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 |
|---|---|---|---|---|---|
4,295 | lean_file_chunk | dfe8da2b27451e13257cec666b13db59f2dec927a24aebb0d2dcef632dff0eda | 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.Algebra.Homology.Additive
public import Mathlib.Algebra.Homology.HomologicalComplexLimits
public import Mathlib.Algebra.Homology.ShortComplex.Exac... |
8,083 | lean_file_chunk | 887dc6560fe7ade762f64fd5871d43bd55cdffe350c1c8a9db91450335140c76 | 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.Algebra.Homology.ExactSequence
/-!
# Exact sequences with four terms
The main definition in this file is `ComposableArrows.Exact.cokerIsoKer`:
g... |
3,422 | lean_file_chunk | 9568eeffa04a9b071ca44450ba560e1030d03c9ddc0990d249c2886060edc6a3 | 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.Algebra.Homology.Bifunctor
public import Mathlib.Algebra.Homology.TotalComplexSymmetry
/-!
# Action of the flip of a bifunctor on homological com... |
5,435 | lean_file_chunk | e47b88ae41cc8025bbe972b227bab04685908d0661d9cd921c1486d398d81b81 | 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.Subobject.Limits
/-!
# Image-to-kernel comparison maps
Whenever `f : A ⟶ B` and `g : B ⟶ C` satisfy `w : f ≫ g = 0`,
we hav... |
7,738 | lean_file_chunk | 5fd92103ade0032b62d99c92bf0583d326751aa2061839dfc1c25e9816b221e9 | 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, Joël Riou
-/
module
public import Mathlib.Algebra.Homology.HomologicalComplex
/-!
# Bicomplexes
Given a category `C` with zero morphisms and two complex shapes
`c₁ : Com... |
1,704 | lean_file_chunk | 5886b5639a72fa4f247f3b6ea74a941f926bcf60e4a433ea78b45a40ac1d77bb | 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.Algebra.Homology.Additive
public import Mathlib.CategoryTheory.Linear.LinearFunctor
/-!
# The category of homological complexes is linear
In thi... |
11,248 | lean_file_chunk | 38540e15c41a07e43e8c8dc0fa3b32362194e035670a4f3ab2d78edb58051f2a | 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.Algebra.Homology.BifunctorFlip
public import Mathlib.Algebra.Homology.Homotopy
/-!
# The action of a bifunctor on homological complexes factors t... |
12,406 | lean_file_chunk | 10001ba4be5419a31d2a10d67ad98fe5a2943be1765739f334dc43ac63b34fa2 | 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.Algebra.Homology.TotalComplex
public import Mathlib.CategoryTheory.GradedObject.Bifunctor
/-!
# The action of a bifunctor on homological complexe... |
10,908 | lean_file_chunk | 962767a3888fba5238e94df5eff4d9c2cdab8a423d5de0f6c7bdb29a0a72b8a7 | 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.Algebra.Homology.HomologicalComplex
/-!
# Homological complexes supported in a single degree
We define `single V j c : V ⥤ HomologicalComp... |
5,313 | lean_file_chunk | 297adc760f952b5195e7a72521504590e1835398de292f14a65083dabbbf2159 | 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.Algebra.Homology.HomologicalComplexLimits
public import Mathlib.Algebra.Homology.Additive
/-! Binary biproducts of homological complexes
In this... |
2,271 | lean_file_chunk | 27b30448f51ddbd1d4b28e7f059d8a4c2ac5e1704dd5cdacc6961fb7e758ed93 | 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.Algebra.Homology.ComplexShape
public import Mathlib.Algebra.Group.Int.Defs
public import Mathlib.Algebra.Group.Nat.Defs
/-!
# Complex shapes with... |
19,567 | lean_file_chunk | 733146572411c5fa50e06595bd0ca5c545636cfef2c12f9ebbf06e8dbbd0c971 | 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.Linear.Basic
public import Mathlib.Algebra.Homology.ComplexShapeSigns
public import Mathlib.Algebra.Homology.HomologicalBicomplex
p... |
2,585 | lean_file_chunk | b01b6940d260f7d10c5ed5be9f71dc1d82cfda1bd1cae7e6709b8b15e9770705 | 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.Algebra.Homology.CommSq
public import Mathlib.CategoryTheory.Limits.Shapes.Pullback.Square
/-!
