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 |
|---|---|---|---|---|---|
17,815 | lean_file_chunk | 9312dac513ee38194521a67fe80476ab3276e52c32b636ebb2e9788a7d5f1fb0 | 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, Devon Tuma, Oliver Nash
-/
module
public import Mathlib.Algebra.Group.Submonoid.Membership
public import Mathlib.Algebra.GroupWithZero.Associated
public import Mathlib.Algebra.G... |
5,041 | lean_file_chunk | 513532a0cda4da16b3215e36eeec174939240210e830213e070e18f7ded65caa | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 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.Group.Subgroup.Pointwise
public import Mathlib.Algebra.GroupWithZero.Submonoid.Pointwise
/-!
# Subgroups in a group with zero
-/
@[e... |
1,782 | lean_file_chunk | 3ae1ec98065eb0807fbc7561044a704411812d89133036b5e0cb775ed26e8b42 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Xavier Roblot
-/
module
public import Mathlib.Algebra.Regular.Basic
public import Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors
/-!
# Torsion-free monoids ... |
6,006 | lean_file_chunk | 8e4b25ae53508c53635a396caf34ef3dc130e071e52a9087d6c7fc40c9f2e67c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Amelia Livingston, Yury Kudryashov,
Neil Strickland, Aaron Anderson
-/
module
public import Mathlib.Algebra.GroupWithZero.Units.Basi... |
1,937 | lean_file_chunk | 7588bb4f6dd5ac4bcb35b874bca9a6128db46bb01b8088d6a82b29036f8a9fa2 | 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.GroupWithZero.Hom
/-! # Isomorphisms of monoids with zero -/
@[expose] public secti... |
12,270 | lean_file_chunk | 473976eaea1d99b96b582ac0bedb97c1716cc52b6774df930e9e744ae61a4f19 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Antoine Chambert-Loir and Filippo Nuccio. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández, Filippo A. E. Nuccio
-/
module
public import Mathlib.Algebra.Group.Subgroup.Pointwise
public import Ma... |
1,669 | lean_file_chunk | ecf561c9f02865f9d7400fd6c690af670e63dae7f7f1eeeeff8784d27c850a93 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 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.GroupWithZero.Defs
public import Mathlib.Algebra.NeZero
/-!
# `NeZero 1` in a nontrivial `MulZeroOneClass`.
This file exists t... |
10,063 | lean_file_chunk | 8ada865fac6c90aed3ddc82cf5f096f63e32fb3f84233c213533e9547d85c85f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
module
public import Mathlib.Algebra.Group.Hom.Basic
public import Mathlib.Algebra.GroupWithZero.Basic
/-!
# Monoid with zero and group with zero homomorphisms
This f... |
23,986 | lean_file_chunk | 0e752c20eb8f7cb20526bb18563808a68b2065b7673ce59307374aa1cd58f73c | 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, Jens Wagemaker
-/
module
public import Mathlib.Algebra.Prime.Lemmas
public import Mathlib.Order.BoundedOrder.Basic
/-!
# Associated elements.
In this file we define ... |
3,903 | lean_file_chunk | 9df9f0c12dc8d666d7bf7e05b2d42c6571bf68668753f3277603cf2e702e2fc9 | lean_source_file | pkg_mathlib_source_file_chunks | constructor
· rintro ⟨x, hx, u, rfl⟩
refine ⟨x * u, ?_, mul_assoc ..⟩
simpa
· rintro ⟨x, ⟨hx, rfl⟩⟩
use x
theorem dvdNotUnit_of_lt {a b : Associates M} (hlt : a < b) : DvdNotUnit a b := by
constructor
· rintro rfl
apply not_lt_of_ge _ hlt
apply dvd_zero
rcases hlt with ⟨⟨x, rfl⟩, ndvd⟩
... |
1,202 | lean_file_chunk | 32eb63a296480045d5f8bc89fdc745ebdd3b5eaf71a1d0c1622524a776e2b02d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 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.ModEq
public import Mathlib.Algebra.Field.Basic
public import Mathlib.Tactic.MinImports
/-!
# Congruence modulo multiples of ... |
2,761 | lean_file_chunk | a9330d1da8aa16d3b002a1b64e6a5dcea1a9379ecdbbddbc8c75466768a1eb6d | 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.Field.Defs
public import Mathlib.Algebra.Ring.Opposite
public import Mathlib.Data.Int.Cast.Lemmas
/-!
