/zero-free-region/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 241. The project gives no license, so trough does not reproduce its text.","title":"Sum nonneg"},{"id":"strongpnt/zero-free-region/part-02/lem-sumrho2ge02","kind":"node","lean":"lem_sumrho2ge02","parent":5881,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-02/index.html#lem-sumrho2ge02","scope":"strongpnt/zero-free-region/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 252. The project gives no license, so trough does not reproduce its text.","title":"Real nonneg"},{"id":"strongpnt/zero-free-region/part-02/lem-explicit2real2","kind":"node","lean":"lem_explicit2Real2","parent":5881,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-02/index.html#lem-explicit2real2","scope":"strongpnt/zero-free-region/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 263. The project gives no license, so trough does not reproduce its text.","title":"Double bound"},{"id":"strongpnt/zero-free-region/part-02/lem-log2olog","kind":"node","lean":"lem_log2Olog","parent":5881,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-02/index.html#lem-log2olog","scope":"strongpnt/zero-free-region/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 272. The project gives no license, so trough does not reproduce its text.","title":"Log compare"},{"id":"strongpnt/zero-free-region/part-02/lem-w2t","kind":"node","lean":"lem_w2t","parent":5881,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-02/index.html#lem-w2t","scope":"strongpnt/zero-free-region/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 281. The project gives no license, so trough does not reproduce its text.","title":"Trivial bound"},{"id":"strongpnt/zero-free-region/part-02/lem-log2olog2","kind":"node","lean":"lem_log2Olog2","parent":5881,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-02/index.html#lem-log2olog2","scope":"strongpnt/zero-free-region/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 288. The project gives no license, so trough does not reproduce its text.","title":"Log compare"},{"id":"strongpnt/zero-free-region/part-02/lem-z2bound","kind":"node","lean":"lem_Z2bound","parent":5881,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-02/index.html#lem-z2bound","scope":"strongpnt/zero-free-region/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 296. The project gives no license, so trough does not reproduce its text.","title":"Shift bound"},{"id":"strongpnt/zero-free-region/part-02/lem-z1split","kind":"node","lean":"lem_Z1split","parent":5881,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-02/index.html#lem-z1split","scope":"strongpnt/zero-free-region/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 306. The project gives no license, so trough does not reproduce its text.","title":"Split sum"},{"id":"strongpnt/zero-free-region/part-02/lem-z1splitge","kind":"node","lean":"lem_Z1splitge","parent":5881,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-02/index.html#lem-z1splitge","scope":"strongpnt/zero-free-region/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 318. The project gives no license, so trough does not reproduce its text.","title":"Split bound"},{"id":"strongpnt/zero-free-region/part-02/lem-1deltatrho0","kind":"node","lean":"lem_1deltatrho0","parent":5881,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-02/index.html#lem-1deltatrho0","scope":"strongpnt/zero-free-region/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 331. The project gives no license, so trough does not reproduce its text.","title":"Difference real"},{"id":"strongpnt/zero-free-region/part-02/lem-1delsigreal","kind":"node","lean":"lem_1delsigReal","parent":5881,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-02/index.html#lem-1delsigreal","scope":"strongpnt/zero-free-region/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 342. The project gives no license, so trough does not reproduce its text.","title":"Real inverse"},{"id":"strongpnt/zero-free-region/part-02/lem-11delsiginr","kind":"node","lean":"lem_11delsiginR","parent":5881,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-02/index.html#lem-11delsiginr","scope":"strongpnt/zero-free-region/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 354. The project gives no license, so trough does not reproduce its text.","title":"Inverse real"},{"id":"strongpnt/zero-free-region/part-02/lem-11delsiginr2","kind":"node","lean":"lem_11delsiginR2","parent":5881,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-02/index.html#lem-11delsiginr2","scope":"strongpnt/zero-free-region/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 362. The project gives no license, so trough does not reproduce its text.","title":"Inverse real"},{"id":"strongpnt/zero-free-region/part-02/lem-rereal","kind":"node","lean":"lem_ReReal","parent":5881,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-02/index.html#lem-rereal","scope":"strongpnt/zero-free-region/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 371. The project gives no license, so trough does not reproduce its text.","title":"Real part"},{"id":"strongpnt/zero-free-region/part-02/lem-1delsigreal2","kind":"node","lean":"lem_1delsigReal2","parent":5881,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-02/index.html#lem-1delsigreal2","scope":"strongpnt/zero-free-region/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 378. The project gives no license, so trough does not reproduce its text.","title":"Real inverse"},{"id":"strongpnt/zero-free-region/part-02/lem-1delsigreal3","kind":"node","lean":"lem_re_inv_one_plus_delta_minus_rho_real","parent":5881,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-02/index.html#lem-1delsigreal3","scope":"strongpnt/zero-free-region/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 389. The project gives no license, so trough does not reproduce its text.","title":"Real inverse"},{"data_url":"scopes/strongpnt/zero-free-region/part-03.json","id":"strongpnt/zero-free-region/part-03","kind":"scope","navigation":"transparent","parent":5859,"result":false,"route":"../roadmap/strongpnt/zero-free-region/part-03/index.html","scope":"strongpnt/zero-free-region/part-03","status":"fully_proved","title":"Sum bound to Real sum"},{"id":"strongpnt/zero-free-region/part-03/lem-z1splitge2","kind":"node","lean":"lem_Z1splitge2","parent":5902,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-03/index.html#lem-z1splitge2","scope":"strongpnt/zero-free-region/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 400. The project gives no license, so trough does not reproduce its text.","title":"Sum bound"},{"id":"strongpnt/zero-free-region/part-03/lem-z1splitge3","kind":"node","lean":"lem_Z1splitge3","parent":5902,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-03/index.html#lem-z1splitge3","scope":"strongpnt/zero-free-region/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 411. The project gives no license, so trough does not reproduce its text.","title":"Real bound"},{"id":"strongpnt/zero-free-region/part-03/lem-rhoinzt","kind":"node","lean":"lem_rho_in_Zt","parent":5902,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-03/index.html#lem-rhoinzt","scope":"strongpnt/zero-free-region/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 422. The project gives no license, so trough does not reproduce its text.","title":"In set"},{"id":"strongpnt/zero-free-region/part-03/lem-z1bound","kind":"node","lean":"Z1bound","parent":5902,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-03/index.html#lem-z1bound","scope":"strongpnt/zero-free-region/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 434. The project gives no license, so trough does not reproduce its text.","title":"Zeta bound"},{"id":"strongpnt/zero-free-region/part-03/lem-z0bound","kind":"node","lean":"Z0bound","parent":5902,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-03/index.html#lem-z0bound","scope":"strongpnt/zero-free-region/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 445. The project gives no license, so trough does not reproduce its text.","title":"At one"},{"id":"strongpnt/zero-free-region/part-03/lem-z0boundre","kind":"node","lean":"Z0boundRe","parent":5902,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-03/index.html#lem-z0boundre","scope":"strongpnt/zero-free-region/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 453. The project gives no license, so trough does not reproduce its text.","title":"Real one"},{"id":"strongpnt/zero-free-region/part-03/lem-z0bound-const","kind":"node","lean":"Z0bound_const","parent":5902,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-03/index.html#lem-z0bound-const","scope":"strongpnt/zero-free-region/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 462. The project gives no license, so trough does not reproduce its text.","title":"One bound"},{"id":"strongpnt/zero-free-region/part-03/lem-z0boundre-const","kind":"node","lean":"Z0boundRe_const","parent":5902,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-03/index.html#lem-z0boundre-const","scope":"strongpnt/zero-free-region/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 472. The project gives no license, so trough does not reproduce its text.","title":"Real bound"},{"id":"strongpnt/zero-free-region/part-03/lem-z0boundre-const2","kind":"node","lean":"Z0boundRe_const2","parent":5902,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-03/index.html#lem-z0boundre-const2","scope":"strongpnt/zero-free-region/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 482. The project gives no license, so trough does not reproduce its text.","title":"Real sum"},{"id":"strongpnt/zero-free-region/part-03/lem-z0boundre-const3","kind":"node","lean":"Z0boundRe_const3","parent":5902,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-03/index.html#lem-z0boundre-const3","scope":"strongpnt/zero-free-region/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 492. The project gives no license, so trough does not reproduce its text.","title":"Real diff"},{"id":"strongpnt/zero-free-region/part-03/lem-z341bounds-const","kind":"node","lean":"Z341bounds_const","parent":5902,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-03/index.html#lem-z341bounds-const","scope":"strongpnt/zero-free-region/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 502. The project gives no license, so trough does not reproduce its text.","title":"Combined bound"},{"id":"strongpnt/zero-free-region/part-03/lem-zeta1zetaseries","kind":"node","lean":"zeta1zetaseries","parent":5902,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-03/index.html#lem-zeta1zetaseries","scope":"strongpnt/zero-free-region/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 517. The project gives no license, so trough does not reproduce its text.","title":"Series form"},{"id":"strongpnt/zero-free-region/part-03/lem-zeta1zetaseriesxy","kind":"node","lean":"zeta1zetaseriesxy","parent":5902,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-03/index.html#lem-zeta1zetaseriesxy","scope":"strongpnt/zero-free-region/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 529. The project gives no license, so trough does not reproduce its text.","title":"Series form"},{"id":"strongpnt/zero-free-region/part-03/lem-zconverges1","kind":"node","lean":"Zconverges1","parent":5902,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-03/index.html#lem-zconverges1","scope":"strongpnt/zero-free-region/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 541. The project gives no license, so trough does not reproduce its text.","title":"Converges"},{"id":"strongpnt/zero-free-region/part-03/lem-rezconverges1","kind":"node","lean":"ReZconverges1","parent":5902,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-03/index.html#lem-rezconverges1","scope":"strongpnt/zero-free-region/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 549. The project gives no license, so trough does not reproduce its text.","title":"Real converges"},{"id":"strongpnt/zero-free-region/part-03/lem-nxy","kind":"node","lean":"lem_nxy","parent":5902,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-03/index.html#lem-nxy","scope":"strongpnt/zero-free-region/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 557. The project gives no license, so trough does not reproduce its text.","title":"Exponent split"},{"id":"strongpnt/zero-free-region/part-03/lem-zeta1zetaseriesxy2","kind":"node","lean":"lem_zeta1zetaseriesxy2","parent":5902,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-03/index.html#lem-zeta1zetaseriesxy2","scope":"strongpnt/zero-free-region/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 565. The project gives no license, so trough does not reproduce its text.","title":"Series split"},{"id":"strongpnt/zero-free-region/part-03/lem-zseriesconverges1","kind":"node","lean":"Zseriesconverges1","parent":5902,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-03/index.html#lem-zseriesconverges1","scope":"strongpnt/zero-free-region/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 576. The project gives no license, so trough does not reproduce its text.","title":"Series converges"},{"id":"strongpnt/zero-free-region/part-03/lem-realnx","kind":"node","lean":"lem_realnx","parent":5902,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-03/index.html#lem-realnx","scope":"strongpnt/zero-free-region/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 584. The project gives no license, so trough does not reproduce its text.","title":"Real terms"},{"id":"strongpnt/zero-free-region/part-03/lem-sumreallambda","kind":"node","lean":"sumRealLambda","parent":5902,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-03/index.html#lem-sumreallambda","scope":"strongpnt/zero-free-region/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 593. The project gives no license, so trough does not reproduce its text.","title":"Real sum"},{"data_url":"scopes/strongpnt/zero-free-region/part-04.json","id":"strongpnt/zero-free-region/part-04","kind":"scope","navigation":"transparent","parent":5859,"result":false,"route":"../roadmap/strongpnt/zero-free-region/part-04/index.html","scope":"strongpnt/zero-free-region/part-04","status":"fully_proved","title":"Series real to Inequality"},{"id":"strongpnt/zero-free-region/part-04/lem-sumrealz","kind":"node","lean":"lem_sumRealZ","parent":5923,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-04/index.html#lem-sumrealz","scope":"strongpnt/zero-free-region/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 601. The project gives no license, so trough does not reproduce its text.","title":"Series real"},{"id":"strongpnt/zero-free-region/part-04/lem-reallambdaxy","kind":"node","lean":"RealLambdaxy","parent":5923,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-04/index.html#lem-reallambdaxy","scope":"strongpnt/zero-free-region/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 609. The project gives no license, so trough does not reproduce its text.","title":"Real factor"},{"id":"strongpnt/zero-free-region/part-04/lem-rezseriesren","kind":"node","lean":"ReZseriesRen","parent":5923,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-04/index.html#lem-rezseriesren","scope":"strongpnt/zero-free-region/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 619. The project gives no license, so trough does not reproduce its text.","title":"Real series"},{"id":"strongpnt/zero-free-region/part-04/lem-rezeta1zetaseries","kind":"node","lean":"Rezeta1zetaseries","parent":5923,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-04/index.html#lem-rezeta1zetaseries","scope":"strongpnt/zero-free-region/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 629. The project gives no license, so trough does not reproduce its text.","title":"Cos form"},{"id":"strongpnt/zero-free-region/part-04/lem-rezetaseries","kind":"node","lean":"Rezetaseries_convergence","parent":5923,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-04/index.html#lem-rezetaseries","scope":"strongpnt/zero-free-region/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 637. The project gives no license, so trough does not reproduce its text.","title":"Cos series"},{"id":"strongpnt/zero-free-region/part-04/lem-rezetaseries2t","kind":"node","lean":"Rezetaseries2t","parent":5923,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-04/index.html#lem-rezetaseries2t","scope":"strongpnt/zero-free-region/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 645. The project gives no license, so trough does not reproduce its text.","title":"Double cos"},{"id":"strongpnt/zero-free-region/part-04/lem-cost0","kind":"node","lean":"lem_cost0","parent":5923,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-04/index.html#lem-cost0","scope":"strongpnt/zero-free-region/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 653. The project gives no license, so trough does not reproduce its text.","title":"Zero cos"},{"id":"strongpnt/zero-free-region/part-04/lem-rezetaseries0","kind":"node","lean":"Rezetaseries0","parent":5923,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-04/index.html#lem-rezetaseries0","scope":"strongpnt/zero-free-region/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 661. The project gives no license, so trough does not reproduce its text.","title":"Zero series"},{"id":"strongpnt/zero-free-region/part-04/lem-rezeta1zetaseries1","kind":"node","lean":"Rezeta1zetaseries1","parent":5923,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-04/index.html#lem-rezeta1zetaseries1","scope":"strongpnt/zero-free-region/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 669. The project gives no license, so trough does not reproduce its text.","title":"Delta series"},{"id":"strongpnt/zero-free-region/part-04/lem-rezeta1zetaseries2","kind":"node","lean":"Rezeta1zetaseries2","parent":5923,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-04/index.html#lem-rezeta1zetaseries2","scope":"strongpnt/zero-free-region/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 680. The project gives no license, so trough does not reproduce its text.","title":"Delta double"},{"id":"strongpnt/zero-free-region/part-04/lem-rezeta1zetaseries0","kind":"node","lean":"Rezeta1zetaseries0","parent":5923,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-04/index.html#lem-rezeta1zetaseries0","scope":"strongpnt/zero-free-region/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 693. The project gives no license, so trough does not reproduce its text.","title":"Delta zero"},{"id":"strongpnt/zero-free-region/part-04/lem-341zseries","kind":"node","lean":"Z341series","parent":5923,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-04/index.html#lem-341zseries","scope":"strongpnt/zero-free-region/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 705. The project gives no license, so trough does not reproduce its text.","title":"341 series"},{"id":"strongpnt/zero-free-region/part-04/lem-341seriesconv","kind":"node","lean":"lem341seriesConv","parent":5923,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-04/index.html#lem-341seriesconv","scope":"strongpnt/zero-free-region/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 719. The project gives no license, so trough does not reproduce its text.","title":"Sum converges"},{"id":"strongpnt/zero-free-region/part-04/lem-341series","kind":"node","lean":"lem341series","parent":5923,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-04/index.html#lem-341series","scope":"strongpnt/zero-free-region/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 734. The project gives no license, so trough does not reproduce its text.","title":"Factor form"},{"id":"strongpnt/zero-free-region/part-04/lem-341seriesconverge","kind":"node","lean":"lem_341seriesConverge","parent":5923,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-04/index.html#lem-341seriesconverge","scope":"strongpnt/zero-free-region/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 748. The project gives no license, so trough does not reproduce its text.","title":"Factor conv"},{"id":"strongpnt/zero-free-region/part-04/lem-341series2","kind":"node","lean":"lem_341series2","parent":5923,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-04/index.html#lem-341series2","scope":"strongpnt/zero-free-region/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 760. The project gives no license, so trough does not reproduce its text.","title":"Series equal"},{"id":"strongpnt/zero-free-region/part-04/lem-lambpostriglogn","kind":"node","lean":"lem_Lambda_pos_trig_sum","parent":5923,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-04/index.html#lem-lambpostriglogn","scope":"strongpnt/zero-free-region/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 773. The project gives no license, so trough does not reproduce its text.","title":"Term nonneg"},{"id":"strongpnt/zero-free-region/part-04/lem-seriespos","kind":"node","lean":"lem_seriespos","parent":5923,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-04/index.html#lem-seriespos","scope":"strongpnt/zero-free-region/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 782. The project gives no license, so trough does not reproduce its text.","title":"Series nonneg"},{"id":"strongpnt/zero-free-region/part-04/lem-341zpos","kind":"node","lean":"Z341pos","parent":5923,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-04/index.html#lem-341zpos","scope":"strongpnt/zero-free-region/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 791. The project gives no license, so trough does not reproduce its text.","title":"Positive sum"},{"id":"strongpnt/zero-free-region/part-04/lem-341tsc","kind":"node","lean":"lem341tsC","parent":5923,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-04/index.html#lem-341tsc","scope":"strongpnt/zero-free-region/part-04","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 803. The project gives no license, so trough does not reproduce its text.","title":"Inequality"},{"data_url":"scopes/strongpnt/zero-free-region/part-05.json","id":"strongpnt/zero-free-region/part-05","kind":"scope","navigation":"transparent","parent":5859,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rearranged to Bound on "],["math","\\(\\zeta'/\\zeta\\)"]],"source":"Rearranged to Bound on $\\zeta'/\\zeta$"},"result":false,"route":"../roadmap/strongpnt/zero-free-region/part-05/index.html","scope":"strongpnt/zero-free-region/part-05","status":"fully_proved","title":"Rearranged to Bound on $\\zeta'/\\zeta$"},{"id":"strongpnt/zero-free-region/part-05/lem-341tsc2","kind":"node","lean":"lem341tsC2","parent":5944,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/index.html#lem-341tsc2","scope":"strongpnt/zero-free-region/part-05","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 814. The project gives no license, so trough does not reproduce its text.","title":"Rearranged"},{"id":"strongpnt/zero-free-region/part-05/lem-341tsc3","kind":"node","lean":"lem341tsC3","parent":5944,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/index.html#lem-341tsc3","scope":"strongpnt/zero-free-region/part-05","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 825. The project gives no license, so trough does not reproduce its text.","title":"Final bound"},{"id":"strongpnt/zero-free-region/part-05/lem-zerofree","kind":"node","lean":"zerofree","parent":5944,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/index.html#lem-zerofree","scope":"strongpnt/zero-free-region/part-05","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 837. The project gives no license, so trough does not reproduce its text.","title":"Zero free"},{"data_url":"scopes/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta.json","id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta","kind":"scope","parent":5944,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bound on "],["math","\\(\\zeta'/\\zeta\\)"]],"source":"Bound on $\\zeta'/\\zeta$"},"result":false,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/index.html","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta","status":"fully_proved","title":"Bound on $\\zeta'/\\zeta$"},{"data_url":"scopes/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01.json","id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01","kind":"scope","navigation":"transparent","parent":5948,"result":false,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/index.html","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01","status":"fully_proved","title":"Delta zeros to Inverse gap"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/def-yt","kind":"node","lean":"Yt","parent":5949,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/index.html#def-yt","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this definition at PNT/PNT4_ZeroFreeRegion.tex, line 849. The project gives no license, so trough does not reproduce its text.","title":"Delta zeros"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/def-deltaz","kind":"node","lean":"deltaz, deltaz_t","parent":5949,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/index.html#def-deltaz","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this definition at PNT/PNT4_ZeroFreeRegion.tex, line 856. The project gives no license, so trough does not reproduce its text.","title":"Delta def"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/lem-delta19","kind":"node","lean":"lem_delta19","parent":5949,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/index.html#lem-delta19","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 860. The project gives no license, so trough does not reproduce its text.","title":"Delta range"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/lem-zfrdelta","kind":"node","lean":"lem_ZFRdelta","parent":5949,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/index.html#lem-zfrdelta","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 869. The project gives no license, so trough does not reproduce its text.","title":"Zero free"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/lem-zfrind","kind":"node","lean":"lem_ZFRinD","parent":5949,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/index.html#lem-zfrind","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 877. The project gives no license, so trough does not reproduce its text.","title":"Disk inclusion"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/lem-zfrnotk","kind":"node","lean":"lem_ZFRnotK","parent":5949,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/index.html#lem-zfrnotk","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 886. The project gives no license, so trough does not reproduce its text.","title":"Not zero"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/lem-zeta-expansion-zfr","kind":"node","lean":"lem_Zeta_Expansion_ZFR","parent":5949,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/index.html#lem-zeta-expansion-zfr","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 895. The project gives no license, so trough does not reproduce its text.","title":"Expansion"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/lem-abszrhorere","kind":"node","lean":"lem_abszrhoReRe","parent":5949,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/index.html#lem-abszrhorere","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 907. The project gives no license, so trough does not reproduce its text.","title":"Distance real"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/lem-rerhotodeltarho","kind":"node","lean":"lem_Rerhotodeltarho","parent":5949,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/index.html#lem-rerhotodeltarho","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 916. The project gives no license, so trough does not reproduce its text.","title":"Real bound"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/lem-dimt2d","kind":"node","lean":"lem_DImt2d","parent":5949,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/index.html#lem-dimt2d","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 925. The project gives no license, so trough does not reproduce its text.","title":"Imag bound"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/lem-dimt2","kind":"node","lean":"lem_DIMt2","parent":5949,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/index.html#lem-dimt2","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 933. The project gives no license, so trough does not reproduce its text.","title":"Imag growth"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/lem-dlogimlog","kind":"node","lean":"lem_DlogImlog","parent":5949,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/index.html#lem-dlogimlog","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 942. The project gives no license, so trough does not reproduce its text.","title":"Log bound"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/lem-d1logtlog","kind":"node","lean":"lem_D1logtlog","parent":5949,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/index.html#lem-d1logtlog","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 951. The project gives no license, so trough does not reproduce its text.","title":"Log compare"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/lem-ddt2dz","kind":"node","lean":"lem_Ddt2dz","parent":5949,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/index.html#lem-ddt2dz","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 960. The project gives no license, so trough does not reproduce its text.","title":"Delta compare"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/lem-deltarhotodeltat","kind":"node","lean":"lem_deltarhotodeltat","parent":5949,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/index.html#lem-deltarhotodeltat","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 969. The project gives no license, so trough does not reproduce its text.","title":"Delta bound"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/lem-rerhotodeltat","kind":"node","lean":"lem_Rerhotodeltat","parent":5949,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/index.html#lem-rerhotodeltat","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 978. The project gives no license, so trough does not reproduce its text.","title":"Real bound"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/lem-rezrerho","kind":"node","lean":"lem_RezRerho","parent":5949,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/index.html#lem-rezrerho","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 987. The project gives no license, so trough does not reproduce its text.","title":"Gap bound"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/lem-abszrhodelta","kind":"node","lean":"lem_abszrhodelta","parent":5949,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/index.html#lem-abszrhodelta","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 996. The project gives no license, so trough does not reproduce its text.","title":"Gap size"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/lem-abszrhodeltanot0","kind":"node","lean":"lem_abszrhodeltanot0","parent":5949,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/index.html#lem-abszrhodeltanot0","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1005. The project gives no license, so trough does not reproduce its text.","title":"Nonzero gap"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/lem-1abszrho","kind":"node","lean":"lem_1abszrho","parent":5949,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01/index.html#lem-1abszrho","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-01","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1014. The project gives no license, so trough does not reproduce its text.","title":"Inverse gap"},{"data_url":"scopes/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02.json","id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02","kind":"scope","navigation":"transparent","parent":5948,"result":false,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/index.html","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02","status":"fully_proved","title":"Order nat to Real diff"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/lem-m-rho-zeta-nat","kind":"node","lean":"lem_m_rho_zeta_nat","parent":5970,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/index.html#lem-m-rho-zeta-nat","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1023. The project gives no license, so trough does not reproduce its text.","title":"Order nat"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/lem-finitekzeta","kind":"node","lean":"lem_finiteKzeta","parent":5970,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/index.html#lem-finitekzeta","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1033. The project gives no license, so trough does not reproduce its text.","title":"Finite set"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/lem-triangle-zfr","kind":"node","lean":"lem_triangle_ZFR","parent":5970,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/index.html#lem-triangle-zfr","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1043. The project gives no license, so trough does not reproduce its text.","title":"Triangle"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/lem-zeta-triangle-zfr","kind":"node","lean":"lem_Zeta_Triangle_ZFR","parent":5970,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/index.html#lem-zeta-triangle-zfr","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1057. The project gives no license, so trough does not reproduce its text.","title":"Triangle"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/lem-sumk1abs","kind":"node","lean":"lem_sumK1abs","parent":5970,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/index.html#lem-sumk1abs","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1069. The project gives no license, so trough does not reproduce its text.","title":"Sum bound"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/lem-sum-m-rho-bound-c","kind":"node","lean":"lem_sum_m_rho_bound_c","parent":5970,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/index.html#lem-sum-m-rho-bound-c","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1079. The project gives no license, so trough does not reproduce its text.","title":"Order bound"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/lem-sum-m-rho-zeta","kind":"node","lean":"lem_sum_m_rho_zeta","parent":5970,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/index.html#lem-sum-m-rho-zeta","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1089. The project gives no license, so trough does not reproduce its text.","title":"Order sum"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/lem-sumkdeltatlogt","kind":"node","lean":"lem_sumKdeltatlogt","parent":5970,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/index.html#lem-sumkdeltatlogt","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1102. The project gives no license, so trough does not reproduce its text.","title":"Sum bound"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/lem-sumklogt2","kind":"node","lean":"lem_sumKlogt2","parent":5970,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/index.html#lem-sumklogt2","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1113. The project gives no license, so trough does not reproduce its text.","title":"Sum bound"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/lem-logderivzetalogt0","kind":"node","lean":"lem_logDerivZetalogt0","parent":5970,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/index.html#lem-logderivzetalogt0","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1123. The project gives no license, so trough does not reproduce its text.","title":"Log bound"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/lem-logderivzetalogt2","kind":"node","lean":"lem_logDerivZetalogt2","parent":5970,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/index.html#lem-logderivzetalogt2","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1133. The project gives no license, so trough does not reproduce its text.","title":"Log bound"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/lem-rhodre4","kind":"node","lean":"lem_rhoDRe4","parent":5970,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/index.html#lem-rhodre4","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1144. The project gives no license, so trough does not reproduce its text.","title":"Real bound"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/lem-drez6dz","kind":"node","lean":"lem_DRez6dz","parent":5970,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/index.html#lem-drez6dz","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1153. The project gives no license, so trough does not reproduce its text.","title":"Real bound"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/lem-yind","kind":"node","lean":"lem_YinD","parent":5970,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/index.html#lem-yind","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1162. The project gives no license, so trough does not reproduce its text.","title":"In disk"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/lem-rhoyzero","kind":"node","lean":"lem_rhoYzero","parent":5970,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/index.html#lem-rhoyzero","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1171. The project gives no license, so trough does not reproduce its text.","title":"Zero set"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/lem-absreabs","kind":"node","lean":"lem_absReabs","parent":5970,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/index.html#lem-absreabs","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1182. The project gives no license, so trough does not reproduce its text.","title":"Real abs"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/lem-zre","kind":"node","lean":"lem_zRe","parent":5970,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/index.html#lem-zre","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1192. The project gives no license, so trough does not reproduce its text.","title":"Real diff"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/lem-zre2","kind":"node","lean":"lem_zRe2","parent":5970,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/index.html#lem-zre2","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1203. The project gives no license, so trough does not reproduce its text.","title":"Real diff"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/lem-rezit","kind":"node","lean":"lem_Rezit","parent":5970,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/index.html#lem-rezit","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1215. The project gives no license, so trough does not reproduce its text.","title":"Real diff"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/lem-zre3","kind":"node","lean":"lem_zRe3","parent":5970,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02/index.html#lem-zre3","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-02","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1224. The project gives no license, so trough does not reproduce its text.","title":"Real diff"},{"data_url":"scopes/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03.json","id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03","kind":"scope","navigation":"transparent","parent":5948,"result":false,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03/index.html","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03","status":"fully_proved","title":"Neg bound to Uniform bound on the\u2026"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03/lem-negleabs","kind":"node","lean":"lem_negleabs","parent":5991,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03/index.html#lem-negleabs","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1235. The project gives no license, so trough does not reproduce its text.","title":"Neg bound"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03/lem-absrez1d","kind":"node","lean":"lem_absrez1d","parent":5991,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03/index.html#lem-absrez1d","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1246. The project gives no license, so trough does not reproduce its text.","title":"Real bound"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03/lem-absrez1d2","kind":"node","lean":"lem_absrez1d2","parent":5991,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03/index.html#lem-absrez1d2","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1257. The project gives no license, so trough does not reproduce its text.","title":"Real bound"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03/lem-absrez1d3","kind":"node","lean":"lem_absrez1d3","parent":5991,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03/index.html#lem-absrez1d3","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1267. The project gives no license, so trough does not reproduce its text.","title":"Real bound"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03/lem-zre4","kind":"node","lean":"lem_zRe4","parent":5991,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03/index.html#lem-zre4","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1278. The project gives no license, so trough does not reproduce its text.","title":"Real bound"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03/lem-kzetaempty","kind":"node","lean":"lem_Kzetaempty","parent":5991,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03/index.html#lem-kzetaempty","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1290. The project gives no license, so trough does not reproduce its text.","title":"Empty set"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03/lem-sumempty","kind":"node","lean":"lem_sumempty","parent":5991,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03/index.html#lem-sumempty","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1301. The project gives no license, so trough does not reproduce its text.","title":"Empty sum"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03/lem-ksumempty","kind":"node","lean":"lem_Ksumempty","parent":5991,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03/index.html#lem-ksumempty","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1314. The project gives no license, so trough does not reproduce its text.","title":"Zero sum"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03/lem-zetacenterbd","kind":"node","lean":"lem_zetacenterbd","parent":5991,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03/index.html#lem-zetacenterbd","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1328. The project gives no license, so trough does not reproduce its text.","title":"Center bound"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03/lem-logderivzetalogt32","kind":"node","lean":"lem_logDerivZetalogt32","parent":5991,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03/index.html#lem-logderivzetalogt32","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1354. The project gives no license, so trough does not reproduce its text.","title":"Log bound"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03/thm-final-result","kind":"node","lean":"thm_final_result","parent":5991,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bound on "],["math","\\(\\zeta'/\\zeta\\)"]],"source":"Bound on $\\zeta'/\\zeta$"},"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03/index.html#thm-final-result","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this theorem at PNT/PNT4_ZeroFreeRegion.tex, line 1365. The project gives no license, so trough does not reproduce its text.","title":"Bound on $\\zeta'/\\zeta$"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03/lem-zetazerofree","kind":"node","lean":"ZetaZeroFree_p","parent":5991,"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03/index.html#lem-zetazerofree","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1376. The project gives no license, so trough does not reproduce its text.","title":"Zero-free region near 1"},{"id":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03/lem-logderivzetabndunif","kind":"node","lean":"LogDerivZetaBndUnif2","parent":5991,"reading_title":{"format":"inline-parts-v1","parts":[["text","Uniform bound on the logarithmic derivative of "],["math","\\(\\zeta\\)"]],"source":"Uniform bound on the logarithmic derivative of $\\zeta$"},"result":true,"route":"../roadmap/strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03/index.html#lem-logderivzetabndunif","scope":"strongpnt/zero-free-region/part-05/bound-on-zeta-zeta/part-03","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this lemma at PNT/PNT4_ZeroFreeRegion.tex, line 1388. The project gives no license, so trough does not reproduce its text.","title":"Uniform bound on the logarithmic derivative of $\\zeta$"},{"data_url":"scopes/strongpnt/strong-pnt.json","id":"strongpnt/strong-pnt","kind":"scope","parent":5340,"result":false,"route":"../roadmap/strongpnt/strong-pnt/index.html","scope":"strongpnt/strong-pnt","status":"fully_proved","title":"Strong PNT"},{"id":"strongpnt/strong-pnt/strong-pnt","kind":"node","lean":"Strong_PNT","parent":6005,"result":true,"route":"../roadmap/strongpnt/strong-pnt/index.html#strong-pnt","scope":"strongpnt/strong-pnt","status":"fully_proved","summary":"The blueprint of Strong PNT (Math Inc) states this theorem at PNT/PNT5_Strong.tex, line 3. The project gives no license, so trough does not reproduce its text.","title":"Strong PNT"},{"data_url":"chapters/formal-conjectures.json","id":"formal-conjectures","kind":"scope","parent":0,"result":false,"route":"../roadmap/formal-conjectures/index.html","scope":"formal-conjectures","status":"can_prove","summary":"Formal Conjectures, from Google DeepMind, states open and solved mathematical problems in Lean, with a site at https://google-deepmind.github.io/formal-conjectures/. This chapter maps it as read at commit 2055e5ad (2026-11-06), Lean toolchain v4.33.1: a part per source folder of `FormalConjectures/`, an article per problem file and, in it, a card per statement.","title":"Formal Conjectures"},{"data_url":"scopes/formal-conjectures/by-subject.json","id":"formal-conjectures/by-subject","kind":"scope","parent":6007,"result":false,"route":"../roadmap/formal-conjectures/by-subject/index.html","scope":"formal-conjectures/by-subject","status":"planned","summary":"The problems by subject: a page per top-level class of the Mathematics Subject Classification that a statement's `AMS` attribute names, titled as Formal Conjectures names it.","title":"By subject"},{"data_url":"scopes/formal-conjectures/by-subject/part-01.json","id":"formal-conjectures/by-subject/part-01","kind":"scope","navigation":"transparent","parent":6008,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/index.html","scope":"formal-conjectures/by-subject/part-01","status":"planned","title":"Mathematical logic and\u2026 to Several complex variables and\u2026"},{"data_url":"scopes/formal-conjectures/by-subject/part-01/03-mathematical-logic-and-foundations.json","id":"formal-conjectures/by-subject/part-01/03-mathematical-logic-and-foundations","kind":"scope","parent":6009,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/03-mathematical-logic-and-foundations/index.html","scope":"formal-conjectures/by-subject/part-01/03-mathematical-logic-and-foundations","status":"planned","summary":"The 30 problems with a statement whose `AMS` attribute names 03, Mathematical logic and foundations, in the Mathematics Subject Classification. A problem with statements in several subjects is listed under each; its article sits in its source folder's part.","title":"Mathematical logic and foundations"},{"id":"formal-conjectures/by-subject/part-01/03-mathematical-logic-and-foundations/part-01","kind":"node","navigation":"transparent","parent":6010,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/03-mathematical-logic-and-foundations/part-01/index.html","scope":"formal-conjectures/by-subject/part-01/03-mathematical-logic-and-foundations","status":"can_state","title":"Erd\u0151s Problem 70 to Erd\u0151s Problem 1177"},{"id":"formal-conjectures/by-subject/part-01/03-mathematical-logic-and-foundations/part-02","kind":"node","navigation":"transparent","parent":6010,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/03-mathematical-logic-and-foundations/part-02/index.html","scope":"formal-conjectures/by-subject/part-01/03-mathematical-logic-and-foundations","status":"can_state","title":"Erd\u0151s Problem 1219 to Tao's Optimization Constant 14\u2026"},{"data_url":"scopes/formal-conjectures/by-subject/part-01/05-combinatorics.json","id":"formal-conjectures/by-subject/part-01/05-combinatorics","kind":"scope","parent":6009,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/05-combinatorics/index.html","scope":"formal-conjectures/by-subject/part-01/05-combinatorics","status":"planned","summary":"The 609 problems with a statement whose `AMS` attribute names 05, Combinatorics, in the Mathematics Subject Classification. A problem with statements in several subjects is listed under each; its article sits in its source folder's part.","title":"Combinatorics"},{"data_url":"scopes/formal-conjectures/by-subject/part-01/05-combinatorics/part-01.json","id":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01","kind":"scope","navigation":"transparent","parent":6013,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/05-combinatorics/part-01/index.html","scope":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01","status":"planned","title":"Erd\u0151s Problem 1 to Erd\u0151s Problem 1159"},{"id":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-01","kind":"node","navigation":"transparent","parent":6014,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-01/index.html","scope":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01","status":"can_state","title":"Erd\u0151s Problem 1 to Erd\u0151s Problem 59"},{"id":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-02","kind":"node","navigation":"transparent","parent":6014,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-02/index.html","scope":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01","status":"can_state","title":"Erd\u0151s Problem 60 to Erd\u0151s Problem 87"},{"id":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-03","kind":"node","navigation":"transparent","parent":6014,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-03/index.html","scope":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01","status":"can_state","title":"Erd\u0151s Problem 88 to Erd\u0151s Problem 147"},{"id":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-04","kind":"node","navigation":"transparent","parent":6014,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-04/index.html","scope":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01","status":"can_state","title":"Erd\u0151s Problem 150 to Erd\u0151s Problem 172"},{"id":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-05","kind":"node","navigation":"transparent","parent":6014,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-05/index.html","scope":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01","status":"can_state","title":"Erd\u0151s Problem 173 to Erd\u0151s Problem 202"},{"id":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-06","kind":"node","navigation":"transparent","parent":6014,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-06/index.html","scope":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01","status":"can_state","title":"Erd\u0151s Problem 203 to Erd\u0151s Problem 307"},{"id":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-07","kind":"node","navigation":"transparent","parent":6014,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-07/index.html","scope":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01","status":"can_state","title":"Erd\u0151s Problem 312 to Erd\u0151s Problem 447"},{"id":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-08","kind":"node","navigation":"transparent","parent":6014,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-08/index.html","scope":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01","status":"can_state","title":"Erd\u0151s Problem 476 to Erd\u0151s Problem 546"},{"id":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-09","kind":"node","navigation":"transparent","parent":6014,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-09/index.html","scope":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01","status":"can_state","title":"Erd\u0151s Problem 547 to Erd\u0151s Problem 569"},{"id":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-10","kind":"node","navigation":"transparent","parent":6014,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-10/index.html","scope":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01","status":"can_state","title":"Erd\u0151s Problem 570 to Erd\u0151s Problem 602"},{"id":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-11","kind":"node","navigation":"transparent","parent":6014,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-11/index.html","scope":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01","status":"can_state","title":"Erd\u0151s Problem 603 to Erd\u0151s Problem 640"},{"id":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-12","kind":"node","navigation":"transparent","parent":6014,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-12/index.html","scope":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01","status":"can_state","title":"Erd\u0151s Problem 641 to Erd\u0151s Problem 703"},{"id":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-13","kind":"node","navigation":"transparent","parent":6014,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-13/index.html","scope":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01","status":"can_state","title":"Erd\u0151s Problem 704 to Erd\u0151s Problem 734"},{"id":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-14","kind":"node","navigation":"transparent","parent":6014,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-14/index.html","scope":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01","status":"can_state","title":"Erd\u0151s Problem 737 to Erd\u0151s Problem 774"},{"id":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-15","kind":"node","navigation":"transparent","parent":6014,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-15/index.html","scope":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01","status":"can_state","title":"Erd\u0151s Problem 775 to Erd\u0151s Problem 816"},{"id":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-16","kind":"node","navigation":"transparent","parent":6014,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-16/index.html","scope":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01","status":"can_state","title":"Erd\u0151s Problem 817 to Erd\u0151s Problem 877"},{"id":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-17","kind":"node","navigation":"transparent","parent":6014,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-17/index.html","scope":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01","status":"can_state","title":"Erd\u0151s Problem 881 to Erd\u0151s Problem 921"},{"id":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-18","kind":"node","navigation":"transparent","parent":6014,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-18/index.html","scope":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01","status":"can_state","title":"Erd\u0151s Problem 922 to Erd\u0151s Problem 1017"},{"id":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-19","kind":"node","navigation":"transparent","parent":6014,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-19/index.html","scope":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01","status":"can_state","title":"Erd\u0151s Problem 1020 to Erd\u0151s Problem 1068"},{"id":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-20","kind":"node","navigation":"transparent","parent":6014,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/05-combinatorics/part-01/part-20/index.html","scope":"formal-conjectures/by-subject/part-01/05-combinatorics/part-01","status":"can_state","title":"Erd\u0151s Problem 1069 to Erd\u0151s Problem 1159"},{"data_url":"scopes/formal-conjectures/by-subject/part-01/05-combinatorics/part-02.json","id":"formal-conjectures/by-subject/part-01/05-combinatorics/part-02","kind":"scope","navigation":"transparent","parent":6013,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/05-combinatorics/part-02/index.html","scope":"formal-conjectures/by-subject/part-01/05-combinatorics/part-02","status":"planned","title":"Erd\u0151s Problem 1161 to Open Quantum Problem 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"],["math","\\(n\\)"],["text"," into "],["math","\\(x + y\\)"],["text","\u2026 to Lam--Litt conjecture"]],"source":"Partitions of $n$ into $x + y$\u2026 to Lam--Litt conjecture"},"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/11-number-theory/part-03/index.html","scope":"formal-conjectures/by-subject/part-01/11-number-theory/part-03","status":"planned","title":"Partitions of $n$ into $x + y$\u2026 to Lam--Litt conjecture"},{"id":"formal-conjectures/by-subject/part-01/11-number-theory/part-03/part-01","kind":"node","navigation":"transparent","parent":6092,"reading_title":{"format":"inline-parts-v1","parts":[["text","Partitions of "],["math","\\(n\\)"],["text"," into "],["math","\\(x + y\\)"],["text","\u2026 to Number of primes "],["math","\\(p < n\\)"],["text"," such\u2026"]],"source":"Partitions of $n$ into $x + y$\u2026 to Number of primes $p < n$ such\u2026"},"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/11-number-theory/part-03/part-01/index.html","scope":"formal-conjectures/by-subject/part-01/11-number-theory/part-03","status":"can_state","title":"Partitions of $n$ into $x + y$\u2026 to Number of primes $p < n$ such\u2026"},{"id":"formal-conjectures/by-subject/part-01/11-number-theory/part-03/part-02","kind":"node","navigation":"transparent","parent":6092,"reading_title":{"format":"inline-parts-v1","parts":[["text","Values "],["math","\\(m = \\sigma(k)\\)"],["text"," ($0 < m\u2026 to Number of integers $0 < k \\le\u2026"]],"source":"Values $m = \\sigma(k)$ ($0 < m\u2026 to Number of integers $0 < k \\le\u2026"},"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/11-number-theory/part-03/part-02/index.html","scope":"formal-conjectures/by-subject/part-01/11-number-theory/part-03","status":"can_state","title":"Values $m = \\sigma(k)$ ($0 < m\u2026 to Number of integers $0 < k \\le\u2026"},{"id":"formal-conjectures/by-subject/part-01/11-number-theory/part-03/part-03","kind":"node","navigation":"transparent","parent":6092,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-01/11-number-theory/part-03/part-03/index.html","scope":"formal-conjectures/by-subject/part-01/11-number-theory/part-03","status":"can_state","title":"Primitive roots of the form k\u00b2\u2026 to Asymptotics of an inner sum\u2026"},{"id":"formal-conjectures/by-subject/part-01/11-number-theory/part-03/part-04","kind":"node","navigation":"transparent","parent":6092,"reading_title":{"format":"inline-parts-v1","parts":[["text","Catalan numbers modulo "],["math","\\(6\\)"],["text"," to Sum of a triangular number, a\u2026"]],"source":"Catalan numbers modulo $6$ to Sum of a triangular number, 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A problem with statements in several subjects is listed under each; its article sits in its source folder's part.","title":"Partial differential equations"},{"id":"formal-conjectures/by-subject/part-02/37-dynamical-systems-and-ergodic-theory","kind":"node","parent":6132,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-02/37-dynamical-systems-and-ergodic-theory/index.html","scope":"formal-conjectures/by-subject/part-02","status":"can_state","summary":"The 9 problems with a statement whose `AMS` attribute names 37, Dynamical systems and ergodic theory, in the Mathematics Subject Classification. A problem with statements in several subjects is listed under each; its article sits in its source folder's part.","title":"Dynamical systems and ergodic theory"},{"id":"formal-conjectures/by-subject/part-02/39-difference-and-functional-equations","kind":"node","parent":6132,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-02/39-difference-and-functional-equations/index.html","scope":"formal-conjectures/by-subject/part-02","status":"can_state","summary":"The 1 problem with a statement whose `AMS` attribute names 39, Difference and functional equations, in the Mathematics Subject Classification. A problem with statements in several subjects is listed under each; its article sits in its source folder's part.","title":"Difference and functional equations"},{"id":"formal-conjectures/by-subject/part-02/40-sequences-series-summability","kind":"node","parent":6132,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-02/40-sequences-series-summability/index.html","scope":"formal-conjectures/by-subject/part-02","status":"can_state","summary":"The 10 problems with a statement whose `AMS` attribute names 40, Sequences, series, summability, in the Mathematics Subject Classification. A problem with statements in several subjects is listed under each; its article sits in its source folder's part.","title":"Sequences, series, summability"},{"id":"formal-conjectures/by-subject/part-02/41-approximations-and-expansions","kind":"node","parent":6132,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-02/41-approximations-and-expansions/index.html","scope":"formal-conjectures/by-subject/part-02","status":"can_state","summary":"The 4 problems with a statement whose `AMS` attribute names 41, Approximations and expansions, in the Mathematics Subject Classification. A problem with statements in several subjects is listed under each; its article sits in its source folder's part.","title":"Approximations and expansions"},{"id":"formal-conjectures/by-subject/part-02/42-harmonic-analysis-on-euclidean-spaces","kind":"node","parent":6132,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-02/42-harmonic-analysis-on-euclidean-spaces/index.html","scope":"formal-conjectures/by-subject/part-02","status":"can_state","summary":"The 15 problems with a statement whose `AMS` attribute names 42, Harmonic analysis on Euclidean spaces, in the Mathematics Subject Classification. A problem with statements in several subjects is listed under each; its article sits in its source folder's part.","title":"Harmonic analysis on Euclidean spaces"},{"id":"formal-conjectures/by-subject/part-02/46-functional-analysis","kind":"node","parent":6132,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-02/46-functional-analysis/index.html","scope":"formal-conjectures/by-subject/part-02","status":"can_state","summary":"The 6 problems with a statement whose `AMS` attribute names 46, Functional analysis, in the Mathematics Subject Classification. A problem with statements in several subjects is listed under each; its article sits in its source folder's part.","title":"Functional analysis"},{"id":"formal-conjectures/by-subject/part-02/47-operator-theory","kind":"node","parent":6132,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-02/47-operator-theory/index.html","scope":"formal-conjectures/by-subject/part-02","status":"can_state","summary":"The 4 problems with a statement whose `AMS` attribute names 47, Operator theory, in the Mathematics Subject Classification. A problem with statements in several subjects is listed under each; its article sits in its source folder's part.","title":"Operator theory"},{"id":"formal-conjectures/by-subject/part-02/49-calculus-of-variations-and-optimal-control-optim","kind":"node","parent":6132,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-02/49-calculus-of-variations-and-optimal-control-optim/index.html","scope":"formal-conjectures/by-subject/part-02","status":"can_state","summary":"The 1 problem with a statement whose `AMS` attribute names 49, Calculus of variations and optimal control; optimization, in the Mathematics Subject Classification. A problem with statements in several subjects is listed under each; its article sits in its source folder's part.","title":"Calculus of variations and optimal control; optimization"},{"data_url":"scopes/formal-conjectures/by-subject/part-02/51-geometry.json","id":"formal-conjectures/by-subject/part-02/51-geometry","kind":"scope","parent":6132,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-02/51-geometry/index.html","scope":"formal-conjectures/by-subject/part-02/51-geometry","status":"planned","summary":"The 30 problems with a statement whose `AMS` attribute names 51, Geometry, in the Mathematics Subject Classification. A problem with statements in several subjects is listed under each; its article sits in its source folder's part.","title":"Geometry"},{"id":"formal-conjectures/by-subject/part-02/51-geometry/part-01","kind":"node","navigation":"transparent","parent":6143,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-02/51-geometry/part-01/index.html","scope":"formal-conjectures/by-subject/part-02/51-geometry","status":"can_state","title":"Erd\u0151s Problem 160 to Integer part of area of a\u2026"},{"id":"formal-conjectures/by-subject/part-02/51-geometry/part-02","kind":"node","navigation":"transparent","parent":6143,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-02/51-geometry/part-02/index.html","scope":"formal-conjectures/by-subject/part-02/51-geometry","status":"can_state","title":"The Erd\u0151s\u2013Ulam problem to Mathoverflow 34145"},{"data_url":"scopes/formal-conjectures/by-subject/part-02/52-convex-and-discrete-geometry.json","id":"formal-conjectures/by-subject/part-02/52-convex-and-discrete-geometry","kind":"scope","parent":6132,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-02/52-convex-and-discrete-geometry/index.html","scope":"formal-conjectures/by-subject/part-02/52-convex-and-discrete-geometry","status":"planned","summary":"The 96 problems with a statement whose `AMS` attribute names 52, Convex and discrete geometry, in the Mathematics Subject Classification. A problem with statements in several subjects is listed under each; its article sits in its source folder's part.","title":"Convex and discrete geometry"},{"id":"formal-conjectures/by-subject/part-02/52-convex-and-discrete-geometry/part-01","kind":"node","navigation":"transparent","parent":6146,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-02/52-convex-and-discrete-geometry/part-01/index.html","scope":"formal-conjectures/by-subject/part-02/52-convex-and-discrete-geometry","status":"can_state","title":"Erd\u0151s Problem 89 to Erd\u0151s Problem 209"},{"id":"formal-conjectures/by-subject/part-02/52-convex-and-discrete-geometry/part-02","kind":"node","navigation":"transparent","parent":6146,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-02/52-convex-and-discrete-geometry/part-02/index.html","scope":"formal-conjectures/by-subject/part-02/52-convex-and-discrete-geometry","status":"can_state","title":"Erd\u0151s Problem 211 to Erd\u0151s Problem 654"},{"id":"formal-conjectures/by-subject/part-02/52-convex-and-discrete-geometry/part-03","kind":"node","navigation":"transparent","parent":6146,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-02/52-convex-and-discrete-geometry/part-03/index.html","scope":"formal-conjectures/by-subject/part-02/52-convex-and-discrete-geometry","status":"can_state","title":"Erd\u0151s Problem 655 to Erd\u0151s Problem 957"},{"id":"formal-conjectures/by-subject/part-02/52-convex-and-discrete-geometry/part-04","kind":"node","navigation":"transparent","parent":6146,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-02/52-convex-and-discrete-geometry/part-04/index.html","scope":"formal-conjectures/by-subject/part-02/52-convex-and-discrete-geometry","status":"can_state","title":"Erd\u0151s Problem 958 to Kakeya problem"},{"id":"formal-conjectures/by-subject/part-02/52-convex-and-discrete-geometry/part-05","kind":"node","navigation":"transparent","parent":6146,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-02/52-convex-and-discrete-geometry/part-05/index.html","scope":"formal-conjectures/by-subject/part-02/52-convex-and-discrete-geometry","status":"can_state","title":"Moser's Worm to VC\u2099 dimension of convex sets in\u2026"},{"id":"formal-conjectures/by-subject/part-02/53-differential-geometry","kind":"node","parent":6132,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-02/53-differential-geometry/index.html","scope":"formal-conjectures/by-subject/part-02","status":"can_state","summary":"The 1 problem with a statement whose `AMS` attribute names 53, Differential geometry, in the Mathematics Subject Classification. A problem with statements in several subjects is listed under each; its article sits in its source folder's part.","title":"Differential geometry"},{"id":"formal-conjectures/by-subject/part-02/54-general-topology","kind":"node","parent":6132,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-02/54-general-topology/index.html","scope":"formal-conjectures/by-subject/part-02","status":"can_state","summary":"The 10 problems with a statement whose `AMS` attribute names 54, General topology, in the Mathematics Subject Classification. A problem with statements in several subjects is listed under each; its article sits in its source folder's part.","title":"General topology"},{"id":"formal-conjectures/by-subject/part-02/57-manifolds-and-cell-complexes","kind":"node","parent":6132,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-02/57-manifolds-and-cell-complexes/index.html","scope":"formal-conjectures/by-subject/part-02","status":"can_state","summary":"The 3 problems with a statement whose `AMS` attribute names 57, Manifolds and cell complexes, in the Mathematics Subject Classification. A problem with statements in several subjects is listed under each; its article sits in its source folder's part.","title":"Manifolds and cell complexes"},{"id":"formal-conjectures/by-subject/part-02/58-global-analysis-analysis-on-manifolds","kind":"node","parent":6132,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-02/58-global-analysis-analysis-on-manifolds/index.html","scope":"formal-conjectures/by-subject/part-02","status":"can_state","summary":"The 1 problem with a statement whose `AMS` attribute names 58, Global analysis, analysis on manifolds, in the Mathematics Subject Classification. A problem with statements in several subjects is listed under each; its article sits in its source folder's part.","title":"Global analysis, analysis on manifolds"},{"data_url":"scopes/formal-conjectures/by-subject/part-02/60-probability-theory-and-stochastic-processes.json","id":"formal-conjectures/by-subject/part-02/60-probability-theory-and-stochastic-processes","kind":"scope","parent":6132,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-02/60-probability-theory-and-stochastic-processes/index.html","scope":"formal-conjectures/by-subject/part-02/60-probability-theory-and-stochastic-processes","status":"planned","summary":"The 23 problems with a statement whose `AMS` attribute names 60, Probability theory and stochastic processes, in the Mathematics Subject Classification. A problem with statements in several subjects is listed under each; its article sits in its source folder's part.","title":"Probability theory and stochastic processes"},{"id":"formal-conjectures/by-subject/part-02/60-probability-theory-and-stochastic-processes/part-01","kind":"node","navigation":"transparent","parent":6156,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-02/60-probability-theory-and-stochastic-processes/part-01/index.html","scope":"formal-conjectures/by-subject/part-02/60-probability-theory-and-stochastic-processes","status":"can_state","title":"Erd\u0151s Problem 395 to Central Binomial Tail Bounds\u2026"},{"id":"formal-conjectures/by-subject/part-02/60-probability-theory-and-stochastic-processes/part-02","kind":"node","navigation":"transparent","parent":6156,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-02/60-probability-theory-and-stochastic-processes/part-02/index.html","scope":"formal-conjectures/by-subject/part-02/60-probability-theory-and-stochastic-processes","status":"can_state","title":"Convexity of Fisher information\u2026 to Mathoverflow 339137"},{"id":"formal-conjectures/by-subject/part-02/68-computer-science","kind":"node","parent":6132,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-02/68-computer-science/index.html","scope":"formal-conjectures/by-subject/part-02","status":"can_state","summary":"The 17 problems with a statement whose `AMS` attribute names 68, Computer science, in the Mathematics Subject Classification. A problem with statements in several subjects is listed under each; its article sits in its source folder's part.","title":"Computer science"},{"id":"formal-conjectures/by-subject/part-02/70-mechanics-of-particles-and-systems","kind":"node","parent":6132,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-02/70-mechanics-of-particles-and-systems/index.html","scope":"formal-conjectures/by-subject/part-02","status":"can_state","summary":"The 1 problem with a statement whose `AMS` attribute names 70, Mechanics of particles and systems, in the Mathematics Subject Classification. A problem with statements in several subjects is listed under each; its article sits in its source folder's part.","title":"Mechanics of particles and systems"},{"id":"formal-conjectures/by-subject/part-02/78-optics-electromagnetic-theory","kind":"node","parent":6132,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-02/78-optics-electromagnetic-theory/index.html","scope":"formal-conjectures/by-subject/part-02","status":"can_state","summary":"The 1 problem with a statement whose `AMS` attribute names 78, Optics, electromagnetic theory, in the Mathematics Subject Classification. A problem with statements in several subjects is listed under each; its article sits in its source folder's part.","title":"Optics, electromagnetic theory"},{"data_url":"scopes/formal-conjectures/by-subject/part-03.json","id":"formal-conjectures/by-subject/part-03","kind":"scope","navigation":"transparent","parent":6008,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-03/index.html","scope":"formal-conjectures/by-subject/part-03","status":"planned","title":"Quantum theory to Information and communication\u2026"},{"id":"formal-conjectures/by-subject/part-03/81-quantum-theory","kind":"node","parent":6162,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-03/81-quantum-theory/index.html","scope":"formal-conjectures/by-subject/part-03","status":"can_state","summary":"The 4 problems with a statement whose `AMS` attribute names 81, Quantum theory, in the Mathematics Subject Classification. A problem with statements in several subjects is listed under each; its article sits in its source folder's part.","title":"Quantum theory"},{"id":"formal-conjectures/by-subject/part-03/91-game-theory-economics-social-and-behavioral-scie","kind":"node","parent":6162,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-03/91-game-theory-economics-social-and-behavioral-scie/index.html","scope":"formal-conjectures/by-subject/part-03","status":"can_state","summary":"The 3 problems with a statement whose `AMS` attribute names 91, Game theory, economics, social and behavioral sciences, in the Mathematics Subject Classification. A problem with statements in several subjects is listed under each; its article sits in its source folder's part.","title":"Game theory, economics, social and behavioral sciences"},{"id":"formal-conjectures/by-subject/part-03/94-information-and-communication-circuits","kind":"node","parent":6162,"result":false,"route":"../roadmap/formal-conjectures/by-subject/part-03/94-information-and-communication-circuits/index.html","scope":"formal-conjectures/by-subject/part-03","status":"can_state","summary":"The 6 problems with a statement whose `AMS` attribute names 94, Information and communication, circuits, in the Mathematics Subject Classification. A problem with statements in several subjects is listed under each; its article sits in its source folder's part.","title":"Information and communication, circuits"},{"data_url":"scopes/formal-conjectures/fc100-subsets.json","id":"formal-conjectures/fc100-subsets","kind":"scope","parent":6007,"result":false,"route":"../roadmap/formal-conjectures/fc100-subsets/index.html","scope":"formal-conjectures/fc100-subsets","status":"planned","summary":"Formal Conjectures' 4 benchmark lists in FormalConjectures/Subsets, a page each.","title":"FC100 subsets"},{"data_url":"scopes/formal-conjectures/fc100-subsets/fc100openset1.json","id":"formal-conjectures/fc100-subsets/fc100openset1","kind":"scope","parent":6166,"result":false,"route":"../roadmap/formal-conjectures/fc100-subsets/fc100openset1/index.html","scope":"formal-conjectures/fc100-subsets/fc100openset1","status":"planned","summary":"*Quoted from FormalConjectures/Subsets/FC100OpenSet1.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0.*","title":"FC100OpenSet1"},{"id":"formal-conjectures/fc100-subsets/fc100openset1/part-01","kind":"node","navigation":"transparent","parent":6167,"result":false,"route":"../roadmap/formal-conjectures/fc100-subsets/fc100openset1/part-01/index.html","scope":"formal-conjectures/fc100-subsets/fc100openset1","status":"can_state","title":"Entries 1 to 20"},{"id":"formal-conjectures/fc100-subsets/fc100openset1/part-02","kind":"node","navigation":"transparent","parent":6167,"result":false,"route":"../roadmap/formal-conjectures/fc100-subsets/fc100openset1/part-02/index.html","scope":"formal-conjectures/fc100-subsets/fc100openset1","status":"can_state","title":"Entries 21 to 40"},{"id":"formal-conjectures/fc100-subsets/fc100openset1/part-03","kind":"node","navigation":"transparent","parent":6167,"result":false,"route":"../roadmap/formal-conjectures/fc100-subsets/fc100openset1/part-03/index.html","scope":"formal-conjectures/fc100-subsets/fc100openset1","status":"can_state","title":"Entries 41 to 60"},{"id":"formal-conjectures/fc100-subsets/fc100openset1/part-04","kind":"node","navigation":"transparent","parent":6167,"result":false,"route":"../roadmap/formal-conjectures/fc100-subsets/fc100openset1/part-04/index.html","scope":"formal-conjectures/fc100-subsets/fc100openset1","status":"can_state","title":"Entries 61 to 80"},{"id":"formal-conjectures/fc100-subsets/fc100openset1/part-05","kind":"node","navigation":"transparent","parent":6167,"result":false,"route":"../roadmap/formal-conjectures/fc100-subsets/fc100openset1/part-05/index.html","scope":"formal-conjectures/fc100-subsets/fc100openset1","status":"can_state","title":"Entries 81 to 100"},{"data_url":"scopes/formal-conjectures/fc100-subsets/fc100openset2.json","id":"formal-conjectures/fc100-subsets/fc100openset2","kind":"scope","parent":6166,"result":false,"route":"../roadmap/formal-conjectures/fc100-subsets/fc100openset2/index.html","scope":"formal-conjectures/fc100-subsets/fc100openset2","status":"planned","summary":"*Quoted from FormalConjectures/Subsets/FC100OpenSet2.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0.*","title":"FC100OpenSet2"},{"id":"formal-conjectures/fc100-subsets/fc100openset2/part-01","kind":"node","navigation":"transparent","parent":6173,"result":false,"route":"../roadmap/formal-conjectures/fc100-subsets/fc100openset2/part-01/index.html","scope":"formal-conjectures/fc100-subsets/fc100openset2","status":"can_state","title":"Entries 1 to 20"},{"id":"formal-conjectures/fc100-subsets/fc100openset2/part-02","kind":"node","navigation":"transparent","parent":6173,"result":false,"route":"../roadmap/formal-conjectures/fc100-subsets/fc100openset2/part-02/index.html","scope":"formal-conjectures/fc100-subsets/fc100openset2","status":"can_state","title":"Entries 21 to 40"},{"id":"formal-conjectures/fc100-subsets/fc100openset2/part-03","kind":"node","navigation":"transparent","parent":6173,"result":false,"route":"../roadmap/formal-conjectures/fc100-subsets/fc100openset2/part-03/index.html","scope":"formal-conjectures/fc100-subsets/fc100openset2","status":"can_state","title":"Entries 41 to 60"},{"id":"formal-conjectures/fc100-subsets/fc100openset2/part-04","kind":"node","navigation":"transparent","parent":6173,"result":false,"route":"../roadmap/formal-conjectures/fc100-subsets/fc100openset2/part-04/index.html","scope":"formal-conjectures/fc100-subsets/fc100openset2","status":"can_state","title":"Entries 61 to 80"},{"id":"formal-conjectures/fc100-subsets/fc100openset2/part-05","kind":"node","navigation":"transparent","parent":6173,"result":false,"route":"../roadmap/formal-conjectures/fc100-subsets/fc100openset2/part-05/index.html","scope":"formal-conjectures/fc100-subsets/fc100openset2","status":"can_state","title":"Entries 81 to 100"},{"data_url":"scopes/formal-conjectures/fc100-subsets/fc100solvedset1.json","id":"formal-conjectures/fc100-subsets/fc100solvedset1","kind":"scope","parent":6166,"result":false,"route":"../roadmap/formal-conjectures/fc100-subsets/fc100solvedset1/index.html","scope":"formal-conjectures/fc100-subsets/fc100solvedset1","status":"planned","summary":"*Quoted from FormalConjectures/Subsets/FC100SolvedSet1.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0.*","title":"FC100SolvedSet1"},{"id":"formal-conjectures/fc100-subsets/fc100solvedset1/part-01","kind":"node","navigation":"transparent","parent":6179,"result":false,"route":"../roadmap/formal-conjectures/fc100-subsets/fc100solvedset1/part-01/index.html","scope":"formal-conjectures/fc100-subsets/fc100solvedset1","status":"can_state","title":"Entries 2 to 22"},{"id":"formal-conjectures/fc100-subsets/fc100solvedset1/part-02","kind":"node","navigation":"transparent","parent":6179,"result":false,"route":"../roadmap/formal-conjectures/fc100-subsets/fc100solvedset1/part-02/index.html","scope":"formal-conjectures/fc100-subsets/fc100solvedset1","status":"can_state","title":"Entries 23 to 43"},{"id":"formal-conjectures/fc100-subsets/fc100solvedset1/part-03","kind":"node","navigation":"transparent","parent":6179,"result":false,"route":"../roadmap/formal-conjectures/fc100-subsets/fc100solvedset1/part-03/index.html","scope":"formal-conjectures/fc100-subsets/fc100solvedset1","status":"can_state","title":"Entries 44 to 63"},{"id":"formal-conjectures/fc100-subsets/fc100solvedset1/part-04","kind":"node","navigation":"transparent","parent":6179,"result":false,"route":"../roadmap/formal-conjectures/fc100-subsets/fc100solvedset1/part-04/index.html","scope":"formal-conjectures/fc100-subsets/fc100solvedset1","status":"can_state","title":"Entries 64 to 84"},{"id":"formal-conjectures/fc100-subsets/fc100solvedset1/part-05","kind":"node","navigation":"transparent","parent":6179,"result":false,"route":"../roadmap/formal-conjectures/fc100-subsets/fc100solvedset1/part-05/index.html","scope":"formal-conjectures/fc100-subsets/fc100solvedset1","status":"can_state","title":"Entries 85 to 100"},{"data_url":"scopes/formal-conjectures/fc100-subsets/fc100solvedset2.json","id":"formal-conjectures/fc100-subsets/fc100solvedset2","kind":"scope","parent":6166,"result":false,"route":"../roadmap/formal-conjectures/fc100-subsets/fc100solvedset2/index.html","scope":"formal-conjectures/fc100-subsets/fc100solvedset2","status":"planned","summary":"*Quoted from FormalConjectures/Subsets/FC100SolvedSet2.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0.*","title":"FC100SolvedSet2"},{"id":"formal-conjectures/fc100-subsets/fc100solvedset2/part-01","kind":"node","navigation":"transparent","parent":6185,"result":false,"route":"../roadmap/formal-conjectures/fc100-subsets/fc100solvedset2/part-01/index.html","scope":"formal-conjectures/fc100-subsets/fc100solvedset2","status":"can_state","title":"Entries 1 to 20"},{"id":"formal-conjectures/fc100-subsets/fc100solvedset2/part-02","kind":"node","navigation":"transparent","parent":6185,"result":false,"route":"../roadmap/formal-conjectures/fc100-subsets/fc100solvedset2/part-02/index.html","scope":"formal-conjectures/fc100-subsets/fc100solvedset2","status":"can_state","title":"Entries 21 to 40"},{"id":"formal-conjectures/fc100-subsets/fc100solvedset2/part-03","kind":"node","navigation":"transparent","parent":6185,"result":false,"route":"../roadmap/formal-conjectures/fc100-subsets/fc100solvedset2/part-03/index.html","scope":"formal-conjectures/fc100-subsets/fc100solvedset2","status":"can_state","title":"Entries 41 to 60"},{"id":"formal-conjectures/fc100-subsets/fc100solvedset2/part-04","kind":"node","navigation":"transparent","parent":6185,"result":false,"route":"../roadmap/formal-conjectures/fc100-subsets/fc100solvedset2/part-04/index.html","scope":"formal-conjectures/fc100-subsets/fc100solvedset2","status":"can_state","title":"Entries 61 to 80"},{"id":"formal-conjectures/fc100-subsets/fc100solvedset2/part-05","kind":"node","navigation":"transparent","parent":6185,"result":false,"route":"../roadmap/formal-conjectures/fc100-subsets/fc100solvedset2/part-05/index.html","scope":"formal-conjectures/fc100-subsets/fc100solvedset2","status":"can_state","title":"Entries 81 to 100"},{"data_url":"scopes/formal-conjectures/erdos-problems.json","id":"formal-conjectures/erdos-problems","kind":"scope","parent":6007,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/index.html","scope":"formal-conjectures/erdos-problems","status":"can_prove","summary":"The 1,011 problem files in FormalConjectures/ErdosProblems that state a research or textbook statement, at commit 2055e5ad, one article each, in sections by hundreds of problem number.","title":"Erd\u0151s problems"},{"data_url":"scopes/formal-conjectures/erdos-problems/1-99.json","id":"formal-conjectures/erdos-problems/1-99","kind":"scope","parent":6191,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/index.html","scope":"formal-conjectures/erdos-problems/1-99","status":"can_prove","summary":"The Erd\u0151s problems numbered 1 to 99 that Formal Conjectures states, one article each.","title":"Problems 1 to 99"},{"data_url":"scopes/formal-conjectures/erdos-problems/1-99/part-01.json","id":"formal-conjectures/erdos-problems/1-99/part-01","kind":"scope","navigation":"transparent","parent":6192,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-01/index.html","scope":"formal-conjectures/erdos-problems/1-99/part-01","status":"can_prove","title":"Erd\u0151s Problem 1 to Erd\u0151s Problem 21"},{"id":"formal-conjectures/erdos-problems/1-99/part-01/erdos-problem-1","kind":"node","lean":"Erdos1.erdos_1, Erdos1.erdos_1.variants.weaker, Erdos1.erdos_1.variants.lb, Erdos1.erdos_1.variants.lb_strong, Erdos1.erdos_1.variants.real, Erdos1.erdos_1.variants.least_N_3, Erdos1.erdos_1.variants.least_N_4, Erdos1.erdos_1.variants.least_N_5, Erdos1.erdos_1.variants.least_N_9","parent":6193,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-01/index.html#erdos-problem-1","scope":"formal-conjectures/erdos-problems/1-99/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/1.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 1"},{"id":"formal-conjectures/erdos-problems/1-99/part-01/erdos-problem-2","kind":"node","lean":"Erdos2.erdos_2","parent":6193,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-01/index.html#erdos-problem-2","scope":"formal-conjectures/erdos-problems/1-99/part-01","status":"can_prove","summary":">","title":"Erd\u0151s Problem 2"},{"id":"formal-conjectures/erdos-problems/1-99/part-01/erdos-problem-3","kind":"node","lean":"Erdos3.erdos_3, Erdos3.erdos_3.variants.kelley_meka, Erdos3.erdos_3.variants.three, Erdos3.erdos_3.variants.green_tao, Erdos3.erdos_3.variants.gowers, Erdos3.erdos_3.variants.leng_sah_sawhney","parent":6193,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-01/index.html#erdos-problem-3","scope":"formal-conjectures/erdos-problems/1-99/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/3.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 3"},{"id":"formal-conjectures/erdos-problems/1-99/part-01/erdos-problem-4","kind":"node","lean":"Erdos4.erdos_4, Erdos4.erdos_4.variants.rankin","parent":6193,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-01/index.html#erdos-problem-4","scope":"formal-conjectures/erdos-problems/1-99/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/4.