chars int64 40 24k | content_type stringclasses 9
values | id stringlengths 13 64 | source stringclasses 5
values | subset stringclasses 31
values | text stringlengths 40 24k |
|---|---|---|---|---|---|
23,994 | lean_file_chunk | b0b8de5561ec887473ba1fb8c655a825f6d5ffe48ec476d84c5c676898b81078 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
module
public import Mathlib.Logic.Relator
public import Mathlib.Tactic.Use
public import Mathlib.Tactic.MkIffOfInductiveProp
public import Mathlib.Tactic.SimpRw
pub... |
6,828 | lean_file_chunk | 280f67f57d979f86cd225493bdcc5aadc4df24df6f2fb83b4462d36841813b7d | lean_source_file | pkg_mathlib_source_file_chunks | reflTransGen_eq_self
theorem ReflTransGen.lift' {p : β → β → Prop} {a b : α} (f : α → β)
(h : ∀ a b, r a b → ReflTransGen p (f a) (f b))
(hab : ReflTransGen r a b) : ReflTransGen p (f a) (f b) := by
simpa [reflTransGen_eq_self] using hab.lift f h
theorem reflTransGen_closed {p : α → α → Prop} :
(∀ a b, ... |
2,744 | lean_file_chunk | 07aad1da0b7e07d42ff2363987948347dfa5c600bc66f40cff28d5789a1bde23 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
module
public import Mathlib.Init
/-!
# Typeclasses for commuting heterogeneous operations
The three classes in this file are for two-argument functions ... |
8,143 | lean_file_chunk | 9411092c45384eb81a2e79e47c8af46d9ab7e4297e1b2b35f155f4d391324937 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
module
public import Mathlib.Data.Finsupp.Lex
public import Mathlib.Data.Finsupp.Multiset
public import Mathlib.Order.GameAdd
/-!
# Termination of a hydra game
This file dea... |
5,595 | lean_file_chunk | b86ce59ce508e55c0cc79b593d4d4220aaed5d40ae82c28be291324a970f38c2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
module
public import Mathlib.Logic.Function.Defs
/-!
# Relator for functions, pairs, sums, and lists.
-/
@[expose] public section
namespace Relator
universe u₁ u₂... |
23,957 | lean_file_chunk | 86a78068e21442e4cc34bac283904298d78551507a80499dc8dce257a07dba30 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.CategoryTheory.Functor.Hom
public import Mathlib.CategoryTheory.Products.Basic
public import Mathlib.Data.ULift
public import Mathlib.Logic.... |
23,954 | lean_file_chunk | b0f4c5bdbe54936880731e689cd292a1a195aa11bbe7f59b50bb6f150dd3493f | lean_source_file | pkg_mathlib_source_file_chunks | hF.has_corepresentation.choose_spec.some
variable {F} in
/-- Any corepresenting object for a corepresentable functor `F` is isomorphic to `coreprX F`. -/
noncomputable def CorepresentableBy.isoCoreprX {Y : C} (e : F.CorepresentableBy Y) :
Y ≅ F.coreprX :=
CorepresentableBy.uniqueUpToIso e (corepresentableBy F)
... |
5,684 | lean_file_chunk | 272b269f0c8d52844a4ab52cf7b88afef05765474909eaaedcbc66e4a2472cde | lean_source_file | pkg_mathlib_source_file_chunks | variable {D : Type*} [Category.{v₁} D] (F : C ⥤ D)
/-- The natural transformation `yoneda.obj X ⟶ F.op ⋙ yoneda.obj (F.obj X)`
when `F : C ⥤ D` and `X : C`. -/
def yonedaMap (X : C) : yoneda.obj X ⟶ F.op ⋙ yoneda.obj (F.obj X) where
app _ := TypeCat.ofHom fun f ↦ F.map f
@[simp]
lemma yonedaMap_app_apply {Y : C} {X... |
7,266 | lean_file_chunk | 18f2cc377c1ad57787b740f58c1352eff814061ad5fc33f647c796dc158ac80f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
module
public import Mathlib.CategoryTheory.Equivalence
/-!