# Field structure on the multiplica... |
3,072 | lean_file_chunk | ca09f5cae028084d5dffee6316ebf7330c2a551d1366abb1b73e773f0d39bca9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
module
public import Mathlib.Algebra.Field.Defs
public import Mathlib.Data.NNRat.Defs
/-!
# The rational numbers form a field
This file contains th... |
1,654 | lean_file_chunk | a0a229814032d4de3a4115ba4efe91d423b9e19b195b8cfe4e488fe816567f48 | 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.Field.Basic
public import Mathlib.Algebra.Ring.TransferInstance
/-!
# Transfer algebraic structures across `Equiv`s
This conti... |
608 | lean_file_chunk | 1a14927656faecac401a9a59e26eea5641348c102502437f0b2baa9dee73bedd | 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, Johan Commelin
-/
module
public import Mathlib.Algebra.Field.Defs
public import Mathlib.Algebra.Ring.NegOnePow
public import Mathlib.Tactic.NormNum
/-! # Integer powers of `-1`... |
3,990 | lean_file_chunk | c9eccf2c420dadd7e5b62c810ece2f25096e3ca3fe2cf5439d2a7515c279a0a8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
module
public import Mathlib.Algebra.Field.Defs
public import Mathlib.Tactic.Common
/-!
# `IsField` predicate
... |
13,765 | lean_file_chunk | ab63c9a98a881229ee1debbadfdd785765bbb454c631b4adee662fd63607859d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
module
public import Mathlib.Algebra.Field.Defs
public import Mathlib.Algebra.Ring.GrindInstances
public import ... |
2,017 | lean_file_chunk | 83dd98114b733554e69201be2ca29a95cb7354e6f17c87a7b699edcf77c8be3f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 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.Field.Defs
public import Mathlib.Algebra.GroupWithZero.ULift
public import Mathlib.Algebra.Ring.ULift
/-!
# Field instances for `UL... |
10,201 | lean_file_chunk | 0281db478090a9e3a419a1eb078d3dd5effd3ac2b55455b5e769f4080606a3a3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Johannes Hölzl, Mario Carneiro, Yaël Dillies
-/
module
public import Mathlib.Algebra.Ring.Defs
public import Mathlib.Data.Rat.Init
/-!
# Division... |
762 | lean_file_chunk | f16b43cc13cb9241cbddfdb48476fb1bf6f6a384aba1cf630c9af24795e3e8d5 | 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.Field.TransferInstance
public import Mathlib.Logic.Small.Defs
/-!
# Transfer field structures from `α` to `Shrink α`
-/
@[expose] ... |
3,408 | lean_file_chunk | 89017ea6ce5f44b95d81cd82bab523e8c392333363d57f0a0414b750bfba7c40 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
module
public import Mathlib.Algebra.Field.Basic
public import Mathlib.Algebra.Ring.GeomSum
/-!
# Partial sums of geometric series in a field
This file determine... |
1,024 | lean_file_chunk | 821132ea5c0e135ec6a1f2cfaf87375d8c6ab987e97061768e88ca58cd743988 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
module
public import Mathlib.Algebra.Field.Defs
public import Mathlib.Algebra.Ring.Int.Parity
/-!
# Results abo... |
1,175 | lean_file_chunk | 884e49b0f6b6f9f6d995d41a6db71dcd9aae7880c729e9c4641a2212c07fbd4b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 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.Field.IsField
public import Mathlib.Algebra.Group.Units.Equiv
public import Mathlib.Algebra.GroupWithZero.Equiv
/-!
# If a semiring is a ... |
7,611 | lean_file_chunk | a0e2fb512f20a661544aafadbcbd6242dfc70bf3603dd49218d41fa496bf20c2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
module
public import Mathlib.Algebra.Field.Opposite
public import Mathlib.Algebra.Module.Opposite
public import Mathlib.Algebra.Order.Archimedean.Basic
public ... |
1,102 | lean_file_chunk | 801666194459e6cc271a35c27340a55e687d05c42cc0d1f98b43d5d2dbe3d7d9 | 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
-/
module
public import Mathlib.Algebra.Field.Basic
public import Mathlib.Data.ZMod.Basic
/-!
# `ZMod p` is a field
-/
@[expose] public section
namespace ZMod
variable (... |
1,642 | lean_file_chunk | 355b52c293678c9e615816f6ed1997d924c10f553b6889793edf09eff50997a4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
module
public import Mathlib.Algebra.Field.Defs
public import Mathlib.Algebra.Ring.MinimalAxioms
/-!