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 4"},{"id":"formal-conjectures/erdos-problems/1-99/part-01/erdos-problem-5","kind":"node","lean":"Erdos5.erdos_5, Erdos5.erdos_5.variants.limit_point_set, Erdos5.erdos_5.variants.westzynthius, Erdos5.erdos_5.variants.goldston_pintz_yildirim, Erdos5.erdos_5.variants.erdos_ricci, Erdos5.erdos_5.variants.hildebrand_maier, Erdos5.erdos_5.variants.hildebrand_maier_measure, Erdos5.erdos_5.variants.pintz, Erdos5.erdos_5.variants.banks_freiberg_maynard, Erdos5.erdos_5.variants.banks_freiberg_maynard_measure, Erdos5.erdos_5.variants.merikoski, Erdos5.erdos_5.variants.merikoski_measure, Erdos5.erdos_5.variants.merikoski_bounded_gaps, Erdos5.erdos_5.variants.dense","parent":6193,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-01/index.html#erdos-problem-5","scope":"formal-conjectures/erdos-problems/1-99/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/5.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 5"},{"id":"formal-conjectures/erdos-problems/1-99/part-01/erdos-problem-6","kind":"node","lean":"Erdos6.erdos_6, Erdos6.erdos_6.variants.increasing, Erdos6.erdos_6.variants.decreasing","parent":6193,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-01/index.html#erdos-problem-6","scope":"formal-conjectures/erdos-problems/1-99/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/6.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 6"},{"id":"formal-conjectures/erdos-problems/1-99/part-01/erdos-problem-7","kind":"node","lean":"Erdos7.erdos_7","parent":6193,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-01/index.html#erdos-problem-7","scope":"formal-conjectures/erdos-problems/1-99/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/7.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 7"},{"id":"formal-conjectures/erdos-problems/1-99/part-01/erdos-problem-9","kind":"node","lean":"Erdos9.erdos_9.variants.infinite, Erdos9.erdos_9","parent":6193,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-01/index.html#erdos-problem-9","scope":"formal-conjectures/erdos-problems/1-99/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/9.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 9"},{"id":"formal-conjectures/erdos-problems/1-99/part-01/erdos-problem-10","kind":"node","lean":"Erdos10.erdos_10, Erdos10.erdos_10.variants.gallagher, Erdos10.erdos_10.variants.granville_soundararajan_odd, Erdos10.erdos_10.variants.grechuk_example, Erdos10.erdos_10.variants.two_pows, Erdos10.erdos_10.variants.grechuk","parent":6193,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-01/index.html#erdos-problem-10","scope":"formal-conjectures/erdos-problems/1-99/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/10.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 10"},{"id":"formal-conjectures/erdos-problems/1-99/part-01/erdos-problem-11","kind":"node","lean":"Erdos11.erdos_11, Erdos11.erdos_11.variants.all_odd, Erdos11.erdos_11.variants.not_four_dvd, Erdos11.erdos_11.variants.two_pow_two, Erdos11.erdos_11.variants.finite_bound1, Erdos11.erdos_11.variants.finite_bound2, Erdos11.erdos_11.variants.granville_soundararajan","parent":6193,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-01/index.html#erdos-problem-11","scope":"formal-conjectures/erdos-problems/1-99/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/11.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 11"},{"id":"formal-conjectures/erdos-problems/1-99/part-01/erdos-problem-12","kind":"node","lean":"Erdos12.isGood_example, Erdos12.erdos_12.parts.i, Erdos12.erdos_12.parts.ii, Erdos12.erdos_12.parts.iii, Erdos12.erdos_12.variants.erdos_sarkozy_density_0, Erdos12.erdos_12.variants.erdos_sarkozy, Erdos12.erdos_12.variants.example, Erdos12.erdos_12.variants.schoen, Erdos12.erdos_12.variants.baier","parent":6193,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-01/index.html#erdos-problem-12","scope":"formal-conjectures/erdos-problems/1-99/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/12.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 12"},{"id":"formal-conjectures/erdos-problems/1-99/part-01/erdos-problem-13","kind":"node","lean":"Erdos13.erdos_13, Erdos13.erdos_13.variants.general","parent":6193,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-01/index.html#erdos-problem-13","scope":"formal-conjectures/erdos-problems/1-99/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/13.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 13"},{"id":"formal-conjectures/erdos-problems/1-99/part-01/erdos-problem-14","kind":"node","lean":"Erdos14.erdos_14.parts.i, Erdos14.erdos_14.parts.ii","parent":6193,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-01/index.html#erdos-problem-14","scope":"formal-conjectures/erdos-problems/1-99/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/14.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 14"},{"id":"formal-conjectures/erdos-problems/1-99/part-01/erdos-problem-15-convergence-of-series-with-prim","kind":"node","lean":"Erdos15.erdos_15","parent":6193,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-01/index.html#erdos-problem-15-convergence-of-series-with-prim","scope":"formal-conjectures/erdos-problems/1-99/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/15.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 15: Convergence of Series with Primes"},{"id":"formal-conjectures/erdos-problems/1-99/part-01/erdos-problem-16","kind":"node","lean":"Erdos16.erdos_16, Erdos16.erdos_16.variant.romanoff, Erdos16.erdos_16.variant.erdos","parent":6193,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-01/index.html#erdos-problem-16","scope":"formal-conjectures/erdos-problems/1-99/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/16.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 16"},{"id":"formal-conjectures/erdos-problems/1-99/part-01/erdos-problem-17","kind":"node","lean":"Erdos17.erdos_17, Erdos17.erdos_17.variants.upper_BES, Erdos17.erdos_17.variants.upper_Elsholtz","parent":6193,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-01/index.html#erdos-problem-17","scope":"formal-conjectures/erdos-problems/1-99/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/17.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 17"},{"id":"formal-conjectures/erdos-problems/1-99/part-01/erdos-problem-18","kind":"node","lean":"Erdos18.factorial_isPractical, Erdos18.erdos_18a, Erdos18.erdos_18b, Erdos18.erdos_18c, Erdos18.erdos_18_upper_bound, Erdos18.erdos_18_vose","parent":6193,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-01/index.html#erdos-problem-18","scope":"formal-conjectures/erdos-problems/1-99/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/18.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 18"},{"id":"formal-conjectures/erdos-problems/1-99/part-01/erdos-problem-19","kind":"node","lean":"Erdos19.EFLConfig.le_chromaticNumber, Erdos19.EFLConfig.colorable_of_sharedColoring, Erdos19.EFLConfig.card_sharedSet_le, Erdos19.erdos_19, Erdos19.erdos_19.variants.lt_ten, Erdos19.erdos_19.variants.kahn, Erdos19.erdos_19.variants.large_n","parent":6193,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-01/index.html#erdos-problem-19","scope":"formal-conjectures/erdos-problems/1-99/part-01","status":"can_prove","summary":"> >","title":"Erd\u0151s Problem 19"},{"id":"formal-conjectures/erdos-problems/1-99/part-01/erdos-problem-20","kind":"node","lean":"Erdos20.erdos_20, Erdos20.erdos_20.variants.erdos_rado_bound","parent":6193,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-01/index.html#erdos-problem-20","scope":"formal-conjectures/erdos-problems/1-99/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/20.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 20"},{"id":"formal-conjectures/erdos-problems/1-99/part-01/erdos-problem-21","kind":"node","lean":"Erdos21.erdos_21, Erdos21.erdos_21.variants.lower, Erdos21.erdos_21.variants.three_n, Erdos21.erdos_21.variants.small_values","parent":6193,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-01/index.html#erdos-problem-21","scope":"formal-conjectures/erdos-problems/1-99/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/21.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 21"},{"data_url":"scopes/formal-conjectures/erdos-problems/1-99/part-02.json","id":"formal-conjectures/erdos-problems/1-99/part-02","kind":"scope","navigation":"transparent","parent":6192,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-02/index.html","scope":"formal-conjectures/erdos-problems/1-99/part-02","status":"can_prove","title":"Erd\u0151s Problem 22 to Erd\u0151s Problem 43"},{"id":"formal-conjectures/erdos-problems/1-99/part-02/erdos-problem-22","kind":"node","lean":"Erdos22.erdos_22, Erdos22.erdos_22.variants.szemeredi_upper, Erdos22.erdos_22.variants.bollobas_erdos_lower, Erdos22.erdos_22.variants.fox_loh_zhao","parent":6214,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-02/index.html#erdos-problem-22","scope":"formal-conjectures/erdos-problems/1-99/part-02","status":"can_prove","summary":"> > >","title":"Erd\u0151s Problem 22"},{"id":"formal-conjectures/erdos-problems/1-99/part-02/erdos-problem-23","kind":"node","lean":"Erdos23.erdos_23.variants.n5, Erdos23.erdos_23.variants.n5_tight, Erdos23.erdos_23","parent":6214,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-02/index.html#erdos-problem-23","scope":"formal-conjectures/erdos-problems/1-99/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/23.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 23"},{"id":"formal-conjectures/erdos-problems/1-99/part-02/erdos-problem-24","kind":"node","lean":"Erdos24.erdos_24","parent":6214,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-02/index.html#erdos-problem-24","scope":"formal-conjectures/erdos-problems/1-99/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/24.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 24"},{"id":"formal-conjectures/erdos-problems/1-99/part-02/erdos-problem-25-logarithmic-density-of-size-dep","kind":"node","lean":"Erdos25.erdos_25","parent":6214,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-02/index.html#erdos-problem-25-logarithmic-density-of-size-dep","scope":"formal-conjectures/erdos-problems/1-99/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/25.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 25: Logarithmic density of size-dependent congruences"},{"id":"formal-conjectures/erdos-problems/1-99/part-02/erdos-problem-26","kind":"node","lean":"Erdos26.erdos_26, Erdos26.erdos_26.variants.rusza, Erdos26.erdos_26.variants.tenenbaum","parent":6214,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-02/index.html#erdos-problem-26","scope":"formal-conjectures/erdos-problems/1-99/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/26.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 26"},{"id":"formal-conjectures/erdos-problems/1-99/part-02/erdos-problem-28","kind":"node","lean":"Erdos28.erdos_28","parent":6214,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-02/index.html#erdos-problem-28","scope":"formal-conjectures/erdos-problems/1-99/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/28.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 28"},{"id":"formal-conjectures/erdos-problems/1-99/part-02/erdos-problem-29","kind":"node","lean":"Erdos29.erdos_29","parent":6214,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-02/index.html#erdos-problem-29","scope":"formal-conjectures/erdos-problems/1-99/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/29.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 29"},{"id":"formal-conjectures/erdos-problems/1-99/part-02/erdos-problem-30","kind":"node","lean":"Erdos30.erdos_30, Erdos30.erdos_30.variants.O_one, Erdos30.erdos_30.variants.erdos_turan, Erdos30.erdos_30.variants.lindstrom, Erdos30.erdos_30.variants.balogh_furedi_roy, Erdos30.erdos_30.variants.obryant, Erdos30.erdos_30.variants.carter_hunter_obryant, Erdos30.erdos_30.variants.singer, Erdos30.erdos_30.variants.isEquivalent_sqrt, Erdos30.erdos_30.variants.elementary_difference_count, Erdos30.erdos_30.variants.distinct_sums_card, Erdos30.erdos_30.variants.distinct_sums_in_range, Erdos30.erdos_30.variants.erdos_turan_counting, Erdos30.erdos_30.variants.lindstrom_weak","parent":6214,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-02/index.html#erdos-problem-30","scope":"formal-conjectures/erdos-problems/1-99/part-02","status":"can_prove","summary":">","title":"Erd\u0151s Problem 30"},{"id":"formal-conjectures/erdos-problems/1-99/part-02/erdos-problem-31","kind":"node","lean":"Erdos31.erdos_31","parent":6214,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-02/index.html#erdos-problem-31","scope":"formal-conjectures/erdos-problems/1-99/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/31.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements 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Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 68"},{"id":"formal-conjectures/erdos-problems/1-99/part-04/erdos-problem-69","kind":"node","lean":"Erdos69.erdos_69, Erdos69.erdos_69.variants.specialisation_of_erdos_257","parent":6256,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-04/index.html#erdos-problem-69","scope":"formal-conjectures/erdos-problems/1-99/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/69.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 69"},{"id":"formal-conjectures/erdos-problems/1-99/part-04/erdos-problem-70","kind":"node","lean":"Erdos70.erdos_70, Erdos70.erdos_70.variants.erdos_rado, Erdos70.erdos_70.variants.omega_times_two_four, Erdos70.erdos_70.variants.omega_three, 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Erdos72.erdos_72.variants.powers_of_two","parent":6256,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-04/index.html#erdos-problem-72","scope":"formal-conjectures/erdos-problems/1-99/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/72.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 72"},{"id":"formal-conjectures/erdos-problems/1-99/part-04/erdos-problem-73","kind":"node","lean":"Erdos73.erdos_73","parent":6256,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-04/index.html#erdos-problem-73","scope":"formal-conjectures/erdos-problems/1-99/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/73.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 73"},{"id":"formal-conjectures/erdos-problems/1-99/part-04/erdos-problem-74","kind":"node","lean":"Erdos74.erdos_74, Erdos74.erdos_74.variants.sqrt","parent":6256,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-04/index.html#erdos-problem-74","scope":"formal-conjectures/erdos-problems/1-99/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/74.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 74"},{"id":"formal-conjectures/erdos-problems/1-99/part-04/erdos-problem-75","kind":"node","lean":"Erdos75.erdos_75","parent":6256,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-04/index.html#erdos-problem-75","scope":"formal-conjectures/erdos-problems/1-99/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/75.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 75"},{"id":"formal-conjectures/erdos-problems/1-99/part-04/erdos-problem-77","kind":"node","lean":"Erdos77.erdos_77","parent":6256,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-04/index.html#erdos-problem-77","scope":"formal-conjectures/erdos-problems/1-99/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/77.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 77"},{"id":"formal-conjectures/erdos-problems/1-99/part-04/erdos-problem-78","kind":"node","lean":"Erdos78.erdos_78, Erdos78.erdos_78.variants.nonconstructive, Erdos78.erdos_78.variants.little_o_sqrt, Erdos78.erdos_78.variants.cohen, Erdos78.erdos_78.variants.li, Erdos78.erdos_78.variants.strongly_explicit, Erdos78.erdos_78.variants.li_strongly_explicit","parent":6256,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-04/index.html#erdos-problem-78","scope":"formal-conjectures/erdos-problems/1-99/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/78.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 78"},{"id":"formal-conjectures/erdos-problems/1-99/part-04/erdos-problem-79","kind":"node","lean":"Erdos79.erdos_79","parent":6256,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-04/index.html#erdos-problem-79","scope":"formal-conjectures/erdos-problems/1-99/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/79.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 79"},{"id":"formal-conjectures/erdos-problems/1-99/part-04/erdos-problem-80","kind":"node","lean":"Erdos80.erdos_80.parts.i, Erdos80.erdos_80.parts.ii","parent":6256,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-04/index.html#erdos-problem-80","scope":"formal-conjectures/erdos-problems/1-99/part-04","status":"can_prove","summary":">","title":"Erd\u0151s Problem 80"},{"id":"formal-conjectures/erdos-problems/1-99/part-04/erdos-problem-81","kind":"node","lean":"Erdos81.erdos_81, Erdos81.erdos_81.variants.split","parent":6256,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-04/index.html#erdos-problem-81","scope":"formal-conjectures/erdos-problems/1-99/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/81.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 81"},{"id":"formal-conjectures/erdos-problems/1-99/part-04/erdos-problem-82","kind":"node","lean":"Erdos82.erdos_82, Erdos82.erdos_82.variants.F_upper_bound","parent":6256,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-04/index.html#erdos-problem-82","scope":"formal-conjectures/erdos-problems/1-99/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/82.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 82"},{"id":"formal-conjectures/erdos-problems/1-99/part-04/erdos-problem-83","kind":"node","lean":"Erdos83.erdos_83, Erdos83.erdos_83.variants.extremal, Erdos83.erdos_83.variants.ahlswede_khachatrian","parent":6256,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-04/index.html#erdos-problem-83","scope":"formal-conjectures/erdos-problems/1-99/part-04","status":"can_prove","summary":"*Quoted 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85"},{"id":"formal-conjectures/erdos-problems/1-99/part-04/erdos-problem-86","kind":"node","lean":"Erdos86.erdos_86","parent":6256,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-04/index.html#erdos-problem-86","scope":"formal-conjectures/erdos-problems/1-99/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/86.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 86"},{"id":"formal-conjectures/erdos-problems/1-99/part-04/erdos-problem-87","kind":"node","lean":"Erdos87.erdos_87.parts.i, Erdos87.erdos_87.parts.ii","parent":6256,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-04/index.html#erdos-problem-87","scope":"formal-conjectures/erdos-problems/1-99/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/87.lean in Formal Conjectures (Google DeepMind) at 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Erdos91.erdos_91.variants.nine","parent":6277,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-05/index.html#erdos-problem-91","scope":"formal-conjectures/erdos-problems/1-99/part-05","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/91.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 91"},{"id":"formal-conjectures/erdos-problems/1-99/part-05/erdos-problem-92","kind":"node","lean":"Erdos92.erdos_92.variants.weak, Erdos92.erdos_92.variants.strong","parent":6277,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-05/index.html#erdos-problem-92","scope":"formal-conjectures/erdos-problems/1-99/part-05","status":"can_prove","summary":">","title":"Erd\u0151s Problem 92"},{"id":"formal-conjectures/erdos-problems/1-99/part-05/erdos-problem-93","kind":"node","lean":"Erdos93.erdos_93","parent":6277,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-05/index.html#erdos-problem-93","scope":"formal-conjectures/erdos-problems/1-99/part-05","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/93.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 93"},{"id":"formal-conjectures/erdos-problems/1-99/part-05/erdos-problem-94","kind":"node","lean":"Erdos94.erdos_94, Erdos94.erdos_94.variants.no_three_on_a_line, Erdos94.erdos_94.variants.regular_ngon","parent":6277,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-05/index.html#erdos-problem-94","scope":"formal-conjectures/erdos-problems/1-99/part-05","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/94.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 94"},{"id":"formal-conjectures/erdos-problems/1-99/part-05/erdos-problem-95","kind":"node","lean":"Erdos95.erdos_95, Erdos95.erdos_95.variants.guth_katz","parent":6277,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-05/index.html#erdos-problem-95","scope":"formal-conjectures/erdos-problems/1-99/part-05","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/95.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 95"},{"id":"formal-conjectures/erdos-problems/1-99/part-05/erdos-problem-96","kind":"node","lean":"Erdos96.erdos_96","parent":6277,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1-99/part-05/index.html#erdos-problem-96","scope":"formal-conjectures/erdos-problems/1-99/part-05","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/96.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 96"},{"id":"formal-conjectures/erdos-problems/1-99/part-05/erdos-problem-97","kind":"node","lean":"Erdos97.erdos_97, Erdos97.erdos_97.variants.three_equidistant, Erdos97.erdos_97.variants.k_equidistant, Erdos97.erdos_97.variants.three_unit_distance, 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Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 100"},{"id":"formal-conjectures/erdos-problems/100-199/part-01/erdos-problem-101","kind":"node","lean":"Erdos101.erdos_101","parent":6290,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/100-199/part-01/index.html#erdos-problem-101","scope":"formal-conjectures/erdos-problems/100-199/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/101.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 101"},{"id":"formal-conjectures/erdos-problems/100-199/part-01/erdos-problem-102","kind":"node","lean":"Erdos102.erdos_102, Erdos102.erdos_102.variants.five, Erdos102.erdos_102.variants.upper_sqrt, Erdos102.erdos_102.variants.hunter","parent":6290,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/100-199/part-01/index.html#erdos-problem-102","scope":"formal-conjectures/erdos-problems/100-199/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/102.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 102"},{"id":"formal-conjectures/erdos-problems/100-199/part-01/erdos-problem-103","kind":"node","lean":"Erdos103.erdos_103, Erdos103.erdos_103.variants.two_le","parent":6290,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/100-199/part-01/index.html#erdos-problem-103","scope":"formal-conjectures/erdos-problems/100-199/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/103.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 103"},{"id":"formal-conjectures/erdos-problems/100-199/part-01/erdos-problem-104","kind":"node","lean":"Erdos104.erdos_104","parent":6290,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/100-199/part-01/index.html#erdos-problem-104","scope":"formal-conjectures/erdos-problems/100-199/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/104.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 104"},{"id":"formal-conjectures/erdos-problems/100-199/part-01/erdos-problem-105","kind":"node","lean":"Erdos105.erdos_105, Erdos105.erdos_105.variants.hickerson, Erdos105.erdos_105.variants.beck_szemeredi_trotter, Erdos105.erdos_105.variants.sub_four","parent":6290,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/100-199/part-01/index.html#erdos-problem-105","scope":"formal-conjectures/erdos-problems/100-199/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/105.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 105"},{"id":"formal-conjectures/erdos-problems/100-199/part-01/erdos-problem-107","kind":"node","lean":"Erdos107.erdos_107, Erdos107.nonempty_cardSet, Erdos107.variants.ersz_bounds, Erdos107.variants.su_bound, Erdos107.variants.hmpt_bound","parent":6290,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/100-199/part-01/index.html#erdos-problem-107","scope":"formal-conjectures/erdos-problems/100-199/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/107.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 107"},{"id":"formal-conjectures/erdos-problems/100-199/part-01/erdos-problem-108","kind":"node","lean":"Erdos108.erdos_108","parent":6290,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/100-199/part-01/index.html#erdos-problem-108","scope":"formal-conjectures/erdos-problems/100-199/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/108.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 108"},{"id":"formal-conjectures/erdos-problems/100-199/part-01/erdos-problem-109","kind":"node","lean":"Erdos109.erdos_109","parent":6290,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/100-199/part-01/index.html#erdos-problem-109","scope":"formal-conjectures/erdos-problems/100-199/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/109.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 109"},{"id":"formal-conjectures/erdos-problems/100-199/part-01/erdos-problem-110","kind":"node","lean":"Erdos110.erdos_110, Erdos110.erdos_110.variants.lambie_hanson, Erdos110.erdos_110.variants.de_bruijn_erdos","parent":6290,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/100-199/part-01/index.html#erdos-problem-110","scope":"formal-conjectures/erdos-problems/100-199/part-01","status":"can_prove","summary":"*Quoted from 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111"},{"id":"formal-conjectures/erdos-problems/100-199/part-01/erdos-problem-113","kind":"node","lean":"Erdos113.erdos_113","parent":6290,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/100-199/part-01/index.html#erdos-problem-113","scope":"formal-conjectures/erdos-problems/100-199/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/113.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 113"},{"id":"formal-conjectures/erdos-problems/100-199/part-01/erdos-problem-114","kind":"node","lean":"Erdos114.erdos_114, Erdos114.erdos_114.variants.large_n, Erdos114.erdos_114.variants.two, Erdos114.erdos_114.variants.one","parent":6290,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/100-199/part-01/index.html#erdos-problem-114","scope":"formal-conjectures/erdos-problems/100-199/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/114.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 114"},{"id":"formal-conjectures/erdos-problems/100-199/part-01/erdos-problem-115","kind":"node","lean":"Erdos115.erdos_115","parent":6290,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/100-199/part-01/index.html#erdos-problem-115","scope":"formal-conjectures/erdos-problems/100-199/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/115.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s 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191"},{"id":"formal-conjectures/erdos-problems/100-199/part-04/erdos-problem-192","kind":"node","lean":"Erdos192.erdos_192","parent":6353,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/100-199/part-04/index.html#erdos-problem-192","scope":"formal-conjectures/erdos-problems/100-199/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/192.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 192"},{"id":"formal-conjectures/erdos-problems/100-199/part-04/erdos-problem-193","kind":"node","lean":"Erdos193.erdos_193, Erdos193.erdos_193_z2","parent":6353,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/100-199/part-04/index.html#erdos-problem-193","scope":"formal-conjectures/erdos-problems/100-199/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/193.lean in Formal Conjectures (Google 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Erdos195.erdos_195.variants.leq_4_bound","parent":6353,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/100-199/part-04/index.html#erdos-problem-195","scope":"formal-conjectures/erdos-problems/100-199/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/195.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 195"},{"id":"formal-conjectures/erdos-problems/100-199/part-04/erdos-problem-196","kind":"node","lean":"Erdos196.erdos_196","parent":6353,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/100-199/part-04/index.html#erdos-problem-196","scope":"formal-conjectures/erdos-problems/100-199/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/196.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 196"},{"id":"formal-conjectures/erdos-problems/100-199/part-04/erdos-problem-197","kind":"node","lean":"Erdos197.erdos_197","parent":6353,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/100-199/part-04/index.html#erdos-problem-197","scope":"formal-conjectures/erdos-problems/100-199/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/197.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 197"},{"data_url":"scopes/formal-conjectures/erdos-problems/100-199/part-05.json","id":"formal-conjectures/erdos-problems/100-199/part-05","kind":"scope","navigation":"transparent","parent":6289,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/100-199/part-05/index.html","scope":"formal-conjectures/erdos-problems/100-199/part-05","status":"can_prove","title":"Erd\u0151s Problem 198 to 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198"},{"id":"formal-conjectures/erdos-problems/100-199/part-05/erdos-problem-199","kind":"node","lean":"Erdos199.erdos_199","parent":6374,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/100-199/part-05/index.html#erdos-problem-199","scope":"formal-conjectures/erdos-problems/100-199/part-05","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/199.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 199"},{"data_url":"scopes/formal-conjectures/erdos-problems/200-299.json","id":"formal-conjectures/erdos-problems/200-299","kind":"scope","parent":6191,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/index.html","scope":"formal-conjectures/erdos-problems/200-299","status":"can_prove","summary":"The Erd\u0151s problems numbered 200 to 299 that Formal Conjectures states, one article each.","title":"Problems 200 to 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218"},{"id":"formal-conjectures/erdos-problems/200-299/part-01/erdos-problem-219","kind":"node","lean":"Erdos219.erdos_219","parent":6378,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-01/index.html#erdos-problem-219","scope":"formal-conjectures/erdos-problems/200-299/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/219.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 219"},{"id":"formal-conjectures/erdos-problems/200-299/part-01/erdos-problem-220","kind":"node","lean":"Erdos220.erdos_220","parent":6378,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-01/index.html#erdos-problem-220","scope":"formal-conjectures/erdos-problems/200-299/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/220.lean in Formal Conjectures (Google DeepMind) at commit 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224"},{"id":"formal-conjectures/erdos-problems/200-299/part-02/erdos-problem-225","kind":"node","lean":"Erdos225.erdos_225","parent":6399,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-02/index.html#erdos-problem-225","scope":"formal-conjectures/erdos-problems/200-299/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/225.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 225"},{"id":"formal-conjectures/erdos-problems/200-299/part-02/erdos-problem-226","kind":"node","lean":"Erdos226.erdos_226","parent":6399,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-02/index.html#erdos-problem-226","scope":"formal-conjectures/erdos-problems/200-299/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/226.lean in Formal Conjectures (Google DeepMind) at commit 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Erdos235.erdos_235.variants.hooley","parent":6399,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-02/index.html#erdos-problem-235","scope":"formal-conjectures/erdos-problems/200-299/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/235.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 235"},{"id":"formal-conjectures/erdos-problems/200-299/part-02/erdos-problem-236","kind":"node","lean":"Erdos236.erdos_236","parent":6399,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-02/index.html#erdos-problem-236","scope":"formal-conjectures/erdos-problems/200-299/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/236.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements 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Erdos238.erdos_238.variants.small_c1","parent":6399,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-02/index.html#erdos-problem-238","scope":"formal-conjectures/erdos-problems/200-299/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/238.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 238"},{"id":"formal-conjectures/erdos-problems/200-299/part-02/erdos-problem-239","kind":"node","lean":"Erdos239.erdos_239","parent":6399,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-02/index.html#erdos-problem-239","scope":"formal-conjectures/erdos-problems/200-299/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/239.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements 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250"},{"id":"formal-conjectures/erdos-problems/200-299/part-03/erdos-problem-251","kind":"node","lean":"Erdos251.erdos_251","parent":6420,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-03/index.html#erdos-problem-251","scope":"formal-conjectures/erdos-problems/200-299/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/251.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 251"},{"id":"formal-conjectures/erdos-problems/200-299/part-03/erdos-problem-252","kind":"node","lean":"Erdos252.erdos_252, Erdos252.erdos_252.variants.k_eq_zero, Erdos252.erdos_252.variants.k_eq_one, Erdos252.erdos_252.variants.k_eq_two, Erdos252.erdos_252.variants.k_eq_three, Erdos252.erdos_252.variants.k_eq_four, Erdos252.erdos_252.variants.k_ge_five, Erdos252.erdos_252.variants.schinzel, 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below.*","title":"Erd\u0151s Problem 253"},{"id":"formal-conjectures/erdos-problems/200-299/part-03/erdos-problem-254","kind":"node","lean":"Erdos254.erdos_254, Erdos254.erdos_254.variants.cassels","parent":6420,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-03/index.html#erdos-problem-254","scope":"formal-conjectures/erdos-problems/200-299/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/254.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 254"},{"id":"formal-conjectures/erdos-problems/200-299/part-03/erdos-problem-257","kind":"node","lean":"Erdos257.erdos_257, Erdos257.erdos_257.variants.tsum_top_eq, Erdos257.erdos_257.variants.tsum_top, Erdos257.erdos_257.variants.summable_reciprocal_support","parent":6420,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-03/index.html#erdos-problem-257","scope":"formal-conjectures/erdos-problems/200-299/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/257.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 257"},{"id":"formal-conjectures/erdos-problems/200-299/part-03/erdos-problem-258","kind":"node","lean":"Erdos258.erdos_258, Erdos258.erdos_258.variants.monotone, Erdos258.erdos_258.variants.constant","parent":6420,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-03/index.html#erdos-problem-258","scope":"formal-conjectures/erdos-problems/200-299/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/258.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 258"},{"id":"formal-conjectures/erdos-problems/200-299/part-03/erdos-problem-259","kind":"node","lean":"Erdos259.erdos_259","parent":6420,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-03/index.html#erdos-problem-259","scope":"formal-conjectures/erdos-problems/200-299/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/259.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 259"},{"id":"formal-conjectures/erdos-problems/200-299/part-03/erdos-problem-260","kind":"node","lean":"Erdos260.erdos_260","parent":6420,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-03/index.html#erdos-problem-260","scope":"formal-conjectures/erdos-problems/200-299/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/260.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 260"},{"id":"formal-conjectures/erdos-problems/200-299/part-03/erdos-problem-261","kind":"node","lean":"Erdos261.erdos_261.variants.borwein_loring, Erdos261.erdos_261.variants.borwein_loring_property, Erdos261.erdos_261.parts.i, Erdos261.erdos_261.variants.le_10000, Erdos261.erdos_261.parts.ii, Erdos261.erdos_261.parts.iii, Erdos261.erdos_261.variants.two_representations","parent":6420,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-03/index.html#erdos-problem-261","scope":"formal-conjectures/erdos-problems/200-299/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/261.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 261"},{"id":"formal-conjectures/erdos-problems/200-299/part-03/erdos-problem-263","kind":"node","lean":"Erdos263.erdos_263.parts.i, Erdos263.erdos_263.parts.ii, Erdos263.erdos_263.variants.folklore, Erdos263.erdos_263.variants.sub_doubly_exponential, Erdos263.erdos_263.variants.super_doubly_exponential, Erdos263.erdos_263.variants.super_doubly_exponential_eventual, Erdos263.erdos_263.variants.doubly_exponential_all_but_countable","parent":6420,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-03/index.html#erdos-problem-263","scope":"formal-conjectures/erdos-problems/200-299/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/263.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 263"},{"id":"formal-conjectures/erdos-problems/200-299/part-03/erdos-problem-264","kind":"node","lean":"Erdos264.erdos_264.parts.i, Erdos264.erdos_264.parts.ii, Erdos264.erdos_264.variants.example, Erdos264.erdos_264.variants.ko_tao_neg, Erdos264.erdos_264.variants.ko_tao_pos","parent":6420,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-03/index.html#erdos-problem-264","scope":"formal-conjectures/erdos-problems/200-299/part-03","status":"can_prove","summary":"*Quoted 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Erdos267.erdos_267.variants.specialization_pow_two, Erdos267.erdos_267.variants.fibonacci_inverse_sum","parent":6420,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-03/index.html#erdos-problem-267","scope":"formal-conjectures/erdos-problems/200-299/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/267.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 267"},{"id":"formal-conjectures/erdos-problems/200-299/part-03/erdos-problem-268","kind":"node","lean":"Erdos268.erdos_268","parent":6420,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-03/index.html#erdos-problem-268","scope":"formal-conjectures/erdos-problems/200-299/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/268.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as 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Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 269"},{"id":"formal-conjectures/erdos-problems/200-299/part-04/erdos-problem-270","kind":"node","lean":"Erdos270.erdos_270, Erdos270.erdos_270.variants.every_value, Erdos270.erdos_270.variants.monotone, Erdos270.erdos_270.variants.linear","parent":6441,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-04/index.html#erdos-problem-270","scope":"formal-conjectures/erdos-problems/200-299/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/270.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 270"},{"id":"formal-conjectures/erdos-problems/200-299/part-04/erdos-problem-272","kind":"node","lean":"Erdos272.erdos_272, Erdos272.erdos_272.variants.isBigO_sq, Erdos272.erdos_272.variants.szabo, Erdos272.erdos_272.variants.szabo_strong","parent":6441,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-04/index.html#erdos-problem-272","scope":"formal-conjectures/erdos-problems/200-299/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/272.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 272"},{"id":"formal-conjectures/erdos-problems/200-299/part-04/erdos-problem-273","kind":"node","lean":"Erdos273.erdos_273, Erdos273.erdos_273.variants.three","parent":6441,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-04/index.html#erdos-problem-273","scope":"formal-conjectures/erdos-problems/200-299/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/273.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 273"},{"id":"formal-conjectures/erdos-problems/200-299/part-04/erdos-problem-274","kind":"node","lean":"Erdos274.erdos_274, Erdos274.erdos_274.variants.abelian, Erdos274.herzog_schonheim","parent":6441,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-04/index.html#erdos-problem-274","scope":"formal-conjectures/erdos-problems/200-299/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/274.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 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2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 280"},{"id":"formal-conjectures/erdos-problems/200-299/part-04/erdos-problem-281","kind":"node","lean":"Erdos281.erdos_281","parent":6441,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-04/index.html#erdos-problem-281","scope":"formal-conjectures/erdos-problems/200-299/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/281.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 281"},{"id":"formal-conjectures/erdos-problems/200-299/part-04/erdos-problem-282","kind":"node","lean":"Erdos282.erdos_282, Erdos282.erdos_282.variants.general, Erdos282.erdos_282.variants.fibonacci, Erdos282.erdos_282.variants.graham, 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Erdos283.erdos_283.variants.van_doorn_quadratic","parent":6441,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-04/index.html#erdos-problem-283","scope":"formal-conjectures/erdos-problems/200-299/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/283.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 283"},{"id":"formal-conjectures/erdos-problems/200-299/part-04/erdos-problem-284","kind":"node","lean":"Erdos284.erdos_284","parent":6441,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-04/index.html#erdos-problem-284","scope":"formal-conjectures/erdos-problems/200-299/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/284.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 284"},{"id":"formal-conjectures/erdos-problems/200-299/part-04/erdos-problem-285","kind":"node","lean":"Erdos285.erdos_285, Erdos285.erdos_285.variants.lb","parent":6441,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-04/index.html#erdos-problem-285","scope":"formal-conjectures/erdos-problems/200-299/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/285.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 285"},{"id":"formal-conjectures/erdos-problems/200-299/part-04/erdos-problem-286","kind":"node","lean":"Erdos286.erdos_286","parent":6441,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-04/index.html#erdos-problem-286","scope":"formal-conjectures/erdos-problems/200-299/part-04","status":"can_prove","summary":"*Quoted from 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2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 290"},{"data_url":"scopes/formal-conjectures/erdos-problems/200-299/part-05.json","id":"formal-conjectures/erdos-problems/200-299/part-05","kind":"scope","navigation":"transparent","parent":6377,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-05/index.html","scope":"formal-conjectures/erdos-problems/200-299/part-05","status":"can_prove","title":"Erd\u0151s Problem 291 to Erd\u0151s Problem 299"},{"id":"formal-conjectures/erdos-problems/200-299/part-05/erdos-problem-291","kind":"node","lean":"Erdos291.erdos_291.parts.i, Erdos291.erdos_291.parts.ii, Erdos291.erdos_291.variants.steinerberger_generalization, Erdos291.erdos_291.variants.shiu_heuristic_asymptotic, Erdos291.erdos_291.variants.shiu_heuristic_density_zero, Erdos291.erdos_291.variants.wu_yan","parent":6462,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-05/index.html#erdos-problem-291","scope":"formal-conjectures/erdos-problems/200-299/part-05","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/291.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 291"},{"id":"formal-conjectures/erdos-problems/200-299/part-05/erdos-problem-292","kind":"node","lean":"Erdos292.erdos_292, Erdos292.erdos_292.variants.martin, Erdos292.erdos_292.variants.mul, Erdos292.erdos_292.variants.prime_pow, Erdos292.erdos_292.variants.two_mul","parent":6462,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-05/index.html#erdos-problem-292","scope":"formal-conjectures/erdos-problems/200-299/part-05","status":"can_prove","summary":"*Quoted from 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Erdos295.erdos_295.variants.erdos_straus","parent":6462,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-05/index.html#erdos-problem-295","scope":"formal-conjectures/erdos-problems/200-299/part-05","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/295.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 295"},{"id":"formal-conjectures/erdos-problems/200-299/part-05/erdos-problem-296","kind":"node","lean":"Erdos296.erdos_296","parent":6462,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-05/index.html#erdos-problem-296","scope":"formal-conjectures/erdos-problems/200-299/part-05","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/296.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 296"},{"id":"formal-conjectures/erdos-problems/200-299/part-05/erdos-problem-298","kind":"node","lean":"Erdos298.erdos_298, Erdos298.erdos_298.variants.natural_density","parent":6462,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-05/index.html#erdos-problem-298","scope":"formal-conjectures/erdos-problems/200-299/part-05","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/298.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 298"},{"id":"formal-conjectures/erdos-problems/200-299/part-05/erdos-problem-299","kind":"node","lean":"Erdos299.erdos_299, Erdos299.erdos_299.variants.density","parent":6462,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/200-299/part-05/index.html#erdos-problem-299","scope":"formal-conjectures/erdos-problems/200-299/part-05","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/299.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 299"},{"data_url":"scopes/formal-conjectures/erdos-problems/300-399.json","id":"formal-conjectures/erdos-problems/300-399","kind":"scope","parent":6191,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/300-399/index.html","scope":"formal-conjectures/erdos-problems/300-399","status":"can_prove","summary":"The Erd\u0151s problems numbered 300 to 399 that Formal Conjectures states, one article each.","title":"Problems 300 to 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in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 308"},{"id":"formal-conjectures/erdos-problems/300-399/part-01/erdos-problem-309","kind":"node","lean":"Erdos309.erdos_309, Erdos309.erdos_309.variants.asymptotic","parent":6471,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/300-399/part-01/index.html#erdos-problem-309","scope":"formal-conjectures/erdos-problems/300-399/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/309.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 309"},{"id":"formal-conjectures/erdos-problems/300-399/part-01/erdos-problem-310","kind":"node","lean":"Erdos310.erdos_310, Erdos310.erdos_310.variants.liu_sawhney","parent":6471,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/300-399/part-01/index.html#erdos-problem-310","scope":"formal-conjectures/erdos-problems/300-399/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/310.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 310"},{"id":"formal-conjectures/erdos-problems/300-399/part-01/erdos-problem-311","kind":"node","lean":"Erdos311.erdos_311, Erdos311.delta_ge_inv_lcm, Erdos311.delta_ge_exp","parent":6471,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/300-399/part-01/index.html#erdos-problem-311","scope":"formal-conjectures/erdos-problems/300-399/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/311.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the 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Erdos313.erdos_313.variants.exists_at_least_eight_primary_pseudoperfect","parent":6471,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/300-399/part-01/index.html#erdos-problem-313","scope":"formal-conjectures/erdos-problems/300-399/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/313.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 313"},{"id":"formal-conjectures/erdos-problems/300-399/part-01/erdos-problem-314","kind":"node","lean":"Erdos314.erdos_314","parent":6471,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/300-399/part-01/index.html#erdos-problem-314","scope":"formal-conjectures/erdos-problems/300-399/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/314.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean 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Erdos326.erdos_326.variants.eq","parent":6492,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/300-399/part-02/index.html#erdos-problem-326","scope":"formal-conjectures/erdos-problems/300-399/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/326.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 326"},{"id":"formal-conjectures/erdos-problems/300-399/part-02/erdos-problem-328","kind":"node","lean":"Erdos328.erdos_328","parent":6492,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/300-399/part-02/index.html#erdos-problem-328","scope":"formal-conjectures/erdos-problems/300-399/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/328.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s 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Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 331"},{"id":"formal-conjectures/erdos-problems/300-399/part-02/erdos-problem-332","kind":"node","lean":"Erdos332.erdos_332","parent":6492,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/300-399/part-02/index.html#erdos-problem-332","scope":"formal-conjectures/erdos-problems/300-399/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/332.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 332"},{"id":"formal-conjectures/erdos-problems/300-399/part-02/erdos-problem-333","kind":"node","lean":"Erdos333.erdos_333","parent":6492,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/300-399/part-02/index.html#erdos-problem-333","scope":"formal-conjectures/erdos-problems/300-399/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/333.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 333"},{"id":"formal-conjectures/erdos-problems/300-399/part-02/erdos-problem-337","kind":"node","lean":"Erdos337.erdos_337, Erdos337.erdos_337.variants.h_fold, Erdos337.erdos_337.variants.ruzsa_turjanyi, 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Erdos338.erdos_338.variants.triangular, Erdos338.erdos_338.variants.hegyvari_hennecart_plagne","parent":6492,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/300-399/part-02/index.html#erdos-problem-338","scope":"formal-conjectures/erdos-problems/300-399/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/338.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 338"},{"id":"formal-conjectures/erdos-problems/300-399/part-02/erdos-problem-340","kind":"node","lean":"Erdos340.erdos_340, Erdos340.erdos_340.variants.isTheta, Erdos340.erdos_340.variants.third, Erdos340.erdos_340.variants.sub_hasPosDensity, Erdos340.erdos_340.variants._22_mem_sub, Erdos340.erdos_340.variants._33_mem_sub, Erdos340.erdos_340.variants.cofinite_sub, Erdos340.erdos_340.variants.co_density_zero_sub","parent":6492,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/300-399/part-02/index.html#erdos-problem-340","scope":"formal-conjectures/erdos-problems/300-399/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/340.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 340"},{"id":"formal-conjectures/erdos-problems/300-399/part-02/erdos-problem-341","kind":"node","lean":"Erdos341.erdos_341","parent":6492,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/300-399/part-02/index.html#erdos-problem-341","scope":"formal-conjectures/erdos-problems/300-399/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/341.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 341"},{"id":"formal-conjectures/erdos-problems/300-399/part-02/erdos-problem-342","kind":"node","lean":"Erdos342.erdos_342.parts.i, Erdos342.erdos_342.parts.ii, Erdos342.erdos_342.parts.iii","parent":6492,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/300-399/part-02/index.html#erdos-problem-342","scope":"formal-conjectures/erdos-problems/300-399/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/342.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 342"},{"id":"formal-conjectures/erdos-problems/300-399/part-02/erdos-problem-344","kind":"node","lean":"Erdos344.erdos_344, Erdos344.erdos_344.variants.sqrt_two, Erdos344.erdos_344.variants.folkman","parent":6492,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/300-399/part-02/index.html#erdos-problem-344","scope":"formal-conjectures/erdos-problems/300-399/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/344.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 344"},{"id":"formal-conjectures/erdos-problems/300-399/part-02/erdos-problem-345","kind":"node","lean":"Erdos345.erdos_345, Erdos345.kthPowers_isAddComplete","parent":6492,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/300-399/part-02/index.html#erdos-problem-345","scope":"formal-conjectures/erdos-problems/300-399/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/345.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean 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346"},{"id":"formal-conjectures/erdos-problems/300-399/part-02/erdos-problem-347","kind":"node","lean":"Erdos347.erdos_347","parent":6492,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/300-399/part-02/index.html#erdos-problem-347","scope":"formal-conjectures/erdos-problems/300-399/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/347.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 347"},{"data_url":"scopes/formal-conjectures/erdos-problems/300-399/part-03.json","id":"formal-conjectures/erdos-problems/300-399/part-03","kind":"scope","navigation":"transparent","parent":6470,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/300-399/part-03/index.html","scope":"formal-conjectures/erdos-problems/300-399/part-03","status":"can_prove","title":"Erd\u0151s Problem 348 to Erd\u0151s Problem 368"},{"id":"formal-conjectures/erdos-problems/300-399/part-03/erdos-problem-348","kind":"node","lean":"Erdos348.erdos_348","parent":6513,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/300-399/part-03/index.html#erdos-problem-348","scope":"formal-conjectures/erdos-problems/300-399/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/348.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 348"},{"id":"formal-conjectures/erdos-problems/300-399/part-03/erdos-problem-349","kind":"node","lean":"Erdos349.erdos_349, Erdos349.complete_for_alpha_in_Ioo_one_to_goldenRatio, Erdos349.exists_t_for_k_disjoint_segments, Erdos349.erdos_349.variants.floor_3_halves_odd, Erdos349.erdos_349.variants.floor_3_halves_even, Erdos349.alpha_gt_two_not_isGoodPair, Erdos349.alpha_le_one_not_isGoodPair, Erdos349.exists_finsetSum_two_pow, 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351"},{"id":"formal-conjectures/erdos-problems/300-399/part-03/erdos-problem-352","kind":"node","lean":"Erdos352.erdos_352","parent":6513,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/300-399/part-03/index.html#erdos-problem-352","scope":"formal-conjectures/erdos-problems/300-399/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/352.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 352"},{"id":"formal-conjectures/erdos-problems/300-399/part-03/erdos-problem-353","kind":"node","lean":"Erdos353.erdos_353, Erdos353.erdos_353.variants.isosceles_triangle, Erdos353.erdos_353.variants.right_angled_triangle, Erdos353.erdos_353.variants.cyclic_quadrilateral, Erdos353.erdos_353.variants.congruent_sides","parent":6513,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/300-399/part-03/index.html#erdos-problem-353","scope":"formal-conjectures/erdos-problems/300-399/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/353.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 353"},{"id":"formal-conjectures/erdos-problems/300-399/part-03/erdos-problem-354","kind":"node","lean":"Erdos354.erdos_354.parts.i, Erdos354.erdos_354.parts.ii","parent":6513,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/300-399/part-03/index.html#erdos-problem-354","scope":"formal-conjectures/erdos-problems/300-399/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/354.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 354"},{"id":"formal-conjectures/erdos-problems/300-399/part-03/erdos-problem-355","kind":"node","lean":"Erdos355.erdos_355","parent":6513,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/300-399/part-03/index.html#erdos-problem-355","scope":"formal-conjectures/erdos-problems/300-399/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/355.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 355"},{"id":"formal-conjectures/erdos-problems/300-399/part-03/erdos-problem-357","kind":"node","lean":"Erdos357.erdos_357.parts.i, Erdos357.erdos_357.parts.ii.bigO_version, Erdos357.erdos_357.parts.ii.bigO_version_symm, Erdos357.erdos_357.parts.ii.bigTheta_version, Erdos357.erdos_357.parts.ii.littleO_version, Erdos357.erdos_357.parts.ii.littleO_version_symm, 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430"},{"id":"formal-conjectures/erdos-problems/400-499/part-02/erdos-problem-431","kind":"node","lean":"Erdos431.erdos_431, Erdos431.erdos_431.variants.ostmann","parent":6585,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-02/index.html#erdos-problem-431","scope":"formal-conjectures/erdos-problems/400-499/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/431.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 431"},{"id":"formal-conjectures/erdos-problems/400-499/part-02/erdos-problem-433","kind":"node","lean":"Erdos433.erdos_433","parent":6585,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-02/index.html#erdos-problem-433","scope":"formal-conjectures/erdos-problems/400-499/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/433.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 433"},{"id":"formal-conjectures/erdos-problems/400-499/part-02/erdos-problem-434","kind":"node","lean":"Erdos434.erdos_434.parts.i, Erdos434.erdos_434.parts.ii","parent":6585,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-02/index.html#erdos-problem-434","scope":"formal-conjectures/erdos-problems/400-499/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/434.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 434"},{"id":"formal-conjectures/erdos-problems/400-499/part-02/erdos-problem-435","kind":"node","lean":"Erdos435.erdos_435","parent":6585,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-02/index.html#erdos-problem-435","scope":"formal-conjectures/erdos-problems/400-499/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/435.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 435"},{"id":"formal-conjectures/erdos-problems/400-499/part-02/erdos-problem-436","kind":"node","lean":"Erdos436.erdos_436.parts.i, Erdos436.erdos_436.parts.ii, Erdos436.erdos_436.variants.two_two, Erdos436.erdos_436.variants.three_two, Erdos436.erdos_436.variants.four_two, Erdos436.erdos_436.variants.five_two, Erdos436.erdos_436.variants.six_two, Erdos436.erdos_436.variants.seven_two, Erdos436.erdos_436.variants.three_three, Erdos436.erdos_436.variants.even_three, Erdos436.erdos_436.variants.ge_four","parent":6585,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-02/index.html#erdos-problem-436","scope":"formal-conjectures/erdos-problems/400-499/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/436.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 436"},{"id":"formal-conjectures/erdos-problems/400-499/part-02/erdos-problem-437","kind":"node","lean":"Erdos437.erdos_437, Erdos437.erdos_437.variants.little_o, Erdos437.erdos_437.variants.tao","parent":6585,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-02/index.html#erdos-problem-437","scope":"formal-conjectures/erdos-problems/400-499/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/437.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 437"},{"id":"formal-conjectures/erdos-problems/400-499/part-02/erdos-problem-438","kind":"node","lean":"Erdos438.erdos_438","parent":6585,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-02/index.html#erdos-problem-438","scope":"formal-conjectures/erdos-problems/400-499/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/438.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 438"},{"id":"formal-conjectures/erdos-problems/400-499/part-02/erdos-problem-440","kind":"node","lean":"Erdos440.erdos_440.parts.i, Erdos440.erdos_440.parts.ii, Erdos440.erdos_440.variants.erdos_szemeredi","parent":6585,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-02/index.html#erdos-problem-440","scope":"formal-conjectures/erdos-problems/400-499/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/440.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 440"},{"id":"formal-conjectures/erdos-problems/400-499/part-02/erdos-problem-441","kind":"node","lean":"Erdos441.erdos_441, Erdos441.erdos_441.variants.chen_dai_infinitely_often, Erdos441.erdos_441.variants.lower_bound, Erdos441.erdos_441.variants.erdos_upper_bound, Erdos441.erdos_441.variants.chen, Erdos441.erdos_441.variants.chen_dai_upper_bound","parent":6585,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-02/index.html#erdos-problem-441","scope":"formal-conjectures/erdos-problems/400-499/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/441.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 441"},{"id":"formal-conjectures/erdos-problems/400-499/part-02/erdos-problem-442","kind":"node","lean":"Erdos442.erdos_442, Erdos442.erdos_442.variants.tao","parent":6585,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-02/index.html#erdos-problem-442","scope":"formal-conjectures/erdos-problems/400-499/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/442.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 442"},{"id":"formal-conjectures/erdos-problems/400-499/part-02/erdos-problem-443","kind":"node","lean":"Erdos443.erdos_443.parts.i, Erdos443.erdos_443.parts.ii","parent":6585,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-02/index.html#erdos-problem-443","scope":"formal-conjectures/erdos-problems/400-499/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/443.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 443"},{"id":"formal-conjectures/erdos-problems/400-499/part-02/erdos-problem-444","kind":"node","lean":"Erdos444.erdos_444","parent":6585,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-02/index.html#erdos-problem-444","scope":"formal-conjectures/erdos-problems/400-499/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/444.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 444"},{"data_url":"scopes/formal-conjectures/erdos-problems/400-499/part-03.json","id":"formal-conjectures/erdos-problems/400-499/part-03","kind":"scope","navigation":"transparent","parent":6563,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-03/index.html","scope":"formal-conjectures/erdos-problems/400-499/part-03","status":"can_prove","title":"Erd\u0151s Problem 445 to Erd\u0151s Problem 470"},{"id":"formal-conjectures/erdos-problems/400-499/part-03/erdos-problem-445","kind":"node","lean":"Erdos445.erdos_445, Erdos445.erdos_445.variants.heilbronn, Erdos445.erdos_445.variants.heath_brown","parent":6606,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-03/index.html#erdos-problem-445","scope":"formal-conjectures/erdos-problems/400-499/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/445.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 445"},{"id":"formal-conjectures/erdos-problems/400-499/part-03/erdos-problem-447","kind":"node","lean":"Erdos447.erdos_447.parts.i, Erdos447.erdos_447.parts.ii","parent":6606,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-03/index.html#erdos-problem-447","scope":"formal-conjectures/erdos-problems/400-499/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/447.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and 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Erdos450.erdos_450.linear_scale_suffices","parent":6606,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-03/index.html#erdos-problem-450","scope":"formal-conjectures/erdos-problems/400-499/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/450.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 450"},{"id":"formal-conjectures/erdos-problems/400-499/part-03/erdos-problem-452","kind":"node","lean":"Erdos452.erdos_452","parent":6606,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-03/index.html#erdos-problem-452","scope":"formal-conjectures/erdos-problems/400-499/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/452.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 452"},{"id":"formal-conjectures/erdos-problems/400-499/part-03/erdos-problem-453","kind":"node","lean":"Erdos453.erdos_453","parent":6606,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-03/index.html#erdos-problem-453","scope":"formal-conjectures/erdos-problems/400-499/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/453.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 453"},{"id":"formal-conjectures/erdos-problems/400-499/part-03/erdos-problem-454","kind":"node","lean":"Erdos454.erdos_454, Erdos454.erdos_454.variants.two_le_limsup","parent":6606,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-03/index.html#erdos-problem-454","scope":"formal-conjectures/erdos-problems/400-499/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/454.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 454"},{"id":"formal-conjectures/erdos-problems/400-499/part-03/erdos-problem-455","kind":"node","lean":"Erdos455.erdos_455, Erdos455.erdos_455.variants.liminf","parent":6606,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-03/index.html#erdos-problem-455","scope":"formal-conjectures/erdos-problems/400-499/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/455.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 455"},{"id":"formal-conjectures/erdos-problems/400-499/part-03/erdos-problem-456","kind":"node","lean":"Erdos456.erdos_456.parts.i, Erdos456.erdos_456.parts.ii, Erdos456.erdos_456.parts.iii, 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below.*","title":"Erd\u0151s Problem 458"},{"id":"formal-conjectures/erdos-problems/400-499/part-03/erdos-problem-459","kind":"node","lean":"Erdos459.erdos_459, Erdos459.erdos_459.variants.upper_tight","parent":6606,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-03/index.html#erdos-problem-459","scope":"formal-conjectures/erdos-problems/400-499/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/459.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 459"},{"id":"formal-conjectures/erdos-problems/400-499/part-03/erdos-problem-461","kind":"node","lean":"Erdos461.erdos_461","parent":6606,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-03/index.html#erdos-problem-461","scope":"formal-conjectures/erdos-problems/400-499/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/461.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 461"},{"id":"formal-conjectures/erdos-problems/400-499/part-03/erdos-problem-462","kind":"node","lean":"Erdos462.erdos_462","parent":6606,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-03/index.html#erdos-problem-462","scope":"formal-conjectures/erdos-problems/400-499/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/462.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 462"},{"id":"formal-conjectures/erdos-problems/400-499/part-03/erdos-problem-463","kind":"node","lean":"Erdos463.erdos_463","parent":6606,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-03/index.html#erdos-problem-463","scope":"formal-conjectures/erdos-problems/400-499/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/463.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 463"},{"id":"formal-conjectures/erdos-problems/400-499/part-03/erdos-problem-464","kind":"node","lean":"Erdos464.erdos_464","parent":6606,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-03/index.html#erdos-problem-464","scope":"formal-conjectures/erdos-problems/400-499/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/464.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 464"},{"id":"formal-conjectures/erdos-problems/400-499/part-03/erdos-problem-466","kind":"node","lean":"Erdos466.erdos_466, Erdos466.erdos_466.variants.graham, Erdos466.erdos_466.variants.sarkozy","parent":6606,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-03/index.html#erdos-problem-466","scope":"formal-conjectures/erdos-problems/400-499/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/466.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 466"},{"id":"formal-conjectures/erdos-problems/400-499/part-03/erdos-problem-467","kind":"node","lean":"Erdos467.erdos_467","parent":6606,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-03/index.html#erdos-problem-467","scope":"formal-conjectures/erdos-problems/400-499/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/467.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 467"},{"id":"formal-conjectures/erdos-problems/400-499/part-03/erdos-problem-469","kind":"node","lean":"Erdos469.erdos_469","parent":6606,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-03/index.html#erdos-problem-469","scope":"formal-conjectures/erdos-problems/400-499/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/469.lean in Formal Conjectures (Google DeepMind) at commit 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472"},{"id":"formal-conjectures/erdos-problems/400-499/part-04/erdos-problem-473","kind":"node","lean":"Erdos473.erdos_473, Erdos473.erdos_473.variants.finite","parent":6627,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-04/index.html#erdos-problem-473","scope":"formal-conjectures/erdos-problems/400-499/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/473.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 473"},{"id":"formal-conjectures/erdos-problems/400-499/part-04/erdos-problem-476","kind":"node","lean":"Erdos476.erdos_476","parent":6627,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-04/index.html#erdos-problem-476","scope":"formal-conjectures/erdos-problems/400-499/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/476.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 476"},{"id":"formal-conjectures/erdos-problems/400-499/part-04/erdos-problem-477","kind":"node","lean":"Erdos477.erdos_477, Erdos477.erdos_477.variants.S_sq, Erdos477.erdos_477.variants.degree_two_dvd_condition_b_ne_zero, Erdos477.erdos_477.variants.X_pow_three, Erdos477.erdos_477.variants.monomial","parent":6627,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-04/index.html#erdos-problem-477","scope":"formal-conjectures/erdos-problems/400-499/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/477.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 477"},{"id":"formal-conjectures/erdos-problems/400-499/part-04/erdos-problem-478","kind":"node","lean":"Erdos478.erdos_478","parent":6627,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-04/index.html#erdos-problem-478","scope":"formal-conjectures/erdos-problems/400-499/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/478.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 478"},{"id":"formal-conjectures/erdos-problems/400-499/part-04/erdos-problem-479","kind":"node","lean":"Erdos479.erdos_479","parent":6627,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-04/index.html#erdos-problem-479","scope":"formal-conjectures/erdos-problems/400-499/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/479.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 479"},{"id":"formal-conjectures/erdos-problems/400-499/part-04/erdos-problem-480","kind":"node","lean":"Erdos480.erdos_480, Erdos480.erdos_480.variants.chung_graham, Erdos480.erdos_480.variants.chung_graham_best_possible","parent":6627,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-04/index.html#erdos-problem-480","scope":"formal-conjectures/erdos-problems/400-499/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/480.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 480"},{"id":"formal-conjectures/erdos-problems/400-499/part-04/erdos-problem-481","kind":"node","lean":"Erdos481.erdos_481","parent":6627,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-04/index.html#erdos-problem-481","scope":"formal-conjectures/erdos-problems/400-499/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/481.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 481"},{"id":"formal-conjectures/erdos-problems/400-499/part-04/erdos-problem-482","kind":"node","lean":"Erdos482.erdos_482, Erdos482.erdos_482.variants.stoll_general, Erdos482.erdos_482.variants.binary_explicit, 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483"},{"id":"formal-conjectures/erdos-problems/400-499/part-04/erdos-problem-484","kind":"node","lean":"Erdos484.erdos_484","parent":6627,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-04/index.html#erdos-problem-484","scope":"formal-conjectures/erdos-problems/400-499/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/484.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 484"},{"id":"formal-conjectures/erdos-problems/400-499/part-04/erdos-problem-485","kind":"node","lean":"Erdos485.erdos_485","parent":6627,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-04/index.html#erdos-problem-485","scope":"formal-conjectures/erdos-problems/400-499/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/485.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 485"},{"id":"formal-conjectures/erdos-problems/400-499/part-04/erdos-problem-486-logarithmic-density-for-sets-a","kind":"node","lean":"Erdos486.erdos_486","parent":6627,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-04/index.html#erdos-problem-486-logarithmic-density-for-sets-a","scope":"formal-conjectures/erdos-problems/400-499/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/486.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 486: Logarithmic density for sets avoiding modular 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2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 488"},{"id":"formal-conjectures/erdos-problems/400-499/part-04/erdos-problem-489","kind":"node","lean":"Erdos489.erdos_489, Erdos489.erdos_489.variants.squarefree","parent":6627,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-04/index.html#erdos-problem-489","scope":"formal-conjectures/erdos-problems/400-499/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/489.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 489"},{"id":"formal-conjectures/erdos-problems/400-499/part-04/erdos-problem-490","kind":"node","lean":"Erdos490.erdos_490, Erdos490.erdos_490.variants.limit","parent":6627,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-04/index.html#erdos-problem-490","scope":"formal-conjectures/erdos-problems/400-499/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/490.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 490"},{"id":"formal-conjectures/erdos-problems/400-499/part-04/erdos-problem-491","kind":"node","lean":"Erdos491.erdos_491","parent":6627,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-04/index.html#erdos-problem-491","scope":"formal-conjectures/erdos-problems/400-499/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/491.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 491"},{"id":"formal-conjectures/erdos-problems/400-499/part-04/erdos-problem-493","kind":"node","lean":"Erdos493.erdos_493","parent":6627,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-04/index.html#erdos-problem-493","scope":"formal-conjectures/erdos-problems/400-499/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/493.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 493"},{"id":"formal-conjectures/erdos-problems/400-499/part-04/erdos-problem-494","kind":"node","lean":"Erdos494.erdos_494.variants.k_eq_2_card_not_pow_two, Erdos494.erdos_494.variants.k_eq_2_card_pow_two, Erdos494.erdos_494.variants.k_eq_3_card_gt_6, Erdos494.erdos_494.variants.k_eq_4_card_gt_12, Erdos494.erdos_494.variants.card_divisible_by_prime_gt_k, Erdos494.erdos_494.variants.k_eq_card, Erdos494.erdos_494.variants.card_eq_2k, 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499"},{"id":"formal-conjectures/erdos-problems/400-499/part-05/erdos-problem-495","kind":"node","lean":"Erdos495.erdos_495","parent":6648,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-05/index.html#erdos-problem-495","scope":"formal-conjectures/erdos-problems/400-499/part-05","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/495.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 495"},{"id":"formal-conjectures/erdos-problems/400-499/part-05/erdos-problem-497","kind":"node","lean":"Erdos497.erdos_497","parent":6648,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-05/index.html#erdos-problem-497","scope":"formal-conjectures/erdos-problems/400-499/part-05","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/497.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 497"},{"id":"formal-conjectures/erdos-problems/400-499/part-05/erdos-problem-498","kind":"node","lean":"Erdos498.erdos_498","parent":6648,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-05/index.html#erdos-problem-498","scope":"formal-conjectures/erdos-problems/400-499/part-05","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/498.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 498"},{"id":"formal-conjectures/erdos-problems/400-499/part-05/erdos-problem-499","kind":"node","lean":"Erdos499.erdos_499, Erdos499.vanDerWaerden, Erdos499.erdos_499.variants.one_le","parent":6648,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/400-499/part-05/index.html#erdos-problem-499","scope":"formal-conjectures/erdos-problems/400-499/part-05","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/499.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 499"},{"data_url":"scopes/formal-conjectures/erdos-problems/500-599.json","id":"formal-conjectures/erdos-problems/500-599","kind":"scope","parent":6191,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/index.html","scope":"formal-conjectures/erdos-problems/500-599","status":"can_prove","summary":"The Erd\u0151s problems numbered 500 to 599 that Formal Conjectures states, one article each.","title":"Problems 500 to 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Erdos504.erdos_504.variants.sendov","parent":6654,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-01/index.html#erdos-problem-504","scope":"formal-conjectures/erdos-problems/500-599/part-01","status":"can_prove","summary":"> > > > >","title":"Erd\u0151s Problem 504"},{"id":"formal-conjectures/erdos-problems/500-599/part-01/erdos-problem-505","kind":"node","lean":"Erdos505.erdos_505, Erdos505.erdos_505.small_dim","parent":6654,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-01/index.html#erdos-problem-505","scope":"formal-conjectures/erdos-problems/500-599/part-01","status":"can_prove","summary":"> > > > > > > >","title":"Erd\u0151s Problem 505"},{"id":"formal-conjectures/erdos-problems/500-599/part-01/erdos-problem-506","kind":"node","lean":"Erdos506.erdos_506, Erdos506.erdos_506.variants.large_n, Erdos506.erdos_506.variants.small_n, 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Erdos507.erdos_507.variants.upper_cpz24","parent":6654,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-01/index.html#erdos-problem-507","scope":"formal-conjectures/erdos-problems/500-599/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/507.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 507"},{"id":"formal-conjectures/erdos-problems/500-599/part-01/erdos-problem-508","kind":"node","lean":"Erdos508.HadwigerNelsonProblem, Erdos508.HadwigerNelsonAtLeastFive, Erdos508.HadwigerNelsonAtLeastSix, Erdos508.HadwigerNelsonAtLeast4, Erdos508.HadwigerNelsonAtMostSeven, Erdos508.HadwigerNelsonAtLeastThree","parent":6654,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-01/index.html#erdos-problem-508","scope":"formal-conjectures/erdos-problems/500-599/part-01","status":"can_prove","summary":"> 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FormalConjectures/ErdosProblems/520.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 520"},{"id":"formal-conjectures/erdos-problems/500-599/part-01/erdos-problem-521","kind":"node","lean":"Erdos521.erdos_521","parent":6654,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-01/index.html#erdos-problem-521","scope":"formal-conjectures/erdos-problems/500-599/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/521.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 521"},{"id":"formal-conjectures/erdos-problems/500-599/part-01/erdos-problem-522","kind":"node","lean":"Erdos522.erdos_522, Erdos522.erdos_522.variants.zero_one, Erdos522.erdos_522.variants.number_real_roots, 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FormalConjectures/ErdosProblems/525.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 525"},{"data_url":"scopes/formal-conjectures/erdos-problems/500-599/part-02.json","id":"formal-conjectures/erdos-problems/500-599/part-02","kind":"scope","navigation":"transparent","parent":6653,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-02/index.html","scope":"formal-conjectures/erdos-problems/500-599/part-02","status":"can_prove","title":"Erd\u0151s Problem 532 to Erd\u0151s Problem 552"},{"id":"formal-conjectures/erdos-problems/500-599/part-02/erdos-problem-532","kind":"node","lean":"Erdos532.erdos_532","parent":6675,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-02/index.html#erdos-problem-532","scope":"formal-conjectures/erdos-problems/500-599/part-02","status":"can_prove","summary":"*Quoted from 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FormalConjectures/ErdosProblems/538.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 538"},{"id":"formal-conjectures/erdos-problems/500-599/part-02/erdos-problem-539","kind":"node","lean":"Erdos539.erdos_539, Erdos539.erdos_539.variants.sq, Erdos539.erdos_539.variants.sq_isBigO, Erdos539.erdos_539.variants.isBigO_sq, Erdos539.erdos_539.variants.sq_cube_root, Erdos539.erdos_539.variants.isBigO_sq_cube_root, Erdos539.erdos_539.variants.sq_cube_root_isBigO, Erdos539.erdos_539.variants.limit","parent":6675,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-02/index.html#erdos-problem-539","scope":"formal-conjectures/erdos-problems/500-599/part-02","status":"can_prove","summary":"> > >","title":"Erd\u0151s Problem 539"},{"id":"formal-conjectures/erdos-problems/500-599/part-02/erdos-problem-540","kind":"node","lean":"Erdos540.erdos_540","parent":6675,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-02/index.html#erdos-problem-540","scope":"formal-conjectures/erdos-problems/500-599/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/540.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 540"},{"id":"formal-conjectures/erdos-problems/500-599/part-02/erdos-problem-541","kind":"node","lean":"Erdos541.erdos_541, Erdos541.erdos_541.variants.general_moduli, Erdos541.erdos_541.variants.large_primes","parent":6675,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-02/index.html#erdos-problem-541","scope":"formal-conjectures/erdos-problems/500-599/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/541.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 541"},{"id":"formal-conjectures/erdos-problems/500-599/part-02/erdos-problem-542","kind":"node","lean":"Erdos542.erdos_542.parts.i, Erdos542.erdos_542.variants.sharp, Erdos542.erdos_542.parts.ii, Erdos542.erdos_542.variants.schinzel_szekeres, Erdos542.erdos_542.variants.log_power, Erdos542.erdos_542.variants.chen, Erdos542.erdos_542.variants.one_add_little_o, Erdos542.erdos_542.variants.only_two","parent":6675,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-02/index.html#erdos-problem-542","scope":"formal-conjectures/erdos-problems/500-599/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/542.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 542"},{"id":"formal-conjectures/erdos-problems/500-599/part-02/erdos-problem-544","kind":"node","lean":"Erdos544.erdos_544.parts.i, Erdos544.erdos_544.parts.ii","parent":6675,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-02/index.html#erdos-problem-544","scope":"formal-conjectures/erdos-problems/500-599/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/544.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 544"},{"id":"formal-conjectures/erdos-problems/500-599/part-02/erdos-problem-545","kind":"node","lean":"Erdos545.erdos_545","parent":6675,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-02/index.html#erdos-problem-545","scope":"formal-conjectures/erdos-problems/500-599/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/545.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 545"},{"id":"formal-conjectures/erdos-problems/500-599/part-02/erdos-problem-546","kind":"node","lean":"Erdos546.erdos_546","parent":6675,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-02/index.html#erdos-problem-546","scope":"formal-conjectures/erdos-problems/500-599/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/546.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 546"},{"id":"formal-conjectures/erdos-problems/500-599/part-02/erdos-problem-547","kind":"node","lean":"Erdos547.erdos_547, Erdos547.erdos_547.variants.sufficiently_large","parent":6675,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-02/index.html#erdos-problem-547","scope":"formal-conjectures/erdos-problems/500-599/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/547.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 547"},{"id":"formal-conjectures/erdos-problems/500-599/part-02/erdos-problem-548","kind":"node","lean":"Erdos548.erdos_548","parent":6675,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-02/index.html#erdos-problem-548","scope":"formal-conjectures/erdos-problems/500-599/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/548.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 548"},{"id":"formal-conjectures/erdos-problems/500-599/part-02/erdos-problem-549","kind":"node","lean":"Erdos549.erdos_549","parent":6675,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-02/index.html#erdos-problem-549","scope":"formal-conjectures/erdos-problems/500-599/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/549.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 549"},{"id":"formal-conjectures/erdos-problems/500-599/part-02/erdos-problem-550","kind":"node","lean":"Erdos550.erdos_550","parent":6675,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-02/index.html#erdos-problem-550","scope":"formal-conjectures/erdos-problems/500-599/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/550.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 550"},{"id":"formal-conjectures/erdos-problems/500-599/part-02/erdos-problem-551","kind":"node","lean":"Erdos551.erdos_551, Erdos551.erdos_551.variants.sufficiently_large","parent":6675,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-02/index.html#erdos-problem-551","scope":"formal-conjectures/erdos-problems/500-599/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/551.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 551"},{"id":"formal-conjectures/erdos-problems/500-599/part-02/erdos-problem-552","kind":"node","lean":"Erdos552.erdos_552.parts.i, Erdos552.erdos_552.parts.ii, Erdos552.erdos_552.variants.succ_le_add_two","parent":6675,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-02/index.html#erdos-problem-552","scope":"formal-conjectures/erdos-problems/500-599/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/552.