# 2-commutative squares of functors
Similarly to `Mathlib/CategoryTheory/CommSq.lean`, which defines the notion o... |
23,936 | lean_file_chunk | bcbeb2981a168544d6c7065cdafbc9f49ea71e6e7e3fc5f555762883dcd07436 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.StrictInitial
public import Mathlib.CategoryTheory.Limits.Types.Coproducts
public import Mathlib.CategoryTheory.L... |
5,057 | lean_file_chunk | 198b3008ca86d4b24414f2f147af5ed5e1e707f569185b6fe963a9a5258883b2 | lean_source_file | pkg_mathlib_source_file_chunks | isPullback_initial_to_of_cofan_isVanKampen (isVanKampen_finiteCoproducts hc) i j e
lemma FinitaryExtensive.isPullback_initial_to_sigma_ι [FinitaryExtensive C] {ι : Type*} [Finite ι]
(X : ι → C) (i j : ι) (e : i ≠ j) :
IsPullback (initial.to _) (initial.to _) (Sigma.ι X i) (Sigma.ι X j) :=
FinitaryExtensive.i... |
17,228 | lean_file_chunk | a2d0635cb63b34ca480d1096a151c70f251547c820f5b29f738bf74f5d7715c5 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
module
public import Mathlib.CategoryTheory.Adjunction.Basic
public import Mathlib.CategoryTheory.Category.Preorder
public import Mathlib.CategoryTheory.IsomorphismClass... |
12,205 | lean_file_chunk | 2088698ae14e38b3ab37d030b4bfd85bd64171fe1bc74e76aa5d3afe7172337d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.CategoryTheory.Category.ULift
public import Mathlib.CategoryTheory.EqToHom
public import Mathlib.CategoryTheory.Skeletal
public import Mathl... |
8,658 | lean_file_chunk | dfd5d0f7a21d769bae9e019064a95a020c87ed15347d1840e735fc3c4ba221aa | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Robin Carlier
-/
module
public import Mathlib.CategoryTheory.Groupoid
public import Mathlib.CategoryTheory.Types.Basic
public import Mathlib.CategoryTheory.Whiskering
publ... |
5,062 | lean_file_chunk | 00a8cbb492dc4628b496e3cb109a46ffd78d494c12c6fa24a11818639a04fac1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
module
public import Mathlib.CategoryTheory.Yoneda
/-!
# `IsRepresentedBy` predicate
In this file we define the predicate `IsRepresentedBy`: A presheaf `F` is ... |
2,263 | lean_file_chunk | 07c17fb490ef2c00e97f4b1354349b02722a98cd0adf748c8df571c57b4342be | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks
/-!
## Locally directed gluing
We say that a diagram of sets is "locally directed" if for any `V,... |
1,783 | lean_file_chunk | 123d044bce14ac8aeb227963ae78cffbc97f8f0d573fce6fdc6341c60afea348 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.CategoryTheory.Category.Cat
public import Mathlib.CategoryTheory.Groupoid
public import Mathlib.CategoryTheory.Types.Basic
/-!
# Obje... |
3,660 | lean_file_chunk | 0324e797596170a5855bb5c2b4f441c7538b60c921318339ce62ac42b15a132d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.CategoryTheory.Iso
/-!
# Conjugate morphisms by isomorphisms
We define
`CategoryTheory.Iso.homCongr : (X ≅ X₁) → (Y ≅ Y₁) → (X ⟶ Y) ... |
12,293 | lean_file_chunk | f64fabd4e4f66e98e9f51ebb137c515a74be46de75a11dfd7a6fe2cb1eea1887 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Kim Morrison
-/
module
public import Mathlib.CategoryTheory.Opposites
public import Mathlib.CategoryTheory.Groupoid
/-!
# Facts about epimorphisms and monomorphisms.
The d... |
5,156 | lean_file_chunk | 0915bb03df379acb7895849ae57b2f377160e05d33bd33f8b3ab70ec89f1767c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jack McKoen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jack McKoen
-/
module
public import Mathlib.CategoryTheory.Comma.Arrow
public import Mathlib.CategoryTheory.EpiMono
/-!
# Retracts
Defines retracts of objects and morphisms.
-/
@[exp... |
15,057 | lean_file_chunk | f42fcdc479b39f779d139264d6e2d19f5134dc8c23862f67d976e0abf2c7eb80 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Kyle Miller, Adam Topaz, Rémi Bottinelli, Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Adam Topaz, Rémi Bottinelli, Junyan Xu
-/
module
public import Mathlib.Topology.Category.TopCat.Limits.Konig
/-!
# Cofiltered system... |
1,447 | lean_file_chunk | 04454a8645e343289c786905c711379750a931c02cc6610176fa7c1d7454521a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
module
public import Mathlib.CategoryTheory.EpiMono
/-!
# Balanced categories
A category is called balanced if any morphism that is both monic and epic is an isomorp... |
17,781 | lean_file_chunk | 94cb357f6a4862b98f3bf6290eb072ac2463708a1bfb39199c459e04b179f92e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Joël Riou
-/
module
public import Mathlib.CategoryTheory.Shift.Basic
public import Mathlib.Data.Set.Subsingleton
public import Mathlib.Algebra.Group.Int.Defs
/-!
# The ca... |
3,234 | lean_file_chunk | 66d34d93c7463cb8449589e535b7cddc7363c9356f2e68bc641393173a67f916 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
module
public import Mathlib.CategoryTheory.Functor.Const
public import Mathlib.CategoryTheory.Discrete.Basic
public import Mathlib.Data.ULift
/-!