# Relation between pullback/pushout squares and ke... |
23,971 | lean_file_chunk | 7ce0a165c9130c52c72dfa1803a0a17a97520afe2cb70a23cf053aa82902edd2 | 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.GradedObject.Associator
public import Mathlib.CategoryTheory.Linear.LinearFunctor
public import Mathlib.Algebra.Homology.Bifunctor
... |
12,301 | lean_file_chunk | fbc3142d401708802cb5e4cb26ac58653e08fb79e8b8d2a996c08e5db3e3421c | lean_source_file | pkg_mathlib_source_file_chunks | lemma d₃_eq_zero (i₁ : ι₁) (i₂ : ι₂) (i₃ : ι₃) (j : ι₄) (h : ¬ c₃.Rel i₃ (c₃.next i₃)) :
d₃ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j = 0 := by
dsimp [d₃]
rw [shape _ _ _ h, Functor.map_zero, Functor.map_zero, zero_comp, smul_zero]
lemma d₃_eq (i₁ : ι₁) (i₂ : ι₂) {i₃ i₃' : ι₃} (h₃ : c₃.Rel i₃ i₃') (j : ι₄) :
d₃... |
4,179 | lean_file_chunk | 0c112cb110e3c5a8515162c3d50d5f640ad41170d59c118ce7a0373cdcc2177e | 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.Abelian.Refinements
public import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
/-!
# Refinements
This file contains l... |
14,985 | lean_file_chunk | 04ff0a71453853eb00ad47f78961f48ccb097fad0df97c5737f170e4b56c07f8 | 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.Algebra.Homology.ShortComplex.HomologicalComplex
public import Mathlib.Algebra.Homology.ShortComplex.SnakeLemma
public import Mathlib.Algebra.Homo... |
17,547 | lean_file_chunk | 4e3c8fbe0f0c01b1e457b80f53a6f0eaca40fbdab1a32425116911b9624908e9 | 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.Algebra.Homology.HomotopyCofiber
public import Mathlib.Algebra.Homology.HomotopyCategory
public import Mathlib.Algebra.Homology.QuasiIso
public im... |
3,396 | lean_file_chunk | 29a1d9310ab6ae42e6fb9badd3a30329c5131483fe10e25292b96ff7021966de | 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, Joël Riou
-/
module
public import Mathlib.Algebra.Homology.QuasiIso
public import Mathlib.CategoryTheory.Abelian.FunctorCategory
/-!
# Complexes in functor categories... |
6,445 | lean_file_chunk | 24c405f86aa1adfc10a34d52e626bd0ceba9cc59cff5fa8faa0a3b4530b82637 | 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.Algebra.Homology.TotalComplex
/-! The symmetry of the total complex of a bicomplex
Let `K : HomologicalComplex₂ C c₁ c₂` be a bicomplex. If we a... |
15,720 | lean_file_chunk | ce91419721365fd0e63c32d2f4feac43c8e5dc1e59e17e2c10664c82f8161a54 | 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.Algebra.Homology.ComplexShape
public import Mathlib.Algebra.Ring.NegOnePow
public import Mathlib.CategoryTheory.GradedObject.Trifunctor
/-! Signs... |
10,734 | lean_file_chunk | df4c763ce318b10b68953b2396da78f76e7dcd3e5b5f04e06234cafa7ba80a4f | 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.Algebra.Homology.Homotopy
public import Mathlib.Algebra.Homology.Linear
public import Mathlib.CategoryTheory.MorphismProperty.IsInvertedBy
p... |
2,722 | lean_file_chunk | 0801e4e92c540bface75cf92b78c8c17f310042b1ff4ea2220f3aa479fc7e42e | 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.Abelian.GrothendieckCategory.Basic
public import Mathlib.CategoryTheory.Generator.HomologicalComplex
public import Mathlib.Algebra.... |
17,345 | lean_file_chunk | 1dbea6c6448d1a69e644630ed01a4e1e18366d8e52c4d76aeb533d272109f0af | 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.Algebra.Homology.HomotopyCategory.Shift
public import Mathlib.Algebra.Homology.TotalComplex
public import Mathlib.Tactic.Linarith
/-!
# Behaviour... |
8,306 | lean_file_chunk | 39d2b4cf2a0e690a3c66acfafcb06b3c8a187aa6345bfa470def0aa4e746d973 | 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.Algebra.Homology.Single
public import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
public import Mathlib.CategoryTheory.Limits.Preserves.Fini... |
5,022 | lean_file_chunk | dbbb18a7d57b676ce58ce520af1cdfb0ccc35c4cfd75faed824b0a07b9fc2e0f | 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.Algebra.Homology.HomologicalComplex
public import Mathlib.CategoryTheory.DifferentialObject
/-!