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 552"},{"data_url":"scopes/formal-conjectures/erdos-problems/500-599/part-03.json","id":"formal-conjectures/erdos-problems/500-599/part-03","kind":"scope","navigation":"transparent","parent":6653,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-03/index.html","scope":"formal-conjectures/erdos-problems/500-599/part-03","status":"can_prove","title":"Erd\u0151s Problem 554 to Erd\u0151s Problem 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554"},{"id":"formal-conjectures/erdos-problems/500-599/part-03/erdos-problem-555","kind":"node","lean":"Erdos555.erdos_555","parent":6696,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-03/index.html#erdos-problem-555","scope":"formal-conjectures/erdos-problems/500-599/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/555.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 555"},{"id":"formal-conjectures/erdos-problems/500-599/part-03/erdos-problem-556","kind":"node","lean":"Erdos556.erdos_556, Erdos556.erdos_556.variants.luczak_asymptotic, Erdos556.erdos_556.variants.luczak_asymptotic_even, Erdos556.erdos_556.variants.odd_lower_bound, Erdos556.erdos_556.variants.kss_large_odd, Erdos556.erdos_556.variants.benevides_skokan_large_even","parent":6696,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-03/index.html#erdos-problem-556","scope":"formal-conjectures/erdos-problems/500-599/part-03","status":"can_prove","summary":"> > >","title":"Erd\u0151s Problem 556"},{"id":"formal-conjectures/erdos-problems/500-599/part-03/erdos-problem-557","kind":"node","lean":"Erdos557.erdos_557, Erdos557.erdos_557.variants.explicit, Erdos557.erdos_557.variants.tight, Erdos557.erdos_557.variants.tight_odd","parent":6696,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-03/index.html#erdos-problem-557","scope":"formal-conjectures/erdos-problems/500-599/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/557.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 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below.*","title":"Erd\u0151s Problem 561"},{"id":"formal-conjectures/erdos-problems/500-599/part-03/erdos-problem-562","kind":"node","lean":"Erdos562.erdos_562","parent":6696,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-03/index.html#erdos-problem-562","scope":"formal-conjectures/erdos-problems/500-599/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/562.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 562"},{"id":"formal-conjectures/erdos-problems/500-599/part-03/erdos-problem-563","kind":"node","lean":"Erdos563.erdos_563","parent":6696,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-03/index.html#erdos-problem-563","scope":"formal-conjectures/erdos-problems/500-599/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/563.lean in Formal 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564"},{"id":"formal-conjectures/erdos-problems/500-599/part-03/erdos-problem-566","kind":"node","lean":"Erdos566.erdos_566","parent":6696,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-03/index.html#erdos-problem-566","scope":"formal-conjectures/erdos-problems/500-599/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/566.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 566"},{"id":"formal-conjectures/erdos-problems/500-599/part-03/erdos-problem-567","kind":"node","lean":"Erdos567.erdos_567.parts.i, Erdos567.erdos_567.parts.ii, Erdos567.erdos_567.parts.iii","parent":6696,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-03/index.html#erdos-problem-567","scope":"formal-conjectures/erdos-problems/500-599/part-03","status":"can_prove","summary":"> > > >","title":"Erd\u0151s Problem 567"},{"id":"formal-conjectures/erdos-problems/500-599/part-03/erdos-problem-568","kind":"node","lean":"Erdos568.erdos_568","parent":6696,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-03/index.html#erdos-problem-568","scope":"formal-conjectures/erdos-problems/500-599/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/568.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 568"},{"id":"formal-conjectures/erdos-problems/500-599/part-03/erdos-problem-569","kind":"node","lean":"Erdos569.erdos_569","parent":6696,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-03/index.html#erdos-problem-569","scope":"formal-conjectures/erdos-problems/500-599/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/569.lean in Formal Conjectures (Google DeepMind) at commit 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570"},{"id":"formal-conjectures/erdos-problems/500-599/part-03/erdos-problem-571","kind":"node","lean":"Erdos571.erdos_571","parent":6696,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-03/index.html#erdos-problem-571","scope":"formal-conjectures/erdos-problems/500-599/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/571.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 571"},{"id":"formal-conjectures/erdos-problems/500-599/part-03/erdos-problem-572","kind":"node","lean":"Erdos572.erdos_572","parent":6696,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-03/index.html#erdos-problem-572","scope":"formal-conjectures/erdos-problems/500-599/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/572.lean in Formal Conjectures (Google DeepMind) at commit 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Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 578"},{"data_url":"scopes/formal-conjectures/erdos-problems/500-599/part-04.json","id":"formal-conjectures/erdos-problems/500-599/part-04","kind":"scope","navigation":"transparent","parent":6653,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-04/index.html","scope":"formal-conjectures/erdos-problems/500-599/part-04","status":"can_prove","title":"Erd\u0151s Problem 579 to Erd\u0151s Problem 599"},{"id":"formal-conjectures/erdos-problems/500-599/part-04/erdos-problem-579","kind":"node","lean":"Erdos579.erdos_579, Erdos579.erdos_579.variants.ehss_large_delta","parent":6717,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-04/index.html#erdos-problem-579","scope":"formal-conjectures/erdos-problems/500-599/part-04","status":"can_prove","summary":"*Quoted from 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581"},{"id":"formal-conjectures/erdos-problems/500-599/part-04/erdos-problem-582","kind":"node","lean":"Erdos582.erdos_582","parent":6717,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-04/index.html#erdos-problem-582","scope":"formal-conjectures/erdos-problems/500-599/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/582.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 582"},{"id":"formal-conjectures/erdos-problems/500-599/part-04/erdos-problem-583","kind":"node","lean":"Erdos583.erdos_583","parent":6717,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-04/index.html#erdos-problem-583","scope":"formal-conjectures/erdos-problems/500-599/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/583.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 583"},{"id":"formal-conjectures/erdos-problems/500-599/part-04/erdos-problem-586","kind":"node","lean":"Erdos586.erdos_586","parent":6717,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-04/index.html#erdos-problem-586","scope":"formal-conjectures/erdos-problems/500-599/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/586.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 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Erdos588.erdos_588.variants.grunbaum_order","parent":6717,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-04/index.html#erdos-problem-588","scope":"formal-conjectures/erdos-problems/500-599/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/588.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 588"},{"id":"formal-conjectures/erdos-problems/500-599/part-04/erdos-problem-590","kind":"node","lean":"Erdos590.erdos_590, Erdos590.erdos_590.variants.two, Erdos590.erdos_590.variants.ge_three_false, Erdos590.erdos_590.variants.finite_cardinal","parent":6717,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-04/index.html#erdos-problem-590","scope":"formal-conjectures/erdos-problems/500-599/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/590.lean in Formal 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Erdos596.erdos_596.variants.K4_K3_finite_ramsey","parent":6717,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-04/index.html#erdos-problem-596","scope":"formal-conjectures/erdos-problems/500-599/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/596.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 596"},{"id":"formal-conjectures/erdos-problems/500-599/part-04/erdos-problem-597","kind":"node","lean":"Erdos597.erdos_597, Erdos597.erdos_597.variants.finite, Erdos597.erdos_597.variants.erdos_hajnal, Erdos597.erdos_597.variants.baumgartner","parent":6717,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-04/index.html#erdos-problem-597","scope":"formal-conjectures/erdos-problems/500-599/part-04","status":"can_prove","summary":">","title":"Erd\u0151s Problem 597"},{"id":"formal-conjectures/erdos-problems/500-599/part-04/erdos-problem-598","kind":"node","lean":"Erdos598.erdos_598","parent":6717,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-04/index.html#erdos-problem-598","scope":"formal-conjectures/erdos-problems/500-599/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/598.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 598"},{"id":"formal-conjectures/erdos-problems/500-599/part-04/erdos-problem-599","kind":"node","lean":"Erdos599.erdos_599, Erdos599.erdos_599.variants.aharoni_berger","parent":6717,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/500-599/part-04/index.html#erdos-problem-599","scope":"formal-conjectures/erdos-problems/500-599/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/599.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 599"},{"data_url":"scopes/formal-conjectures/erdos-problems/600-699.json","id":"formal-conjectures/erdos-problems/600-699","kind":"scope","parent":6191,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/index.html","scope":"formal-conjectures/erdos-problems/600-699","status":"can_prove","summary":"The Erd\u0151s problems numbered 600 to 699 that Formal Conjectures states, one article each.","title":"Problems 600 to 699"},{"data_url":"scopes/formal-conjectures/erdos-problems/600-699/part-01.json","id":"formal-conjectures/erdos-problems/600-699/part-01","kind":"scope","navigation":"transparent","parent":6735,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-01/index.html","scope":"formal-conjectures/erdos-problems/600-699/part-01","status":"can_prove","title":"Erd\u0151s Problem 600 to 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Erdos601.erdos_601.variants.at_omega","parent":6736,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-01/index.html#erdos-problem-601","scope":"formal-conjectures/erdos-problems/600-699/part-01","status":"can_prove","summary":">","title":"Erd\u0151s Problem 601"},{"id":"formal-conjectures/erdos-problems/600-699/part-01/erdos-problem-602","kind":"node","lean":"Erdos602.erdos_602, Erdos602.erdos_602.variants.disjoint, Erdos602.erdos_602.variants.countable_index, Erdos602.erdos_602.variants.single_set, Erdos602.erdos_602.variants.empty_index, Erdos602.erdos_602.variants.unique_index, Erdos602.erdos_602.variants.two_sets, Erdos602.disjoint_without_infinite_claim.disproof","parent":6736,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-01/index.html#erdos-problem-602","scope":"formal-conjectures/erdos-problems/600-699/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/602.lean in Formal 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Erdos604.erdos_604.variants.average","parent":6736,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-01/index.html#erdos-problem-604","scope":"formal-conjectures/erdos-problems/600-699/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/604.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 604"},{"id":"formal-conjectures/erdos-problems/600-699/part-01/erdos-problem-609","kind":"node","lean":"Erdos609.erdos_609","parent":6736,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-01/index.html#erdos-problem-609","scope":"formal-conjectures/erdos-problems/600-699/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/609.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements 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2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 614"},{"id":"formal-conjectures/erdos-problems/600-699/part-01/erdos-problem-615","kind":"node","lean":"Erdos615.erdos_615, Erdos615.erdos_615.variants.fox_loh_zhao, Erdos615.erdos_615.variants.sudakov","parent":6736,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-01/index.html#erdos-problem-615","scope":"formal-conjectures/erdos-problems/600-699/part-01","status":"can_prove","summary":"> > >","title":"Erd\u0151s Problem 615"},{"id":"formal-conjectures/erdos-problems/600-699/part-01/erdos-problem-616","kind":"node","lean":"Erdos616.erdos_616","parent":6736,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-01/index.html#erdos-problem-616","scope":"formal-conjectures/erdos-problems/600-699/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/616.lean in Formal Conjectures (Google DeepMind) at 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617"},{"id":"formal-conjectures/erdos-problems/600-699/part-01/erdos-problem-618","kind":"node","lean":"Erdos618.erdos_618","parent":6736,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-01/index.html#erdos-problem-618","scope":"formal-conjectures/erdos-problems/600-699/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/618.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 618"},{"id":"formal-conjectures/erdos-problems/600-699/part-01/erdos-problem-619","kind":"node","lean":"Erdos619.erdos_619, Erdos619.erdos_619.variants.add_edges_diam_three, Erdos619.erdos_619.variants.h_three_le, Erdos619.erdos_619.variants.h_five_le, Erdos619.erdos_619.variants.h_three_lower","parent":6736,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-01/index.html#erdos-problem-619","scope":"formal-conjectures/erdos-problems/600-699/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/619.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 619"},{"id":"formal-conjectures/erdos-problems/600-699/part-01/erdos-problem-621","kind":"node","lean":"Erdos621.erdos_621","parent":6736,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-01/index.html#erdos-problem-621","scope":"formal-conjectures/erdos-problems/600-699/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/621.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements 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Problem 641"},{"id":"formal-conjectures/erdos-problems/600-699/part-02/erdos-problem-642","kind":"node","lean":"Erdos642.erdos_642, Erdos642.erdos_642.variants.upper_bound_n_pow_three_halves, Erdos642.erdos_642.variants.upper_bound_n_log_pow_eight","parent":6757,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-02/index.html#erdos-problem-642","scope":"formal-conjectures/erdos-problems/600-699/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/642.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 642"},{"id":"formal-conjectures/erdos-problems/600-699/part-02/erdos-problem-643","kind":"node","lean":"Erdos643.erdos_643.parts.i, Erdos643.erdos_643.parts.ii, Erdos643.erdos_643.variants.exact_t_ge_four","parent":6757,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-02/index.html#erdos-problem-643","scope":"formal-conjectures/erdos-problems/600-699/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/643.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 643"},{"id":"formal-conjectures/erdos-problems/600-699/part-02/erdos-problem-644","kind":"node","lean":"Erdos644.erdos_644.variants.elementary_upper_bound, Erdos644.erdos_644.parts.i, Erdos644.erdos_644.parts.ii","parent":6757,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-02/index.html#erdos-problem-644","scope":"formal-conjectures/erdos-problems/600-699/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/644.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 644"},{"id":"formal-conjectures/erdos-problems/600-699/part-02/erdos-problem-645","kind":"node","lean":"Erdos645.erdos_645","parent":6757,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-02/index.html#erdos-problem-645","scope":"formal-conjectures/erdos-problems/600-699/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/645.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 645"},{"id":"formal-conjectures/erdos-problems/600-699/part-02/erdos-problem-646","kind":"node","lean":"Erdos646.erdos_646","parent":6757,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-02/index.html#erdos-problem-646","scope":"formal-conjectures/erdos-problems/600-699/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/646.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 646"},{"id":"formal-conjectures/erdos-problems/600-699/part-02/erdos-problem-647","kind":"node","lean":"Erdos647.erdos_647, Erdos647.erdos_647.variants.twenty_four, Erdos647.erdos_647.variants.lim, Erdos647.erdos_647.variants.infinite","parent":6757,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-02/index.html#erdos-problem-647","scope":"formal-conjectures/erdos-problems/600-699/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/647.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 647"},{"id":"formal-conjectures/erdos-problems/600-699/part-02/erdos-problem-648","kind":"node","lean":"Erdos648.erdos_648","parent":6757,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-02/index.html#erdos-problem-648","scope":"formal-conjectures/erdos-problems/600-699/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/648.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 648"},{"id":"formal-conjectures/erdos-problems/600-699/part-02/erdos-problem-649","kind":"node","lean":"Erdos649.erdos_649, Erdos649.erdos_649.variants.finite, Erdos649.erdos_649.variants.no_solution_two_seven, Erdos649.erdos_649.variants.tong, Erdos649.erdos_649.variants.tong_question, Erdos649.erdos_649.variants.sampaio, Erdos649.erdos_649.variants.rmm_2020","parent":6757,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-02/index.html#erdos-problem-649","scope":"formal-conjectures/erdos-problems/600-699/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/649.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 649"},{"id":"formal-conjectures/erdos-problems/600-699/part-02/erdos-problem-650","kind":"node","lean":"Erdos650.erdos_650.parts.i, Erdos650.erdos_650.parts.ii, Erdos650.erdos_650.variants.erdos_suranyi, Erdos650.erdos_650.variants.erdos_selfridge","parent":6757,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-02/index.html#erdos-problem-650","scope":"formal-conjectures/erdos-problems/600-699/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/650.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 650"},{"id":"formal-conjectures/erdos-problems/600-699/part-02/erdos-problem-651","kind":"node","lean":"Erdos651.nonempty_cardSet, Erdos651.erdos_651, Erdos651.erdos_651.variants.pohoata_zakharov, Erdos651.erdos_651.variants.subexponential, Erdos651.erdos_651.variants.planar, Erdos651.erdos_651.variants.antitone","parent":6757,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-02/index.html#erdos-problem-651","scope":"formal-conjectures/erdos-problems/600-699/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/651.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 651"},{"id":"formal-conjectures/erdos-problems/600-699/part-02/erdos-problem-652","kind":"node","lean":"Erdos652.erdos_652, Erdos652.erdos_652.variants.first, Erdos652.erdos_652.variants.product, Erdos652.erdos_652.variants.elekes, Erdos652.erdos_652.variants.mathialagan, Erdos652.erdos_652.variants.kth_lower_bound, Erdos652.erdos_652.variants.order_sqrt","parent":6757,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-02/index.html#erdos-problem-652","scope":"formal-conjectures/erdos-problems/600-699/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/652.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 652"},{"id":"formal-conjectures/erdos-problems/600-699/part-02/erdos-problem-653","kind":"node","lean":"Erdos653.erdos_653","parent":6757,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-02/index.html#erdos-problem-653","scope":"formal-conjectures/erdos-problems/600-699/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/653.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 653"},{"id":"formal-conjectures/erdos-problems/600-699/part-02/erdos-problem-654","kind":"node","lean":"Erdos654.erdos_654.parts.i, Erdos654.erdos_654.parts.ii, Erdos654.erdos_654.variants.trivial_bound, Erdos654.erdos_654.variants.feng, Erdos654.erdos_654.variants.general_position, Erdos654.erdos_654.variants.general_position_one_sub_eps, Erdos654.erdos_654.variants.erdos_pach","parent":6757,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-02/index.html#erdos-problem-654","scope":"formal-conjectures/erdos-problems/600-699/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/654.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 654"},{"id":"formal-conjectures/erdos-problems/600-699/part-02/erdos-problem-655","kind":"node","lean":"Erdos655.erdos_655, Erdos655.erdos_655.variants.general_position","parent":6757,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-02/index.html#erdos-problem-655","scope":"formal-conjectures/erdos-problems/600-699/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/655.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 655"},{"data_url":"scopes/formal-conjectures/erdos-problems/600-699/part-03.json","id":"formal-conjectures/erdos-problems/600-699/part-03","kind":"scope","navigation":"transparent","parent":6735,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-03/index.html","scope":"formal-conjectures/erdos-problems/600-699/part-03","status":"can_prove","title":"Erd\u0151s Problem 656 to Erd\u0151s Problem 677"},{"id":"formal-conjectures/erdos-problems/600-699/part-03/erdos-problem-656","kind":"node","lean":"Erdos656.erdos_656","parent":6778,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-03/index.html#erdos-problem-656","scope":"formal-conjectures/erdos-problems/600-699/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/656.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 656"},{"id":"formal-conjectures/erdos-problems/600-699/part-03/erdos-problem-657","kind":"node","lean":"Erdos657.erdos_657, Erdos657.erdos_657.variants.higher_dim, Erdos657.erdos_657.variants.dim_one, Erdos657.erdos_657.variants.dim_one_lower, Erdos657.erdos_657.variants.dim_one_upper, Erdos657.erdos_657.variants.straus","parent":6778,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-03/index.html#erdos-problem-657","scope":"formal-conjectures/erdos-problems/600-699/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/657.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 657"},{"id":"formal-conjectures/erdos-problems/600-699/part-03/erdos-problem-658","kind":"node","lean":"Erdos658.erdos_658","parent":6778,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-03/index.html#erdos-problem-658","scope":"formal-conjectures/erdos-problems/600-699/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/658.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 658"},{"id":"formal-conjectures/erdos-problems/600-699/part-03/erdos-problem-659","kind":"node","lean":"Erdos659.erdos_659","parent":6778,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-03/index.html#erdos-problem-659","scope":"formal-conjectures/erdos-problems/600-699/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/659.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 659"},{"id":"formal-conjectures/erdos-problems/600-699/part-03/erdos-problem-660","kind":"node","lean":"Erdos660.erdos_660, Erdos660.erdos_660.variants.altman_planar, Erdos660.erdos_660.variants.Er75f","parent":6778,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-03/index.html#erdos-problem-660","scope":"formal-conjectures/erdos-problems/600-699/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/660.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 660"},{"id":"formal-conjectures/erdos-problems/600-699/part-03/erdos-problem-661","kind":"node","lean":"Erdos661.erdos_661, Erdos661.erdos_661.variants.grid_upper_bound, Erdos661.erdos_661.variants.le_minimalDistinctDistances, Erdos661.erdos_661.variants.little_o_distinct_distances, Erdos661.erdos_661.variants.lenz","parent":6778,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-03/index.html#erdos-problem-661","scope":"formal-conjectures/erdos-problems/600-699/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/661.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 661"},{"id":"formal-conjectures/erdos-problems/600-699/part-03/erdos-problem-663","kind":"node","lean":"Erdos663.erdos_663","parent":6778,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-03/index.html#erdos-problem-663","scope":"formal-conjectures/erdos-problems/600-699/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/663.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 663"},{"id":"formal-conjectures/erdos-problems/600-699/part-03/erdos-problem-664","kind":"node","lean":"Erdos664.erdos_664, Erdos664.erdos_664.variants.two_fifths, Erdos664.erdos_664.variants.block_design","parent":6778,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-03/index.html#erdos-problem-664","scope":"formal-conjectures/erdos-problems/600-699/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/664.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 664"},{"id":"formal-conjectures/erdos-problems/600-699/part-03/erdos-problem-665","kind":"node","lean":"Erdos665.erdos_665","parent":6778,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-03/index.html#erdos-problem-665","scope":"formal-conjectures/erdos-problems/600-699/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/665.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 665"},{"id":"formal-conjectures/erdos-problems/600-699/part-03/erdos-problem-666","kind":"node","lean":"Erdos666.erdos_666","parent":6778,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-03/index.html#erdos-problem-666","scope":"formal-conjectures/erdos-problems/600-699/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/666.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 666"},{"id":"formal-conjectures/erdos-problems/600-699/part-03/erdos-problem-667","kind":"node","lean":"Erdos667.erdos_667, Erdos667.erdos_667.variants.monotoneOn, Erdos667.erdos_667.variants.endpoint, Erdos667.erdos_667.variants.ramsey_bounds","parent":6778,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-03/index.html#erdos-problem-667","scope":"formal-conjectures/erdos-problems/600-699/part-03","status":"can_prove","summary":">","title":"Erd\u0151s Problem 667"},{"id":"formal-conjectures/erdos-problems/600-699/part-03/erdos-problem-668","kind":"node","lean":"Erdos668.erdos_668.parts.i, Erdos668.erdos_668.parts.ii, Erdos668.erdos_668.variants.four_points","parent":6778,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-03/index.html#erdos-problem-668","scope":"formal-conjectures/erdos-problems/600-699/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/668.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 668"},{"id":"formal-conjectures/erdos-problems/600-699/part-03/erdos-problem-669","kind":"node","lean":"Erdos669.erdos_669.parts.i, Erdos669.erdos_669.parts.ii, Erdos669.erdos_669.variants.exactly_le_atLeast, Erdos669.erdos_669.variants.k_two, Erdos669.erdos_669.variants.burr_grunbaum_sloane_exactly, Erdos669.erdos_669.variants.burr_grunbaum_sloane_atLeast, Erdos669.erdos_669.variants.upper_bound, Erdos669.erdos_669.variants.limit_le","parent":6778,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-03/index.html#erdos-problem-669","scope":"formal-conjectures/erdos-problems/600-699/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/669.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 669"},{"id":"formal-conjectures/erdos-problems/600-699/part-03/erdos-problem-670","kind":"node","lean":"Erdos670.erdos_670, Erdos670.erdos_670.variants.dim_one, Erdos670.erdos_670.variants.choose_two, Erdos670.erdos_670.variants.ho, Erdos670.erdos_670.variants.uniform_in_dim","parent":6778,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-03/index.html#erdos-problem-670","scope":"formal-conjectures/erdos-problems/600-699/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/670.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 670"},{"id":"formal-conjectures/erdos-problems/600-699/part-03/erdos-problem-672","kind":"node","lean":"Erdos672.erdos_672, Erdos672.erdos_672.variants.euler, Erdos672.erdos_672.variants.oblath","parent":6778,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-03/index.html#erdos-problem-672","scope":"formal-conjectures/erdos-problems/600-699/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/672.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 672"},{"id":"formal-conjectures/erdos-problems/600-699/part-03/erdos-problem-673","kind":"node","lean":"Erdos673.erdos_673.parts.i, Erdos673.erdos_673.parts.ii, Erdos673.erdos_673.variants.average, Erdos673.erdos_673.variants.tao","parent":6778,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-03/index.html#erdos-problem-673","scope":"formal-conjectures/erdos-problems/600-699/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/673.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 673"},{"id":"formal-conjectures/erdos-problems/600-699/part-03/erdos-problem-674","kind":"node","lean":"Erdos674.erdos_674, Erdos674.erdos_674.variants.infinite","parent":6778,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-03/index.html#erdos-problem-674","scope":"formal-conjectures/erdos-problems/600-699/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/674.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 674"},{"id":"formal-conjectures/erdos-problems/600-699/part-03/erdos-problem-675","kind":"node","lean":"Erdos675.erdos_675.parts.i, Erdos675.erdos_675.parts.ii, Erdos675.erdos_675.parts.iii, Erdos675.erdos_675.variants.squarefree, Erdos675.erdos_675.variants.brun","parent":6778,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-03/index.html#erdos-problem-675","scope":"formal-conjectures/erdos-problems/600-699/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/675.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 675"},{"id":"formal-conjectures/erdos-problems/600-699/part-03/erdos-problem-676","kind":"node","lean":"Erdos676.erdos_676","parent":6778,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-03/index.html#erdos-problem-676","scope":"formal-conjectures/erdos-problems/600-699/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/676.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 676"},{"id":"formal-conjectures/erdos-problems/600-699/part-03/erdos-problem-677","kind":"node","lean":"Erdos677.erdos_677","parent":6778,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-03/index.html#erdos-problem-677","scope":"formal-conjectures/erdos-problems/600-699/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/677.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 677"},{"data_url":"scopes/formal-conjectures/erdos-problems/600-699/part-04.json","id":"formal-conjectures/erdos-problems/600-699/part-04","kind":"scope","navigation":"transparent","parent":6735,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-04/index.html","scope":"formal-conjectures/erdos-problems/600-699/part-04","status":"can_prove","title":"Erd\u0151s Problem 678 to Erd\u0151s Problem 698"},{"id":"formal-conjectures/erdos-problems/600-699/part-04/erdos-problem-678","kind":"node","lean":"Erdos678.erdos_678, Erdos678.erdos_678.variants.infinitely_many_triples, Erdos678.erdos_678.variants.not_infinitely_many_pairs","parent":6799,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-04/index.html#erdos-problem-678","scope":"formal-conjectures/erdos-problems/600-699/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/678.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 678"},{"id":"formal-conjectures/erdos-problems/600-699/part-04/erdos-problem-679","kind":"node","lean":"Erdos679.erdos_679.parts.i, Erdos679.erdos_679.parts.ii","parent":6799,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-04/index.html#erdos-problem-679","scope":"formal-conjectures/erdos-problems/600-699/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/679.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 679"},{"id":"formal-conjectures/erdos-problems/600-699/part-04/erdos-problem-680","kind":"node","lean":"Erdos680.erdos_680.parts.i, Erdos680.erdos_680.parts.ii","parent":6799,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-04/index.html#erdos-problem-680","scope":"formal-conjectures/erdos-problems/600-699/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/680.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 680"},{"id":"formal-conjectures/erdos-problems/600-699/part-04/erdos-problem-681","kind":"node","lean":"Erdos681.erdos_681","parent":6799,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-04/index.html#erdos-problem-681","scope":"formal-conjectures/erdos-problems/600-699/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/681.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 681"},{"id":"formal-conjectures/erdos-problems/600-699/part-04/erdos-problem-682","kind":"node","lean":"Erdos682.erdos_682, Erdos682.erdos_682.variants.exceptional_count, Erdos682.erdos_682.variants.dickson_counterexample","parent":6799,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-04/index.html#erdos-problem-682","scope":"formal-conjectures/erdos-problems/600-699/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/682.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 682"},{"id":"formal-conjectures/erdos-problems/600-699/part-04/erdos-problem-683","kind":"node","lean":"Erdos683.erdos_683, Erdos683.erdos_683.variant.sylvester_schur, Erdos683.erdos_683.variant.erdos_log, Erdos683.erdos_683.variant.exp_sqrt","parent":6799,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-04/index.html#erdos-problem-683","scope":"formal-conjectures/erdos-problems/600-699/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/683.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 683"},{"id":"formal-conjectures/erdos-problems/600-699/part-04/erdos-problem-684","kind":"node","lean":"Erdos684.erdos_684.upper_bound, Erdos684.erdos_684.lower_bound","parent":6799,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-04/index.html#erdos-problem-684","scope":"formal-conjectures/erdos-problems/600-699/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/684.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 684"},{"id":"formal-conjectures/erdos-problems/600-699/part-04/erdos-problem-685","kind":"node","lean":"Erdos685.erdos_685.parts.i, Erdos685.erdos_685.parts.ii, Erdos685.erdos_685.variants.trivial_lower_bound","parent":6799,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-04/index.html#erdos-problem-685","scope":"formal-conjectures/erdos-problems/600-699/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/685.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 685"},{"id":"formal-conjectures/erdos-problems/600-699/part-04/erdos-problem-686","kind":"node","lean":"Erdos686.erdos_686, Erdos686.erdos_686.variants.square, Erdos686.erdos_686.variants.four, Erdos686.erdos_686.variants.four_two, Erdos686.erdos_686.variants.four_three, Erdos686.erdos_686.variants.nine, Erdos686.erdos_686.variants.twenty_five, Erdos686.erdos_686.variants.non_square","parent":6799,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-04/index.html#erdos-problem-686","scope":"formal-conjectures/erdos-problems/600-699/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/686.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 686"},{"id":"formal-conjectures/erdos-problems/600-699/part-04/erdos-problem-687","kind":"node","lean":"Erdos687.erdos_687.parts.i, Erdos687.erdos_687.parts.ii, Erdos687.erdos_687.parts.iii, Erdos687.erdos_687.variants.iwaniec, Erdos687.erdos_687.variants.fgkmt, Erdos687.erdos_687.variants.maier_pomerance","parent":6799,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-04/index.html#erdos-problem-687","scope":"formal-conjectures/erdos-problems/600-699/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/687.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 687"},{"id":"formal-conjectures/erdos-problems/600-699/part-04/erdos-problem-688","kind":"node","lean":"Erdos688.erdos_688.parts.i, Erdos688.erdos_688.parts.ii, Erdos688.erdos_688.variants.lglglg_over_lglg_is_big_o","parent":6799,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-04/index.html#erdos-problem-688","scope":"formal-conjectures/erdos-problems/600-699/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/688.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 688"},{"id":"formal-conjectures/erdos-problems/600-699/part-04/erdos-problem-689","kind":"node","lean":"Erdos689.erdos_689","parent":6799,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-04/index.html#erdos-problem-689","scope":"formal-conjectures/erdos-problems/600-699/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/689.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 689"},{"id":"formal-conjectures/erdos-problems/600-699/part-04/erdos-problem-690","kind":"node","lean":"Erdos690.erdos_690, Erdos690.erdos_690.variants.hasDensity, Erdos690.erdos_690.variants.cambie_unimodal, Erdos690.erdos_690.variants.cambie_not_unimodal, Erdos690.erdos_690.variants.large_k","parent":6799,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-04/index.html#erdos-problem-690","scope":"formal-conjectures/erdos-problems/600-699/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/690.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 690"},{"id":"formal-conjectures/erdos-problems/600-699/part-04/erdos-problem-691","kind":"node","lean":"Erdos691.erdos_691","parent":6799,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-04/index.html#erdos-problem-691","scope":"formal-conjectures/erdos-problems/600-699/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/691.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 691"},{"id":"formal-conjectures/erdos-problems/600-699/part-04/erdos-problem-692","kind":"node","lean":"Erdos692.erdos_692.parts.i, Erdos692.erdos_692.parts.ii, Erdos692.erdos_692.variants.erdos_upper_bound, Erdos692.erdos_692.variants.cambie_three, Erdos692.erdos_692.variants.cambie_local_maxima","parent":6799,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-04/index.html#erdos-problem-692","scope":"formal-conjectures/erdos-problems/600-699/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/692.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 692"},{"id":"formal-conjectures/erdos-problems/600-699/part-04/erdos-problem-694","kind":"node","lean":"Erdos694.erdos_694, Erdos694.erdos_694.variants.carmichael, Erdos694.erdos_694.variants.inf_unique","parent":6799,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-04/index.html#erdos-problem-694","scope":"formal-conjectures/erdos-problems/600-699/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/694.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 694"},{"id":"formal-conjectures/erdos-problems/600-699/part-04/erdos-problem-695","kind":"node","lean":"Erdos695.erdos_695, Erdos695.erdos_695.variants.upperBound","parent":6799,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-04/index.html#erdos-problem-695","scope":"formal-conjectures/erdos-problems/600-699/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/695.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 695"},{"id":"formal-conjectures/erdos-problems/600-699/part-04/erdos-problem-696","kind":"node","lean":"Erdos696.erdos_696","parent":6799,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-04/index.html#erdos-problem-696","scope":"formal-conjectures/erdos-problems/600-699/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/696.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 696"},{"id":"formal-conjectures/erdos-problems/600-699/part-04/erdos-problem-697","kind":"node","lean":"Erdos697.density_exists, Erdos697.erdos_697.variants.delta_lt, Erdos697.erdos_697.parts.i, Erdos697.erdos_697.parts.ii","parent":6799,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-04/index.html#erdos-problem-697","scope":"formal-conjectures/erdos-problems/600-699/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/697.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 697"},{"id":"formal-conjectures/erdos-problems/600-699/part-04/erdos-problem-698","kind":"node","lean":"Erdos698.erdos_698, Erdos698.erdos_698.variants.erdos_szekeres, Erdos698.erdos_698.variants.erdos_szekeres_sharp, Erdos698.erdos_698.variants.bergman","parent":6799,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-04/index.html#erdos-problem-698","scope":"formal-conjectures/erdos-problems/600-699/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/698.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 698"},{"data_url":"scopes/formal-conjectures/erdos-problems/600-699/part-05.json","id":"formal-conjectures/erdos-problems/600-699/part-05","kind":"scope","navigation":"transparent","parent":6735,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-05/index.html","scope":"formal-conjectures/erdos-problems/600-699/part-05","status":"can_prove","title":"Erd\u0151s Problem 699"},{"id":"formal-conjectures/erdos-problems/600-699/part-05/erdos-problem-699","kind":"node","lean":"Erdos699.sylvester_schur, Erdos699.erdos_699, Erdos699.erdos_szekeres_strengthening","parent":6820,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/600-699/part-05/index.html#erdos-problem-699","scope":"formal-conjectures/erdos-problems/600-699/part-05","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/699.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 699"},{"data_url":"scopes/formal-conjectures/erdos-problems/700-799.json","id":"formal-conjectures/erdos-problems/700-799","kind":"scope","parent":6191,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/index.html","scope":"formal-conjectures/erdos-problems/700-799","status":"can_prove","summary":"The Erd\u0151s problems numbered 700 to 799 that Formal Conjectures states, one article each.","title":"Problems 700 to 799"},{"data_url":"scopes/formal-conjectures/erdos-problems/700-799/part-01.json","id":"formal-conjectures/erdos-problems/700-799/part-01","kind":"scope","navigation":"transparent","parent":6822,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-01/index.html","scope":"formal-conjectures/erdos-problems/700-799/part-01","status":"can_prove","title":"Erd\u0151s Problem 700 to Erd\u0151s Problem 722"},{"id":"formal-conjectures/erdos-problems/700-799/part-01/erdos-problem-700","kind":"node","lean":"Erdos700.erdos_700.parts.i, Erdos700.erdos_700.variants.prime_power_divisor, Erdos700.erdos_700.parts.ii, Erdos700.erdos_700.parts.iii, Erdos700.erdos_700.variants.prime_pow, Erdos700.erdos_700.variants.prime_mul","parent":6823,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-01/index.html#erdos-problem-700","scope":"formal-conjectures/erdos-problems/700-799/part-01","status":"can_prove","summary":"> >","title":"Erd\u0151s Problem 700"},{"id":"formal-conjectures/erdos-problems/700-799/part-01/erdos-problem-701","kind":"node","lean":"Erdos701.erdos_701","parent":6823,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-01/index.html#erdos-problem-701","scope":"formal-conjectures/erdos-problems/700-799/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/701.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 701"},{"id":"formal-conjectures/erdos-problems/700-799/part-01/erdos-problem-702","kind":"node","lean":"Erdos702.erdos_702, Erdos702.erdos_702.variants.all_n_false","parent":6823,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-01/index.html#erdos-problem-702","scope":"formal-conjectures/erdos-problems/700-799/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/702.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 702"},{"id":"formal-conjectures/erdos-problems/700-799/part-01/erdos-problem-703","kind":"node","lean":"Erdos703.erdos_703, Erdos703.erdos_703.variants.zero, Erdos703.erdos_703.variants.frankl_furedi","parent":6823,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-01/index.html#erdos-problem-703","scope":"formal-conjectures/erdos-problems/700-799/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/703.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 703"},{"id":"formal-conjectures/erdos-problems/700-799/part-01/erdos-problem-704","kind":"node","lean":"Erdos704.erdos_704.parts.i, Erdos704.erdos_704.parts.ii, Erdos704.erdos_704.variants.cube_colouring, 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2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 706"},{"id":"formal-conjectures/erdos-problems/700-799/part-01/erdos-problem-707-embedding-sidon-sets-in-perfec","kind":"node","lean":"Erdos707.erdos_707, Erdos707.erdos_707.variants.prime_power, Erdos707.erdos_707.variants.prime, Erdos707.erdos_707.variants.counterexample_prime, Erdos707.erdos_707.variants.counterexample_mian_chowla, Erdos707.erdos_707.variants.counterexample_hall, Erdos707.erdos_707.variants.perfect_difference_set_size_bound, Erdos707.erdos_707.variants.singer_construction, Erdos707.erdos_707.variants.example_sidon_set, Erdos707.erdos_707.variants.example_embedding, Erdos707.erdos_707.variants.small_sidon_sets","parent":6823,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-01/index.html#erdos-problem-707-embedding-sidon-sets-in-perfec","scope":"formal-conjectures/erdos-problems/700-799/part-01","status":"can_prove","summary":"> >","title":"Erd\u0151s Problem 707: Embedding Sidon Sets in Perfect Difference Sets"},{"id":"formal-conjectures/erdos-problems/700-799/part-01/erdos-problem-709","kind":"node","lean":"Erdos709.erdos_709.variants.singleton, Erdos709.erdos_709, Erdos709.erdos_709.variants.ersu_lower, Erdos709.erdos_709.variants.ersu_upper, Erdos709.erdos_709.variants.van_doorn_lower","parent":6823,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-01/index.html#erdos-problem-709","scope":"formal-conjectures/erdos-problems/700-799/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/709.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 709"},{"id":"formal-conjectures/erdos-problems/700-799/part-01/erdos-problem-710","kind":"node","lean":"Erdos710.erdos_710, Erdos710.erdos_710.variants.lower_bound, 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Lean statements below.*","title":"Erd\u0151s Problem 711"},{"id":"formal-conjectures/erdos-problems/700-799/part-01/erdos-problem-712","kind":"node","lean":"Erdos712.erdos_712, Erdos712.erdos_712.variants.limit_exists, Erdos712.erdos_712.variants.graph_case, Erdos712.erdos_712.variants.graph_case_div_sq","parent":6823,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-01/index.html#erdos-problem-712","scope":"formal-conjectures/erdos-problems/700-799/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/712.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 712"},{"id":"formal-conjectures/erdos-problems/700-799/part-01/erdos-problem-713","kind":"node","lean":"Erdos713.erdos_713.parts.i, 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in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 717"},{"id":"formal-conjectures/erdos-problems/700-799/part-01/erdos-problem-719","kind":"node","lean":"Erdos719.erdos_719","parent":6823,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-01/index.html#erdos-problem-719","scope":"formal-conjectures/erdos-problems/700-799/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/719.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 719"},{"id":"formal-conjectures/erdos-problems/700-799/part-01/erdos-problem-720","kind":"node","lean":"Erdos720.erdos_720.parts.i, Erdos720.erdos_720.parts.ii, Erdos720.erdos_720.parts.iii, 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FormalConjectures/ErdosProblems/734.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 734"},{"id":"formal-conjectures/erdos-problems/700-799/part-02/erdos-problem-735","kind":"node","lean":"Erdos735.erdos_735","parent":6844,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-02/index.html#erdos-problem-735","scope":"formal-conjectures/erdos-problems/700-799/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/735.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 735"},{"id":"formal-conjectures/erdos-problems/700-799/part-02/erdos-problem-737","kind":"node","lean":"Erdos737.erdos_737, 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are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 740: Infinitary version of chromatic number and odd cycles"},{"id":"formal-conjectures/erdos-problems/700-799/part-02/erdos-problem-741","kind":"node","lean":"Erdos741.erdos_741.parts.i, Erdos741.erdos_741.variants.lower, Erdos741.erdos_741.variants.exact_density, Erdos741.erdos_741.parts.ii","parent":6844,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-02/index.html#erdos-problem-741","scope":"formal-conjectures/erdos-problems/700-799/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/741.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 741"},{"id":"formal-conjectures/erdos-problems/700-799/part-02/erdos-problem-742","kind":"node","lean":"Erdos742.erdos_742, Erdos742.variants.plesnik_bound, Erdos742.variants.fan_bound, Erdos742.variants.furedi_bound","parent":6844,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-02/index.html#erdos-problem-742","scope":"formal-conjectures/erdos-problems/700-799/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/742.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 742"},{"id":"formal-conjectures/erdos-problems/700-799/part-02/erdos-problem-743","kind":"node","lean":"Erdos743.erdos_743, Erdos743.erdos_743.variants.stars_or_paths, Erdos743.erdos_743.variants.all_but_two_stars, Erdos743.erdos_743.variants.fishburn, Erdos743.erdos_743.variants.bollobas, Erdos743.erdos_743.variants.bounded_degree, Erdos743.erdos_743.variants.almost_linear_degree, Erdos743.erdos_743.variants.largest_trees","parent":6844,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-02/index.html#erdos-problem-743","scope":"formal-conjectures/erdos-problems/700-799/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/743.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 743"},{"id":"formal-conjectures/erdos-problems/700-799/part-02/erdos-problem-745-second-largest-component-of-a","kind":"node","lean":"Erdos745.erdos_745, Erdos745.erdos_745.variants.supercritical, Erdos745.erdos_745.variants.noncritical_asymptotic","parent":6844,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-02/index.html#erdos-problem-745-second-largest-component-of-a","scope":"formal-conjectures/erdos-problems/700-799/part-02","status":"can_prove","summary":"> > >","title":"Erd\u0151s Problem 745: Second-largest component of a random graph"},{"id":"formal-conjectures/erdos-problems/700-799/part-02/erdos-problem-747","kind":"node","lean":"Erdos747.erdos_747","parent":6844,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-02/index.html#erdos-problem-747","scope":"formal-conjectures/erdos-problems/700-799/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/747.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 747"},{"data_url":"scopes/formal-conjectures/erdos-problems/700-799/part-03.json","id":"formal-conjectures/erdos-problems/700-799/part-03","kind":"scope","navigation":"transparent","parent":6822,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-03/index.html","scope":"formal-conjectures/erdos-problems/700-799/part-03","status":"can_prove","title":"Erd\u0151s Problem 748 to Erd\u0151s Problem 772"},{"id":"formal-conjectures/erdos-problems/700-799/part-03/erdos-problem-748","kind":"node","lean":"Erdos748.erdos_748, Erdos748.erdos_748.variants.lower_bound, Erdos748.erdos_748.variants.green_sapozhenko, Erdos748.erdos_748.variants.asymptotic","parent":6865,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-03/index.html#erdos-problem-748","scope":"formal-conjectures/erdos-problems/700-799/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/748.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 748"},{"id":"formal-conjectures/erdos-problems/700-799/part-03/erdos-problem-749","kind":"node","lean":"Erdos749.erdos_749","parent":6865,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-03/index.html#erdos-problem-749","scope":"formal-conjectures/erdos-problems/700-799/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/749.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 749"},{"id":"formal-conjectures/erdos-problems/700-799/part-03/erdos-problem-750","kind":"node","lean":"Erdos750.erdos_750.variants.stiebitz, Erdos750.erdos_750, Erdos750.erdos_750.variants.epsilon, 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(Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 751"},{"id":"formal-conjectures/erdos-problems/700-799/part-03/erdos-problem-753","kind":"node","lean":"Erdos753.erdos_753, Erdos753.erdos_753.variants.alon","parent":6865,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-03/index.html#erdos-problem-753","scope":"formal-conjectures/erdos-problems/700-799/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/753.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 753"},{"id":"formal-conjectures/erdos-problems/700-799/part-03/erdos-problem-755","kind":"node","lean":"Erdos755.erdos_755, Erdos755.erdos_755.variants.any_size_cdl","parent":6865,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-03/index.html#erdos-problem-755","scope":"formal-conjectures/erdos-problems/700-799/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/755.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 755"},{"id":"formal-conjectures/erdos-problems/700-799/part-03/erdos-problem-756","kind":"node","lean":"Erdos756.erdos_756, Erdos756.erdos_756.variants.bhowmick, Erdos756.erdos_756.variants.bhowmick_general","parent":6865,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-03/index.html#erdos-problem-756","scope":"formal-conjectures/erdos-problems/700-799/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/756.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 756"},{"id":"formal-conjectures/erdos-problems/700-799/part-03/erdos-problem-757","kind":"node","lean":"Erdos757.erdos_757, Erdos757.erdos_757.variants.lowerBound, Erdos757.erdos_757.variants.upperBound","parent":6865,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-03/index.html#erdos-problem-757","scope":"formal-conjectures/erdos-problems/700-799/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/757.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 757"},{"id":"formal-conjectures/erdos-problems/700-799/part-03/erdos-problem-760","kind":"node","lean":"Erdos760.erdos_760, Erdos760.erdos_760.variants.erdos_gimbel","parent":6865,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-03/index.html#erdos-problem-760","scope":"formal-conjectures/erdos-problems/700-799/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/760.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 760"},{"id":"formal-conjectures/erdos-problems/700-799/part-03/erdos-problem-761","kind":"node","lean":"Erdos761.erdos_761.parts.i, Erdos761.erdos_761.parts.ii, Erdos761.erdos_761.variants.parts_ii_imp_i","parent":6865,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-03/index.html#erdos-problem-761","scope":"formal-conjectures/erdos-problems/700-799/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/761.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 761"},{"id":"formal-conjectures/erdos-problems/700-799/part-03/erdos-problem-762","kind":"node","lean":"Erdos762.erdos_762, Erdos762.erdos_762.variants.bounded_clique_number, Erdos762.erdos_762.variants.steiner","parent":6865,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-03/index.html#erdos-problem-762","scope":"formal-conjectures/erdos-problems/700-799/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/762.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 762"},{"id":"formal-conjectures/erdos-problems/700-799/part-03/erdos-problem-763","kind":"node","lean":"Erdos763.erdos_763, Erdos763.erdos_763.variants.erdos_fuchs, Erdos763.erdos_763.variants.montgomery_vaughan","parent":6865,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-03/index.html#erdos-problem-763","scope":"formal-conjectures/erdos-problems/700-799/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/763.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 763"},{"id":"formal-conjectures/erdos-problems/700-799/part-03/erdos-problem-764","kind":"node","lean":"Erdos764.erdos_764, Erdos764.erdos_764.variants.vaughan","parent":6865,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-03/index.html#erdos-problem-764","scope":"formal-conjectures/erdos-problems/700-799/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/764.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 764"},{"id":"formal-conjectures/erdos-problems/700-799/part-03/erdos-problem-765","kind":"node","lean":"Erdos765.erdos_765, Erdos765.erdos_765.variants.second_term","parent":6865,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-03/index.html#erdos-problem-765","scope":"formal-conjectures/erdos-problems/700-799/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/765.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 765"},{"id":"formal-conjectures/erdos-problems/700-799/part-03/erdos-problem-766","kind":"node","lean":"Erdos766.erdos_766.parts.i, Erdos766.erdos_766.parts.ii, Erdos766.erdos_766.variants.dirac_erdos, Erdos766.erdos_766.variants.minExtremalNumber_estimate, 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Erdos770.erdos_770.variants.three","parent":6865,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-03/index.html#erdos-problem-770","scope":"formal-conjectures/erdos-problems/700-799/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/770.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 770"},{"id":"formal-conjectures/erdos-problems/700-799/part-03/erdos-problem-771","kind":"node","lean":"Erdos771.erdos_771, Erdos771.erdos_771.variants.erdos_graham_lower, Erdos771.erdos_771.variants.alon_freiman_upper","parent":6865,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-03/index.html#erdos-problem-771","scope":"formal-conjectures/erdos-problems/700-799/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/771.lean in Formal Conjectures (Google DeepMind) at 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773"},{"id":"formal-conjectures/erdos-problems/700-799/part-04/erdos-problem-774","kind":"node","lean":"Erdos774.erdos_774","parent":6886,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-04/index.html#erdos-problem-774","scope":"formal-conjectures/erdos-problems/700-799/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/774.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 774"},{"id":"formal-conjectures/erdos-problems/700-799/part-04/erdos-problem-775","kind":"node","lean":"Erdos775.erdos_775, Erdos775.erdos_775.variants.gao, Erdos775.erdos_775.variants.spencer, Erdos775.erdos_775.variants.moon_moser","parent":6886,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-04/index.html#erdos-problem-775","scope":"formal-conjectures/erdos-problems/700-799/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/775.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 775"},{"id":"formal-conjectures/erdos-problems/700-799/part-04/erdos-problem-777","kind":"node","lean":"Erdos777.erdos_777.parts.i, Erdos777.erdos_777.parts.ii, Erdos777.erdos_777.parts.iii","parent":6886,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-04/index.html#erdos-problem-777","scope":"formal-conjectures/erdos-problems/700-799/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/777.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 777"},{"id":"formal-conjectures/erdos-problems/700-799/part-04/erdos-problem-779","kind":"node","lean":"Erdos779.erdos_779","parent":6886,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-04/index.html#erdos-problem-779","scope":"formal-conjectures/erdos-problems/700-799/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/779.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 779"},{"id":"formal-conjectures/erdos-problems/700-799/part-04/erdos-problem-780","kind":"node","lean":"Erdos780.erdos_780, Erdos780.erdos_780.variants.sharp","parent":6886,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-04/index.html#erdos-problem-780","scope":"formal-conjectures/erdos-problems/700-799/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/780.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 780"},{"id":"formal-conjectures/erdos-problems/700-799/part-04/erdos-problem-781","kind":"node","lean":"Erdos781.erdos_781.parts.i, Erdos781.erdos_781.parts.ii, Erdos781.erdos_781.variants.brown_erdos_freedman","parent":6886,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-04/index.html#erdos-problem-781","scope":"formal-conjectures/erdos-problems/700-799/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/781.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 781"},{"id":"formal-conjectures/erdos-problems/700-799/part-04/erdos-problem-782","kind":"node","lean":"Erdos782.erdos_782.parts.i, Erdos782.erdos_782.parts.ii, Erdos782.erdos_782.variants.no_four_term_ap, Erdos782.erdos_782.variants.dietmann_elsholtz","parent":6886,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-04/index.html#erdos-problem-782","scope":"formal-conjectures/erdos-problems/700-799/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/782.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 782"},{"id":"formal-conjectures/erdos-problems/700-799/part-04/erdos-problem-785","kind":"node","lean":"Erdos785.erdos_785, Erdos785.erdos_785.variants.danzer, Erdos785.erdos_785.variants.sarkozy_szemeredi, Erdos785.erdos_785.variants.chen_fang, Erdos785.erdos_785.variants.narkiewicz, Erdos785.erdos_785.variants.ruzsa_upper_bound, Erdos785.erdos_785.variants.ruzsa_lower_bound, Erdos785.erdos_785.variants.chen_fang_limsup, Erdos785.erdos_785.variants.chen_conjecture, Erdos785.erdos_785.variants.chen_fang_sharp","parent":6886,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-04/index.html#erdos-problem-785","scope":"formal-conjectures/erdos-problems/700-799/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/785.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 785"},{"id":"formal-conjectures/erdos-problems/700-799/part-04/erdos-problem-786","kind":"node","lean":"Erdos786.erdos_786.parts.i, Erdos786.erdos_786.parts.ii, Erdos786.erdos_786.parts.i.example, 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Erdos787.erdos_787.variants.beker","parent":6886,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-04/index.html#erdos-problem-787","scope":"formal-conjectures/erdos-problems/700-799/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/787.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 787"},{"id":"formal-conjectures/erdos-problems/700-799/part-04/erdos-problem-789","kind":"node","lean":"Erdos789.erdos_789, Erdos789.erdos_789.variants.sq, Erdos789.erdos_789.variants.isBigO_sq, Erdos789.erdos_789.variants.sq_isBigO, Erdos789.erdos_789.variants.sqrt_loglog_div_log_isBigO, Erdos789.erdos_789.variants.cube_root_linearithmic_isBigO, Erdos789.erdos_789.variants.isBigO_cube_root_linearithmic, Erdos789.erdos_789.variants.cube_root_linearithmic","parent":6886,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-04/index.html#erdos-problem-789","scope":"formal-conjectures/erdos-problems/700-799/part-04","status":"can_prove","summary":"> > >","title":"Erd\u0151s Problem 789"},{"id":"formal-conjectures/erdos-problems/700-799/part-04/erdos-problem-793","kind":"node","lean":"Erdos793.erdos_793, Erdos793.erdos_793.variants.constant","parent":6886,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-04/index.html#erdos-problem-793","scope":"formal-conjectures/erdos-problems/700-799/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/793.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 793"},{"id":"formal-conjectures/erdos-problems/700-799/part-04/erdos-problem-794","kind":"node","lean":"Erdos794.erdos_794, Erdos794.erdos_794.variants.balogh, Erdos794.erdos_794.variants.harris, Erdos794.erdos_794.variants.frankl_furedi","parent":6886,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-04/index.html#erdos-problem-794","scope":"formal-conjectures/erdos-problems/700-799/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/794.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 794"},{"id":"formal-conjectures/erdos-problems/700-799/part-04/erdos-problem-796","kind":"node","lean":"Erdos796.erdos_796","parent":6886,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-04/index.html#erdos-problem-796","scope":"formal-conjectures/erdos-problems/700-799/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/796.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 796"},{"id":"formal-conjectures/erdos-problems/700-799/part-04/erdos-problem-798","kind":"node","lean":"Erdos798.erdos_798, Erdos798.erdos_798.variants.lower_bound, Erdos798.erdos_798.variants.upper_bound","parent":6886,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/700-799/part-04/index.html#erdos-problem-798","scope":"formal-conjectures/erdos-problems/700-799/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/798.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 798"},{"data_url":"scopes/formal-conjectures/erdos-problems/800-899.json","id":"formal-conjectures/erdos-problems/800-899","kind":"scope","parent":6191,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/800-899/index.html","scope":"formal-conjectures/erdos-problems/800-899","status":"can_prove","summary":"The Erd\u0151s problems numbered 800 to 899 that Formal Conjectures states, one article each.","title":"Problems 800 to 899"},{"data_url":"scopes/formal-conjectures/erdos-problems/800-899/part-01.json","id":"formal-conjectures/erdos-problems/800-899/part-01","kind":"scope","navigation":"transparent","parent":6903,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/800-899/part-01/index.html","scope":"formal-conjectures/erdos-problems/800-899/part-01","status":"can_prove","title":"Erd\u0151s Problem 800 to Erd\u0151s Problem 823"},{"id":"formal-conjectures/erdos-problems/800-899/part-01/erdos-problem-800","kind":"node","lean":"Erdos800.erdos_800","parent":6904,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/800-899/part-01/index.html#erdos-problem-800","scope":"formal-conjectures/erdos-problems/800-899/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/800.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 800"},{"id":"formal-conjectures/erdos-problems/800-899/part-01/erdos-problem-801","kind":"node","lean":"Erdos801.erdos_801","parent":6904,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/800-899/part-01/index.html#erdos-problem-801","scope":"formal-conjectures/erdos-problems/800-899/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/801.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 801"},{"id":"formal-conjectures/erdos-problems/800-899/part-01/erdos-problem-802","kind":"node","lean":"Erdos802.erdos_802, Erdos802.erdos_802.variants.aeks, Erdos802.erdos_802.variants.shearer, Erdos802.erdos_802.variants.r_eq_three, 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Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 803"},{"id":"formal-conjectures/erdos-problems/800-899/part-01/erdos-problem-804","kind":"node","lean":"Erdos804.erdos_804.parts.i, Erdos804.erdos_804.parts.ii, Erdos804.erdos_804.variants.alon_sudakov","parent":6904,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/800-899/part-01/index.html#erdos-problem-804","scope":"formal-conjectures/erdos-problems/800-899/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/804.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 804"},{"id":"formal-conjectures/erdos-problems/800-899/part-01/erdos-problem-805","kind":"node","lean":"Erdos805.erdos_805.parts.i, Erdos805.erdos_805.parts.ii, Erdos805.erdos_805.variants.alon_sudakov, 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at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 806"},{"id":"formal-conjectures/erdos-problems/800-899/part-01/erdos-problem-808","kind":"node","lean":"Erdos808.erdos_808, Erdos808.erdos_808.variants.positive, Erdos808.erdos_808.variants.ars_lower_bound","parent":6904,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/800-899/part-01/index.html#erdos-problem-808","scope":"formal-conjectures/erdos-problems/800-899/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/808.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 808"},{"id":"formal-conjectures/erdos-problems/800-899/part-01/erdos-problem-810","kind":"node","lean":"Erdos810.erdos_810","parent":6904,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/800-899/part-01/index.html#erdos-problem-810","scope":"formal-conjectures/erdos-problems/800-899/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/810.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 810"},{"id":"formal-conjectures/erdos-problems/800-899/part-01/erdos-problem-812","kind":"node","lean":"Erdos812.erdos_812.parts.i, Erdos812.erdos_812.parts.ii, Erdos812.erdos_812.variants.lower_bound","parent":6904,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/800-899/part-01/index.html#erdos-problem-812","scope":"formal-conjectures/erdos-problems/800-899/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/812.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 812"},{"id":"formal-conjectures/erdos-problems/800-899/part-01/erdos-problem-813","kind":"node","lean":"Erdos813.erdos_813, Erdos813.erdos_813.variants.erdos_hajnal, Erdos813.erdos_813.variants.bucic_sudakov, Erdos813.erdos_813.variants.lower, Erdos813.erdos_813.variants.upper","parent":6904,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/800-899/part-01/index.html#erdos-problem-813","scope":"formal-conjectures/erdos-problems/800-899/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/813.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 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Erdos816.erdos_816.variants.complete_bipartite","parent":6904,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/800-899/part-01/index.html#erdos-problem-816","scope":"formal-conjectures/erdos-problems/800-899/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/816.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 816"},{"id":"formal-conjectures/erdos-problems/800-899/part-01/erdos-problem-817","kind":"node","lean":"Erdos817.erdos_817, Erdos817.erdos_817.variants.bdd_power","parent":6904,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/800-899/part-01/index.html#erdos-problem-817","scope":"formal-conjectures/erdos-problems/800-899/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/817.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 817"},{"id":"formal-conjectures/erdos-problems/800-899/part-01/erdos-problem-818","kind":"node","lean":"Erdos818.erdos_818, Erdos818.erdos_818.variants.solymosi","parent":6904,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/800-899/part-01/index.html#erdos-problem-818","scope":"formal-conjectures/erdos-problems/800-899/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/818.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 818"},{"id":"formal-conjectures/erdos-problems/800-899/part-01/erdos-problem-819","kind":"node","lean":"Erdos819.erdos_819, Erdos819.erdos_819.variants.erdos_freud_lower, 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Erdos891.erdos_891.variants.schinzel, Erdos891.erdos_891.variants.case_k_2, Erdos891.erdos_891.variants.weisenberg","parent":6967,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/800-899/part-04/index.html#erdos-problem-891","scope":"formal-conjectures/erdos-problems/800-899/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/891.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 891"},{"data_url":"scopes/formal-conjectures/erdos-problems/800-899/part-05.json","id":"formal-conjectures/erdos-problems/800-899/part-05","kind":"scope","navigation":"transparent","parent":6903,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/800-899/part-05/index.html","scope":"formal-conjectures/erdos-problems/800-899/part-05","status":"can_prove","title":"Erd\u0151s Problem 893 to Erd\u0151s Problem 899"},{"id":"formal-conjectures/erdos-problems/800-899/part-05/erdos-problem-893","kind":"node","lean":"Erdos893.erdos_893, Erdos893.erdos_893.variants.unbounded","parent":6988,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/800-899/part-05/index.html#erdos-problem-893","scope":"formal-conjectures/erdos-problems/800-899/part-05","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/893.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 893"},{"id":"formal-conjectures/erdos-problems/800-899/part-05/erdos-problem-894","kind":"node","lean":"Erdos894.erdos_894","parent":6988,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/800-899/part-05/index.html#erdos-problem-894","scope":"formal-conjectures/erdos-problems/800-899/part-05","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/894.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 894"},{"id":"formal-conjectures/erdos-problems/800-899/part-05/erdos-problem-895","kind":"node","lean":"Erdos895.erdos_895, Erdos895.erdos_895.variants.eighteen, Erdos895.erdos_895.variants.hindman","parent":6988,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/800-899/part-05/index.html#erdos-problem-895","scope":"formal-conjectures/erdos-problems/800-899/part-05","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/895.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 895"},{"id":"formal-conjectures/erdos-problems/800-899/part-05/erdos-problem-897","kind":"node","lean":"Erdos897.erdos_897.parts.i, Erdos897.erdos_897.parts.ii, Erdos897.erdos_897.variants.log_growth, 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statements below.*","title":"Erd\u0151s Problem 898"},{"id":"formal-conjectures/erdos-problems/800-899/part-05/erdos-problem-899","kind":"node","lean":"Erdos899.erdos_899","parent":6988,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/800-899/part-05/index.html#erdos-problem-899","scope":"formal-conjectures/erdos-problems/800-899/part-05","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/899.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 899"},{"data_url":"scopes/formal-conjectures/erdos-problems/900-999.json","id":"formal-conjectures/erdos-problems/900-999","kind":"scope","parent":6191,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/index.html","scope":"formal-conjectures/erdos-problems/900-999","status":"can_prove","summary":"The Erd\u0151s problems numbered 900 to 999 that Formal Conjectures states, 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below.*","title":"Erd\u0151s Problem 900"},{"id":"formal-conjectures/erdos-problems/900-999/part-01/erdos-problem-901","kind":"node","lean":"Erdos901.erdos_901","parent":6996,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-01/index.html#erdos-problem-901","scope":"formal-conjectures/erdos-problems/900-999/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/901.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 901"},{"id":"formal-conjectures/erdos-problems/900-999/part-01/erdos-problem-902","kind":"node","lean":"Erdos902.erdos_902, Erdos902.erdos_902.variants.f_one, Erdos902.erdos_902.variants.f_two, Erdos902.erdos_902.variants.erdos_lower, Erdos902.erdos_902.variants.erdos_upper, Erdos902.erdos_902.variants.szekeres_f_three, Erdos902.erdos_902.variants.szekeres_lower","parent":6996,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-01/index.html#erdos-problem-902","scope":"formal-conjectures/erdos-problems/900-999/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/902.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 902"},{"id":"formal-conjectures/erdos-problems/900-999/part-01/erdos-problem-903","kind":"node","lean":"Erdos903.erdos_903","parent":6996,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-01/index.html#erdos-problem-903","scope":"formal-conjectures/erdos-problems/900-999/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/903.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 903"},{"id":"formal-conjectures/erdos-problems/900-999/part-01/erdos-problem-904","kind":"node","lean":"Erdos904.erdos_904","parent":6996,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-01/index.html#erdos-problem-904","scope":"formal-conjectures/erdos-problems/900-999/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/904.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 904"},{"id":"formal-conjectures/erdos-problems/900-999/part-01/erdos-problem-905","kind":"node","lean":"Erdos905.erdos_905","parent":6996,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-01/index.html#erdos-problem-905","scope":"formal-conjectures/erdos-problems/900-999/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/905.lean in Formal 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FormalConjectures/ErdosProblems/910.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 910"},{"id":"formal-conjectures/erdos-problems/900-999/part-01/erdos-problem-911","kind":"node","lean":"Erdos911.erdos_911","parent":6996,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-01/index.html#erdos-problem-911","scope":"formal-conjectures/erdos-problems/900-999/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/911.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 911"},{"id":"formal-conjectures/erdos-problems/900-999/part-01/erdos-problem-912","kind":"node","lean":"Erdos912.erdos_912.variants.selfridge, Erdos912.erdos_912, 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913"},{"id":"formal-conjectures/erdos-problems/900-999/part-01/erdos-problem-914","kind":"node","lean":"Erdos914.erdos_914, Erdos914.erdos_914.variants.equitable_colouring","parent":6996,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-01/index.html#erdos-problem-914","scope":"formal-conjectures/erdos-problems/900-999/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/914.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 914"},{"id":"formal-conjectures/erdos-problems/900-999/part-01/erdos-problem-915","kind":"node","lean":"Erdos915.erdos_915, Erdos915.erdos_915.variants.edge_disjoint","parent":6996,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-01/index.html#erdos-problem-915","scope":"formal-conjectures/erdos-problems/900-999/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/915.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 915"},{"id":"formal-conjectures/erdos-problems/900-999/part-01/erdos-problem-917","kind":"node","lean":"Erdos917.erdos_917.parts.i, Erdos917.erdos_917.parts.ii, Erdos917.erdos_917.parts.iii, Erdos917.erdos_917.variants.lower_construction, Erdos917.erdos_917.variants.not_asymptotic, Erdos917.erdos_917.variants.three_dvd, Erdos917.erdos_917.variants.dirac, Erdos917.erdos_917.variants.erdos_three_dvd, Erdos917.erdos_917.variants.stiebitz_upper, Erdos917.erdos_917.variants.luo_ma_yang","parent":6996,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-01/index.html#erdos-problem-917","scope":"formal-conjectures/erdos-problems/900-999/part-01","status":"can_prove","summary":">","title":"Erd\u0151s Problem 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920"},{"id":"formal-conjectures/erdos-problems/900-999/part-01/erdos-problem-921","kind":"node","lean":"Erdos921.erdos_921","parent":6996,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-01/index.html#erdos-problem-921","scope":"formal-conjectures/erdos-problems/900-999/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/921.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 921"},{"id":"formal-conjectures/erdos-problems/900-999/part-01/erdos-problem-922","kind":"node","lean":"Erdos922.erdos_922","parent":6996,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-01/index.html#erdos-problem-922","scope":"formal-conjectures/erdos-problems/900-999/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/922.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 922"},{"id":"formal-conjectures/erdos-problems/900-999/part-01/erdos-problem-923","kind":"node","lean":"Erdos923.erdos_923","parent":6996,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-01/index.html#erdos-problem-923","scope":"formal-conjectures/erdos-problems/900-999/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/923.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 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Erdos961.erdos_961.variants.jutila_ramachandra_shorey_upper_bound","parent":7038,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-03/index.html#erdos-problem-961","scope":"formal-conjectures/erdos-problems/900-999/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/961.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 961"},{"id":"formal-conjectures/erdos-problems/900-999/part-03/erdos-problem-962","kind":"node","lean":"Erdos962.erdos_962, Erdos962.erdos_962.variants.tang_lower_bound, Erdos962.erdos_962.variants.tao_upper_bound","parent":7038,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-03/index.html#erdos-problem-962","scope":"formal-conjectures/erdos-problems/900-999/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/962.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 962"},{"id":"formal-conjectures/erdos-problems/900-999/part-03/erdos-problem-963","kind":"node","lean":"Erdos963.erdos_963.parts.i, Erdos963.erdos_963.parts.ii, Erdos963.erdos_963.variants.greedy, Erdos963.erdos_963.variants.positive","parent":7038,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-03/index.html#erdos-problem-963","scope":"formal-conjectures/erdos-problems/900-999/part-03","status":"can_prove","summary":">","title":"Erd\u0151s Problem 963"},{"id":"formal-conjectures/erdos-problems/900-999/part-03/erdos-problem-964","kind":"node","lean":"Erdos964.erdos_964, Erdos964.erdos_964.variants.eberhard","parent":7038,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-03/index.html#erdos-problem-964","scope":"formal-conjectures/erdos-problems/900-999/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/964.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 964"},{"id":"formal-conjectures/erdos-problems/900-999/part-03/erdos-problem-965","kind":"node","lean":"Erdos965.erdos_965, Erdos965.erdos_965.variants.generalization","parent":7038,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-03/index.html#erdos-problem-965","scope":"formal-conjectures/erdos-problems/900-999/part-03","status":"can_prove","summary":">","title":"Erd\u0151s Problem 965"},{"id":"formal-conjectures/erdos-problems/900-999/part-03/erdos-problem-966","kind":"node","lean":"Erdos966.erdos_966","parent":7038,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-03/index.html#erdos-problem-966","scope":"formal-conjectures/erdos-problems/900-999/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/966.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 966"},{"id":"formal-conjectures/erdos-problems/900-999/part-03/erdos-problem-967","kind":"node","lean":"Erdos967.erdos_967, Erdos967.erdos_967.variants.yip, Erdos967.erdos_967.variants.finite, Erdos967.erdos_967.variants.two_three_five, Erdos967.erdos_967.variants.tauberian","parent":7038,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-03/index.html#erdos-problem-967","scope":"formal-conjectures/erdos-problems/900-999/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/967.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 967"},{"id":"formal-conjectures/erdos-problems/900-999/part-03/erdos-problem-968","kind":"node","lean":"Erdos968.erdos_968, Erdos968.erdos_968.variants.sum_abs_diff_isTheta_log_sq, Erdos968.erdos_968.variants.decreasing_steps_pos_lower_density, Erdos968.erdos_968.variants.infinite_increasingTriples, Erdos968.erdos_968.variants.infinite_decreasingTriples","parent":7038,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-03/index.html#erdos-problem-968","scope":"formal-conjectures/erdos-problems/900-999/part-03","status":"can_prove","summary":"> > > > > >","title":"Erd\u0151s Problem 968"},{"id":"formal-conjectures/erdos-problems/900-999/part-03/erdos-problem-969","kind":"node","lean":"Erdos969.erdos_969, Erdos969.erdos_969.variants.prime_number_theorem, Erdos969.erdos_969.variants.evelyn_linfoot","parent":7038,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-03/index.html#erdos-problem-969","scope":"formal-conjectures/erdos-problems/900-999/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/969.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 969"},{"id":"formal-conjectures/erdos-problems/900-999/part-03/erdos-problem-970","kind":"node","lean":"Erdos970.erdos_970, Erdos970.erdos_970.variants.iterated_log","parent":7038,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-03/index.html#erdos-problem-970","scope":"formal-conjectures/erdos-problems/900-999/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/970.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 970"},{"data_url":"scopes/formal-conjectures/erdos-problems/900-999/part-04.json","id":"formal-conjectures/erdos-problems/900-999/part-04","kind":"scope","navigation":"transparent","parent":6995,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-04/index.html","scope":"formal-conjectures/erdos-problems/900-999/part-04","status":"can_prove","title":"Erd\u0151s Problem 971 to Erd\u0151s Problem 996"},{"id":"formal-conjectures/erdos-problems/900-999/part-04/erdos-problem-971","kind":"node","lean":"Erdos971.erdos_971, Erdos971.erdos_971.variants.infinite_sequence, Erdos971.erdos_971.variants.many_small","parent":7059,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-04/index.html#erdos-problem-971","scope":"formal-conjectures/erdos-problems/900-999/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/971.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 971"},{"id":"formal-conjectures/erdos-problems/900-999/part-04/erdos-problem-972","kind":"node","lean":"Erdos972.erdos_972","parent":7059,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-04/index.html#erdos-problem-972","scope":"formal-conjectures/erdos-problems/900-999/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/972.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 972"},{"id":"formal-conjectures/erdos-problems/900-999/part-04/erdos-problem-973","kind":"node","lean":"Erdos973.erdos_973, Erdos973.erdos_973.variants.luo_yang_zhu, Erdos973.erdos_973.variants.le_one, Erdos973.erdos_973.variants.m2_bounds, Erdos973.erdos_973.variants.tang","parent":7059,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-04/index.html#erdos-problem-973","scope":"formal-conjectures/erdos-problems/900-999/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/973.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 973"},{"id":"formal-conjectures/erdos-problems/900-999/part-04/erdos-problem-974","kind":"node","lean":"Erdos974.erdos_974, Erdos974.erdos_974.variants.two_tuples","parent":7059,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-04/index.html#erdos-problem-974","scope":"formal-conjectures/erdos-problems/900-999/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/974.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 974"},{"id":"formal-conjectures/erdos-problems/900-999/part-04/erdos-problem-975","kind":"node","lean":"Erdos975.erdos_975, Erdos975.erdos_975.variants.upper_bound, Erdos975.erdos_975.variants.lower_bound, Erdos975.erdos_975.variants.quadratic, Erdos975.erdos_975.variants.n2_plus_1_strong, Erdos975.erdos_975.variants.n2_plus_1","parent":7059,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-04/index.html#erdos-problem-975","scope":"formal-conjectures/erdos-problems/900-999/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/975.