# The c... |
2,695 | lean_file_chunk | a1c53353105f973a1ff6c822859108edab83badcc432ec2b5203ddc66be2997e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
module
public import Mathlib.CategoryTheory.Subobject.Lattice
public import Mathlib.CategoryTheory.Limits.Shapes.StrongEpi
/-!
# Extremal epimorphisms
An extremal epimorphis... |
4,279 | lean_file_chunk | 239d90df1731e434fafa8e851318d2ecca92bc276645c66ed4f3a3a9ccd7aa59 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
module
public import Mathlib.Tactic.CategoryTheory.Reassoc
/-!
# Natural transformations
Defines natural transformations... |
6,488 | lean_file_chunk | 76d6962caa6a30a08701e957312667c9001b103a3cbf09a2feae9ee266e26d03 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
public import Mathlib.CategoryTheory.Limits.Shapes.Terminal
public import Mathlib.CategoryTheory... |
903 | lean_file_chunk | 3fd5b51de454cdab577a2d330d540087e0da1f6b74aaad3456d91a827c0637fe | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.Tactic.CategoryTheory.Elementwise
public import Mathlib.CategoryTheory.ConcreteCategory.Basic
/-!
# Use the `elementwise` attribute to crea... |
1,577 | lean_file_chunk | b617eaeb73c0e6929e38f3da0a16ab672b438f34e5398a115576d4a4e56c261a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.CategoryTheory.EssentialImage
public import Mathlib.CategoryTheory.Types.Basic
public import Mathlib.Logic.UnivLE
/-!
# Universe inequaliti... |
5,605 | lean_file_chunk | 949a96fd48974febbb89d58e529b25960b6abc83d4f824449befa9cc1b72ac40 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Alvaro Belmonte. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alvaro Belmonte, Joël Riou
-/
module
public import Mathlib.CategoryTheory.EqToHom
public import Mathlib.CategoryTheory.Pi.Basic
public import Mathlib.Data.ULift
public import Mathlib... |
7,368 | lean_file_chunk | 00e9138bed7ff88755f10d9ad33cc269f5260ca65348facc1214d37ecaef18e0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Kim Morrison, David Wärn
-/
module
public import Mathlib.CategoryTheory.ObjectProperty.FullSubcategory
public import Mathlib.CategoryTheory.Products.Basic
public import Math... |
12,262 | lean_file_chunk | 7664b33d19745a96d1c3f53293ad1b04e88794d92cced7d5e411a6e4207217f0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
module
public import Mathlib.CategoryTheory.NatIso
public import Mathlib.CategoryTheory.EqToHom
public import Mathlib.CategoryTheory.Groupoid
/-!
# Quotient category
Const... |
8,552 | lean_file_chunk | 0f2d8865190258a1b6d666a472166c80f50e3c5f63fc39966f4abf25da893986 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Kim Morrison
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
public import Mathlib.CategoryTheory.Limits.Shapes.Kernels
public import Mathlib.C... |
3,717 | lean_file_chunk | 22bc145871a0fddf552756d918cc9464f208b3b01130ce07e87bbea7d83325c6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Algebra.Group.Units.Equiv
public import Mathlib.CategoryTheory.Endomorphism
public import Mathlib.CategoryTheory.HomCongr
/-!
# Conju... |
121 | lean_file_chunk | 1c620afbc6c975382b2b970fd1f30c63f6e13337e3f1cf61682530b7daa6a7ef | lean_source_file | pkg_mathlib_source_file_chunks | module -- shake: keep-all
public import Mathlib.CategoryTheory.Adhesive.Basic
deprecated_module (since := "2026-02-02") |
17,311 | lean_file_chunk | 88f18167ccf6fed8b735fdfedc0ae91aa2820d997bf4a3618befe999917550e4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.Tactic.CategoryTheory.Elementwise
public import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
public import Mathlib.CategoryTheory.Limit... |
7,435 | lean_file_chunk | 8c011eb7f58204657824ab29250b6e30bb87d9a997da45fc526d369808423405 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
module
public import Mathlib.Data.List.Chain
public import Mathlib.CategoryTheory.IsConnected
public import Mathlib.CategoryTheory.Sigma.Basic
public import Mathlib.Cate... |
10,933 | lean_file_chunk | 0ed2f9765cf15ea652b6420be9cf585a31111c76fdcfae7ce1f6c932e83ac139 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
module
public import Mathlib.CategoryTheory.Comma.Arrow
public import Mathlib.CategoryTheory.CommSq
/-!