# Homological complexes are differential gr... |
20,883 | lean_file_chunk | 221d1f8f99eac1d6e51717825fddaf20a5d611ff620a5363c63b6a29d6ac428b | 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.Algebra.Homology.HomologicalComplexBiprod
public import Mathlib.Algebra.Homology.Homotopy
public import Mathlib.CategoryTheory.MorphismProperty.Is... |
7,637 | lean_file_chunk | 96bbe6896143a988029d0bc3900e38b59bbc3490e3ad8a65760e8cecd9066875 | 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.Algebra.Homology.HasNoLoop
public import Mathlib.Algebra.Homology.Single
public import Mathlib.CategoryTheory.Yoneda
/-!
# A homological complex ... |
5,186 | lean_file_chunk | 8e66265934c3ba7a9826fe0c6ea04078cf022bdca30f484bb7a6481f80a7b802 | 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.Algebra.Homology.HomologySequence
public import Mathlib.Algebra.Homology.ShortComplex.ConcreteCategory
/-!
# Homology of complexes in concrete ca... |
23,927 | lean_file_chunk | fda06498de67000dd6a9c118d1952b919068273ecae4b9141301c360837c0776 | 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.Algebra.Homology.Linear
public import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
public import Mathlib.Tactic.Abel
/-!
# Chai... |
7,344 | lean_file_chunk | 7b4cf9cf90b50db027ee591566f765712e5f3095cb983092eedb7d10d70dcce8 | lean_source_file | pkg_mathlib_source_file_chunks | (P.xNextIso h).inv ≫ (mkCoinductiveAux₂ e zero comm_zero one comm_one succ i).2.1 =
(mkCoinductiveAux₂ e zero comm_zero one comm_one succ j).1 ≫ (Q.xPrevIso h).hom := by
subst j
rcases i with (_ | _ | i) <;> simp [mkCoinductiveAux₂]
/-- A constructor for a `Homotopy e 0`, for `e` a chain map between `ℕ`-inde... |
7,095 | lean_file_chunk | 9faebfeba76873e9fd4209a070fa4705c2ad0d80bf2a1173dde3c18593259561 | 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.Algebra.Homology.HomologySequence
public import Mathlib.Algebra.Homology.QuasiIso
public import Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four
... |
7,307 | lean_file_chunk | 55adb93e3813da888dbda4ed3d2d772fd2d07f18961cdbe1ce272eb193f63399 | 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.Algebra.Homology.ShortComplex.Basic
public import Mathlib.CategoryTheory.Preadditive.Biproducts
public import Mathlib.CategoryTheory.Limits.Shapes... |
8,506 | lean_file_chunk | b8ff7b81406c9e48e96633e01f0c558ab2e363d101ad52eca0f16bc6f3f416e3 | 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.Algebra.Homology.Single
public import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
/-!
# The homology of single complexes
The main de... |
9,964 | lean_file_chunk | 77e0903c0c662c82c2632d1734ee6b0eba2992749ca5c055dc783c05f7b8c233 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
module
public import Mathlib.Algebra.Lie.OfAssociative
/-!
# Jordan rings
Let `A` be a non-unital, non-associative ring. Then `A` is said to be a (commutat... |
7,184 | lean_file_chunk | 64009d78d03af38926830352ad3b9ada716a68c9b62477fa7077ddce21380d08 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
module
public import Mathlib.Algebra.Algebra.Equiv
public import Mathlib.Algebra.Module.Opposite
public import Mathlib.Algebra.Ring.Opposite
/-!
# Algebra structures on t... |
8,953 | lean_file_chunk | 2ec26d6092642e9b47837f4ea98f2ff4be6b19da55dc0812225408483502109d | 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, Yury Kudryashov
-/
module
public import Mathlib.Algebra.Algebra.Tower
/-!
# The `RestrictScalars` type alias
See the documentation attached to the `RestrictScalars` definitio... |
6,562 | lean_file_chunk | 5a5b35a60023b2e017ca5eb753ee793653a0dfa94ca2607f26edcde96bd6ff9d | 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, Yury Kudryashov
-/
module
public import Mathlib.Algebra.Algebra.NonUnitalHom
public import Mathlib.LinearAlgebra.TensorProduct.Map
/-!
# Facts about algebras involving bilinear... |
2,538 | lean_file_chunk | 1771fc18172b970b3a506540c845794e97ce2dd7b795c55d5be43d0130a77694 | 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, Yury Kudryashov
-/
module
public import Mathlib.Algebra.Algebra.Defs
public import Mathlib.Algebra.Module.Equiv.Defs
public import Mathlib.Data.Rat.Cast.CharZero
/-!