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 975"},{"id":"formal-conjectures/erdos-problems/900-999/part-04/erdos-problem-976","kind":"node","lean":"Erdos976.erdos_976, Erdos976.erdos_976.parts.i, Erdos976.erdos_976.parts.ii, Erdos976.erdos_976.variants.nagell_ricci, Erdos976.erdos_976.variants.erdos, Erdos976.erdos_976.variants.tenenbaum","parent":7059,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-04/index.html#erdos-problem-976","scope":"formal-conjectures/erdos-problems/900-999/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/976.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 976"},{"id":"formal-conjectures/erdos-problems/900-999/part-04/erdos-problem-977","kind":"node","lean":"Erdos977.erdos_977, Erdos977.erdos_977.variants.schinzel, Erdos977.erdos_977.variants.stewart, Erdos977.erdos_977.variants.factorial, Erdos977.erdos_977.variants.lai","parent":7059,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-04/index.html#erdos-problem-977","scope":"formal-conjectures/erdos-problems/900-999/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/977.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 977"},{"id":"formal-conjectures/erdos-problems/900-999/part-04/erdos-problem-978","kind":"node","lean":"Erdos978.erdos_978.variants.sub_one, Erdos978.erdos_978.parts.i, Erdos978.erdos_978.variants.sub_two, Erdos978.erdos_978.variants.allow_fixed_divisors, Erdos978.erdos_978.parts.ii, Erdos978.erdos_978.parts.iii","parent":7059,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-04/index.html#erdos-problem-978","scope":"formal-conjectures/erdos-problems/900-999/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/978.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 978"},{"id":"formal-conjectures/erdos-problems/900-999/part-04/erdos-problem-979","kind":"node","lean":"Erdos979.erdos_979, Erdos979.erdos_979.variants.k2, Erdos979.erdos_979.variants.k3","parent":7059,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-04/index.html#erdos-problem-979","scope":"formal-conjectures/erdos-problems/900-999/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/979.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 979"},{"id":"formal-conjectures/erdos-problems/900-999/part-04/erdos-problem-980","kind":"node","lean":"Erdos980.erdos_980, Erdos980.erdos_980.variants.erdos_two","parent":7059,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-04/index.html#erdos-problem-980","scope":"formal-conjectures/erdos-problems/900-999/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/980.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 980"},{"id":"formal-conjectures/erdos-problems/900-999/part-04/erdos-problem-981","kind":"node","lean":"Erdos981.erdos_981, Erdos981.erdos_981.variants.first_time","parent":7059,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-04/index.html#erdos-problem-981","scope":"formal-conjectures/erdos-problems/900-999/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/981.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 981"},{"id":"formal-conjectures/erdos-problems/900-999/part-04/erdos-problem-982","kind":"node","lean":"Erdos982.erdos_982, Erdos982.erdos_982.variants.concyclic","parent":7059,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-04/index.html#erdos-problem-982","scope":"formal-conjectures/erdos-problems/900-999/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/982.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 982"},{"id":"formal-conjectures/erdos-problems/900-999/part-04/erdos-problem-983","kind":"node","lean":"Erdos983.erdos_983.parts.i, Erdos983.erdos_983.parts.ii, Erdos983.erdos_983.variants.le_primeCounting, Erdos983.erdos_983.variants.upper_bound, Erdos983.erdos_983.variants.erdos_straus","parent":7059,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-04/index.html#erdos-problem-983","scope":"formal-conjectures/erdos-problems/900-999/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/983.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 983"},{"id":"formal-conjectures/erdos-problems/900-999/part-04/erdos-problem-985","kind":"node","lean":"Erdos985.erdos_985, Erdos985.erdos_985.variants.two_three_five_primitive_root","parent":7059,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-04/index.html#erdos-problem-985","scope":"formal-conjectures/erdos-problems/900-999/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/985.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 985"},{"id":"formal-conjectures/erdos-problems/900-999/part-04/erdos-problem-986","kind":"node","lean":"Erdos986.erdos_986","parent":7059,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-04/index.html#erdos-problem-986","scope":"formal-conjectures/erdos-problems/900-999/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/986.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 986"},{"id":"formal-conjectures/erdos-problems/900-999/part-04/erdos-problem-987","kind":"node","lean":"Erdos987.erdos_987.parts.i, Erdos987.erdos_987.variants.sup_limsup_infty, Erdos987.erdos_987.variants.log_lower_bound, Erdos987.erdos_987.variants.sqrt_lower_bound, Erdos987.erdos_987.variants.linear_upper_bound_weak, Erdos987.erdos_987.variants.linear_upper_bound_clunie, Erdos987.erdos_987.variants.linear_upper_bound, Erdos987.erdos_987.variants.sqrt_log_upper_bound, Erdos987.erdos_987.parts.ii, Erdos987.erdos_987.variants.finite_distinct_points","parent":7059,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-04/index.html#erdos-problem-987","scope":"formal-conjectures/erdos-problems/900-999/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/987.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 987"},{"id":"formal-conjectures/erdos-problems/900-999/part-04/erdos-problem-990","kind":"node","lean":"Erdos990.erdos_990, Erdos990.erdos_990.variants.erdos_turan, Erdos990.erdos_990.variants.hayman, 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docstrings and Lean statements below.*","title":"Erd\u0151s Problem 994"},{"id":"formal-conjectures/erdos-problems/900-999/part-04/erdos-problem-995","kind":"node","lean":"Erdos995.erdos_995, Erdos995.erdos_995.variants.erdos_lower_bound","parent":7059,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-04/index.html#erdos-problem-995","scope":"formal-conjectures/erdos-problems/900-999/part-04","status":"can_prove","summary":"> > > > >","title":"Erd\u0151s Problem 995"},{"id":"formal-conjectures/erdos-problems/900-999/part-04/erdos-problem-996","kind":"node","lean":"Erdos996.erdos_996, Erdos996.erdos_996.variants.log2","parent":7059,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-04/index.html#erdos-problem-996","scope":"formal-conjectures/erdos-problems/900-999/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/996.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 996"},{"data_url":"scopes/formal-conjectures/erdos-problems/900-999/part-05.json","id":"formal-conjectures/erdos-problems/900-999/part-05","kind":"scope","navigation":"transparent","parent":6995,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-05/index.html","scope":"formal-conjectures/erdos-problems/900-999/part-05","status":"can_prove","title":"Erd\u0151s Problem 997 to Erd\u0151s Problem 999"},{"id":"formal-conjectures/erdos-problems/900-999/part-05/erdos-problem-997","kind":"node","lean":"Erdos997.erdos_997, Erdos997.erdos_997.variants.lacunary, Erdos997.erdos_997.variants.irrational","parent":7080,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-05/index.html#erdos-problem-997","scope":"formal-conjectures/erdos-problems/900-999/part-05","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/997.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 997"},{"id":"formal-conjectures/erdos-problems/900-999/part-05/erdos-problem-999","kind":"node","lean":"Erdos999.erdos_999, Erdos999.erdos_999.variants.easy_direction, Erdos999.erdos_999.variants.erdos","parent":7080,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/900-999/part-05/index.html#erdos-problem-999","scope":"formal-conjectures/erdos-problems/900-999/part-05","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/999.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 999"},{"data_url":"scopes/formal-conjectures/erdos-problems/1000-1099.json","id":"formal-conjectures/erdos-problems/1000-1099","kind":"scope","parent":6191,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/1000-1099/index.html","scope":"formal-conjectures/erdos-problems/1000-1099","status":"can_prove","summary":"The Erd\u0151s problems numbered 1000 to 1099 that Formal Conjectures states, one article each.","title":"Problems 1000 to 1099"},{"data_url":"scopes/formal-conjectures/erdos-problems/1000-1099/part-01.json","id":"formal-conjectures/erdos-problems/1000-1099/part-01","kind":"scope","navigation":"transparent","parent":7083,"result":false,"route":"../roadmap/formal-conjectures/erdos-problems/1000-1099/part-01/index.html","scope":"formal-conjectures/erdos-problems/1000-1099/part-01","status":"can_prove","title":"Erd\u0151s Problem 1000 to Erd\u0151s Problem 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Erdos1001.erdos_1001.variants.est_formula","parent":7084,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1000-1099/part-01/index.html#erdos-problem-1001","scope":"formal-conjectures/erdos-problems/1000-1099/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/1001.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 1001"},{"id":"formal-conjectures/erdos-problems/1000-1099/part-01/erdos-problem-1002","kind":"node","lean":"Erdos1002.erdos_1002, Erdos1002.erdos_1002.variants.kesten","parent":7084,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1000-1099/part-01/index.html#erdos-problem-1002","scope":"formal-conjectures/erdos-problems/1000-1099/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/1002.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 1002"},{"id":"formal-conjectures/erdos-problems/1000-1099/part-01/erdos-problem-1003","kind":"node","lean":"Erdos1003.erdos_1003, Erdos1003.erdos_1003.variants.Icc, Erdos1003.erdos_1003.variants.eps87","parent":7084,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1000-1099/part-01/index.html#erdos-problem-1003","scope":"formal-conjectures/erdos-problems/1000-1099/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/1003.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 1003"},{"id":"formal-conjectures/erdos-problems/1000-1099/part-01/erdos-problem-1004","kind":"node","lean":"Erdos1004.erdos_1004, Erdos1004.erdos_1004.variants.le_of_isDistinctTotientRun","parent":7084,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1000-1099/part-01/index.html#erdos-problem-1004","scope":"formal-conjectures/erdos-problems/1000-1099/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/1004.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 1004"},{"id":"formal-conjectures/erdos-problems/1000-1099/part-01/erdos-problem-1005","kind":"node","lean":"Erdos1005.erdos_1005, Erdos1005.erdos_1005.variants.constant","parent":7084,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1000-1099/part-01/index.html#erdos-problem-1005","scope":"formal-conjectures/erdos-problems/1000-1099/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/1005.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 1005"},{"id":"formal-conjectures/erdos-problems/1000-1099/part-01/erdos-problem-1007","kind":"node","lean":"Erdos1007.erdos_1007, Erdos1007.erdos_1007.variants.dimension_four_extremal, Erdos1007.erdos_1007.variants.dimension_five, Erdos1007.erdos_1007.variants.dimension_five_extremal","parent":7084,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1000-1099/part-01/index.html#erdos-problem-1007","scope":"formal-conjectures/erdos-problems/1000-1099/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/1007.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 1007"},{"id":"formal-conjectures/erdos-problems/1000-1099/part-01/erdos-problem-1008","kind":"node","lean":"Erdos1008.erdos_1008, Erdos1008.erdos_1008.variants.three_quarters, Erdos1008.erdos_1008.variants.folkman, Erdos1008.erdos_1008.variants.lower_bound","parent":7084,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1000-1099/part-01/index.html#erdos-problem-1008","scope":"formal-conjectures/erdos-problems/1000-1099/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/1008.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 1008"},{"id":"formal-conjectures/erdos-problems/1000-1099/part-01/erdos-problem-1009","kind":"node","lean":"Erdos1009.erdos_1009, Erdos1009.erdos_1009.variants.sauer","parent":7084,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1000-1099/part-01/index.html#erdos-problem-1009","scope":"formal-conjectures/erdos-problems/1000-1099/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/1009.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 1009"},{"id":"formal-conjectures/erdos-problems/1000-1099/part-01/erdos-problem-1010","kind":"node","lean":"Erdos1010.erdos_1010, Erdos1010.erdos_1010.variants.rademacher, Erdos1010.erdos_1010.variants.erdos","parent":7084,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1000-1099/part-01/index.html#erdos-problem-1010","scope":"formal-conjectures/erdos-problems/1000-1099/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/1010.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 1010"},{"id":"formal-conjectures/erdos-problems/1000-1099/part-01/erdos-problem-1013","kind":"node","lean":"Erdos1013.erdos_1013.parts.i, Erdos1013.erdos_1013.parts.ii, Erdos1013.erdos_1013.variants.graver_yackel, 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commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 1014"},{"id":"formal-conjectures/erdos-problems/1000-1099/part-01/erdos-problem-1017","kind":"node","lean":"Erdos1017.erdos_1017.lower_bound, Erdos1017.erdos_1017.upper_bound","parent":7084,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1000-1099/part-01/index.html#erdos-problem-1017","scope":"formal-conjectures/erdos-problems/1000-1099/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/1017.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 1017"},{"id":"formal-conjectures/erdos-problems/1000-1099/part-01/erdos-problem-1020","kind":"node","lean":"Erdos1020.erdos_1020","parent":7084,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1000-1099/part-01/index.html#erdos-problem-1020","scope":"formal-conjectures/erdos-problems/1000-1099/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/1020.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 1020"},{"id":"formal-conjectures/erdos-problems/1000-1099/part-01/erdos-problem-1021","kind":"node","lean":"Erdos1021.erdos_1021, Erdos1021.erdos_1021.variants.janzer","parent":7084,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1000-1099/part-01/index.html#erdos-problem-1021","scope":"formal-conjectures/erdos-problems/1000-1099/part-01","status":"can_prove","summary":"*Quoted from 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Erdos1186.erdos_1186.variants.prs_conjecture","parent":7238,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1100-1199/part-04/index.html#erdos-problem-1186","scope":"formal-conjectures/erdos-problems/1100-1199/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/1186.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 1186"},{"id":"formal-conjectures/erdos-problems/1100-1199/part-04/erdos-problem-1187","kind":"node","lean":"Erdos1187.erdos_1187.parts.i, Erdos1187.erdos_1187.parts.ii","parent":7238,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1100-1199/part-04/index.html#erdos-problem-1187","scope":"formal-conjectures/erdos-problems/1100-1199/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/1187.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as 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Erdos1189.erdos_1189.variants.bbmst","parent":7238,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1100-1199/part-04/index.html#erdos-problem-1189","scope":"formal-conjectures/erdos-problems/1100-1199/part-04","status":"can_prove","summary":">","title":"Erd\u0151s Problem 1189"},{"id":"formal-conjectures/erdos-problems/1100-1199/part-04/erdos-problem-1190","kind":"node","lean":"Erdos1190.erdos_1190, Erdos1190.erdos_1190.variants.lt_one, Erdos1190.erdos_1190.variants.tendsto_zero, Erdos1190.erdos_1190.variants.lower_bound, Erdos1190.erdos_1190.variants.upper_bound","parent":7238,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1100-1199/part-04/index.html#erdos-problem-1190","scope":"formal-conjectures/erdos-problems/1100-1199/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/1190.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 1190"},{"id":"formal-conjectures/erdos-problems/1100-1199/part-04/erdos-problem-1191","kind":"node","lean":"Erdos1191.erdos_1191.parts.i, Erdos1191.erdos_1191.parts.ii, Erdos1191.erdos_1191.variants.liminf_le","parent":7238,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1100-1199/part-04/index.html#erdos-problem-1191","scope":"formal-conjectures/erdos-problems/1100-1199/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/1191.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 1191"},{"id":"formal-conjectures/erdos-problems/1100-1199/part-04/erdos-problem-1192","kind":"node","lean":"Erdos1192.erdos_1192, Erdos1192.erdos_1192.variants.renyi, 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in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 1204"},{"id":"formal-conjectures/erdos-problems/1200-1299/erdos-problem-1205","kind":"node","lean":"Erdos1205.erdos_1205, Erdos1205.erdos_1205.variants.upper_bound","parent":7256,"result":true,"route":"../roadmap/formal-conjectures/erdos-problems/1200-1299/index.html#erdos-problem-1205","scope":"formal-conjectures/erdos-problems/1200-1299","status":"can_prove","summary":"*Quoted from FormalConjectures/ErdosProblems/1205.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Erd\u0151s Problem 1205"},{"id":"formal-conjectures/erdos-problems/1200-1299/erdos-problem-1206","kind":"node","lean":"Erdos1206.erdos_1206.parts.i, 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(1966)"},{"id":"formal-conjectures/wikipedia/part-04/idoneal-numbers-completeness-conjecture","kind":"node","lean":"Idoneal.idoneal_numbers_completeness","parent":7702,"result":true,"route":"../roadmap/formal-conjectures/wikipedia/part-04/index.html#idoneal-numbers-completeness-conjecture","scope":"formal-conjectures/wikipedia/part-04","status":"can_prove","summary":">","title":"Idoneal numbers completeness conjecture"},{"id":"formal-conjectures/wikipedia/part-04/inscribed-square-problem","kind":"node","lean":"InscribedSquare.inscribed_square_problem, InscribedSquare.inscribed_rectangle_problem, InscribedSquare.exists_inscribed_rectangle, InscribedSquare.exists_inscribed_rectangle_of_smooth, InscribedSquare.exists_inscribed_square_of_C2","parent":7702,"result":true,"route":"../roadmap/formal-conjectures/wikipedia/part-04/index.html#inscribed-square-problem","scope":"formal-conjectures/wikipedia/part-04","status":"can_prove","summary":">","title":"Inscribed square 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numbers"},{"id":"formal-conjectures/wikipedia/part-05/jacobian-conjecture","kind":"node","lean":"JacobianConjecture.jacobian_conjecture, JacobianConjecture.jacobian_conjecture_two_variables","parent":7723,"result":true,"route":"../roadmap/formal-conjectures/wikipedia/part-05/index.html#jacobian-conjecture","scope":"formal-conjectures/wikipedia/part-05","status":"can_prove","summary":"*Quoted from FormalConjectures/Wikipedia/JacobianConjecture.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Jacobian conjecture"},{"id":"formal-conjectures/wikipedia/part-05/jacobson-conjecture","kind":"node","lean":"Jacobson.jacobson_conjecture, Jacobson.jacobson_conjecture_of_comm_ring, 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>","title":"Local uniformization"},{"id":"formal-conjectures/wikipedia/part-05/lonely-runner-conjecture","kind":"node","lean":"LonelyRunnerConjecture.lonely_runner_conjecture, LonelyRunnerConjecture.lonely_runner_conjecture.variants.tao_2017","parent":7723,"result":true,"route":"../roadmap/formal-conjectures/wikipedia/part-05/index.html#lonely-runner-conjecture","scope":"formal-conjectures/wikipedia/part-05","status":"can_prove","summary":"*Quoted from FormalConjectures/Wikipedia/LonelyRunnerConjecture.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Lonely runner conjecture"},{"id":"formal-conjectures/wikipedia/part-05/lovaszs-conjecture-on-hamiltonian-paths-in-verte","kind":"node","lean":"LovaszConjecture.lovasz_conjecture, LovaszConjecture.lovasz_conjecture.variants.babai, 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Primes"},{"data_url":"scopes/formal-conjectures/greens-open-problems.json","id":"formal-conjectures/greens-open-problems","kind":"scope","parent":6007,"result":false,"route":"../roadmap/formal-conjectures/greens-open-problems/index.html","scope":"formal-conjectures/greens-open-problems","status":"can_prove","summary":"The 65 problem files in FormalConjectures/GreensOpenProblems that state a research or textbook statement, at commit 2055e5ad, one article each.","title":"Green's open problems"},{"data_url":"scopes/formal-conjectures/greens-open-problems/part-01.json","id":"formal-conjectures/greens-open-problems/part-01","kind":"scope","navigation":"transparent","parent":7823,"result":false,"route":"../roadmap/formal-conjectures/greens-open-problems/part-01/index.html","scope":"formal-conjectures/greens-open-problems/part-01","status":"can_prove","title":"Ben Green's Open Problem 1 to Green's Open Problem 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Green5.green_5.variants.gowers_sl_two","parent":7824,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-01/index.html#ben-greens-open-problem-5","scope":"formal-conjectures/greens-open-problems/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/5.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Ben Green's Open Problem 5"},{"id":"formal-conjectures/greens-open-problems/part-01/ben-greens-open-problem-7","kind":"node","lean":"Green7.green_7.variants.positive_density","parent":7824,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-01/index.html#ben-greens-open-problem-7","scope":"formal-conjectures/greens-open-problems/part-01","status":"can_prove","summary":">","title":"Ben Green's Open Problem 7"},{"id":"formal-conjectures/greens-open-problems/part-01/greens-open-problem-9","kind":"node","lean":"Green9.green_9_i, Green9.green_9_ii, Green9.green_9_iii","parent":7824,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-01/index.html#greens-open-problem-9","scope":"formal-conjectures/greens-open-problems/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/9.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Green's Open Problem 9"},{"id":"formal-conjectures/greens-open-problems/part-01/greens-open-problem-12","kind":"node","lean":"Green12.green_12","parent":7824,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-01/index.html#greens-open-problem-12","scope":"formal-conjectures/greens-open-problems/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/12.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Green's Open Problem 12"},{"id":"formal-conjectures/greens-open-problems/part-01/ben-greens-open-problem-14","kind":"node","lean":"Green14.green_14_polynomial, Green14.green_14_polynomial_k_eq_3, Green14.green_14_quadratic, Green14.green_14_lower_bound_green, Green14.green_14_lower_bound_hunter, Green14.green_14_lower_bound_brown_landman_robertson, Green14.green_14_lower_bound_li_shu, Green14.green_14_upper_bound_schoen, Green14.green_14_upper_bound_kelley_meka, Green14.green_14_variant_2r2, Green14.W_3_3, Green14.W_3_4, Green14.W_3_5, Green14.W_3_6, Green14.W_3_7, Green14.W_3_8, Green14.W_3_9, Green14.W_3_10, Green14.W_3_11, Green14.W_3_12, Green14.W_3_13, Green14.W_3_14, Green14.W_3_15, Green14.W_3_16, Green14.W_3_17, Green14.W_3_18, Green14.W_3_19, Green14.W_3_20_lower, Green14.W_3_20, Green14.W_3_21_lower, Green14.W_3_21_eq, Green14.W_3_22_lower, Green14.W_3_22_eq, Green14.W_3_23_lower, Green14.W_3_23_eq, Green14.W_3_24_lower, Green14.W_3_24_eq, Green14.W_3_25_lower, Green14.W_3_25_eq, Green14.W_3_26_lower, Green14.W_3_26_eq, Green14.W_3_27_lower, Green14.W_3_27_eq, Green14.W_3_28_lower, Green14.W_3_28_eq, Green14.W_3_29_lower, Green14.W_3_29_eq, Green14.W_3_30_lower, Green14.W_3_30_eq, Green14.W_3_31_lower, Green14.W_3_32_lower, Green14.W_3_33_lower, Green14.W_3_34_lower, Green14.W_3_35_lower, Green14.W_3_36_lower, Green14.W_3_37_lower, Green14.W_3_38_lower, Green14.W_3_39_lower","parent":7824,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-01/index.html#ben-greens-open-problem-14","scope":"formal-conjectures/greens-open-problems/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/14.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Ben Green's Open Problem 14"},{"id":"formal-conjectures/greens-open-problems/part-01/greens-open-problem-15","kind":"node","lean":"Green15.green_15, Green15.green_15_ap4","parent":7824,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-01/index.html#greens-open-problem-15","scope":"formal-conjectures/greens-open-problems/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/15.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Green's Open Problem 15"},{"id":"formal-conjectures/greens-open-problems/part-01/ben-greens-open-problem-16","kind":"node","lean":"Green16.green_16, Green16.green_16_lower_bound, Green16.green_16_upper_bound, Green16.zhao_question","parent":7824,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-01/index.html#ben-greens-open-problem-16","scope":"formal-conjectures/greens-open-problems/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/16.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Ben Green's Open Problem 16"},{"id":"formal-conjectures/greens-open-problems/part-01/ben-greens-open-problem-18","kind":"node","lean":"Green18.green_18, Green18.green_18.bmz_corners","parent":7824,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-01/index.html#ben-greens-open-problem-18","scope":"formal-conjectures/greens-open-problems/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/18.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Ben Green's Open Problem 18"},{"id":"formal-conjectures/greens-open-problems/part-01/ben-greens-open-problem-19","kind":"node","lean":"Green19.green_19, Green19.green_19.lower, Green19.green_19.upper","parent":7824,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-01/index.html#ben-greens-open-problem-19","scope":"formal-conjectures/greens-open-problems/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/19.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Ben Green's Open Problem 19"},{"id":"formal-conjectures/greens-open-problems/part-01/ben-greens-open-problem-21","kind":"node","lean":"Green21.green_21, Green21.green_21.variants.fox_kleitman, Green21.green_21.variants.fox_kleitman_sharp, Green21.green_21.variants.fox_kleitman_modular, Green21.green_21.variants.milicevic","parent":7824,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-01/index.html#ben-greens-open-problem-21","scope":"formal-conjectures/greens-open-problems/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/21.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Ben Green's Open Problem 21"},{"id":"formal-conjectures/greens-open-problems/part-01/greens-open-problem-22","kind":"node","lean":"Green22.green_22, Green22.green_22.variants.green_sawhney_bound, Green22.green_22.variants.moreira_infinite, Green22.green_22.variants.lower_nine","parent":7824,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-01/index.html#greens-open-problem-22","scope":"formal-conjectures/greens-open-problems/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/22.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Green's Open Problem 22"},{"id":"formal-conjectures/greens-open-problems/part-01/greens-open-problem-23","kind":"node","lean":"Green23.green_23","parent":7824,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-01/index.html#greens-open-problem-23","scope":"formal-conjectures/greens-open-problems/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/23.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Green's Open Problem 23"},{"id":"formal-conjectures/greens-open-problems/part-01/greens-open-problem-24","kind":"node","lean":"Green24.green_24, Green24.variants.upper_trivial, Green24.variants.upper_HL, Green24.variants.lower_HL, Green24.variants.conjecture","parent":7824,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-01/index.html#greens-open-problem-24","scope":"formal-conjectures/greens-open-problems/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/24.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Green's Open Problem 24"},{"id":"formal-conjectures/greens-open-problems/part-01/greens-open-problem-25","kind":"node","lean":"Green25.green_25, Green25.green_25.upper, Green25.green_25.lower, Green25.green_25.variants.lower_ess89, Green25.green_25.variants.upper_ess89, Green25.green_25.variants.upper_pow_23_40","parent":7824,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-01/index.html#greens-open-problem-25","scope":"formal-conjectures/greens-open-problems/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/25.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Green's Open Problem 25"},{"id":"formal-conjectures/greens-open-problems/part-01/greens-open-problem-26","kind":"node","lean":"Green26.green_26, Green26.green_26.variants.yu25, Green26.green_26.variants.alm91, Green26.green_26.variants.open","parent":7824,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-01/index.html#greens-open-problem-26","scope":"formal-conjectures/greens-open-problems/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/26.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Green's Open Problem 26"},{"id":"formal-conjectures/greens-open-problems/part-01/greens-open-problem-27","kind":"node","lean":"Green27.green_27.equivalent, Green27.green_27.lower, Green27.green_27.upper, Green27.green_27.variants.lower_be23, Green27.green_27.variants.lower_cy26, Green27.green_27.variants.upper_be23, Green27.green_27.variants.previous_lower, Green27.green_27.variants.previous_upper","parent":7824,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-01/index.html#greens-open-problem-27","scope":"formal-conjectures/greens-open-problems/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/27.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Green's Open Problem 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Green36.green_36.variants.cks05","parent":7845,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-02/index.html#greens-open-problem-36","scope":"formal-conjectures/greens-open-problems/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/36.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Green's Open Problem 36"},{"id":"formal-conjectures/greens-open-problems/part-02/ben-greens-open-problem-37","kind":"node","lean":"Green37.green_37, Green37.green_37_asymptotic, Green37.green_37_theta, Green37.green_37_lower_bound","parent":7845,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-02/index.html#ben-greens-open-problem-37","scope":"formal-conjectures/greens-open-problems/part-02","status":"can_prove","summary":">","title":"Ben Green's Open Problem 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Green39.green_39.variant_theta","parent":7845,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-02/index.html#greens-open-problem-39","scope":"formal-conjectures/greens-open-problems/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/39.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Green's Open Problem 39"},{"id":"formal-conjectures/greens-open-problems/part-02/ben-greens-open-problem-40","kind":"node","lean":"Green40.green_40, Green40.green_40.sanity_f_one, Green40.green_40.upper_bound, Green40.green_40.f_eq_one_for_all, Green40.green_40.f_two_eq_one, Green40.green_40.upper_bound_f_two, Green40.green_40.variants.arbitrary_subsets, Green40.green_40.variants.arbitrary_subsets_sanity_f_tilde_two, Green40.green_40.f_tilde_le_f, Green40.green_40.variants.all_n","parent":7845,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-02/index.html#ben-greens-open-problem-40","scope":"formal-conjectures/greens-open-problems/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/40.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Ben Green's Open Problem 40"},{"id":"formal-conjectures/greens-open-problems/part-02/ben-greens-open-problem-41","kind":"node","lean":"Green41.minCopies_set_nonempty, Green41.green_41, Green41.green_41.variants.double_exponential_bound, Green41.green_41.variants.polynomial_bound, Green41.green_41.variants.kravitz_leng","parent":7845,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-02/index.html#ben-greens-open-problem-41","scope":"formal-conjectures/greens-open-problems/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/41.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Ben Green's Open Problem 41"},{"id":"formal-conjectures/greens-open-problems/part-02/greens-open-problem-42","kind":"node","lean":"Green42.green_42, Green42.green_42.variants.dimension_one, Green42.green_42.variants.dimension_eight, Green42.green_42.variants.dimension_twenty_four","parent":7845,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-02/index.html#greens-open-problem-42","scope":"formal-conjectures/greens-open-problems/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/42.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Green's Open Problem 42"},{"id":"formal-conjectures/greens-open-problems/part-02/ben-greens-open-problem-43","kind":"node","lean":"Green43.green_43, Green43.green_43.variants.alon_peres, Green43.green_43.variants.range_N051_N052, Green43.green_43.variants.range_N051_N052_linear, Green43.green_43.variants.general_exponent, Green43.green_43.variants.general_exponent_lt_half","parent":7845,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-02/index.html#ben-greens-open-problem-43","scope":"formal-conjectures/greens-open-problems/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/43.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Ben Green's Open Problem 43"},{"id":"formal-conjectures/greens-open-problems/part-02/greens-open-problem-44","kind":"node","lean":"Green44.green_44, Green44.green_44.variants.less_than_sqrt","parent":7845,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-02/index.html#greens-open-problem-44","scope":"formal-conjectures/greens-open-problems/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/44.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Green's Open Problem 44"},{"id":"formal-conjectures/greens-open-problems/part-02/ben-greens-open-problem-45","kind":"node","lean":"Green45.green_45","parent":7845,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-02/index.html#ben-greens-open-problem-45","scope":"formal-conjectures/greens-open-problems/part-02","status":"can_prove","summary":"> >","title":"Ben Green's Open Problem 45"},{"id":"formal-conjectures/greens-open-problems/part-02/ben-greens-open-problem-46","kind":"node","lean":"Green46.green_46.improve_lower, Green46.green_46.improve_upper, Green46.green_46.improve_upper_conjectured, Green46.green_46.variants.lower, Green46.green_46.variants.upper","parent":7845,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-02/index.html#ben-greens-open-problem-46","scope":"formal-conjectures/greens-open-problems/part-02","status":"can_prove","summary":"> >","title":"Ben Green's Open Problem 46"},{"id":"formal-conjectures/greens-open-problems/part-02/greens-open-problem-47","kind":"node","lean":"Green47.green_47, Green47.green_47.variants.finite_exceptions","parent":7845,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-02/index.html#greens-open-problem-47","scope":"formal-conjectures/greens-open-problems/part-02","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/47.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Green's Open Problem 47"},{"id":"formal-conjectures/greens-open-problems/part-02/greens-open-problem-49","kind":"node","lean":"Green49.green_49","parent":7845,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-02/index.html#greens-open-problem-49","scope":"formal-conjectures/greens-open-problems/part-02","status":"can_prove","summary":">","title":"Green's Open Problem 49"},{"id":"formal-conjectures/greens-open-problems/part-02/ben-greens-open-problem-50","kind":"node","lean":"Green50.green_50","parent":7845,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-02/index.html#ben-greens-open-problem-50","scope":"formal-conjectures/greens-open-problems/part-02","status":"can_prove","summary":"> >","title":"Ben Green's Open Problem 50"},{"data_url":"scopes/formal-conjectures/greens-open-problems/part-03.json","id":"formal-conjectures/greens-open-problems/part-03","kind":"scope","navigation":"transparent","parent":7823,"result":false,"route":"../roadmap/formal-conjectures/greens-open-problems/part-03/index.html","scope":"formal-conjectures/greens-open-problems/part-03","status":"can_prove","title":"Green's Open Problem 51 to Green's Open Problem 85"},{"id":"formal-conjectures/greens-open-problems/part-03/greens-open-problem-51","kind":"node","lean":"Green51.green_51, Green51.green_51.lower, Green51.green_51.upper, Green51.green_51.one_half, Green51.green_51.lower_ap, Green51.green_51.upper_ap","parent":7866,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-03/index.html#greens-open-problem-51","scope":"formal-conjectures/greens-open-problems/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/51.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Green's Open Problem 51"},{"id":"formal-conjectures/greens-open-problems/part-03/greens-open-problem-52","kind":"node","lean":"Green52.green_52, Green52.green_52_log","parent":7866,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-03/index.html#greens-open-problem-52","scope":"formal-conjectures/greens-open-problems/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/52.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Green's Open Problem 52"},{"id":"formal-conjectures/greens-open-problems/part-03/greens-open-problem-53","kind":"node","lean":"Green53.green_53","parent":7866,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-03/index.html#greens-open-problem-53","scope":"formal-conjectures/greens-open-problems/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/53.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Green's Open Problem 53"},{"id":"formal-conjectures/greens-open-problems/part-03/ben-greens-open-problem-54","kind":"node","lean":"Green54.green_54, Green54.green_54_known_case","parent":7866,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-03/index.html#ben-greens-open-problem-54","scope":"formal-conjectures/greens-open-problems/part-03","status":"can_prove","summary":"> >","title":"Ben Green's Open Problem 54"},{"id":"formal-conjectures/greens-open-problems/part-03/ben-greens-open-problem-57","kind":"node","lean":"Green57.green_57.variants.z3_functional, Green57.green_57.variants.z3, Green57.green_57","parent":7866,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-03/index.html#ben-greens-open-problem-57","scope":"formal-conjectures/greens-open-problems/part-03","status":"can_prove","summary":"> > > > > >","title":"Ben Green's Open Problem 57"},{"id":"formal-conjectures/greens-open-problems/part-03/ben-greens-open-problem-58","kind":"node","lean":"Green58.green_58","parent":7866,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-03/index.html#ben-greens-open-problem-58","scope":"formal-conjectures/greens-open-problems/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/58.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Ben Green's Open Problem 58"},{"id":"formal-conjectures/greens-open-problems/part-03/ben-greens-open-problem-60","kind":"node","lean":"Green60.green_60","parent":7866,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-03/index.html#ben-greens-open-problem-60","scope":"formal-conjectures/greens-open-problems/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/60.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Ben Green's Open Problem 60"},{"id":"formal-conjectures/greens-open-problems/part-03/ben-greens-open-problem-61","kind":"node","lean":"Green61.green_61","parent":7866,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-03/index.html#ben-greens-open-problem-61","scope":"formal-conjectures/greens-open-problems/part-03","status":"can_prove","summary":">","title":"Ben Green's Open Problem 61"},{"id":"formal-conjectures/greens-open-problems/part-03/ben-greens-open-problem-62","kind":"node","lean":"Green62.green_62","parent":7866,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-03/index.html#ben-greens-open-problem-62","scope":"formal-conjectures/greens-open-problems/part-03","status":"can_prove","summary":"> >","title":"Ben Green's Open Problem 62"},{"id":"formal-conjectures/greens-open-problems/part-03/greens-open-problem-64","kind":"node","lean":"Green64.green_64, Green64.green_64.variants.p_sub_one","parent":7866,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-03/index.html#greens-open-problem-64","scope":"formal-conjectures/greens-open-problems/part-03","status":"can_prove","summary":">","title":"Green's Open Problem 64"},{"id":"formal-conjectures/greens-open-problems/part-03/ben-greens-open-problem-65","kind":"node","lean":"Green65.green_65, Green65.green_65.variants.prime_minus_one","parent":7866,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-03/index.html#ben-greens-open-problem-65","scope":"formal-conjectures/greens-open-problems/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/65.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Ben Green's Open Problem 65"},{"id":"formal-conjectures/greens-open-problems/part-03/ben-greens-open-problem-66","kind":"node","lean":"Green66.green_66, Green66.green_66.variants.trivial_bound","parent":7866,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-03/index.html#ben-greens-open-problem-66","scope":"formal-conjectures/greens-open-problems/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/66.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, 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68"},{"id":"formal-conjectures/greens-open-problems/part-03/ben-greens-open-problem-70","kind":"node","lean":"Green70.green_70, Green70.green_70.variants.five_visible, Green70.green_70.variants.empty_pentagon","parent":7866,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-03/index.html#ben-greens-open-problem-70","scope":"formal-conjectures/greens-open-problems/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/70.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Ben Green's Open Problem 70"},{"id":"formal-conjectures/greens-open-problems/part-03/ben-greens-open-problem-72","kind":"node","lean":"Green72.allowedSetSize_le, Green72.NoKInLine, Green72.green_72, Green72.green_72.variants.eventually, Green72.no_three_in_line_le, Green72.no_k_in_line_big","parent":7866,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-03/index.html#ben-greens-open-problem-72","scope":"formal-conjectures/greens-open-problems/part-03","status":"can_prove","summary":"> > >","title":"Ben Green's Open Problem 72"},{"id":"formal-conjectures/greens-open-problems/part-03/ben-greens-open-problem-75","kind":"node","lean":"Green75.green_75, Green75.green_75.variants.exists_constant, Green75.green_75.variants.pinchasi, Green75.green_75.variants.alon","parent":7866,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-03/index.html#ben-greens-open-problem-75","scope":"formal-conjectures/greens-open-problems/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/75.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Ben Green's Open Problem 75"},{"id":"formal-conjectures/greens-open-problems/part-03/ben-greens-open-problem-76","kind":"node","lean":"Green76.green_76, Green76.green_76.variants.exists_constant, Green76.green_76.variants.weak_net, Green76.green_76.variants.subset, Green76.green_76.variants.subset_log_log","parent":7866,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-03/index.html#ben-greens-open-problem-76","scope":"formal-conjectures/greens-open-problems/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/76.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Ben Green's Open Problem 76"},{"id":"formal-conjectures/greens-open-problems/part-03/ben-greens-open-problem-77","kind":"node","lean":"Green77.green_77","parent":7866,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-03/index.html#ben-greens-open-problem-77","scope":"formal-conjectures/greens-open-problems/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/77.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Ben Green's Open Problem 77"},{"id":"formal-conjectures/greens-open-problems/part-03/ben-greens-open-problem-82","kind":"node","lean":"Green82.green_82, Green82.green_82_lower_bound, Green82.green_82_upper_bound","parent":7866,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-03/index.html#ben-greens-open-problem-82","scope":"formal-conjectures/greens-open-problems/part-03","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/82.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Ben Green's Open Problem 82"},{"id":"formal-conjectures/greens-open-problems/part-03/greens-open-problem-85","kind":"node","lean":"Green85.green_85, Green85.green_85_loose","parent":7866,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-03/index.html#greens-open-problem-85","scope":"formal-conjectures/greens-open-problems/part-03","status":"can_prove","summary":">","title":"Green's Open Problem 85"},{"data_url":"scopes/formal-conjectures/greens-open-problems/part-04.json","id":"formal-conjectures/greens-open-problems/part-04","kind":"scope","navigation":"transparent","parent":7823,"result":false,"route":"../roadmap/formal-conjectures/greens-open-problems/part-04/index.html","scope":"formal-conjectures/greens-open-problems/part-04","status":"can_prove","title":"Green's Open Problem 87 to Ben Green's Open Problem 97"},{"id":"formal-conjectures/greens-open-problems/part-04/greens-open-problem-87","kind":"node","lean":"Green87.green_87.variants.exists_tendsto, Green87.green_87.decreasing, Green87.green_87.variants.order, Green87.green_87.asymptotic","parent":7887,"result":true,"route":"../roadmap/formal-conjectures/greens-open-problems/part-04/index.html#greens-open-problem-87","scope":"formal-conjectures/greens-open-problems/part-04","status":"can_prove","summary":"*Quoted from FormalConjectures/GreensOpenProblems/87.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"Green's Open Problem 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statements below.*","title":"Furstenberg's `times p, times q` conjectures"},{"id":"formal-conjectures/arxiv/part-01/the-circulant-hadamard-conjecture","kind":"node","lean":"CirculantHadamard.circulant_hadamard_conjecture","parent":8018,"result":true,"route":"../roadmap/formal-conjectures/arxiv/part-01/index.html#the-circulant-hadamard-conjecture","scope":"formal-conjectures/arxiv/part-01","status":"can_prove","summary":">","title":"The circulant Hadamard conjecture"},{"id":"formal-conjectures/arxiv/part-01/an-arithmetic-sum-associated-with-the-classical","kind":"node","lean":"Arxiv.\u00ab2501.03234\u00bb.conjecture_1_1, Arxiv.\u00ab2501.03234\u00bb.conjecture_4_1, Arxiv.\u00ab2501.03234\u00bb.conjecture_4_2, Arxiv.\u00ab2501.03234\u00bb.conjecture_4_3, Arxiv.\u00ab2501.03234\u00bb.conjecture_4_4","parent":8018,"result":true,"route":"../roadmap/formal-conjectures/arxiv/part-01/index.html#an-arithmetic-sum-associated-with-the-classical","scope":"formal-conjectures/arxiv/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/Arxiv/2501.03234/ArithmeticSumS.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"An Arithmetic Sum Associated with the Classical Theta Function"},{"id":"formal-conjectures/arxiv/part-01/a-conjecture-by-margulis-on-matrix-groups","kind":"node","lean":"Margulis.conjecture_1_1, Margulis.huang_shi_theorem_1_2","parent":8018,"result":true,"route":"../roadmap/formal-conjectures/arxiv/part-01/index.html#a-conjecture-by-margulis-on-matrix-groups","scope":"formal-conjectures/arxiv/part-01","status":"can_prove","summary":"*Quoted from FormalConjectures/Arxiv/2504.17644/Margulis.lean in Formal Conjectures (Google DeepMind) at commit 2055e5ad, Apache-2.0, as are the docstrings and Lean statements below.*","title":"A conjecture by Margulis on matrix groups"},{"id":"formal-conjectures/arxiv/part-01/txgraffiti-conjecture-1-independence-annihilatio","kind":"node","lean":"Arxiv.\u00ab2507.17780\u00bb.tx_graffiti_conjecture_1","parent":8018,"result":true,"route":"../roadmap/formal-conjectures/arxiv/part-01/index.html#txgraffiti-conjecture-1-independence-annihilatio","scope":"formal-conjectures/arxiv/part-01","status":"can_prove","summary":"> > > >","title":"TxGraffiti Conjecture 1: independence, annihilation, and residue"},{"data_url":"scopes/formal-conjectures/arxiv/part-02.json","id":"formal-conjectures/arxiv/part-02","kind":"scope","navigation":"transparent","parent":8017,"result":false,"route":"../roadmap/formal-conjectures/arxiv/part-02/index.html","scope":"formal-conjectures/arxiv/part-02","status":"can_prove","title":"TxGraffiti Conjecture 2: zero\u2026 to The Dixmier 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21.149"},{"data_url":"scopes/formal-conjectures/millennium-problems.json","id":"formal-conjectures/millennium-problems","kind":"scope","parent":6007,"result":false,"route":"../roadmap/formal-conjectures/millennium-problems/index.html","scope":"formal-conjectures/millennium-problems","status":"can_prove","summary":"The 5 problem files in FormalConjectures/Millennium that state a research or textbook statement, at commit 2055e5ad, one article each.","title":"Millennium problems"},{"id":"formal-conjectures/millennium-problems/the-birch-and-swinnerton-dyer-bsd-conjecture","kind":"node","lean":"WeierstrassCurve.Affine.NumberField.weakBSD, WeierstrassCurve.Affine.NumberField.strongBSD, WeierstrassCurve.Affine.Rat.weakBSD, 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>","title":"Riemann Hypothesis and its generalizations"},{"data_url":"scopes/formal-conjectures/optimization-constants.json","id":"formal-conjectures/optimization-constants","kind":"scope","parent":6007,"result":false,"route":"../roadmap/formal-conjectures/optimization-constants/index.html","scope":"formal-conjectures/optimization-constants","status":"can_prove","summary":"The 5 problem files in FormalConjectures/OptimizationConstants that state a research or textbook statement, at commit 2055e5ad, one article each.","title":"Optimization constants"},{"id":"formal-conjectures/optimization-constants/taos-optimization-constant-1a-an-autocorrelation","kind":"node","lean":"Constant1a.c1a_lower_bound, Constant1a.c1a_upper_bound, Constant1a.mem_Ico_c1a, Constant1a.mem_Ioc_c1a, 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undecidable"},{"data_url":"scopes/formal-conjectures/open-quantum-problems.json","id":"formal-conjectures/open-quantum-problems","kind":"scope","parent":6007,"result":false,"route":"../roadmap/formal-conjectures/open-quantum-problems/index.html","scope":"formal-conjectures/open-quantum-problems","status":"can_prove","summary":"The 3 problem files in FormalConjectures/OpenQuantumProblems that state a research or textbook statement, at commit 2055e5ad, one article each.","title":"Open quantum problems"},{"id":"formal-conjectures/open-quantum-problems/open-quantum-problem-13-mutually-unbiased-bases","kind":"node","lean":"OpenQuantumProblem13.mutuallyUnbiasedBases_dim2, OpenQuantumProblem13.mutuallyUnbiasedBases_dim6_bounds, OpenQuantumProblem13.mutuallyUnbiasedBases_dim6_bounds.variants.upper_five, OpenQuantumProblem13.mutuallyUnbiasedBases_dim6, OpenQuantumProblem13.mutuallyUnbiasedBases_dim10, OpenQuantumProblem13.mutuallyUnbiasedBases_dim12, 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states"},{"data_url":"scopes/formal-conjectures/hilberts-problems.json","id":"formal-conjectures/hilberts-problems","kind":"scope","parent":6007,"result":false,"route":"../roadmap/formal-conjectures/hilberts-problems/index.html","scope":"formal-conjectures/hilberts-problems","status":"can_prove","summary":"The 2 problem files in FormalConjectures/HilbertProblems that state a research or textbook statement, at commit 2055e5ad, one article each.","title":"Hilbert's problems"},{"id":"formal-conjectures/hilberts-problems/hilberts-fifth-problem-and-the-hilbert-smith-con","kind":"node","lean":"Hilbert5.hilbert_smith_conjecture, Hilbert5.hilbert_smith_conjecture.variants.riemannian, Hilbert5.hilbert_smith_conjecture.variants.dimension_three, Hilbert5.hilbert_smith_padic_formulation, 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`:=`.*","title":"WeierstrassCurve.isModularModelOfLevel_conductorLevel_of_not_cube_dvd_of_modRepIsIrreducible_of_not_sq_dvd"},{"id":"anthropic-flt/4-modularity/weierstrasscurve-ismodularmodeloflevel-of-patchi","kind":"node","lean":"WeierstrassCurve.isModularModelOfLevel_of_patchingDatum","parent":8153,"result":true,"route":"../roadmap/anthropic-flt/4-modularity/index.html#weierstrasscurve-ismodularmodeloflevel-of-patchi","scope":"anthropic-flt/4-modularity","status":"fully_proved","summary":"*Quoted from Theorems/Thm_WeierstrassCurve_isModularModelOfLevel_of_patchingDatum.lean, line 11 of anthropics/fermats-last-theorem at commit 6e837e75, Apache-2.0: the file's text from the end of its imports and `attribute` lines to the top-level `:=`.*","title":"WeierstrassCurve.isModularModelOfLevel_of_patchingDatum"},{"id":"anthropic-flt/4-modularity/weierstrasscurve-nonempty-deformationringdata-or","kind":"node","lean":"WeierstrassCurve.nonempty_deformationRingData_ordinaryCondition_of_isSemistableModel","parent":8153,"result":true,"route":"../roadmap/anthropic-flt/4-modularity/index.html#weierstrasscurve-nonempty-deformationringdata-or","scope":"anthropic-flt/4-modularity","status":"fully_proved","summary":"*Quoted from Theorems/Thm_WeierstrassCurve_nonempty_deformationRingData_ordinaryCondition_of_isSemistableModel.lean, line 21 of anthropics/fermats-last-theorem at commit 6e837e75, Apache-2.0: the file's text from the end of its imports and `attribute` lines to the top-level `:=`.*","title":"WeierstrassCurve.nonempty_deformationRingData_ordinaryCondition_of_isSemistableModel"},{"id":"anthropic-flt/4-modularity/weierstrasscurve-threefiveauxiliarycurveexists","kind":"node","lean":"WeierstrassCurve.threeFiveAuxiliaryCurveExists","parent":8153,"result":true,"route":"../roadmap/anthropic-flt/4-modularity/index.html#weierstrasscurve-threefiveauxiliarycurveexists","scope":"anthropic-flt/4-modularity","status":"fully_proved","summary":"*Quoted from Theorems/Thm_WeierstrassCurve_threeFiveAuxiliaryCurveExists.lean, line 8 of anthropics/fermats-last-theorem at commit 6e837e75, Apache-2.0: the file's text from the end of its imports and `attribute` lines to the top-level `:=`.*","title":"WeierstrassCurve.threeFiveAuxiliaryCurveExists"},{"data_url":"scopes/anthropic-flt/5-level-lowering.json","id":"anthropic-flt/5-level-lowering","kind":"scope","parent":8132,"result":false,"route":"../roadmap/anthropic-flt/5-level-lowering/index.html","scope":"anthropic-flt/5-level-lowering","status":"fully_proved","summary":"*Quoted from the route of the proof in PROOF-PATH.md of anthropics/fermats-last-theorem at commit 6e837e75, Apache-2.0.*","title":"5. Level lowering"},{"id":"anthropic-flt/5-level-lowering/freypackage-level-lowering-to-two","kind":"node","lean":"FreyPackage.level_lowering_to_two","parent":8174,"result":true,"route":"../roadmap/anthropic-flt/5-level-lowering/index.html#freypackage-level-lowering-to-two","scope":"anthropic-flt/5-level-lowering","status":"fully_proved","summary":"*Quoted from Theorems/Thm_FreyPackage_level_lowering_to_two.lean, line 131 of anthropics/fermats-last-theorem at commit 6e837e75, Apache-2.0: the file's text from the end of its imports and `attribute` lines to the top-level `:=`.*","title":"Level lowering"},{"id":"anthropic-flt/5-level-lowering/freypackage-modularrepofconductorlevel","kind":"node","lean":"FreyPackage.modularRepOfConductorLevel","parent":8174,"result":true,"route":"../roadmap/anthropic-flt/5-level-lowering/index.html#freypackage-modularrepofconductorlevel","scope":"anthropic-flt/5-level-lowering","status":"fully_proved","summary":"*Quoted from Theorems/Thm_FreyPackage_modularRepOfConductorLevel.lean, line 131 of anthropics/fermats-last-theorem at commit 6e837e75, Apache-2.0: the file's text from the end of its imports and `attribute` lines to the top-level `:=`.*","title":"FreyPackage.modularRepOfConductorLevel"},{"id":"anthropic-flt/5-level-lowering/freypackage-level-lowering-to-two-of-conductorle","kind":"node","lean":"FreyPackage.level_lowering_to_two_of_conductorLevel","parent":8174,"result":true,"route":"../roadmap/anthropic-flt/5-level-lowering/index.html#freypackage-level-lowering-to-two-of-conductorle","scope":"anthropic-flt/5-level-lowering","status":"fully_proved","summary":"*Quoted from Theorems/Thm_FreyPackage_level_lowering_to_two_of_conductorLevel.lean, line 91 of anthropics/fermats-last-theorem at commit 6e837e75, Apache-2.0: the file's text from the end of its imports and `attribute` lines to the top-level `:=`.*","title":"FreyPackage.level_lowering_to_two_of_conductorLevel"},{"id":"anthropic-flt/5-level-lowering/freypackage-level-lowering-odd-prime-of-conducto","kind":"node","lean":"FreyPackage.level_lowering_odd_prime_of_conductorLevel","parent":8174,"result":true,"route":"../roadmap/anthropic-flt/5-level-lowering/index.html#freypackage-level-lowering-odd-prime-of-conducto","scope":"anthropic-flt/5-level-lowering","status":"fully_proved","summary":"*Quoted from Theorems/Thm_FreyPackage_level_lowering_odd_prime_of_conductorLevel.lean, line 90 of anthropics/fermats-last-theorem at commit 6e837e75, Apache-2.0: the file's text from the end of its imports and `attribute` lines to the top-level `:=`.*","title":"FreyPackage.level_lowering_odd_prime_of_conductorLevel"},{"id":"anthropic-flt/5-level-lowering/freypackage-level-lowering-at-p-of-conductorleve","kind":"node","lean":"FreyPackage.level_lowering_at_p_of_conductorLevel","parent":8174,"result":true,"route":"../roadmap/anthropic-flt/5-level-lowering/index.html#freypackage-level-lowering-at-p-of-conductorleve","scope":"anthropic-flt/5-level-lowering","status":"fully_proved","summary":"*Quoted from Theorems/Thm_FreyPackage_level_lowering_at_p_of_conductorLevel.lean, line 70 of anthropics/fermats-last-theorem at commit 6e837e75, Apache-2.0: the file's text from the end of its imports and `attribute` lines to the top-level `:=`.*","title":"FreyPackage.level_lowering_at_p_of_conductorLevel"},{"id":"anthropic-flt/5-level-lowering/freycurve-ispeuramifieeat-odd-of-integralform","kind":"node","lean":"FreyCurve.isPeuRamifieeAt_odd_of_integralForm","parent":8174,"result":true,"route":"../roadmap/anthropic-flt/5-level-lowering/index.html#freycurve-ispeuramifieeat-odd-of-integralform","scope":"anthropic-flt/5-level-lowering","status":"fully_proved","summary":"*Quoted from Theorems/Thm_FreyCurve_isPeuRamifieeAt_odd_of_integralForm.lean, line 8 of anthropics/fermats-last-theorem at commit 6e837e75, Apache-2.0: the file's text from the end of its imports and `attribute` lines to the top-level `:=`.*","title":"FreyCurve.isPeuRamifieeAt_odd_of_integralForm"},{"id":"anthropic-flt/5-level-lowering/cuspform-newformbadprimecoeff","kind":"node","lean":"CuspForm.newformBadPrimeCoeff","parent":8174,"result":true,"route":"../roadmap/anthropic-flt/5-level-lowering/index.html#cuspform-newformbadprimecoeff","scope":"anthropic-flt/5-level-lowering","status":"fully_proved","summary":"*Quoted from Theorems/Thm_CuspForm_newformBadPrimeCoeff.lean, line 7 of anthropics/fermats-last-theorem at commit 6e837e75, Apache-2.0: the file's text from the end of its imports and `attribute` lines to the top-level `:=`.*","title":"CuspForm.newformBadPrimeCoeff"},{"id":"anthropic-flt/5-level-lowering/freypackage-eigenformresidualattachmentatfamily","kind":"node","lean":"FreyPackage.eigenformResidualAttachmentAtFamily","parent":8174,"result":true,"route":"../roadmap/anthropic-flt/5-level-lowering/index.html#freypackage-eigenformresidualattachmentatfamily","scope":"anthropic-flt/5-level-lowering","status":"fully_proved","summary":"*Quoted from Theorems/Thm_FreyPackage_eigenformResidualAttachmentAtFamily.lean, line 24 of anthropics/fermats-last-theorem at commit 6e837e75, Apache-2.0: the file's text from the end of its imports and `attribute` lines to the top-level `:=`.*","title":"FreyPackage.eigenformResidualAttachmentAtFamily"},{"id":"anthropic-flt/5-level-lowering/freypackage-freygaloisrep-isunramifiedat","kind":"node","lean":"FreyPackage.freyGaloisRep_isUnramifiedAt","parent":8174,"result":true,"route":"../roadmap/anthropic-flt/5-level-lowering/index.html#freypackage-freygaloisrep-isunramifiedat","scope":"anthropic-flt/5-level-lowering","status":"fully_proved","summary":"*Quoted from Theorems/Thm_FreyPackage_freyGaloisRep_isUnramifiedAt.lean, line 15 of anthropics/fermats-last-theorem at commit 6e837e75, Apache-2.0: the file's text from the end of its imports and `attribute` lines to the top-level `:=`.*","title":"FreyPackage.freyGaloisRep_isUnramifiedAt"},{"id":"anthropic-flt/5-level-lowering/weierstrasscurve-isresiduallymodularoflevel-div","kind":"node","lean":"WeierstrassCurve.isResiduallyModularOfLevel_div_of_isUnramifiedAt_sqf","parent":8174,"result":true,"route":"../roadmap/anthropic-flt/5-level-lowering/index.html#weierstrasscurve-isresiduallymodularoflevel-div","scope":"anthropic-flt/5-level-lowering","status":"fully_proved","summary":"*Quoted from Theorems/Thm_WeierstrassCurve_isResiduallyModularOfLevel_div_of_isUnramifiedAt_sqf.lean, line 87 of anthropics/fermats-last-theorem at commit 6e837e75, Apache-2.0: the file's text from the end of its imports and `attribute` lines to the top-level `:=`.*","title":"WeierstrassCurve.isResiduallyModularOfLevel_div_of_isUnramifiedAt_sqf"},{"data_url":"scopes/anthropic-flt/6-s2-0-2-0.json","id":"anthropic-flt/6-s2-0-2-0","kind":"scope","parent":8132,"result":false,"route":"../roadmap/anthropic-flt/6-s2-0-2-0/index.html","scope":"anthropic-flt/6-s2-0-2-0","status":"fully_proved","summary":"*Quoted from the route of the proof in PROOF-PATH.md of anthropics/fermats-last-theorem at commit 6e837e75, Apache-2.0.*","title":"6. S\u2082(\u0393\u2080(2)) = 0"},{"id":"anthropic-flt/6-s2-0-2-0/modularform-s2-gamma0-2-eq-zero","kind":"node","lean":"ModularForm.S2_Gamma0_2_eq_zero","parent":8185,"result":true,"route":"../roadmap/anthropic-flt/6-s2-0-2-0/index.html#modularform-s2-gamma0-2-eq-zero","scope":"anthropic-flt/6-s2-0-2-0","status":"fully_proved","summary":"*Quoted from Theorems/Thm_ModularForm_S2_Gamma0_2_eq_zero.lean, line 13 of anthropics/fermats-last-theorem at commit 6e837e75, Apache-2.0: the file's text from the end of its imports and `attribute` lines to the top-level `:=`.*","title":"S\u2082(\u0393\u2080(2)) = 0"},{"id":"anthropic-flt/6-s2-0-2-0/modularform-s2-gamma0-one-eq-zero","kind":"node","lean":"ModularForm.S2_Gamma0_one_eq_zero","parent":8185,"result":true,"route":"../roadmap/anthropic-flt/6-s2-0-2-0/index.html#modularform-s2-gamma0-one-eq-zero","scope":"anthropic-flt/6-s2-0-2-0","status":"fully_proved","summary":"*Quoted from Theorems/Thm_ModularForm_S2_Gamma0_one_eq_zero.lean, line 13 of anthropics/fermats-last-theorem at commit 6e837e75, Apache-2.0: the file's text from the end of its imports and `attribute` lines to the top-level `:=`.*","title":"ModularForm.S2_Gamma0_one_eq_zero"},{"id":"anthropic-flt/exact-strength-of-the-named-steps","kind":"node","parent":8132,"result":false,"route":"../roadmap/anthropic-flt/exact-strength-of-the-named-steps/index.html","scope":"anthropic-flt","status":"can_state","summary":"*Quoted from the section \u201cExact strength of the named steps\u201d of PROOF-PATH.md of anthropics/fermats-last-theorem at commit 6e837e75, Apache-2.0: each \u201cProved:\u201d clause, cut with \u2026 where the text goes on, and entries that say neither what is proved nor what is not, whole.*","title":"Exact strength of the named steps"},{"data_url":"chapters/cfsg.json","id":"cfsg","kind":"scope","parent":0,"result":false,"route":"../roadmap/cfsg/index.html","scope":"cfsg","status":"fully_proved","summary":"AxiomMath/CFSG is a Lean proof of the classification of finite simple groups, by Axiom Math. Its README says:","title":"Classification of Finite Simple Groups (Axiom)"},{"data_url":"scopes/cfsg/case-division.json","id":"cfsg/case-division","kind":"scope","parent":8189,"result":false,"route":"../roadmap/cfsg/case-division/index.html","scope":"cfsg/case-division","status":"fully_proved","summary":"The top theorem, the module that reduces it to 9 cases, and the theorem the top theorem's proof gives for each case, in the order of the reduction's hypotheses. Each entry for a case names the proposition it proves.","title":"The case division"},{"id":"cfsg/case-division/top-theorem","kind":"node","lean":"CFSG.classification_finite_simple_groups","parent":8190,"result":true,"route":"../roadmap/cfsg/case-division/index.html#top-theorem","scope":"cfsg/case-division","status":"fully_proved","summary":"The proof's top theorem, quoted from CFSG/Statement.lean in AxiomMath/CFSG at commit 69cb972f, Apache-2.0:","title":"CFSG.classification_finite_simple_groups"},{"id":"cfsg/case-division/reduction","kind":"node","lean":"CFSG.Assembly.GroupProp, CFSG.Assembly.KProper, CFSG.Assembly.OddOrderStatement, CFSG.Assembly.AbelianStatement, CFSG.Assembly.DisconnectedStatement, CFSG.Assembly.LowTwoRankStatement, CFSG.Assembly.DichotomyStatement, CFSG.Assembly.ComponentStatement, CFSG.Assembly.QuasithinStatement, CFSG.Assembly.RankThreeStatement, CFSG.Assembly.LargeEvenStatement, CFSG.Assembly.charTwo_classification, CFSG.Assembly.kProper_classification, CFSG.Assembly.classification","parent":8190,"result":true,"route":"../roadmap/cfsg/case-division/index.html#reduction","scope":"cfsg/case-division","status":"fully_proved","summary":"The module `CFSG.Assembly.Reduction` defines each case of the proof's case division as a proposition and derives the classification from them. Its 14 declarations, in file order, quoted from CFSG/Assembly/Reduction.lean in AxiomMath/CFSG at commit 69cb972f, Apache-2.0; each definition is shown whole:","title":"CFSG.Assembly.Reduction"},{"id":"cfsg/case-division/case-hodd-oddorderstatement","kind":"node","lean":"CFSG.FeitThompson.isSolvable_of_odd_natCard","parent":8190,"result":true,"route":"../roadmap/cfsg/case-division/index.html#case-hodd-oddorderstatement","scope":"cfsg/case-division","status":"fully_proved","summary":"The case `hodd : OddOrderStatement` of `CFSG.Assembly.classification`. What it asserts, defined in CFSG/Assembly/Reduction.lean, line 16:","title":"CFSG.FeitThompson.isSolvable_of_odd_natCard"},{"id":"cfsg/case-division/case-hab-abelianstatement","kind":"node","lean":"CFSG.Assembly.abelian_simple","parent":8190,"result":true,"route":"../roadmap/cfsg/case-division/index.html#case-hab-abelianstatement","scope":"cfsg/case-division","status":"fully_proved","summary":"The case `hab : AbelianStatement` of `CFSG.Assembly.classification`. What it asserts, defined in CFSG/Assembly/Reduction.lean, line 19:","title":"CFSG.Assembly.abelian_simple"},{"id":"cfsg/case-division/case-hdisc-disconnectedstatement","kind":"node","lean":"CFSG.Assembly.disconnected_involution_graph","parent":8190,"result":true,"route":"../roadmap/cfsg/case-division/index.html#case-hdisc-disconnectedstatement","scope":"cfsg/case-division","status":"fully_proved","summary":"The case `hdisc : DisconnectedStatement` of `CFSG.Assembly.classification`. What it asserts, defined in CFSG/Assembly/Reduction.lean, line 22:","title":"CFSG.Assembly.disconnected_involution_graph"},{"id":"cfsg/case-division/case-hlow-lowtworankstatement","kind":"node","lean":"CFSG.Assembly.low_two_rank","parent":8190,"result":true,"route":"../roadmap/cfsg/case-division/index.html#case-hlow-lowtworankstatement","scope":"cfsg/case-division","status":"fully_proved","summary":"The case `hlow : LowTwoRankStatement` of `CFSG.Assembly.classification`. What it asserts, defined in CFSG/Assembly/Reduction.lean, line 26:","title":"CFSG.Assembly.low_two_rank"},{"id":"cfsg/case-division/case-hdich-dichotomystatement","kind":"node","lean":"CFSG.Assembly.component_dichotomy","parent":8190,"result":true,"route":"../roadmap/cfsg/case-division/index.html#case-hdich-dichotomystatement","scope":"cfsg/case-division","status":"fully_proved","summary":"The case `hdich : DichotomyStatement` of `CFSG.Assembly.classification`. What it asserts, defined in CFSG/Assembly/Reduction.lean, line 30:","title":"CFSG.Assembly.component_dichotomy"},{"id":"cfsg/case-division/case-hcomp-componentstatement","kind":"node","lean":"CFSG.Assembly.component_type","parent":8190,"result":true,"route":"../roadmap/cfsg/case-division/index.html#case-hcomp-componentstatement","scope":"cfsg/case-division","status":"fully_proved","summary":"The case `hcomp : ComponentStatement` of `CFSG.Assembly.classification`. What it asserts, defined in CFSG/Assembly/Reduction.lean, line 34:","title":"CFSG.Assembly.component_type"},{"id":"cfsg/case-division/case-hqt-quasithinstatement","kind":"node","lean":"CFSG.Assembly.quasithin","parent":8190,"result":true,"route":"../roadmap/cfsg/case-division/index.html#case-hqt-quasithinstatement","scope":"cfsg/case-division","status":"fully_proved","summary":"The case `hqt : QuasithinStatement` of `CFSG.Assembly.classification`. What it asserts, defined in CFSG/Assembly/Reduction.lean, line 38:","title":"CFSG.Assembly.quasithin"},{"id":"cfsg/case-division/case-hr3-rankthreestatement","kind":"node","lean":"CFSG.Assembly.rank_three_even","parent":8190,"result":true,"route":"../roadmap/cfsg/case-division/index.html#case-hr3-rankthreestatement","scope":"cfsg/case-division","status":"fully_proved","summary":"The case `hr3 : RankThreeStatement` of `CFSG.Assembly.classification`. What it asserts, defined in CFSG/Assembly/Reduction.lean, line 42:","title":"CFSG.Assembly.rank_three_even"},{"id":"cfsg/case-division/case-hle-largeevenstatement","kind":"node","lean":"CFSG.Assembly.large_even","parent":8190,"result":true,"route":"../roadmap/cfsg/case-division/index.html#case-hle-largeevenstatement","scope":"cfsg/case-division","status":"fully_proved","summary":"The case `hle : LargeEvenStatement` of `CFSG.Assembly.classification`. What it asserts, defined in CFSG/Assembly/Reduction.lean, line 46:","title":"CFSG.Assembly.large_even"},{"data_url":"scopes/cfsg/milestones.json","id":"cfsg/milestones","kind":"scope","parent":8189,"result":false,"route":"../roadmap/cfsg/milestones/index.html","scope":"cfsg/milestones","status":"fully_proved","summary":"The 110 modules directly under `CFSG/Milestones/`, by file name, one article each; CFSG/Milestones.lean imports 108 of them. 31 milestones are imported by no case theorem's file, directly or through other files.","title":"Milestones"},{"data_url":"scopes/cfsg/milestones/part-01.json","id":"cfsg/milestones/part-01","kind":"scope","navigation":"transparent","parent":8202,"result":false,"route":"../roadmap/cfsg/milestones/part-01/index.html","scope":"cfsg/milestones/part-01","status":"fully_proved","title":"AT_alternating_automorphism_perm\u2026 to CLS_large_even_classification"},{"id":"cfsg/milestones/part-01/at-alternating-automorphism-permutation","kind":"node","lean":"CFSG.Milestones.alternating_automorphism_permutation","parent":8203,"result":true,"route":"../roadmap/cfsg/milestones/part-01/index.html#at-alternating-automorphism-permutation","scope":"cfsg/milestones/part-01","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.AT_alternating_automorphism_permutation`, one of those CFSG/Milestones.lean imports.","title":"AT_alternating_automorphism_permutation"},{"id":"cfsg/milestones/part-01/bi-brauer-characterization","kind":"node","lean":"CFSG.Milestones.bi_brauer_characterization","parent":8203,"result":true,"route":"../roadmap/cfsg/milestones/part-01/index.html#bi-brauer-characterization","scope":"cfsg/milestones/part-01","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.BI_Brauer_characterization`, one of those CFSG/Milestones.lean imports.","title":"BI_Brauer_characterization"},{"id":"cfsg/milestones/part-01/bl-block-section-column-orthogonality","kind":"node","lean":"CFSG.Milestones.block_section_orthogonality","parent":8203,"result":true,"route":"../roadmap/cfsg/milestones/part-01/index.html#bl-block-section-column-orthogonality","scope":"cfsg/milestones/part-01","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.BL_block_section_column_orthogonality`, one of those CFSG/Milestones.lean imports.","title":"BL_block_section_column_orthogonality"},{"id":"cfsg/milestones/part-01/bt-principal-block-brauer-correspondence","kind":"node","lean":"CFSG.Milestones.brauer_third_main_theorem","parent":8203,"result":true,"route":"../roadmap/cfsg/milestones/part-01/index.html#bt-principal-block-brauer-correspondence","scope":"cfsg/milestones/part-01","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.BT_principal_block_Brauer_correspondence`, one of those CFSG/Milestones.lean imports.","title":"BT_principal_block_Brauer_correspondence"},{"id":"cfsg/milestones/part-01/cat-converse","kind":"node","lean":"CFSG.Milestones.cat_converse","parent":8203,"result":true,"route":"../roadmap/cfsg/milestones/part-01/index.html#cat-converse","scope":"cfsg/milestones/part-01","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CAT_converse`, one of those CFSG/Milestones.lean imports.","title":"CAT_converse"},{"id":"cfsg/milestones/part-01/cat-native-realization","kind":"node","lean":"CFSG.Milestones.cat_native_realization_cyclic, CFSG.Milestones.cat_native_realization_alternating, CFSG.Milestones.cat_native_realization_lie, CFSG.Milestones.native_carrier_finite_simple_of_ne_B, CFSG.Milestones.cat_native_realization_cyclic', CFSG.Milestones.cat_native_realization_alternating', CFSG.Milestones.cat_native_realization_sporadic, CFSG.Milestones.cat_native_realization","parent":8203,"result":true,"route":"../roadmap/cfsg/milestones/part-01/index.html#cat-native-realization","scope":"cfsg/milestones/part-01","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CAT_native_realization`, one of those CFSG/Milestones.lean imports.","title":"CAT_native_realization"},{"id":"cfsg/milestones/part-01/cfsg-classification","kind":"node","lean":"CFSG.Milestones.cfsg_classification, CFSG.Milestones.cfsg_classification_and_realization, CFSG.Milestones.cfsg_classification_native_and_realization_of_certificates, CFSG.Milestones.cfsg_classification_native_and_realization","parent":8203,"result":true,"route":"../roadmap/cfsg/milestones/part-01/index.html#cfsg-classification","scope":"cfsg/milestones/part-01","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CFSG_classification`, one of those CFSG/Milestones.lean imports.","title":"CFSG_classification"},{"id":"cfsg/milestones/part-01/cfsg-exhaustiveness","kind":"node","lean":"CFSG.Milestones.cfsg_exhaustiveness","parent":8203,"result":true,"route":"../roadmap/cfsg/milestones/part-01/index.html#cfsg-exhaustiveness","scope":"cfsg/milestones/part-01","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CFSG_exhaustiveness`, one of those CFSG/Milestones.lean imports.","title":"CFSG_exhaustiveness"},{"id":"cfsg/milestones/part-01/cls-k-proper-simple-classification","kind":"node","lean":"CFSG.Milestones.cls_k_proper_simple_classification, CFSG.Milestones.cls_k_proper_simple_classification_catalog","parent":8203,"result":true,"route":"../roadmap/cfsg/milestones/part-01/index.html#cls-k-proper-simple-classification","scope":"cfsg/milestones/part-01","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CLS_K_proper_simple_classification`, one of those CFSG/Milestones.lean imports.","title":"CLS_K_proper_simple_classification"},{"id":"cfsg/milestones/part-01/cls-qtke-classification","kind":"node","lean":"CFSG.Milestones.cls_qtke_classification_even_characteristic, CFSG.Milestones.cls_qtke_classification","parent":8203,"result":true,"route":"../roadmap/cfsg/milestones/part-01/index.html#cls-qtke-classification","scope":"cfsg/milestones/part-01","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CLS_QTKE_classification`, one of those CFSG/Milestones.lean imports.","title":"CLS_QTKE_classification"},{"id":"cfsg/milestones/part-01/cls-abelian-simple-catalog","kind":"node","lean":"CFSG.Milestones.cls_abelian_simple_catalog","parent":8203,"result":true,"route":"../roadmap/cfsg/milestones/part-01/index.html#cls-abelian-simple-catalog","scope":"cfsg/milestones/part-01","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CLS_abelian_simple_catalog`, one of those CFSG/Milestones.lean imports.","title":"CLS_abelian_simple_catalog"},{"id":"cfsg/milestones/part-01/cls-characteristic-two-classification","kind":"node","lean":"CFSG.Milestones.cls_characteristic_two_classification","parent":8203,"result":true,"route":"../roadmap/cfsg/milestones/part-01/index.html#cls-characteristic-two-classification","scope":"cfsg/milestones/part-01","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CLS_characteristic_two_classification`, one of those CFSG/Milestones.lean imports.","title":"CLS_characteristic_two_classification"},{"id":"cfsg/milestones/part-01/cls-component-classification","kind":"node","lean":"CFSG.Milestones.cls_component_classification","parent":8203,"result":true,"route":"../roadmap/cfsg/milestones/part-01/index.html#cls-component-classification","scope":"cfsg/milestones/part-01","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CLS_component_classification`, one of those CFSG/Milestones.lean imports.","title":"CLS_component_classification"},{"id":"cfsg/milestones/part-01/cls-component-dichotomy","kind":"node","lean":"CFSG.Milestones.cls_component_dichotomy","parent":8203,"result":true,"route":"../roadmap/cfsg/milestones/part-01/index.html#cls-component-dichotomy","scope":"cfsg/milestones/part-01","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CLS_component_dichotomy`, one of those CFSG/Milestones.lean imports.","title":"CLS_component_dichotomy"},{"id":"cfsg/milestones/part-01/cls-component-free-centralizer-quotient-characte","kind":"node","lean":"CFSG.Milestones.cls_component_free_centralizer_quotient_characteristic_two","parent":8203,"result":true,"route":"../roadmap/cfsg/milestones/part-01/index.html#cls-component-free-centralizer-quotient-characte","scope":"cfsg/milestones/part-01","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CLS_component_free_centralizer_quotient_characteristic_two`, one of those CFSG/Milestones.lean imports.","title":"CLS_component_free_centralizer_quotient_characteristic_two"},{"id":"cfsg/milestones/part-01/cls-component-free-involution-odd-cores-trivial","kind":"node","lean":"CFSG.Milestones.cls_component_free_involution_odd_cores_trivial","parent":8203,"result":true,"route":"../roadmap/cfsg/milestones/part-01/index.html#cls-component-free-involution-odd-cores-trivial","scope":"cfsg/milestones/part-01","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CLS_component_free_involution_odd_cores_trivial`, one of those CFSG/Milestones.lean imports.","title":"CLS_component_free_involution_odd_cores_trivial"},{"id":"cfsg/milestones/part-01/cls-connectivity-macwilliams-four-generator","kind":"node","lean":"CFSG.Milestones.cls_connectivity_macwilliams_four_generator","parent":8203,"result":true,"route":"../roadmap/cfsg/milestones/part-01/index.html#cls-connectivity-macwilliams-four-generator","scope":"cfsg/milestones/part-01","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CLS_connectivity_MacWilliams_four_generator`, one of those CFSG/Milestones.lean imports.","title":"CLS_connectivity_MacWilliams_four_generator"},{"id":"cfsg/milestones/part-01/cls-disconnected-simple-involution-graph-classif","kind":"node","lean":"CFSG.Milestones.cls_disconnected_simple_involution_graph_classification","parent":8203,"result":true,"route":"../roadmap/cfsg/milestones/part-01/index.html#cls-disconnected-simple-involution-graph-classif","scope":"cfsg/milestones/part-01","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CLS_disconnected_simple_involution_graph_classification`, one of those CFSG/Milestones.lean imports.","title":"CLS_disconnected_simple_involution_graph_classification"},{"id":"cfsg/milestones/part-01/cls-involution-centralizers-characteristic-two-t","kind":"node","lean":"CFSG.Milestones.cls_involution_centralizers_characteristic_two_transfer","parent":8203,"result":true,"route":"../roadmap/cfsg/milestones/part-01/index.html#cls-involution-centralizers-characteristic-two-t","scope":"cfsg/milestones/part-01","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CLS_involution_centralizers_characteristic_two_transfer`, one of those CFSG/Milestones.lean imports.","title":"CLS_involution_centralizers_characteristic_two_transfer"},{"id":"cfsg/milestones/part-01/cls-large-even-classification","kind":"node","lean":"CFSG.Milestones.cls_large_even_classification","parent":8203,"result":true,"route":"../roadmap/cfsg/milestones/part-01/index.html#cls-large-even-classification","scope":"cfsg/milestones/part-01","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CLS_large_even_classification`, one of those CFSG/Milestones.lean imports.","title":"CLS_large_even_classification"},{"data_url":"scopes/cfsg/milestones/part-02.json","id":"cfsg/milestones/part-02","kind":"scope","navigation":"transparent","parent":8202,"result":false,"route":"../roadmap/cfsg/milestones/part-02/index.html","scope":"cfsg/milestones/part-02","status":"fully_proved","title":"CLS_low_two_rank_classification to CL_LIT_SQTK_simple_list"},{"id":"cfsg/milestones/part-02/cls-low-two-rank-classification","kind":"node","lean":"CFSG.Milestones.cls_low_two_rank_classification","parent":8224,"result":true,"route":"../roadmap/cfsg/milestones/part-02/index.html#cls-low-two-rank-classification","scope":"cfsg/milestones/part-02","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CLS_low_two_rank_classification`, one of those CFSG/Milestones.lean imports.","title":"CLS_low_two_rank_classification"},{"id":"cfsg/milestones/part-02/cls-rank-three-even-classification","kind":"node","lean":"CFSG.Milestones.cls_rank_three_even_classification","parent":8224,"result":true,"route":"../roadmap/cfsg/milestones/part-02/index.html#cls-rank-three-even-classification","scope":"cfsg/milestones/part-02","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CLS_rank_three_even_classification`, one of those CFSG/Milestones.lean imports.","title":"CLS_rank_three_even_classification"},{"id":"cfsg/milestones/part-02/cl-lit-as-vol-i-background","kind":"node","lean":"CFSG.Milestones.cl_lit_as_vol_i_background_1, CFSG.Milestones.cl_lit_as_vol_i_background_2, CFSG.Milestones.cl_lit_as_vol_i_background_3, CFSG.Milestones.cl_lit_as_vol_i_background_4, CFSG.Milestones.cl_lit_as_vol_i_background_partial, CFSG.Milestones.cl_lit_as_vol_i_background","parent":8224,"result":true,"route":"../roadmap/cfsg/milestones/part-02/index.html#cl-lit-as-vol-i-background","scope":"cfsg/milestones/part-02","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CL_LIT_AS_vol_I_background`, one of those CFSG/Milestones.lean imports.","title":"CL_LIT_AS_vol_I_background"},{"id":"cfsg/milestones/part-02/cl-lit-aschbacher-1983-uniqueness-case-rank-thre","kind":"node","lean":"CFSG.Milestones.cl_lit_aschbacher_1983_uniqueness_case_rank_three","parent":8224,"result":true,"route":"../roadmap/cfsg/milestones/part-02/index.html#cl-lit-aschbacher-1983-uniqueness-case-rank-thre","scope":"cfsg/milestones/part-02","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CL_LIT_Aschbacher_1983_uniqueness_case_rank_three`, one of those CFSG/Milestones.lean imports.","title":"CL_LIT_Aschbacher_1983_uniqueness_case_rank_three"},{"id":"cfsg/milestones/part-02/cl-lit-aschbacher-component-type-theorem-5","kind":"node","lean":"CFSG.Milestones.cl_lit_aschbacher_component_type_theorem_5","parent":8224,"result":true,"route":"../roadmap/cfsg/milestones/part-02/index.html#cl-lit-aschbacher-component-type-theorem-5","scope":"cfsg/milestones/part-02","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CL_LIT_Aschbacher_component_type_theorem_5`, one of those CFSG/Milestones.lean imports.","title":"CL_LIT_Aschbacher_component_type_theorem_5"},{"id":"cfsg/milestones/part-02/cl-lit-aschbacher-proper-two-generated-core","kind":"node","lean":"CFSG.Milestones.cl_lit_aschbacher_proper_two_generated_core","parent":8224,"result":true,"route":"../roadmap/cfsg/milestones/part-02/index.html#cl-lit-aschbacher-proper-two-generated-core","scope":"cfsg/milestones/part-02","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CL_LIT_Aschbacher_proper_two_generated_core`, one of those CFSG/Milestones.lean imports.","title":"CL_LIT_Aschbacher_proper_two_generated_core"},{"id":"cfsg/milestones/part-02/cl-lit-borel-tits-local-parabolic","kind":"node","lean":"CFSG.Milestones.cl_lit_borel_tits_local_parabolic, CFSG.Milestones.cl_lit_borel_tits_local_parabolic_all_models","parent":8224,"result":true,"route":"../roadmap/cfsg/milestones/part-02/index.html#cl-lit-borel-tits-local-parabolic","scope":"cfsg/milestones/part-02","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CL_LIT_Borel_Tits_local_parabolic`, one of those CFSG/Milestones.lean imports.","title":"CL_LIT_Borel_Tits_local_parabolic"},{"id":"cfsg/milestones/part-02/cl-lit-curtis-tits-presentation-theorem","kind":"node","lean":"CFSG.Milestones.cl_lit_curtis_tits_presentation_theorem","parent":8224,"result":true,"route":"../roadmap/cfsg/milestones/part-02/index.html#cl-lit-curtis-tits-presentation-theorem","scope":"cfsg/milestones/part-02","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CL_LIT_Curtis_Tits_presentation_theorem`, one of those CFSG/Milestones.lean imports.","title":"CL_LIT_Curtis_Tits_presentation_theorem"},{"id":"cfsg/milestones/part-02/cl-lit-fischer-3t-existence-uniqueness","kind":"node","lean":"CFSG.Milestones.cl_lit_fischer_3t_existence_uniqueness, CFSG.Milestones.cl_lit_fischer_3t_existence_uniqueness_full","parent":8224,"result":true,"route":"../roadmap/cfsg/milestones/part-02/index.html#cl-lit-fischer-3t-existence-uniqueness","scope":"cfsg/milestones/part-02","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CL_LIT_Fischer_3T_existence_uniqueness`, one of those CFSG/Milestones.lean imports.","title":"CL_LIT_Fischer_3T_existence_uniqueness"},{"id":"cfsg/milestones/part-02/cl-lit-fischer-three-transposition-theorem","kind":"node","lean":"CFSG.Milestones.cl_lit_fischer_three_transposition_theorem, CFSG.Milestones.cl_lit_fischer_three_transposition_theorem_pinned, CFSG.Milestones.cl_lit_fischer_three_transposition_theorem_recognition","parent":8224,"result":true,"route":"../roadmap/cfsg/milestones/part-02/index.html#cl-lit-fischer-three-transposition-theorem","scope":"cfsg/milestones/part-02","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CL_LIT_Fischer_three_transposition_theorem`, one of those CFSG/Milestones.lean imports.","title":"CL_LIT_Fischer_three_transposition_theorem"},{"id":"cfsg/milestones/part-02/cl-lit-gh74-sectional-two-rank-four","kind":"node","lean":"CFSG.Milestones.cl_lit_gh74_sectional_two_rank_four, CFSG.Milestones.cl_lit_gh74_sectional_two_rank_four_models","parent":8224,"result":true,"route":"../roadmap/cfsg/milestones/part-02/index.html#cl-lit-gh74-sectional-two-rank-four","scope":"cfsg/milestones/part-02","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CL_LIT_GH74_sectional_two_rank_four`, one of those CFSG/Milestones.lean imports.","title":"CL_LIT_GH74_sectional_two_rank_four"},{"id":"cfsg/milestones/part-02/cl-lit-gl83-theorem-c-preuniqueness-rank-three","kind":"node","lean":"CFSG.Milestones.cl_lit_gl83_theorem_c_preuniqueness_rank_three","parent":8224,"result":true,"route":"../roadmap/cfsg/milestones/part-02/index.html#cl-lit-gl83-theorem-c-preuniqueness-rank-three","scope":"cfsg/milestones/part-02","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CL_LIT_GL83_Theorem_C_preuniqueness_rank_three`, one of those CFSG/Milestones.lean