# The category of commutative squares
In this file, we define a bundl... |
6,276 | lean_file_chunk | a6cb8677defeae6602668e3a0e276b0b3995940637af8f28e90564f554ec1e2f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Kim Morrison, Simon Hudon
-/
module
public import Mathlib.Algebra.Group.Action.Defs
public import Mathlib.Algebra.Group.Equiv.Defs
public import Mathlib.Algebra.Grou... |
2,225 | lean_file_chunk | e0bf4deeee8961420a181d5e959117f13546feebaedf2b1dace53c6ad002e34d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Andrea Laretto. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrea Laretto, Fernando Chu
-/
module
public import Mathlib.CategoryTheory.Opposites
/-!
# Dinatural transformations
Dinatural transformations are special kinds of transformations b... |
9,899 | lean_file_chunk | f91078c1df843f7d7e1e5b99b27c942dcb19cc5fc8f525858804b372c8332488 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
module
public import Mathlib.CategoryTheory.EssentiallySmall
/-!
# The Yoneda functor for locally small categories
Let `C` be a locally `w`-small category. We define the Yon... |
6,504 | lean_file_chunk | 48d9b01f141df809694730b239bb884191bca40a47296e2fb8ca50a36a671fc7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
module
public import Mathlib.CategoryTheory.Elements
public import Mathlib.CategoryTheory.IsConnected
public import Mathlib.CategoryTheory.SingleObj
public import Mathlib.Gr... |
7,968 | lean_file_chunk | 1d66914aa38a0d31a9440ddb224180cb083ff424fb9faa63c180a94e9f71e16e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
module
public import Mathlib.CategoryTheory.NatIso
public import Mathlib.CategoryTheory.ObjectProperty.ClosedUnderIsomorphisms
public import Mathlib.CategoryTheory.Objec... |
3,343 | lean_file_chunk | 7f0a18259bb3b8cd47c64f4427add8eb9184dd0c1530fcf3af0ee9c31622a32b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Reid Barton, Joël Riou
-/
module
public import Mathlib.CategoryTheory.Functor.FullyFaithful
/-!
# Induced categories and full subcategories
Given a category `D` and a fu... |
7,739 | lean_file_chunk | 36b0085a1417c922bca8fd97f99104ae9a5706826f38b4bd64e43fdefec4ff49 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
module
public import Mathlib.CategoryTheory.Comma.Arrow
public import Mathlib.SetTheory.Cardinal.HasCardinalLT
/-!
# Representatives of small categories
Given a type `Ω : Ty... |
7,630 | lean_file_chunk | 18ece605a13e3e298dcf3133cc84feae271efc665721a43a82f82757312fcb05 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.CategoryTheory.Endomorphism
public import Mathlib.CategoryTheory.FinCategory.Basic
public import Mathlib.CategoryTheory.Category.Cat
p... |
23,982 | lean_file_chunk | a9bf7d0bac11e4e6d4cc0e7ba6e65306b0b12bdaa37711ccb09a4cdff38e1e0c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stephen Morgan, Kim Morrison
-/
module
public import Mathlib.CategoryTheory.Equivalence
/-!
# Opposite categories
We provide a category instance on `Cᵒᵖ`.
The morphisms `X ⟶ Y` are de... |
3,477 | lean_file_chunk | ace82e7a125fe6436de7f19a0feafa85149b696ce11fc7db22abc04d07fb8445 | lean_source_file | pkg_mathlib_source_file_chunks | counitIso := (NatIso.unop e.counitIso).symm
functor_unitIso_comp X := by
apply Quiver.Hom.op_inj
simp
/-- An equivalence between `C` and `Dᵒᵖ` gives an equivalence between `Cᵒᵖ` and `D`. -/
@[simps!] def leftOp (e : C ≌ Dᵒᵖ) : Cᵒᵖ ≌ D := e.op.trans (opOpEquivalence D)
/-- An equivalence between `Cᵒᵖ` and `D... |
1,981 | lean_file_chunk | 08384f68d9b7347eeebd32b11a25abade2c60a2cb177b443b18b4ecf1f89965c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Kim Morrison, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
module
public import Mathlib.CategoryTheory.Functor.Category
public import Mathlib.CategoryTheory.Iso
/-!
# Thin categories
A thin category ... |
22,299 | lean_file_chunk | d5cb2c84cd8d4fcc009d837a963cb45b132aad1fad295ecb62731d111067eac8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Sina Hazratpour
-/
module
public import Mathlib.CategoryTheory.Category.Cat.AsSmall
public import Mathlib.CategoryTheory.Elements
public import Mathlib.CategoryTheory.Comm... |
4,678 | lean_file_chunk | f2fd36458a9f884cf2e6c9f30b9b64d2052aff9cbbe5e663cd980100c524ab66 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Sina Hazratpour. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sina Hazratpour
-/
module
public import Mathlib.CategoryTheory.Functor.FullyFaithful
public import Mathlib.CategoryTheory.MorphismProperty.Composition
/-!
