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 |
|---|---|---|---|---|---|
5,470 | lean_file_chunk | 3dde1eaf7f779aef1c389e56875f21d8bc5bf5aabaddc4ba53803f0d4d7b1ea0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Hanting Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hanting Zhang
-/
module
public import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
public import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
/-! # Pointwise instances o... |
10,711 | lean_file_chunk | 24f0e0a313e2351911fadf4b4ec6a0ccc974b019224bdbd7d6473830aba3e7e8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
module
public import Mathlib.LinearAlgebra.AffineSpace.Combination
/-!
# Centroid of a Finite Set of Points in Affine Space
This file defines the centroid of a finite ... |
3,419 | lean_file_chunk | 95c82533870df03fd11800bffcde1256dec0af11c2cceb7c9a7ff0f68f4079f4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Paul Reichert. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul Reichert
-/
module
public import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
/-!
# Affine map restrictions
This file defines restrictions of affine maps.
## Main def... |
7,499 | lean_file_chunk | 83e1af23cc79ca2a220d85ab758bb973feb0fae97fd0f7855b8b4ae4d0881782 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.LinearAlgebra.AffineSpace.AffineMap
public import Mathlib.Tactic.Field
public import Mathlib.Tactic.FieldSimp
public import Mathlib.Ta... |
1,652 | lean_file_chunk | 9ca054c1b2728ce2716142c39c4ab085da060e3427e02491cfba02db0e03805b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
module
public import Mathlib.Algebra.AddTorsor.Defs
/-!
# Affine space
In this file we introduce the following notation:
* `AffineSpace V P` is an alternative notatio... |
14,564 | lean_file_chunk | 8d653f1c42650cbad71e03266565ace4b0a6a81775d72429874e92f3b13d3c2f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 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.CharP.Invertible
public import Mathlib.Algebra.Order.Module.Synonym
public import Mathlib.LinearAlgebra.AffineSpace.Midpoint
p... |
1,775 | lean_file_chunk | f864dc4f1ea01d46d182257f4a186acc6a15480acc9e01f913da9ecc442efdba | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 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.CharP.Invertible
public import Mathlib.LinearAlgebra.AffineSpace.Midpoint
/-!
# Midpoint of a segment for characteristic zero... |
15,432 | lean_file_chunk | 715bc0f539a87992d598e9fb57bb1ee23c97a828a289c76b1f7714598aa47ee0 | 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.LinearAlgebra.AffineSpace.Centroid
public import Mathlib.LinearAlgebra.AffineSpace.Independent
public import Mathlib.LinearAlgebra.AffineSpace... |
9,334 | lean_file_chunk | ceeca7c0414c61b11d475db495e11fb72a2146b98b4d84421af67683fcee70f8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
module
public import Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
/-!
# Ceva's theorem.
This file proves various versions of Ceva's theorem.
## References
* https... |
23,979 | lean_file_chunk | 07f5a19a4e2d24b97dfea614b2d7c9e6e05898179c76b4b5f095b356b49e80e9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
module
public import Mathlib.FieldTheory.Finiteness
public import Mathlib.LinearAlgebra.AffineSpace.Basis
public import Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
p... |
16,500 | lean_file_chunk | 3e36428f42ce3cd7691305c6ed5cc75eacca6d6716ee0bb487385fc4255786d7 | lean_source_file | pkg_mathlib_source_file_chunks | Collinear k (Set.range p) ↔ ¬AffineIndependent k p := by
rw [collinear_iff_finrank_le_one,
finrank_vectorSpan_le_iff_not_affineIndependent k p (Fintype.card_fin 3)]
/-- Three points are affinely independent if and only if they are not collinear. -/
theorem affineIndependent_iff_not_collinear_set {p₁ p₂ p₃ : P} :... |
23,628 | lean_file_chunk | 740c58a564f204c1536ab67b7097c5f2d5f133418dc200e4da9e8b1de68bfea2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.LinearAlgebra.AffineSpace.AffineMap
public import Mathlib.LinearAlgebra.GeneralLinearGroup.Basic
/-!
# Affine equivalences
In this f... |
9,706 | lean_file_chunk | 1e0963d43edba3fe81b715fe6a9f211505ec305ba8ee1e2062c5d090ce8cb2f2 | 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.LinearAlgebra.RootSystem.IsValuedIn
/-!
# Invariant and root-positive bilinear forms on root pairings
This file contains basic results... |
21,877 | lean_file_chunk | 0218d7d0174d87c8c4159a248af9d14ff85827bca1886cf00b49b1e6de81c603 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
module
public import Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas
public import Mathlib.Order.Interval.Set.OrdConnectedLinear
/-!