imports.","title":"CL_LIT_GL83_Theorem_C_preuniqueness_rank_three"},{"id":"cfsg/milestones/part-02/cl-lit-gms-monster-uniqueness","kind":"node","lean":"CFSG.Milestones.cl_lit_gms_monster_uniqueness, CFSG.Milestones.cl_lit_gms_monster_uniqueness_of_two","parent":8224,"result":true,"route":"../roadmap/cfsg/milestones/part-02/index.html#cl-lit-gms-monster-uniqueness","scope":"cfsg/milestones/part-02","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CL_LIT_GMS_Monster_uniqueness`, one of those CFSG/Milestones.lean imports.","title":"CL_LIT_GMS_Monster_uniqueness"},{"id":"cfsg/milestones/part-02/cl-lit-gilman-solomon-maximal-unbalancing-reduct","kind":"node","lean":"CFSG.Milestones.cl_lit_gilman_solomon_maximal_unbalancing_reduction, CFSG.Milestones.cl_lit_gilman_solomon_maximal_unbalancing_reduction_native, CFSG.Milestones.cl_lit_gilman_solomon_maximal_unbalancing_reduction_models","parent":8224,"result":true,"route":"../roadmap/cfsg/milestones/part-02/index.html#cl-lit-gilman-solomon-maximal-unbalancing-reduct","scope":"cfsg/milestones/part-02","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CL_LIT_Gilman_Solomon_maximal_unbalancing_reduction`, one of those CFSG/Milestones.lean imports.","title":"CL_LIT_Gilman_Solomon_maximal_unbalancing_reduction"},{"id":"cfsg/milestones/part-02/cl-lit-goldschmidt-strongly-closed-abelian","kind":"node","lean":"CFSG.Milestones.cl_lit_goldschmidt_strongly_closed_abelian","parent":8224,"result":true,"route":"../roadmap/cfsg/milestones/part-02/index.html#cl-lit-goldschmidt-strongly-closed-abelian","scope":"cfsg/milestones/part-02","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CL_LIT_Goldschmidt_strongly_closed_abelian`, one of those CFSG/Milestones.lean imports.","title":"CL_LIT_Goldschmidt_strongly_closed_abelian"},{"id":"cfsg/milestones/part-02/cl-lit-goldschmidt-strongly-closed-abelian-model","kind":"node","lean":"CFSG.Milestones.cl_lit_goldschmidt_strongly_closed_abelian_models","parent":8224,"result":true,"route":"../roadmap/cfsg/milestones/part-02/index.html#cl-lit-goldschmidt-strongly-closed-abelian-model","scope":"cfsg/milestones/part-02","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CL_LIT_Goldschmidt_strongly_closed_abelian_models`, which CFSG/Milestones.lean does not import.","title":"CL_LIT_Goldschmidt_strongly_closed_abelian_models"},{"id":"cfsg/milestones/part-02/cl-lit-gorenstein-walter-l-balance","kind":"node","lean":"CFSG.Milestones.cl_lit_gorenstein_walter_l_balance","parent":8224,"result":true,"route":"../roadmap/cfsg/milestones/part-02/index.html#cl-lit-gorenstein-walter-l-balance","scope":"cfsg/milestones/part-02","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CL_LIT_Gorenstein_Walter_L_balance`, one of those CFSG/Milestones.lean imports.","title":"CL_LIT_Gorenstein_Walter_L_balance"},{"id":"cfsg/milestones/part-02/cl-lit-griess-schur-multipliers-lie-type","kind":"node","lean":"CFSG.Milestones.cl_lit_griess_schur_multipliers_lie_type, CFSG.Milestones.cl_lit_griess_schur_multipliers_all_models, CFSG.Milestones.cl_lit_griess_schur_multipliers_all_models_statement, CFSG.Milestones.cl_lit_griess_schur_multipliers_central_covers, CFSG.Milestones.cl_lit_griess_schur_multipliers_all_native_models, CFSG.Milestones.cl_lit_griess_schur_multipliers_native_central_covers, CFSG.Milestones.cl_lit_griess_schur_multipliers_central_covers_z2, CFSG.Milestones.cl_lit_griess_schur_multipliers_native_central_covers_z2, CFSG.Milestones.cl_lit_griess_schur_multipliers_native_central_covers_unrestricted, CFSG.Milestones.cl_lit_griess_schur_multipliers_native_central_covers_z2_unrestricted, CFSG.Milestones.cl_lit_griess_schur_multipliers_g2_four_root_lifts","parent":8224,"result":true,"route":"../roadmap/cfsg/milestones/part-02/index.html#cl-lit-griess-schur-multipliers-lie-type","scope":"cfsg/milestones/part-02","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CL_LIT_Griess_Schur_multipliers_Lie_type`, one of those CFSG/Milestones.lean imports.","title":"CL_LIT_Griess_Schur_multipliers_Lie_type"},{"id":"cfsg/milestones/part-02/cl-lit-monster-existence-griess-conway","kind":"node","lean":"CFSG.Milestones.cl_lit_monster_existence_griess_conway_1, CFSG.Milestones.cl_lit_monster_existence_griess_conway_2, CFSG.Milestones.cl_lit_monster_existence_griess_conway","parent":8224,"result":true,"route":"../roadmap/cfsg/milestones/part-02/index.html#cl-lit-monster-existence-griess-conway","scope":"cfsg/milestones/part-02","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CL_LIT_Monster_existence_Griess_Conway`, one of those CFSG/Milestones.lean imports.","title":"CL_LIT_Monster_existence_Griess_Conway"},{"id":"cfsg/milestones/part-02/cl-lit-sqtk-simple-list","kind":"node","lean":"CFSG.Milestones.cl_lit_sqtk_simple_list_section","parent":8224,"result":true,"route":"../roadmap/cfsg/milestones/part-02/index.html#cl-lit-sqtk-simple-list","scope":"cfsg/milestones/part-02","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CL_LIT_SQTK_simple_list`, one of those CFSG/Milestones.lean imports.","title":"CL_LIT_SQTK_simple_list"},{"data_url":"scopes/cfsg/milestones/part-03.json","id":"cfsg/milestones/part-03","kind":"scope","navigation":"transparent","parent":8202,"result":false,"route":"../roadmap/cfsg/milestones/part-03/index.html","scope":"cfsg/milestones/part-03","status":"fully_proved","title":"CL_LIT_SQTK_simple_listBase to CT_rank_two_standard_recognition\u2026"},{"id":"cfsg/milestones/part-03/cl-lit-sqtk-simple-listbase","kind":"node","lean":"CFSG.Milestones.SQTKSimpleList, CFSG.Milestones.cl_lit_sqtk_simple_list, CFSG.Milestones.cl_lit_sqtk_simple_list_simple","parent":8245,"result":true,"route":"../roadmap/cfsg/milestones/part-03/index.html#cl-lit-sqtk-simple-listbase","scope":"cfsg/milestones/part-03","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CL_LIT_SQTK_simple_listBase`, which CFSG/Milestones.lean does not import.","title":"CL_LIT_SQTK_simple_listBase"},{"id":"cfsg/milestones/part-03/cl-lit-steinberg-presentation-theorem","kind":"node","lean":"CFSG.Milestones.cl_lit_steinberg_presentation_theorem","parent":8245,"result":true,"route":"../roadmap/cfsg/milestones/part-03/index.html#cl-lit-steinberg-presentation-theorem","scope":"cfsg/milestones/part-03","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CL_LIT_Steinberg_presentation_theorem`, one of those CFSG/Milestones.lean imports.","title":"CL_LIT_Steinberg_presentation_theorem"},{"id":"cfsg/milestones/part-03/cl-lit-stroth-leon-sims-baby-monster-recognition","kind":"node","lean":"CFSG.Milestones.cl_lit_stroth_leon_sims_baby_monster_recognition, CFSG.Milestones.cl_lit_stroth_leon_sims_baby_monster_recognition_models","parent":8245,"result":true,"route":"../roadmap/cfsg/milestones/part-03/index.html#cl-lit-stroth-leon-sims-baby-monster-recognition","scope":"cfsg/milestones/part-03","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CL_LIT_Stroth_Leon_Sims_Baby_Monster_recognition`, one of those CFSG/Milestones.lean imports.","title":"CL_LIT_Stroth_Leon_Sims_Baby_Monster_recognition"},{"id":"cfsg/milestones/part-03/cl-lit-timmesfeld-1978-large-extraspecial","kind":"node","lean":"CFSG.Milestones.cl_lit_timmesfeld_1978_large_extraspecial, CFSG.Milestones.cl_lit_timmesfeld_1978_large_extraspecial_models","parent":8245,"result":true,"route":"../roadmap/cfsg/milestones/part-03/index.html#cl-lit-timmesfeld-1978-large-extraspecial","scope":"cfsg/milestones/part-03","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CL_LIT_Timmesfeld_1978_large_extraspecial`, one of those CFSG/Milestones.lean imports.","title":"CL_LIT_Timmesfeld_1978_large_extraspecial"},{"id":"cfsg/milestones/part-03/cl-lit-timmesfeld-root-involutions","kind":"node","lean":"CFSG.Milestones.cl_lit_timmesfeld_root_involutions, CFSG.Milestones.cl_lit_timmesfeld_root_involutions_models","parent":8245,"result":true,"route":"../roadmap/cfsg/milestones/part-03/index.html#cl-lit-timmesfeld-root-involutions","scope":"cfsg/milestones/part-03","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CL_LIT_Timmesfeld_root_involutions`, one of those CFSG/Milestones.lean imports.","title":"CL_LIT_Timmesfeld_root_involutions"},{"id":"cfsg/milestones/part-03/cl-lit-walter-abelian-sylow-two-simple","kind":"node","lean":"CFSG.Milestones.cl_lit_walter_abelian_sylow_two_simple, CFSG.Milestones.cl_lit_walter_abelian_sylow_two_simple_models","parent":8245,"result":true,"route":"../roadmap/cfsg/milestones/part-03/index.html#cl-lit-walter-abelian-sylow-two-simple","scope":"cfsg/milestones/part-03","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CL_LIT_Walter_abelian_Sylow_two_simple`, one of those CFSG/Milestones.lean imports.","title":"CL_LIT_Walter_abelian_Sylow_two_simple"},{"id":"cfsg/milestones/part-03/cl-lit-classical-involution-theorem","kind":"node","lean":"CFSG.Milestones.cl_lit_classical_involution_theorem_models, CFSG.Milestones.cl_lit_classical_involution_theorem","parent":8245,"result":true,"route":"../roadmap/cfsg/milestones/part-03/index.html#cl-lit-classical-involution-theorem","scope":"cfsg/milestones/part-03","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CL_LIT_classical_involution_theorem`, one of those CFSG/Milestones.lean imports.","title":"CL_LIT_classical_involution_theorem"},{"id":"cfsg/milestones/part-03/cl-lit-signalizer-functor-theorem","kind":"node","lean":"CFSG.Milestones.cl_lit_signalizer_functor_theorem_1, CFSG.Milestones.cl_lit_signalizer_functor_theorem_2","parent":8245,"result":true,"route":"../roadmap/cfsg/milestones/part-03/index.html#cl-lit-signalizer-functor-theorem","scope":"cfsg/milestones/part-03","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CL_LIT_signalizer_functor_theorem`, one of those CFSG/Milestones.lean imports.","title":"CL_LIT_signalizer_functor_theorem"},{"id":"cfsg/milestones/part-03/ctb-kproper-component-local-b","kind":"node","lean":"CFSG.Milestones.ctb_kproper_component_local_b","parent":8245,"result":true,"route":"../roadmap/cfsg/milestones/part-03/index.html#ctb-kproper-component-local-b","scope":"cfsg/milestones/part-03","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CTB_KProper_component_local_B`, one of those CFSG/Milestones.lean imports.","title":"CTB_KProper_component_local_B"},{"id":"cfsg/milestones/part-03/ctgms-bender-1971-even-rank-one-model-and-strong","kind":"node","lean":"CFSG.Milestones.ctgms_bender_1971_even_rank_one_model_and_strong_embedding_input_1, CFSG.Milestones.ctgms_bender_1971_even_rank_one_model_and_strong_embedding_input_2, CFSG.Milestones.ctgms_bender_1971_even_rank_one_model_and_strong_embedding_input_3, CFSG.Milestones.ctgms_bender_1971_even_rank_one_model_and_strong_embedding_input_4","parent":8245,"result":true,"route":"../roadmap/cfsg/milestones/part-03/index.html#ctgms-bender-1971-even-rank-one-model-and-strong","scope":"cfsg/milestones/part-03","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CTGMS_Bender_1971_even_rank_one_model_and_strong_embedding_input`, one of those CFSG/Milestones.lean imports.","title":"CTGMS_Bender_1971_even_rank_one_model_and_strong_embedding_input"},{"id":"cfsg/milestones/part-03/ctss-sporadic-standard-centerless-m12-j2-and-oth","kind":"node","lean":"CFSG.Milestones.ctss_sporadic_standard_centerless_m12_j2_and_other_quotients_frontier, CFSG.Milestones.ctss_sporadic_standard_centerless_m12_j2_and_other_quotients_frontier_models","parent":8245,"result":true,"route":"../roadmap/cfsg/milestones/part-03/index.html#ctss-sporadic-standard-centerless-m12-j2-and-oth","scope":"cfsg/milestones/part-03","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CTSS_sporadic_standard_centerless_M12_J2_and_other_quotients_frontier`, one of those CFSG/Milestones.lean imports.","title":"CTSS_sporadic_standard_centerless_M12_J2_and_other_quotients_frontier"},{"id":"cfsg/milestones/part-03/ctu-kproper-simple-unbalanced-recognition","kind":"node","lean":"CFSG.Milestones.ctu_relativeU_catalog, CFSG.Milestones.ctu_kproper_simple_unbalanced_recognition, CFSG.Milestones.ctu_kproper_simple_unbalanced_recognition_models","parent":8245,"result":true,"route":"../roadmap/cfsg/milestones/part-03/index.html#ctu-kproper-simple-unbalanced-recognition","scope":"cfsg/milestones/part-03","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CTU_KProper_simple_unbalanced_recognition`, one of those CFSG/Milestones.lean imports.","title":"CTU_KProper_simple_unbalanced_recognition"},{"id":"cfsg/milestones/part-03/ct-seitz-centerless-reduction-frontier","kind":"node","lean":"CFSG.Milestones.ct_seitz_centerless_reduction_frontier","parent":8245,"result":true,"route":"../roadmap/cfsg/milestones/part-03/index.html#ct-seitz-centerless-reduction-frontier","scope":"cfsg/milestones/part-03","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CT_Seitz_centerless_reduction_frontier`, one of those CFSG/Milestones.lean imports.","title":"CT_Seitz_centerless_reduction_frontier"},{"id":"cfsg/milestones/part-03/ct-catalog-nonabelian-type-partition","kind":"node","lean":"CFSG.Blueprint.suzukiModel_admissible, CFSG.Blueprint.oddLieType_or_evenLieType_of_lieModel, CFSG.Milestones.ct_catalog_nonabelian_type_partition_converse, CFSG.Milestones.ct_catalog_nonabelian_type_partition","parent":8245,"result":true,"route":"../roadmap/cfsg/milestones/part-03/index.html#ct-catalog-nonabelian-type-partition","scope":"cfsg/milestones/part-03","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CT_catalog_nonabelian_type_partition`, one of those CFSG/Milestones.lean imports.","title":"CT_catalog_nonabelian_type_partition"},{"id":"cfsg/milestones/part-03/ct-even-small-lie-standard-frontier","kind":"node","lean":"CFSG.Milestones.ct_even_small_lie_standard_frontier, CFSG.Milestones.ct_even_small_lie_standard_frontier_models","parent":8245,"result":true,"route":"../roadmap/cfsg/milestones/part-03/index.html#ct-even-small-lie-standard-frontier","scope":"cfsg/milestones/part-03","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CT_even_small_Lie_standard_frontier`, one of those CFSG/Milestones.lean imports.","title":"CT_even_small_Lie_standard_frontier"},{"id":"cfsg/milestones/part-03/ct-even-standard-residual-reduction","kind":"node","lean":"CFSG.Milestones.ct_even_standard_residual_reduction","parent":8245,"result":true,"route":"../roadmap/cfsg/milestones/part-03/index.html#ct-even-standard-residual-reduction","scope":"cfsg/milestones/part-03","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CT_even_standard_residual_reduction`, one of those CFSG/Milestones.lean imports.","title":"CT_even_standard_residual_reduction"},{"id":"cfsg/milestones/part-03/ct-initial-standard-component-frontier","kind":"node","lean":"CFSG.Milestones.ct_initial_standard_component_frontier","parent":8245,"result":true,"route":"../roadmap/cfsg/milestones/part-03/index.html#ct-initial-standard-component-frontier","scope":"cfsg/milestones/part-03","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CT_initial_standard_component_frontier`, one of those CFSG/Milestones.lean imports.","title":"CT_initial_standard_component_frontier"},{"id":"cfsg/milestones/part-03/ct-odd-standard-recognition-frontier","kind":"node","lean":"CFSG.Milestones.ct_odd_standard_recognition_frontier, CFSG.Milestones.ct_odd_standard_recognition_frontier_models","parent":8245,"result":true,"route":"../roadmap/cfsg/milestones/part-03/index.html#ct-odd-standard-recognition-frontier","scope":"cfsg/milestones/part-03","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CT_odd_standard_recognition_frontier`, one of those CFSG/Milestones.lean imports.","title":"CT_odd_standard_recognition_frontier"},{"id":"cfsg/milestones/part-03/ct-quaternion-component-recognition-frontier","kind":"node","lean":"CFSG.Milestones.ct_quaternion_component_recognition_frontier","parent":8245,"result":true,"route":"../roadmap/cfsg/milestones/part-03/index.html#ct-quaternion-component-recognition-frontier","scope":"cfsg/milestones/part-03","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CT_quaternion_component_recognition_frontier`, one of those CFSG/Milestones.lean imports.","title":"CT_quaternion_component_recognition_frontier"},{"id":"cfsg/milestones/part-03/ct-rank-two-standard-recognition-frontier","kind":"node","lean":"CFSG.Milestones.ct_rank_two_standard_recognition_frontier, CFSG.Milestones.ct_rank_two_standard_recognition_frontier_models","parent":8245,"result":true,"route":"../roadmap/cfsg/milestones/part-03/index.html#ct-rank-two-standard-recognition-frontier","scope":"cfsg/milestones/part-03","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CT_rank_two_standard_recognition_frontier`, one of those CFSG/Milestones.lean imports.","title":"CT_rank_two_standard_recognition_frontier"},{"data_url":"scopes/cfsg/milestones/part-04.json","id":"cfsg/milestones/part-04","kind":"scope","navigation":"transparent","parent":8202,"result":false,"route":"../roadmap/cfsg/milestones/part-04/index.html","scope":"cfsg/milestones/part-04","status":"fully_proved","title":"CT_standard_recognition_assembly to LC_GL_embedding_theorem_D2_rank_\u2026"},{"id":"cfsg/milestones/part-04/ct-standard-recognition-assembly","kind":"node","lean":"CFSG.Milestones.ct_standard_recognition_assembly","parent":8266,"result":true,"route":"../roadmap/cfsg/milestones/part-04/index.html#ct-standard-recognition-assembly","scope":"cfsg/milestones/part-04","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.CT_standard_recognition_assembly`, one of those CFSG/Milestones.lean imports.","title":"CT_standard_recognition_assembly"},{"id":"cfsg/milestones/part-04/fam-m-finiteness","kind":"node","lean":"CFSG.Milestones.fam_m_finiteness","parent":8266,"result":true,"route":"../roadmap/cfsg/milestones/part-04/index.html#fam-m-finiteness","scope":"cfsg/milestones/part-04","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.FAM_M_finiteness`, one of those CFSG/Milestones.lean imports.","title":"FAM_M_finiteness"},{"id":"cfsg/milestones/part-04/fam-m-simplicity","kind":"node","lean":"CFSG.Milestones.fam_m_simplicity","parent":8266,"result":true,"route":"../roadmap/cfsg/milestones/part-04/index.html#fam-m-simplicity","scope":"cfsg/milestones/part-04","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.FAM_M_simplicity`, one of those CFSG/Milestones.lean imports.","title":"FAM_M_simplicity"},{"id":"cfsg/milestones/part-04/fam-th-simplicity","kind":"node","lean":"CFSG.Milestones.fam_th_simplicity","parent":8266,"result":true,"route":"../roadmap/cfsg/milestones/part-04/index.html#fam-th-simplicity","scope":"cfsg/milestones/part-04","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.FAM_Th_simplicity`, one of those CFSG/Milestones.lean imports.","title":"FAM_Th_simplicity"},{"id":"cfsg/milestones/part-04/fam-classical-model-membership","kind":"node","lean":"CFSG.Milestones.ClassicalRep, CFSG.Milestones.fam_classical_model_membership_alltags, CFSG.Milestones.statedClassicalModel_tag_mem, CFSG.Milestones.fam_classical_model_membership","parent":8266,"result":true,"route":"../roadmap/cfsg/milestones/part-04/index.html#fam-classical-model-membership","scope":"cfsg/milestones/part-04","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.FAM_classical_model_membership`, one of those CFSG/Milestones.lean imports.","title":"FAM_classical_model_membership"},{"id":"cfsg/milestones/part-04/fam-sporadic-converse","kind":"node","lean":"CFSG.Milestones.fam_sporadic_converse_tauCeti, CFSG.Milestones.fam_sporadic_converse","parent":8266,"result":true,"route":"../roadmap/cfsg/milestones/part-04/index.html#fam-sporadic-converse","scope":"cfsg/milestones/part-04","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.FAM_sporadic_converse`, one of those CFSG/Milestones.lean imports.","title":"FAM_sporadic_converse"},{"id":"cfsg/milestones/part-04/fam-sporadic-finiteness","kind":"node","lean":"CFSG.Milestones.fam_sporadic_finiteness_tauCeti, CFSG.Milestones.fam_sporadic_finiteness","parent":8266,"result":true,"route":"../roadmap/cfsg/milestones/part-04/index.html#fam-sporadic-finiteness","scope":"cfsg/milestones/part-04","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.FAM_sporadic_finiteness`, one of those CFSG/Milestones.lean imports.","title":"FAM_sporadic_finiteness"},{"id":"cfsg/milestones/part-04/fam-sporadic-native-converse","kind":"node","lean":"CFSG.Milestones.fam_sporadic_native_converse","parent":8266,"result":true,"route":"../roadmap/cfsg/milestones/part-04/index.html#fam-sporadic-native-converse","scope":"cfsg/milestones/part-04","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.FAM_sporadic_native_converse`, one of those CFSG/Milestones.lean imports.","title":"FAM_sporadic_native_converse"},{"id":"cfsg/milestones/part-04/fam-sporadic-representatives","kind":"node","lean":"CFSG.Milestones.fam_sporadic_representatives_definition","parent":8266,"result":true,"route":"../roadmap/cfsg/milestones/part-04/index.html#fam-sporadic-representatives","scope":"cfsg/milestones/part-04","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.FAM_sporadic_representatives`, one of those CFSG/Milestones.lean imports.","title":"FAM_sporadic_representatives"},{"id":"cfsg/milestones/part-04/frob-kernel-existence","kind":"node","lean":"CFSG.Milestones.frobenius_kernel, CFSG.Milestones.frobenius_kernel_existence","parent":8266,"result":true,"route":"../roadmap/cfsg/milestones/part-04/index.html#frob-kernel-existence","scope":"cfsg/milestones/part-04","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.FROB_kernel_existence`, one of those CFSG/Milestones.lean imports.","title":"FROB_kernel_existence"},{"id":"cfsg/milestones/part-04/gw-original-maximal-odd-s-invariant-r-and-normal","kind":"node","lean":"CFSG.Milestones.gw_original_maximal_odd_s_invariant_r_and_normalizer_core","parent":8266,"result":true,"route":"../roadmap/cfsg/milestones/part-04/index.html#gw-original-maximal-odd-s-invariant-r-and-normal","scope":"cfsg/milestones/part-04","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.GW_original_maximal_odd_S_invariant_R_and_normalizer_core`, one of those CFSG/Milestones.lean imports.","title":"GW_original_maximal_odd_S_invariant_R_and_normalizer_core"},{"id":"cfsg/milestones/part-04/gw-original-normalizer-r-pure-projective-shared","kind":"node","lean":"CFSG.Milestones.gw_original_normalizer_r_pure_projective_shared_fourgroup_and_index_divisor","parent":8266,"result":true,"route":"../roadmap/cfsg/milestones/part-04/index.html#gw-original-normalizer-r-pure-projective-shared","scope":"cfsg/milestones/part-04","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.GW_original_normalizer_R_pure_projective_shared_fourgroup_and_index_divisor`, one of those CFSG/Milestones.lean imports.","title":"GW_original_normalizer_R_pure_projective_shared_fourgroup_and_index_divisor"},{"id":"cfsg/milestones/part-04/gw-original-normalizer-coset-involution-fixedpoi","kind":"node","lean":"CFSG.Milestones.gw_original_normalizer_coset_involution_fixedpoints_secondary_torus_divisibility","parent":8266,"result":true,"route":"../roadmap/cfsg/milestones/part-04/index.html#gw-original-normalizer-coset-involution-fixedpoi","scope":"cfsg/milestones/part-04","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.GW_original_normalizer_coset_involution_fixedpoints_secondary_torus_divisibility`, one of those CFSG/Milestones.lean imports.","title":"GW_original_normalizer_coset_involution_fixedpoints_secondary_torus_divisibility"},{"id":"cfsg/milestones/part-04/gw-original-post-proposition-sixteen-all-prime-s","kind":"node","lean":"CFSG.Milestones.gw_original_post_proposition_sixteen_all_prime_stationarity_and_no_growth","parent":8266,"result":true,"route":"../roadmap/cfsg/milestones/part-04/index.html#gw-original-post-proposition-sixteen-all-prime-s","scope":"cfsg/milestones/part-04","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.GW_original_post_proposition_sixteen_all_prime_stationarity_and_no_growth`, one of those CFSG/Milestones.lean imports.","title":"GW_original_post_proposition_sixteen_all_prime_stationarity_and_no_growth"},{"id":"cfsg/milestones/part-04/gw-original-post-proposition-sixteen-canonical-s","kind":"node","lean":"CFSG.Milestones.gw_original_post_proposition_sixteen_canonical_stationary_prime_normalizers, CFSG.Milestones.gw_original_post_proposition_sixteen_canonical_stationary_prime_normalizers_full","parent":8266,"result":true,"route":"../roadmap/cfsg/milestones/part-04/index.html#gw-original-post-proposition-sixteen-canonical-s","scope":"cfsg/milestones/part-04","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.GW_original_post_proposition_sixteen_canonical_stationary_prime_normalizers`, one of those CFSG/Milestones.lean imports.","title":"GW_original_post_proposition_sixteen_canonical_stationary_prime_normalizers"},{"id":"cfsg/milestones/part-04/gw-original-q-five-index-three-character-kernel","kind":"node","lean":"CFSG.Milestones.gw_original_q_five_index_three_character_kernel_order_contradiction, CFSG.Milestones.gw_original_q_five_index_three_character_kernel_order_contradiction_full","parent":8266,"result":true,"route":"../roadmap/cfsg/milestones/part-04/index.html#gw-original-q-five-index-three-character-kernel","scope":"cfsg/milestones/part-04","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.GW_original_q_five_index_three_character_kernel_order_contradiction`, one of those CFSG/Milestones.lean imports.","title":"GW_original_q_five_index_three_character_kernel_order_contradiction"},{"id":"cfsg/milestones/part-04/lc-gl-gf2-recognition-rank-four","kind":"node","lean":"CFSG.Milestones.lc_gl_gf2_recognition_rank_four","parent":8266,"result":true,"route":"../roadmap/cfsg/milestones/part-04/index.html#lc-gl-gf2-recognition-rank-four","scope":"cfsg/milestones/part-04","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.LC_GL_GF2_recognition_rank_four`, one of those CFSG/Milestones.lean imports.","title":"LC_GL_GF2_recognition_rank_four"},{"id":"cfsg/milestones/part-04/lc-gl-constructed-pretrichotomy-rank-four","kind":"node","lean":"CFSG.Milestones.lc_gl_constructed_pretrichotomy_rank_four","parent":8266,"result":true,"route":"../roadmap/cfsg/milestones/part-04/index.html#lc-gl-constructed-pretrichotomy-rank-four","scope":"cfsg/milestones/part-04","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.LC_GL_constructed_pretrichotomy_rank_four`, one of those CFSG/Milestones.lean imports.","title":"LC_GL_constructed_pretrichotomy_rank_four"},{"id":"cfsg/milestones/part-04/lc-gl-embedding-theorem-d1-rank-four","kind":"node","lean":"CFSG.Milestones.lc_gl_embedding_theorem_d1_rank_four","parent":8266,"result":true,"route":"../roadmap/cfsg/milestones/part-04/index.html#lc-gl-embedding-theorem-d1-rank-four","scope":"cfsg/milestones/part-04","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.LC_GL_embedding_theorem_D1_rank_four`, one of those CFSG/Milestones.lean imports.","title":"LC_GL_embedding_theorem_D1_rank_four"},{"id":"cfsg/milestones/part-04/lc-gl-embedding-theorem-d2-rank-four","kind":"node","lean":"CFSG.Milestones.lc_gl_embedding_theorem_d2_rank_four","parent":8266,"result":true,"route":"../roadmap/cfsg/milestones/part-04/index.html#lc-gl-embedding-theorem-d2-rank-four","scope":"cfsg/milestones/part-04","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.LC_GL_embedding_theorem_D2_rank_four`, one of those CFSG/Milestones.lean imports.","title":"LC_GL_embedding_theorem_D2_rank_four"},{"data_url":"scopes/cfsg/milestones/part-05.json","id":"cfsg/milestones/part-05","kind":"scope","navigation":"transparent","parent":8202,"result":false,"route":"../roadmap/cfsg/milestones/part-05/index.html","scope":"cfsg/milestones/part-05","status":"fully_proved","title":"LC_GL_local_rank_bounds to QT_primitive_regular_cyclic_Burn\u2026"},{"id":"cfsg/milestones/part-05/lc-gl-local-rank-bounds","kind":"node","lean":"CFSG.Milestones.lc_gl_local_rank_bounds","parent":8287,"result":true,"route":"../roadmap/cfsg/milestones/part-05/index.html#lc-gl-local-rank-bounds","scope":"cfsg/milestones/part-05","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.LC_GL_local_rank_bounds`, one of those CFSG/Milestones.lean imports.","title":"LC_GL_local_rank_bounds"},{"id":"cfsg/milestones/part-05/lc-gl-preuniqueness-exclusion-rank-four","kind":"node","lean":"CFSG.Milestones.lc_gl_preuniqueness_exclusion_rank_four","parent":8287,"result":true,"route":"../roadmap/cfsg/milestones/part-05/index.html#lc-gl-preuniqueness-exclusion-rank-four","scope":"cfsg/milestones/part-05","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.LC_GL_preuniqueness_exclusion_rank_four`, one of those CFSG/Milestones.lean imports.","title":"LC_GL_preuniqueness_exclusion_rank_four"},{"id":"cfsg/milestones/part-05/lc-gl-standard-type-recognition-rank-four","kind":"node","lean":"CFSG.Milestones.lc_gl_standard_type_lie_char_two_recognition_rank_four, CFSG.Milestones.lc_gl_standard_type_recognition_rank_four","parent":8287,"result":true,"route":"../roadmap/cfsg/milestones/part-05/index.html#lc-gl-standard-type-recognition-rank-four","scope":"cfsg/milestones/part-05","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.LC_GL_standard_type_recognition_rank_four`, one of those CFSG/Milestones.lean imports.","title":"LC_GL_standard_type_recognition_rank_four"},{"id":"cfsg/milestones/part-05/lc-gl-theorem-a","kind":"node","lean":"CFSG.Milestones.lc_gl_theorem_a","parent":8287,"result":true,"route":"../roadmap/cfsg/milestones/part-05/index.html#lc-gl-theorem-a","scope":"cfsg/milestones/part-05","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.LC_GL_theorem_A`, one of 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`CFSG.Milestones.LR_semidihedral_simple_recognition`, one of those CFSG/Milestones.lean imports.","title":"LR_semidihedral_simple_recognition"},{"id":"cfsg/milestones/part-05/lr-wreathed-simple-recognition","kind":"node","lean":"CFSG.Milestones.lr_wreathed_simple_recognition_1, CFSG.Milestones.lr_wreathed_simple_recognition_2","parent":8287,"result":true,"route":"../roadmap/cfsg/milestones/part-05/index.html#lr-wreathed-simple-recognition","scope":"cfsg/milestones/part-05","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.LR_wreathed_simple_recognition`, one of those CFSG/Milestones.lean imports.","title":"LR_wreathed_simple_recognition"},{"id":"cfsg/milestones/part-05/oo-odd-order-solvable","kind":"node","lean":"CFSG.Milestones.odd_order_solvable","parent":8287,"result":true,"route":"../roadmap/cfsg/milestones/part-05/index.html#oo-odd-order-solvable","scope":"cfsg/milestones/part-05","status":"fully_proved","summary":"The milestone module 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`CFSG.Milestones.QT_primitive_regular_cyclic_Burnside_Schur`, one of those CFSG/Milestones.lean imports.","title":"QT_primitive_regular_cyclic_Burnside_Schur"},{"data_url":"scopes/cfsg/milestones/part-06.json","id":"cfsg/milestones/part-06","kind":"scope","navigation":"transparent","parent":8202,"result":false,"route":"../roadmap/cfsg/milestones/part-06/index.html","scope":"cfsg/milestones/part-06","status":"fully_proved","title":"QT_unique_Shult_simple_strongly_\u2026 to ZJ_glauberman_replacement"},{"id":"cfsg/milestones/part-06/qt-unique-shult-simple-strongly-closed-ti-models","kind":"node","lean":"CFSG.Milestones.qt_unique_shult_simple_strongly_closed_ti_models","parent":8308,"result":true,"route":"../roadmap/cfsg/milestones/part-06/index.html#qt-unique-shult-simple-strongly-closed-ti-models","scope":"cfsg/milestones/part-06","status":"fully_proved","summary":"The milestone module `CFSG.Milestones.QT_unique_Shult_simple_strongly_closed_TI_models`, one of those CFSG/Milestones.lean 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As its README describes, each is imported from a permissively licensed repository into a directory of its own, brought to Lean Pool's Lean and Mathlib, and merged once it passes Lean Pool's CI and an LLM review. This chapter maps its 334 projects at commit 2b205bf9 (2026-11-08), built with Lean v4.35.0 and Mathlib at 56a00585. Each project has a card,\u2026","title":"Lean Pool"},{"data_url":"scopes/lean-pool/part-01.json","id":"lean-pool/part-01","kind":"scope","navigation":"transparent","parent":8319,"result":false,"route":"../roadmap/lean-pool/part-01/index.html","scope":"lean-pool/part-01","status":"can_state","title":"MSC2020 classes 03 to 35"},{"data_url":"scopes/lean-pool/part-01/03-mathematical-logic-and-foundations.json","id":"lean-pool/part-01/03-mathematical-logic-and-foundations","kind":"scope","parent":8320,"result":false,"route":"../roadmap/lean-pool/part-01/03-mathematical-logic-and-foundations/index.html","scope":"lean-pool/part-01/03-mathematical-logic-and-foundations","status":"can_prove","summary":"The Lean Pool projects whose card's first MSC2020 code is in class 03, Mathematical logic and foundations: 21 articles, one per card.","title":"Mathematical logic and 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Sequences"},{"id":"lean-pool/part-02/52-convex-and-discrete-geometry/erdos132-three-chain","kind":"node","lean":"Erdos132ThreeChain.nonDiameterSqDists_ne_chain","parent":8601,"result":true,"route":"../roadmap/lean-pool/part-02/52-convex-and-discrete-geometry/index.html#erdos132-three-chain","scope":"lean-pool/part-02/52-convex-and-discrete-geometry","status":"fully_proved","summary":"> > > > > > > > > >","title":"Planar Point Sets Whose Non-Diameter Distances Form a Geometric 3-Chain"},{"id":"lean-pool/part-02/52-convex-and-discrete-geometry/rupert","kind":"node","lean":"Cube.rupert, Tetrahedron.rupert, TriakisTetrahedron.rupert","parent":8601,"result":true,"route":"../roadmap/lean-pool/part-02/52-convex-and-discrete-geometry/index.html#rupert","scope":"lean-pool/part-02/52-convex-and-discrete-geometry","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/rupert.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"The Rupert Problem for convex polyhedra"},{"id":"lean-pool/part-02/52-convex-and-discrete-geometry/sphere-packing","kind":"node","lean":"PackingBounds.sharpFullCohnElkiesManuscriptConclusions","parent":8601,"result":true,"route":"../roadmap/lean-pool/part-02/52-convex-and-discrete-geometry/index.html#sphere-packing","scope":"lean-pool/part-02/52-convex-and-discrete-geometry","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/sphere-packing.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Sharp asymptotic upper bounds for sphere packing"},{"data_url":"scopes/lean-pool/part-02/53-differential-geometry.json","id":"lean-pool/part-02/53-differential-geometry","kind":"scope","parent":8561,"result":false,"route":"../roadmap/lean-pool/part-02/53-differential-geometry/index.html","scope":"lean-pool/part-02/53-differential-geometry","status":"fully_proved","summary":"The Lean Pool projects whose card's first MSC2020 code is in class 53, Differential geometry: 2 articles, one per card.","title":"Differential geometry"},{"id":"lean-pool/part-02/53-differential-geometry/nashembedding","kind":"node","lean":"NashEmbeddingTheorem.nash_isometric_embedding","parent":8617,"result":true,"route":"../roadmap/lean-pool/part-02/53-differential-geometry/index.html#nashembedding","scope":"lean-pool/part-02/53-differential-geometry","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/nashembedding.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Nash's smooth isometric embedding theorem"},{"id":"lean-pool/part-02/53-differential-geometry/planar-midpoint-rigidity","kind":"node","lean":"PlanarMidpoint.planar_dual_midpoint_rigidity","parent":8617,"result":true,"route":"../roadmap/lean-pool/part-02/53-differential-geometry/index.html#planar-midpoint-rigidity","scope":"lean-pool/part-02/53-differential-geometry","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/planar-midpoint-rigidity.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Planar Euclidean dual-geodesic midpoint rigidity"},{"data_url":"scopes/lean-pool/part-02/55-algebraic-topology.json","id":"lean-pool/part-02/55-algebraic-topology","kind":"scope","parent":8561,"result":false,"route":"../roadmap/lean-pool/part-02/55-algebraic-topology/index.html","scope":"lean-pool/part-02/55-algebraic-topology","status":"fully_proved","summary":"The Lean Pool projects whose card's first MSC2020 code is in class 55, Algebraic topology: 3 articles, one per card.","title":"Algebraic topology"},{"id":"lean-pool/part-02/55-algebraic-topology/directed-topology-lean-4","kind":"node","lean":"DirectedSpace, Dipath, DirectedVanKampen.directed_van_kampen","parent":8620,"result":true,"route":"../roadmap/lean-pool/part-02/55-algebraic-topology/index.html#directed-topology-lean-4","scope":"lean-pool/part-02/55-algebraic-topology","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/directed-topology-lean-4.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Directed Topology in Lean 4"},{"id":"lean-pool/part-02/55-algebraic-topology/eilenberg-zilber","kind":"node","lean":"LeanPool.EilenbergZilber.BisimplicialObject.eilenbergZilber","parent":8620,"result":true,"route":"../roadmap/lean-pool/part-02/55-algebraic-topology/index.html#eilenberg-zilber","scope":"lean-pool/part-02/55-algebraic-topology","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/eilenberg-zilber.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Eilenberg\u2013Zilber theorem for bisimplicial objects"},{"id":"lean-pool/part-02/55-algebraic-topology/whitehead-theorem","kind":"node","lean":"WhiteheadTheorem","parent":8620,"result":true,"route":"../roadmap/lean-pool/part-02/55-algebraic-topology/index.html#whitehead-theorem","scope":"lean-pool/part-02/55-algebraic-topology","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/whitehead-theorem.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Whitehead's theorem for CW-complexes"},{"data_url":"scopes/lean-pool/part-02/57-manifolds-and-cell-complexes.json","id":"lean-pool/part-02/57-manifolds-and-cell-complexes","kind":"scope","parent":8561,"result":false,"route":"../roadmap/lean-pool/part-02/57-manifolds-and-cell-complexes/index.html","scope":"lean-pool/part-02/57-manifolds-and-cell-complexes","status":"fully_proved","summary":"The Lean Pool projects whose card's first MSC2020 code is in class 57, Manifolds and cell complexes: 3 articles, one per card.","title":"Manifolds and cell complexes"},{"id":"lean-pool/part-02/57-manifolds-and-cell-complexes/1mfld","kind":"node","lean":"OneMfld.homeomorph_circle_or_unitInterval","parent":8624,"result":true,"route":"../roadmap/lean-pool/part-02/57-manifolds-and-cell-complexes/index.html#1mfld","scope":"lean-pool/part-02/57-manifolds-and-cell-complexes","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/1mfld.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"The classification of compact 1-manifolds"},{"id":"lean-pool/part-02/57-manifolds-and-cell-complexes/classification-of-surfaces","kind":"node","lean":"LeanEval.Topology.ClassificationOfSurfaces.classification_of_surfaces","parent":8624,"result":true,"route":"../roadmap/lean-pool/part-02/57-manifolds-and-cell-complexes/index.html#classification-of-surfaces","scope":"lean-pool/part-02/57-manifolds-and-cell-complexes","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/classification-of-surfaces.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Classification of Compact Surfaces"},{"id":"lean-pool/part-02/57-manifolds-and-cell-complexes/schoenflies","kind":"node","lean":"Schoenflies.jordan_schoenflies_of_homeomorph","parent":8624,"result":true,"route":"../roadmap/lean-pool/part-02/57-manifolds-and-cell-complexes/index.html#schoenflies","scope":"lean-pool/part-02/57-manifolds-and-cell-complexes","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/schoenflies.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Jordan\u2013Sch\u00f6nflies theorem"},{"data_url":"scopes/lean-pool/part-02/58-global-analysis.json","id":"lean-pool/part-02/58-global-analysis","kind":"scope","parent":8561,"result":false,"route":"../roadmap/lean-pool/part-02/58-global-analysis/index.html","scope":"lean-pool/part-02/58-global-analysis","status":"fully_proved","summary":"The Lean Pool projects whose card's first MSC2020 code is in class 58, \u201cGlobal analysis, analysis on manifolds\u201d: 1 article.","title":"Global analysis, analysis on manifolds"},{"id":"lean-pool/part-02/58-global-analysis/stationary-harmonic-maps","kind":"node","lean":"LeanStationaryHarmonicMaps.StationaryHarmonicMap.StationaryW12LocMap","parent":8628,"result":true,"route":"../roadmap/lean-pool/part-02/58-global-analysis/index.html#stationary-harmonic-maps","scope":"lean-pool/part-02/58-global-analysis","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/stationary-harmonic-maps.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Monotonicity Formula for Stationary Harmonic Maps"},{"data_url":"scopes/lean-pool/part-02/60-probability-theory-and-stochastic-processes.json","id":"lean-pool/part-02/60-probability-theory-and-stochastic-processes","kind":"scope","parent":8561,"result":false,"route":"../roadmap/lean-pool/part-02/60-probability-theory-and-stochastic-processes/index.html","scope":"lean-pool/part-02/60-probability-theory-and-stochastic-processes","status":"fully_proved","summary":"The Lean Pool projects whose card's first MSC2020 code is in class 60, Probability theory and stochastic processes: 15 articles, one per card.","title":"Probability theory and stochastic processes"},{"id":"lean-pool/part-02/60-probability-theory-and-stochastic-processes/agree-to-disagree","kind":"node","lean":"AgreeToDisagree.agreeToDisagree, AgreeToDisagree.agreeToDisagree_beliefs","parent":8630,"result":true,"route":"../roadmap/lean-pool/part-02/60-probability-theory-and-stochastic-processes/index.html#agree-to-disagree","scope":"lean-pool/part-02/60-probability-theory-and-stochastic-processes","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/agree-to-disagree.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Aumann's Agreement Theorem"},{"id":"lean-pool/part-02/60-probability-theory-and-stochastic-processes/concentration-inequalities","kind":"node","lean":"Contrib.Freedman.subgamma_bernstein_tail","parent":8630,"result":true,"route":"../roadmap/lean-pool/part-02/60-probability-theory-and-stochastic-processes/index.html#concentration-inequalities","scope":"lean-pool/part-02/60-probability-theory-and-stochastic-processes","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/concentration-inequalities.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Bennett--Bernstein, Freedman, and Hoeffding concentration inequalities"},{"id":"lean-pool/part-02/60-probability-theory-and-stochastic-processes/burkholder","kind":"node","lean":"MeasureTheory.Lp_Burkholder_inequality_martingaleTransform","parent":8630,"result":true,"route":"../roadmap/lean-pool/part-02/60-probability-theory-and-stochastic-processes/index.html#burkholder","scope":"lean-pool/part-02/60-probability-theory-and-stochastic-processes","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/burkholder.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Burkholder Martingale Transform Inequality"},{"id":"lean-pool/part-02/60-probability-theory-and-stochastic-processes/cramer-wold","kind":"node","lean":"LeanPool.CramerWold.euclideanSpace_measure_eq_of_forall_inner_map_eq","parent":8630,"result":true,"route":"../roadmap/lean-pool/part-02/60-probability-theory-and-stochastic-processes/index.html#cramer-wold","scope":"lean-pool/part-02/60-probability-theory-and-stochastic-processes","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/cramer-wold.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"The Cramer-Wold theorem"},{"id":"lean-pool/part-02/60-probability-theory-and-stochastic-processes/feige","kind":"node","lean":"Feige.sharp_unit_slack_feige_complete","parent":8630,"result":true,"route":"../roadmap/lean-pool/part-02/60-probability-theory-and-stochastic-processes/index.html#feige","scope":"lean-pool/part-02/60-probability-theory-and-stochastic-processes","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/feige.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Feige's sharp unit-slack inequality"},{"id":"lean-pool/part-02/60-probability-theory-and-stochastic-processes/gaussian-moments-counterexamples","kind":"node","lean":"GaussianMomentsCounterexamples.not_GMC_of_three_le","parent":8630,"result":true,"route":"../roadmap/lean-pool/part-02/60-probability-theory-and-stochastic-processes/index.html#gaussian-moments-counterexamples","scope":"lean-pool/part-02/60-probability-theory-and-stochastic-processes","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/gaussian-moments-counterexamples.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Gaussian Moments Conjecture 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networks"},{"id":"lean-pool/part-02/60-probability-theory-and-stochastic-processes/lean-malliavin","kind":"node","lean":"PalomarClarkOcone.generated_clark_ocone","parent":8630,"result":true,"route":"../roadmap/lean-pool/part-02/60-probability-theory-and-stochastic-processes/index.html#lean-malliavin","scope":"lean-pool/part-02/60-probability-theory-and-stochastic-processes","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/lean-malliavin.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Malliavin calculus and the Clark\u2013Ocone 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PoissonDeclumping.maxLeafHeight","parent":8630,"result":true,"route":"../roadmap/lean-pool/part-02/60-probability-theory-and-stochastic-processes/index.html#rare-subtree-patterns","scope":"lean-pool/part-02/60-probability-theory-and-stochastic-processes","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/rare-subtree-patterns.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Rare subtree patterns in conditioned Bienaym\u00e9 trees"},{"id":"lean-pool/part-02/60-probability-theory-and-stochastic-processes/superdiffusion","kind":"node","lean":"Algsuperdiff.superdiffusivity_with_finite_moments","parent":8630,"result":true,"route":"../roadmap/lean-pool/part-02/60-probability-theory-and-stochastic-processes/index.html#superdiffusion","scope":"lean-pool/part-02/60-probability-theory-and-stochastic-processes","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/superdiffusion.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Superdiffusion and anomalous regularization in self-similar random flows"},{"id":"lean-pool/part-02/60-probability-theory-and-stochastic-processes/semicirclelaw","kind":"node","lean":"LeanPool.SemicircleLaw.integral_semicirclePDFReal_eq_one","parent":8630,"result":true,"route":"../roadmap/lean-pool/part-02/60-probability-theory-and-stochastic-processes/index.html#semicirclelaw","scope":"lean-pool/part-02/60-probability-theory-and-stochastic-processes","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/semicirclelaw.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Wigner Semicircle Distribution"},{"data_url":"scopes/lean-pool/part-02/62-statistics.json","id":"lean-pool/part-02/62-statistics","kind":"scope","parent":8561,"result":false,"route":"../roadmap/lean-pool/part-02/62-statistics/index.html","scope":"lean-pool/part-02/62-statistics","status":"fully_proved","summary":"The Lean Pool projects whose card's first MSC2020 code is in class 62, Statistics: 2 articles, one per card.","title":"Statistics"},{"id":"lean-pool/part-02/62-statistics/cencov-petz","kind":"node","lean":"LeanPool.CencovPetz.MonotoneMetricFamily.eq_smul_fisher_of_continuous","parent":8646,"result":true,"route":"../roadmap/lean-pool/part-02/62-statistics/index.html#cencov-petz","scope":"lean-pool/part-02/62-statistics","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/cencov-petz.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Finite \u010cencov-Petz Uniqueness"},{"id":"lean-pool/part-02/62-statistics/rl-theory-in-lean","kind":"node","lean":"StochasticMatrix.stationary_distribution_exists","parent":8646,"result":true,"route":"../roadmap/lean-pool/part-02/62-statistics/index.html#rl-theory-in-lean","scope":"lean-pool/part-02/62-statistics","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/rl-theory-in-lean.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"RL Theory in Lean"},{"data_url":"scopes/lean-pool/part-02/65-numerical-analysis.json","id":"lean-pool/part-02/65-numerical-analysis","kind":"scope","parent":8561,"result":false,"route":"../roadmap/lean-pool/part-02/65-numerical-analysis/index.html","scope":"lean-pool/part-02/65-numerical-analysis","status":"fully_proved","summary":"The Lean Pool projects whose card's first MSC2020 code is in class 65, Numerical analysis: 3 articles, one per card.","title":"Numerical analysis"},{"id":"lean-pool/part-02/65-numerical-analysis/runge-kutta-order-conditions","kind":"node","lean":"RungeKuttaOrderConditions.Butcher.gauss_order6_rat","parent":8649,"result":true,"route":"../roadmap/lean-pool/part-02/65-numerical-analysis/index.html#runge-kutta-order-conditions","scope":"lean-pool/part-02/65-numerical-analysis","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/runge-kutta-order-conditions.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Certified Runge-Kutta Order Conditions"},{"id":"lean-pool/part-02/65-numerical-analysis/flean","kind":"node","lean":"Flean.Float, toFloat, toRat, to_float_to_rat, roundf_close","parent":8649,"result":true,"route":"../roadmap/lean-pool/part-02/65-numerical-analysis/index.html#flean","scope":"lean-pool/part-02/65-numerical-analysis","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/flean.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Flean: Floating-Point Numbers in Lean"},{"id":"lean-pool/part-02/65-numerical-analysis/computablereal","kind":"node","lean":"Computable\u211dSeq, Computable\u211d, IsComputable","parent":8649,"result":true,"route":"../roadmap/lean-pool/part-02/65-numerical-analysis/index.html#computablereal","scope":"lean-pool/part-02/65-numerical-analysis","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/computablereal.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Verified interval-Cauchy real arithmetic"},{"data_url":"scopes/lean-pool/part-02/68-computer-science.json","id":"lean-pool/part-02/68-computer-science","kind":"scope","parent":8561,"result":false,"route":"../roadmap/lean-pool/part-02/68-computer-science/index.html","scope":"lean-pool/part-02/68-computer-science","status":"fully_proved","summary":"The Lean Pool projects whose card's first MSC2020 code is in class 68, Computer science: 21 articles, one per card.","title":"Computer science"},{"data_url":"scopes/lean-pool/part-02/68-computer-science/part-01.json","id":"lean-pool/part-02/68-computer-science/part-01","kind":"scope","navigation":"transparent","parent":8653,"result":false,"route":"../roadmap/lean-pool/part-02/68-computer-science/part-01/index.html","scope":"lean-pool/part-02/68-computer-science/part-01","status":"fully_proved","title":"2-Coloring Cycles in One Round to Sharp list-scheduling 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the card LeanPool/projects/aaronson-ambainis.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"The Aaronson\u2013Ambainis influence conjecture"},{"id":"lean-pool/part-02/68-computer-science/part-01/quantum-query-foundations","kind":"node","lean":"QuantumQueryComplexity.boundedErrorQQuery_characterized_by_advPM","parent":8654,"result":true,"route":"../roadmap/lean-pool/part-02/68-computer-science/part-01/index.html#quantum-query-foundations","scope":"lean-pool/part-02/68-computer-science/part-01","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/quantum-query-foundations.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Adversary bounds and the polynomial method for quantum 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LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Block sensitivity can exceed spectral sensitivity squared"},{"id":"lean-pool/part-02/68-computer-science/part-01/circuit-complexity","kind":"node","lean":"CircuitComplexity.shannon_lower_bound_circuit","parent":8654,"result":true,"route":"../roadmap/lean-pool/part-02/68-computer-science/part-01/index.html#circuit-complexity","scope":"lean-pool/part-02/68-computer-science/part-01","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/circuit-complexity.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Circuit Complexity in Lean 4"},{"id":"lean-pool/part-02/68-computer-science/part-01/lean4-itree","kind":"node","lean":"Lean4Itree.ITree, Lean4Itree.ITree.ieq_iff_eq, 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JL"},{"id":"lean-pool/part-02/68-computer-science/part-01/matrix-multiplication-all-fields","kind":"node","lean":"MatrixMultiplicationAllFields.omega_le_nine_quarters_all_fields","parent":8654,"result":true,"route":"../roadmap/lean-pool/part-02/68-computer-science/part-01/index.html#matrix-multiplication-all-fields","scope":"lean-pool/part-02/68-computer-science/part-01","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/matrix-multiplication-all-fields.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Matrix multiplication exponent at most 9/4 over every field"},{"id":"lean-pool/part-02/68-computer-science/part-01/mriscx","kind":"node","lean":"hoareTripleUp, S_SEQ, proof_otp","parent":8654,"result":true,"route":"../roadmap/lean-pool/part-02/68-computer-science/part-01/index.html#mriscx","scope":"lean-pool/part-02/68-computer-science/part-01","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/mriscx.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"MRiscX"},{"id":"lean-pool/part-02/68-computer-science/part-01/gap-cvp","kind":"node","lean":"GapCVP.Comparator.gapCVP400IsNPHard","parent":8654,"result":true,"route":"../roadmap/lean-pool/part-02/68-computer-science/part-01/index.html#gap-cvp","scope":"lean-pool/part-02/68-computer-science/part-01","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/gap-cvp.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Polynomial-factor hardness of the closest vector problem"},{"id":"lean-pool/part-02/68-computer-science/part-01/pumping-cfg","kind":"node","lean":"Language.IsContextFree.pumping, ChomskyNormalFormGrammar.pumping","parent":8654,"result":true,"route":"../roadmap/lean-pool/part-02/68-computer-science/part-01/index.html#pumping-cfg","scope":"lean-pool/part-02/68-computer-science/part-01","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/pumping-cfg.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Pumping Lemma for Context-Free Grammars"},{"id":"lean-pool/part-02/68-computer-science/part-01/pure-nock","kind":"node","lean":"Nock.Verb.runProgram_adequate_complete, Nock.NPR.collision_prob_le","parent":8654,"result":true,"route":"../roadmap/lean-pool/part-02/68-computer-science/part-01/index.html#pure-nock","scope":"lean-pool/part-02/68-computer-science/part-01","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/pure-nock.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Pure Nock semantics, trace certificates, and polynomial 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the card LeanPool/projects/single-copy-stabilizer-learning.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Exact adaptive single-copy stabilizer learning"},{"id":"lean-pool/part-02/81-quantum-theory/lean-quantumalg","kind":"node","lean":"QuantumAlg.GroverSearch.main_success_probability","parent":8679,"result":true,"route":"../roadmap/lean-pool/part-02/81-quantum-theory/index.html#lean-quantumalg","scope":"lean-pool/part-02/81-quantum-theory","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/lean-quantumalg.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Lean-QuantumAlg"},{"id":"lean-pool/part-02/81-quantum-theory/low-weight-pauli-dynamics","kind":"node","lean":"Lean4LPD.PauliString.pauliNorm_layerStep_error_le_model_of_source_regime","parent":8679,"result":true,"route":"../roadmap/lean-pool/part-02/81-quantum-theory/index.html#low-weight-pauli-dynamics","scope":"lean-pool/part-02/81-quantum-theory","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/low-weight-pauli-dynamics.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Low-weight Pauli dynamics and truncation error"},{"id":"lean-pool/part-02/81-quantum-theory/osforgff","kind":"node","lean":"OSforGFF.gaussianFreeField_satisfies_all_OS_axioms","parent":8679,"result":true,"route":"../roadmap/lean-pool/part-02/81-quantum-theory/index.html#osforgff","scope":"lean-pool/part-02/81-quantum-theory","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/osforgff.yaml of LeanPool/lean-pool at commit 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optimization"},{"id":"lean-pool/part-02/90-operations-research/hsd-interior-point-lp","kind":"node","lean":"HSDInteriorPointLP.YTM_fixed_local_theory_from_paper","parent":8688,"result":true,"route":"../roadmap/lean-pool/part-02/90-operations-research/index.html#hsd-interior-point-lp","scope":"lean-pool/part-02/90-operations-research","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/hsd-interior-point-lp.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Homogeneous self-dual interior-point method for linear programming"},{"id":"lean-pool/part-02/90-operations-research/parameter_free_gradient_p","kind":"node","lean":"V7.main","parent":8688,"result":true,"route":"../roadmap/lean-pool/part-02/90-operations-research/index.html#parameter_free_gradient_p","scope":"lean-pool/part-02/90-operations-research","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/parameter_free_gradient_p.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Parameter-Free Gradient Minimization in l_p Geometry"},{"id":"lean-pool/part-02/90-operations-research/persistent-sos","kind":"node","lean":"PersistentSOS.persistent_failure, PersistentSOS.unique_persistent_failure","parent":8688,"result":true,"route":"../roadmap/lean-pool/part-02/90-operations-research/index.html#persistent-sos","scope":"lean-pool/part-02/90-operations-research","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/persistent-sos.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Persistent failure of quartic sum-of-squares exactness"},{"data_url":"scopes/lean-pool/part-02/91-game-theory.json","id":"lean-pool/part-02/91-game-theory","kind":"scope","parent":8561,"result":false,"route":"../roadmap/lean-pool/part-02/91-game-theory/index.html","scope":"lean-pool/part-02/91-game-theory","status":"can_prove","summary":"The Lean Pool projects whose card's first MSC2020 code is in class 91, \u201cGame theory, economics, finance, and other social and behavioral sciences\u201d: 5 articles, one per card.","title":"Game theory, economics, finance, and other social and behavioral sciences"},{"id":"lean-pool/part-02/91-game-theory/apportionment","kind":"node","lean":"Apportionment.balinski_young","parent":8695,"result":true,"route":"../roadmap/lean-pool/part-02/91-game-theory/index.html#apportionment","scope":"lean-pool/part-02/91-game-theory","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/apportionment.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Apportionmentlib"},{"id":"lean-pool/part-02/91-game-theory/brouwer","kind":"node","lean":"ExistsNashEq, Brouwer, Brouwer_Product, IndexedLOrder.Scarf, FinGame","parent":8695,"result":true,"route":"../roadmap/lean-pool/part-02/91-game-theory/index.html#brouwer","scope":"lean-pool/part-02/91-game-theory","status":"can_prove","summary":"*Quoted from the card LeanPool/projects/brouwer.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Existence of Nash equilibria via Brouwer's fixed-point theorem"},{"id":"lean-pool/part-02/91-game-theory/misere-games","kind":"node","lean":"MisereGames.Form.Promain.of_universe","parent":8695,"result":true,"route":"../roadmap/lean-pool/part-02/91-game-theory/index.html#misere-games","scope":"lean-pool/part-02/91-game-theory","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/misere-games.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Misere combinatorial games"},{"id":"lean-pool/part-02/91-game-theory/aiyagari-equilibrium","kind":"node","lean":"LeanEconomics.aiyagariTauchen_equilibriumRate_unique_Ioo","parent":8695,"result":true,"route":"../roadmap/lean-pool/part-02/91-game-theory/index.html#aiyagari-equilibrium","scope":"lean-pool/part-02/91-game-theory","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/aiyagari-equilibrium.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Normalized capped log-household clearing and dynamic programming"},{"id":"lean-pool/part-02/91-game-theory/scott-measurement","kind":"node","lean":"Scott1964.MeasurementStructures.theorem_4_1_vector","parent":8695,"result":true,"route":"../roadmap/lean-pool/part-02/91-game-theory/index.html#scott-measurement","scope":"lean-pool/part-02/91-game-theory","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/scott-measurement.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Scott's measurement representation theorems"},{"data_url":"scopes/lean-pool/part-02/94-information-and-communication-theory.json","id":"lean-pool/part-02/94-information-and-communication-theory","kind":"scope","parent":8561,"result":false,"route":"../roadmap/lean-pool/part-02/94-information-and-communication-theory/index.html","scope":"lean-pool/part-02/94-information-and-communication-theory","status":"fully_proved","summary":"The Lean Pool projects whose card's first MSC2020 code is in class 94, \u201cInformation and communication theory, circuits\u201d: 5 articles, one per card.","title":"Information and communication theory, circuits"},{"id":"lean-pool/part-02/94-information-and-communication-theory/metric-codes","kind":"node","lean":"MetricCodes.Johnson.main_binary_theorem","parent":8701,"result":true,"route":"../roadmap/lean-pool/part-02/94-information-and-communication-theory/index.html#metric-codes","scope":"lean-pool/part-02/94-information-and-communication-theory","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/metric-codes.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Improved asymptotic bounds for binary and spherical codes"},{"id":"lean-pool/part-02/94-information-and-communication-theory/microwave-networks","kind":"node","lean":"MiLAC.HybridCouplersPhaseShifters.implementable_dft","parent":8701,"result":true,"route":"../roadmap/lean-pool/part-02/94-information-and-communication-theory/index.html#microwave-networks","scope":"lean-pool/part-02/94-information-and-communication-theory","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/microwave-networks.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Microwave network synthesis and power-of-two DFT implementation"},{"id":"lean-pool/part-02/94-information-and-communication-theory/shannon-1948-formalization","kind":"node","lean":"LeanPool.Shannon1948Formalization.entropyNat_unique","parent":8701,"result":true,"route":"../roadmap/lean-pool/part-02/94-information-and-communication-theory/index.html#shannon-1948-formalization","scope":"lean-pool/part-02/94-information-and-communication-theory","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/shannon-1948-formalization.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Shannon Entropy Characterization"},{"id":"lean-pool/part-02/94-information-and-communication-theory/channel-capacity","kind":"node","lean":"ChannelCapacity.exists_unique_capacity_achieving_prior_of_finite","parent":8701,"result":true,"route":"../roadmap/lean-pool/part-02/94-information-and-communication-theory/index.html#channel-capacity","scope":"lean-pool/part-02/94-information-and-communication-theory","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/channel-capacity.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"Uniqueness of Shannon Capacity-Achieving Priors"},{"id":"lean-pool/part-02/94-information-and-communication-theory/zhang-yeung-inequality","kind":"node","lean":"ZhangYeung.zhangYeung","parent":8701,"result":true,"route":"../roadmap/lean-pool/part-02/94-information-and-communication-theory/index.html#zhang-yeung-inequality","scope":"lean-pool/part-02/94-information-and-communication-theory","status":"fully_proved","summary":"*Quoted from the card LeanPool/projects/zhang-yeung-inequality.yaml of LeanPool/lean-pool at commit 2b205bf9, Apache-2.0.*","title":"The Zhang-Yeung non-Shannon information inequality"},{"data_url":"chapters/formal-frontier.json","id":"formal-frontier","kind":"scope","parent":0,"result":false,"route":"../roadmap/formal-frontier/index.html","scope":"formal-frontier","status":"can_prove","summary":"Formal Frontier publishes Lean libraries written by AI agents (\u201cFormal Frontier Agents\u201d, as their formalization.yaml files name the authors), with a dashboard at https://formalfrontier.github.io. This chapter maps the 44 libraries in trough's catalog, each read at a pinned commit: a part per subject and a section per library. A library's page quotes its formalization.yaml (its description, review and automation\u2026","title":"Formal Frontier"},{"data_url":"scopes/formal-frontier/commutative-algebra.json","id":"formal-frontier/commutative-algebra","kind":"scope","parent":8707,"result":false,"route":"../roadmap/formal-frontier/commutative-algebra/index.html","scope":"formal-frontier/commutative-algebra","status":"can_prove","summary":"The Formal Frontier libraries on commutative algebra, one section each.","title":"Commutative algebra"},{"data_url":"scopes/formal-frontier/commutative-algebra/adic-modules.json","id":"formal-frontier/commutative-algebra/adic-modules","kind":"scope","parent":8708,"result":false,"route":"../roadmap/formal-frontier/commutative-algebra/adic-modules/index.html","scope":"formal-frontier/commutative-algebra/adic-modules","status":"fully_proved","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/adic-modules at commit 86f9b35e, Apache-2.0.*","title":"Adic modules"},{"id":"formal-frontier/commutative-algebra/adic-modules/middle-exactness-of-completion-for-a-noetherian","kind":"node","lean":"AdicCompletion.map_exact_of_isNoetherian_middle","parent":8709,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/adic-modules/index.html#middle-exactness-of-completion-for-a-noetherian","scope":"formal-frontier/commutative-algebra/adic-modules","status":"fully_proved","summary":"For every ideal I of a commutative ring R and R-linear maps f : M \u2192 N and g : N \u2192 P, if the image of f is the kernel of g, g is surjective and N is Noetherian over R, then the induced maps on their inverse-limit I-adic completions are exact at the completion of N.","title":"Middle exactness of completion for a Noetherian middle module"},{"id":"formal-frontier/commutative-algebra/algebraic-direct-limits","kind":"node","parent":8708,"result":false,"route":"../roadmap/formal-frontier/commutative-algebra/algebraic-direct-limits/index.html","scope":"formal-frontier/commutative-algebra","status":"can_state","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/algebraic-direct-limits at commit e5d0dea1, Apache-2.0.*","title":"Algebraic direct limits"},{"data_url":"scopes/formal-frontier/commutative-algebra/coherent-modules.json","id":"formal-frontier/commutative-algebra/coherent-modules","kind":"scope","parent":8708,"result":false,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/index.html","scope":"formal-frontier/commutative-algebra/coherent-modules","status":"fully_proved","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/coherent-modules at commit b670b34e, Apache-2.0.*","title":"Coherent modules"},{"data_url":"scopes/formal-frontier/commutative-algebra/coherent-modules/part-01.json","id":"formal-frontier/commutative-algebra/coherent-modules/part-01","kind":"scope","navigation":"transparent","parent":8712,"result":false,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-01/index.html","scope":"formal-frontier/commutative-algebra/coherent-modules/part-01","status":"fully_proved","summary":"*Quoted from the headline results in README.md of FormalFrontier/coherent-modules at commit b670b34e, Apache-2.0.*","title":"Coherence via kernels of maps\u2026 to Inverse images with nilpotent\u2026"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-01/coherence-via-kernels-of-maps-from-finite-free-m","kind":"node","lean":"Module.IsCoherent.iff_finite_and_fg_ker","parent":8713,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-01/index.html#coherence-via-kernels-of-maps-from-finite-free-m","scope":"formal-frontier/commutative-algebra/coherent-modules/part-01","status":"fully_proved","summary":"For any unital, possibly noncommutative ring R and any left R-module M, M is coherent (finitely generated with every finitely generated submodule finitely presented) if and only if M is finitely generated and the kernel of every R-linear map R\u207f \u2192 M is finitely generated for every natural number n, including zero.","title":"Coherence via kernels of maps from finite free modules"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-01/finiteness-projectivity-and-flatness-under-injec","kind":"node","lean":"Module.Finite.iff_finite_tensorProduct_of_injective, Module.FinitePresentation.iff_finitePresentation_tensorProduct_of_injective, Module.finite_projective_of_tensorProduct_of_injective, Module.Flat.of_flat_tensorProduct_of_injective, Module.Flat.iff_flat_tensorProduct_of_injective","parent":8713,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-01/index.html#finiteness-projectivity-and-flatness-under-injec","scope":"formal-frontier/commutative-algebra/coherent-modules/part-01","status":"fully_proved","summary":"Let R be a commutative ring, S an R-algebra and M an additive commutative monoid with its given R-module structure. Assume the canonical unit N \u2192 S \u2297[R] N is injective for every additive commutative R-semimodule N in Type (max uR uM). For a possibly noncommutative semiring S, S \u2297[R] M is finitely generated over S if and only if M is finitely generated over R. For a possibly noncommutative ring S, the corresponding\u2026","title":"Finiteness, projectivity and flatness under injective tensor units"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-01/internal-krull-schmidt-decompositions-and-cancel","kind":"node","lean":"Submodule.exists_fin_isInternal_indecomposable_of_isNoetherian, Submodule.exists_fin_isInternal_indecomposable_of_isArtinian, DirectSum.IsInternal.exists_equiv_of_linearEquiv_of_isLocalRing_end, DirectSum.IsInternal.exists_equiv_of_linearEquiv_of_indecomposable_of_isFiniteLength, LinearEquiv.nonempty_of_prod_of_isFiniteLength","parent":8713,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-01/index.html#internal-krull-schmidt-decompositions-and-cancel","scope":"formal-frontier/commutative-algebra/coherent-modules/part-01","status":"fully_proved","summary":"Over any possibly noncommutative ring, a Noetherian module or an Artinian module is a finite internal direct sum of nonzero indecomposable submodules, with an explicit A i \u2260 \u22a5 fact paired with each summand's categorical indecomposability. Two supplied finite internal decompositions of linearly equivalent modules have matching summands up to linear equivalence when all summands have local endomorphism rings.\u2026","title":"Internal Krull\u2013Schmidt decompositions and cancellation"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-01/coherence-of-square-zero-extensions-and-finite-s","kind":"node","lean":"TrivSqZeroExt.isCoherentRing_iff_finite_of_divisionRing, TrivSqZeroExt.isCoherentRing_finsupp_iff_finite, TrivSqZeroExt.finsupp_faithfullyFlat_and_not_isCoherentRing, TrivSqZeroExt.finite_of_isCoherentRing_of_injective_smul, TrivSqZeroExt.isCoherentRing_iff_finite, TrivSqZeroExt.isCoherentRing","parent":8713,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-01/index.html#coherence-of-square-zero-extensions-and-finite-s","scope":"formal-frontier/commutative-algebra/coherent-modules/part-01","status":"fully_proved","summary":"For a division ring D and a bimodule M in independent universes, the trivial square-zero extension D \u22c9 M is left coherent if and only if M is finitely generated on the left. Both compatible module actions are retained, without centrality. For the stock pointwise left module \u03b9 \u2192\u2080 D and any compatible right action, this is equivalent to finiteness of \u03b9, including the empty index type. For a field k and infinite \u03b9,\u2026","title":"Coherence of square-zero extensions and finite-support idealizations"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-01/additive-intertwining-by-ranges","kind":"node","lean":"OrthogonalIdempotents.semiconj_iff_mapsTo_range","parent":8713,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-01/index.html#additive-intertwining-by-ranges","scope":"formal-frontier/commutative-algebra/coherent-modules/part-01","status":"fully_proved","summary":"An additive map between commutative monoids intertwines a finite family of source endomorphisms summing to the identity with an orthogonal idempotent target family if and only if it maps each source range into its corresponding target range. No source orthogonality, target completeness, subtraction or cancellation is required, and the monoid and index universes are independent. The arbitrary-index range-action\u2026","title":"Additive intertwining by ranges"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-01/fibrewise-additive-intertwining","kind":"node","lean":"OrthogonalIdempotents.semiconj_sum_fiberwise_of_mapsTo_range","parent":8713,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-01/index.html#fibrewise-additive-intertwining","scope":"formal-frontier/commutative-algebra/coherent-modules/part-01","status":"fully_proved","summary":"For additive commutative monoids `M`, `N`, let `P : I \u2192 AddMonoid.End M` have finite index set and sum to the identity, and let `Q : J \u2192 AddMonoid.End N` be an orthogonal idempotent family, with decidable equality on `J`. For any `q : I \u2192 J` and additive `h : M \u2192+ N`, sending `range (P i)` into `range (Q (q i))` for every `i` implies `h \u2218 (\u2211 i with q i = j, P i) = Q j \u2218 h` for every `j`. This implication uses no\u2026","title":"Fibrewise additive intertwining"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-01/finite-coarsening-of-complete-orthogonal-idempot","kind":"node","lean":"CompleteOrthogonalIdempotents.sum_fiberwise","parent":8713,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-01/index.html#finite-coarsening-of-complete-orthogonal-idempot","scope":"formal-frontier/commutative-algebra/coherent-modules/part-01","status":"fully_proved","summary":"Over any possibly noncommutative `Semiring A`, summing a finite complete orthogonal family over the fibres of an arbitrary map `q : I \u2192 J` gives a complete orthogonal family indexed by `J`. Both index types have `Fintype` instances, with decidable equality needed only on `J`; the three universes are independent. The map need not be surjective: unused labels have value zero. Empty index types and zero semirings are\u2026","title":"Finite coarsening of complete orthogonal idempotents"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-01/dual-endomorphisms","kind":"node","lean":"Module.End.dualMapAlgHom","parent":8713,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-01/index.html#dual-endomorphisms","scope":"formal-frontier/commutative-algebra/coherent-modules/part-01","status":"fully_proved","summary":"Over a commutative semiring `R`, `Module.End.dualMapAlgHom` bundles dualization as an algebra homomorphism from the opposite endomorphism algebra of any `R`-semimodule. For a possibly noncommutative `R`-algebra `A`, `Module.Dual.rightRegularAlgHom` gives named operators `F_a(\u03c6)(x) = \u03c6(a * x)`, with `F_a.comp F_b = F_(b * a)` and the original pointwise `R`-scalar structure on the dual. No finiteness or freeness is\u2026","title":"Dual endomorphisms"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-01/tensor-dual-frobenius-criterion","kind":"node","lean":"Module.Dual.exists_nondegenerate_mul_of_tensorProduct_equiv","parent":8713,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-01/index.html#tensor-dual-frobenius-criterion","scope":"formal-frontier/commutative-algebra/coherent-modules/part-01","status":"fully_proved","summary":"For a finite-dimensional, possibly noncommutative or zero algebra `A` over a field `K`, `Module.Dual.exists_nondegenerate_mul_of_tensorProduct_equiv` produces a functional `\u03bb` with both-sided nondegenerate pairing `\u03bb(a * x)` from a subspace `C` of the original dual and an equivalence `e : C \u2297[K] A \u2243\u2097[K] Module.Dual K A` satisfying `e (c \u2297\u209c a) x = c (a * x)`. The algebra needs only `Semiring A`; finiteness of `C` is\u2026","title":"Tensor\u2013dual Frobenius criterion"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-01/annihilator-quotients-with-two-scalar-actions","kind":"node","lean":"isNoetherianRing_quotient_annihilator_of_finite","parent":8713,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-01/index.html#annihilator-quotients-with-two-scalar-actions","scope":"formal-frontier/commutative-algebra/coherent-modules/part-01","status":"fully_proved","summary":"Suppose a ring `A` and a semiring `C` act on the same additive commutative monoid `M`, with commuting given actions. If `M` is finitely generated over `C` and Noetherian over `A`, then `A` modulo the annihilator of its original action is left Noetherian; see `isNoetherianRing_quotient_annihilator_of_finite`. If `M` is an additive commutative group, finite generation over `C` and coherence over `A` instead give a\u2026","title":"Annihilator quotients with two scalar actions"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-01/noetherian-semilinear-hom","kind":"node","lean":"isNoetherian_semilinearMap","parent":8713,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-01/index.html#noetherian-semilinear-hom","scope":"formal-frontier/commutative-algebra/coherent-modules/part-01","status":"fully_proved","summary":"For semirings `R`, `R\u2082`, a ring `S`, a finitely generated `R`-semimodule `M`, and a Noetherian `S`-module `N` with a commuting `R\u2082`-action, `isNoetherian_semilinearMap` makes `M \u2192\u209b\u2097[\u03c3] N` Noetherian for the canonical pointwise `S`-action, for every `\u03c3 : R \u2192+* R\u2082`. The five universes are independent; no surjectivity of `\u03c3`, scalar commutativity or ambient Noetherianity is required. `isNoetherian_linearMap_of_finite`\u2026","title":"Noetherian semilinear Hom"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-01/hom-and-end-of-matrix-columns","kind":"node","lean":"LinearMap.matrixModuleHomAddEquiv","parent":8713,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-01/index.html#hom-and-end-of-matrix-columns","scope":"formal-frontier/commutative-algebra/coherent-modules/part-01","status":"fully_proved","summary":"Over an arbitrary possibly noncommutative semiring `R`, finite-sum matrix actions retain the given coefficient-module structures. For a finite nonempty index `I`, `LinearMap.matrixModuleHomAddEquiv` identifies coefficient-linear maps additively with matrix-linear maps of columns; its inverse is characterized at every coordinate. `Module.matrixModuleEndRingEquiv` preserves the usual order of composition. For the\u2026","title":"Hom and End of matrix columns"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-01/projective-matrix-column-semimodules","kind":"node","lean":"Matrix.Module.projective_iff","parent":8713,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-01/index.html#projective-matrix-column-semimodules","scope":"formal-frontier/commutative-algebra/coherent-modules/part-01","status":"fully_proved","summary":"For a semiring `R`, an additive commutative monoid `M` with its given `R`-module structure, and a finite type `I` with decidable equality, `Matrix.Module.projective_iff` characterizes projectivity of the canonical `Matrix I I R`-module on `I \u2192 M` by `IsEmpty I \u2228 Module.Projective R M`. In particular, for inhabited `I`, projectivity is equivalent to that of `M` over `R`. No coefficient commutativity or additive\u2026","title":"Projective matrix-column semimodules"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-01/left-ideals-of-finite-matrix-semirings","kind":"node","lean":"Submodule.matrixRowsOrderIso","parent":8713,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-01/index.html#left-ideals-of-finite-matrix-semirings","scope":"formal-frontier/commutative-algebra/coherent-modules/part-01","status":"fully_proved","summary":"For `[Semiring R]`, `[Fintype I]` and `[DecidableEq I]`, including empty `I`, `Submodule.matrixRowsOrderIso` identifies left ideals of `Matrix I I R` with `R`-submodules of the row module `I \u2192 R`. Left Artinianity and left Noetherianity of the matrix semiring are equivalent to the corresponding conditions on this row module, not to unrestricted conditions on the coefficient semiring.","title":"Left ideals of finite matrix semirings"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-01/coherence-of-finite-matrix-rings","kind":"node","lean":"Matrix.instIsCoherentRing","parent":8713,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-01/index.html#coherence-of-finite-matrix-rings","scope":"formal-frontier/commutative-algebra/coherent-modules/part-01","status":"fully_proved","summary":"For an arbitrary, possibly noncommutative or zero `Ring R`, a finite type `I` with its given decidable equality, and the canonical regular left modules, `Matrix.instIsCoherentRing` gives coherence of `Matrix I I R` from coherence of `R`, including empty `I`. For nonempty `I`, `IsCoherentRing.of_matrix` reflects coherence and `Matrix.isCoherentRing_iff` gives the equivalence. The ring and index universes are\u2026","title":"Coherence of finite matrix rings"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-01/left-regular-elements-in-left-artinian-rings","kind":"node","lean":"IsLeftRegular.isUnit_of_isArtinianRing","parent":8713,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-01/index.html#left-regular-elements-in-left-artinian-rings","scope":"formal-frontier/commutative-algebra/coherent-modules/part-01","status":"fully_proved","summary":"For an arbitrary unital `Ring A` with `[IsArtinianRing A]`, `IsLeftRegular.isUnit_of_isArtinianRing` makes every left-regular element a unit. Artinianity concerns the original left regular module; no opposite-Artinian, commutativity or nontriviality hypothesis is required, and the carrier universe is arbitrary. The reverse implication is already available through Mathlib's `IsUnit.isRegular`.","title":"Left-regular elements in left Artinian rings"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-01/closed-semimodule-categories","kind":"node","lean":"SemimoduleCat.tensorLeftHomAdjunction","parent":8713,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-01/index.html#closed-semimodule-categories","scope":"formal-frontier/commutative-algebra/coherent-modules/part-01","status":"fully_proved","summary":"For a `CommSemiring R`, `SemimoduleCat.tensorLeftHomAdjunction` makes tensoring with `M` left adjoint to its wrapped-morphism Hom functor. The Hom functor allows an independent carrier universe; the monoidal closed structure uses the common scalar and carrier universe of the tensor category. Currying, uncurrying, evaluation and coevaluation have pure-tensor formulas. Over a commutative ring,\u2026","title":"Closed semimodule categories"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-01/kernels-over-semirings","kind":"node","lean":"SemimoduleCat.kernelIsLimit","parent":8713,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-01/index.html#kernels-over-semirings","scope":"formal-frontier/commutative-algebra/coherent-modules/part-01","status":"fully_proved","summary":"For an arbitrary `Semiring R`, `SemimoduleCat.kernelIsLimit` makes the existing kernel submodule, with its subtype inclusion, a categorical kernel. `SemimoduleCat.hasKernels_semimoduleCat` gives kernels in `SemimoduleCat.{v} R`, with independent coefficient and carrier universes. Over a ring, `ModuleCat.kernelCone_semimodule_hom_apply` shows that the comparison with Mathlib's module kernel preserves underlying\u2026","title":"Kernels over semirings"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-01/sums-of-nilpotent-left-ideals","kind":"node","lean":"Ideal.sup_pow_add_sub_one_le","parent":8713,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-01/index.html#sums-of-nilpotent-left-ideals","scope":"formal-frontier/commutative-algebra/coherent-modules/part-01","status":"fully_proved","summary":"For arbitrary left ideals `I`, `J` of a possibly noncommutative `Semiring R`, `Ideal.sup_pow_add_sub_one_le` gives `(I \u2294 J) ^ (m + n - 1) \u2264 I ^ m * \u22a4 \u2294 J ^ n * \u22a4` for all natural `m`, `n`. Global witnesses `I ^ m = 0` and `J ^ n = 0` therefore give `(I \u2294 J) ^ (m + n - 1) = 0`; `Ideal.isNilpotent_sup` makes the sum of two globally nilpotent left ideals nilpotent. No commutativity, two-sidedness or positivity of the\u2026","title":"Sums of nilpotent left ideals"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-01/inverse-images-with-nilpotent-kernel","kind":"node","lean":"Ideal.pow_comap_eq_zero_of_ker_pow_eq_zero","parent":8713,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-01/index.html#inverse-images-with-nilpotent-kernel","scope":"formal-frontier/commutative-algebra/coherent-modules/part-01","status":"fully_proved","summary":"For a unital semiring homomorphism `f` and a target left ideal `J`, `Ideal.pow_comap_eq_zero_of_ker_pow_eq_zero` gives `J.comap f ^ (m * n) = 0` from the global witnesses `J ^ m = 0` and `(RingHom.ker f) ^ n = 0`. The companion `Ideal.isNilpotent_comap` gives global nilpotence of the inverse image. Both semirings may be noncommutative, all natural exponents including zero are allowed, and the source, target and\u2026","title":"Inverse images with nilpotent kernel"},{"data_url":"scopes/formal-frontier/commutative-algebra/coherent-modules/part-02.json","id":"formal-frontier/commutative-algebra/coherent-modules/part-02","kind":"scope","navigation":"transparent","parent":8712,"result":false,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-02/index.html","scope":"formal-frontier/commutative-algebra/coherent-modules/part-02","status":"fully_proved","summary":"*Quoted from the headline results in README.md of FormalFrontier/coherent-modules at commit b670b34e, Apache-2.0.