# Minimal Axioms for a Field
This file defines constructors to def... |
23,941 | lean_file_chunk | 0331d11804a3d3fa7eac15ca70350c3bfc8864e3000d9f7bd8c506f6c8977b97 | 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, Jireh Loreaux
-/
module
public import Mathlib.Algebra.Algebra.Subalgebra.Lattice
public import Mathlib.Algebra.Algebra.Tower
public import Mathlib.Algebra.Star.Module
publ... |
13,152 | lean_file_chunk | ea4d9c1cadf0085d059424ecd6ebef2110fcce95b70446083b129bef8ecaaeb9 | lean_source_file | pkg_mathlib_source_file_chunks | @[deprecated isMulCommutative_adjoin (since := "2026-03-11")]
abbrev adjoinCommSemiringOfComm {s : Set A}
(hcomm : ∀ a ∈ s, ∀ b ∈ s, a * b = b * a)
(hcomm_star : ∀ a ∈ s, ∀ b ∈ s, a * star b = star b * a) :
CommSemiring (adjoin R s) :=
have := isMulCommutative_adjoin R hcomm hcomm_star
inferInstance
in... |
4,785 | lean_file_chunk | 134b64432ce9aa81a3259da238d5ddf3423082cdde955da474ae27a0bb9fc1b3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Monica Omar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Monica Omar
-/
module
public import Mathlib.Algebra.Central.Basic
public import Mathlib.Algebra.Ring.Action.ConjAct
public import Mathlib.Algebra.Star.StarAlgHom
public import Mathlib.Al... |
9,841 | lean_file_chunk | e7e97bbc68f886cd9f50a5ff21f5b9f3ceebbd3b89085bebd189fb0547eabb1d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Frédéric Dupuis
-/
module
public import Mathlib.Algebra.Star.SelfAdjoint
public import Mathlib.Algebra.Module.Basic
public import Mathlib.Algebra.Module.Equiv.Defs
public im... |
1,064 | lean_file_chunk | a2bcb7a153586b1f122b3e994141e025b69a13bd7e84548192b86db3b45a2e14 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 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.Field.Opposite
public import Mathlib.Algebra.Star.Basic
public import Mathlib.Data.NNRat.Defs
public import Mathlib.Data.Rat.Cast.De... |
1,384 | lean_file_chunk | aea67d7d1fa6b35c5bb386e6a158a1b41986895bb5d5633ad3b59d7a28c20944 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Monica Omar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Monica Omar
-/
module
public import Mathlib.Algebra.Star.Module
public import Mathlib.LinearAlgebra.TensorProduct.Map
/-!
# The star structure on tensor products
This file defines the ... |
22,402 | lean_file_chunk | 475b080682ff288b45696e69bc8798c08ed488fd2a9eb0975267b86cab0dd78f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 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.Group.Subgroup.Defs
public import Mathlib.Algebra.Module.Defs
public import Mathlib.Algebra.Star.Rat
/-!
# Self-adjoint, skew... |
1,648 | lean_file_chunk | 8c7b3ffeb93ad10ee979288aaa50d3329d677a8c355368ea174579e92ce913f1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 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.Ring.Prod
public import Mathlib.Algebra.Star.Basic
/-!
# Basic Results about Star on Product Type
This file provides basic results a... |
4,168 | lean_file_chunk | f00cded08593fc9f28c61dfcf8040b51dc76560088671ae454fc1376851fa39f | 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.Star.Basic
public import Mathlib.Data.Set.Finite.Basic
public import Mathlib.Data.Set.Lattice.Image
public import Mathlib.Algebra.... |
23,993 | lean_file_chunk | b9cd8f2275995a95f93af00bc42b6eb54489857e5d24867c98c054e1a9077ca0 | 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.Equiv
public import Mathlib.Algebra.Algebra.NonUnitalHom
public import Mathlib.Algebra.Algebra.Prod
public import Mathlib.... |
6,782 | lean_file_chunk | 341816fa035ed8930b626de2684c9dc632693849f65a90c9c3e114a0a070e667 | lean_source_file | pkg_mathlib_source_file_chunks | /-- The inverse of a star algebra isomorphism is a star algebra isomorphism. -/
@[symm]
nonrec def symm (e : A ≃⋆ₐ[R] B) : B ≃⋆ₐ[R] A :=
{ e.symm with
map_smul' := fun r b => by
simpa only [apply_inv_apply, inv_apply_apply] using
congr_arg (inv e) (map_smul e r (inv e b)).symm }
/-- See Note [custo... |
2,363 | lean_file_chunk | 07634a1cd26965bca1d8c83aae3518976320bc828b7570afd702d3f3be268e95 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 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.Star.Basic
public import Mathlib.Algebra.Notation.Pi.Defs
public import Mathlib.Algebra.Ring.Pi
/-!