# Chains of roots
Given roots `α` and... |
23,958 | lean_file_chunk | 247f45f98f48695c62313208d4e477c63a9c387eebc64bdb77c28d7f7f69d9fd | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
module
public import Mathlib.LinearAlgebra.RootSystem.Chain
public import Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas
public import Mathlib.LinearAlgebra.RootSystem.IsV... |
3,451 | lean_file_chunk | 67a4c4018bf59c04b3ad4574f8aa685ad6c71cc6238a372dbb031c30850edc28 | lean_source_file | pkg_mathlib_source_file_chunks | `RootPairing.Base.IsPos.induction_on_add` will be more useful. -/
lemma exists_eq_sum_and_forall_sum_mem_of_isPos {i : ι} (hi : b.IsPos i) :
∃ n, ∃ f : Fin n → ι,
range f ⊆ b.support ∧
P.root i = ∑ m, P.root (f m) ∧
∀ m, ∑ m' ≤ m, P.root (f m') ∈ range P.root := by
apply hi.induction_on_add (fun... |
12,302 | lean_file_chunk | 38608c1a6ebada27a27fbdd0af5ff6e9cfaabe82d1b2f2a098703f75c0ec62d8 | 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, Deepro Choudhury, Scott Carnahan
-/
module
public import Mathlib.LinearAlgebra.RootSystem.Defs
public import Mathlib.LinearAlgebra.RootSystem.Finite.Nondegenerate
/-!
# Roo... |
9,250 | lean_file_chunk | 00ee7923cbfcde905206a5368982a1c8f5ec1a1e412b5e195f4fd7f8609354c4 | 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.LinearAlgebra.RootSystem.Hom
public import Mathlib.RepresentationTheory.Basic
/-!
# The Weyl group of a root pairing
This file defines ... |
23,992 | lean_file_chunk | a1d8b07cf85e9727cfcc366cf06d8812b43b0a199510ce0badb71ff8e5d8061a | 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, Deepro Choudhury, Scott Carnahan
-/
module
public import Mathlib.LinearAlgebra.PerfectPairing.Basic
public import Mathlib.LinearAlgebra.Reflection
/-!
# Root data and root ... |
4,035 | lean_file_chunk | 109abd043d9ef380674a821d1556a77f79dc60abf0ecb61ba8846f0458dc56f4 | lean_source_file | pkg_mathlib_source_file_chunks | simp [reflection_apply, h.1]
lemma IsOrthogonal.reflection_apply_right (h : IsOrthogonal P j i) :
P.reflection j (P.root i) = P.root i :=
h.symm.reflection_apply_left
lemma IsOrthogonal.coreflection_apply_left (h : IsOrthogonal P i j) :
P.coreflection j (P.coroot i) = P.coroot i :=
h.flip.reflection_apply... |
15,471 | lean_file_chunk | 31785dc4d6abc575f9b4ab3afa6c199db93d480b110b201b5c0828873c2878d2 | 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, Oliver Nash
-/
module
public import Mathlib.Algebra.Algebra.Rat
public import Mathlib.Algebra.Module.Submodule.Invariant
public import Mathlib.LinearAlgebra.PerfectPai... |
7,137 | lean_file_chunk | a78dc533fbd4ed33ab9def7b0d44c612071be6574b79edf9e34eb96ab0aaa60d | 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.LinearAlgebra.RootSystem.Defs
/-!
# Root pairings made from bilinear forms
A common construction of root systems is given by taking the... |
14,633 | lean_file_chunk | 649eda38275af7bc83b53ef061fb9def06a2cc1d690ef63bdeff3a62ee021385 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 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.CharZero.Infinite
public import Mathlib.Algebra.Module.Submodule.Union
public import Mathlib.LinearAlgebra.Matrix.BilinearForm
public ... |
13,467 | lean_file_chunk | 65dd58c7657fe436396fb647c5cd354936386cf38c4ef37f94af503821257cbd | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
module
public import Mathlib.LinearAlgebra.RootSystem.RootPositive
public import Mathlib.LinearAlgebra.RootSystem.WeylGroup
public import Mathlib.RepresentationTheory.Subm... |
4,986 | lean_file_chunk | 0b68c5631eccd2e3a91baeccaf5904838a5f2411e3e23ec02211c85d09422efc | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 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.Rat
public import Mathlib.LinearAlgebra.PerfectPairing.Restrict
public import Mathlib.LinearAlgebra.RootSystem.IsValuedIn
/-!... |
1,797 | lean_file_chunk | bdd484f74c4585fba11ec89657e2fead44f481ae2675f041858818ef0480d185 | 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.LinearAlgebra.RootSystem.Hom
public import Mathlib.CategoryTheory.Category.Basic
/-!