*","title":"Relative generation by\u2026 to Finite generation from the\u2026"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-02/relative-generation-by-nilpotent-ideals","kind":"node","lean":"Submodule.le_sup_pow_smul_of_le_sup_smul","parent":8734,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-02/index.html#relative-generation-by-nilpotent-ideals","scope":"formal-frontier/commutative-algebra/coherent-modules/part-02","status":"fully_proved","summary":"For a left ideal `I` of a possibly noncommutative semiring `R` and submodules `N`, `P` of an additive commutative `R`-semimodule, `Submodule.le_sup_pow_smul_of_le_sup_smul` propagates `P \u2264 N \u2294 I \u2022 P` to `P \u2264 N \u2294 (I ^ n) \u2022 P` for every natural `n`. Under global nilpotence `IsNilpotent I`, the companion `Submodule.le_of_le_sup_smul_of_isNilpotent` gives `P \u2264 N`. The carrier universes are independent, and no finite\u2026","title":"Relative generation by nilpotent ideals"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-02/extension-of-products-and-powers-of-left-ideals","kind":"node","lean":"Ideal.map_mul_of_isTwoSided","parent":8734,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-02/index.html#extension-of-products-and-powers-of-left-ideals","scope":"formal-frontier/commutative-algebra/coherent-modules/part-02","status":"fully_proved","summary":"For any unital homomorphism `f : A \u2192+* B` between semirings in independent universes, `Ideal.map_mul_of_isTwoSided` gives `(I * J).map f = I.map f * J.map f` whenever `I.map f` is two-sided. The companion `Ideal.map_pow_of_isTwoSided` gives `(I ^ n).map f = (I.map f) ^ n` for every natural exponent, including zero. Neither semiring must be commutative, and `f` need not be surjective.","title":"Extension of products and powers of left ideals"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-02/annihilators-after-localization","kind":"node","lean":"IsLocalizedModule.map_annihilator","parent":8734,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-02/index.html#annihilators-after-localization","scope":"formal-frontier/commutative-algebra/coherent-modules/part-02","status":"fully_proved","summary":"For commutative semirings `R`, `A`, a submonoid `T : Submonoid R`, an `R`-algebra `A` realizing localization at `T`, and a standard localized-module map `f : M \u2192\u2097[R] N`, `IsLocalizedModule.map_annihilator` gives","title":"Annihilators after localization"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-02/coatomicity-of-two-sided-ideals","kind":"node","lean":"TwoSidedIdeal.isCoatomic","parent":8734,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-02/index.html#coatomicity-of-two-sided-ideals","scope":"formal-frontier/commutative-algebra/coherent-modules/part-02","status":"fully_proved","summary":"For any unital, possibly nonassociative ring `R`, `TwoSidedIdeal.isCoatomic` proves `IsCoatomic (TwoSidedIdeal R)`: every proper two-sided ideal lies below a maximal proper two-sided ideal. No nontriviality, commutativity or finiteness assumption is needed. Use this theorem as a local instance with Mathlib's coatom consumers. The simple-quotient example combines it with the congruence correspondence to obtain a\u2026","title":"Coatomicity of two-sided ideals"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-02/radical-colons-and-pointwise-annihilators","kind":"node","lean":"Ideal.IsRadical.colon","parent":8734,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-02/index.html#radical-colons-and-pointwise-annihilators","scope":"formal-frontier/commutative-algebra/coherent-modules/part-02","status":"fully_proved","summary":"For a radical ideal `I` in a commutative semiring `R` and an arbitrary subset `s : Set R`, `Ideal.IsRadical.colon` proves `(I.colon s).IsRadical`; use it as `hI.colon s`. Taking `I = \u22a5` in a reduced commutative semiring shows that the ideal of scalars `a` with `a * b = 0` for every `b \u2208 s` is radical. No finiteness, nonemptiness or nontriviality assumption is needed.","title":"Radical colons and pointwise annihilators"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-02/finitely-generated-ideal-colons-and-localization","kind":"node","lean":"IsLocalization.comap_map_colon","parent":8734,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-02/index.html#finitely-generated-ideal-colons-and-localization","scope":"formal-frontier/commutative-algebra/coherent-modules/part-02","status":"fully_proved","summary":"For commutative semirings `R`, `S`, a submonoid `T : Submonoid R` and an `R`-algebra `S` satisfying `[IsLocalization T S]`, `IsLocalization.comap_map_colon` gives","title":"Finitely generated ideal colons and localization"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-02/unit-ideal-scalar-families","kind":"node","lean":"Ideal.existsUnique_smul_eq_of_span_range_eq_top","parent":8734,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-02/index.html#unit-ideal-scalar-families","scope":"formal-frontier/commutative-algebra/coherent-modules/part-02","status":"fully_proved","summary":"For an arbitrary family of scalars generating the unit ideal in a semiring `S`, `Ideal.existsUnique_smul_eq_of_span_range_eq_top` gives a unique common parameter for compatible elements of an `S`-semimodule whose scalar actions commute. The index type need not be finite or nonempty, and no subtraction or additive cancellation is needed.","title":"Unit-ideal scalar families"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-02/coprime-scalar-rows","kind":"node","lean":"IsCoprime.existsUnique_smul_eq_pair","parent":8734,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-02/index.html#coprime-scalar-rows","scope":"formal-frontier/commutative-algebra/coherent-modules/part-02","status":"fully_proved","summary":"For a commutative semiring `S`, an additive commutative `S`-semimodule `M`, and B\u00e9zout-coprime scalars `a`, `b`, `IsCoprime.existsUnique_smul_eq_pair` gives a unique `k` with `a \u2022 k = x` and `b \u2022 k = y` whenever `b \u2022 x = a \u2022 y`. The companion `smul_mem_range_smul_iff` reflects membership in the range of multiplication by `a` after multiplying by `b`. The two universes are independent; no subtraction, additive\u2026","title":"Coprime scalar rows"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-02/finite-coprime-kernel-decompositions","kind":"node","lean":"RingHom.linearEquivPiKer","parent":8734,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-02/index.html#finite-coprime-kernel-decompositions","scope":"formal-frontier/commutative-algebra/coherent-modules/part-02","status":"fully_proved","summary":"Let a commutative semiring `A` act through a representation `\u03c4 : A \u2192+* Module.End R M`, where `R` is any semiring and `M` is an additive commutative `R`-semimodule. For a finite labelled pairwise B\u00e9zout-coprime family `a`, the sole annihilation premise `\u03c4 (\u220f i, a i) = 0` gives `RingHom.linearEquivPiKer`, an explicit equivalence with the product of the labelled kernels. Its inverse sums the kernel inclusions, and\u2026","title":"Finite coprime-kernel decompositions"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-02/finite-generation-under-pointwise-unimodular-cle","kind":"node","lean":"Submodule.fg_of_fg_comap_of_one_mem_span_smul_mem_range","parent":8734,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-02/index.html#finite-generation-under-pointwise-unimodular-cle","scope":"formal-frontier/commutative-algebra/coherent-modules/part-02","status":"fully_proved","summary":"For arbitrary, possibly noncommutative semirings `R`, `S` and `f : M \u2192\u209b\u2097[\u03c6] N` between additive commutative left semimodules, `Submodule.fg_of_fg_comap_of_one_mem_span_smul_mem_range` proves an `S`-submodule `P` finitely generated if `P.comap f` is finitely generated over `R` and, for every `n \u2208 P`, the clearers `{a : S | \u2203 m : M, f m = a \u2022 n}` generate the unit left ideal. The companion\u2026","title":"Finite generation under pointwise unimodular clearing"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-02/semilinear-noetherian-ascent","kind":"node","lean":"isNoetherian_of_left_inv_smul_mem_range","parent":8734,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-02/index.html#semilinear-noetherian-ascent","scope":"formal-frontier/commutative-algebra/coherent-modules/part-02","status":"fully_proved","summary":"For arbitrary semirings `R`, `S`, additive commutative semimodules `M`, `N` and `f : M \u2192\u209b\u2097[\u03c6] N`, `isNoetherian_of_left_inv_smul_mem_range` transfers `IsNoetherian R M` to `IsNoetherian S N` if each `n` admits `m, a, b` with `b * a = 1` and `f m = a \u2022 n`. The four universes are independent; neither `\u03c6` nor `f` must be surjective. The `isNoetherian_of_unit_smul_mem_range` corollary retains the familiar unit-clearing\u2026","title":"Semilinear Noetherian ascent"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-02/artin-rees-for-noetherian-modules","kind":"node","lean":"Ideal.exists_pow_inf_eq_pow_smul_of_isNoetherian","parent":8734,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-02/index.html#artin-rees-for-noetherian-modules","scope":"formal-frontier/commutative-algebra/coherent-modules/part-02","status":"fully_proved","summary":"For an arbitrary commutative ring `R`, ideal `I` and submodule `K` of a Noetherian `R`-module `M`, `Ideal.exists_pow_inf_eq_pow_smul_of_isNoetherian` gives a constant `c` such that `(I ^ n \u2022 \u22a4) \u2293 K = I ^ (n - c) \u2022 ((I ^ c \u2022 \u22a4) \u2293 K)` for every `n \u2265 c`. Neither Noetherianity of `R` nor finite generation of `I` is assumed. The ring and module universes are independent. The proof uses Mathlib's full Artin\u2013Rees theorem\u2026","title":"Artin\u2013Rees for Noetherian modules"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-02/submodule-adic-cofinality","kind":"node","lean":"Ideal.HasSubmoduleAdicCofinality","parent":8734,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-02/index.html#submodule-adic-cofinality","scope":"formal-frontier/commutative-algebra/coherent-modules/part-02","status":"fully_proved","summary":"For an arbitrary ideal `I` of a commutative ring `R`, `Ideal.HasSubmoduleAdicCofinality` expresses `\u2200 n, \u2203 m, N \u2293 (I ^ m \u2022 \u22a4) \u2264 I ^ n \u2022 N` for every submodule `N` of every finite `R`-module. This is Fujiwara--Kato's (AP); restricting to finitely generated submodules gives the existing `Ideal.HasFiniteSubmoduleAdicCofinality` condition (APf). Both `.cofinal` eliminators allow an independent module universe.\u2026","title":"Submodule adic cofinality"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-02/power-killed-submodule-criteria","kind":"node","lean":"Ideal.hasSubmoduleAdicCofinality_iff_forall_exists_inf_pow_smul_eq_bot","parent":8734,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-02/index.html#power-killed-submodule-criteria","scope":"formal-frontier/commutative-algebra/coherent-modules/part-02","status":"fully_proved","summary":"For an ideal `I` of a commutative ring `R`, `Ideal.hasSubmoduleAdicCofinality_iff_forall_exists_inf_pow_smul_eq_bot` characterizes (AP) by `\u2203 m, N \u2293 (I ^ m \u2022 \u22a4) = \u22a5` whenever `\u2203 k, I ^ k \u2022 N = \u22a5`, for every submodule `N` of every finite `R`-module. The corresponding (APf) criterion tests only finitely generated `N`. Both `.exists_inf_pow_smul_eq_bot` consumers allow an independent module universe. A single common\u2026","title":"Power-killed submodule criteria"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-02/descent-of-adic-cofinality","kind":"node","lean":"Ideal.HasSubmoduleAdicCofinality.of_map_of_faithfullyFlat","parent":8734,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-02/index.html#descent-of-adic-cofinality","scope":"formal-frontier/commutative-algebra/coherent-modules/part-02","status":"fully_proved","summary":"For commutative rings `R`, `A`, `Ideal.HasSubmoduleAdicCofinality.of_map_of_faithfullyFlat` descends (AP) from `I.map (algebraMap R A)` to `I` along a faithfully flat algebra. Alternatively, `Ideal.HasSubmoduleAdicCofinality.of_map_of_linearRetraction` uses an `R`-linear map `p : A \u2192\u2097[R] R` with `p 1 = 1`, without a flatness hypothesis; `p` need not preserve multiplication. Both have corresponding\u2026","title":"Descent of adic cofinality"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-02/adic-cofinality-on-finite-products","kind":"node","lean":"Ideal.hasSubmoduleAdicCofinality_iff_forall_map_evalRingHom","parent":8734,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-02/index.html#adic-cofinality-on-finite-products","scope":"formal-frontier/commutative-algebra/coherent-modules/part-02","status":"fully_proved","summary":"For an arbitrary ideal `I` of `\u03a0 i, R i`, with `[Finite \u03b9]` and commutative rings `R i`, `Ideal.hasSubmoduleAdicCofinality_iff_forall_map_evalRingHom` gives (AP) for `I` iff each projection image `I.map (Pi.evalRingHom R i)` has (AP). The companion `hasFiniteSubmoduleAdicCofinality_iff_forall_map_evalRingHom` does the same for (APf). The characteristic criteria also cover unary `Ideal.pi` and binary `Ideal.prod`,\u2026","title":"Adic cofinality on finite products"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-02/cofinal-ideal-powers","kind":"node","lean":"Ideal.hasSubmoduleAdicCofinality_iff_of_pow_le","parent":8734,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-02/index.html#cofinal-ideal-powers","scope":"formal-frontier/commutative-algebra/coherent-modules/part-02","status":"fully_proved","summary":"For ideals `I, J` of a commutative ring with `I ^ a \u2264 J` and `J ^ b \u2264 I` for some natural numbers `a, b`, `Ideal.hasSubmoduleAdicCofinality_iff_of_pow_le` makes (AP) equivalent for `I` and `J`; the companion `hasFiniteSubmoduleAdicCofinality_iff_of_pow_le` does the same for (APf). The witnesses may be zero. In particular, replacing an ideal by any positive power preserves both conditions. The shared pointwise\u2026","title":"Cofinal ideal powers"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-02/semilinear-transport-of-adic-cofinality","kind":"node","lean":"Submodule.adic_cofinality_iff_of_semilinearEquiv","parent":8734,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-02/index.html#semilinear-transport-of-adic-cofinality","scope":"formal-frontier/commutative-algebra/coherent-modules/part-02","status":"fully_proved","summary":"For semirings `R`, `S`, a coefficient equivalence `\u03c3 : R \u2243+* S` and a compatible semilinear equivalence `e : M \u2243\u209b\u2097[(\u03c3 : R \u2192+* S)] P` between additive commutative semimodules with their given actions, `Submodule.adic_cofinality_iff_of_semilinearEquiv` transports `\u2200 n, \u2203 m, N \u2293 (I ^ m \u2022 \u22a4) \u2264 I ^ n \u2022 N` to the condition for `I.map \u03c3.toRingHom` and `N.map e.toLinearMap`. It allows arbitrary left ideals and submodules\u2026","title":"Semilinear transport of adic cofinality"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-02/krull-intersection-from-adic-cofinality","kind":"node","lean":"Ideal.HasFiniteSubmoduleAdicCofinality.mem_iInf_pow_smul_top_iff","parent":8734,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-02/index.html#krull-intersection-from-adic-cofinality","scope":"formal-frontier/commutative-algebra/coherent-modules/part-02","status":"fully_proved","summary":"For a finite module `M` and an ideal `I` satisfying finite-submodule adic cofinality, `Ideal.HasFiniteSubmoduleAdicCofinality.mem_iInf_pow_smul_top_iff` characterizes membership in `\u2a05 n, I ^ n \u2022 \u22a4` by a scalar `a \u2208 I` fixing the element. The relative bound `I \u2264 (Module.annihilator R M).jacobson` makes this intersection zero. For any submodule `N` with finite quotient, membership in `\u2a05 n, N \u2294 I ^ n \u2022 \u22a4` is\u2026","title":"Krull intersection from adic cofinality"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-02/finite-generation-from-the-first-adic-quotient","kind":"node","lean":"Submodule.fg_iff_finite_quotient_smul","parent":8734,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-02/index.html#finite-generation-from-the-first-adic-quotient","scope":"formal-frontier/commutative-algebra/coherent-modules/part-02","status":"fully_proved","summary":"For an arbitrary submodule `N` of a finite original `R`-module `M`, `Submodule.fg_iff_finite_quotient_smul` detects `N.FG` on the intrinsic quotient `N / I N` over `R`, assuming APf and `I \u2264 (Module.annihilator R M).jacobson`. This is not the quotient by `N \u2229 I M`, and neither ideal-FG nor a Noetherian, finite-presentation, flatness or completeness hypothesis is required.","title":"Finite generation from the first adic quotient"},{"data_url":"scopes/formal-frontier/commutative-algebra/coherent-modules/part-03.json","id":"formal-frontier/commutative-algebra/coherent-modules/part-03","kind":"scope","navigation":"transparent","parent":8712,"result":false,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-03/index.html","scope":"formal-frontier/commutative-algebra/coherent-modules/part-03","status":"fully_proved","summary":"*Quoted from the headline results in README.md of FormalFrontier/coherent-modules at commit b670b34e, Apache-2.0.*","title":"Finite presentation from\u2026 to Faithfully flat tensor products"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-03/finite-presentation-from-cofinal-ideal-quotients","kind":"node","lean":"Ideal.finitePresentation_iff_forall_quotient_smul","parent":8755,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-03/index.html#finite-presentation-from-cofinal-ideal-quotients","scope":"formal-frontier/commutative-algebra/coherent-modules/part-03","status":"fully_proved","summary":"For a finite module `M` over a commutative ring `R` and an ideal `I` satisfying full submodule adic cofinality and `I \u2264 Ideal.jacobson \u22a5`, `Ideal.finitePresentation_iff_forall_quotient_smul` detects finite presentation over `R` on all reductions `M / J j M` over `R / J j`, for any family `J : \u03b9 \u2192 Ideal R` with `\u2200 n : \u2115, \u2203 j, J j \u2264 I ^ n`. The positive-powers criterion\u2026","title":"Finite presentation from cofinal ideal quotients"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-03/essentially-finite-type-scalar-extension","kind":"node","lean":"Algebra.EssFiniteType.isNoetherian_tensorProduct","parent":8755,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-03/index.html#essentially-finite-type-scalar-extension","scope":"formal-frontier/commutative-algebra/coherent-modules/part-03","status":"fully_proved","summary":"For commutative rings `R`, `S`, an essentially finite-type `R`-algebra structure on `S`, and a Noetherian `R`-module `M`, `Algebra.EssFiniteType.isNoetherian_tensorProduct` makes `S \u2297[R] M` Noetherian over `S` with its canonical left-factor action. The three universes are independent, and `M` need only be an additive commutative monoid with its given module structure. Neither ambient ring must be Noetherian; no\u2026","title":"Essentially finite-type scalar extension"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-03/coherent-modules-and-kernels","kind":"node","lean":"Module.IsCoherent","parent":8755,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-03/index.html#coherent-modules-and-kernels","scope":"formal-frontier/commutative-algebra/coherent-modules/part-03","status":"fully_proved","summary":"For a ring and an additive commutative monoid with its given module structure, `Module.IsCoherent` means finite generation together with finite presentation of every finitely generated submodule. `Module.IsCoherent.iff_finite_and_fg_ker` characterizes this by finitely generated kernels of *all* linear maps from finite-rank free modules, including rank zero. The same leaf proves closure under finitely generated\u2026","title":"Coherent modules and kernels"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-03/coherent-homology","kind":"node","lean":"CategoryTheory.ShortComplex.isCoherent_homology","parent":8755,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-03/index.html#coherent-homology","scope":"formal-frontier/commutative-algebra/coherent-modules/part-03","status":"fully_proved","summary":"Over any possibly noncommutative `Ring R`, a three-term complex `S : CategoryTheory.ShortComplex (ModuleCat.{v} R)` with finite left term `S.X\u2081` and coherent middle and right terms `S.X\u2082`, `S.X\u2083` has coherent homology, by `CategoryTheory.ShortComplex.isCoherent_homology`. The conclusion concerns the chosen categorical `S.homology` with its given `R`-action. Neither the left term nor `R` is assumed coherent, and no\u2026","title":"Coherent homology"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-03/coherence-along-semilinear-retractions","kind":"node","lean":"Module.IsCoherent.of_semilinear_retract","parent":8755,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-03/index.html#coherence-along-semilinear-retractions","scope":"formal-frontier/commutative-algebra/coherent-modules/part-03","status":"fully_proved","summary":"For arbitrary rings `R`, `S` and additive commutative monoids `N`, `M` with their given `R`- and `S`-module structures, let `\u03c3 : R \u2192+* S`, `\u03c4 : S \u2192+* R`, `f : N \u2192\u209b\u2097[\u03c3] M` and `g : M \u2192\u209b\u2097[\u03c4] N` satisfy `\u03c4 \u2218 \u03c3 = id` and `g \u2218 f = id`. Then `Module.IsCoherent.of_semilinear_retract` descends coherence of `M` over `S` to coherence of `N` over `R`. The four carrier universes are independent; no additional scalar action,\u2026","title":"Coherence along semilinear retractions"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-03/coherent-dependent-direct-sums","kind":"node","lean":"Module.IsCoherent.directSum_iff","parent":8755,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-03/index.html#coherent-dependent-direct-sums","scope":"formal-frontier/commutative-algebra/coherent-modules/part-03","status":"fully_proved","summary":"For an arbitrary index type, a possibly noncommutative ring `R`, and additive commutative monoids `M i` with their given `R`-module structures, `Module.IsCoherent.directSum_iff` gives","title":"Coherent dependent direct sums"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-03/coherence-over-product-coefficient-rings","kind":"node","lean":"Module.IsCoherent.of_piRing","parent":8755,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-03/index.html#coherence-over-product-coefficient-rings","scope":"formal-frontier/commutative-algebra/coherent-modules/part-03","status":"fully_proved","summary":"For any dependent family of possibly noncommutative rings `R i` and additive commutative monoids `M i` with their given `R i`-module structures, `Module.IsCoherent.of_piRing` reflects coherence of the product to each coordinate. For `[Finite \u03b9]`, `Module.IsCoherent.piRing_iff` gives the converse as well:","title":"Coherence over product coefficient rings"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-03/finite-presentation-over-product-coefficient-rin","kind":"node","lean":"Module.FinitePresentation.of_piRing","parent":8755,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-03/index.html#finite-presentation-over-product-coefficient-rin","scope":"formal-frontier/commutative-algebra/coherent-modules/part-03","status":"fully_proved","summary":"For any dependent family of possibly noncommutative rings `R i` and additive commutative monoids `M i` with their given `R i`-module structures, `Module.FinitePresentation.of_piRing` reflects finite presentation of `\u03a0 i, M i` over `\u03a0 i, R i` to each coordinate. This direction allows an arbitrary index type. For `[Finite \u03b9]`, `Module.FinitePresentation.piRing_iff` gives the converse as well:","title":"Finite presentation over product coefficient rings"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-03/finite-presentation-along-scalar-towers","kind":"node","lean":"Module.FinitePresentation.trans_of_tower","parent":8755,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-03/index.html#finite-presentation-along-scalar-towers","scope":"formal-frontier/commutative-algebra/coherent-modules/part-03","status":"fully_proved","summary":"For possibly noncommutative rings `R`, `S` and an additive commutative monoid `N` with its given `R`- and `S`-module structures and towers `R \u2192 S \u2192 S` and `R \u2192 S \u2192 N`, `Module.FinitePresentation.trans_of_tower` gives finite presentation of `N` over `R` from finite presentation of `S` over `R` and of `N` over `S`. Conversely, `of_restrictScalars` gives finite presentation over `S` from finite presentation over `R`\u2026","title":"Finite presentation along scalar towers"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-03/coherence-along-quotient-coefficients","kind":"node","lean":"Module.isCoherent_iff_of_bijective_of_fg_ker","parent":8755,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-03/index.html#coherence-along-quotient-coefficients","scope":"formal-frontier/commutative-algebra/coherent-modules/part-03","status":"fully_proved","summary":"For possibly noncommutative rings `R`, `S` and additive commutative monoids `N`, `M` with their given `R`- and `S`-module structures, `Module.isCoherent_iff_of_bijective_of_fg_ker` gives `Module.IsCoherent R N \u2194 Module.IsCoherent S M` along a bijective semilinear map `f : N \u2192\u209b\u2097[\u03c3] M`, provided `\u03c3 : R \u2192+* S` is surjective and `(RingHom.ker \u03c3).FG`. Kernel finite generation is as a left ideal, not merely as a\u2026","title":"Coherence along quotient coefficients"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-03/coherence-under-scalar-restriction","kind":"node","lean":"Module.FinitePresentation.isCoherent_restrictScalars","parent":8755,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-03/index.html#coherence-under-scalar-restriction","scope":"formal-frontier/commutative-algebra/coherent-modules/part-03","status":"fully_proved","summary":"For possibly noncommutative rings `R` and `S`, if `S` is coherent as an `R`-module, then `Module.FinitePresentation.isCoherent_restrictScalars` makes every finitely presented `S`-module `N` coherent for its given `R`-action. The additive commutative group `N` has its given module structures and scalar towers `R \u2192 S \u2192 S` and `R \u2192 S \u2192 N`; the three universes are independent. Under these hypotheses, coherence over\u2026","title":"Coherence under scalar restriction"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-03/hom-tensor-and-categories","kind":"node","lean":"Module.IsCoherent.semilinearMap","parent":8755,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-03/index.html#hom-tensor-and-categories","scope":"formal-frontier/commutative-algebra/coherent-modules/part-03","status":"fully_proved","summary":"For semirings `R`, `R\u2082`, a ring `S`, and any `\u03c3 : R \u2192+* R\u2082`, `Module.IsCoherent.semilinearMap` makes `M \u2192\u209b\u2097[\u03c3] N` coherent over `S` when the additive commutative group `M` is finitely presented over `R` and the additive commutative group `N` is coherent over `S`, with commuting given `R\u2082`- and `S`-actions on `N`. The five universes are independent and the Hom action is pointwise. No `S`-action on `M`, `R`-action on\u2026","title":"Hom, tensor, and categories"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-03/coherent-right-duals","kind":"node","lean":"Module.FinitePresentation.isCoherent_dual","parent":8755,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-03/index.html#coherent-right-duals","scope":"formal-frontier/commutative-algebra/coherent-modules/part-03","status":"fully_proved","summary":"For a possibly noncommutative `Ring R` and a finitely presented left `R`-module `M` with its given `AddCommMonoid` and module structure, `Module.FinitePresentation.isCoherent_dual` gives `Module.IsCoherent R\u1d50\u1d52\u1d56 (Module.Dual R M)` assuming only right coherence `[IsCoherentRing R\u1d50\u1d52\u1d56]` of the ring. The dual uses the canonical right action `(q \u2022 f) x = f x * MulOpposite.unop q` for `q : R\u1d50\u1d52\u1d56`, induced by right\u2026","title":"Coherent right duals"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-03/finite-module-tensor-zero-detection","kind":"node","lean":"Module.forall_maximal_smul_top_ne_top_iff_forall_finite_tensorProduct","parent":8755,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-03/index.html#finite-module-tensor-zero-detection","scope":"formal-frontier/commutative-algebra/coherent-modules/part-03","status":"fully_proved","summary":"For an arbitrary additive commutative `R`-module `P` over a `CommRing R`, `Module.forall_maximal_smul_top_ne_top_iff_forall_finite_tensorProduct` characterizes zero-object detection on all finite modules by `m \u2022 (\u22a4 : Submodule R P) \u2260 \u22a4` for every maximal ideal `m`. `Module.subsingleton_tensorProduct_iff_of_finite` gives `Subsingleton (P \u2297[R] M) \u2194 Subsingleton M` for a single finite module in an independent\u2026","title":"Finite-module tensor zero detection"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-03/ideal-action-on-finite-coefficients","kind":"node","lean":"Submodule.FG.ideal_smul_eq_self_iff_sup_annihilator_eq_top","parent":8755,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-03/index.html#ideal-action-on-finite-coefficients","scope":"formal-frontier/commutative-algebra/coherent-modules/part-03","status":"fully_proved","summary":"For an FG submodule `N` of an arbitrary additive commutative module over a `CommRing R`, `Submodule.FG.ideal_smul_eq_self_iff_sup_annihilator_eq_top` says `I \u2022 N = N \u2194 I \u2294 N.annihilator = \u22a4` for every ideal `I`. For finite `P`, `Module.forall_maximal_smul_top_ne_top_iff_annihilator_le_jacobson` characterizes proper image under every maximal ideal by `Module.annihilator R P \u2264 Ideal.jacobson \u22a5`, with no flatness or\u2026","title":"Ideal action on finite coefficients"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-03/semilinear-invariance-of-flatness","kind":"node","lean":"Module.Flat.semilinearEquiv_iff","parent":8755,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-03/index.html#semilinear-invariance-of-flatness","scope":"formal-frontier/commutative-algebra/coherent-modules/part-03","status":"fully_proved","summary":"For equivalent commutative semirings `e : R \u2243+* S` and a compatible semilinear equivalence from an additive commutative `R`-module `M` to an additive commutative `S`-module `N`, `Module.Flat.semilinearEquiv_iff` gives `Module.Flat R M \u2194 Module.Flat S N` for both given scalar actions. `Module.Flat.of_semilinearEquiv e E` takes flatness of `N` to flatness of `M`. The four carrier universes are independent. Over\u2026","title":"Semilinear invariance of flatness"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-03/annihilators-under-finite-limit-preserving-scala","kind":"node","lean":"Module.annihilator_tensorProduct_eq_map_of_finite_of_preservesFiniteLimits","parent":8755,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-03/index.html#annihilators-under-finite-limit-preserving-scala","scope":"formal-frontier/commutative-algebra/coherent-modules/part-03","status":"fully_proved","summary":"For a `CommSemiring R`, a possibly noncommutative `Semiring S` with `Algebra R S`, and a finitely generated semimodule `M`, `Module.annihilator_tensorProduct_eq_map_of_finite_of_preservesFiniteLimits` gives","title":"Annihilators under finite-limit-preserving scalar extension"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-03/annihilator-contraction-under-faithfully-flat-ex","kind":"node","lean":"Module.FaithfullyFlat.annihilator_tensorProduct_eq","parent":8755,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-03/index.html#annihilator-contraction-under-faithfully-flat-ex","scope":"formal-frontier/commutative-algebra/coherent-modules/part-03","status":"fully_proved","summary":"Over a `CommRing R`, `Module.FaithfullyFlat.annihilator_tensorProduct_eq` says tensoring an arbitrary `R`-module `M` with a faithfully flat module `P` preserves its annihilator over `R`. Here `P` is only a module group, while `M` needs only an additive commutative monoid structure. For a faithfully flat, possibly noncommutative ring algebra `S`, `Module.comap_annihilator_tensorProduct_of_faithfullyFlat` identifies\u2026","title":"Annihilator contraction under faithfully flat extension"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-03/faithfully-flat-tensor-products","kind":"node","lean":"Module.FaithfullyFlat.tensorProduct_iff","parent":8755,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-03/index.html#faithfully-flat-tensor-products","scope":"formal-frontier/commutative-algebra/coherent-modules/part-03","status":"fully_proved","summary":"Over a `CommRing R`, for additive commutative groups `C` and `D` with `R`-module structures, `Module.FaithfullyFlat.tensorProduct_iff` says that `C \u2297[R] D` is faithfully flat over `R` if and only if both `C` and `D` are faithfully flat over `R`. The ring and the two module universes are independent; no finiteness, projectivity or nontriviality assumption is needed.","title":"Faithfully flat tensor products"},{"data_url":"scopes/formal-frontier/commutative-algebra/coherent-modules/part-04.json","id":"formal-frontier/commutative-algebra/coherent-modules/part-04","kind":"scope","navigation":"transparent","parent":8712,"result":false,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-04/index.html","scope":"formal-frontier/commutative-algebra/coherent-modules/part-04","status":"fully_proved","summary":"*Quoted from the headline results in README.md of FormalFrontier/coherent-modules at commit b670b34e, Apache-2.0.*","title":"Faithfully flat dependent\u2026 to Finite-type ideal/module\u2026"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-04/faithfully-flat-dependent-direct-sums","kind":"node","lean":"Module.FaithfullyFlat.directSum_iff","parent":8775,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-04/index.html#faithfully-flat-dependent-direct-sums","scope":"formal-frontier/commutative-algebra/coherent-modules/part-04","status":"fully_proved","summary":"For arbitrary families of additive commutative groups with module structures over a `CommRing R`, `Module.FaithfullyFlat.directSum_iff` characterizes faithful flatness of the direct sum by flatness of every component and a nontrivial quotient-tensor fiber in some component for each proper ideal of `R`. The detecting component may depend on the ideal; the components need not individually be faithfully flat. The\u2026","title":"Faithfully flat dependent direct sums"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-04/finitely-generated-ideals-and-dependent-products","kind":"node","lean":"Submodule.smul_pi_of_fg","parent":8775,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-04/index.html#finitely-generated-ideals-and-dependent-products","scope":"formal-frontier/commutative-algebra/coherent-modules/part-04","status":"fully_proved","summary":"Over a `CommSemiring R`, `Submodule.smul_pi_of_fg` says that multiplication by a finitely generated ideal commutes with the full dependent product of submodules of additive commutative `R`-modules. The index type is arbitrary, including empty, and the semiring, index and carrier universes are independent.","title":"Finitely generated ideals and dependent products"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-04/faithful-flatness-after-base-change-covering-max","kind":"node","lean":"Module.FaithfullyFlat.iff_flat_and_baseChange_of_map_maximal_ne_top","parent":8775,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-04/index.html#faithful-flatness-after-base-change-covering-max","scope":"formal-frontier/commutative-algebra/coherent-modules/part-04","status":"fully_proved","summary":"For commutative rings `R` and `S` with an `R`-algebra structure on `S`, suppose every maximal ideal `m` of `R` satisfies `m.map (algebraMap R S) \u2260 \u22a4`. For any additive commutative group `M` with an `R`-module structure, `Module.FaithfullyFlat.iff_flat_and_baseChange_of_map_maximal_ne_top` characterizes faithful flatness over `R` by original flatness `Module.Flat R M` and faithful flatness over `S` of `S \u2297[R] M`,\u2026","title":"Faithful flatness after base change covering maximal ideals"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-04/faithful-flatness-of-unital-algebras","kind":"node","lean":"Module.FaithfullyFlat.algebra_iff_flat_and_fg_proper_ideal","parent":8775,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-04/index.html#faithful-flatness-of-unital-algebras","scope":"formal-frontier/commutative-algebra/coherent-modules/part-04","status":"fully_proved","summary":"For a `CommRing R` and a unital `R`-algebra `A` whose underlying ring may be noncommutative, `Module.FaithfullyFlat.algebra_iff_flat_and_fg_proper_ideal` characterizes faithful flatness by flatness and `I \u2022 (\u22a4 : Submodule R A) \u2260 \u22a4` for every finitely generated proper ideal `I`. The algebra structure is material; this finitely generated ideal criterion is not asserted for arbitrary modules.","title":"Faithful flatness of unital algebras"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-04/faithfully-flat-products-of-algebras","kind":"node","lean":"Module.FaithfullyFlat.pi_algebra_iff","parent":8775,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-04/index.html#faithfully-flat-products-of-algebras","scope":"formal-frontier/commutative-algebra/coherent-modules/part-04","status":"fully_proved","summary":"Over a `CommRing R`, for an arbitrary dependent family of unital, possibly noncommutative `R`-algebras `A i`, `Module.FaithfullyFlat.pi_algebra_iff` characterizes faithful flatness of `\u03a0 i, A i` by flatness of the whole product and a nontrivial fiber `(R \u29f8 I) \u2297[R] A i` in some component for every finitely generated proper ideal `I`. The detecting component may depend on the ideal. Componentwise flatness cannot\u2026","title":"Faithfully flat products of algebras"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-04/products-of-localizations-at-submonoids","kind":"node","lean":"Module.FaithfullyFlat.pi_localization_iff","parent":8775,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-04/index.html#products-of-localizations-at-submonoids","scope":"formal-frontier/commutative-algebra/coherent-modules/part-04","status":"fully_proved","summary":"For a `CommRing R`, arbitrary submonoids `S i` and commutative `R`-algebras `A i` realizing the localizations at `S i`, `Module.FaithfullyFlat.pi_localization_iff` characterizes faithful flatness of `\u03a0 i, A i` by flatness of the whole product and, for every finitely generated proper ideal `I`, disjointness of `I` from some `S i`. These are products with the common diagonal `R`-action; neither finite nor nonempty\u2026","title":"Products of localizations at submonoids"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-04/families-of-principal-localizations","kind":"node","lean":"Module.FaithfullyFlat.directSum_away_iff","parent":8775,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-04/index.html#families-of-principal-localizations","scope":"formal-frontier/commutative-algebra/coherent-modules/part-04","status":"fully_proved","summary":"For a `CommRing R` and any family `s : \u03b9 \u2192 R`, `Module.FaithfullyFlat.directSum_away_iff` characterizes faithful flatness of `\u2a01 i, Localization.Away (s i)` by `Ideal.span (Set.range s) = \u22a4`, without finite or nonempty indexing. For finite index types, `Module.FaithfullyFlat.pi_away_iff` gives the same unit-span criterion for `\u03a0 i, Localization.Away (s i)`. The localizations need not individually be faithfully flat;\u2026","title":"Families of principal localizations"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-04/faithfully-flat-coherence-descent","kind":"node","lean":"Module.IsCoherent.of_isCoherent_tensorProduct_of_faithfullyFlat","parent":8775,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-04/index.html#faithfully-flat-coherence-descent","scope":"formal-frontier/commutative-algebra/coherent-modules/part-04","status":"fully_proved","summary":"`Module.IsCoherent.of_isCoherent_tensorProduct_of_faithfullyFlat` descends coherence of the canonical scalar extension `S \u2297[R] M` to the original module `M`. The scalar algebra `S` may be noncommutative; no finiteness hypothesis on `M` or module-finiteness of `S` is needed. The regular-module consequence `IsCoherentRing.of_faithfullyFlat` descends ring coherence. The original module's additive monoid and action are\u2026","title":"Faithfully flat coherence descent"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-04/universal-coherence","kind":"node","lean":"IsUniversallyCoherent","parent":8775,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-04/index.html#universal-coherence","scope":"formal-frontier/commutative-algebra/coherent-modules/part-04","status":"fully_proved","summary":"`IsUniversallyCoherent` asserts coherence for finitely presented commutative algebras in an independently chosen target universe; the library derives base coherence at `v = u` and supplies the Noetherian case. Finitely presented algebra extensions preserve universal coherence with independent base, extension and target universes. In particular, polynomial rings over universally coherent rings are universally\u2026","title":"Universal coherence"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-04/associative-square-zero-multiplication","kind":"node","lean":"TrivSqZeroExt.mul_assoc_iff","parent":8775,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-04/index.html#associative-square-zero-multiplication","scope":"formal-frontier/commutative-algebra/coherent-modules/part-04","status":"fully_proved","summary":"For a type `R` with multiplication and an additive monoid `M` with distributive left and right actions, `TrivSqZeroExt.mul_assoc_iff` characterizes associativity by coefficient associativity, composition of each action (in reversed coefficient order on the right), and commutation of the two actions. Addition need not commute, the carrier universes are independent, and no coefficient zero, one, or identity law for\u2026","title":"Associative square-zero multiplication"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-04/commutativity-of-square-zero-extensions","kind":"node","lean":"TrivSqZeroExt.isMulCommutative_iff","parent":8775,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-04/index.html#commutativity-of-square-zero-extensions","scope":"formal-frontier/commutative-algebra/coherent-modules/part-04","status":"fully_proved","summary":"For `Mul R`, `AddZeroClass M`, `IsAddCommutative M`, and two given zero-preserving scalar actions `[SMulZeroClass R M]` and `[SMulZeroClass R\u1d50\u1d52\u1d56 M]`, `TrivSqZeroExt.isMulCommutative_iff` characterizes `IsMulCommutative (TrivSqZeroExt R M)` by `IsMulCommutative R \u2227 IsCentralScalar R M`. It retains the original multiplication, addition, zero and both actions, with independent carrier universes. No additive\u2026","title":"Commutativity of square-zero extensions"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-04/square-zero-kernels-and-projection-base-change","kind":"node","lean":"TrivSqZeroExt.kerIdealLinearEquiv","parent":8775,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-04/index.html#square-zero-kernels-and-projection-base-change","scope":"formal-frontier/commutative-algebra/coherent-modules/part-04","status":"fully_proved","summary":"For a commutative semiring `R` and an additive commutative `R`-module `M`, `TrivSqZeroExt.kerIdealLinearEquiv` identifies the projection kernel with `M`. If `M` is finite over `R`, `kerIdeal_finite` makes the kernel finite over the whole extension. `kerIdealBaseChange` identifies the projection-base-changed kernel with `M` without a finiteness assumption; its pure-tensor and inverse formulas are simp lemmas.","title":"Square-zero kernels and projection base change"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-04/full-left-ideal-lattice-of-square-zero-extension","kind":"node","lean":"TrivSqZeroExt.idealOrderIsoWithTop","parent":8775,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-04/index.html#full-left-ideal-lattice-of-square-zero-extension","scope":"formal-frontier/commutative-algebra/coherent-modules/part-04","status":"fully_proved","summary":"For a possibly noncommutative division semiring `D : Type u` and an additive commutative group `M : Type v` with its given actions `[Module D M]`, `[Module D\u1d50\u1d52\u1d56 M]` and `[SMulCommClass D D\u1d50\u1d52\u1d56 M]`, `TrivSqZeroExt.idealOrderIsoWithTop` identifies `Ideal (TrivSqZeroExt D M)` with `WithTop (Submodule D M)` as ordered lattices. The two carrier universes are independent. A coefficient submodule `N` corresponds to\u2026","title":"Full left-ideal lattice of square-zero extensions"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-04/module-length-and-square-zero-extensions","kind":"node","lean":"LinearMap.length_eq_of_bijective","parent":8775,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-04/index.html#module-length-and-square-zero-extensions","scope":"formal-frontier/commutative-algebra/coherent-modules/part-04","status":"fully_proved","summary":"For modules over arbitrary rings, `LinearMap.length_eq_of_bijective` preserves length under a bijective semilinear map whose scalar homomorphism is surjective, without requiring that scalar homomorphism to be injective. The rings and modules have four independent universes, and infinite lengths are included. Its algebra-map specialization recovers Mathlib's surjective-algebra comparison under the same hypotheses.\u2026","title":"Module length and square-zero extensions"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-04/opposites-of-square-zero-extensions","kind":"node","lean":"TrivSqZeroExt.opEquiv","parent":8775,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-04/index.html#opposites-of-square-zero-extensions","scope":"formal-frontier/commutative-algebra/coherent-modules/part-04","status":"fully_proved","summary":"`TrivSqZeroExt.opEquiv` identifies `(TrivSqZeroExt R M)\u1d50\u1d52\u1d56` with `TrivSqZeroExt R\u1d50\u1d52\u1d56 M` by `op (r, m) \u21a6 (op r, m)`. The target uses the original right action on the left and the original left action pulled back by double `unop` on the right. The equivalence needs only `Add R`, `Mul R`, `AddCommMagma M`, and the two given scalar actions. For a division ring `D` and an `AddCommGroup M` with commuting bimodule\u2026","title":"Opposites of square-zero extensions"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-04/left-noetherian-square-zero-extensions","kind":"node","lean":"TrivSqZeroExt.isNoetherianRing_iff","parent":8775,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-04/index.html#left-noetherian-square-zero-extensions","scope":"formal-frontier/commutative-algebra/coherent-modules/part-04","status":"fully_proved","summary":"For a possibly noncommutative `Ring R` and an `AddCommMonoid M` with commuting left and right `R`-module actions, `TrivSqZeroExt.isNoetherianRing_iff` characterizes `IsNoetherianRing (TrivSqZeroExt R M)` by `IsNoetherianRing R \u2227 Module.Finite R M`. Finiteness here is as a left `R`-module. The companion `TrivSqZeroExt.isNoetherianRing_iff_isNoetherian` replaces `Module.Finite R M` by `IsNoetherian R M`. Centrality\u2026","title":"Left Noetherian square-zero extensions"},{"id":"formal-frontier/commutative-algebra/coherent-modules/part-04/finite-type-ideal-module-transfer","kind":"node","lean":"Ideal.ideal_property_iff_module_property","parent":8775,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/coherent-modules/part-04/index.html#finite-type-ideal-module-transfer","scope":"formal-frontier/commutative-algebra/coherent-modules/part-04","status":"fully_proved","summary":"Over a commutative base ring, `Ideal.ideal_property_iff_module_property` equates a property on all finitely generated ideals with one on all finite modules over finite-type algebras. The property must be invariant under compatible algebra/module isomorphisms and descend along every compatible surjective square-zero algebra map for finite source modules. The base universe is independent of the common algebra/module\u2026","title":"Finite-type ideal/module transfer"},{"id":"formal-frontier/commutative-algebra/finite-etale-algebras","kind":"node","parent":8708,"result":false,"route":"../roadmap/formal-frontier/commutative-algebra/finite-etale-algebras/index.html","scope":"formal-frontier/commutative-algebra","status":"can_state","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/finite-etale-algebras at commit 4627c3ab, Apache-2.0.*","title":"Finite \u00e9tale algebras"},{"data_url":"scopes/formal-frontier/commutative-algebra/formal-power-series.json","id":"formal-frontier/commutative-algebra/formal-power-series","kind":"scope","parent":8708,"result":false,"route":"../roadmap/formal-frontier/commutative-algebra/formal-power-series/index.html","scope":"formal-frontier/commutative-algebra/formal-power-series","status":"planned","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/formal-power-series at commit 29bdab8c, Apache-2.0.*","title":"Formal power series"},{"id":"formal-frontier/commutative-algebra/formal-power-series/main-results","kind":"node","parent":8794,"result":false,"route":"../roadmap/formal-frontier/commutative-algebra/formal-power-series/main-results/index.html","scope":"formal-frontier/commutative-algebra/formal-power-series","status":"can_state","summary":"The main results that the formalization.yaml of FormalFrontier/formal-power-series lists under `status.main_results` at commit 29bdab8c, each that the scan finds declared in the file the yaml gives, with its line. They are listed here, not made articles: in several Formal Frontier libraries this list is close to the whole API. The notes are the yaml's.","title":"Main results"},{"data_url":"scopes/formal-frontier/commutative-algebra/graded-rings.json","id":"formal-frontier/commutative-algebra/graded-rings","kind":"scope","parent":8708,"result":false,"route":"../roadmap/formal-frontier/commutative-algebra/graded-rings/index.html","scope":"formal-frontier/commutative-algebra/graded-rings","status":"fully_proved","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/graded-rings at commit 6a459e8e, Apache-2.0.*","title":"Graded rings"},{"id":"formal-frontier/commutative-algebra/graded-rings/quotient-rees-algebra-of-a-regular-principal-ide","kind":"node","lean":"reesAlgebra.principalPolynomialEquiv","parent":8796,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/graded-rings/index.html#quotient-rees-algebra-of-a-regular-principal-ide","scope":"formal-frontier/commutative-algebra/graded-rings","status":"fully_proved","summary":"For any commutative ring R and a in R such that multiplication by a is injective, the quotient Rees_R((a))/(a)Rees_R((a)) is isomorphic as an R/(a)-algebra to (R/(a))[T].","title":"Quotient Rees algebra of a regular principal ideal"},{"id":"formal-frontier/commutative-algebra/graded-rings/finite-weighted-normal-forms-for-rees-modules","kind":"node","lean":"Ideal.Filtration.submodule_fg_iff_exists_weighted_normal_form","parent":8796,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/graded-rings/index.html#finite-weighted-normal-forms-for-rees-modules","scope":"formal-frontier/commutative-algebra/graded-rings","status":"fully_proved","summary":"For any ideal I of a commutative ring R and any R-module M, a natural I-filtration has finitely generated Rees module exactly when one finite family of elements x_i and natural weights w_i satisfies H.N n = sum_i I^(n-w_i) \u2022 R x_i for every natural n, with truncated natural subtraction.","title":"Finite weighted normal forms for Rees modules"},{"id":"formal-frontier/commutative-algebra/graded-rings/finite-generation-of-associated-graded-modules","kind":"node","lean":"Ideal.AssociatedGradedModule.finite_iff","parent":8796,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/graded-rings/index.html#finite-generation-of-associated-graded-modules","scope":"formal-frontier/commutative-algebra/graded-rings","status":"fully_proved","summary":"For any ideal I of a commutative ring R and any R-module M, let C = Rees_R(I)/I Rees_R(I). The associated graded module gr_I(M) is finitely generated over C if and only if M/IM is finitely generated over R.","title":"Finite generation of associated graded modules"},{"id":"formal-frontier/commutative-algebra/graded-rings/polynomial-scalar-extension-over-semirings","kind":"node","lean":"PolynomialModule.Extension.polynomialModule, PolynomialModule.Extension.coeff_smul, PolynomialModule.Extension.universalProperty, PolynomialModule.Extension.coordinate, PolynomialModule.Extension.polynomialLinearEquivPolynomialModule","parent":8796,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/graded-rings/index.html#polynomial-scalar-extension-over-semirings","scope":"formal-frontier/commutative-algebra/graded-rings","status":"fully_proved","summary":"For any possibly noncommutative semiring R and any left R-module M with additive commutative-monoid structure, finitely supported coefficient sequences form a canonical R[X]-module. Constant polynomials act by the original coefficient action and X shifts degree. This module represents C-semilinear maps from M and is natural under semilinear coefficient maps. Under the original commutative-ring and additive-group\u2026","title":"Polynomial scalar extension over semirings"},{"data_url":"scopes/formal-frontier/commutative-algebra/graded-rings/headline-results.json","id":"formal-frontier/commutative-algebra/graded-rings/headline-results","kind":"scope","parent":8796,"result":false,"route":"../roadmap/formal-frontier/commutative-algebra/graded-rings/headline-results/index.html","scope":"formal-frontier/commutative-algebra/graded-rings/headline-results","status":"fully_proved","summary":"*Quoted from the headline results in README.md of FormalFrontier/graded-rings at commit 6a459e8e, Apache-2.0.*","title":"Headline results"},{"id":"formal-frontier/commutative-algebra/graded-rings/headline-results/polynomial-covers-and-equivalences-for-principal","kind":"node","lean":"reesAlgebra.principalPolynomialHom","parent":8801,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/graded-rings/headline-results/index.html#polynomial-covers-and-equivalences-for-principal","scope":"formal-frontier/commutative-algebra/graded-rings/headline-results","status":"fully_proved","summary":"For any commutative ring `R` and `a : R`, evaluation at the class of `a t` defines a surjective `R/(a)`-algebra map `(R/(a))[T] \u2192 C`, where `C` is the Rees algebra modulo its coefficient ideal. If `a` is regular, this same map is an `R/(a)`-algebra equivalence, including unit generators and the zero ring. Composition with augmentation is evaluation at zero (`reesAlgebra.principalPolynomialHom`,\u2026","title":"Polynomial covers and equivalences for principal ideals"},{"id":"formal-frontier/commutative-algebra/graded-rings/main-results","kind":"node","parent":8796,"result":false,"route":"../roadmap/formal-frontier/commutative-algebra/graded-rings/main-results/index.html","scope":"formal-frontier/commutative-algebra/graded-rings","status":"can_state","summary":"The main results that the formalization.yaml of FormalFrontier/graded-rings lists under `status.main_results` at commit 6a459e8e, each that the scan finds declared in the file the yaml gives, with its line. They are listed here, not made articles: in several Formal Frontier libraries this list is close to the whole API. The notes are the yaml's.","title":"Main results"},{"data_url":"scopes/formal-frontier/commutative-algebra/integral-closure.json","id":"formal-frontier/commutative-algebra/integral-closure","kind":"scope","parent":8708,"result":false,"route":"../roadmap/formal-frontier/commutative-algebra/integral-closure/index.html","scope":"formal-frontier/commutative-algebra/integral-closure","status":"fully_proved","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/integral-closure at commit 8f0b15a5, Apache-2.0.*","title":"Integral closure"},{"id":"formal-frontier/commutative-algebra/integral-closure/integral-domains-over-henselian-local-rings-are","kind":"node","lean":"HenselianLocalRing.isLocalRing_of_isIntegral, RingHom.isLocalRing_of_henselian_of_isIntegral","parent":8804,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/integral-closure/index.html#integral-domains-over-henselian-local-rings-are","scope":"formal-frontier/commutative-algebra/integral-closure","status":"fully_proved","summary":"A nonzero commutative domain integral over a commutative henselian local ring is local, without injectivity or module-finiteness assumptions.","title":"Integral domains over henselian local rings are local"},{"id":"formal-frontier/commutative-algebra/integral-closure/integral-closedness-descends-along-linear-retrac","kind":"node","lean":"IsIntegrallyClosed.of_linearRetract","parent":8804,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/integral-closure/index.html#integral-closedness-descends-along-linear-retrac","scope":"formal-frontier/commutative-algebra/integral-closure","status":"fully_proved","summary":"An injective algebra map into an integrally closed domain with a linear left inverse gives integral closedness of its source.","title":"Integral closedness descends along linear retractions"},{"id":"formal-frontier/commutative-algebra/integral-closure/normal-domain-descent-from-arbitrary-field-scala","kind":"node","lean":"Algebra.TensorProduct.isDomain_left, Algebra.TensorProduct.isIntegrallyClosed_left","parent":8804,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/integral-closure/index.html#normal-domain-descent-from-arbitrary-field-scala","scope":"formal-frontier/commutative-algebra/integral-closure","status":"fully_proved","summary":"If scalar extension of a commutative algebra by any field extension is an integrally closed domain, then the original algebra is an integrally closed domain.","title":"Normal-domain descent from arbitrary field scalar extensions"},{"id":"formal-frontier/commutative-algebra/integral-closure/prime-spectrum-embeddings-from-powers-in-the-ima","kind":"node","lean":"PrimeSpectrum.isEmbedding_comap_of_forall_pow_mem_range","parent":8804,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/integral-closure/index.html#prime-spectrum-embeddings-from-powers-in-the-ima","scope":"formal-frontier/commutative-algebra/integral-closure","status":"fully_proved","summary":"A homomorphism of commutative semirings for which each target element has a positive power in its image induces a topological embedding on prime spectra.","title":"Prime-spectrum embeddings from powers in the image"},{"id":"formal-frontier/commutative-algebra/integral-closure/finiteness-from-powers-of-algebra-generators","kind":"node","lean":"Subalgebra.finite_of_adjoin_eq_top_of_forall_exists_pow_mem","parent":8804,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/integral-closure/index.html#finiteness-from-powers-of-algebra-generators","scope":"formal-frontier/commutative-algebra/integral-closure","status":"fully_proved","summary":"Let `R` and `A` be commutative semirings with the given `R`-algebra structure. If a finite set generates `A` as an `R`-algebra and `C : Subalgebra R A` contains a positive power of each chosen generator, then the original `A` is finite as a module over `C` through the canonical inclusion action. Exponents may vary with the generator; no field, Noetherianity, finite type of `C/R`, or ring structure is assumed. The\u2026","title":"Finiteness from powers of algebra generators"},{"id":"formal-frontier/commutative-algebra/integral-closure/integrality-is-unchanged-by-positive-powers","kind":"node","lean":"RingHom.IsIntegralElem.pow","parent":8804,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/integral-closure/index.html#integrality-is-unchanged-by-positive-powers","scope":"formal-frontier/commutative-algebra/integral-closure","status":"fully_proved","summary":"`RingHom.IsIntegralElem.pow` shows that every natural power of an integral element is integral, including exponent zero. Conversely, `of_pow` and the simp lemma `pow_iff` require `0 < n`: integrality of `x ^ n` then implies, and is equivalent to, integrality of `x`.","title":"Integrality is unchanged by positive powers"},{"id":"formal-frontier/commutative-algebra/integral-closure/positive-powers-in-the-range-imply-an-integral-h","kind":"node","lean":"RingHom.isIntegral_of_forall_pow_mem_range","parent":8804,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/integral-closure/index.html#positive-powers-in-the-range-imply-an-integral-h","scope":"formal-frontier/commutative-algebra/integral-closure","status":"fully_proved","summary":"`RingHom.isIntegral_of_forall_pow_mem_range` proves `f.IsIntegral` if each `x : S` has some positive power in `Set.range f`. Each element may have its own exponent; neither a uniform exponent nor surjectivity is required.","title":"Positive powers in the range imply an integral homomorphism"},{"id":"formal-frontier/commutative-algebra/integral-closure/main-results","kind":"node","parent":8804,"result":false,"route":"../roadmap/formal-frontier/commutative-algebra/integral-closure/main-results/index.html","scope":"formal-frontier/commutative-algebra/integral-closure","status":"can_state","summary":"The main results that the formalization.yaml of FormalFrontier/integral-closure lists under `status.main_results` at commit 8f0b15a5, each that the scan finds declared in the file the yaml gives, with its line. They are listed here, not made articles: in several Formal Frontier libraries this list is close to the whole API. The notes are the yaml's.","title":"Main results"},{"data_url":"scopes/formal-frontier/commutative-algebra/minimal-primes.json","id":"formal-frontier/commutative-algebra/minimal-primes","kind":"scope","parent":8708,"result":false,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/index.html","scope":"formal-frontier/commutative-algebra/minimal-primes","status":"can_prove","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/minimal-primes at commit 45952e34, Apache-2.0.*","title":"Minimal primes"},{"id":"formal-frontier/commutative-algebra/minimal-primes/prime-spectra-of-products-of-semifields","kind":"node","lean":"PrimeSpectrum.piHomeomorphUltrafilter","parent":8813,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/index.html#prime-spectra-of-products-of-semifields","scope":"formal-frontier/commutative-algebra/minimal-primes","status":"fully_proved","summary":"The prime spectrum of an arbitrary dependent product of semifields is canonically homeomorphic to the Stone space of ultrafilters on its index type.","title":"Prime spectra of products of semifields"},{"id":"formal-frontier/commutative-algebra/minimal-primes/ideals-in-products-of-division-semirings","kind":"node","lean":"Ideal.piIdeal_piFilter","parent":8813,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/index.html#ideals-in-products-of-division-semirings","scope":"formal-frontier/commutative-algebra/minimal-primes","status":"fully_proved","summary":"Every left ideal of an arbitrary dependent product of division semirings is the zero-set ideal of its characteristic-test filter.","title":"Ideals in products of division semirings"},{"id":"formal-frontier/commutative-algebra/minimal-primes/nontrivial-dependent-reduced-products","kind":"node","lean":"Filter.Product.nontrivial_iff","parent":8813,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/index.html#nontrivial-dependent-reduced-products","scope":"formal-frontier/commutative-algebra/minimal-primes","status":"fully_proved","summary":"For every filter and independently universed dependent type family, the total-section reduced product is nontrivial exactly when a total section exists and nontrivial fibres occur frequently.","title":"Nontrivial dependent reduced products"},{"id":"formal-frontier/commutative-algebra/minimal-primes/krull-dimension-and-chain-cardinalities-of-lexic","kind":"node","lean":"Order.krullDim_toLex_sigma, Order.krullDim_toLex_sum, Set.chainHeight_sigma_lex","parent":8813,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/index.html#krull-dimension-and-chain-cardinalities-of-lexic","scope":"formal-frontier/commutative-algebra/minimal-primes","status":"fully_proved","summary":"For arbitrary subsets of independently universed dependent preorders whose active indices form a finite strict chain, the stock lexicographic Sigma dimension is the active-index dimension plus the sum of active fiber dimensions. Empty support and infinite fiber dimensions are included. For two arbitrary preorder subsets, the tagged lexicographic union has the maximum of their dimensions and their sum plus one. For\u2026","title":"Krull dimension and chain cardinalities of lexicographic sums"},{"id":"formal-frontier/commutative-algebra/minimal-primes/dimension-and-point-ranks-of-finite-dependent-le","kind":"node","lean":"Order.krullDim_toLex_pi, Order.height_toLex_pi, Order.coheight_toLex_pi, Pi.Lex.isStrictOrderOfLT, Pi.Lex.partialOrderOfLT, Pi.Lex.partialOrderOfLT_lt, Pi.Lex.partialOrderOfLT_le_iff, Pi.Lex.partialOrderOfLT_eq","parent":8813,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/index.html#dimension-and-point-ranks-of-finite-dependent-le","scope":"formal-frontier/commutative-algebra/minimal-primes","status":"can_prove","summary":"For arbitrary subsets of independently universed given preorders over a finite linearly ordered index, lexicographic dimension is the WithBot map of ENat decrement applied to the product of the shifted fibre dimensions. Product and addition are in WithBot ENat, with only the decrement in ENat. Empty index, empty fibres and infinite dimensions are included. The canonical partial order is equality or strict\u2026","title":"Dimension and point ranks of finite dependent lexicographic products"},{"data_url":"scopes/formal-frontier/commutative-algebra/minimal-primes/headline-results.json","id":"formal-frontier/commutative-algebra/minimal-primes/headline-results","kind":"scope","parent":8813,"result":false,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/headline-results/index.html","scope":"formal-frontier/commutative-algebra/minimal-primes/headline-results","status":"fully_proved","summary":"*Quoted from the headline results in README.md of FormalFrontier/minimal-primes at commit 45952e34, Apache-2.0.*","title":"Headline results"},{"data_url":"scopes/formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01.json","id":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01","kind":"scope","navigation":"transparent","parent":8819,"result":false,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/index.html","scope":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01","status":"fully_proved","summary":"*Quoted from the headline results in README.md of FormalFrontier/minimal-primes at commit 45952e34, Apache-2.0.*","title":"Successors of finite product\u2026 to Common finite bases of\u2026"},{"id":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/successors-of-finite-product-expansions","kind":"node","lean":"ENat.WithBot.succ_sum_powerset_nonempty","parent":8820,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/index.html#successors-of-finite-product-expansions","scope":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01","status":"fully_proved","summary":"For arbitrary `\u03b9 : Type u`, `J : Finset \u03b9` and `d : \u03b9 \u2192 WithBot \u2115\u221e`, `ENat.WithBot.succ_sum_powerset_nonempty` states that the bottom-aware successor of the sum of `\u220f i \u2208 U, d i` over nonempty subsets `U` of `J` equals `\u220f i \u2208 J, WithBot.succ (d i)`. The equality is in `\u2115\u221e`; the subset products and their sum are in `WithBot \u2115\u221e`. Empty `J` and arbitrary bottom and top values are included, with no decidable-equality\u2026","title":"Successors of finite product expansions"},{"id":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/dimension-from-finite-coordinate-restriction-ima","kind":"node","lean":"Order.krullDim_pi_eq_iSup_range_restrict","parent":8820,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/index.html#dimension-from-finite-coordinate-restriction-ima","scope":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01","status":"fully_proved","summary":"For arbitrary `\u03b9 : Type u` and independently universed given preorders `\u03b2 : \u03b9 \u2192 Type v`, `Order.krullDim_pi_eq_iSup_range_restrict` states","title":"Dimension from finite coordinate-restriction images"},{"id":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/chain-cardinalities-of-unions","kind":"node","lean":"Set.chainHeight_union_le","parent":8820,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/index.html#chain-cardinalities-of-unions","scope":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01","status":"fully_proved","summary":"For an arbitrary relation `r` and subsets `s`, `t` of any type, `Set.chainHeight_union_le` gives `(s \u222a t).chainHeight r \u2264 s.chainHeight r + t.chainHeight r`. The pieces may overlap or be empty; neither relation laws nor finite heights or attainment are required. The proof restricts each comparable subset to its two intersections and uses `ENat` cardinality subadditivity.","title":"Chain cardinalities of unions"},{"id":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/chain-cardinalities-of-exclusive-piece-unions","kind":"node","lean":"Set.chainHeight_iUnion_of_not_rel","parent":8820,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/index.html#chain-cardinalities-of-exclusive-piece-unions","scope":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01","status":"fully_proved","summary":"For an arbitrary relation `r` on `\u03b1 : Type u` and a family `s : \u03b9 \u2192 Set \u03b1` with independent `\u03b9 : Sort v`, `Set.chainHeight_iUnion_of_not_rel` gives `(\u22c3 i, s i).chainHeight r = \u2a06 i, (s i).chainHeight r` when points in `s i \\ s j` and `s j \\ s i` cannot compare for every ordered pair `i,j`. For each pair, the no-cross condition concerns only its opposite exclusive pieces; all loops remain unrestricted. The family\u2026","title":"Chain cardinalities of exclusive-piece unions"},{"id":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/krull-dimension-of-exclusive-piece-unions","kind":"node","lean":"Order.krullDim_iUnion_of_not_lt","parent":8820,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/index.html#krull-dimension-of-exclusive-piece-unions","scope":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01","status":"fully_proved","summary":"For a given preorder on `\u03b1 : Type u`, an arbitrary independent `\u03b9 : Sort v`, and `s : \u03b9 \u2192 Set \u03b1`, `Order.krullDim_iUnion_of_not_lt` states `Order.krullDim (\u22c3 i, s i) = \u2a06 i, Order.krullDim (s i)` when points in `s i \\ s j` and `s j \\ s i` satisfy `\u00ac a < b` for every ordered pair `i,j`. All membership subtypes inherit the given preorder. No antisymmetry, inhabitation, finiteness, directedness or attainment is assumed.","title":"Krull dimension of exclusive-piece unions"},{"id":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/dimension-comparison-and-arbitrary-unions","kind":"node","lean":"Order.krullDim_le_krullDim_iff_chainHeight_le_chainHeight","parent":8820,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/index.html#dimension-comparison-and-arbitrary-unions","scope":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01","status":"fully_proved","summary":"For arbitrary subsets of independently universed given preorders, `Order.krullDim_le_krullDim_iff_chainHeight_le_chainHeight` states `\u03ba(s) \u2264 \u03ba(t) \u2194 H(s) \u2264 H(t)`, where `\u03ba` is the dimension of the membership subtype in `WithBot \u2115\u221e` and `H` is ambient strict-chain vertex height in `\u2115\u221e`. There is no map between the carriers, and each subtype inherits its given preorder. Empty sets have values bottom and zero;\u2026","title":"Dimension comparison and arbitrary unions"},{"id":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/krull-dimension-under-finite-covers","kind":"node","lean":"Order.krullDim_le_iSup_of_subset_biUnion_finset","parent":8820,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/index.html#krull-dimension-under-finite-covers","scope":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01","status":"fully_proved","summary":"For one given preorder on `\u03b1 : Type u`, an independent index `\u03b9 : Type v`, `I : Finset \u03b9`, a family `s : \u03b9 \u2192 Set \u03b1`, and `t \u2286 \u22c3 i \u2208 I, s i`, `Order.krullDim_le_iSup_of_subset_biUnion_finset` states","title":"Krull dimension under finite covers"},{"id":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/chain-cardinalities-of-disjoint-sums","kind":"node","lean":"Set.chainHeight_sum_liftRel","parent":8820,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/index.html#chain-cardinalities-of-disjoint-sums","scope":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01","status":"fully_proved","summary":"For arbitrary relations `r\u03b1`, `r\u03b2` on independently universed types and arbitrary sets `s`, `t`, `Set.chainHeight_sum_liftRel` gives chain height `max (s.chainHeight r\u03b1) (t.chainHeight r\u03b2)` for the union of the two tagged images under `Sum.LiftRel r\u03b1 r\u03b2`. No cross-tag points compare. `Set.chainHeight_sum` gives the same formula for ordinary `\u03b1 \u2295 \u03b2` less-than, induced by precisely the two given `LT` records. Neither\u2026","title":"Chain cardinalities of disjoint sums"},{"id":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/finite-localization-near-a-minimal-prime","kind":"node","lean":"Algebra.FiniteType.exists_finite_awayMap_of_mem_minimalPrimes","parent":8820,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/index.html#finite-localization-near-a-minimal-prime","scope":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01","status":"fully_proved","summary":"For a finite-type algebra over any commutative ring, module finiteness of the residue-field fibre at a minimal prime gives finiteness after inverting an element outside that prime. The map is the canonical localization of the given algebra map. `Algebra.FiniteType.exists_finite_awayMap_of_mem_minimalPrimes` requires no reducedness or Noetherian hypothesis and includes the zero algebra. The supporting\u2026","title":"Finite localization near a minimal prime"},{"id":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/directed-suprema-of-relations-on-a-fixed-set","kind":"node","lean":"Set.scottContinuous_chainHeight","parent":8820,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/index.html#directed-suprema-of-relations-on-a-fixed-set","scope":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01","status":"fully_proved","summary":"For any type `\u03b1` and fixed subset `s`, `Set.scottContinuous_chainHeight` states that `r \u21a6 s.chainHeight r` is Scott continuous for the pointwise implication order on relations `\u03b1 \u2192 \u03b1 \u2192 Prop`. No order or relation laws on `\u03b1` are needed. Scott continuity preserves suprema of nonempty directed families, not arbitrary joins or bottom.","title":"Directed suprema of relations on a fixed set"},{"id":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/directed-unions-chain-cardinality-and-order-dime","kind":"node","lean":"Set.chainHeight_iUnion_of_directed","parent":8820,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/index.html#directed-unions-chain-cardinality-and-order-dime","scope":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01","status":"fully_proved","summary":"For an inclusion-directed family `s : \u03b9 \u2192 Set \u03b1`, with independent carrier and index universes and an arbitrary index `\u03b9 : Sort v`, `Set.chainHeight_iUnion_of_directed` identifies the `ENat` chain height of the union with the supremum of the piece heights for any relation on `\u03b1`. With only a preorder on `\u03b1`, `Order.krullDim_iUnion_of_directed` gives the analogous equality for the induced subtype orders in\u2026","title":"Directed unions: chain cardinality and order dimension"},{"id":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/chain-cardinalities-of-lexicographic-products","kind":"node","lean":"Set.chainHeight_prod_lex","parent":8820,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/index.html#chain-cardinalities-of-lexicographic-products","scope":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01","status":"fully_proved","summary":"For arbitrary relations `r\u03b1`, `r\u03b2` on independently universed types and arbitrary sets `s`, `t`, `Set.chainHeight_prod_lex` proves that the chain height of `s \u00d7\u02e2 t` for `Prod.Lex r\u03b1 r\u03b2` is the product of the factor chain heights, provided `r\u03b1` is irreflexive on `s`. Neither relation needs to be transitive; the second may have loops. `Set.chainHeight_toLex_prod` transports this formula to the canonical less-than\u2026","title":"Chain cardinalities of lexicographic products"},{"id":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/chain-cardinalities-of-lexicographic-sums","kind":"node","lean":"Set.chainHeight_sum_lex","parent":8820,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/index.html#chain-cardinalities-of-lexicographic-sums","scope":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01","status":"fully_proved","summary":"For arbitrary relations `r\u03b1`, `r\u03b2` on independently universed types and arbitrary sets `s`, `t`, `Set.chainHeight_sum_lex` proves that the union of the two injection images has chain height `s.chainHeight r\u03b1 + t.chainHeight r\u03b2` for `Sum.Lex r\u03b1 r\u03b2`. Within each tag the factor relation is used, and every left-tagged point precedes every right-tagged point. No irreflexivity, transitivity, nonemptiness or finiteness is\u2026","title":"Chain cardinalities of lexicographic sums"},{"id":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/krull-dimension-of-lexicographic-products","kind":"node","lean":"Order.krullDim_toLex_prod","parent":8820,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/index.html#krull-dimension-of-lexicographic-products","scope":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01","status":"fully_proved","summary":"For arbitrary subsets `s`, `t` of independently universed preorders, `Order.krullDim_toLex_prod` states `\u03ba(toLex '' (s \u00d7\u02e2 t)) = \u03ba(s) * \u03ba(t) + \u03ba(s) + \u03ba(t)` in `WithBot \u2115\u221e`. The carrier is the membership subtype of that image in `\u03b1 \u00d7\u2097 \u03b2`, with the `Prod.Lex` preorder of the given factor preorders. Ties in the first coordinate require equality, not preorder equivalence. No antisymmetry, nonemptiness, finiteness or\u2026","title":"Krull dimension of lexicographic products"},{"id":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/semimodules-annihilators-and-faithful-actions-on","kind":"node","lean":"Filter.Product.module","parent":8820,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/index.html#semimodules-annihilators-and-faithful-actions-on","scope":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01","status":"fully_proved","summary":"For independent families `R i`, `M i` of semirings and additive commutative monoids, with given `Module (R i) (M i)` records and an arbitrary filter `F`, `Filter.Product.module` supplies pointwise semimodule data on the existing `F.Product M` over the existing scalar semiring `F.Product R`. The data are named rather than globally registered: the given fibre actions stay fixed even when scalar and vector carriers\u2026","title":"Semimodules, annihilators and faithful actions on dependent reduced products"},{"id":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/common-finite-bases-of-dependent-reduced-product","kind":"node","lean":"Filter.Product.basis","parent":8820,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01/index.html#common-finite-bases-of-dependent-reduced-product","scope":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-01","status":"fully_proved","summary":"For the same given semiring and semimodule records, a family of given bases `b i : Module.Basis J (R i) (M i)` with one common finite index `J` induces `Filter.Product.basis` of `F.Product M` over the existing scalar semiring `F.Product R`. Its vector at `j` is represented by `fun i => b i j`, and coefficients are the existing varying maps of the fibre coefficients, on represented and opaque vectors alike. For\u2026","title":"Common finite bases of dependent reduced products"},{"data_url":"scopes/formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02.json","id":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02","kind":"scope","navigation":"transparent","parent":8819,"result":false,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02/index.html","scope":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02","status":"fully_proved","summary":"*Quoted from the headline results in README.md of FormalFrontier/minimal-primes at commit 45952e34, Apache-2.0.*","title":"Exactness of dependent\u2026 to Irreducible closed rectangles\u2026"},{"id":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02/exactness-of-dependent-reduced-product-maps","kind":"node","lean":"Filter.Product.map_exact_iff","parent":8837,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02/index.html#exactness-of-dependent-reduced-product-maps","scope":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02","status":"fully_proved","summary":"For arbitrary functions `f i : X i \u2192 Y i` and `g i : Y i \u2192 Z i` in independent universes, an arbitrary filter `F`, given `Zero (Z i)` records and `Nonempty (\u2200 i, X i)`, `Filter.Product.map_exact_iff` characterizes `Function.Exact` of the two existing `Product.map` functions by `\u2200\u1da0 i in F, Function.Exact (f i) (g i)`. This is one eventual locus of whole fibre exactness, not exactness at every index or separately\u2026","title":"Exactness of dependent reduced-product maps"},{"id":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02/coordinate-maps-and-zero-reflection-of-dependent","kind":"node","lean":"Filter.Product.map","parent":8837,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02/index.html#coordinate-maps-and-zero-reflection-of-dependent","scope":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02","status":"fully_proved","summary":"For any filter `D` and independently universed dependent type families `R` and `S`, coordinate functions `\u03c6 i : R i \u2192 S i` induce the canonical `Filter.Product.map` from `D.Product R` to `D.Product S`. Its representative law uniquely characterizes it, with identity and composition laws and agreement with the existing semiring and constant-family germ maps. No algebra, points or proper-filter hypothesis is needed\u2026","title":"Coordinate maps and zero-reflection of dependent filter products"},{"id":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02/reindexing-dependent-filter-products","kind":"node","lean":"Filter.Product.compTendsto","parent":8837,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02/index.html#reindexing-dependent-filter-products","scope":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02","status":"fully_proved","summary":"For independently universed indices and fibres, a map `u : \u03b9 \u2192 \u03ba` with `Tendsto u F G` induces `Filter.Product.compTendsto` from `G.Product R` to `F.Product (fun i => R (u i))`. It sends a section to its precomposition, respects identity and composition, and agrees with the existing `Filter.Germ.compTendsto` on constant families. No algebra, total-section existence or proper-filter assumption is needed. The full\u2026","title":"Reindexing dependent filter products"},{"id":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02/small-products-of-dependent-filter-products","kind":"node","lean":"Filter.Product.piComparison","parent":8837,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02/index.html#small-products-of-dependent-filter-products","scope":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02","status":"fully_proved","summary":"For an arbitrary filter `D` and `R : \u03b9 \u2192 \u03ba \u2192 Type w`, with independent row, coordinate and fibre universes, `Filter.Product.piComparison` maps the class of a matrix to its family of column classes. It is surjective for arbitrary `\u03ba`. If `[CardinalInterFilter D c]` and `HasCardinalLT \u03ba c`, with `c` in the row universe, `Filter.Product.piEquivOfHasCardinalLT` makes this same comparison an equivalence. The bound is\u2026","title":"Small products of dependent filter products"},{"id":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02/centers-of-dependent-filter-products","kind":"node","lean":"Filter.Product.centerEquiv","parent":8837,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02/index.html#centers-of-dependent-filter-products","scope":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02","status":"fully_proved","summary":"For any filter and an independently universed family with multiplication, a section represents an element of the full intrinsic center exactly when it is central eventually. With a two-sided identity in each fibre, `Filter.Product.centerEquiv` identifies the product of the fibre centers with the center of the product. Neither associativity nor a proper filter is required. The representative API selects a total\u2026","title":"Centers of dependent filter products"},{"id":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02/reduced-filter-germs","kind":"node","lean":"Filter.Germ.isReduced_iff_eventually","parent":8837,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02/index.html#reduced-filter-germs","scope":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02","status":"fully_proved","summary":"For independent index and value universes, any filter `D`, and a value type `M` with only zero and natural-number powers, `Filter.Germ.isReduced_iff_eventually` characterizes reducedness of `Germ D M` by eventual reducedness of `M`. Equivalently, `D = \u22a5` or `M` is reduced. No power laws or properness assumption are needed. The same module characterizes nilpotence by one exponent whose power is zero eventually, and\u2026","title":"Reduced filter germs"},{"id":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02/no-zero-divisors-in-ultrafilter-products","kind":"node","lean":"Filter.Product.noZeroDivisors_iff_eventually","parent":8837,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02/index.html#no-zero-divisors-in-ultrafilter-products","scope":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02","status":"fully_proved","summary":"For an ultrafilter `U` and an independently universed dependent family `R`, `Filter.Product.noZeroDivisors_iff_eventually` characterizes no zero divisors in the product by eventual no zero divisors in the fibres. It requires only `Zero` and `Mul` on each fibre and uses the existing `Con.hasMul` and `Con.zero` operations on the existing product carrier. No associativity, unit, absorption by zero, addition,\u2026","title":"No zero divisors in ultrafilter products"},{"id":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02/integral-pullback-of-maximal-spectra","kind":"node","lean":"MaximalSpectrum.comapOfIsIntegral","parent":8837,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02/index.html#integral-pullback-of-maximal-spectra","scope":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02","status":"fully_proved","summary":"For an integral homomorphism `f : R \u2192+* S` of commutative rings in independent universes, `MaximalSpectrum.comapOfIsIntegral` maps `MaximalSpectrum S` to `MaximalSpectrum R` by ideal contraction. Its image is exactly the maximal ideals containing `RingHom.ker f`, and it is surjective when `f` is injective. The API gives the underlying-ideal equation, agreement with prime-spectrum pullback, identity and\u2026","title":"Integral pullback of maximal spectra"},{"id":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02/irreducible-closed-rectangles-and-topological-pr","kind":"node","lean":"topologicalKrullDim_prod","parent":8837,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02/index.html#irreducible-closed-rectangles-and-topological-pr","scope":"formal-frontier/commutative-algebra/minimal-primes/headline-results/part-02","status":"fully_proved","summary":"For spaces `X : Type u` and `Y : Type v` with their given topologies, `topologicalKrullDim_prod` states","title":"Irreducible closed rectangles and topological product dimensions"},{"id":"formal-frontier/commutative-algebra/minimal-primes/main-results","kind":"node","parent":8813,"result":false,"route":"../roadmap/formal-frontier/commutative-algebra/minimal-primes/main-results/index.html","scope":"formal-frontier/commutative-algebra/minimal-primes","status":"can_state","summary":"The main results that the formalization.yaml of FormalFrontier/minimal-primes lists under `status.main_results` at commit 45952e34, each that the scan finds declared in the file the yaml gives, with its line. They are listed here, not made articles: in several Formal Frontier libraries this list is close to the whole API. The notes are the yaml's.","title":"Main results"},{"data_url":"scopes/formal-frontier/commutative-algebra/multivariate-polynomials.json","id":"formal-frontier/commutative-algebra/multivariate-polynomials","kind":"scope","parent":8708,"result":false,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/index.html","scope":"formal-frontier/commutative-algebra/multivariate-polynomials","status":"fully_proved","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/multivariate-polynomials at commit 724d9804, Apache-2.0.*","title":"Multivariate polynomials"},{"data_url":"scopes/formal-frontier/commutative-algebra/multivariate-polynomials/part-01.json","id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01","kind":"scope","navigation":"transparent","parent":8848,"result":false,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-01/index.html","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01","status":"fully_proved","summary":"*Quoted from the headline results in README.md of FormalFrontier/multivariate-polynomials at commit 724d9804, Apache-2.0.