# Wide subcategories
A wi... |
2,334 | lean_file_chunk | 42a8d0c7567655b26171a735fdf0f61d0c11d38af08aad9d920524572a3b19b2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
module
public import Mathlib.CategoryTheory.Discrete.Basic
/-!
# The empty category
Defines a category structure on `PEmpty`, and the unique functor `PEm... |
13,718 | lean_file_chunk | c7dd786e320ef242dae262c6b21142f10cf67db53bf83fc54b25c93e29aa40bb | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Kim Morrison
-/
module
public import Mathlib.CategoryTheory.Opposites
/-!
# Morphisms from equations between objects.
When working categorically, sometimes one encounters ... |
16,851 | lean_file_chunk | 1dff5e7a80e632a4156f5baf2034ffa3e06c655dcb24e42e2b398fe905437d01 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.Tactic.CategoryTheory.IsoReassoc
public import Mathlib.CategoryTheory.Functor.Category
public import Mathlib.CategoryTheory.Functor.FullyFai... |
17,327 | lean_file_chunk | 997d7c8d92cf7cd4e5ad6b875a277e17ec38b343289f819bb8cea595f8f21b77 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
module
public import Mathlib.Tactic.CategoryTheory.Reassoc
/-!
# Isomorphisms
This file defines isomorphisms between obj... |
9,332 | lean_file_chunk | 82e810c5f11a32ba0b18f959a8a03564d65c3949658e9be05cacdafc067529fa | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
module
public import Mathlib.CategoryTheory.Functor.Category
/-!
# Natural isomorphisms
For the most part, natural isomo... |
23,994 | lean_file_chunk | 1730a48cbb08dcf37f60bd00670f0f8130185be31cd39a7f17504a296c274ac0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
module
public import Mathlib.CategoryTheory.Functor.FullyFaithful
public import Mathlib.CategoryTheory.ObjectProperty.Full... |
9,259 | lean_file_chunk | e5358de747ee826c5297a762b8a95d2d2f36186a06b9c7da6686a023dffd5aca | lean_source_file | pkg_mathlib_source_file_chunks | e.fullyFaithfulFunctor.full
instance full_inverse (e : C ≌ E) : e.inverse.Full :=
e.fullyFaithfulInverse.full
/-- If `e : C ≌ D` is an equivalence of categories, and `iso : e.functor ≅ G` is
an isomorphism, then there is an equivalence of categories whose functor is `G`. -/
@[simps!]
def changeFunctor (e : C ≌ D) {... |
21,308 | lean_file_chunk | d1c1689c3f596342eda390dfdb46cf8efbf500b5fe338a67cf0b9222e1b7387e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Jakob von Raumer
-/
module
public import Mathlib.Data.List.Chain
public import Mathlib.CategoryTheory.PUnit
public import Mathlib.CategoryTheory.Groupoid
public import Mat... |
12,784 | lean_file_chunk | e29e5276f00227cb812ad6943d42b30659123e548bfc8e44e9440da8664c954b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.CategoryTheory.Comma.StructuredArrow.Basic
public import Mathlib.CategoryTheory.Category.Cat
public import Mathlib.CategoryTheory.Essentiall... |
10,566 | lean_file_chunk | 0a36f90b9d0b10de092f9ee7516107c17158e0120995f71fba0c5b97442cf57d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.Algebra.Group.Basic
public import Mathlib.Data.Int.Cast.Defs
public import Mathlib.CategoryTheory.Shift.Basic
public import Mathlib.Category... |
1,421 | lean_file_chunk | aa8fcf860d7479872937d083c591d4e4248fce04241c8f85940194775f88f1ac | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.CategoryTheory.Subobject.ArtinianObject
public import Mathlib.CategoryTheory.Subobject.NoetherianObject
/-!
# Artinian and Noetherian categ... |
3,793 | lean_file_chunk | bd8ae0b2295c7833c30fdcf5218642c73138750fd79091614fe297a76e78aed3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
module
public import Mathlib.CategoryTheory.EssentiallySmall
public import Mathlib.CategoryTheory.FinCategory.Basic
public import Mathlib.Data.Fintype.EquivFin
p... |
10,452 | lean_file_chunk | b6ea0c74f4dde5c36d01101b89a6a6f8aae8928a049651c90351552e1177c51e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Adam Topaz
-/
module
public import Mathlib.CategoryTheory.ConcreteCategory.Forget
public import Mathlib.CategoryTheory.Endomorphism
public import Mathlib.CategoryTheory.Skel... |
8,520 | lean_file_chunk | 8ca32bc091ba66db2ba39857ebeb728778160353702cefaef24d39a93406a891 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Joël Riou
-/
module
public import Mathlib.CategoryTheory.Comma.Arrow
/-!