# Further ... |
2,302 | lean_file_chunk | 72c2aac3a093c0664307727af6640b4d120d4cf8dbed18487309ea3f81b4927c | 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.Algebra.Algebra.Equiv
public import Mathlib.Algebra.Ring.TransferInstance
/-!
# Transfer algebraic structures across `Equiv`s
This con... |
4,552 | lean_file_chunk | 0e7719d0b63610373501e12b2b183eef7b634f80ae6897faca20486e293ab085 | 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, Yury Kudryashov
-/
module
public import Mathlib.Algebra.Algebra.Equiv
public import Mathlib.Algebra.Algebra.Hom
public import Mathlib.Algebra.Module.Prod
/-!
# The R-algebra st... |
15,052 | lean_file_chunk | e66924d9478290242d5807c4de6af3fa543ca404061d289702fe8bb86ff208f5 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Anne Baanen
-/
module
public import Mathlib.Algebra.Algebra.Equiv
public import Mathlib.LinearAlgebra.Span.Basic
/-!
# Towers of algebras
In this file we prove basic facts ab... |
562 | lean_file_chunk | 0bb1d458705b6be09345574dd3380d00517504feb26b5792cd1d5ea1f2c51c7c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul Lezeau, Eric Wieser
-/
module
public import Mathlib.Algebra.Algebra.Basic
public import Mathlib.RingTheory.SimpleRing.Basic
/-!
# Facts about algebras when the coefficient ring is ... |
8,699 | lean_file_chunk | 8ce36b4156f558d12c4b246f4f16e6478847c002841e0a54f034cff6f44669f4 | 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, Yury Kudryashov
-/
module
public import Mathlib.Algebra.Algebra.Equiv
public import Mathlib.Algebra.Algebra.Opposite
public import Mathlib.Algebra.Algebra.Prod
/-!
# The R-alge... |
21,878 | lean_file_chunk | b5044916ecbd96fbff64c81df676ea5d83688fb448fcda065ced56d40a10b88b | 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, Yury Kudryashov
-/
module
public import Mathlib.Algebra.Algebra.Defs
public import Mathlib.Algebra.Module.Equiv.Basic
public import Mathlib.Algebra.Module.Submodule.Ker
public i... |
23,997 | lean_file_chunk | 159ca28a82956b97ed378f199c5644f2b6144a818dea0fa3690b5bbcdc6ef1ba | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 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.Algebra.Bilinear
public import Mathlib.Algebra.Algebra.Opposite
public import Mathlib.Algebra.Group.Pointwise.Finset.Basic
public import M... |
12,600 | lean_file_chunk | 5b5948e27461e0eb2e725223bc0745b3d4585fba5a5f5b461d1609fbb49e9677 | lean_source_file | pkg_mathlib_source_file_chunks | {n : ℕ} {x : A} (hx : x ∈ M ^ n) : C n x hx := by
induction n generalizing x with
| zero =>
rw [pow_zero] at hx
obtain ⟨r, rfl⟩ := mem_one.mp hx
exact algebraMap r
| succ n n_ih =>
revert hx
simp_rw [pow_succ]
exact fun hx ↦ Submodule.mul_induction_on' (fun m hm x ih => mul_mem _ _ hm (n_i... |
15,737 | lean_file_chunk | 42682a355876dfb9732615f84715657570467aba0663f39714e31a4dcf71d4f0 | 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, Yury Kudryashov
-/
module
public import Mathlib.Algebra.Module.LinearMap.Defs
/-!
# Algebras over commutative semirings
In this file we define associative unital `Algebra`s ov... |
784 | lean_file_chunk | 88a475884d097e868b6d5ed94b15c447b5534d5e8d2d199db512207b34346739 | 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.Algebra.Algebra.TransferInstance
public import Mathlib.Algebra.Ring.Shrink
/-!