# Basic Results about Star on Pi ... |
2,532 | lean_file_chunk | 4aad1465c1d5d8c27fb7b339e34e5d50f8de4efc50e09baaaf088f12f212eb65 | 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.Star.Basic
public import Mathlib.Algebra.Star.Pointwise
public import Mathlib.Algebra.Group.Center
/-! # `Set.center`, `Set.centr... |
19,805 | lean_file_chunk | eca254f3b8df2d86065dcca533fb417c28586516f30995e83d9106ac7b76d88a | 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.Action.Opposite
public import Mathlib.Algebra.Group.Action.Units
public import Mathlib.Algebra.Group.Invertible.Defs
public im... |
5,225 | lean_file_chunk | a3fea2f06220d1562612c03e1e94cb4e97870445b9f0527509e603059c65abe0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 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.Star.NonUnitalSubsemiring
public import Mathlib.Algebra.Ring.Subsemiring.Basic
/-!
# Star subrings
A *-subring is a su... |
4,480 | lean_file_chunk | fc466f10a3ab03251f597fa058814382f7d6873910fe143a8de8dd18ceeab152 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 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.Ring.Defs
public import Mathlib.Algebra.Group.Subsemigroup.Basic
public import Mathlib.RingTheory.NonUnitalSubsemiring.B... |
23,978 | lean_file_chunk | 9eb2ee2bbf060247e82bd01cc37ff24c20fa3eff5361a4316822b0b895be1fad | 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.NonUnitalSubalgebra
public import Mathlib.Algebra.Star.StarAlgHom
public import Mathlib.Algebra.Star.Center
public import ... |
23,973 | lean_file_chunk | d69cd58cf66e9ee428b51da214cd6e4e71b3039a9a68b418169ebcbbceb3d5da | lean_source_file | pkg_mathlib_source_file_chunks | /-- The `NonUnitalStarSubalgebra` obtained from `S : NonUnitalSubalgebra R A` by taking the
smallest non-unital subalgebra containing both `S` and `star S`. -/
def starClosure (S : NonUnitalSubalgebra R A) : NonUnitalStarSubalgebra R A where
toNonUnitalSubalgebra := S ⊔ star S
star_mem' {a} ha := by
simpa [← me... |
3,999 | lean_file_chunk | 2f22cfa0c7c3d6cf02bb888a19046be5bcd2b586fd06b7dee04b45925db1581a | lean_source_file | pkg_mathlib_source_file_chunks | variable [CommSemiring R] [StarRing R]
variable [NonUnitalSemiring A] [StarRing A] [Module R A]
variable [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A]
variable (R) in
lemma adjoin_le_centralizer_centralizer (s : Set A) :
adjoin R s ≤ centralizer R (centralizer R s) := by
rw [← toNonUnitalSubalgebr... |
8,121 | lean_file_chunk | d40ddde0bec06cced3323242195628369ac75022a4e8a15a3ccce4c2fce9c575 | 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.CharP.Invertible
public import Mathlib.Algebra.Order.Star.Basic
public import Mathlib.Data.Real.Sqrt
public import Mathlib.Data.Real... |
12,767 | lean_file_chunk | a4ba19b0de3775f2e06779b0304a360813c49a876d51ecbb52a5abc36c16342b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Monica Omar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Monica Omar
-/
module
public import Mathlib.Algebra.Algebra.Bilinear
public import Mathlib.Algebra.WithConv
public import Mathlib.Algebra.Star.Pi
public import Mathlib.Algebra.Star.SelfA... |
1,805 | lean_file_chunk | 31b647cc878614490b31b1edffb85bb55da45d8f2dcf889391b162656ea72e3e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 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.RingQuot
public import Mathlib.Algebra.Star.Basic
/-!
# The *-ring structure on suitable quotients of a *-ring.
-/
public section
n... |
5,562 | lean_file_chunk | 45cd126ace0939a5d0034807e18937660a99ddb13e77a8ea623b53bc54654ef2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 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.Ring.CentroidHom
public import Mathlib.Algebra.Star.StarRingHom
public import Mathlib.Algebra.Star.Subsemiring
public im... |
17,834 | lean_file_chunk | 1d9a050fc672ab962fa5ef85f6ebaf576556b59cdf21da281e5884364bb0dc77 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam, Frédéric Dupuis
-/
module
public import Mathlib.Algebra.Group.Submonoid.Operations
public import Mathlib.Algebra.Star.SelfAdjoint
public import Mathlib.Algebra.Algebra... |
5,802 | lean_file_chunk | 6337cc8bb53ed10c9813eaa514bd4622f2476ff89726fbe31e2cc27a4e2028d8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Monica Omar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Monica Omar
-/
module
public import Mathlib.Algebra.Star.SelfAdjoint
public import Mathlib.Algebra.Group.Idempotent
public import Mathlib.Algebra.Ring.Idempotent
/-!