# Commutative squares
This file provides an API for commutative squares in categories.
If `top`,... |
23,981 | lean_file_chunk | 95ba7e36f9688c07e429712f6e3a88c030005570b7b6b3399b05a4b1add8941c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
module
public import Mathlib.Algebra.Order.Group.Units
public import Mathlib.Algebra.Order.Monoid.LocallyFiniteOrder
public import Mathlib.Data.Int.Interval
public... |
697 | lean_file_chunk | fd1d2ccff4be2eadd5d151202c373db2ed44751dfe6b94c5871f7000b5696bbb | lean_source_file | pkg_mathlib_source_file_chunks | WithBot.denselyOrdered_iff
instance {X : Type*} [Preorder X] [NoMinOrder X] [DenselyOrdered X] :
DenselyOrdered (WithZero X) :=
WithZero.denselyOrdered_iff.mpr inferInstance
lemma Int.not_denselyOrdered : ¬ DenselyOrdered ℤ :=
(LinearOrderedAddCommGroup.discrete_iff_not_denselyOrdered ℤ).mp ⟨.refl _⟩
lemma n... |
2,470 | lean_file_chunk | a99d4f7f7a902dfb1b51c4974a2fb9227759552243f0363d7b7607c209b6e991 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Hang Lu Su. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Fabrizio Barroero, Stefano Francaviglia,
Francesco Milizia, Valerio Proietti, Hang Lu Su, Lawrence Wu
-/
module
public import Mathlib.Algebra.Group.Subgroup.Basic
publi... |
16,797 | lean_file_chunk | d0618b52abd2d1a03da60101d00feb6c431105cc80814f0d4326e37d32f6d5b0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
module
public import Mathlib.GroupTheory.Perm.Cycle.Type
public import Mathlib.GroupTheory.SpecificGroups.Cyclic
/-!
# p-groups
This file contains a p... |
13,482 | lean_file_chunk | a0ec763ad8aba61923a1fd0d8a585b773dfa1af8d28cc8b56a56aa9e2a6f7548 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Joachim Breitner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joachim Breitner
-/
module
public import Mathlib.GroupTheory.OrderOfElement
public import Mathlib.Data.Nat.GCD.BigOperators
public import Mathlib.Order.SupIndep
/-!
# Canonical hom... |
23,994 | lean_file_chunk | 3ee4a29a6f14f4a8b68e80a55db5567e3d2f8945e0d2f677d97a7a4feff96d93 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Joachim Breitner
-/
module
public import Mathlib.Algebra.Group.Action.End
public import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
public import Mathlib.Algebra.Group.Su... |
15,533 | lean_file_chunk | f4bf4532828d39d09948a6858723d9d8c1086516ab4bdb26b5cc121271d1d395 | lean_source_file | pkg_mathlib_source_file_chunks | @[simp]
theorem toList_head? {i j} (w : NeWord M i j) : w.toList.head? = Option.some ⟨i, w.head⟩ := by
fun_induction toList with grind [head]
@[simp]
theorem toList_getLast? {i j} (w : NeWord M i j) : w.toList.getLast? = Option.some ⟨j, w.last⟩ := by
rw [← Option.mem_def]
induction w
· rw [Option.mem_def]
... |
11,628 | lean_file_chunk | 15f4a3674cee903d9eae3fbf481a7ba0d6e0203ec62d77f5bac2e2ea228e8f83 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Inna Capdeboscq, Damiano Testa
-/
module
public import Mathlib.Algebra.Group.Subgroup.Pointwise
public import Mathlib.GroupTheory.QuotientGroup.Defs
public import Mathlib.GroupTheory.... |
23,976 | lean_file_chunk | 976300ca21e40943900e63cb0852688d8bcc42380be0cbae0ed9b0d17ac8b064 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Ines Wright, Joachim Breitner
-/
module
public import Mathlib.GroupTheory.Solvable
public import Mathlib.GroupTheory.Sylow
public import Mathlib.Algebra.Group.Subgroup.O... |
17,755 | lean_file_chunk | d4460bced026e89b62e61158e574305b11f37a36855d3bdd021731adaceddee9 | lean_source_file | pkg_mathlib_source_file_chunks | (hf : Function.Surjective f) [h : IsNilpotent G] :
Group.nilpotencyClass G' ≤ Group.nilpotencyClass G := by
have := nilpotent_of_surjective _ hf
rw [nilpotencyClass_def, nilpotencyClass_def]
classical apply Nat.find_mono
intro n hn
rw [eq_top_iff]
calc
⊤ = f.range := symm (f.range_eq_top_of_surjecti... |
20,603 | lean_file_chunk | d23379febba9c1a973dd80f4924d530342a86ffa92cdbafd962ba6d58f041fad | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
module
public import Mathlib.Algebra.Module.NatInt
public import Mathlib.GroupTheory.Abelianization.Defs
public import Mathlib.GroupTheory.FreeGroup.Basic
public import Mathli... |
15,931 | lean_file_chunk | 6ba8cfddafb9be457afdf2f61100b9a15268ffbdd65fe224367827d0995321e5 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Weiyi Wang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Weiyi Wang
-/
module
public import Mathlib.Algebra.Module.LocalizedModule.Basic
public import Mathlib.Algebra.Order.Module.Archimedean
public import Mathlib.Algebra.Order.Monoid.PNat
publ... |
9,435 | lean_file_chunk | bc8c78e504869d23c34f03e96197614c49b4f7f30621c2ecc9c7d535340c2f05 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Jordan Brown, Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jordan Brown, Thomas Browning, Patrick Lutz
-/
module
public import Mathlib.Data.Fin.VecNotation
public import Mathlib.GroupTheory.Abelianization.Defs
pub... |
23,973 | lean_file_chunk | 188084529188a52f7445446000d6bbbeee25766795d9d025ad849ba042584c54 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
module
public import Mathlib.GroupTheory.Index
/-!