*","title":"Heights of homogeneous\u2026 to Jacobson radicals of\u2026"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01/heights-of-homogeneous-polynomial-evaluations","kind":"node","lean":"Height.exists_logHeight_eval_sub_mul_bound","parent":8849,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-01/index.html#heights-of-homogeneous-polynomial-evaluations","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01","status":"fully_proved","summary":"Over any field with `Height.AdmissibleAbsValues`, a homogeneous polynomial family of common degree `d`, with finite input and output coordinate types and no nonzero common zero over an algebraic closure, has evaluation height uniformly comparable to the input height raised to `d`. More precisely, `Height.exists_logHeight_eval_sub_mul_bound` bounds the difference between evaluation logarithmic height and `d` times\u2026","title":"Heights of homogeneous polynomial evaluations"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01/homogeneous-polynomial-maps-on-projective-coordi","kind":"node","lean":"MvPolynomial.projectivize","parent":8849,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-01/index.html#homogeneous-polynomial-maps-on-projective-coordi","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01","status":"fully_proved","summary":"`MvPolynomial.projectivize` sends a nonzero tuple's class to the class of its polynomial evaluation tuple. The coefficients may lie in any commutative semiring equipped with a homomorphism to the field of points. A common homogeneity degree and nonvanishing on nonzero input tuples make this map well-defined; its values on these classes characterize it uniquely. Coordinate variables induce the identity, and\u2026","title":"Homogeneous polynomial maps on projective coordinate spaces"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01/polynomial-closed-points-and-algebraic-closure-o","kind":"node","lean":"Polynomial.algAutOrbitEquivMaximalSpectrum","parent":8849,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-01/index.html#polynomial-closed-points-and-algebraic-closure-o","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01","status":"fully_proved","summary":"For independently universed fields `K` and `L` with their given `Algebra K L` and `IsAlgClosure K L`, `Polynomial.algAutOrbitEquivMaximalSpectrum` identifies the existing quotient of `L` by its `K`-algebra automorphism orbits with `MaximalSpectrum K[X]`. The ideal of the orbit of `a` is the kernel of `aeval a`, equivalently the ideal generated by `minpoly K a`. No perfectness or separability is assumed. This is a\u2026","title":"Polynomial closed points and algebraic-closure orbits"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01/infinitely-many-maximal-ideals-of-polynomial-rin","kind":"node","lean":"MaximalSpectrum.infinite_polynomial","parent":8849,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-01/index.html#infinitely-many-maximal-ideals-of-polynomial-rin","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01","status":"fully_proved","summary":"Every nontrivial commutative ring `R` has infinitely many maximal ideals in `R[X]`, including when `R` is a finite field, by `MaximalSpectrum.infinite_polynomial`. For any commutative ring `R` and arbitrary variable type `\u03c3`, the maximal spectrum of `MvPolynomial \u03c3 R` is infinite exactly when `R` is nontrivial and either `\u03c3` is nonempty or the maximal spectrum of `R` is infinite, by\u2026","title":"Infinitely many maximal ideals of polynomial rings"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01/factorization-uniqueness-from-a-regular-crossed","kind":"node","lean":"Polynomial.eq_and_eq_of_mul_eq_mul_of_isRegular_resultant","parent":8849,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-01/index.html#factorization-uniqueness-from-a-regular-crossed","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01","status":"fully_proved","summary":"Over any commutative ring, let `G,G'` have degree at most `m` and `H,H'` degree at most `n`, with `G.coeff m = G'.coeff m` and `H.coeff n = H'.coeff n`. If `G * H = G' * H'` and `Polynomial.resultant G' H m n` is regular, then `G = G'` and `H = H'`, by `Polynomial.eq_and_eq_of_mul_eq_mul_of_isRegular_resultant`. Neither monicity nor positive nominal degrees are required, and the ring may have zero divisors.","title":"Factorization uniqueness from a regular crossed resultant"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01/nonconstant-coefficients-of-polynomial-units","kind":"node","lean":"MvPolynomial.isAddUnit_coeff_of_isUnit","parent":8849,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-01/index.html#nonconstant-coefficients-of-polynomial-units","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01","status":"fully_proved","summary":"Over any commutative semiring `R`, every coefficient at a nonzero exponent of a unit in `MvPolynomial \u03c3 R` is additively invertible, by `MvPolynomial.isAddUnit_coeff_of_isUnit`. The variable type is arbitrary, including empty and infinite types.","title":"Nonconstant coefficients of polynomial units"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01/polynomial-units-over-commutative-semirings","kind":"node","lean":"MvPolynomial.isUnit_iff_coeff_isNilpotent_isAddUnit","parent":8849,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-01/index.html#polynomial-units-over-commutative-semirings","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01","status":"fully_proved","summary":"For any commutative semiring `R` and arbitrary variable type `\u03c3`, a polynomial in `MvPolynomial \u03c3 R` is a unit if and only if its constant coefficient is a unit and its coefficient at every nonzero exponent is both nilpotent and additively invertible, by `MvPolynomial.isUnit_iff_coeff_isNilpotent_isAddUnit`. The coefficient and variable types have independent universes; empty variable types and the zero semiring\u2026","title":"Polynomial units over commutative semirings"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01/prime-coefficient-extensions-over-commutative-se","kind":"node","lean":"MvPolynomial.isPrime_map_C_of_subtractive","parent":8849,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-01/index.html#prime-coefficient-extensions-over-commutative-se","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01","status":"fully_proved","summary":"For a commutative semiring `R`, any variable type `\u03c3` and ideal `I`, primality of `I` together with `\u2200 a b : R, a \u2208 I \u2192 a + b \u2208 I \u2192 b \u2208 I` implies primality of `I.map MvPolynomial.C`, by `MvPolynomial.isPrime_map_C_of_subtractive`. For nonempty `\u03c3`, these conditions are also necessary, by `MvPolynomial.isPrime_map_C_iff`.","title":"Prime coefficient extensions over commutative semirings"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01/maximal-coefficient-extensions-over-commutative","kind":"node","lean":"MvPolynomial.isMaximal_map_C_iff","parent":8849,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-01/index.html#maximal-coefficient-extensions-over-commutative","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01","status":"fully_proved","summary":"For a commutative semiring `R`, any variable type `\u03c3` and ideal `I`, `(I.map MvPolynomial.C).IsMaximal \u2194 I.IsMaximal \u2227 IsEmpty \u03c3`, by `MvPolynomial.isMaximal_map_C_iff`. With nonempty `\u03c3`, no coefficient extension is maximal. For proper `I`, the constant-coefficient preimage is a strict intermediate ideal.","title":"Maximal coefficient extensions over commutative semirings"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01/finite-coefficient-product-equations-determine-w","kind":"node","lean":"MvPolynomial.eval\u2082Hom_coefficientProduct_eq_iff","parent":8849,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-01/index.html#finite-coefficient-product-equations-determine-w","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01","status":"fully_proved","summary":"For independent commutative semirings `C` and `R`, an arbitrary homomorphism `\u03c6 : C \u2192+* R`, nominal sizes `m,n`, seeds `g,h : Polynomial C` and arbitrary target coefficient point, `MvPolynomial.eval\u2082Hom_coefficientProduct_eq_iff` identifies the finite evaluated equations with equality of the entire ordinary perturbed product to `T`. It assumes target-side bounds `(g.map \u03c6).natDegree \u2264 m`, `(h.map \u03c6).natDegree \u2264 n`,\u2026","title":"Finite coefficient-product equations determine whole products"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01/radicals-of-coefficient-ideal-extensions","kind":"node","lean":"MvPolynomial.radical_map_C","parent":8849,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-01/index.html#radicals-of-coefficient-ideal-extensions","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01","status":"fully_proved","summary":"For any commutative ring `R`, arbitrary variable type `\u03c3` and ideal `I : Ideal R`, `MvPolynomial.radical_map_C` says that extending `I` along the constant-polynomial map commutes with radicals. `MvPolynomial.mem_radical_map_C_iff` characterizes radical membership coefficientwise: a polynomial lies in the radical of the extended ideal exactly when every coefficient lies in `I.radical`.","title":"Radicals of coefficient-ideal extensions"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01/scalar-annihilators-and-non-zero-divisors","kind":"node","lean":"MvPolynomial.mem_nonZeroDivisors_iff","parent":8849,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-01/index.html#scalar-annihilators-and-non-zero-divisors","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01","status":"fully_proved","summary":"For any commutative semiring `R` and arbitrary variable type `\u03c3`, `MvPolynomial.mem_nonZeroDivisors_iff` characterizes non-zero-divisors by the absence of a nonzero scalar in the original coefficient semiring annihilating every coefficient. No individual coefficient need be a non-zero-divisor. `MvPolynomial.eq_zero_of_mul_eq_zero_of_smul` shows that a polynomial with no nonzero scalar annihilator has no nonzero\u2026","title":"Scalar annihilators and non-zero-divisors"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01/the-degree-zero-homogeneous-component","kind":"node","lean":"MvPolynomial.gradeZeroAlgEquiv","parent":8849,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-01/index.html#the-degree-zero-homogeneous-component","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01","status":"fully_proved","summary":"For any commutative semiring `R` and arbitrary variable type `\u03c3`, `MvPolynomial.gradeZeroAlgEquiv` identifies `R` as an `R`-algebra with Mathlib's original degree-zero homogeneous component. Its forward map is the inherited coefficient algebra map, with ambient polynomial `C r`; its inverse is `constantCoeff`. The construction uses independent universes and commutes with coefficient homomorphisms in the ambient\u2026","title":"The degree-zero homogeneous component"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01/coefficient-ideals-over-commutative-semirings","kind":"node","lean":"MvPolynomial.coeffsIn_eq_restrictScalars_map_C","parent":8849,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-01/index.html#coefficient-ideals-over-commutative-semirings","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01","status":"fully_proved","summary":"For any commutative semiring `R`, ideal `I` and arbitrary variable type `\u03c3`, `MvPolynomial.coeffsIn_eq_restrictScalars_map_C` identifies the extension `Ideal.map C I`, restricted to the canonical coefficient action, with Mathlib's coefficient submodule `coeffsIn \u03c3 I`. Polynomial membership is therefore coefficientwise membership, and contraction along `C` recovers `I`. `MvPolynomial.ker_map_eq_map_C` identifies the\u2026","title":"Coefficient ideals over commutative semirings"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01/square-witnesses-for-failure-of-subtractivity","kind":"node","lean":"MvPolynomial.subtractiveWitness","parent":8849,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-01/index.html#square-witnesses-for-failure-of-subtractivity","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01","status":"fully_proved","summary":"Over any commutative semiring `R`, for `i : \u03c3` and `a b : R`, `MvPolynomial.subtractiveWitness` is `C a + C b * X i + C (a + b) * X i ^ 2`, with linear coefficient `b`. Its square belongs to `I.map C` if `a \u2208 I` and `a + b \u2208 I`, while `b \u2209 I` excludes the witness itself. Thus a radical (in particular prime) extension over nonempty variables forces subtractivity. Coefficient and variable universes are independent;\u2026","title":"Square witnesses for failure of subtractivity"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01/powers-of-augmented-coefficient-ideals","kind":"node","lean":"MvPolynomial.mem_pow_comap_constantCoeff_iff_coeff_mem_pow","parent":8849,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-01/index.html#powers-of-augmented-coefficient-ideals","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01","status":"fully_proved","summary":"For any commutative semiring `R`, ideal `I`, natural number `n`, arbitrary variable type `\u03c3`, and polynomial `p`, `MvPolynomial.mem_pow_comap_constantCoeff_iff_coeff_mem_pow` characterizes membership in `(I.comap constantCoeff) ^ n`: the coefficient at each exponent `d` belongs to `I ^ (n - d.sum (fun _ k => k))`. Subtraction is truncated in `\u2115`, so degrees at least `n` impose no restriction, rather than vanishing.\u2026","title":"Powers of augmented coefficient ideals"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01/primary-coefficient-ideals","kind":"node","lean":"MvPolynomial.isPrimary_map_C_of_subtractive","parent":8849,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-01/index.html#primary-coefficient-ideals","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01","status":"fully_proved","summary":"Over any commutative semiring `R`, `MvPolynomial.isPrimary_map_C_of_subtractive` extends a primary ideal `I` satisfying `a \u2208 I \u2192 a + b \u2208 I \u2192 b \u2208 I` to a primary coefficient ideal, for every variable type `\u03c3`. When `\u03c3` is nonempty, `MvPolynomial.isPrimary_map_C_iff_subtractive` proves that primaryness and this subtractivity condition are also necessary. For a commutative ring, subtractivity is automatic, and\u2026","title":"Primary coefficient ideals"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01/supplied-primary-families-and-deletion-irredunda","kind":"node","lean":"MvPolynomial.primary_family_map_C_iff_of_subtractive","parent":8849,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-01/index.html#supplied-primary-families-and-deletion-irredunda","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01","status":"fully_proved","summary":"Over a commutative semiring `R`, for an ideal `I` and family `Q : \u03b9 \u2192 Ideal R`, `MvPolynomial.primary_family_map_C_iff_of_subtractive` assumes `a \u2208 Q i \u2192 a + b \u2208 Q i \u2192 b \u2208 Q i` for each component. For every variable type `\u03c3`, it equates `I.map C = \u2a05 i, (Q i).map C` with all mapped components primary to `I = \u2a05 i, Q i` with all original components primary. For nonempty `\u03c3`,\u2026","title":"Supplied primary families and deletion irredundance"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01/nilpotent-coefficients-and-irreducible-polynomia","kind":"node","lean":"MvPolynomial.isNilpotent_iff_coeff","parent":8849,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-01/index.html#nilpotent-coefficients-and-irreducible-polynomia","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01","status":"fully_proved","summary":"For any commutative semiring `R` and arbitrary variable type `\u03c3`, `MvPolynomial.isNilpotent_iff_coeff` characterizes nilpotent polynomials by nilpotence of every coefficient. `PrimeSpectrum.irreducibleSpace_mvPolynomial_iff` states that `PrimeSpectrum (MvPolynomial \u03c3 R)` is irreducible exactly when `PrimeSpectrum R` is irreducible. Coefficients and variables have independent universes; empty or infinite variables\u2026","title":"Nilpotent coefficients and irreducible polynomial spectra"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01/jacobson-radicals-of-coefficient-ideals","kind":"node","lean":"Ideal.jacobson_map_mvPolynomial_C","parent":8849,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-01/index.html#jacobson-radicals-of-coefficient-ideals","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-01","status":"fully_proved","summary":"For any commutative ring `R`, ideal `I` and nonempty variable type `\u03c3`, `Ideal.jacobson_map_mvPolynomial_C` identifies `Ideal.jacobson (I.map C)` with `I.radical.map C`. In particular, the Jacobson radical of zero in `MvPolynomial \u03c3 R` is its nilradical. Coefficient and variable universes are independent; no Jacobson-ring, Noetherian, properness or nontriviality hypothesis is needed. The integer example separates\u2026","title":"Jacobson radicals of coefficient ideals"},{"data_url":"scopes/formal-frontier/commutative-algebra/multivariate-polynomials/part-02.json","id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02","kind":"scope","navigation":"transparent","parent":8848,"result":false,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-02/index.html","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02","status":"fully_proved","title":"Integral elements over\u2026 to Fraction-ring renaming is\u2026"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02/integral-elements-over-coefficient-rings","kind":"node","lean":"MvPolynomial.isIntegral_iff_coeff_isNilpotent","parent":8870,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-02/index.html#integral-elements-over-coefficient-rings","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02","status":"fully_proved","summary":"For any commutative ring `R` and arbitrary variable type `\u03c3`, `MvPolynomial.isIntegral_iff_coeff_isNilpotent` characterizes `IsIntegral R p` by nilpotence of every nonconstant coefficient. For reduced `R`, `MvPolynomial.isIntegral_iff_eq_C_coeff_zero` specializes this to `p = C (p.coeff 0)`. The coefficient and variable universes are independent; trivial coefficient rings and empty or infinite variable types are\u2026","title":"Integral elements over coefficient rings"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02/integral-closure-in-arbitrarily-many-variables","kind":"node","lean":"MvPolynomial.isIntegrallyClosed","parent":8870,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-02/index.html#integral-closure-in-arbitrarily-many-variables","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02","status":"fully_proved","summary":"For every integrally closed commutative domain `R` and arbitrary variable type `\u03b9`, `MvPolynomial.isIntegrallyClosed` makes the ordinary polynomial ring `MvPolynomial \u03b9 R` integrally closed. The coefficient and variable universes are independent, and the variables may be empty or infinite. No Noetherian or unique-factorization hypothesis is imposed on `R`.","title":"Integral closure in arbitrarily many variables"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02/flatness-of-commutative-ring-variable-renaming","kind":"node","lean":"MvPolynomial.rename_flat_iff","parent":8870,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-02/index.html#flatness-of-commutative-ring-variable-renaming","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02","status":"fully_proved","summary":"For every commutative ring `R` and arbitrary `f : \u03c3 \u2192 \u03c4`, with coefficient and variable types in independent universes, `MvPolynomial.rename_flat_iff` characterizes flatness of the full ring homomorphism `MvPolynomial.rename (R := R) f` by injectivity of `f` or subsingleton coefficients. The scalar action is induced by this original polynomial-ring map, not just by `R` or by its image. Injective maps give free\u2026","title":"Flatness of commutative-ring variable renaming"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02/canonical-unique-variable-degree","kind":"node","lean":"MvPolynomial.natDegree_uniqueAlgEquiv","parent":8870,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-02/index.html#canonical-unique-variable-degree","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02","status":"fully_proved","summary":"For any commutative semiring `R`, unique variable type `\u03c3` (`[Unique \u03c3]`), and `p : MvPolynomial \u03c3 R`, `MvPolynomial.natDegree_uniqueAlgEquiv` identifies `Polynomial.natDegree (MvPolynomial.uniqueAlgEquiv R \u03c3 p)` with `p.totalDegree`. The coefficient and variable universes are independent. No nonzero-polynomial or nontrivial-coefficient premise is needed: both natural-number degrees assign zero to the zero\u2026","title":"Canonical unique-variable degree"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02/injectivity-and-surjectivity-of-variable-renamin","kind":"node","lean":"MvPolynomial.injective_rename_iff","parent":8870,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-02/index.html#injectivity-and-surjectivity-of-variable-renamin","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02","status":"fully_proved","summary":"For any commutative semiring `R` and any map `f : \u03c3 \u2192 \u03c4`, with the coefficient, source-variable and target-variable types in independent universes, the canonical renaming map from the full polynomial semiring `MvPolynomial \u03c3 R` to the full polynomial semiring `MvPolynomial \u03c4 R` is injective exactly when `f` is injective or `R` is subsingleton (`MvPolynomial.injective_rename_iff`), and is surjective exactly when `f`\u2026","title":"Injectivity and surjectivity of variable renaming"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02/formal-smoothness-of-variable-renaming","kind":"node","lean":"MvPolynomial.rename_formallySmooth_iff_injective","parent":8870,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-02/index.html#formal-smoothness-of-variable-renaming","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02","status":"fully_proved","summary":"Over any commutative ring `R`, `MvPolynomial.rename_formallySmooth_iff_injective` characterizes formal smoothness of the full source polynomial-ring homomorphism by injectivity of that homomorphism. Equivalently, `MvPolynomial.rename_formallySmooth_iff` says that the variable map is injective or the coefficients are subsingleton. The scalar action is induced by the original renaming, not just by `R`. The\u2026","title":"Formal smoothness of variable renaming"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02/smoothness-of-variable-renaming","kind":"node","lean":"MvPolynomial.rename_smooth_iff","parent":8870,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-02/index.html#smoothness-of-variable-renaming","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02","status":"fully_proved","summary":"Over any commutative ring `R`, `MvPolynomial.rename_smooth_iff` says that the canonical homomorphism on full polynomial rings is smooth exactly when the variable map is injective with finite complement, or `R` is subsingleton. This is stock `RingHom.Smooth`, including finite presentation, for the source-polynomial action induced by renaming. Coefficients and both variable types have independent universes; no\u2026","title":"Smoothness of variable renaming"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02/formal-unramifiedness-of-variable-renaming","kind":"node","lean":"MvPolynomial.formallyUnramified_rename_iff_surjective","parent":8870,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-02/index.html#formal-unramifiedness-of-variable-renaming","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02","status":"fully_proved","summary":"Over any commutative ring `R`, `MvPolynomial.formallyUnramified_rename_iff_surjective` says that `MvPolynomial.rename f`, as a homomorphism from the full source polynomial ring, is formally unramified exactly when it is surjective: equivalently, when `f` is surjective or `R` is subsingleton. The coefficient and variable types may lie in independent universes, and neither variable type must be finite or inhabited.\u2026","title":"Formal unramifiedness of variable renaming"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02/integrality-of-polynomial-variable-renaming","kind":"node","lean":"MvPolynomial.isIntegral_rename_iff","parent":8870,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-02/index.html#integrality-of-polynomial-variable-renaming","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02","status":"fully_proved","summary":"Over any commutative ring `R`, `MvPolynomial.isIntegral_rename_iff` states that renaming between the full polynomial rings is integral exactly when the variable map is surjective or the coefficient ring is subsingleton. See the integrality section for the precise map, generality and applications.","title":"Integrality of polynomial variable renaming"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02/algebraicity-of-polynomial-variable-renaming","kind":"node","lean":"MvPolynomial.isAlgebraic_rename_iff","parent":8870,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-02/index.html#algebraicity-of-polynomial-variable-renaming","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02","status":"fully_proved","summary":"Over any commutative ring `R`, `MvPolynomial.isAlgebraic_rename_iff` states that the full target polynomial ring is algebraic over the full source via the specified renaming exactly when the coefficients are nontrivial and the variable map is surjective or noninjective. A collision gives a nonzero constant annihilator, even when a target variable is missed; zero coefficients give no such polynomial, even for\u2026","title":"Algebraicity of polynomial variable renaming"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02/finite-presentation-of-commutative-ring-variable","kind":"node","lean":"MvPolynomial.finitePresentation_rename_iff","parent":8870,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-02/index.html#finite-presentation-of-commutative-ring-variable","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02","status":"fully_proved","summary":"`MvPolynomial.finitePresentation_rename_iff` states the relative finite-presentation criterion: subsingleton coefficients, or finitely many missed target variables together with finitely many source collision participants. Both collision endpoints are counted. See the finite-presentation section for the coefficient-ring boundary and infinite-variable examples.","title":"Finite presentation of commutative-ring variable renaming"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02/zero-variable-presentations-and-semiring-renamin","kind":"node","lean":"Algebra.FinitePresentation.of_surjective_algebraMap","parent":8870,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-02/index.html#zero-variable-presentations-and-semiring-renamin","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02","status":"fully_proved","summary":"`Algebra.FinitePresentation.of_surjective_algebraMap` gives a zero-variable presentation for a surjective scalar map from a commutative semiring to any semiring, with finitely generated ideal kernel. For commutative-semiring coefficients with subsingleton additive units, renaming reflects zero and every surjective variable map is finitely presented over its full source polynomial semiring. No cancellation, order,\u2026","title":"Zero-variable presentations and semiring renaming"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02/variable-renaming-has-an-exact-relative-finite-t","kind":"node","lean":"MvPolynomial.finiteType_rename_iff","parent":8870,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-02/index.html#variable-renaming-has-an-exact-relative-finite-t","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02","status":"fully_proved","summary":"`MvPolynomial.finiteType_rename_iff` characterizes finite generation of the target polynomial algebra over the source polynomial semiring: the coefficients are subsingleton or the set of missed target variables is finite. See the relative finite-type section for the algebra structure, generality and infinite-variable applications.","title":"Variable renaming has an exact relative finite-type criterion"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02/variable-renaming-has-an-exact-relative-essentia","kind":"node","lean":"MvPolynomial.essFiniteType_rename_iff","parent":8870,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-02/index.html#variable-renaming-has-an-exact-relative-essentia","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02","status":"fully_proved","summary":"`MvPolynomial.essFiniteType_rename_iff` characterizes `RingHom.EssFiniteType` of renaming for arbitrary `f : \u03c3 \u2192 \u03c4` over a commutative ring `R`, with independent coefficient and variable universes: `R` is subsingleton or `Set.Finite ((Set.range f)\u1d9c)`. Scalars are `MvPolynomial \u03c3 R`, acting on `MvPolynomial \u03c4 R` by `p \u2022 q = rename f p * q`. Essential finite type is distinct from finite type, but their criteria\u2026","title":"Variable renaming has an exact relative essentially finite-type criterion"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02/module-finiteness-of-variable-renaming-has-a-sha","kind":"node","lean":"MvPolynomial.moduleFinite_rename_iff","parent":8870,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-02/index.html#module-finiteness-of-variable-renaming-has-a-sha","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02","status":"fully_proved","summary":"For any commutative semiring `R` and any map `f : \u03c3 \u2192 \u03c4`, with all three types in independent universes, `MvPolynomial.moduleFinite_rename_iff` states that the full target polynomial semiring is a finite module over the full source polynomial semiring under `p \u2022 q = MvPolynomial.rename f p * q` exactly when `f` is surjective or `R` is subsingleton. This is stronger than finite type: over nontrivial coefficients,\u2026","title":"Module finiteness of variable renaming has a sharper criterion"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02/quasi-finiteness-of-commutative-ring-variable-re","kind":"node","lean":"MvPolynomial.rename_quasiFinite_iff","parent":8870,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-02/index.html#quasi-finiteness-of-commutative-ring-variable-re","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02","status":"fully_proved","summary":"For any commutative ring `R` and arbitrary `f : \u03c3 \u2192 \u03c4`, with all three types in independent universes, `MvPolynomial.rename_quasiFinite_iff` states that the original full polynomial-ring map is quasi-finite exactly when `f` is surjective or `R` is subsingleton. It uses Mathlib's residue-field-fiber convention, without a `FiniteType` hypothesis, and the full source-polynomial action. Module finiteness supplies the\u2026","title":"Quasi-finiteness of commutative-ring variable renaming"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02/finite-coefficient-aggregation-has-an-exact-inje","kind":"node","lean":"Finsupp.mapDomain_injective_iff","parent":8870,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-02/index.html#finite-coefficient-aggregation-has-an-exact-inje","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02","status":"fully_proved","summary":"For types `\u03b1`, `\u03b2` and `M` in independent universes, an additive commutative monoid `M`, and any index map `f : \u03b1 \u2192 \u03b2`, `Finsupp.mapDomain_injective_iff` characterizes injectivity of `Finsupp.mapDomain f : (\u03b1 \u2192\u2080 M) \u2192 (\u03b2 \u2192\u2080 M)`: it holds exactly when `f` is injective or `M` is subsingleton. The map adds coefficients whose indices have the same image; cancellation, a ring structure and nontrivial coefficients are not\u2026","title":"Finite coefficient aggregation has an exact injectivity criterion"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02/joint-degree-and-support-profiles-recover-bivari","kind":"node","lean":"Finsupp.mapDomain_degree_support_hasLeftInverse_iff","parent":8870,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-02/index.html#joint-degree-and-support-profiles-recover-bivari","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02","status":"fully_proved","summary":"For any additive commutative monoid `M` in an arbitrary universe, `Finsupp.mapDomain_degree_support_hasLeftInverse_iff` proves that pushing coefficients along `d \u21a6 (d.degree, d.support.card)`, for `d : Fin 2 \u2192\u2080 \u2115`, admits a total left inverse exactly when `M` is subsingleton. The profile groups coefficients jointly by total degree and the number of variables with positive exponent. The decoder may be any total\u2026","title":"Joint degree-and-support profiles recover bivariate coefficients only for trivial coefficients"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02/variable-renaming-kernels-have-an-exact-presenta","kind":"node","lean":"MvPolynomial.ker_rename_eq_span_X_sub_X","parent":8870,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-02/index.html#variable-renaming-kernels-have-an-exact-presenta","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02","status":"fully_proved","summary":"For any commutative ring `R` and any map `f : \u03c3 \u2192 \u03c4`, with all three types in independent universes, `MvPolynomial.ker_rename_eq_span_X_sub_X` identifies the kernel of `MvPolynomial.rename (R := R) f` with the ideal generated by `X i - X j` for pairs satisfying `f i = f j`. Moreover, `MvPolynomial.ker_rename_fg_iff` says that this ideal is finitely generated exactly when `R` is subsingleton or the set\u2026","title":"Variable-renaming kernels have an exact presentation and finite-generation criterion"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02/fraction-ring-renaming-is-surjective-equivalentl","kind":"node","lean":"MvPolynomial.surjective_fractionRing_map_rename_iff","parent":8870,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-02/index.html#fraction-ring-renaming-is-surjective-equivalentl","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-02","status":"fully_proved","summary":"For any integral domain `R` (`[CommRing R] [IsDomain R]`) and injective `\u03c1 : \u03c4 \u2192 \u03c3`, `MvPolynomial.surjective_fractionRing_map_rename_iff` characterizes surjectivity of the induced fraction-ring map by `Function.Surjective \u03c1`. Its source `L` and target `K` may be any commutative-ring fraction-ring realizations with their `Algebra` and `IsFractionRing` instances, without `Field` instances. The five universes of `R`,\u2026","title":"Fraction-ring renaming is surjective, equivalently integral, exactly when it covers all variables"},{"data_url":"scopes/formal-frontier/commutative-algebra/multivariate-polynomials/part-03.json","id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03","kind":"scope","navigation":"transparent","parent":8848,"result":false,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-03/index.html","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03","status":"fully_proved","summary":"*Quoted from the headline results in README.md of FormalFrontier/multivariate-polynomials at commit 724d9804, Apache-2.0.*","title":"Omitted variables are\u2026 to Comparison with Lagrange\u2026"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03/omitted-variables-are-algebraically-independent","kind":"node","lean":"MvPolynomial.algebraicIndependent_fractionRing_map_rename_compl","parent":8891,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-03/index.html#omitted-variables-are-algebraically-independent","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03","status":"fully_proved","summary":"For a commutative domain `R`, an injective map `\u03c1 : \u03c4 \u2192 \u03c3`, and arbitrary commutative-ring fraction rings `L` of `MvPolynomial \u03c4 R` and `K` of `MvPolynomial \u03c3 R`, `MvPolynomial.algebraicIndependent_fractionRing_map_rename_compl` proves that the images of `X i` in `K`, indexed by `{i : \u03c3 // i \u2209 Set.range \u03c1}`, are algebraically independent over `L`. The `L`-algebra structure on `K` must be induced by the same\u2026","title":"Omitted variables are algebraically independent over the source fraction ring"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03/omitted-variables-form-a-transcendence-basis-ove","kind":"node","lean":"MvPolynomial.isAlgebraic_adjoin_fractionRing_map_rename_compl","parent":8891,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-03/index.html#omitted-variables-form-a-transcendence-basis-ove","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03","status":"fully_proved","summary":"In the same five-universe commutative-domain setting and under the full canonical fraction-renaming scalar action, `MvPolynomial.isAlgebraic_adjoin_fractionRing_map_rename_compl` proves that `K` is algebraic over the `L`-subalgebra generated by the complementary localized variables, and `MvPolynomial.isTranscendenceBasis_fractionRing_map_rename_compl` identifies that family as a transcendence basis over `L`. Under\u2026","title":"Omitted variables form a transcendence basis over the source fraction ring"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03/the-image-of-evaluating-one-polynomial-at-two-po","kind":"node","lean":"MvPolynomial.mem_range_eval_prod_iff","parent":8891,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-03/index.html#the-image-of-evaluating-one-polynomial-at-two-po","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03","status":"fully_proved","summary":"Over any commutative ring `R`, for assignments `a b : \u03c3 \u2192 R` with independent coefficient and variable universes, `MvPolynomial.mem_range_eval_prod_iff` characterizes value pairs `(r, s) : R \u00d7 R` of one common polynomial by `r - s \u2208 Ideal.span (Set.range (fun i => a i - b i))`, without a finite-variable, field or comaximality assumption. The examples contrast same-parity pairs at the one-variable integer points `0`\u2026","title":"The image of evaluating one polynomial at two points"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03/finite-simultaneous-polynomial-evaluation-has-an","kind":"node","lean":"MvPolynomial.surjective_eval_family_iff","parent":8891,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-03/index.html#finite-simultaneous-polynomial-evaluation-has-an","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03","status":"fully_proved","summary":"For any commutative ring `R`, arbitrary variable type `\u03c3`, finite point-index type `\u03b9` and family `a : \u03b9 \u2192 \u03c3 \u2192 R`, `MvPolynomial.surjective_eval_family_iff` states that `p \u21a6 (i \u21a6 MvPolynomial.eval (a i) p)` is surjective exactly when the coordinate differences `a i s - a j s`, for all `s : \u03c3`, generate the unit ideal for every pair of distinct indices `i \u2260 j`. The three types have independent universes. Empty\u2026","title":"Finite simultaneous polynomial evaluation has an exact surjectivity criterion"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03/evaluation-kernel-powers-are-detected-by-transla","kind":"node","lean":"MvPolynomial.mem_pow_ker_eval_addUnits_iff","parent":8891,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-03/index.html#evaluation-kernel-powers-are-detected-by-transla","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03","status":"fully_proved","summary":"For any commutative semiring `R` and point `a : \u03c3 \u2192 AddUnits R`, `MvPolynomial.mem_pow_ker_eval_addUnits_iff` characterizes membership in every `n`th evaluation-kernel power by vanishing of all coefficients of total degree `< n` after the translation `p(X + a)`. Over a commutative ring, `MvPolynomial.mem_pow_ker_eval_iff` allows arbitrary points `a : \u03c3 \u2192 R`. Both results allow arbitrary variable types, independent\u2026","title":"Evaluation-kernel powers are detected by translated coefficients"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03/hilbert-basis-theorem-for-left-noetherian-rings","kind":"node","lean":"Polynomial.isNoetherianRing_of_isNoetherianRing","parent":8891,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-03/index.html#hilbert-basis-theorem-for-left-noetherian-rings","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03","status":"fully_proved","summary":"`Polynomial.isNoetherianRing_of_isNoetherianRing` proves `IsNoetherianRing (Polynomial R)` for any `R : Type u` with `[Ring R]` and `[IsNoetherianRing R]`. The polynomial variable is central; coefficient commutativity, nontriviality and absence of zero divisors are not required. On commutative rings, `Polynomial.isNoetherianRing_of_isNoetherianRing_eq_isNoetherianRing` identifies this proof with Mathlib's\u2026","title":"Hilbert basis theorem for left-Noetherian rings"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03/artinian-polynomial-semiring-criteria","kind":"node","lean":"Polynomial.isArtinianRing_iff","parent":8891,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-03/index.html#artinian-polynomial-semiring-criteria","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03","status":"fully_proved","summary":"For any `R : Type u` with `[Semiring R]`, including noncommutative coefficients, `Polynomial.isArtinianRing_iff` proves `IsArtinianRing (Polynomial R) \u2194 Subsingleton R`. For arbitrary variable types `\u03c3 : Type u` and coefficients `R : Type v` with `[CommSemiring R]`, `MvPolynomial.isArtinianRing_iff` proves `IsArtinianRing (MvPolynomial \u03c3 R) \u2194 IsArtinianRing R \u2227 (IsEmpty \u03c3 \u2228 Subsingleton R)`. Both use polynomial\u2026","title":"Artinian polynomial-semiring criteria"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03/univariate-coefficient-module-noetherianity-and","kind":"node","lean":"Polynomial.isNoetherian_iff","parent":8891,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-03/index.html#univariate-coefficient-module-noetherianity-and","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03","status":"fully_proved","summary":"For any `R : Type u` with `[Semiring R]`, `Polynomial.isNoetherian_iff` and `Polynomial.isArtinian_iff` give `IsNoetherian R (Polynomial R) \u2194 Subsingleton R` and `IsArtinian R (Polynomial R) \u2194 Subsingleton R`, respectively. Both use the canonical left coefficient action `r \u2022 p = Polynomial.C r * p`, not polynomial multiplication as a self-module. The universe is arbitrary; no commutativity, ring, domain or field\u2026","title":"Univariate coefficient-module Noetherianity and Artinianity each characterize trivial coefficients"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03/radical-coefficient-ideal-extensions-are-charact","kind":"node","lean":"MvPolynomial.isRadical_map_C_iff","parent":8891,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-03/index.html#radical-coefficient-ideal-extensions-are-charact","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03","status":"fully_proved","summary":"For any commutative semiring `R`, ideal `I`, and nonempty variable type `\u03c3`, `MvPolynomial.isRadical_map_C_iff` says that `I.map C` is radical exactly when `I` is radical and satisfies `a \u2208 I \u2192 a + b \u2208 I \u2192 b \u2208 I`. The sufficiency theorem also allows empty variable types. The coefficient and variable universes are independent; no properness, ring, domain, reducedness or finiteness hypothesis is needed. The examples\u2026","title":"Radical coefficient-ideal extensions are characterized by subtractivity"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03/coefficient-and-variable-ideals-have-an-exact-fi","kind":"node","lean":"MvPolynomial.ideal_map_C_sup_span_X_image_fg_iff","parent":8891,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-03/index.html#coefficient-and-variable-ideals-have-an-exact-fi","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03","status":"fully_proved","summary":"For any commutative semiring `R`, coefficient ideal `I`, and variable subset `S`, `MvPolynomial.ideal_map_C_sup_span_X_image_fg_iff` characterizes finite generation of `Ideal.map C I \u2294 Ideal.span (X '' S)`: `I` must be finitely generated, and either `S` is finite or `I` is the unit ideal. The coefficient and variable universes are independent; no ring, domain, subtractivity or finite-ambient-variable hypothesis is\u2026","title":"Coefficient-and-variable ideals have an exact finite-generation criterion"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03/the-variable-ideal-is-not-finitely-generated","kind":"node","lean":"MvPolynomial.idealOfVars_not_fg","parent":8891,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-03/index.html#the-variable-ideal-is-not-finitely-generated","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03","status":"fully_proved","summary":"For a nontrivial commutative semiring and infinitely many variables, `MvPolynomial.idealOfVars_not_fg` proves non-finite generation. A finite union of polynomial supports misses some variable; evaluation separating that variable gives a contradiction. Mathlib supplies the complementary *finite*-variable ideal theorem. This library does not prove the motivating complete short exact sequence.","title":"The variable ideal is not finitely generated"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03/antichain-coefficients-of-monomial-ideals","kind":"node","lean":"MvPolynomial.monomialIdealCoeffMap","parent":8891,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-03/index.html#antichain-coefficients-of-monomial-ideals","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03","status":"fully_proved","summary":"For any commutative semiring `R`, arbitrary variable type `\u03c3` and antichain `S : Set (\u03c3 \u2192\u2080 \u2115)`, `MvPolynomial.monomialIdealCoeffMap` restricts coefficients from the stock ideal generated by the monic monomials with exponents in `S` to `S \u2192\u2080 R`. It is surjective and semilinear over `constantCoeff`, for the full polynomial-semiring action on the ideal and the coefficient action on the target. Its coordinate and\u2026","title":"Antichain coefficients of monomial ideals"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03/variable-classes-form-the-augmentation-cotangent","kind":"node","lean":"MvPolynomial.idealOfVarsCotangentBasis","parent":8891,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-03/index.html#variable-classes-form-the-augmentation-cotangent","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03","status":"fully_proved","summary":"For any `(\u03c3 : Type u)` and `(R : Type v) [CommRing R]`, `MvPolynomial.idealOfVarsCotangentBasis` is a `\u03c3`-indexed `R`-basis of the stock variable-ideal cotangent `I/I\u00b2`. Its vectors are the classes of `X i`, with inherited coefficient action `r \u2022 [p] = [C r * p]`. The `_apply` and `_repr_X` lemmas characterize those vectors, and `idealOfVarsCotangentBasis_repr_toCotangent` computes each coordinate as\u2026","title":"Variable classes form the augmentation cotangent basis"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03/degree-controlled-finite-interpolation","kind":"node","lean":"MvPolynomial.exists_eval_eq_totalDegree_le_of_pairwise_span_eq_top","parent":8891,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-03/index.html#degree-controlled-finite-interpolation","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03","status":"fully_proved","summary":"Over any commutative ring `R`, let `s : Finset \u03b9` select points `a : \u03b9 \u2192 \u03c3 \u2192 R`. If, for every distinct `i, j \u2208 s`, the coordinate differences `a i x - a j x` span the unit ideal, then any prescribed values `v : \u03b9 \u2192 R` are attained on `s` by a polynomial of total degree at most `s.card - 1` (`MvPolynomial.exists_eval_eq_totalDegree_le_of_pairwise_span_eq_top`). Both the variable type and ambient label type may be\u2026","title":"Degree-controlled finite interpolation"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03/comparison-with-lagrange-interpolation","kind":"node","lean":"MvPolynomial.uniqueAlgEquiv_eq_interpolate_of_eval_eq","parent":8891,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/part-03/index.html#comparison-with-lagrange-interpolation","scope":"formal-frontier/commutative-algebra/multivariate-polynomials/part-03","status":"fully_proved","summary":"Over a field, for a unique variable type, a polynomial taking prescribed values at distinct nodes and having total degree strictly below the number of nodes maps to Mathlib's Lagrange interpolant under the canonical single-variable equivalence (`MvPolynomial.uniqueAlgEquiv_eq_interpolate_of_eval_eq`). The univariate degree bound follows from the total-degree bound. On a nonempty selection, the bound `s.card - 1`\u2026","title":"Comparison with Lagrange interpolation"},{"id":"formal-frontier/commutative-algebra/multivariate-polynomials/main-results","kind":"node","parent":8848,"result":false,"route":"../roadmap/formal-frontier/commutative-algebra/multivariate-polynomials/main-results/index.html","scope":"formal-frontier/commutative-algebra/multivariate-polynomials","status":"can_state","summary":"The main results that the formalization.yaml of FormalFrontier/multivariate-polynomials lists under `status.main_results` at commit 724d9804, each that the scan finds declared in the file the yaml gives, with its line. They are listed here, not made articles: in several Formal Frontier libraries this list is close to the whole API. The notes are the yaml's.","title":"Main results"},{"id":"formal-frontier/commutative-algebra/smooth-regularity","kind":"node","parent":8708,"result":false,"route":"../roadmap/formal-frontier/commutative-algebra/smooth-regularity/index.html","scope":"formal-frontier/commutative-algebra","status":"can_state","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/smooth-regularity at commit fbc0ee82, Apache-2.0.*","title":"Smooth regularity"},{"data_url":"scopes/formal-frontier/commutative-algebra/toric-ideals.json","id":"formal-frontier/commutative-algebra/toric-ideals","kind":"scope","parent":8708,"result":false,"route":"../roadmap/formal-frontier/commutative-algebra/toric-ideals/index.html","scope":"formal-frontier/commutative-algebra/toric-ideals","status":"fully_proved","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/toric-ideals at commit 593116e0, Apache-2.0.*","title":"Toric ideals"},{"id":"formal-frontier/commutative-algebra/toric-ideals/uniform-monomial-evaluation-kernels","kind":"node","lean":"MvPolynomial.uniform_monomial_kernel","parent":8909,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/toric-ideals/index.html#uniform-monomial-evaluation-kernels","scope":"formal-frontier/commutative-algebra/toric-ideals","status":"fully_proved","summary":"For any map from a finite variable type to a commutative monoid, one finite set of same-product exponent pairs gives coefficient-one monomial differences spanning the kernel of monomial evaluation in MvPolynomial over every commutative coefficient ring, including empty variable types and zero rings, with all three universes independent.","title":"Uniform monomial-evaluation kernels"},{"id":"formal-frontier/commutative-algebra/toric-ideals/monomial-conductors","kind":"node","lean":"AddSubmonoid.exists_relative_root_conductor","parent":8909,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/toric-ideals/index.html#monomial-conductors","scope":"formal-frontier/commutative-algebra/toric-ideals","status":"fully_proved","summary":"Every finitely generated additive submonoid S of finitely supported natural exponent vectors contains c such that c+n belongs to S whenever n is a difference of elements of S and has a positive multiple in S.","title":"Monomial conductors"},{"id":"formal-frontier/commutative-algebra/toric-ideals/own-fraction-normalization-of-monomial-algebras","kind":"node","lean":"MvPolynomial.map_integralClosure_adjoin_monomial","parent":8909,"result":true,"route":"../roadmap/formal-frontier/commutative-algebra/toric-ideals/index.html#own-fraction-normalization-of-monomial-algebras","scope":"formal-frontier/commutative-algebra/toric-ideals","status":"fully_proved","summary":"For a monomial algebra in arbitrarily many variables over an integrally closed commutative coefficient domain, its normalization in its own fraction ring has canonical polynomial-fraction-field image generated by monomials whose exponents are differences of submonoid elements and have a positive multiple in the submonoid.","title":"Own-fraction normalization of monomial algebras"},{"id":"formal-frontier/commutative-algebra/toric-ideals/main-results","kind":"node","parent":8909,"result":false,"route":"../roadmap/formal-frontier/commutative-algebra/toric-ideals/main-results/index.html","scope":"formal-frontier/commutative-algebra/toric-ideals","status":"can_state","summary":"The main results that the formalization.yaml of FormalFrontier/toric-ideals lists under `status.main_results` at commit 593116e0, each that the scan finds declared in the file the yaml gives, with its line. They are listed here, not made articles: in several Formal Frontier libraries this list is close to the whole API. The notes are the yaml's.","title":"Main results"},{"data_url":"scopes/formal-frontier/ring-theory.json","id":"formal-frontier/ring-theory","kind":"scope","parent":8707,"result":false,"route":"../roadmap/formal-frontier/ring-theory/index.html","scope":"formal-frontier/ring-theory","status":"fully_proved","summary":"The Formal Frontier libraries on ring theory, one section each.","title":"Ring theory"},{"data_url":"scopes/formal-frontier/ring-theory/group-rings.json","id":"formal-frontier/ring-theory/group-rings","kind":"scope","parent":8914,"result":false,"route":"../roadmap/formal-frontier/ring-theory/group-rings/index.html","scope":"formal-frontier/ring-theory/group-rings","status":"fully_proved","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/group-rings at commit 8e90e656, Apache-2.0.*","title":"Group rings"},{"id":"formal-frontier/ring-theory/group-rings/convolution-support","kind":"node","lean":"MonoidAlgebra.one_lt_card_support_coeff_mul","parent":8915,"result":true,"route":"../roadmap/formal-frontier/ring-theory/group-rings/index.html#convolution-support","scope":"formal-frontier/ring-theory/group-rings","status":"fully_proved","summary":"Over a no-zero-divisor semiring and a magma with two unique products, a product of input support cardinalities greater than one forces the convolution support cardinality to be greater than one. Coefficients need not commute, and no monoid or nontriviality assumption is imposed. See `MonoidAlgebra.one_lt_card_support_coeff_mul` and the precise support bound.","title":"Convolution support"},{"data_url":"scopes/formal-frontier/ring-theory/group-rings/main-results.json","id":"formal-frontier/ring-theory/group-rings/main-results","kind":"scope","parent":8915,"result":false,"route":"../roadmap/formal-frontier/ring-theory/group-rings/main-results/index.html","scope":"formal-frontier/ring-theory/group-rings/main-results","status":"planned","summary":"The main results that the formalization.yaml of FormalFrontier/group-rings lists under `status.main_results` at commit 8e90e656, each that the scan finds declared in the file the yaml gives, with its line. They are listed here, not made articles: in several Formal Frontier libraries this list is close to the whole API. The notes are the yaml's.","title":"Main results"},{"id":"formal-frontier/ring-theory/group-rings/main-results/part-01","kind":"node","parent":8917,"result":false,"route":"../roadmap/formal-frontier/ring-theory/group-rings/main-results/part-01/index.html","scope":"formal-frontier/ring-theory/group-rings/main-results","status":"can_state","summary":"*Quoted from the main results in formalization.yaml of FormalFrontier/group-rings at commit 8e90e656, Apache-2.0.*","title":"Main results 1 to 100"},{"id":"formal-frontier/ring-theory/group-rings/main-results/part-02","kind":"node","parent":8917,"result":false,"route":"../roadmap/formal-frontier/ring-theory/group-rings/main-results/part-02/index.html","scope":"formal-frontier/ring-theory/group-rings/main-results","status":"can_state","summary":"*Quoted from the main results in formalization.yaml of FormalFrontier/group-rings at commit 8e90e656, Apache-2.0.*","title":"Main results 101 to 117"},{"data_url":"scopes/formal-frontier/ring-theory/projective-modules.json","id":"formal-frontier/ring-theory/projective-modules","kind":"scope","parent":8914,"result":false,"route":"../roadmap/formal-frontier/ring-theory/projective-modules/index.html","scope":"formal-frontier/ring-theory/projective-modules","status":"fully_proved","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/projective-modules at commit 27f95a8b, Apache-2.0.*","title":"Projective modules"},{"id":"formal-frontier/ring-theory/projective-modules/sylvesters-determinant-identity-for-finite-proje","kind":"node","lean":"Module.determinant_one_add_comp_comm","parent":8920,"result":true,"route":"../roadmap/formal-frontier/ring-theory/projective-modules/index.html#sylvesters-determinant-identity-for-finite-proje","scope":"formal-frontier/ring-theory/projective-modules","status":"fully_proved","summary":"For finite projective modules P and Q over a commutative ring R and R-linear maps f : P \u2192 Q and g : Q \u2192 P, det(1 + g \u2218 f) = det(1 + f \u2218 g).","title":"Sylvester's determinant identity for finite projective modules"},{"id":"formal-frontier/ring-theory/projective-modules/tensor-product-determinants-and-rank-powers","kind":"node","lean":"Module.determinant_tensorProduct_map, Module.rankPower_tensorProduct","parent":8920,"result":true,"route":"../roadmap/formal-frontier/ring-theory/projective-modules/index.html#tensor-product-determinants-and-rank-powers","scope":"formal-frontier/ring-theory/projective-modules","status":"fully_proved","summary":"For finite projective R-modules P and Q over a commutative ring R, det(f \u2297 g) = rankPower_Q(det(f)) \u00b7 rankPower_P(det(g)) for all endomorphisms f of P and g of Q, and the rank-power map of P \u2297 Q is rankPower_P \u2218 rankPower_Q.","title":"Tensor-product determinants and rank powers"},{"id":"formal-frontier/ring-theory/projective-modules/projective-modules-lift-across-nilpotent-ideals","kind":"node","lean":"Module.exists_projective_lift, Module.exists_finite_projective_lift","parent":8920,"result":true,"route":"../roadmap/formal-frontier/ring-theory/projective-modules/index.html#projective-modules-lift-across-nilpotent-ideals","scope":"formal-frontier/ring-theory/projective-modules","status":"fully_proved","summary":"For any ring R and nilpotent two-sided ideal I, every projective R/I-module is isomorphic to the canonical reduction of a projective R-module. Finitely generated inputs admit finitely generated lifts. Neither commutativity nor a finite-generation assumption on I is needed.","title":"Projective modules lift across nilpotent ideals"},{"data_url":"scopes/formal-frontier/ring-theory/projective-modules/headline-results.json","id":"formal-frontier/ring-theory/projective-modules/headline-results","kind":"scope","parent":8920,"result":false,"route":"../roadmap/formal-frontier/ring-theory/projective-modules/headline-results/index.html","scope":"formal-frontier/ring-theory/projective-modules/headline-results","status":"fully_proved","summary":"*Quoted from the headline results in README.md of FormalFrontier/projective-modules at commit 27f95a8b, Apache-2.0.*","title":"Headline results"},{"id":"formal-frontier/ring-theory/projective-modules/headline-results/algebra-valued-pairing-matrices","kind":"node","lean":"LinearMap.toMatrix\u2082_compl\u2081\u2082_algebra","parent":8924,"result":true,"route":"../roadmap/formal-frontier/ring-theory/projective-modules/headline-results/index.html#algebra-valued-pairing-matrices","scope":"formal-frontier/ring-theory/projective-modules/headline-results","status":"fully_proved","summary":"For finite based modules over a commutative semiring `R` and a semiring `R`-algebra `S`, `LinearMap.toMatrix\u2082_compl\u2081\u2082_algebra` expresses composition of an `S`-valued `R`-bilinear map by coefficient-matrix multiplication after `algebraMap R S`. The target algebra need not be commutative, finite or projective over `R`; the identity-algebra formula recovers the scalar case. For integer-bilinear pairings into a\u2026","title":"Algebra-valued pairing matrices"},{"id":"formal-frontier/ring-theory/projective-modules/headline-results/normalized-pairing-regulators","kind":"node","lean":"LinearMap.pairingRegulator","parent":8924,"result":true,"route":"../roadmap/formal-frontier/ring-theory/projective-modules/headline-results/index.html#normalized-pairing-regulators","scope":"formal-frontier/ring-theory/projective-modules/headline-results","status":"fully_proved","summary":"For finitely generated abelian groups of equal integer-module rank and a real-valued integer-bilinear pairing, `LinearMap.pairingRegulator` is the absolute determinant on independent full frames divided by the product of their spans' indices in the original groups, including torsion. Frame invariance and arbitrary-frame evaluation require neither freeness of the original groups nor nondegeneracy; zero determinants\u2026","title":"Normalized pairing regulators"},{"id":"formal-frontier/ring-theory/projective-modules/headline-results/nilpotence-from-characteristic-polynomials","kind":"node","lean":"Module.isNilpotent_of_isNilpotent_charpoly_sub_charpoly_zero","parent":8924,"result":true,"route":"../roadmap/formal-frontier/ring-theory/projective-modules/headline-results/index.html#nilpotence-from-characteristic-polynomials","scope":"formal-frontier/ring-theory/projective-modules/headline-results","status":"fully_proved","summary":"For a finite projective module over any commutative ring, `Module.isNilpotent_of_isNilpotent_charpoly_sub_charpoly_zero` shows that a nilpotent difference between the characteristic polynomials of an endomorphism and zero forces the endomorphism to be nilpotent. Equality is a special case. The underlying `Module.charpoly_zero_dvd_X_pow` needs no constant-rank, faithfulness or global-monicity assumption. The given\u2026","title":"Nilpotence from characteristic polynomials"},{"id":"formal-frontier/ring-theory/projective-modules/headline-results/frobenius-functionals-and-self-injectivity","kind":"node","lean":"Module.injective_of_nondegenerate_mul","parent":8924,"result":true,"route":"../roadmap/formal-frontier/ring-theory/projective-modules/headline-results/index.html#frobenius-functionals-and-self-injectivity","scope":"formal-frontier/ring-theory/projective-modules/headline-results","status":"fully_proved","summary":"For a finite-dimensional algebra `A` over a field `K`, `Module.injective_of_nondegenerate_mul` proves `Module.Injective A A` from a given linear functional `l : A \u2192\u2097[K] K` whose multiplication pairing `l (x * y)` is nondegenerate. The algebra may be noncommutative, the scalar and algebra universes are independent, and no normalization such as `l 1 = 1` is required. The resulting injectivity applies to arbitrary\u2026","title":"Frobenius functionals and self-injectivity"},{"data_url":"scopes/formal-frontier/ring-theory/projective-modules/main-results.json","id":"formal-frontier/ring-theory/projective-modules/main-results","kind":"scope","parent":8920,"result":false,"route":"../roadmap/formal-frontier/ring-theory/projective-modules/main-results/index.html","scope":"formal-frontier/ring-theory/projective-modules/main-results","status":"planned","summary":"The main results that the formalization.yaml of FormalFrontier/projective-modules lists under `status.main_results` at commit 27f95a8b, each that the scan finds declared in the file the yaml gives, with its line. They are listed here, not made articles: in several Formal Frontier libraries this list is close to the whole API. The notes are the yaml's.","title":"Main results"},{"id":"formal-frontier/ring-theory/projective-modules/main-results/part-01","kind":"node","parent":8929,"result":false,"route":"../roadmap/formal-frontier/ring-theory/projective-modules/main-results/part-01/index.html","scope":"formal-frontier/ring-theory/projective-modules/main-results","status":"can_state","summary":"*Quoted from the main results in formalization.yaml of FormalFrontier/projective-modules at commit 27f95a8b, Apache-2.0.*","title":"Main results 1 to 100"},{"id":"formal-frontier/ring-theory/projective-modules/main-results/part-02","kind":"node","parent":8929,"result":false,"route":"../roadmap/formal-frontier/ring-theory/projective-modules/main-results/part-02/index.html","scope":"formal-frontier/ring-theory/projective-modules/main-results","status":"can_state","summary":"*Quoted from the main results in formalization.yaml of FormalFrontier/projective-modules at commit 27f95a8b, Apache-2.0.*","title":"Main results 101 to 200"},{"id":"formal-frontier/ring-theory/projective-modules/main-results/part-03","kind":"node","parent":8929,"result":false,"route":"../roadmap/formal-frontier/ring-theory/projective-modules/main-results/part-03/index.html","scope":"formal-frontier/ring-theory/projective-modules/main-results","status":"can_state","summary":"*Quoted from the main results in formalization.yaml of FormalFrontier/projective-modules at commit 27f95a8b, Apache-2.0.*","title":"Main results 201 to 210"},{"data_url":"scopes/formal-frontier/ring-theory/stable-range.json","id":"formal-frontier/ring-theory/stable-range","kind":"scope","parent":8914,"result":false,"route":"../roadmap/formal-frontier/ring-theory/stable-range/index.html","scope":"formal-frontier/ring-theory/stable-range","status":"planned","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/stable-range at commit 4d01f179, Apache-2.0.*","title":"Stable range"},{"id":"formal-frontier/ring-theory/stable-range/main-results","kind":"node","parent":8933,"result":false,"route":"../roadmap/formal-frontier/ring-theory/stable-range/main-results/index.html","scope":"formal-frontier/ring-theory/stable-range","status":"can_state","summary":"The main results that the formalization.yaml of FormalFrontier/stable-range lists under `status.main_results` at commit 4d01f179, each that the scan finds declared in the file the yaml gives, with its line. They are listed here, not made articles: in several Formal Frontier libraries this list is close to the whole API. The notes are the yaml's.","title":"Main results"},{"data_url":"scopes/formal-frontier/algebraic-geometry.json","id":"formal-frontier/algebraic-geometry","kind":"scope","parent":8707,"result":false,"route":"../roadmap/formal-frontier/algebraic-geometry/index.html","scope":"formal-frontier/algebraic-geometry","status":"can_prove","summary":"The Formal Frontier libraries on algebraic geometry, one section each.","title":"Algebraic geometry"},{"data_url":"scopes/formal-frontier/algebraic-geometry/algebraic-groups.json","id":"formal-frontier/algebraic-geometry/algebraic-groups","kind":"scope","parent":8935,"result":false,"route":"../roadmap/formal-frontier/algebraic-geometry/algebraic-groups/index.html","scope":"formal-frontier/algebraic-geometry/algebraic-groups","status":"can_prove","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/algebraic-groups at commit cad2dc41, Apache-2.0.*","title":"Algebraic groups"},{"id":"formal-frontier/algebraic-geometry/algebraic-groups/closed-linear-embeddings-of-affine-algebraic-gro","kind":"node","lean":"AlgebraicGeometry.exists_generalLinearGroupHom_isClosedImmersion","parent":8936,"result":true,"route":"../roadmap/formal-frontier/algebraic-geometry/algebraic-groups/index.html#closed-linear-embeddings-of-affine-algebraic-gro","scope":"formal-frontier/algebraic-geometry/algebraic-groups","status":"fully_proved","summary":"Every affine group scheme locally of finite type over a field K admits a group-scheme morphism into GL_n over K for some natural number n whose underlying scheme morphism is a closed immersion. Nonreduced group schemes are included, with no representation or basis supplied.","title":"Closed linear embeddings of affine algebraic groups"},{"id":"formal-frontier/algebraic-geometry/algebraic-groups/point-representations-and-comodules","kind":"node","lean":"AlgebraicGeometry.PointRepresentation.rightComoduleEquiv","parent":8936,"result":true,"route":"../roadmap/formal-frontier/algebraic-geometry/algebraic-groups/index.html#point-representations-and-comodules","scope":"formal-frontier/algebraic-geometry/algebraic-groups","status":"fully_proved","summary":"For a commutative Hopf algebra over a commutative ring and a fixed arbitrary module, natural point representations on the module's scalar extensions, tested on commutative base algebras in any universe containing the Hopf algebra, correspond bijectively to right-comodule structures on that module.","title":"Point representations and comodules"},{"id":"formal-frontier/algebraic-geometry/algebraic-groups/finite-dimensional-subcoalgebra-containment","kind":"node","lean":"Coalgebra.exists_finite_subcoalgebra_of_finite","parent":8936,"result":true,"route":"../roadmap/formal-frontier/algebraic-geometry/algebraic-groups/index.html#finite-dimensional-subcoalgebra-containment","scope":"formal-frontier/algebraic-geometry/algebraic-groups","status":"fully_proved","summary":"Every finite subset of a coalgebra over a field is contained in a finite-dimensional subcoalgebra.","title":"Finite-dimensional subcoalgebra containment"},{"id":"formal-frontier/algebraic-geometry/algebraic-groups/finite-hopf-algebras-and-right-hopf-modules","kind":"node","lean":"HopfAlgebra.exists_nondegenerate_mul, Coalgebra.RightComodule.coinvariantsTensorEquiv","parent":8936,"result":true,"route":"../roadmap/formal-frontier/algebraic-geometry/algebraic-groups/index.html#finite-hopf-algebras-and-right-hopf-modules","scope":"formal-frontier/algebraic-geometry/algebraic-groups","status":"can_prove","summary":"Every finite-dimensional Hopf algebra H over a field K admits a K-linear functional \u03bb whose multiplication pairing (a, x) \u21a6 \u03bb(a * x) is nondegenerate in both arguments. The Hopf algebra need not be commutative or cocommutative, and no restriction on the characteristic or assumption of antipode bijectivity is required. Over a commutative semiring, a specified right comodule over a possibly noncommutative semiring\u2026","title":"Finite Hopf algebras and right Hopf modules"},{"id":"formal-frontier/algebraic-geometry/algebraic-groups/bidual-coactions-and-the-antipode-square","kind":"node","lean":"Coalgebra.RightComodule.dual_dual_coaction_eval, Coalgebra.RightComodule.dual_dual_eq_transport","parent":8936,"result":true,"route":"../roadmap/formal-frontier/algebraic-geometry/algebraic-groups/index.html#bidual-coactions-and-the-antipode-square","scope":"formal-frontier/algebraic-geometry/algebraic-groups","status":"fully_proved","summary":"For a specified right comodule on a finite projective module over a commutative semiring, canonical bidual evaluation identifies its bidual with the original coaction twisted on coefficients by the antipode square. The coefficient Hopf algebra need not be commutative and its antipode need not be invertible. For commutative coefficients the twist is identity.","title":"Bidual coactions and the antipode square"},{"id":"formal-frontier/algebraic-geometry/algebraic-groups/surjectivity-and-faithfully-flat-coordinates","kind":"node","lean":"BialgHom.faithfullyFlat_of_injective_of_isReduced","parent":8936,"result":true,"route":"../roadmap/formal-frontier/algebraic-geometry/algebraic-groups/index.html#surjectivity-and-faithfully-flat-coordinates","scope":"formal-frontier/algebraic-geometry/algebraic-groups","status":"fully_proved","summary":"For homomorphisms of given affine group schemes locally of finite type, dominance over an algebraically closed field implies surjectivity on all scheme points; over any field, schematic dominance suffices. Over any field with reduced target `H`, dominance or injectivity of the compatible coordinate map `A \u2192 B` gives `Module.FaithfullyFlat A B` for its given algebra structure. The coordinate presentations may have\u2026","title":"Surjectivity and faithfully flat coordinates"},{"data_url":"scopes/formal-frontier/algebraic-geometry/algebraic-groups/main-results.json","id":"formal-frontier/algebraic-geometry/algebraic-groups/main-results","kind":"scope","parent":8936,"result":false,"route":"../roadmap/formal-frontier/algebraic-geometry/algebraic-groups/main-results/index.html","scope":"formal-frontier/algebraic-geometry/algebraic-groups/main-results","status":"planned","summary":"The main results that the formalization.yaml of FormalFrontier/algebraic-groups lists under `status.main_results` at commit cad2dc41, each that the scan finds declared in the file the yaml gives, with its line. They are listed here, not made articles: in several Formal Frontier libraries this list is close to the whole API. The notes are the yaml's.","title":"Main results"},{"id":"formal-frontier/algebraic-geometry/algebraic-groups/main-results/part-01","kind":"node","parent":8943,"result":false,"route":"../roadmap/formal-frontier/algebraic-geometry/algebraic-groups/main-results/part-01/index.html","scope":"formal-frontier/algebraic-geometry/algebraic-groups/main-results","status":"can_state","summary":"*Quoted from the main results in formalization.yaml of FormalFrontier/algebraic-groups at commit cad2dc41, Apache-2.0.*","title":"Main results 1 to 100"},{"id":"formal-frontier/algebraic-geometry/algebraic-groups/main-results/part-02","kind":"node","parent":8943,"result":false,"route":"../roadmap/formal-frontier/algebraic-geometry/algebraic-groups/main-results/part-02/index.html","scope":"formal-frontier/algebraic-geometry/algebraic-groups/main-results","status":"can_state","summary":"*Quoted from the main results in formalization.yaml of FormalFrontier/algebraic-groups at commit cad2dc41, Apache-2.0.*","title":"Main results 101 to 200"},{"id":"formal-frontier/algebraic-geometry/algebraic-groups/main-results/part-03","kind":"node","parent":8943,"result":false,"route":"../roadmap/formal-frontier/algebraic-geometry/algebraic-groups/main-results/part-03/index.html","scope":"formal-frontier/algebraic-geometry/algebraic-groups/main-results","status":"can_state","summary":"*Quoted from the main results in formalization.yaml of FormalFrontier/algebraic-groups at commit cad2dc41, Apache-2.0.*","title":"Main results 201 to 300"},{"id":"formal-frontier/algebraic-geometry/algebraic-groups/main-results/part-04","kind":"node","parent":8943,"result":false,"route":"../roadmap/formal-frontier/algebraic-geometry/algebraic-groups/main-results/part-04/index.html","scope":"formal-frontier/algebraic-geometry/algebraic-groups/main-results","status":"can_state","summary":"*Quoted from the main results in formalization.yaml of FormalFrontier/algebraic-groups at commit cad2dc41, Apache-2.0.*","title":"Main results 301 to 328"},{"data_url":"scopes/formal-frontier/algebraic-geometry/projective-space.json","id":"formal-frontier/algebraic-geometry/projective-space","kind":"scope","parent":8935,"result":false,"route":"../roadmap/formal-frontier/algebraic-geometry/projective-space/index.html","scope":"formal-frontier/algebraic-geometry/projective-space","status":"planned","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/projective-space at commit 9dbd208f, Apache-2.0.*","title":"Projective space"},{"id":"formal-frontier/algebraic-geometry/projective-space/main-results","kind":"node","parent":8948,"result":false,"route":"../roadmap/formal-frontier/algebraic-geometry/projective-space/main-results/index.html","scope":"formal-frontier/algebraic-geometry/projective-space","status":"can_state","summary":"The main results that the formalization.yaml of FormalFrontier/projective-space lists under `status.main_results` at commit 9dbd208f, each that the scan finds declared in the file the yaml gives, with its line. They are listed here, not made articles: in several Formal Frontier libraries this list is close to the whole API. The notes are the yaml's.","title":"Main results"},{"data_url":"scopes/formal-frontier/algebraic-geometry/ringed-spaces.json","id":"formal-frontier/algebraic-geometry/ringed-spaces","kind":"scope","parent":8935,"result":false,"route":"../roadmap/formal-frontier/algebraic-geometry/ringed-spaces/index.html","scope":"formal-frontier/algebraic-geometry/ringed-spaces","status":"fully_proved","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/ringed-spaces at commit 052015f0, Apache-2.0.*","title":"Ringed spaces"},{"id":"formal-frontier/algebraic-geometry/ringed-spaces/continuous-function-spaces","kind":"node","lean":"ContinuousFunctions.ringedSpace","parent":8950,"result":true,"route":"../roadmap/formal-frontier/algebraic-geometry/ringed-spaces/index.html#continuous-function-spaces","scope":"formal-frontier/algebraic-geometry/ringed-spaces","status":"fully_proved","summary":"A topological commutative ring `R` gives a `ContinuousFunctions.ringedSpace` with stalk evaluation and full maps induced by continuous maps; open embeddings give ringed-space open immersions. For a T1 topological field `K` with a topological division-ring structure, its stalks are local and the maps and open immersions are locally ringed. See the continuous-functions guide.","title":"Continuous-function spaces"},{"id":"formal-frontier/algebraic-geometry/ringed-spaces/finite-regularity-scalar-functions","kind":"node","lean":"FiniteRegularityFunctions.sheaf","parent":8950,"result":true,"route":"../roadmap/formal-frontier/algebraic-geometry/ringed-spaces/index.html#finite-regularity-scalar-functions","scope":"formal-frontier/algebraic-geometry/ringed-spaces","status":"fully_proved","summary":"For a charted space over a nontrivially normed field, `FiniteRegularityFunctions.sheaf` gives chosen-chart `C^r` scalar sections for `r : \u2115` and local stalks without an `IsManifold` or nonempty-carrier assumption. Smooth sections weaken to finite order; order zero is naturally isomorphic to the existing continuous scalar sheaf, including compatible stalk evaluation. Finite-`C^r` maps pull sections back and induce\u2026","title":"Finite-regularity scalar functions"},{"data_url":"scopes/formal-frontier/algebraic-geometry/scheme-properties.json","id":"formal-frontier/algebraic-geometry/scheme-properties","kind":"scope","parent":8935,"result":false,"route":"../roadmap/formal-frontier/algebraic-geometry/scheme-properties/index.html","scope":"formal-frontier/algebraic-geometry/scheme-properties","status":"fully_proved","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/scheme-properties at commit 09db7d88, Apache-2.0.*","title":"Scheme properties"},{"id":"formal-frontier/algebraic-geometry/scheme-properties/finiteness-near-all-component-generic-points","kind":"node","lean":"AlgebraicGeometry.Scheme.Hom.exists_isFinite_morphismRestrict_of_discreteTopology_genericPoints, AlgebraicGeometry.Scheme.Hom.exists_isFinite_morphismRestrict_of_finite_irreducibleComponents","parent":8953,"result":true,"route":"../roadmap/formal-frontier/algebraic-geometry/scheme-properties/index.html#finiteness-near-all-component-generic-points","scope":"formal-frontier/algebraic-geometry/scheme-properties","status":"fully_proved","summary":"A locally finite type, quasi-compact morphism with finite ordinary fibres over every component generic point becomes finite on the whole preimage of a dense open containing all those points whenever their canonical subspace topology is discrete, in particular when the target has finitely many irreducible components.","title":"Finiteness near all component generic points"},{"data_url":"scopes/formal-frontier/algebraic-geometry/scheme-properties/main-results.json","id":"formal-frontier/algebraic-geometry/scheme-properties/main-results","kind":"scope","parent":8953,"result":false,"route":"../roadmap/formal-frontier/algebraic-geometry/scheme-properties/main-results/index.html","scope":"formal-frontier/algebraic-geometry/scheme-properties/main-results","status":"planned","summary":"The main results that the formalization.yaml of FormalFrontier/scheme-properties lists under `status.main_results` at commit 09db7d88, each that the scan finds declared in the file the yaml gives, with its line. They are listed here, not made articles: in several Formal Frontier libraries this list is close to the whole API. The notes are the yaml's.","title":"Main results"},{"id":"formal-frontier/algebraic-geometry/scheme-properties/main-results/part-01","kind":"node","parent":8955,"result":false,"route":"../roadmap/formal-frontier/algebraic-geometry/scheme-properties/main-results/part-01/index.html","scope":"formal-frontier/algebraic-geometry/scheme-properties/main-results","status":"can_state","summary":"*Quoted from the main results in formalization.yaml of FormalFrontier/scheme-properties at commit 09db7d88, Apache-2.0.*","title":"Main results 1 to 100"},{"id":"formal-frontier/algebraic-geometry/scheme-properties/main-results/part-02","kind":"node","parent":8955,"result":false,"route":"../roadmap/formal-frontier/algebraic-geometry/scheme-properties/main-results/part-02/index.html","scope":"formal-frontier/algebraic-geometry/scheme-properties/main-results","status":"can_state","summary":"*Quoted from the main results in formalization.yaml of FormalFrontier/scheme-properties at commit 09db7d88, Apache-2.0.*","title":"Main results 101 to 149"},{"data_url":"scopes/formal-frontier/number-theory.json","id":"formal-frontier/number-theory","kind":"scope","parent":8707,"result":false,"route":"../roadmap/formal-frontier/number-theory/index.html","scope":"formal-frontier/number-theory","status":"can_prove","summary":"The Formal Frontier libraries on number theory, one section each.","title":"Number theory"},{"data_url":"scopes/formal-frontier/number-theory/dedekind-ideal-arithmetic.json","id":"formal-frontier/number-theory/dedekind-ideal-arithmetic","kind":"scope","parent":8958,"result":false,"route":"../roadmap/formal-frontier/number-theory/dedekind-ideal-arithmetic/index.html","scope":"formal-frontier/number-theory/dedekind-ideal-arithmetic","status":"can_prove","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/dedekind-ideal-arithmetic at commit c7630759, Apache-2.0.*","title":"Dedekind ideal arithmetic"},{"id":"formal-frontier/number-theory/dedekind-ideal-arithmetic/ideal-norms-factorization-and-principal-primes","kind":"node","lean":"Ideal.absNorm_lt_absNorm_of_lt, Ideal.eq_of_le_of_absNorm_eq, FractionalIdeal.absNorm_eq_finprod_count, FractionalIdeal.exists_mem_count_spanSingleton_eq, Ideal.exists_principal_prime_over_of_norm_natAbs_eq","parent":8959,"result":true,"route":"../roadmap/formal-frontier/number-theory/dedekind-ideal-arithmetic/index.html#ideal-norms-factorization-and-principal-primes","scope":"formal-frontier/number-theory/dedekind-ideal-arithmetic","status":"fully_proved","summary":"In an infinite commutative Dedekind domain with finite quotients, a nonzero smaller ideal has strictly larger absolute norm, and nested ideals of equal absolute norm coincide. Over a Dedekind domain finite free over the integers, each nonzero fractional-ideal norm is the finite-support product of height-one prime norms raised to signed integer exponents; the fraction field needs no number-field structure. In an\u2026","title":"Ideal norms, factorization and principal primes"},{"id":"formal-frontier/number-theory/dedekind-ideal-arithmetic/class-groups-under-localization","kind":"node","lean":"ClassGroup.extendedHom_surjective_of_isLocalization, ClassGroup.isPrincipalIdealRing_of_generator_of_isLocalization, ClassGroup.uniqueFactorizationMonoid_of_generator_of_isLocalization","parent":8959,"result":true,"route":"../roadmap/formal-frontier/number-theory/dedekind-ideal-arithmetic/index.html#class-groups-under-localization","scope":"formal-frontier/number-theory/dedekind-ideal-arithmetic","status":"fully_proved","summary":"Class extension is surjective for torsion-free localizations of Dedekind domains, and a supplied class generator killed by extension makes the target a principal ideal ring and a unique factorization domain.","title":"Class groups under localization"},{"id":"formal-frontier/number-theory/dedekind-ideal-arithmetic/section-indices-and-rational-frame-coordinates","kind":"node","lean":"Module.Invertible.finite_quotient_span_singleton, Module.Invertible.exists_nonzero_smul_eq, LinearEquiv.fibreCoordinate, LinearEquiv.coefficientIdeal, LinearEquiv.coefficientMap_injective","parent":8959,"result":true,"route":"../roadmap/formal-frontier/number-theory/dedekind-ideal-arithmetic/index.html#section-indices-and-rational-frame-coordinates","scope":"formal-frontier/number-theory/dedekind-ideal-arithmetic","status":"can_prove","summary":"Nonzero sections of invertible modules over commutative domains with finite quotients span finite-index submodules; any two nonzero sections of one invertible module over a commutative domain admit a common nonzero scalar multiple. A supplied frame on K tensor_R M extends to an F-linear coordinate on F tensor_R M through compatible algebras over commutative semirings R and K, with F not required commutative,\u2026","title":"Section indices and rational-frame coordinates"},{"id":"formal-frontier/number-theory/dedekind-ideal-arithmetic/intrinsic-heights-of-normed-invertible-modules","kind":"node","lean":"NumberField.ArchimedeanNorms.height_eq, NumberField.ArchimedeanNorms.height_pos, NumberField.ArchimedeanNorms.normalizedLogHeight_eq, NumberField.ArchimedeanNorms.height_tensor, NumberField.ArchimedeanNorms.height_rescale, NumberField.ArchimedeanNorms.height_comap, NumberField.ArchimedeanNorms.height_eq_rationalFrame, NumberField.ArchimedeanNorms.normalizedLogHeight_eq_rationalFrame, NumberField.ArchimedeanNorms.height_eq_finitePlace","parent":8959,"result":true,"route":"../roadmap/formal-frontier/number-theory/dedekind-ideal-arithmetic/index.html#intrinsic-heights-of-normed-invertible-modules","scope":"formal-frontier/number-theory/dedekind-ideal-arithmetic","status":"fully_proved","summary":"For a number field K, definite seminorms on the archimedean completion fibres of an invertible module over its ring of integers define a positive height equal to the section index divided by the weighted norm product for every nonzero integral section. Ordinary norms have weight one at real places and two at complex places. The normalized logarithmic height is (log index minus the weighted log norm sum) divided by\u2026","title":"Intrinsic heights of normed invertible modules"},{"id":"formal-frontier/number-theory/dedekind-ideal-arithmetic/adic-fibre-seminorms-and-integral-spans","kind":"node","lean":"Module.adicIntegralSpan, Module.adicIntegralSpanEquiv, Module.mem_adicIntegralSpan_self_iff, LinearEquiv.adicSeminorm, Module.adicSeminorm, Module.adicSeminorm_apply, Module.adicSeminorm_unique, Module.adicSeminorm_add_le_max, Module.adicSeminorm_self","parent":8959,"result":true,"route":"../roadmap/formal-frontier/number-theory/dedekind-ideal-arithmetic/index.html#adic-fibre-seminorms-and-integral-spans","scope":"formal-frontier/number-theory/dedekind-ideal-arithmetic","status":"fully_proved","summary":"Let R be an infinite Dedekind domain with finite quotients, K its fraction field, p a height-one prime and F = p.adicCompletion K with its selected residue-cardinality norm. An invertible R-module M has a canonical definite ultrametric F-seminorm on F tensor_R M. For every rational frame e and every fibre vector x, its value is q raised to FractionalIdeal.count K p e.coefficientIdeal times the norm of\u2026","title":"Adic fibre seminorms and integral spans"},{"id":"formal-frontier/number-theory/dedekind-ideal-arithmetic/main-results","kind":"node","parent":8959,"result":false,"route":"../roadmap/formal-frontier/number-theory/dedekind-ideal-arithmetic/main-results/index.html","scope":"formal-frontier/number-theory/dedekind-ideal-arithmetic","status":"can_state","summary":"The main results that the formalization.yaml of FormalFrontier/dedekind-ideal-arithmetic lists under `status.main_results` at commit c7630759, each that the scan finds declared in the file the yaml gives, with its line. They are listed here, not made articles: in several Formal Frontier libraries this list is close to the whole API. The notes are the yaml's.","title":"Main results"},{"data_url":"scopes/formal-frontier/number-theory/iwasawa-modules.json","id":"formal-frontier/number-theory/iwasawa-modules","kind":"scope","parent":8958,"result":false,"route":"../roadmap/formal-frontier/number-theory/iwasawa-modules/index.html","scope":"formal-frontier/number-theory/iwasawa-modules","status":"fully_proved","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/iwasawa-modules at commit e71db96b, Apache-2.0.