# Star projections
... |
1,502 | lean_file_chunk | f065c83e1534f670c8d2de72352b1e36bb00003977115192e17cc38a6999a427 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 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.BigOperators.Finsupp.Basic
public import Mathlib.Algebra.Star.SelfAdjoint
/-! # Big-operators lemmas about `star` algebraic operation... |
14,503 | lean_file_chunk | acd4da8ffadd210830d092c665d1efd9887b5b33b2bd0e29acac366ade7f5e9b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 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.Ring.Defs
public import Mathlib.Algebra.Star.Basic
/-!
# Morphisms of star rings
This file defines a new type of morph... |
9,644 | lean_file_chunk | 815d7d92e162b52a82d9bcbf72486c64bf004a60bd2222b09abf903bb6993f38 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 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.Star.Basic
/-!
# Morphisms of star monoids
This file defines the type of morphisms `StarMonoidHom` between monoids `A` and `B` wh... |
2,351 | lean_file_chunk | 86706da331d6df86980fdfa321ae0c7edc22dfa430cba8d71ebb4d2dd112d424 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 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.Star.Basic
public import Mathlib.Algebra.FreeAlgebra
/-!
# A *-algebra structure on the free algebra.
Reversing words gives a *-stru... |
3,358 | lean_file_chunk | da387336302f41c16b60a9b9fe0182b815eb5c0871a057091d54ba86471fc972 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 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.BigOperators.Pi
public import Mathlib.Algebra.Star.Pi
public import Mathlib.Algebra.Group.Pointwise.Set.Basic
/-!
# Conjugation-neg... |
5,134 | lean_file_chunk | 45820d4c951b28fb3630da9174d6c35f135ca786afb7c421824a43bd914b2b64 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Xavier Généreux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Généreux, María Inés de Frutos Fernández
-/
module
public import Mathlib.Algebra.Algebra.Defs
public import Mathlib.Algebra.SkewMonoidAlgebra.Basic
/-!
# Univariate skew polyn... |
16,308 | lean_file_chunk | 3cf1039f0773fb628b0be05b8a91e6908533ff688c8efd961828129517b816bd | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 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.AddGroupWithTop
public import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
public import Mathlib.Algebra.Order.Monoid.U... |
5,038 | lean_file_chunk | cbf3bf00ac61960075f21ed8cfaa7dc84db803904e5d0a1f961490f71e51d770 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
module
public import Mathlib.Data.List.MinMax
public import Mathlib.Algebra.Tropical.Basic
public import Mathlib.Order.ConditionallyCompleteLattice.Finset
public i... |
2,958 | lean_file_chunk | 6ec3b35ac5de95eb005a57956647cedd2d74098e8ccf7b1546346c3127d56a43 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 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.Tropical.Basic
public import Mathlib.Order.ConditionallyCompleteLattice.Basic
/-!
# Order on tropical algebraic structure
T... |
3,154 | lean_file_chunk | f5d8030c4afc18c06be39ae276175341ed0c6fd88cbfede66629b49e64883f89 | 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.FreeMonoid.Basic
public import Mathlib.Algebra.Group.TypeTags.Basic
/-!
# `List.count` as a bundled homomorphism
In this fil... |
18,508 | lean_file_chunk | e6d2c0c1a7dd2f4e650a82ac2d935f13c5e439daac4e09f56413920134b06ac0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Yury Kudryashov
-/
module
public import Mathlib.Algebra.Group.Action.Defs
public import Mathlib.Algebra.Group.Units.Defs
public import Mathlib.Algebra.Group.Equiv.Defs
publi... |
1,157 | lean_file_chunk | 6551f656d24fd62d48c9bfc43b409cf890661078afb6309dec2346427dfd2303 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Hannah Fechtner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hannah Fechtner
-/
module
public import Mathlib.Algebra.FreeMonoid.Basic
public import Mathlib.Data.Finset.Lattice.Lemmas
/-!
# The finite set of symbols in a FreeMonoid element
Th... |
1,028 | lean_file_chunk | 113739b535a87776b11574c920d53e3131396a7ae99d5c03a8d630204b2f6c1d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 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.FreeMonoid.Basic
public import Mathlib.Algebra.Group.UniqueProds.Basic
public import Mathlib.Algebra.Order.Group.Nat
/-!