# Transfer module and algebra structures from `α` to `Shrink... |
23,979 | lean_file_chunk | d0f3b30488dba05dbfc3a59feb188a4667dd5092d3bdff0ab7bc78b4b6ed3ac1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
module
public import Mathlib.Algebra.Algebra.NonUnitalHom
public import Mathlib.Data.Set.UnionLift
public import Mathlib.LinearAlgebra.Span.Basic
public import Mathlib... |
23,940 | lean_file_chunk | fad045aab4a260a395a71dae897806c302c2b53da23a0f04c518b7f2858bd0d1 | lean_source_file | pkg_mathlib_source_file_chunks | (mem : ∀ (x) (hx : x ∈ s), p x (subset_adjoin R hx))
(add : ∀ x y hx hy, p x hx → p y hy → p (x + y) (add_mem hx hy)) (zero : p 0 (zero_mem _))
(mul : ∀ x y hx hy, p x hx → p y hy → p (x * y) (mul_mem hx hy))
(smul : ∀ r x hx, p x hx → p (r • x) (SMulMemClass.smul_mem r hx))
{x} (hx : x ∈ adjoin R s) : ... |
570 | lean_file_chunk | 00ad9d633eee59fdb4bde4d2bb8bdbe982e8191ce38ff6f92760ad347cda56fc | lean_source_file | pkg_mathlib_source_file_chunks | x ∈ nonUnitalSubalgebraOfNonUnitalSubsemiring S ↔ x ∈ S :=
Iff.rfl
end Nat
section Int
variable {R : Type*} [NonUnitalNonAssocRing R]
/-- A non-unital subring is a non-unital `ℤ`-subalgebra. -/
def nonUnitalSubalgebraOfNonUnitalSubring (S : NonUnitalSubring R) : NonUnitalSubalgebra ℤ R where
toNonUnitalSubsemir... |
23,952 | lean_file_chunk | 6d4e715b935816c7be3300551e4c6290bcb596b82d5102ac605b748d2dfa7817 | 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, Yury Kudryashov
-/
module
public import Mathlib.Algebra.Algebra.Hom
public import Mathlib.Algebra.Ring.Action.Group
/-!
# Isomorphisms of `R`-algebras
This file defines bundle... |
5,066 | lean_file_chunk | 09b352b6938b025ad32df189aa03e305c7298ef3e6887d15168a3d1ae727c9c8 | lean_source_file | pkg_mathlib_source_file_chunks | right_inv := (fun x ↦ show (↑(f * f⁻¹) : S →ₐ[R] S) x = x by rw [mul_inv_cancel]; rfl) }
invFun := fun f ↦ ⟨f, f.symm, f.comp_symm, f.symm_comp⟩
map_mul' := fun _ _ ↦ rfl
/-- See also `Finite.algHom` -/
instance _root_.Finite.algEquiv [Finite (A₁ →ₐ[R] A₂)] : Finite (A₁ ≃ₐ[R] A₂) :=
Finite.of_injective _ AlgEqui... |
1,065 | lean_file_chunk | b2d88fc0938a77f4748ce937251683a8462cf8e07450b2d70099621f6db31a38 | 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, Yury Kudryashov
-/
module
public import Mathlib.Algebra.Algebra.Defs
public import Mathlib.Data.Rat.Cast.Defs
/-!
# Facts about `algebraMap` when the coefficient ring is a fiel... |
23,995 | lean_file_chunk | 7eae1509d6f4102168e7d722c0166789100bd76148cbec3438933eb78cb2f989 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
module
public import Mathlib.Algebra.Algebra.Defs
public import Mathlib.Algebra.Algebra.NonUnitalHom
public import Mathlib.Algebra.Star.Module
public import Mathlib.Al... |
8,150 | lean_file_chunk | f00ae5bff3f3b4901b501fb361f8d437c4a304378ad5792da1dc023711289fef | lean_source_file | pkg_mathlib_source_file_chunks | [FunLike F (Unitization R A) C] [AlgHomClass F R (Unitization R A) C] {φ ψ : F}
(h : ∀ a : A, φ a = ψ a) : φ = ψ :=
algHom_ext h (fun r => by simp only [AlgHomClass.commutes])
/-- See note [partially-applied ext lemmas] -/
@[ext 1100]
theorem algHom_ext' {φ ψ : Unitization R A →ₐ[R] C}
(h :
φ.toNonUnit... |
6,623 | lean_file_chunk | 6bd59adb3971ad35dc5710c9e491a0907f4f7c02a04699ebca50dc46dc3384da | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
module
public import Mathlib.Algebra.Algebra.Bilinear
public import Mathlib.LinearAlgebra.TensorProduct.Tower
public import Mathlib.RingTheory.Localization.FractionRing
pu... |
5,583 | lean_file_chunk | c363b6ac739f8f02858b88e7052563bb3edf228e774f60cd4bec313a8435b646 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
module
public import Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
public import Mathlib.Algebra.Order.Star.Basic
public import Mathlib.Algebra.Order.Module.Defs
... |
2,504 | lean_file_chunk | 7636c9c605324d8791f051371b71477cf6bdf2bc022274ab63c089d0668789a5 | 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
-/
module
public import Mathlib.Algebra.Algebra.Defs
public import Mathlib.Data.ZMod.Basic
/-!