# Complements
In this file we define the complement of a subgroup.
## Main definitions
- `Subgroup.IsComplem... |
6,036 | lean_file_chunk | d0ae4af3b62f0f676f3b96618a0541c7c1cf2c5143eb160060c28c6c3a132447 | lean_source_file | pkg_mathlib_source_file_chunks | (toRightFun hT g : G) * g⁻¹ ∈ H :=
(congr_arg (· ∈ H) (by rw [mul_inv_rev, inv_inv])).mp (H.inv_mem (mul_inv_toRightFun_mem hT g))
@[to_additive]
theorem encard_left [H.FiniteIndex] (h : IsComplement S H) : S.encard = H.index := by
rw [← h.finite_left.cast_ncard_eq, h.ncard_left]
@[to_additive]
theorem encard_rig... |
3,174 | lean_file_chunk | 0f1aa5736d16e801744525403fb3d4dcfce44bf668fe391b4dd616429fa3788c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
module
public import Mathlib.GroupTheory.Finiteness
public import Mathlib.SetTheory.Cardinal.Finite
/-!
# Rank of a group
This file defines the rank of a group, ... |
3,677 | lean_file_chunk | edf1cfbcb4182ee558f24c6ab592fff1d2f7fdc18ab62a02eff9b54d1d77eadc | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
module
public import Mathlib.GroupTheory.Perm.Cycle.Type
/-!
# Fixed-point-free automorphisms
This file defines fixed-point-free automorphisms and proves some ba... |
3,551 | lean_file_chunk | f263fd36387dd6cb9c7d1ef6b4cbfb2d2b898af75a1831d69d6f36dcd8e1c0d6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Rodriguez
-/
module
public import Mathlib.Algebra.BigOperators.Finprod
public import Mathlib.Algebra.Group.ConjFinite
public import Mathlib.Algebra.Group.Subgroup... |
19,569 | lean_file_chunk | dfcdb9f7d4ce984bb993a34f21eaca72520149b662bb7d6b919d450ed2138029 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Antoine Chambert-Loir, Richard Copley. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, Richard Copley
-/
module
public import Mathlib.Algebra.Order.Ring.Rat
public import Mathlib.GroupTheory.Complement
public import Mathlib.... |
23,951 | lean_file_chunk | 8d2eb79292d31f02e0a7709d01d428a7a7c595a334e8574f3c18cc02b2f0c34d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
module
public import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
public import Mathlib.Algebra.Group.Subgroup.ZPowers.Basic
public import Mathlib.Algebra.Gro... |
13,045 | lean_file_chunk | babe499218caa55e29a5ea116d1c0ea059269ed83517311f933c807c2bcd58a2 | lean_source_file | pkg_mathlib_source_file_chunks | [Finite <| MulAction.orbitRel.Quotient G X] (hi : H.index ≠ 0) :
Finite <| MulAction.orbitRel.Quotient H X := by
have := fintypeOfIndexNeZero hi
exact MulAction.finite_quotient_of_finite_quotient_of_finite_quotient
@[to_additive]
lemma finite_quotient_of_pretransitive_of_index_ne_zero {X : Type*} [MulAction G ... |
5,758 | lean_file_chunk | 3bb132ecb475aad7f67811061c9d4ab7812ef42a313b8b0b14964c5c9b241a82 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Heather Macbeth, Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Patrick Massot
-/
module
public import Mathlib.Algebra.Group.Subgroup.Order
public import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Alge... |
7,319 | lean_file_chunk | 31f5b6700df945cf16b1c785ef3949c88419cca42fd004aa2d3ce135905e8474 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
module
public import Mathlib.Data.Real.Basic
public import Mathlib.GroupTheory.Finiteness
public import Mathlib.GroupTheory.Index
public import Mathlib.NumberTheory.He... |
8,327 | lean_file_chunk | ed2282e85c1364f8fb3828ae6e1b20f9b99ee9150c6b39f3c04e642cf7a1d0d2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Boyang Hu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Boyang Hu
-/
module
public import Mathlib.GroupTheory.Abelianization.Defs
public import Mathlib.GroupTheory.Sylow
public import Mathlib.GroupTheory.Transfer
public import Mathlib.Data.ZMod... |
4,249 | lean_file_chunk | dd60c280b22d99248494388c56659898d70860233e3b03c660928dc55a51e3f2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau, Robert Y. Lewis
-/
module
public import Mathlib.Algebra.Group.Defs
/-!