# Free m... |
8,123 | lean_file_chunk | afc90e4916eb367d6c1dcf09fa1733bd2c341392146ca492c54b195ac21418fc | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 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.DirectSum.Module
public import Mathlib.Algebra.Lie.OfAssociative
public import Mathlib.Algebra.Lie.Ideal
public import Mathlib.Algebra... |
6,156 | lean_file_chunk | 16f58bf193520dc68bbdacc43c4bdbd0d714c050b5fedba85b40af539c74dad2 | 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, Jonathan Reich
-/
module
public import Mathlib.Algebra.Lie.Free
public import Mathlib.Algebra.Lie.Quotient
public import Mathlib.Data.Matrix.Cartan
/-!
# Serre construction... |
23,974 | lean_file_chunk | 27f5f70ac0b4dddc0c9d9cd93c3e52e4abc625bdb2e72a7236f297d1fe147b59 | 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.Lie.Basic
public import Mathlib.RingTheory.Artinian.Module
/-!
# Lie subalgebras
This file defines Lie subalgebras of a Lie algebra ... |
2,059 | lean_file_chunk | f15ac80620df73018c85d1b09bc443ad6f64bfb47e7d07be8eeb8920d05f0c12 | lean_source_file | pkg_mathlib_source_file_chunks | /-- An injective Lie algebra morphism is an equivalence onto its range. -/
noncomputable def ofInjective (f : L₁ →ₗ⁅R⁆ L₂) (h : Function.Injective f) : L₁ ≃ₗ⁅R⁆ f.range :=
{ LinearEquiv.ofInjective (f : L₁ →ₗ[R] L₂) <| by rwa [LieHom.coe_toLinearMap] with
map_lie' {x y} := SetCoe.ext <| f.map_lie x y }
@[simp]
t... |
6,391 | lean_file_chunk | d150bf07c5eef38833070995036e371cc27712cb307e903815f70af20f7d31e7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
module
public import Mathlib.Algebra.Lie.Abelian
/-!
# Lie algebra cohomology in low degree
This file defines low degree cochains of Lie algebras with coefficients ... |
6,034 | lean_file_chunk | 6a1f89b7e8699a3fa69ccbed5c45230815919e998bb8c9fe35aa888a124017c4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 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.Lie.OfAssociative
public import Mathlib.Algebra.RingQuot
public import Mathlib.LinearAlgebra.TensorAlgebra.Basic
/-!
# Universal enve... |
4,965 | lean_file_chunk | 3b4dcbb5a9e81cad3e9c9b2471132d36b0014ebd5cc7997a9136834bb19d4411 | 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.Lie.Nilpotent
public import Mathlib.Algebra.Lie.Normalizer
/-!
# Cartan subalgebras
Cartan subalgebras are one of the most important... |
6,180 | lean_file_chunk | 333608f05e81b41e3844b685479eb75cccc0fbbfe5cf82535d6aa87f1ebbeb52 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 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.Lie.Engel
public import Mathlib.Algebra.Lie.Normalizer
public import Mathlib.Algebra.Lie.OfAssociative
public import Mathlib.Alg... |
3,960 | lean_file_chunk | b7edc883e7af49e933db693821c98be799d16a3eb18cfdc3a8fff813f20fc63a | 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.Lie.OfAssociative
public import Mathlib.LinearAlgebra.Matrix.Reindex
public import Mathlib.LinearAlgebra.Matrix.ToLinearEquiv
/-!
# L... |
23,966 | lean_file_chunk | b68ab085c287852154dc27d36427fb9d34795e2517ef224985aa76cad32b8044 | 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.Lie.Solvable
public import Mathlib.Algebra.Lie.Quotient
public import Mathlib.Algebra.Lie.Normalizer
public import Mathlib.Algebra.Ord... |
14,431 | lean_file_chunk | d5c8e2d9adad8bfea6e5bcb2615f1c64ec7c30541f16264c5c8102460bd5e4ae | lean_source_file | pkg_mathlib_source_file_chunks | use f x, ⟨g m_n, ih (Submodule.mem_map_of_mem h₁)⟩
simp [hfg x m_n, h₂]
obtain ⟨y, n, hn⟩ := this
rw [← hn]
apply LieSubmodule.lie_mem_lie
· simp
· exact SetLike.coe_mem n
variable [LieModule R L₂ M₂] (hg_inj : Injective g)
include hg_inj hfg in
theorem Function.Injective.lieModuleIsNilpoten... |
17,793 | lean_file_chunk | ffd09894afffc2c35afe55cd3aa4cba1e9eb26b62148798b29576cd0d935cee8 | 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.Lie.Abelian
public import Mathlib.Algebra.Lie.BaseChange
public import Mathlib.Algebra.Lie.IdealOperations
public import Mathlib.Order... |
11,967 | lean_file_chunk | 4063394c8ffd542b2b8c6396f44b02fe35158ee92c12671f2492853230af1c31 | 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.Lie.Ideal
/-!