# The `ZMod n`-algebra structure on rings whose characteristic divides `... |
16,547 | lean_file_chunk | 5192f54f359a930f693f1cc33423cbb5d9d9fc27f80506eb69ae0bd8275c30eb | 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, Yury Kudryashov
-/
module
public import Mathlib.Algebra.Algebra.Basic
/-!
# Homomorphisms of `R`-algebras
This file defines bundled homomorphisms of `R`-algebras.
## Main def... |
16,419 | lean_file_chunk | 50bf113659cd372340e88d1d79d6e61125da59729b24183c6bd57f346571070b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
module
public import Mathlib.Algebra.Algebra.Hom
public import Mathlib.Algebra.GroupWithZero.Action.Prod
/-!
# Morphisms of non-unital algebras
This file defines morphis... |
10,562 | lean_file_chunk | 9349b02f545340e1c28764cb4163942446568125fc74a39d9a5aed1f1d047db5 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
module
public import Mathlib.Data.Seq.Defs
public import Mathlib.Algebra.Field.Defs
/-!
# Basic Definitions/Theorems for Continued Fractions
## Summary
We def... |
17,462 | lean_file_chunk | 42192678117318b1ff879ea886871ee43b5dd60231782b600778f94c0bda1287 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
module
public import Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence
public import Mathlib.Algebra.ContinuedFractions.TerminatedStable
public import Mat... |
4,923 | lean_file_chunk | 3c2f30474c5045effe608733993f0f26a079b477dcd95ce62ef828a05572c497 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
module
public import Mathlib.Algebra.ContinuedFractions.Basic
public import Mathlib.Algebra.GroupWithZero.Basic
public import Mathlib.Data.Seq.Basic
/-!
# Basic... |
3,226 | lean_file_chunk | 90c726de0dbcf89cf17d956056bba3b04bea3b9bf6cddd249d5c8accdf1e4b0a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
module
public import Mathlib.Algebra.ContinuedFractions.Translations
/-!
# Recurrence Lemmas for the Continuants (`conts`) Function of Continued Fractions
## S... |
2,756 | lean_file_chunk | ea004953345cd3edaab003f6253e0af232a0684a9454faf32890b9542d818c46 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
module
public import Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence
public import Mathlib.Algebra.ContinuedFractions.TerminatedStable
public import Mat... |
3,404 | lean_file_chunk | b709fd6faef53e041085e2c2e3bb1e5b4e82ff77b639e46a13223bb9cfc35887 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
module
public import Mathlib.Algebra.ContinuedFractions.Translations
/-!
# Stabilisation of gcf Computations Under Termination
## Summary
We show that the con... |
2,343 | lean_file_chunk | 4e6507a70e1597f1b493fac823ae662fe282745b5028258710398853e8403ef7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Apurva Nakade. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Apurva Nakade
-/
module
public import Mathlib.Algebra.Module.RingHom
public import Mathlib.Algebra.Order.Module.Defs
public import Mathlib.Algebra.Order.Nonneg.Basic
/-!
# Modules ove... |
4,384 | lean_file_chunk | 2583fc072ad809c373d07f513a4c76b5873d94ef855706554c3f3be33c709212 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
module
public import Mathlib.Algebra.Order.GroupWithZero.Canonical
public import Mathlib.Algebra.Order.Nonneg.Basic
public import Mathlib.Algebra.Order.Nonneg.La... |
8,864 | lean_file_chunk | 7015cb33d7b8a0d8913efbefef0054e4ec71b1ebbde8ca4213a190389b39fef6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
module
public import Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic
public import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
public import Mathlib.Algebra.O... |
1,379 | lean_file_chunk | 7de530c6e8a4ca43a842dea6e8df302be7a46380a584486dd543e9305b235512 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
module
public import Mathlib.Algebra.Order.Floor.Defs
public import Mathlib.Algebra.Order.Nonneg.Basic
/-!
# Nonnegative elements are archimedean
This file def... |
4,166 | lean_file_chunk | 8f59d98648715b7a470a189c1b0cddd5c0571d855f2680f9e5f05dde020a15ec | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
module
public import Mathlib.Algebra.Field.Basic
public import Mathlib.Algebra.Order.Field.Canonical
public import Mathlib.Algebra.Order.Nonneg.Ring
public impor... |
2,512 | lean_file_chunk | 145ca1f357bf42784be404c85f2c4e151b9f7d2aadd76cf5776071760ebaf09e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
module
public import Mathlib.Order.CompleteLatticeIntervals
public import Mathlib.Order.LatticeIntervals
/-!