*","title":"Iwasawa modules"},{"id":"formal-frontier/number-theory/iwasawa-modules/pseudo-nullity-and-short-exact-sequences","kind":"node","lean":"Module.IsPseudoNull","parent":8966,"result":true,"route":"../roadmap/formal-frontier/number-theory/iwasawa-modules/index.html#pseudo-nullity-and-short-exact-sequences","scope":"formal-frontier/number-theory/iwasawa-modules","status":"fully_proved","summary":"Over any commutative ring, `Module.IsPseudoNull` says a module's support avoids primes of height at most one; equivalently, each such localization vanishes. Submodules, quotients and extensions preserve pseudo-nullity, and in a short exact sequence the middle term is pseudo-null exactly when both endpoints are (exact-sequence criterion). The predicate needs no finite-generation, Noetherianity or domain hypothesis.","title":"Pseudo-nullity and short exact sequences"},{"id":"formal-frontier/number-theory/iwasawa-modules/main-results","kind":"node","parent":8966,"result":false,"route":"../roadmap/formal-frontier/number-theory/iwasawa-modules/main-results/index.html","scope":"formal-frontier/number-theory/iwasawa-modules","status":"can_state","summary":"The main results that the formalization.yaml of FormalFrontier/iwasawa-modules lists under `status.main_results` at commit e71db96b, each that the scan finds declared in the file the yaml gives, with its line. They are listed here, not made articles: in several Formal Frontier libraries this list is close to the whole API. The notes are the yaml's.","title":"Main results"},{"id":"formal-frontier/number-theory/norm-forms","kind":"node","parent":8958,"result":false,"route":"../roadmap/formal-frontier/number-theory/norm-forms/index.html","scope":"formal-frontier/number-theory","status":"can_state","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/norm-forms at commit 4b2893fb, Apache-2.0.*","title":"Norm forms"},{"data_url":"scopes/formal-frontier/number-theory/polynomial-root-stability.json","id":"formal-frontier/number-theory/polynomial-root-stability","kind":"scope","parent":8958,"result":false,"route":"../roadmap/formal-frontier/number-theory/polynomial-root-stability/index.html","scope":"formal-frontier/number-theory/polynomial-root-stability","status":"fully_proved","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/polynomial-root-stability at commit efade497, Apache-2.0.*","title":"Polynomial root stability"},{"id":"formal-frontier/number-theory/polynomial-root-stability/polynomial-root-scaling","kind":"node","lean":"Polynomial.continuous_eval_scaleRoots","parent":8970,"result":true,"route":"../roadmap/formal-frontier/number-theory/polynomial-root-stability/index.html#polynomial-root-scaling","scope":"formal-frontier/number-theory/polynomial-root-stability","status":"fully_proved","summary":"For a fixed polynomial `p : R[X]` over a topological semiring `R`, evaluation of Mathlib's root-scaling operation is jointly continuous by `Polynomial.continuous_eval_scaleRoots`: `(t, z) \u21a6 (p.scaleRoots t).eval z`. The polynomial is fixed; no continuity across changes of degree is asserted, and multiplication need not be commutative.","title":"Polynomial root scaling"},{"id":"formal-frontier/number-theory/polynomial-root-stability/polynomial-scaling-in-valuation-balls","kind":"node","lean":"Valuation.map_eval_sub_eq_of_mul_map_sub_lt_map_derivative","parent":8970,"result":true,"route":"../roadmap/formal-frontier/number-theory/polynomial-root-stability/index.html#polynomial-scaling-in-valuation-balls","scope":"formal-frontier/number-theory/polynomial-root-stability","status":"fully_proved","summary":"Let `R : Type u` be a commutative ring and `\u0393\u2080 : Type w` a `LinearOrderedCommMonoidWithZero`, with independent universes, and let `v : Valuation R \u0393\u2080`. For `f : R[X]`, `M : \u0393\u2080` and `a : R`, assume `\u2200 n > 0, v (f.coeff n) \u2264 M` and `v a \u2264 1`; the constant coefficient is unrestricted. Set `D = v (f.derivative.eval a)` and `B = {b | M * v (b - a) < D}`. `Valuation.map_eval_sub_eq_of_mul_map_sub_lt_map_derivative` gives\u2026","title":"Polynomial scaling in valuation balls"},{"id":"formal-frontier/number-theory/polynomial-root-stability/root-uniqueness-in-an-ultrametric-normed-ring","kind":"node","lean":"Polynomial.eq_of_eval_eq_zero_of_mul_norm_sub_lt_norm_derivative","parent":8970,"result":true,"route":"../roadmap/formal-frontier/number-theory/polynomial-root-stability/index.html#root-uniqueness-in-an-ultrametric-normed-ring","scope":"formal-frontier/number-theory/polynomial-root-stability","status":"fully_proved","summary":"Let `R` be an ultrametric normed commutative ring with multiplicative norm, let `f : R[X]` satisfy `\u2200 n, \u2016f.coeff n\u2016 \u2264 M`, and let `y` be a root with `\u2016y\u2016 \u2264 1`. Then every root `x` satisfying `M * \u2016x - y\u2016 < \u2016f.derivative.eval y\u2016` equals `y`. The criterion `Polynomial.eq_of_eval_eq_zero_of_mul_norm_sub_lt_norm_derivative` requires neither division nor completeness, and does not assume `\u2016x\u2016 \u2264 1`: the derivative bound\u2026","title":"Root uniqueness in an ultrametric normed ring"},{"id":"formal-frontier/number-theory/polynomial-root-stability/quantitative-resultant-controlled-factorization","kind":"node","lean":"Polynomial.exists_factorization_of_supNorm_lt_resultant_sq","parent":8970,"result":true,"route":"../roadmap/formal-frontier/number-theory/polynomial-root-stability/index.html#quantitative-resultant-controlled-factorization","scope":"formal-frontier/number-theory/polynomial-root-stability","status":"fully_proved","summary":"Let `K` be a complete ultrametric normed field, and let `f, g\u2080, h\u2080 : K[X]`. Write `m = g\u2080.natDegree`, `n = h\u2080.natDegree`, `e = f - g\u2080 * h\u2080`, `\u03b4 = e.supNorm` and `r = \u2016g\u2080.resultant h\u2080\u2016`, using the natural-degree resultant. Assume `g\u2080.supNorm \u2264 1`, `h\u2080.supNorm \u2264 1`, `e.degree < (m + n : WithBot \u2115)` and `\u03b4 < r\u00b2`. Then `Polynomial.exists_factorization_of_supNorm_lt_resultant_sq` gives `f = g * h`, preserving each\u2026","title":"Quantitative resultant-controlled factorization"},{"id":"formal-frontier/number-theory/polynomial-root-stability/main-results","kind":"node","parent":8970,"result":false,"route":"../roadmap/formal-frontier/number-theory/polynomial-root-stability/main-results/index.html","scope":"formal-frontier/number-theory/polynomial-root-stability","status":"can_state","summary":"The main results that the formalization.yaml of FormalFrontier/polynomial-root-stability lists under `status.main_results` at commit efade497, each that the scan finds declared in the file the yaml gives, with its line. They are listed here, not made articles: in several Formal Frontier libraries this list is close to the whole API. The notes are the yaml's.","title":"Main results"},{"data_url":"scopes/formal-frontier/number-theory/quadratic-algebras.json","id":"formal-frontier/number-theory/quadratic-algebras","kind":"scope","parent":8958,"result":false,"route":"../roadmap/formal-frontier/number-theory/quadratic-algebras/index.html","scope":"formal-frontier/number-theory/quadratic-algebras","status":"planned","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/quadratic-algebras at commit 16a5b4c5, Apache-2.0.*","title":"Quadratic algebras"},{"id":"formal-frontier/number-theory/quadratic-algebras/main-results","kind":"node","parent":8976,"result":false,"route":"../roadmap/formal-frontier/number-theory/quadratic-algebras/main-results/index.html","scope":"formal-frontier/number-theory/quadratic-algebras","status":"can_state","summary":"The main results that the formalization.yaml of FormalFrontier/quadratic-algebras lists under `status.main_results` at commit 16a5b4c5, each that the scan finds declared in the file the yaml gives, with its line. They are listed here, not made articles: in several Formal Frontier libraries this list is close to the whole API. The notes are the yaml's.","title":"Main results"},{"data_url":"scopes/formal-frontier/number-theory/valuation-integers.json","id":"formal-frontier/number-theory/valuation-integers","kind":"scope","parent":8958,"result":false,"route":"../roadmap/formal-frontier/number-theory/valuation-integers/index.html","scope":"formal-frontier/number-theory/valuation-integers","status":"can_prove","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/valuation-integers at commit 212f1a04, Apache-2.0.*","title":"Valuation integers"},{"id":"formal-frontier/number-theory/valuation-integers/prescribed-finite-separable-residue-extensions","kind":"node","lean":"Valuation.Integers.exists_extension_residueField_equiv, Valuation.Integers.residueExtensionResidueEquiv","parent":8978,"result":true,"route":"../roadmap/formal-frontier/number-theory/valuation-integers/index.html#prescribed-finite-separable-residue-extensions","scope":"formal-frontier/number-theory/valuation-integers","status":"can_prove","summary":"Every finite separable extension of the original residue field of a valuation ring is realized by a compatible finite simple field extension of the same degree, with a base-compatible residue isomorphism carrying the chosen power-basis generator to the residue of an integral generator.","title":"Prescribed finite separable residue extensions"},{"id":"formal-frontier/number-theory/valuation-integers/residue-degree-bounded-by-field-degree","kind":"node","lean":"ValuationRing.finite_residueField, ValuationRing.finrank_residueField_le","parent":8978,"result":true,"route":"../roadmap/formal-frontier/number-theory/valuation-integers/index.html#residue-degree-bounded-by-field-degree","scope":"formal-frontier/number-theory/valuation-integers","status":"fully_proved","summary":"If R is a valuation domain with fraction field F, L is a finite field extension of F, and a local R-algebra S embeds into L over R with compatible original residue-field actions, then the residue field of S is finite over that of R and its degree is at most the degree of L over F.","title":"Residue degree bounded by field degree"},{"id":"formal-frontier/number-theory/valuation-integers/unique-residue-and-field-homomorphism-lifts","kind":"node","lean":"ValuationRing.existsUnique_residueAlgHom_lift_of_finrank_eq, ValuationRing.existsUnique_fieldAlgHom_lift_of_finrank_eq, ValuationRing.fieldAlgHom_eq_id_of_residue_eq_id, ValuationRing.fieldAlgHom_eq_comp_of_residue_eq_comp","parent":8978,"result":true,"route":"../roadmap/formal-frontier/number-theory/valuation-integers/index.html#unique-residue-and-field-homomorphism-lifts","scope":"formal-frontier/number-theory/valuation-integers","status":"fully_proved","summary":"Over a Henselian valuation domain R with fraction field F, if local commutative R-algebras S and T embed in finite field extensions L/F and E/F with compatible given ambient and residue scalar towers, [L : F] = [\u03baS : \u03baR], [E : F] = [\u03baT : \u03baR] and \u03baS/\u03baR separable, then every \u03baR-algebra homomorphism \u03c6 from \u03baS to \u03baT lifts uniquely to an R-algebra homomorphism \u03c8 from S to T satisfying residue T (\u03c8 s) = \u03c6 (residue S s)\u2026","title":"Unique residue and field homomorphism lifts"},{"id":"formal-frontier/number-theory/valuation-integers/a-sharp-hensel-criterion-at-arbitrary-valuation","kind":"node","lean":"Valuation.exists_root_of_map_eval_lt_map_derivative_sq","parent":8978,"result":true,"route":"../roadmap/formal-frontier/number-theory/valuation-integers/index.html#a-sharp-hensel-criterion-at-arbitrary-valuation","scope":"formal-frontier/number-theory/valuation-integers","status":"fully_proved","summary":"Over a Henselian valuation ring, the strict inequality v(f(a)) < v(f'(a))\u00b2 gives a root with exact error/displacement identity and unchanged derivative value, unique throughout the original derivative ball.","title":"A sharp Hensel criterion at arbitrary valuation rank"},{"id":"formal-frontier/number-theory/valuation-integers/unit-norms-over-henselian-valuation-domains","kind":"node","lean":"ValuationRing.exists_isUnit_norm_eq_of_sub_one_mem, ValuationRing.exists_isUnit_norm_eq","parent":8978,"result":true,"route":"../roadmap/formal-frontier/number-theory/valuation-integers/index.html#unit-norms-over-henselian-valuation-domains","scope":"formal-frontier/number-theory/valuation-integers","status":"fully_proved","summary":"Let R be a Henselian valuation domain with fraction field F and S a local commutative R-algebra embedded in a finite extension L/F, with compatible given ambient and residue algebras. If field degree equals residue degree and the residue extension is separable, every u with u - 1 in an ideal I contained in the maximal ideal of R is the norm of a unit s of S with s - 1 in the extension of I to S. If the base residue\u2026","title":"Unit norms over Henselian valuation domains"},{"data_url":"scopes/formal-frontier/group-theory.json","id":"formal-frontier/group-theory","kind":"scope","parent":8707,"result":false,"route":"../roadmap/formal-frontier/group-theory/index.html","scope":"formal-frontier/group-theory","status":"can_prove","summary":"The Formal Frontier libraries on group theory, one section each.","title":"Group theory"},{"data_url":"scopes/formal-frontier/group-theory/general-linear-groups.json","id":"formal-frontier/group-theory/general-linear-groups","kind":"scope","parent":8984,"result":false,"route":"../roadmap/formal-frontier/group-theory/general-linear-groups/index.html","scope":"formal-frontier/group-theory/general-linear-groups","status":"can_prove","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/general-linear-groups at commit a2199cf0, Apache-2.0.*","title":"General linear groups"},{"id":"formal-frontier/group-theory/general-linear-groups/principal-unit-quotients-and-jacobson-lifting","kind":"node","lean":"Ideal.principalUnitsQuotientEquiv, Ideal.principalUnitsToQuotient_ker, Ideal.principalUnitsQuotientEquiv_symm_mk","parent":8985,"result":true,"route":"../roadmap/formal-frontier/group-theory/general-linear-groups/index.html#principal-unit-quotients-and-jacobson-lifting","scope":"formal-frontier/group-theory/general-linear-groups","status":"fully_proved","summary":"For left ideals I, J in any ring R with I^2 \u2264 J and I \u2264 Ring.jacobson R, the subgroup U(I) = {u : R\u02e3 | u - 1 \u2208 I} modulo the kernel of u \u21a6 [u - 1] is canonically isomorphic to the additive quotient I / (I \u2229 J), with inverse [x] \u21a6 [1 + x], without two-sidedness or J \u2264 I.","title":"Principal-unit quotients and Jacobson lifting"},{"id":"formal-frontier/group-theory/general-linear-groups/square-zero-unit-kernels-and-conjugation","kind":"node","lean":"RingHom.unitsKerEquivOfSquareZero, RingHom.unitsKerEquivOfSquareZero_conj_apply_coe","parent":8985,"result":true,"route":"../roadmap/formal-frontier/group-theory/general-linear-groups/index.html#square-zero-unit-kernels-and-conjugation","scope":"formal-frontier/group-theory/general-linear-groups","status":"fully_proved","summary":"For any ring homomorphism f : R \u2192+* S from a possibly noncommutative ring to a semiring with (ker f)^2 = 0, the units mapping to 1 are isomorphic to the additive group ker f by u \u21a6 u - 1 and x \u21a6 1 + x, and conjugation by any g : R\u02e3 corresponds to x \u21a6 g * x * g\u207b\u00b9, without a surjectivity hypothesis.","title":"Square-zero unit kernels and conjugation"},{"id":"formal-frontier/group-theory/general-linear-groups/nilpotent-principal-unit-groups","kind":"node","lean":"Ideal.isNilpotent_stabilizer_quotient_one_of_pow_eq_bot, Ideal.nilpotencyClass_stabilizer_quotient_one_le_of_pow_eq_bot","parent":8985,"result":true,"route":"../roadmap/formal-frontier/group-theory/general-linear-groups/index.html#nilpotent-principal-unit-groups","scope":"formal-frontier/group-theory/general-linear-groups","status":"fully_proved","summary":"For any left ideal I of a possibly noncommutative ring R and any natural N with I^N = 0, the principal-unit subgroup {u : R\u02e3 | u - 1 \u2208 I} is nilpotent of class at most N - 1, using natural subtraction and including the trivial-subgroup cases N = 0 and N = 1.","title":"Nilpotent principal-unit groups"},{"id":"formal-frontier/group-theory/general-linear-groups/positive-principal-unit-layer-commutator-pairing","kind":"node","lean":"Ideal.powLayerCommutator, Ideal.principalUnitsPowLayerCommutator, Ideal.principalUnitsPowLayerCommutator_mk_mk, Ideal.principalUnitsPowLayerSymbol_commutator","parent":8985,"result":true,"route":"../roadmap/formal-frontier/group-theory/general-linear-groups/index.html#positive-principal-unit-layer-commutator-pairing","scope":"formal-frontier/group-theory/general-linear-groups","status":"fully_proved","summary":"For a left ideal I in any ring R and positive degrees a, b, the bracket xy - yx descends to an integer-bilinear pairing of successive ideal layers in degree a+b, and its restriction along the injective principal-unit symbols is the pairing represented by the ordered commutator of original-ring units, without two-sidedness, Jacobson containment or symbol surjectivity.","title":"Positive principal-unit layer commutator pairings"},{"id":"formal-frontier/group-theory/general-linear-groups/functorial-symbols-and-jacobson-equivalences","kind":"node","lean":"Ideal.principalUnitsPowLayerSymbol_map, Ideal.powLayerMap_commutator, Ideal.principalUnitsPowLayerMap_commutator, Ideal.principalUnitsPowLayerEquiv, Ideal.principalUnitsPowLayerEquiv_symm_map, Ideal.principalUnitsPowLayerSymbol_surjective_of_forall_isNilpotent, Ideal.principalUnitsPowLayerSymbol_surjective_of_pow_eq_bot","parent":8985,"result":true,"route":"../roadmap/formal-frontier/group-theory/general-linear-groups/index.html#functorial-symbols-and-jacobson-equivalences","scope":"formal-frontier/group-theory/general-linear-groups","status":"can_prove","summary":"On positive layers, unital ring maps carrying left ideals into left ideals induce integer-linear maps preserving symbols and commutator pairings, while Jacobson-contained ideals have symbol equivalences with natural inverses when both ideals satisfy the radical condition, and pointwise-nil or power-nilpotent left ideals have surjective symbols.","title":"Functorial symbols and Jacobson equivalences"},{"id":"formal-frontier/group-theory/general-linear-groups/scalar-unit-reflection-and-canonical-quasi-regul","kind":"node","lean":"RingHom.isQuasiregular_ker","parent":8985,"result":true,"route":"../roadmap/formal-frontier/group-theory/general-linear-groups/index.html#scalar-unit-reflection-and-canonical-quasi-regul","scope":"formal-frontier/group-theory/general-linear-groups","status":"fully_proved","summary":"For `f : R \u2192+* S` from a possibly noncommutative ring to a semiring, `IsLocalHom f` (reflection of units) makes the canonical two-sided kernel `TwoSidedIdeal.ker f` quasi-regular, without surjectivity, by `RingHom.isQuasiregular_ker`. For a surjective such map, unit reflection is equivalent to kernel quasi-regularity by `RingHom.isLocalHom_iff_isQuasiregular_ker`; its reverse direction is also available through\u2026","title":"Scalar unit reflection and canonical quasi-regular kernels"},{"data_url":"scopes/formal-frontier/group-theory/general-linear-groups/main-results.json","id":"formal-frontier/group-theory/general-linear-groups/main-results","kind":"scope","parent":8985,"result":false,"route":"../roadmap/formal-frontier/group-theory/general-linear-groups/main-results/index.html","scope":"formal-frontier/group-theory/general-linear-groups/main-results","status":"planned","summary":"The main results that the formalization.yaml of FormalFrontier/general-linear-groups lists under `status.main_results` at commit a2199cf0, each that the scan finds declared in the file the yaml gives, with its line. They are listed here, not made articles: in several Formal Frontier libraries this list is close to the whole API. The notes are the yaml's.","title":"Main results"},{"id":"formal-frontier/group-theory/general-linear-groups/main-results/part-01","kind":"node","parent":8992,"result":false,"route":"../roadmap/formal-frontier/group-theory/general-linear-groups/main-results/part-01/index.html","scope":"formal-frontier/group-theory/general-linear-groups/main-results","status":"can_state","summary":"*Quoted from the main results in formalization.yaml of FormalFrontier/general-linear-groups at commit a2199cf0, Apache-2.0.*","title":"Main results 1 to 100"},{"id":"formal-frontier/group-theory/general-linear-groups/main-results/part-02","kind":"node","parent":8992,"result":false,"route":"../roadmap/formal-frontier/group-theory/general-linear-groups/main-results/part-02/index.html","scope":"formal-frontier/group-theory/general-linear-groups/main-results","status":"can_state","summary":"*Quoted from the main results in formalization.yaml of FormalFrontier/general-linear-groups at commit a2199cf0, Apache-2.0.*","title":"Main results 101 to 200"},{"id":"formal-frontier/group-theory/general-linear-groups/main-results/part-03","kind":"node","parent":8992,"result":false,"route":"../roadmap/formal-frontier/group-theory/general-linear-groups/main-results/part-03/index.html","scope":"formal-frontier/group-theory/general-linear-groups/main-results","status":"can_state","summary":"*Quoted from the main results in formalization.yaml of FormalFrontier/general-linear-groups at commit a2199cf0, Apache-2.0.*","title":"Main results 201 to 221"},{"data_url":"scopes/formal-frontier/group-theory/group-theory.json","id":"formal-frontier/group-theory/group-theory","kind":"scope","parent":8984,"result":false,"route":"../roadmap/formal-frontier/group-theory/group-theory/index.html","scope":"formal-frontier/group-theory/group-theory","status":"fully_proved","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/group-theory at commit cad97f1a, Apache-2.0.*","title":"Group theory"},{"id":"formal-frontier/group-theory/group-theory/local-compactness-of-homomorphism-spaces","kind":"node","lean":"ContinuousMonoidHom.locallyCompactSpace_of_fg","parent":8996,"result":true,"route":"../roadmap/formal-frontier/group-theory/group-theory/index.html#local-compactness-of-homomorphism-spaces","scope":"formal-frontier/group-theory/group-theory","status":"fully_proved","summary":"For every finitely generated discrete group G and locally compact Hausdorff topological group D, the space of continuous homomorphisms G \u2192 D with the compact-open topology is locally compact.","title":"Local compactness of homomorphism spaces"},{"data_url":"scopes/formal-frontier/group-theory/group-theory/headline-results.json","id":"formal-frontier/group-theory/group-theory/headline-results","kind":"scope","parent":8996,"result":false,"route":"../roadmap/formal-frontier/group-theory/group-theory/headline-results/index.html","scope":"formal-frontier/group-theory/group-theory/headline-results","status":"fully_proved","summary":"*Quoted from the headline results in README.md of FormalFrontier/group-theory at commit cad97f1a, Apache-2.0.*","title":"Headline results"},{"id":"formal-frontier/group-theory/group-theory/headline-results/no-small-subgroups-under-open-homomorphisms","kind":"node","lean":"NoSmallSubgroups.of_isOpenMap","parent":8998,"result":true,"route":"../roadmap/formal-frontier/group-theory/group-theory/headline-results/index.html#no-small-subgroups-under-open-homomorphisms","scope":"formal-frontier/group-theory/group-theory/headline-results","status":"fully_proved","summary":"Let `G` be a topological group, `H` a group with a topology, and `f : G \u2192* H` an open homomorphism whose kernel has the discrete topology induced from `G`. If `G` has no small subgroups, then so does `H`; neither continuity of `f`, continuous operations on `H`, nor surjectivity is required. If `f` is also continuous at the identity, `G` has no small subgroups if and only if `H` does. See\u2026","title":"No small subgroups under open homomorphisms"},{"data_url":"scopes/formal-frontier/group-theory/group-theory/main-results.json","id":"formal-frontier/group-theory/group-theory/main-results","kind":"scope","parent":8996,"result":false,"route":"../roadmap/formal-frontier/group-theory/group-theory/main-results/index.html","scope":"formal-frontier/group-theory/group-theory/main-results","status":"planned","summary":"The main results that the formalization.yaml of FormalFrontier/group-theory lists under `status.main_results` at commit cad97f1a, each that the scan finds declared in the file the yaml gives, with its line. They are listed here, not made articles: in several Formal Frontier libraries this list is close to the whole API. The notes are the yaml's.","title":"Main results"},{"id":"formal-frontier/group-theory/group-theory/main-results/part-01","kind":"node","parent":9000,"result":false,"route":"../roadmap/formal-frontier/group-theory/group-theory/main-results/part-01/index.html","scope":"formal-frontier/group-theory/group-theory/main-results","status":"can_state","summary":"*Quoted from the main results in formalization.yaml of FormalFrontier/group-theory at commit cad97f1a, Apache-2.0.*","title":"Main results 1 to 100"},{"id":"formal-frontier/group-theory/group-theory/main-results/part-02","kind":"node","parent":9000,"result":false,"route":"../roadmap/formal-frontier/group-theory/group-theory/main-results/part-02/index.html","scope":"formal-frontier/group-theory/group-theory/main-results","status":"can_state","summary":"*Quoted from the main results in formalization.yaml of FormalFrontier/group-theory at commit cad97f1a, Apache-2.0.*","title":"Main results 101 to 194"},{"id":"formal-frontier/group-theory/profinite-groups","kind":"node","parent":8984,"result":false,"route":"../roadmap/formal-frontier/group-theory/profinite-groups/index.html","scope":"formal-frontier/group-theory","status":"can_state","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/profinite-groups at commit bf258c54, Apache-2.0.*","title":"Profinite groups"},{"id":"formal-frontier/group-theory/supernatural-numbers","kind":"node","parent":8984,"result":false,"route":"../roadmap/formal-frontier/group-theory/supernatural-numbers/index.html","scope":"formal-frontier/group-theory","status":"can_state","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/supernatural-numbers at commit cb444a40, Apache-2.0.*","title":"Supernatural numbers"},{"data_url":"scopes/formal-frontier/homological-algebra-and-category-theory.json","id":"formal-frontier/homological-algebra-and-category-theory","kind":"scope","parent":8707,"result":false,"route":"../roadmap/formal-frontier/homological-algebra-and-category-theory/index.html","scope":"formal-frontier/homological-algebra-and-category-theory","status":"planned","summary":"The Formal Frontier libraries on homological algebra and category theory, one section each.","title":"Homological algebra and category theory"},{"data_url":"scopes/formal-frontier/homological-algebra-and-category-theory/continuous-group-cohomology.json","id":"formal-frontier/homological-algebra-and-category-theory/continuous-group-cohomology","kind":"scope","parent":9005,"result":false,"route":"../roadmap/formal-frontier/homological-algebra-and-category-theory/continuous-group-cohomology/index.html","scope":"formal-frontier/homological-algebra-and-category-theory/continuous-group-cohomology","status":"planned","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/continuous-group-cohomology at commit 4434e3f2, Apache-2.0.*","title":"Continuous group cohomology"},{"data_url":"scopes/formal-frontier/homological-algebra-and-category-theory/continuous-group-cohomology/main-results.json","id":"formal-frontier/homological-algebra-and-category-theory/continuous-group-cohomology/main-results","kind":"scope","parent":9006,"result":false,"route":"../roadmap/formal-frontier/homological-algebra-and-category-theory/continuous-group-cohomology/main-results/index.html","scope":"formal-frontier/homological-algebra-and-category-theory/continuous-group-cohomology/main-results","status":"planned","summary":"The main results that the formalization.yaml of FormalFrontier/continuous-group-cohomology lists under `status.main_results` at commit 4434e3f2, each that the scan finds declared in the file the yaml gives, with its line. They are listed here, not made articles: in several Formal Frontier libraries this list is close to the whole API. The notes are the yaml's.","title":"Main results"},{"id":"formal-frontier/homological-algebra-and-category-theory/continuous-group-cohomology/main-results/part-01","kind":"node","parent":9007,"result":false,"route":"../roadmap/formal-frontier/homological-algebra-and-category-theory/continuous-group-cohomology/main-results/part-01/index.html","scope":"formal-frontier/homological-algebra-and-category-theory/continuous-group-cohomology/main-results","status":"can_state","summary":"*Quoted from the main results in formalization.yaml of FormalFrontier/continuous-group-cohomology at commit 4434e3f2, Apache-2.0.*","title":"Main results 1 to 100"},{"id":"formal-frontier/homological-algebra-and-category-theory/continuous-group-cohomology/main-results/part-02","kind":"node","parent":9007,"result":false,"route":"../roadmap/formal-frontier/homological-algebra-and-category-theory/continuous-group-cohomology/main-results/part-02/index.html","scope":"formal-frontier/homological-algebra-and-category-theory/continuous-group-cohomology/main-results","status":"can_state","summary":"*Quoted from the main results in formalization.yaml of FormalFrontier/continuous-group-cohomology at commit 4434e3f2, Apache-2.0.*","title":"Main results 101 to 200"},{"id":"formal-frontier/homological-algebra-and-category-theory/continuous-group-cohomology/main-results/part-03","kind":"node","parent":9007,"result":false,"route":"../roadmap/formal-frontier/homological-algebra-and-category-theory/continuous-group-cohomology/main-results/part-03/index.html","scope":"formal-frontier/homological-algebra-and-category-theory/continuous-group-cohomology/main-results","status":"can_state","summary":"*Quoted from the main results in formalization.yaml of FormalFrontier/continuous-group-cohomology at commit 4434e3f2, Apache-2.0.*","title":"Main results 201 to 205"},{"data_url":"scopes/formal-frontier/homological-algebra-and-category-theory/finite-group-tate-cohomology.json","id":"formal-frontier/homological-algebra-and-category-theory/finite-group-tate-cohomology","kind":"scope","parent":9005,"result":false,"route":"../roadmap/formal-frontier/homological-algebra-and-category-theory/finite-group-tate-cohomology/index.html","scope":"formal-frontier/homological-algebra-and-category-theory/finite-group-tate-cohomology","status":"planned","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/finite-group-tate-cohomology at commit c0c9d563, Apache-2.0.*","title":"Finite-group Tate cohomology"},{"id":"formal-frontier/homological-algebra-and-category-theory/finite-group-tate-cohomology/main-results","kind":"node","parent":9011,"result":false,"route":"../roadmap/formal-frontier/homological-algebra-and-category-theory/finite-group-tate-cohomology/main-results/index.html","scope":"formal-frontier/homological-algebra-and-category-theory/finite-group-tate-cohomology","status":"can_state","summary":"The main results that the formalization.yaml of FormalFrontier/finite-group-tate-cohomology lists under `status.main_results` at commit c0c9d563, each that the scan finds declared in the file the yaml gives, with its line. They are listed here, not made articles: in several Formal Frontier libraries this list is close to the whole API. The notes are the yaml's.","title":"Main results"},{"data_url":"scopes/formal-frontier/homological-algebra-and-category-theory/homological-algebra.json","id":"formal-frontier/homological-algebra-and-category-theory/homological-algebra","kind":"scope","parent":9005,"result":false,"route":"../roadmap/formal-frontier/homological-algebra-and-category-theory/homological-algebra/index.html","scope":"formal-frontier/homological-algebra-and-category-theory/homological-algebra","status":"planned","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/homological-algebra at commit 8bcaebe9, Apache-2.0.*","title":"Homological algebra"},{"data_url":"scopes/formal-frontier/homological-algebra-and-category-theory/homological-algebra/main-results.json","id":"formal-frontier/homological-algebra-and-category-theory/homological-algebra/main-results","kind":"scope","parent":9013,"result":false,"route":"../roadmap/formal-frontier/homological-algebra-and-category-theory/homological-algebra/main-results/index.html","scope":"formal-frontier/homological-algebra-and-category-theory/homological-algebra/main-results","status":"planned","summary":"The main results that the formalization.yaml of FormalFrontier/homological-algebra lists under `status.main_results` at commit 8bcaebe9, each that the scan finds declared in the file the yaml gives, with its line. They are listed here, not made articles: in several Formal Frontier libraries this list is close to the whole API. The notes are the yaml's.","title":"Main results"},{"id":"formal-frontier/homological-algebra-and-category-theory/homological-algebra/main-results/part-01","kind":"node","parent":9014,"result":false,"route":"../roadmap/formal-frontier/homological-algebra-and-category-theory/homological-algebra/main-results/part-01/index.html","scope":"formal-frontier/homological-algebra-and-category-theory/homological-algebra/main-results","status":"can_state","summary":"*Quoted from the main results in formalization.yaml of FormalFrontier/homological-algebra at commit 8bcaebe9, Apache-2.0.*","title":"Main results 1 to 100"},{"id":"formal-frontier/homological-algebra-and-category-theory/homological-algebra/main-results/part-02","kind":"node","parent":9014,"result":false,"route":"../roadmap/formal-frontier/homological-algebra-and-category-theory/homological-algebra/main-results/part-02/index.html","scope":"formal-frontier/homological-algebra-and-category-theory/homological-algebra/main-results","status":"can_state","summary":"*Quoted from the main results in formalization.yaml of FormalFrontier/homological-algebra at commit 8bcaebe9, Apache-2.0.*","title":"Main results 101 to 200"},{"id":"formal-frontier/homological-algebra-and-category-theory/homological-algebra/main-results/part-03","kind":"node","parent":9014,"result":false,"route":"../roadmap/formal-frontier/homological-algebra-and-category-theory/homological-algebra/main-results/part-03/index.html","scope":"formal-frontier/homological-algebra-and-category-theory/homological-algebra/main-results","status":"can_state","summary":"*Quoted from the main results in formalization.yaml of FormalFrontier/homological-algebra at commit 8bcaebe9, Apache-2.0.*","title":"Main results 201 to 218"},{"data_url":"scopes/formal-frontier/homological-algebra-and-category-theory/morphism-properties.json","id":"formal-frontier/homological-algebra-and-category-theory/morphism-properties","kind":"scope","parent":9005,"result":false,"route":"../roadmap/formal-frontier/homological-algebra-and-category-theory/morphism-properties/index.html","scope":"formal-frontier/homological-algebra-and-category-theory/morphism-properties","status":"planned","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/morphism-properties at commit 1666ed81, Apache-2.0.*","title":"Morphism properties"},{"id":"formal-frontier/homological-algebra-and-category-theory/morphism-properties/main-results","kind":"node","parent":9018,"result":false,"route":"../roadmap/formal-frontier/homological-algebra-and-category-theory/morphism-properties/main-results/index.html","scope":"formal-frontier/homological-algebra-and-category-theory/morphism-properties","status":"can_state","summary":"The main results that the formalization.yaml of FormalFrontier/morphism-properties lists under `status.main_results` at commit 1666ed81, each that the scan finds declared in the file the yaml gives, with its line. They are listed here, not made articles: in several Formal Frontier libraries this list is close to the whole API. The notes are the yaml's.","title":"Main results"},{"data_url":"scopes/formal-frontier/homological-algebra-and-category-theory/sheaf-cohomology.json","id":"formal-frontier/homological-algebra-and-category-theory/sheaf-cohomology","kind":"scope","parent":9005,"result":false,"route":"../roadmap/formal-frontier/homological-algebra-and-category-theory/sheaf-cohomology/index.html","scope":"formal-frontier/homological-algebra-and-category-theory/sheaf-cohomology","status":"planned","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/sheaf-cohomology at commit 85cf087f, Apache-2.0.*","title":"Sheaf cohomology"},{"id":"formal-frontier/homological-algebra-and-category-theory/sheaf-cohomology/main-results","kind":"node","parent":9020,"result":false,"route":"../roadmap/formal-frontier/homological-algebra-and-category-theory/sheaf-cohomology/main-results/index.html","scope":"formal-frontier/homological-algebra-and-category-theory/sheaf-cohomology","status":"can_state","summary":"The main results that the formalization.yaml of FormalFrontier/sheaf-cohomology lists under `status.main_results` at commit 85cf087f, each that the scan finds declared in the file the yaml gives, with its line. They are listed here, not made articles: in several Formal Frontier libraries this list is close to the whole API. The notes are the yaml's.","title":"Main results"},{"id":"formal-frontier/homological-algebra-and-category-theory/simplicial-objects","kind":"node","parent":9005,"result":false,"route":"../roadmap/formal-frontier/homological-algebra-and-category-theory/simplicial-objects/index.html","scope":"formal-frontier/homological-algebra-and-category-theory","status":"can_state","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/simplicial-objects at commit e210890f, Apache-2.0.*","title":"Simplicial objects"},{"id":"formal-frontier/homological-algebra-and-category-theory/simplicial-rings","kind":"node","parent":9005,"result":false,"route":"../roadmap/formal-frontier/homological-algebra-and-category-theory/simplicial-rings/index.html","scope":"formal-frontier/homological-algebra-and-category-theory","status":"can_state","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/simplicial-rings at commit 72878574, Apache-2.0.*","title":"Simplicial rings"},{"data_url":"scopes/formal-frontier/topology.json","id":"formal-frontier/topology","kind":"scope","parent":8707,"result":false,"route":"../roadmap/formal-frontier/topology/index.html","scope":"formal-frontier/topology","status":"can_prove","summary":"The Formal Frontier libraries on topology, one section each.","title":"Topology"},{"data_url":"scopes/formal-frontier/topology/fundamental-groupoids.json","id":"formal-frontier/topology/fundamental-groupoids","kind":"scope","parent":9024,"result":false,"route":"../roadmap/formal-frontier/topology/fundamental-groupoids/index.html","scope":"formal-frontier/topology/fundamental-groupoids","status":"can_prove","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/fundamental-groupoids at commit ce38a38f, Apache-2.0.*","title":"Fundamental groupoids"},{"id":"formal-frontier/topology/fundamental-groupoids/based-van-kampen-for-intersection-closed-open-co","kind":"node","lean":"FundamentalGroup.existsUnique_hom_of_isOpenCover_of_infClosed, FundamentalGroup.isColimitOfIsOpenCover","parent":9025,"result":true,"route":"../roadmap/formal-frontier/topology/fundamental-groupoids/index.html#based-van-kampen-for-intersection-closed-open-co","scope":"formal-frontier/topology/fundamental-groupoids","status":"can_prove","summary":"For an arbitrarily indexed, binary-intersection-closed open cover of X by path-connected neighborhoods of a common point x, homomorphisms from the local fundamental groups to any group, compatible along every inclusion, determine a unique homomorphism from \u03c0\u2081(X,x), and the inclusion cocone is a colimit in GrpCat at the space universe, with independent space, index and target-group universes.","title":"Based van Kampen for intersection-closed open covers"},{"id":"formal-frontier/topology/fundamental-groupoids/the-fundamental-group-of-the-circle-is-z","kind":"node","lean":"Circle.fundamentalGroupEquivInt, Circle.fundamentalGroupEquivInt_loop, AddCircle.fundamentalGroupEquivInt, AddCircle.fundamentalGroupEquivInt_loop","parent":9025,"result":true,"route":"../roadmap/formal-frontier/topology/fundamental-groupoids/index.html#the-fundamental-group-of-the-circle-is-z","scope":"formal-frontier/topology/fundamental-groupoids","status":"fully_proved","summary":"The complex unit circle has \u03c0\u2081(Circle,1) \u2243* Multiplicative \u2124 with its counterclockwise generator sent to +1, while for every nonzero real period T the equivalence \u03c0\u2081(AddCircle T,0) \u2243* Multiplicative \u2124 sends the projected real segment from 0 to Tn to n for every integer n.","title":"The fundamental group of the circle is \u2124"},{"id":"formal-frontier/topology/fundamental-groupoids/covering-fibers-cosets-properness-and-homotopy-g","kind":"node","lean":"IsCoveringMap.isProperMap_iff_finite_fibers, IsCoveringMap.monodromyQuotientEquiv, IsCoveringMap.isProperMap_iff_finiteIndex, IsCoveringMap.injective_homotopyGroup_mapFun, IsCoveringMap.bijective_homotopyGroup_mapFun, IsCoveringMap.homotopyGroupMulEquiv","parent":9025,"result":true,"route":"../roadmap/formal-frontier/topology/fundamental-groupoids/index.html#covering-fibers-cosets-properness-and-homotopy-g","scope":"formal-frontier/topology/fundamental-groupoids","status":"can_prove","summary":"For a covering p : E \u2192 B between arbitrary topological spaces in independent universes, properness is equivalent to finite point fibers and, when E is path connected with chosen e\u2080, monodromy identifies the fiber over p(e\u2080) with the left cosets of im(\u03c0\u2081(E,e\u2080) \u2192 \u03c0\u2081(B,p(e\u2080))) and properness is equivalent to finite index of that image subgroup, while for every e : E postcomposition on based homotopy classes of N-cubes\u2026","title":"Covering fibers, cosets, properness and homotopy groups"},{"id":"formal-frontier/topology/fundamental-groupoids/fundamental-groups-of-based-disk-attachments","kind":"node","lean":"FundamentalGroup.surjective_map_diskAttachmentInclusion, FundamentalGroup.ker_map_diskAttachmentInclusion, FundamentalGroup.diskAttachmentFundamentalGroupQuotientEquiv, FundamentalGroup.diskAttachmentFundamentalGroupHomEquiv","parent":9025,"result":true,"route":"../roadmap/formal-frontier/topology/fundamental-groupoids/index.html#fundamental-groups-of-based-disk-attachments","scope":"formal-frontier/topology/fundamental-groupoids","status":"can_prove","summary":"For arbitrary based disk attachments, the fundamental group is the quotient by the normal closure of the positive attaching loop classes, compatibly with inclusion and with homomorphisms into every group.","title":"Fundamental groups of based disk attachments"},{"id":"formal-frontier/topology/fundamental-groupoids/based-source-and-positive-degree-homotopy-fiber","kind":"node","lean":"BasedTopCat.range_homotopyFiberProjection, BasedTopCat.range_homotopyFiberLoopInclusion, BasedTopCat.range_homotopyFiberLoopMap, HomotopyGroup.homotopyFiberSequence, HomotopyGroup.homotopyFiberSequenceMap, HomotopyGroup.homotopyFiberSequence_exact, ContinuousMap.HomotopyFiber.loopInclusion, HomotopyGroup.homotopyFiberConnectingHom, HomotopyGroup.range_map_eq_ker_homotopyFiberConnectingHom, HomotopyGroup.homotopyFiberConnectingHom_naturality, HomotopyGroup.homotopyFiberConnectingHom_fin_class","parent":9025,"result":true,"route":"../roadmap/formal-frontier/topology/fundamental-groupoids/index.html#based-source-and-positive-degree-homotopy-fiber","scope":"formal-frontier/topology/fundamental-groupoids","status":"can_prove","summary":"For every original based source (Z,z0), topological E,B in independent Z,E,B universes, f : C(E,B), e0 : E, b : B and h : f e0 = b, let F be the ordinary homotopy fiber, p its projection, J the forward-loop inclusion and \u03a9f the loop map; on whole based hom-sets [Z,-] relative to {z0}, transported by ordinary ULift, range p_* = f_*\u207b\u00b9{c_B}, range J_* = p_*\u207b\u00b9{c_E} and range (\u03a9f)_* = J_*\u207b\u00b9{c_F} for constant classes c_X\u2026","title":"Based-source and positive-degree homotopy-fiber exactness"},{"data_url":"scopes/formal-frontier/topology/fundamental-groupoids/headline-results.json","id":"formal-frontier/topology/fundamental-groupoids/headline-results","kind":"scope","parent":9025,"result":false,"route":"../roadmap/formal-frontier/topology/fundamental-groupoids/headline-results/index.html","scope":"formal-frontier/topology/fundamental-groupoids/headline-results","status":"fully_proved","summary":"*Quoted from the headline results in README.md of FormalFrontier/fundamental-groupoids at commit ce38a38f, Apache-2.0.*","title":"Headline results"},{"id":"formal-frontier/topology/fundamental-groupoids/headline-results/compactly-generated-compact-open-currying","kind":"node","lean":"Homeomorph.compactlyGeneratedCurry","parent":9031,"result":true,"route":"../roadmap/formal-frontier/topology/fundamental-groupoids/headline-results/index.html#compactly-generated-compact-open-currying","scope":"formal-frontier/topology/fundamental-groupoids/headline-results","status":"fully_proved","summary":"For original `UCompactlyGeneratedSpace.{u}` spaces `P` and `A` and any topological space `Y`, currying is a homeomorphism `\u03ba_u C_co(\u03ba_u (P \u00d7 A), Y) \u2243\u209c \u03ba_u C_co(P, \u03ba_u C_co(A, Y))`. Here `C_co` has the ordinary compact-open topology and `\u03ba_u` denotes refinement by compact Hausdorff probes in universe `u`. The product uses the original factor topologies, and the right-hand compact-open space uses the refined inner\u2026","title":"Compactly generated compact-open currying"},{"data_url":"scopes/formal-frontier/topology/ideal-completion.json","id":"formal-frontier/topology/ideal-completion","kind":"scope","parent":9024,"result":false,"route":"../roadmap/formal-frontier/topology/ideal-completion/index.html","scope":"formal-frontier/topology/ideal-completion","status":"fully_proved","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/ideal-completion at commit e316d02a, Apache-2.0.*","title":"Ideal completion"},{"id":"formal-frontier/topology/ideal-completion/principal-embedding","kind":"node","lean":"Order.Ideal.principalHom","parent":9033,"result":true,"route":"../roadmap/formal-frontier/topology/ideal-completion/index.html#principal-embedding","scope":"formal-frontier/topology/ideal-completion","status":"fully_proved","summary":"`Order.Ideal.principalHom` sends `a` to its principal ideal as a bounded lattice homomorphism; `principalHom_injective` shows it is injective. Its operation laws are available through the bundled homomorphism.","title":"Principal embedding"},{"id":"formal-frontier/topology/ideal-completion/prime-ideals-and-boolean-valued-maps","kind":"node","lean":"Order.Ideal.primeEquivBoolHom","parent":9033,"result":true,"route":"../roadmap/formal-frontier/topology/ideal-completion/index.html#prime-ideals-and-boolean-valued-maps","scope":"formal-frontier/topology/ideal-completion","status":"fully_proved","summary":"For `[Lattice A]` and `[BoundedOrder A]`, `Order.Ideal.primeEquivBoolHom` identifies prime order ideals with bounded lattice homomorphisms `A \u2192 Bool`. The characteristic map sends precisely the ideal to `false`; conversely `BoundedLatticeHom.falseIdeal` is its prime false fiber. Every such map is surjective. This includes the one-element lattice, for which both sides of the equivalence are empty.","title":"Prime ideals and Boolean-valued maps"},{"data_url":"scopes/formal-frontier/topology/metrization.json","id":"formal-frontier/topology/metrization","kind":"scope","parent":9024,"result":false,"route":"../roadmap/formal-frontier/topology/metrization/index.html","scope":"formal-frontier/topology/metrization","status":"fully_proved","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/metrization at commit 9c7cf0d8, Apache-2.0.*","title":"Metrization"},{"id":"formal-frontier/topology/metrization/sigma-locally-finite-basis-criterion","kind":"node","lean":"TopologicalSpace.metrizableSpace_of_sigmaLocallyFinite_isTopologicalBasis","parent":9036,"result":true,"route":"../roadmap/formal-frontier/topology/metrization/index.html#sigma-locally-finite-basis-criterion","scope":"formal-frontier/topology/metrization","status":"fully_proved","summary":"A T\u2083 space is metrizable if it has an open basis in countably many locally finite indexed levels. Each level may have an arbitrary index type; local finiteness counts the original indices, including repetitions.","title":"Sigma-locally-finite basis criterion"},{"id":"formal-frontier/topology/metrization/metric-converse","kind":"node","lean":"TopologicalSpace.exists_sigmaLocallyFinite_isTopologicalBasis","parent":9036,"result":true,"route":"../roadmap/formal-frontier/topology/metrization/index.html#metric-converse","scope":"formal-frontier/topology/metrization","status":"fully_proved","summary":"Every metrizable space has an open basis in countably many locally finite levels, each indexed by the points of the space.","title":"Metric converse"},{"id":"formal-frontier/topology/metrization/closed-metrizable-neighbourhood-cover-criterion","kind":"node","lean":"TopologicalSpace.metrizableSpace_of_locallyFinite_isClosed_interior_cover","parent":9036,"result":true,"route":"../roadmap/formal-frontier/topology/metrization/index.html#closed-metrizable-neighbourhood-cover-criterion","scope":"formal-frontier/topology/metrization","status":"fully_proved","summary":"A space is metrizable if it has a locally finite family of closed subsets whose interiors cover the space and each of which is metrizable in its original subtype topology. No ambient separation assumption is required.","title":"Closed metrizable neighbourhood-cover criterion"},{"id":"formal-frontier/topology/metrization/main-results","kind":"node","parent":9036,"result":false,"route":"../roadmap/formal-frontier/topology/metrization/main-results/index.html","scope":"formal-frontier/topology/metrization","status":"can_state","summary":"The main results that the formalization.yaml of FormalFrontier/metrization lists under `status.main_results` at commit 9c7cf0d8, each that the scan finds declared in the file the yaml gives, with its line. They are listed here, not made articles: in several Formal Frontier libraries this list is close to the whole API. The notes are the yaml's.","title":"Main results"},{"data_url":"scopes/formal-frontier/topology/restricted-product-duality.json","id":"formal-frontier/topology/restricted-product-duality","kind":"scope","parent":9024,"result":false,"route":"../roadmap/formal-frontier/topology/restricted-product-duality/index.html","scope":"formal-frontier/topology/restricted-product-duality","status":"fully_proved","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/restricted-product-duality at commit df6cc0bb, Apache-2.0.*","title":"Restricted-product duality"},{"id":"formal-frontier/topology/restricted-product-duality/compact-pontryagin-biduality","kind":"node","lean":"PontryaginDual.evalEquivOfCompact","parent":9041,"result":true,"route":"../roadmap/formal-frontier/topology/restricted-product-duality/index.html#compact-pontryagin-biduality","scope":"formal-frontier/topology/restricted-product-duality","status":"fully_proved","summary":"Canonical evaluation is a topological group isomorphism from every compact Hausdorff commutative topological group to its Circle-valued Pontryagin bidual.","title":"Compact Pontryagin biduality"},{"data_url":"scopes/formal-frontier/topology/spectral-stone-duality.json","id":"formal-frontier/topology/spectral-stone-duality","kind":"scope","parent":9024,"result":false,"route":"../roadmap/formal-frontier/topology/spectral-stone-duality/index.html","scope":"formal-frontier/topology/spectral-stone-duality","status":"fully_proved","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/spectral-stone-duality at commit 3d9e22b2, Apache-2.0.*","title":"Spectral Stone duality"},{"id":"formal-frontier/topology/spectral-stone-duality/specialization-and-soberification-dimensions","kind":"node","lean":"topologicalKrullDim_eq_krullDim_specialization, topologicalKrullDim_irreducibleCloseds","parent":9043,"result":true,"route":"../roadmap/formal-frontier/topology/spectral-stone-duality/index.html#specialization-and-soberification-dimensions","scope":"formal-frontier/topology/spectral-stone-duality","status":"fully_proved","summary":"For quasi-sober spaces, topological Krull dimension equals the order-theoretic Krull dimension of the original-point specialization preorder, and canonical soberification preserves topological Krull dimension for every topological space.","title":"Specialization and soberification dimensions"},{"id":"formal-frontier/topology/spectral-stone-duality/closed-points-among-generizations","kind":"node","lean":"isClosed_singleton_nhdsKer_iff","parent":9043,"result":true,"route":"../roadmap/formal-frontier/topology/spectral-stone-duality/index.html#closed-points-among-generizations","scope":"formal-frontier/topology/spectral-stone-duality","status":"fully_proved","summary":"In any `T\u2080` space, the original point is the unique closed point of its generization subspace `nhdsKer {x}`: `isClosed_singleton_nhdsKer_iff` characterizes every closed singleton in that subspace. No sobriety or spectrality is needed. See closed generizations.","title":"Closed points among generizations"},{"data_url":"scopes/formal-frontier/topology/sphere-topology.json","id":"formal-frontier/topology/sphere-topology","kind":"scope","parent":9024,"result":false,"route":"../roadmap/formal-frontier/topology/sphere-topology/index.html","scope":"formal-frontier/topology/sphere-topology","status":"fully_proved","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/sphere-topology at commit 672d46f9, Apache-2.0.*","title":"Sphere topology"},{"data_url":"scopes/formal-frontier/topology/sphere-topology/part-01.json","id":"formal-frontier/topology/sphere-topology/part-01","kind":"scope","navigation":"transparent","parent":9046,"result":false,"route":"../roadmap/formal-frontier/topology/sphere-topology/part-01/index.html","scope":"formal-frontier/topology/sphere-topology/part-01","status":"fully_proved","summary":"*Quoted from the headline results in README.md of FormalFrontier/sphere-topology at commit 672d46f9, Apache-2.0.*","title":"Brouwer fixed points on complex\u2026 to Fundamental groups of\u2026"},{"id":"formal-frontier/topology/sphere-topology/part-01/brouwer-fixed-points-on-complex-disks","kind":"node","lean":"Complex.exists_fixedPoint_closedBall_of_boundary_mapsTo, Complex.exists_fixedPoint_closedBall","parent":9047,"result":true,"route":"../roadmap/formal-frontier/topology/sphere-topology/part-01/index.html#brouwer-fixed-points-on-complex-disks","scope":"formal-frontier/topology/sphere-topology/part-01","status":"fully_proved","summary":"For c : \u2102 and r \u2265 0, every continuous complex-valued map on Metric.closedBall c r has a fixed point if it maps the sphere boundary into that ball, without any interior image bound.","title":"Brouwer fixed points on complex disks"},{"id":"formal-frontier/topology/sphere-topology/part-01/hairy-ball-theorem-for-the-metric-two-sphere","kind":"node","lean":"SphereTopology.sphere2_exists_zero_of_orthogonal_field","parent":9047,"result":true,"route":"../roadmap/formal-frontier/topology/sphere-topology/part-01/index.html#hairy-ball-theorem-for-the-metric-two-sphere","scope":"formal-frontier/topology/sphere-topology/part-01","status":"fully_proved","summary":"Every continuous map V from the unit sphere in EuclideanSpace \u211d (Fin 3) to EuclideanSpace \u211d (Fin 3) satisfying \u27e8p, V(p)\u27e9 = 0 at every sphere point p vanishes somewhere.","title":"Hairy-ball theorem for the metric two-sphere"},{"id":"formal-frontier/topology/sphere-topology/part-01/free-fundamental-groups-of-circle-bouquets","kind":"node","lean":"CircleBouquet.generatorEquiv, CircleBouquet.generatorEquiv_of","parent":9047,"result":true,"route":"../roadmap/formal-frontier/topology/sphere-topology/part-01/index.html#free-fundamental-groups-of-circle-bouquets","scope":"formal-frontier/topology/sphere-topology/part-01","status":"fully_proved","summary":"For any label type I and independent ambient and label universes, FreeGroup I is multiplicatively equivalent to the fundamental group of the point-wide-pushout CircleBouquet.space I at its common point, sending each free generator to its included counterclockwise complex-unit-circle loop, including for empty I.","title":"Free fundamental groups of circle bouquets"},{"id":"formal-frontier/topology/sphere-topology/part-01/sphere-cogroups-and-sharp-cocommutativity","kind":"node","lean":"Metric.Sphere.cogroup, Metric.Sphere.cogroup_isCommMonObj_iff","parent":9047,"result":true,"route":"../roadmap/formal-frontier/topology/sphere-topology/part-01/index.html#sphere-cogroups-and-sharp-cocommutativity","scope":"formal-frontier/topology/sphere-topology/part-01","status":"fully_proved","summary":"For every n : \u2115 and universe u, Metric.Sphere.cogroup n gives the last-axis-north-based ULift.{u} of the unit sphere in EuclideanSpace \u211d (Fin (n + 2)) a group-object structure in the ordinary opposite based-homotopy category, whose specified ordered-pinch monoid structure is commutative exactly when 0 < n.","title":"Sphere cogroups and sharp cocommutativity"},{"id":"formal-frontier/topology/sphere-topology/part-01/nontrivial-second-homotopy-group-of-the-metric-t","kind":"node","lean":"HomotopyGroup.nontrivial_sphere_two","parent":9047,"result":true,"route":"../roadmap/formal-frontier/topology/sphere-topology/part-01/index.html#nontrivial-second-homotopy-group-of-the-metric-t","scope":"formal-frontier/topology/sphere-topology/part-01","status":"fully_proved","summary":"In every universe u, the second homotopy group of Metric.Sphere.basedSpace.{u} 1, the last-axis-north-based ULift.{u} of the unit sphere in EuclideanSpace \u211d (Fin 3), is nontrivial.","title":"Nontrivial second homotopy group of the metric two-sphere"},{"id":"formal-frontier/topology/sphere-topology/part-01/h-space-fundamental-groups-commute","kind":"node","lean":"FundamentalGroup.mul_comm","parent":9047,"result":true,"route":"../roadmap/formal-frontier/topology/sphere-topology/part-01/index.html#h-space-fundamental-groups-commute","scope":"formal-frontier/topology/sphere-topology/part-01","status":"fully_proved","summary":"`FundamentalGroup.mul_comm` applies at the unit of any H-space, using only its two relative unit homotopies. No associativity, commutativity or strict unit laws for the multiplication are assumed. Continuous multiplication with a strict unit also gives `Path.mul_homotopic_trans`, without associativity or inverses.","title":"H-space fundamental groups commute"},{"id":"formal-frontier/topology/sphere-topology/part-01/last-axis-one-point-compactification-coordinates","kind":"node","lean":"EuclideanSpace.onePointHomeomorphSphere","parent":9047,"result":true,"route":"../roadmap/formal-frontier/topology/sphere-topology/part-01/index.html#last-axis-one-point-compactification-coordinates","scope":"formal-frontier/topology/sphere-topology/part-01","status":"fully_proved","summary":"`EuclideanSpace.onePointHomeomorphSphere` identifies `OnePoint (EuclideanSpace \u211d (Fin n))` with the unit sphere in `EuclideanSpace \u211d (Fin (n+1))`, for every `n`, including zero. Infinity maps to the last unit vector and finite zero to its antipode. Finite coordinates have denominator `\u2016x\u2016\u00b2+4`, and the off-pole inverse has coordinates `2*p_i/(1-p_last)`. The underlying append-zero-last linear isometry also applies\u2026","title":"Last-axis one-point compactification coordinates"},{"id":"formal-frontier/topology/sphere-topology/part-01/finite-generalized-loops-and-based-sphere-map-cl","kind":"node","lean":"GenLoop.sphereHomeomorph","parent":9047,"result":true,"route":"../roadmap/formal-frontier/topology/sphere-topology/part-01/index.html#finite-generalized-loops-and-based-sphere-map-cl","scope":"formal-frontier/topology/sphere-topology/part-01","status":"fully_proved","summary":"For every `n : \u2115`, `GenLoop.sphereHomeomorph` gives a homeomorphism from `GenLoop (Fin n) X x` to `{F : C(S\u207f, X) // F (EuclideanSpace.northPoint n) = x}`, where `S\u207f` is the unit sphere in `EuclideanSpace \u211d (Fin (n+1))`. `HomotopyGroup.equivSphereClasses` gives a natural set equivalence from `HomotopyGroup (Fin n) X x` to the existing `ZerothHomotopy` of this subtype. Inverse representatives precompose with\u2026","title":"Finite generalized loops and based sphere-map classes"},{"id":"formal-frontier/topology/sphere-topology/part-01/homotopy-groups-and-the-sphere-map-group","kind":"node","lean":"HomotopyGroup.mulOppositeEquivSphereHom","parent":9047,"result":true,"route":"../roadmap/formal-frontier/topology/sphere-topology/part-01/index.html#homotopy-groups-and-the-sphere-map-group","scope":"formal-frontier/topology/sphere-topology/part-01","status":"fully_proved","summary":"For `n : \u2115` and any `X : BasedTopCat.{u}`, `HomotopyGroup.mulOppositeEquivSphereHom` gives a multiplicative equivalence from `(HomotopyGroup (Fin (n + 1)) X.right (BasedTopCat.point X))\u1d50\u1d52\u1d56` to the canonical `CategoryTheory.yonedaGrp` evaluation of `Metric.Sphere.cogroup n` at `op (op (Q.obj X))`, where `Q := BasedTopCat.homotopyQuotient`. Its elements are incoming morphisms to the cogroup carrier in the ordinary\u2026","title":"Homotopy groups and the sphere-map group"},{"id":"formal-frontier/topology/sphere-topology/part-01/nontrivial-second-homotopy-group-of-s2","kind":"node","lean":"Metric.Sphere.id_not_nullhomotopic_two","parent":9047,"result":true,"route":"../roadmap/formal-frontier/topology/sphere-topology/part-01/index.html#nontrivial-second-homotopy-group-of-s2","scope":"formal-frontier/topology/sphere-topology/part-01","status":"fully_proved","summary":"`Metric.Sphere.id_not_nullhomotopic_two` shows that the identity of the unit two-sphere in `EuclideanSpace \u211d (Fin 3)`, lifted to any universe, is not homotopic to any constant map. `Metric.Sphere.homotopyQuotient_id_ne_const_two` distinguishes its identity class from the constant class at the last-axis north pole in the based-homotopy category. `HomotopyGroup.nontrivial_sphere_two` proves that the corresponding\u2026","title":"Nontrivial second homotopy group of S\u00b2"},{"id":"formal-frontier/topology/sphere-topology/part-01/canonical-antipodal-quotient-covering","kind":"node","lean":"Metric.Sphere.isQuotientCoveringMap_toProjectivization","parent":9047,"result":true,"route":"../roadmap/formal-frontier/topology/sphere-topology/part-01/index.html#canonical-antipodal-quotient-covering","scope":"formal-frontier/topology/sphere-topology/part-01","status":"fully_proved","summary":"For `E : Type w` with `[NormedAddCommGroup E] [NormedSpace \u211d E]`, `Metric.Sphere.isQuotientCoveringMap_toProjectivization` makes the canonical continuous map `Metric.Sphere.toProjectivization : C(Metric.sphere (0 : E) 1, Projectivization \u211d E)` a quotient covering for the scalar action of `Metric.sphere (0 : \u211d) 1`, whose elements act by identity and negation. The sphere uses the original norm; the target has the\u2026","title":"Canonical antipodal quotient covering"},{"id":"formal-frontier/topology/sphere-topology/part-01/real-projective-fundamental-groups-and-lifted-en","kind":"node","lean":"Projectivization.fundamentalGroupEquivZModTwo","parent":9047,"result":true,"route":"../roadmap/formal-frontier/topology/sphere-topology/part-01/index.html#real-projective-fundamental-groups-and-lifted-en","scope":"formal-frontier/topology/sphere-topology/part-01","status":"fully_proved","summary":"For a real normed space `E` with `2 < Module.rank \u211d E`, `Projectivization.fundamentalGroupEquivZModTwo` identifies `\u03c0\u2081(Projectivization \u211d E, b)` with `Multiplicative (ZMod 2)`. A chosen unit lift over the supplied basepoint fixes the lift description, but not the resulting coordinate. The identity means returning to that lift; `Multiplicative.ofAdd (1 : ZMod 2)` means reaching its antipode. The sphere uses the\u2026","title":"Real projective fundamental groups and lifted endpoints"},{"id":"formal-frontier/topology/sphere-topology/part-01/embeddings-closed-maps-and-neighborhoods-of-poin","kind":"node","lean":"TopCat.isEmbedding_widePushoutDesc_point","parent":9047,"result":true,"route":"../roadmap/formal-frontier/topology/sphere-topology/part-01/index.html#embeddings-closed-maps-and-neighborhoods-of-poin","scope":"formal-frontier/topology/sphere-topology/part-01","status":"fully_proved","summary":"`TopCat.isEmbedding_widePushoutDesc_point` promotes point-preserving component embeddings to an embedding of the canonical wide pushouts; open or closed component embeddings give an open or closed embedding, respectively. Point reflection and component open or closed maps also suffice for the corresponding map property. The index may be arbitrary or empty, and source and target universes are independent, with no\u2026","title":"Embeddings, closed maps and neighborhoods of point-wide-pushouts"},{"id":"formal-frontier/topology/sphere-topology/part-01/point-relative-equivalences-and-contractions","kind":"node","lean":"TopCat.widePushoutPointHomotopyEquiv","parent":9047,"result":true,"route":"../roadmap/formal-frontier/topology/sphere-topology/part-01/index.html#point-relative-equivalences-and-contractions","scope":"formal-frontier/topology/sphere-topology/part-01","status":"fully_proved","summary":"`TopCat.widePushoutPointHomotopyEquiv` descends point-preserving factor maps and supplied relative inverse homotopies. Inverse point preservation follows from the forward maps and left relative homotopies. The chosen relative witnesses compute on the head and every factor. Supplied contractions fixing the attaching points contract the wide pushout, including an empty index family. This is stronger data than\u2026","title":"Point-relative equivalences and contractions"},{"id":"formal-frontier/topology/sphere-topology/part-01/factor-retractions-and-separation-of-point-wide","kind":"node","lean":"TopCat.widePushoutPointRetraction","parent":9047,"result":true,"route":"../roadmap/formal-frontier/topology/sphere-topology/part-01/index.html#factor-retractions-and-separation-of-point-wide","scope":"formal-frontier/topology/sphere-topology/part-01","status":"fully_proved","summary":"`TopCat.widePushoutPointRetraction` retracts the categorical wide pushout of arbitrary chosen-point spaces onto any selected factor, and each factor inclusion is an embedding. The retractions jointly distinguish points, without separation or finiteness assumptions, even for the empty index family. The pushout is T\u2080, T\u2081 or Hausdorff exactly when every factor has that property; factorwise inheritance is available by\u2026","title":"Factor retractions and separation of point-wide-pushouts"},{"id":"formal-frontier/topology/sphere-topology/part-01/charts-around-a-common-subset","kind":"node","lean":"TopCat.widePushoutPointChart","parent":9047,"result":true,"route":"../roadmap/formal-frontier/topology/sphere-topology/part-01/index.html#charts-around-a-common-subset","scope":"formal-frontier/topology/sphere-topology/part-01","status":"fully_proved","summary":"`TopCat.widePushoutPointChart` adjoins one factor to a subset containing the common point. Distinct charts intersect in that subset, and the subset together with the charts covers the pushout, including an empty family. Factorwise path connectedness of its inverse images makes the common subset path connected; a chart then needs path connectedness only of its own whole factor. An open common subset gives open\u2026","title":"Charts around a common subset"},{"id":"formal-frontier/topology/sphere-topology/part-01/fundamental-groups-of-factor-plus-core-charts","kind":"node","lean":"TopCat.widePushoutPointChartFundamentalGroupDesc","parent":9047,"result":true,"route":"../roadmap/formal-frontier/topology/sphere-topology/part-01/index.html#fundamental-groups-of-factor-plus-core-charts","scope":"formal-frontier/topology/sphere-topology/part-01","status":"fully_proved","summary":"`TopCat.widePushoutPointChartFundamentalGroupDesc` glues homomorphisms on an arbitrary path-connected open neighborhood of the common point and its charts, assuming path connectedness of each whole factor. Only agreement on the core is required. The inclusion-induced core and chart restrictions characterize the result, for arbitrary target groups in an independent universe, empty index families and coincident\u2026","title":"Fundamental groups of factor-plus-core charts"},{"data_url":"scopes/formal-frontier/topology/sphere-topology/part-02.json","id":"formal-frontier/topology/sphere-topology/part-02","kind":"scope","navigation":"transparent","parent":9046,"result":false,"route":"../roadmap/formal-frontier/topology/sphere-topology/part-02/index.html","scope":"formal-frontier/topology/sphere-topology/part-02","status":"fully_proved","summary":"*Quoted from the headline results in README.md of FormalFrontier/sphere-topology at commit 672d46f9, Apache-2.0.*","title":"Subfamily retractions and\u2026 to Covering-space lifts of\u2026"},{"id":"formal-frontier/topology/sphere-topology/part-02/subfamily-retractions-and-continuous-factorizati","kind":"node","lean":"TopCat.widePushoutPointSubfamilyRetraction","parent":9065,"result":true,"route":"../roadmap/formal-frontier/topology/sphere-topology/part-02/index.html#subfamily-retractions-and-continuous-factorizati","scope":"formal-frontier/topology/sphere-topology/part-02","status":"fully_proved","summary":"`TopCat.widePushoutPointSubfamilyRetraction` retracts onto any subfamily, with an embedding inclusion whose range includes the common point. This inclusion is closed exactly when the chosen-point singleton in every omitted factor is closed; retained factors need no separation hypothesis. A continuous map factors through that inclusion exactly when its support lies in the subfamily, without separation or compactness\u2026","title":"Subfamily retractions and continuous factorization"},{"id":"formal-frontier/topology/sphere-topology/part-02/strict-finite-stage-factorization-of-compact-hom","kind":"node","lean":"TopCat.exists_finite_widePushout_point_subfamily_homotopyRel","parent":9065,"result":true,"route":"../roadmap/formal-frontier/topology/sphere-topology/part-02/index.html#strict-finite-stage-factorization-of-compact-hom","scope":"formal-frontier/topology/sphere-topology/part-02","status":"fully_proved","summary":"`TopCat.exists_finite_widePushout_point_subfamily_homotopyRel` factors a prescribed compact-image relative homotopy in a point-wide-pushout of T\u2081 factors through a finite subfamily containing any prescribed finite set. Recovery holds at every pair of parameters, including both endpoint maps and arbitrary relative subsets. Compact domains suffice, without source separation, factor compactness or countability. At a\u2026","title":"Strict finite-stage factorization of compact homotopies"},{"id":"formal-frontier/topology/sphere-topology/part-02/arbitrary-index-circle-bouquets","kind":"node","lean":"CircleBouquet.space","parent":9065,"result":true,"route":"../roadmap/formal-frontier/topology/sphere-topology/part-02/index.html#arbitrary-index-circle-bouquets","scope":"formal-frontier/topology/sphere-topology/part-02","status":"fully_proved","summary":"`CircleBouquet.space` is the wide pushout of based complex circles, with a common point even for the empty family and independent index/ambient universes. Inclusions, descent and extensionality give its universal property; retractions detect the positive generators as nonidentity. `CircleBouquet.generatorEquiv` identifies its fundamental group with `FreeGroup I`, sending each free generator to the included\u2026","title":"Arbitrary-index circle bouquets"},{"id":"formal-frontier/topology/sphere-topology/part-02/diagonal-equivalence-for-non-antipodal-pairs","kind":"node","lean":"Metric.Sphere.diagonalHomotopyEquiv","parent":9065,"result":true,"route":"../roadmap/formal-frontier/topology/sphere-topology/part-02/index.html#diagonal-equivalence-for-non-antipodal-pairs","scope":"formal-frontier/topology/sphere-topology/part-02","status":"fully_proved","summary":"In any real normed vector space, `Metric.Sphere.diagonalHomotopyEquiv` has the diagonal as its forward map and first projection as its inverse. Normalized affine interpolation stays in the non-antipodal-pair subspace and fixes the diagonal. No finite-dimensionality or nonemptiness hypothesis is needed.","title":"Diagonal equivalence for non-antipodal pairs"},{"id":"formal-frontier/topology/sphere-topology/part-02/covering-space-lifts-of-simplices","kind":"node","lean":"IsCoveringMap.singularLiftEquiv","parent":9065,"result":true,"route":"../roadmap/formal-frontier/topology/sphere-topology/part-02/index.html#covering-space-lifts-of-simplices","scope":"formal-frontier/topology/sphere-topology/part-02","status":"fully_proved","summary":"For a covering `p : Y \u27f6 X`, `IsCoveringMap.singularLiftEquiv` identifies the lift fiber of a singular simplex with a point fiber by evaluation at any point. Restriction is bijective for every simplex operator, including degeneracies and faces omitting vertex zero. There are no nonemptiness, connectedness, local connectedness, surjectivity or finite- sheet assumptions on the base or total space. The singular map is\u2026","title":"Covering-space lifts of simplices"},{"id":"formal-frontier/topology/sphere-topology/main-results","kind":"node","parent":9046,"result":false,"route":"../roadmap/formal-frontier/topology/sphere-topology/main-results/index.html","scope":"formal-frontier/topology/sphere-topology","status":"can_state","summary":"The main results that the formalization.yaml of FormalFrontier/sphere-topology lists under `status.main_results` at commit 672d46f9, each that the scan finds declared in the file the yaml gives, with its line. They are listed here, not made articles: in several Formal Frontier libraries this list is close to the whole API. The notes are the yaml's.","title":"Main results"},{"data_url":"scopes/formal-frontier/topology/vector-bundles.json","id":"formal-frontier/topology/vector-bundles","kind":"scope","parent":9024,"result":false,"route":"../roadmap/formal-frontier/topology/vector-bundles/index.html","scope":"formal-frontier/topology/vector-bundles","status":"fully_proved","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/vector-bundles at commit e3fb9de3, Apache-2.0.*","title":"Vector bundles"},{"id":"formal-frontier/topology/vector-bundles/compatible-hermitian-metrics","kind":"node","lean":"VectorBundle.continuousHermitianMetric","parent":9072,"result":true,"route":"../roadmap/formal-frontier/topology/vector-bundles/index.html#compatible-hermitian-metrics","scope":"formal-frontier/topology/vector-bundles","status":"fully_proved","summary":"Every vector bundle over an RCLike scalar field with finite-dimensional normed model and paracompact Hausdorff base admits a continuous positive-definite Hermitian fiber metric compatible with the original fiber topologies.","title":"Compatible Hermitian metrics"},{"id":"formal-frontier/topology/vector-bundles/orthogonal-complements-for-a-supplied-metric","kind":"node","lean":"Bundle.ContinuousHermitianMetric.orthogonalBundle, Bundle.ContinuousHermitianMetric.orthogonalAdditionHomeomorph","parent":9072,"result":true,"route":"../roadmap/formal-frontier/topology/vector-bundles/index.html#orthogonal-complements-for-a-supplied-metric","scope":"formal-frontier/topology/vector-bundles","status":"fully_proved","summary":"For a vector bundle over an RCLike scalar field with finite-dimensional normed model and a supplied continuous Hermitian metric, every intrinsic vector bundle of fiber submodules with a normed model and a topologically embedded total inclusion has an orthogonal-complement vector bundle in the original fiber subtype and ambient-induced total topologies, and addition identifies their fiberwise product with its\u2026","title":"Orthogonal complements for a supplied metric"},{"id":"formal-frontier/topology/vector-bundles/fiberwise-linear-isomorphism-criterion","kind":"node","lean":"VectorBundle.homeomorphOfFiberwiseLinearEquivOfIsModuleTopology","parent":9072,"result":true,"route":"../roadmap/formal-frontier/topology/vector-bundles/index.html#fiberwise-linear-isomorphism-criterion","scope":"formal-frontier/topology/vector-bundles","status":"fully_proved","summary":"Over a nontrivially normed field, a family of fiberwise algebraic linear equivalences between two vector bundles on the same topological base induces a base-preserving homeomorphism of their original total spaces whenever its total-space map is continuous, the source normed model is finite-dimensional and the target normed model has the module topology.","title":"Fiberwise linear isomorphism criterion"},{"id":"formal-frontier/topology/vector-bundles/sigma-compactness-of-a-fiber-bundle","kind":"node","lean":"FiberBundle.sigmaCompactSpace_totalSpace_iff","parent":9072,"result":true,"route":"../roadmap/formal-frontier/topology/vector-bundles/index.html#sigma-compactness-of-a-fiber-bundle","scope":"formal-frontier/topology/vector-bundles","status":"fully_proved","summary":"For a fiber bundle with nonempty R\u2081 base and nonempty model, the original total space is sigma-compact if and only if the base and model are both sigma-compact.","title":"Sigma-compactness of a fiber bundle"},{"id":"formal-frontier/topology/vector-bundles/total-space-continuity-of-semilinear-maps-from-f","kind":"node","lean":"LinearMap.continuous_total_of_local_coordinates_of_basis_semilinear","parent":9072,"result":true,"route":"../roadmap/formal-frontier/topology/vector-bundles/index.html#total-space-continuity-of-semilinear-maps-from-f","scope":"formal-frontier/topology/vector-bundles","status":"fully_proved","summary":"Let `p : T \u2192 X` be continuous, with given `R`-module structures and subspace topologies on its equality-subtype fibres, and let `\u03c3 : R \u2192+* S` be a homomorphism of semirings, with a topology on `S`. Coordinate models with finite bases may vary over an arbitrary open cover. If their scalar basis coordinates composed with `\u03c3` and exact inverse vector coordinates are continuous, parameter-continuity of the images of\u2026","title":"Total-space continuity of semilinear maps from finite local bases"},{"id":"formal-frontier/topology/vector-bundles/main-results","kind":"node","parent":9072,"result":false,"route":"../roadmap/formal-frontier/topology/vector-bundles/main-results/index.html","scope":"formal-frontier/topology/vector-bundles","status":"can_state","summary":"The main results that the formalization.yaml of FormalFrontier/vector-bundles lists under `status.main_results` at commit e3fb9de3, each that the scan finds declared in the file the yaml gives, with its line. They are listed here, not made articles: in several Formal Frontier libraries this list is close to the whole API. The notes are the yaml's.","title":"Main results"},{"data_url":"scopes/formal-frontier/analysis.json","id":"formal-frontier/analysis","kind":"scope","parent":8707,"result":false,"route":"../roadmap/formal-frontier/analysis/index.html","scope":"formal-frontier/analysis","status":"fully_proved","summary":"The Formal Frontier libraries on analysis, one section each.","title":"Analysis"},{"data_url":"scopes/formal-frontier/analysis/complex-analysis.json","id":"formal-frontier/analysis/complex-analysis","kind":"scope","parent":9079,"result":false,"route":"../roadmap/formal-frontier/analysis/complex-analysis/index.html","scope":"formal-frontier/analysis/complex-analysis","status":"fully_proved","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/complex-analysis at commit c22c803e, Apache-2.0.*","title":"Complex analysis"},{"id":"formal-frontier/analysis/complex-analysis/finite-dimensional-differentiability-implies-ana","kind":"node","lean":"DifferentiableOn.analyticOnNhd_of_finiteDimensional","parent":9080,"result":true,"route":"../roadmap/formal-frontier/analysis/complex-analysis/index.html#finite-dimensional-differentiability-implies-ana","scope":"formal-frontier/analysis/complex-analysis","status":"fully_proved","summary":"A complex Fr\u00e9chet-differentiable map on an open subset of a finite-dimensional complex normed space is analytic near every point of that set, when its target is a complete complex normed space; dimension zero is included.","title":"Finite-dimensional differentiability implies analyticity"},{"id":"formal-frontier/analysis/complex-analysis/differentiability-analyticity-equivalence","kind":"node","lean":"analyticOnNhd_iff_differentiableOn_of_finiteDimensional","parent":9080,"result":true,"route":"../roadmap/formal-frontier/analysis/complex-analysis/index.html#differentiability-analyticity-equivalence","scope":"formal-frontier/analysis/complex-analysis","status":"fully_proved","summary":"On such an open set, analyticity near every point is equivalent to complex Fr\u00e9chet differentiability, under the same source and target hypotheses.","title":"Differentiability\u2013analyticity equivalence"},{"id":"formal-frontier/analysis/complex-analysis/polydisc-cauchy-formula","kind":"node","lean":"DifferentiableOn.torusIntegral_cauchy","parent":9080,"result":true,"route":"../roadmap/formal-frontier/analysis/complex-analysis/index.html#polydisc-cauchy-formula","scope":"formal-frontier/analysis/complex-analysis","status":"fully_proved","summary":"A complex-differentiable map on a positive-radius closed coordinate polydisc, valued in a complete complex normed space, agrees at interior points with its normalized torus boundary integral.","title":"Polydisc Cauchy formula"},{"id":"formal-frontier/analysis/complex-analysis/polynomial-boundary-degree-counts-interior-roots","kind":"node","lean":"Polynomial.degree_normalizedCircleMap","parent":9080,"result":true,"route":"../roadmap/formal-frontier/analysis/complex-analysis/index.html#polynomial-boundary-degree-counts-interior-roots","scope":"formal-frontier/analysis/complex-analysis","status":"fully_proved","summary":"A complex polynomial with no zeros on the unit circle gives an unbased continuous circle self-map by normalizing its values. Its degree equals the number of roots strictly inside the unit disk, counted with multiset multiplicity. Nonmonic polynomials, repeated roots and nonzero constants of arbitrary phase are included.","title":"Polynomial boundary degree counts interior roots"},{"id":"formal-frontier/analysis/complex-analysis/exterior-polynomial-root-scaling-homotopy","kind":"node","lean":"Polynomial.normalizedCircleMapScaleRootsHomotopy","parent":9080,"result":true,"route":"../roadmap/formal-frontier/analysis/complex-analysis/index.html#exterior-polynomial-root-scaling-homotopy","scope":"formal-frontier/analysis/complex-analysis","status":"fully_proved","summary":"If `p.eval w \u2260 0` whenever `1 \u2264 \u2016w\u2016`, root scaling over the real unit interval joins the normalized leading monomial to the canonical normalized boundary map. Its characteristic value is `normalize ((p.scaleRoots (t : \u2102)).eval z)`. The leading coefficient's arbitrary phase is retained; the corresponding degree statement is `Circle.degree (p.normalizedCircleMap _) = (p.natDegree : \u2124)`.","title":"Exterior polynomial root-scaling homotopy"},{"id":"formal-frontier/analysis/complex-analysis/compact-scalar-kernel-operators","kind":"node","lean":"ContinuousMap.kernelOperator","parent":9080,"result":true,"route":"../roadmap/formal-frontier/analysis/complex-analysis/index.html#compact-scalar-kernel-operators","scope":"formal-frontier/analysis/complex-analysis","status":"fully_proved","summary":"A continuous scalar kernel on `X \u00d7 Y`, with compact `Y` and a finite Borel measure, gives a compact complex-linear map from `L\u00b2(Y, \u03bc)` to `C(X, \u2102)`. The parameter space `X` is arbitrary; compactness of `X` is needed only for the sup-norm bound. The construction uses a generic transpose of an operator-norm-continuous field, compact whenever its normed target is proper.","title":"Compact scalar-kernel operators"},{"id":"formal-frontier/analysis/complex-analysis/detecting-finite-eigenspaces-of-compact-positive","kind":"node","lean":"ContinuousLinearMap.exists_eigenvector_comp_adjoint_detected","parent":9080,"result":true,"route":"../roadmap/formal-frontier/analysis/complex-analysis/index.html#detecting-finite-eigenspaces-of-compact-positive","scope":"formal-frontier/analysis/complex-analysis","status":"fully_proved","summary":"On an arbitrary complete real or complex Hilbert space, compact `T` and continuous linear `B` with `B.comp T \u2260 0` give a nonzero eigenvalue of `T.comp T.adjoint` whose finite-dimensional eigenspace contains a vector `v` with `B v \u2260 0`. No separability or commutation hypothesis is needed.","title":"Detecting finite eigenspaces of compact positive squares"},{"data_url":"scopes/formal-frontier/analysis/sequence-growth.json","id":"formal-frontier/analysis/sequence-growth","kind":"scope","parent":9079,"result":false,"route":"../roadmap/formal-frontier/analysis/sequence-growth/index.html","scope":"formal-frontier/analysis/sequence-growth","status":"fully_proved","summary":"*Quoted from the project description in formalization.yaml of FormalFrontier/sequence-growth at commit ccf81c71, Apache-2.0.*","title":"Sequence growth"},{"id":"formal-frontier/analysis/sequence-growth/nat-divisiongrowth-affine-lower-bound","kind":"node","lean":"Nat.DivisionGrowth.affine_lower_bound","parent":9088,"result":true,"route":"../roadmap/formal-frontier/analysis/sequence-growth/index.html#nat-divisiongrowth-affine-lower-bound","scope":"formal-frontier/analysis/sequence-growth","status":"fully_proved","summary":"`Nat.DivisionGrowth.affine_lower_bound`: if `0 < s < n` and `\u2200 m, n * (u m / s) \u2264 u (m + 1)`, then for every `m : \u2115`, with `\u03bb = (n : \u211a) / s` and `C = (n : \u211a) * ((s - 1 : \u2115) : \u211a) / ((n : \u211a) - s)`, `\u03bb ^ m * ((u 0 : \u211a) - C) + C \u2264 (u m : \u211a)`. There is no seed condition.","title":"`Nat.DivisionGrowth.affine_lower_bound`"},{"id":"formal-frontier/analysis/sequence-growth/nat-divisiongrowth-eventually-dominates","kind":"node","lean":"Nat.DivisionGrowth.eventually_dominates","parent":9088,"result":true,"route":"../roadmap/formal-frontier/analysis/sequence-growth/index.html#nat-divisiongrowth-eventually-dominates","scope":"formal-frontier/analysis/sequence-growth","status":"fully_proved","summary":"`Nat.DivisionGrowth.eventually_dominates`: under the same recurrence, `0 < s`, `0 < b`, `s * b < n`, and the sufficient natural seed `n * (s - 1) < (n - s) * u 0`, for each `K : \u2115` there is `M : \u2115` such that every `m \u2265 M` satisfies `K * b ^ m < u m`.","title":"`Nat.DivisionGrowth.eventually_dominates`"},{"id":"formal-frontier/analysis/sequence-growth/main-results","kind":"node","parent":9088,"result":false,"route":"../roadmap/formal-frontier/analysis/sequence-growth/main-results/index.html","scope":"formal-frontier/analysis/sequence-growth","status":"can_state","summary":"The main results that the formalization.yaml of FormalFrontier/sequence-growth lists under `status.main_results` at commit ccf81c71, each that the scan finds declared in the file the yaml gives, with its line. They are listed here, not made articles: in several Formal Frontier libraries this list is close to the whole API. The notes are the yaml's.","title":"Main 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