# Ideal operations for Lie algebras
Given a Lie module `M` over a Lie algebra `L`, there is a natural action of the Li... |
7,799 | lean_file_chunk | 23254a681659fbce07f4867f0ad64834a968432ef3f140774699f776750318d9 | 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.Lie.Abelian
public import Mathlib.LinearAlgebra.TensorProduct.Tower
/-!
# Tensor products of Lie modules
Tensor products of Lie modu... |
15,495 | lean_file_chunk | fdb3c27e63a8ad86fc7825c9719267ca2474fc2816317658c98050c46d24b13f | 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.Lie.OfAssociative
public import Mathlib.Algebra.Lie.IdealOperations
/-!
# Trivial Lie modules and Abelian Lie algebras
The action of... |
11,551 | lean_file_chunk | c485062c75a09da0f4f3e70d0c8c128e485755e8c56b5f7fd34045cb6a7e4cf0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Lucas Whitfield. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lucas Whitfield, Johan Commelin
-/
module
public import Mathlib.Algebra.Lie.Weights.Basic
public import Mathlib.RingTheory.Finiteness.Nilpotent
/-!
# Lie's theorem for Solvable Lie ... |
7,165 | lean_file_chunk | 53969c301d6f6a451d39b95cdd9729530db9c60e1b4cd4ff0faf8968d142c3b7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 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.Lie.EngelSubalgebra
public import Mathlib.Algebra.Lie.OfAssociative
public import Mathlib.Algebra.Module.LinearMap.Polynomial
pu... |
6,878 | lean_file_chunk | 3b669a0c4e7a22cd2e8920b39d96d8d0366adb3292053a933f12aa6557dfaada | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Leonid Ryvkin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonid Ryvkin
-/
module
public import Mathlib.Algebra.Lie.Ideal
/-! ### Products of Lie algebras
This file defines the Lie algebra structure the Product of two Lie algebras
## Main d... |
3,244 | lean_file_chunk | e0c34d591b83a65c926d2e01709110862b315aa2502a93b35057c4bfb50afd93 | 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.NonUnitalHom
public import Mathlib.Algebra.Lie.Basic
/-!
# Lie algebras as non-unital, non-associative algebras
The definiti... |
8,368 | lean_file_chunk | 6f93babfd95b6ce9f8cf1a894455c9c26bd0b7b7de04937f8fe08cd186254a38 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 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.Lie.Semisimple.Defs
public import Mathlib.LinearAlgebra.BilinearForm.Orthogonal
/-!
# Lie algebras with non-degenerate invarian... |
14,080 | lean_file_chunk | b686824d9a925d7514edf0f3c501d313fab78c3faf5e902e97b0a695f4279ee1 | 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.Lie.Basic
public import Mathlib.Algebra.Lie.Subalgebra
public import Mathlib.Algebra.Lie.Submodule
public import Mathlib.Algebra.Algeb... |
6,227 | lean_file_chunk | 0daee14790ce041537239ec736b55a3f05e95eb01e72c599abcad9d4eaa5296e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
module
public import Mathlib.Algebra.DirectSum.Decomposition
public import Mathlib.Algebra.Lie.Derivation.Basic
/-!
# Graded Lie algebras
This file defines typecla... |
19,538 | lean_file_chunk | 434dcdb5ff376cfcfc39b98a9fa67bfb2e658bc9a40cbbac5a7c4f0034597cf2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
module
public import Mathlib.Algebra.Exact
public import Mathlib.Algebra.Lie.Cochain
/-!
# Extensions of Lie algebras
This file defines extensions of Lie algebras, ... |
6,902 | lean_file_chunk | ce52ebad6bde71c726bdfd2f576137ba43d329f58c98bc5dc8bf8a50a4f2338f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 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.Lie.Matrix
public import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
public import Mathlib.Tactic.NoncommRing
/-!