# Lattice structures on the type of nonnegative ele... |
4,635 | lean_file_chunk | 2663cf8d4a8b87904eb8f6fbc89027a9fb4212c8b2dddaa5778faf2af866e0d1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 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.Order.Ring.Defs
public import Mathlib.Algebra.Ring.InjSurj
public import Mathlib.Tactic.FastInstance
/-!
# Algebraic structur... |
1,054 | lean_file_chunk | 26450389ce7aeb02ad9d460246ee6570fb7310d61a3beea81bebce5e13b77c4e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 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.Field.Defs
public import Mathlib.Algebra.Order.Positive.Ring
/-!
# Algebraic structures on the set of positive numbers
In th... |
1,948 | lean_file_chunk | e733aa02925f11673b0feb8ea7dd7a89fa7c719a71dccf9cdd37e37425ef2212 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
module
public import Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
public import Mathlib.Algebra.Order.Monoid.Canon... |
903 | lean_file_chunk | bb03c4e5c56437e0a5d2ce7b676359a4c53507ff11fc7ec832c11cf3bd039b5e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
module
public import Mathlib.Algebra.Order.Group.Synonym
public import Mathlib.Algebra.Order.Monoid.Unbundled.OrderD... |
1,639 | lean_file_chunk | 5b4d23bce461d485f25c668ef780d0b14b8d0dd567b4a5880930cd66024b2cb9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
module
public import Mathlib.Algebra.Order.GroupWithZero.Canonical
public import Mathlib.Algebra.Order.Monoid.Unbund... |
5,053 | lean_file_chunk | 94bcba54e4286cc1edc500b90423ab4307fb02b36f93274be327de8498acc908 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 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.Group.Prod
public import Mathlib.Algebra.Order.Hom.Monoid
public import Mathlib.Data.Prod.Lex
public import Mathlib.Order.Prod... |
2,444 | lean_file_chunk | 56f9659bd6725d4e3e7d2a2cdfc4f64326328a6985a432f01e1fcb8769924999 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
module
public import Mathlib.Algebra.Group.Prod
public import Mathlib.Algebra.Order.Group.Synonym
public import Math... |
3,450 | lean_file_chunk | b2a509f558b44b800390cdde9dcdff91fa5db9a3fb3c0753ab0a6e0005432b16 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
module
public import Mathlib.Algebra.Order.Monoid.Defs
public import Mathlib.Order.Hom.Basic
/-!
# Ordered monoids
... |
680 | lean_file_chunk | 2caf0ac63fd06ad5f199430099e4b3cac48de9b64cebcf31669a9befdb2c37ac | 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.GroupWithZero.NonZeroDivisors
public import Mathlib.Algebra.Order.Hom.Monoid
public import Mathlib.Data.PNat.Basic
/-!
# Equivalence be... |
3,233 | lean_file_chunk | 5918f763f2803f9fe5993d4939de537338cbc73ac9da5188a74f45ac5b638cbe | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl, Yuyang Zhao
-/
module
public import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
public import Mathlib.Algebra.Order.Z... |
4,975 | lean_file_chunk | dfce76e6b6323699c6490404df714b60efe2b98abdd94ad8c5c7cd82dacc9da2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl, Yuyang Zhao
-/
module
public import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
/-!
# Ordered monoids
This file pro... |
2,857 | lean_file_chunk | ba2e6c49d474a7f0799646b1822b91b5122f4e515f39899dcf2c44c4a57701ed | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
module
public import Mathlib.Algebra.Group.Submonoid.Defs
public import Mathlib.Algebra.Order.Monoid.Basic
public import Mathlib.Order.Interval.Set.Defs
/-!
# Ordered... |
8,452 | lean_file_chunk | 3673bdd612ab72f2b233ce72a393e7bf1eb88be106bceecc928e56e74d036482 | 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.Algebra.Group.Subgroup.Ker
public import Mathlib.Algebra.Order.Group.Units
public import Mathlib.Algebra.Order.Hom.MonoidWithZero
public impor... |
1,811 | lean_file_chunk | 28bb8839a2815ca665dd829a67107d2f188a46f36a0eeb71a17a0e973494f39b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
module
public import Mathlib.Algebra.Order.Monoid.Unbundled.TypeTags
public import Mathlib.Algebra.Order.Monoid.Cano... |
2,066 | lean_file_chunk | 99eceeda80b2db91e3d45f010f5fbd3e59013819f09359a0cabdba301ac17a9a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
module
public import Mathlib.Order.Hom.Basic
public import Mathlib.Algebra.Group.Units.Defs
/-!