# Eckmann-Hilton argument
The Eckmann-Hilton argument says that if a type carries two mo... |
10,354 | lean_file_chunk | 71f0e45b714ca8a47bdc5716f355fe0fece21881f8ebf76d1f6d3ca03f37d03e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
module
public import Mathlib.Algebra.Group.ULift
public import Mathlib.Algebra.GroupWithZero.Subgroup
public import Mathlib.Algebra.Module.NatInt
public import Mathlib.G... |
11,284 | lean_file_chunk | e8d8fe6c3ce6b70c3d3ef354539b89159eea01e8642a7d81b148d220e08e880d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
module
public import Mathlib.Algebra.Group.Subgroup.Pointwise
public import Mathlib.GroupTheory.Coset.Basic
/-!
# Double cosets
This file defines double cosets for... |
13,238 | lean_file_chunk | 722127c1a1f5c285149e6bd3ded845405751a10f8a909e51b01dd30878d63d1f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
module
public import Mathlib.GroupTheory.Transfer
/-!
# The Schur-Zassenhaus Theorem
In this file we prove the Schur-Zassenhaus theorem.
## Main results
- `Sub... |
6,661 | lean_file_chunk | eaea5a01d1d8babd7b74c728c1fd8852922bf8bfa839fba191b777d40f24e5c7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Michael Howes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Howes, Newell Jensen
-/
module
public import Mathlib.Algebra.Group.Subgroup.Basic
public import Mathlib.GroupTheory.FreeGroup.Basic
public import Mathlib.GroupTheory.QuotientGr... |
8,077 | lean_file_chunk | 0dc9f0792ff08113772bc3176cd4279224f137a2421a842c9bf3f8d21bdb63e3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
module
public import Mathlib.Algebra.BigOperators.Fin
public import Mathlib.GroupTheory.Abelianization.Finite
public import Mathlib.GroupTheory.SpecificGroups.Dihe... |
23,990 | lean_file_chunk | 86e8b767067702aaeb8fbed5c8959c591999b879c9f03ee5065acdf14d2c6fe1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
module
public import Mathlib.Algebra.Order.Archimedean.Basic
public import Mathlib.Data.SetLike.Fintype
public import Mathlib.GroupTheory.PGroup
public ... |
14,988 | lean_file_chunk | ea5db1ce8ef3db5515524bfd40241bdfdb8a2a1a5ec9bc052d1783916483634e | lean_source_file | pkg_mathlib_source_file_chunks | mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Finite G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Nat.card H = p ^ n) :
Nat.card (normalizer H ⧸ H.comap (normalizer (H : Set G)).subtype) ≡
Nat.card (G ⧸ H) [MOD p] := by
rw [← Nat.card_congr (fixed... |
537 | lean_file_chunk | 914312541f087026ddd7580ad904e073329d401f5fb10710576b316455f8c485 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
module
public import Mathlib.Algebra.Regular.Basic
public import Mathlib.Data.Fintype.Card
/-!
# Finite monoids are Dedekind-finite
-/
@[expose] public section
instance (M)... |
2,835 | lean_file_chunk | aa31dc7e69314acbe7d94d07032368011d2fe9d397007351ba32d644830aa4f1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
module
public import Mathlib.GroupTheory.FiniteIndexNormalSubgroup
/-!
# Residually Finite Groups
In this file we define residually finite groups and prove some ... |
14,168 | lean_file_chunk | 4a90c4797229b77a896e800c8187e6be913180c7f5a1afe2e513e1b9ed7d5fb3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Julian Berman. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Berman
-/
module
public import Mathlib.GroupTheory.PGroup
public import Mathlib.LinearAlgebra.Quotient.Defs
/-!
# Torsion groups
This file defines torsion groups, i.e. groups ... |
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