# Lie algebras of... |
23,961 | lean_file_chunk | 6481bc7a9c2bd0130ac585b9f0e2a396aadcec5565c6111d3807b0fc62810346 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 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.Module.Submodule.Equiv
public import Mathlib.Algebra.Module.Equiv.Basic
public import Mathlib.Algebra.Module.Rat
public import Mathlib... |
9,317 | lean_file_chunk | 711a526db43dd439ac579b5cedf59a4589694a6616b78d6263bdb96abe7a9867 | lean_source_file | pkg_mathlib_source_file_chunks | map_smulₛₗ _ _ _ := by rw [← coe_toLinearMap, map_smulₛₗ]
@[simp]
theorem map_lie (f : M →ₗ⁅R,L⁆ N) (x : L) (m : M) : f ⁅x, m⁆ = ⁅x, f m⁆ :=
LieModuleHom.map_lie' f
variable [LieAlgebra R L] [LieModule R L N] [LieModule R L P] in
theorem map_lie₂ (f : M →ₗ⁅R,L⁆ N →ₗ[R] P) (x : L) (m : M) (n : N) :
⁅x, f m n⁆ = ... |
7,900 | lean_file_chunk | 95af83ed244e9cecb9881986cb1e365793b628e872469e525290ddfa68f4ca9f | 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.Lie.Ideal
public import Mathlib.Algebra.Lie.OfAssociative
public import Mathlib.LinearAlgebra.Isomorphisms
public import Mathlib.RingT... |
17,855 | lean_file_chunk | 65d6baa9b25170a1a073538f48b6644a6bc83e4b67c573f714e49859b477af2a | 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.Lie.Submodule
/-!
# Lie Ideals
This file defines Lie ideals, which are Lie submodules of a Lie algebra over itself.
They are defined... |
9,557 | lean_file_chunk | 99a0b247e99273358e27245d8bd7cd0a3c9cdeb6cc04e969f8456b143f75da43 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 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.Lie.OfAssociative
public import Mathlib.LinearAlgebra.Eigenspace.Basic
public import Mathlib.LinearAlgebra.FreeModule.StrongRankCondit... |
17,441 | lean_file_chunk | 494984e7727dad06092e6b3b229eaaf1bf74cadef58b791438a0ccee36595455 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 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.Lie.CartanSubalgebra
public import Mathlib.Algebra.Lie.Rank
/-!
# Existence of Cartan subalgebras
In this file we prove existe... |
7,559 | lean_file_chunk | 46d6a5528a74f13610ed29082d9b0e0aaf9ce8b827cf6b4ae7a3f4c77cdbeefd | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
module
public import Mathlib.Algebra.Group.EvenFunction
public import Mathlib.Algebra.Lie.Cochain
public import Mathlib.Algebra.Lie.InvariantForm
public import Mathl... |
5,376 | lean_file_chunk | 2304af545583fe29544bebd11e794d6423275a48daf85feea31334daa7641ed6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Leonid Ryvkin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonid Ryvkin
-/
module
public import Mathlib.Algebra.Lie.Derivation.Basic
public import Mathlib.Algebra.Lie.Extension
public import Mathlib.Algebra.Lie.Prod
/-!
# Semi-direct product... |
6,746 | lean_file_chunk | 9fe790fe058fe0cfa33ee99df6a435ff7f5d7f610060aabdf905eabfaa693dc6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 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.Lie.InvariantForm
public import Mathlib.Algebra.Lie.Semisimple.Basic
public import Mathlib.Algebra.Lie.TraceForm
/-!
# Lie algebras w... |
13,074 | lean_file_chunk | e7505f862d7f8835a1bcdf293672366e55f00947be21c54856e0285512085c9b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 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.Lie.AdjointAction.Basic
public import Mathlib.Algebra.Lie.Nilpotent
public import Mathlib.Algebra.Lie.Normalizer
/-!
# Engel's theore... |
23,972 | lean_file_chunk | 57faba5c363f8e9ef16648c97848692e9ff34619938901f20e79db07074400f6 | 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.Lie.Subalgebra
public import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie sub... |
11,060 | lean_file_chunk | fda2d3f53edd2531f52b46db0f7382de9e85c132cee5f79310eb4aa0eb102d17 | lean_source_file | pkg_mathlib_source_file_chunks | have := Submodule.coe_iSup_of_directed _ hdir.directed_val
simp_rw [← iSup_toSubmodule, Set.iUnion_coe_set, coe_toSubmodule] at this
rw [← this, SetLike.coe_set_eq, sSup_eq_iSup, iSup_subtype]
@[simp]
lemma sSup_image_lieSpan_singleton : sSup ((fun x ↦ lieSpan R L {x}) '' N) = N := by
refine le_antisymm (sSup_le... |
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