# Units in ordered ... |
788 | lean_file_chunk | 9d8691fc3bdbfc0e53f1fe0ae1d1ab9e2c63d9c64b45395d6f841497d9fcaae4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Paul Lezeau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul Lezeau
-/
module
public import Mathlib.Algebra.GroupWithZero.Associated
public import Mathlib.Algebra.Order.Monoid.Canonical.Defs
/-!
# Order on associates
This file shows that di... |
23,983 | lean_file_chunk | fee6db83612c91dfaa188f1713ff82301f221a54b87421f1668eca31aad028d2 | 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.BigOperators.Group.Finset.Basic
public import Mathlib.Algebra.Group.Subgroup.Lattice
public import Mathlib.Algebra.Order.Archimedean.Bas... |
13,068 | lean_file_chunk | f2c502f2eaba116aaa8c60b22bbd4e9434035b567d49437f1a084481952f24a4 | lean_source_file | pkg_mathlib_source_file_chunks | refine Set.ssubset_iff_subset_ne.mpr ⟨by simpa [subsemigroup] using hst.le, ?_⟩
contrapose! hst with heq
apply le_of_eq
simpa [mk_surjective, subsemigroup] using heq
@[to_additive]
theorem subgroup_antitone : Antitone (subgroup (M := M)) := by
intro s t hst
obtain rfl | hs := eq_or_ne s ⊤
· rw [eq_top_iff.... |
21,392 | lean_file_chunk | ee8844501e62077abb50babc0d8327fbcdf54cf70dd001e03454284bdc6ac512 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
module
public import Mathlib.Algebra.Field.Defs
public import Mathlib.Algebra.Order.Archimedean.Defs
public import Mathlib.Algebra.Order.Floor.Semiring
public import... |
3,470 | lean_file_chunk | c85bc3bfc7dc9166a2aa87fb85b3fb373a12cc123c8b9c14f5584c5d57814f1e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
module
public import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Order.Group.OrderIso
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
/-!
# Definition... |
1,207 | lean_file_chunk | f643440d24226428631512da74ed76c35e1fc2c0c10c99d876a7c21edb229f25 | 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.Group.Submonoid.Defs
public import Mathlib.Algebra.Order.Archimedean.Basic
/-!
# Submonoids of archimedean monoids
This file... |
3,175 | lean_file_chunk | 0ec7ef3e0ad344fd54a966989db7c22e2a131a026c296abf6350b5d41dfa6c54 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Damien Thomine. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damien Thomine
-/
module
public import Mathlib.Algebra.BigOperators.Group.Finset.Indicator
public import Mathlib.Algebra.Order.Archimedean.Basic
public import Mathlib.Algebra.Order.Bi... |
2,434 | lean_file_chunk | 523b985b53fd714e3a1318cb9a27d9ff3a3a993195b006b1e08289b985be063c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Alex J. Best, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Yaël Dillies
-/
module
public import Mathlib.Algebra.Order.Archimedean.Basic
public import Mathlib.Algebra.Order.Hom.Ring
/-!
### Uniqueness of ring homomor... |
23,970 | lean_file_chunk | d5fe3a8838b9981865847f95ceca3bf781308f4a659515bd3507b608a502b679 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.Algebra.Group.Equiv.Defs
public import Mathlib.Algebra.Group.Hom.Basic
public import Mathlib.Algebra.Order.Group.Unbundled.Basic
public impo... |
1,200 | lean_file_chunk | 8c65a0f719e4355b5f4577414ad3464002818f2f3deef241a20c79749c6d4d6d | lean_source_file | pkg_mathlib_source_file_chunks | e.symm ∘ g = f ↔ g = e ∘ f :=
e.toEquiv.symm_comp_eq f g
@[to_additive]
lemma lt_symm_apply (e : α ≃*o β) {x : α} {y : β} : x < e.symm y ↔ e x < y :=
e.toOrderIso.lt_symm_apply
@[to_additive]
lemma symm_apply_lt (e : α ≃*o β) {x : α} {y : β} : e.symm y < x ↔ y < e x :=
e.toOrderIso.symm_apply_lt
variable (f)
... |
13,687 | lean_file_chunk | 66e5d0c42f492ea81d423f039c5af1abf3beb299b4363e165f2b187c270fd865 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Alex J. Best, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Yaël Dillies
-/
module
public import Mathlib.Algebra.Order.Hom.MonoidWithZero
public import Mathlib.Algebra.Ring.Equiv
/-!
# Ordered ring homomorphisms
Hom... |
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