# The category of root pairings
This file defines ... |
16,963 | lean_file_chunk | a2c0b06a9574d01ab94d18da64dd97be726df9a868828d90146fb7f1b9883fa8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 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.Group.Irreducible.Indecomposable
public import Mathlib.Algebra.Module.LinearMap.Rat
public import Mathlib.Algebra.Module.Submodule.Uni... |
23,964 | lean_file_chunk | dfdec780300102fd20b1cbefa2bac1657505a204b39f93669c73ef13d5820d7d | 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.LinearAlgebra.RootSystem.Basic
public import Mathlib.LinearAlgebra.RootSystem.Defs
/-!
# Morphisms of root pairings
This file defines m... |
3,575 | lean_file_chunk | 49a888c06d3c9160ce23735b7fd59d4e577507f3ac51bbaf718d1a7e2d15b302 | lean_source_file | pkg_mathlib_source_file_chunks | exact Equiv.ext (congrArg Hom.weightMap h') h (congrArg Hom.indexEquiv h')
apply Hom.coweightHom_injective P
exact MulOpposite.unop_inj.mp h
lemma coweightHom_op {P : RootPairing ι R M N} (g : Aut P) :
MulOpposite.unop (coweightHom P g) = coweightEquiv P P g :=
rfl
@[simp]
lemma coweightEquiv_inv {P : RootP... |
14,031 | lean_file_chunk | 26f4929088254749b208654232cce62d68b4f3f822af71d2b14de7fde974ba6f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash, Scott Carnahan
-/
module
public import Mathlib.LinearAlgebra.RootSystem.IsValuedIn
/-!
# Reduced root pairings
This file contains basic definitions and results related to ... |
10,313 | lean_file_chunk | 9c02c60d56da1587be5dbb5f8370e5820920144686145bee37ae3f5a8d2b7286 | 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.LinearAlgebra.PiTensorProduct
public import Mathlib.Logic.Equiv.Fin.Basic
public import Mathlib.Algebra.DirectSum.Algebra
/-!
# Tensor power ... |
5,729 | lean_file_chunk | 8ac90bb4dc97296581bf904d42ec83e4dc7736c482ce9f6592694f744e9c4721 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
module
public import Mathlib.LinearAlgebra.PiTensorProduct
public import Mathlib.Tactic.SuppressCompilation
/-!
# Symmetric tensor power of a semimodule over a commutative se... |
2,446 | lean_file_chunk | f8e63a7f7f082b1f6074357ba2e7af5f6ad6a3cf6df2fedf1b33813c86cae5c1 | 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.LinearAlgebra.Dual.Defs
public import Mathlib.LinearAlgebra.TensorPower.Basic
/-!
# The pairing between the tensor power of the dual and the tens... |
4,513 | lean_file_chunk | d9874c7e0e20fa94cd261c91b86bacee2da110b59e1ffb2c8ed5018c986dcfee | 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.Epi
public import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
public import Mathlib.LinearAlgebra.Finsupp.LinearCombin... |
23,958 | lean_file_chunk | 557fee6a4e0b819d1b8c4fc04ac0596ac74bfb1de81a10d81b86e3edb7aded6b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
module
public import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
public import ... |
13,788 | lean_file_chunk | aef760cebdf7403a6cd5e4b77d2cd58ed51b1398a94ca5d805ee830a0ead87ff | lean_source_file | pkg_mathlib_source_file_chunks | lemma comap_covBy_of_surjective {f : M →ₛₗ[τ₁₂] M₂} (hf : Surjective f)
{p q : Submodule R₂ M₂} (h : p ⋖ q) :
p.comap f ⋖ q.comap f := by
refine ⟨lt_of_le_of_ne (comap_mono h.1.le) ((comap_injective_of_surjective hf).ne h.1.ne), ?_⟩
intro N h₁ h₂
refine h.2 (lt_map_of_comap_lt_of_surjective hf h₁) ?_
rw... |
23,991 | lean_file_chunk | 4c74e38deade71f3f436dfc1aeac6508cbae052e74af665395852e9b848e058b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
module
public import Mathlib.Algebra.Module.Submodule.Lattice
public import Mathl... |
3,177 | lean_file_chunk | 1392498fdc49454fd5e1115fcfb6fccfbcaa71ace99e8cfb6d2512255386dc1a | lean_source_file | pkg_mathlib_source_file_chunks | exact ⟨mem_sup_right haT', htT.trans (le_sup_left (a := span R _))⟩
end
end AddCommMonoid
section AddCommGroup
variable [Ring R] [AddCommGroup M] [Module R M] {ι : Type*} [DecidableEq ι] {i j : ι}
lemma sup_toAddSubgroup (p p' : Submodule R M) :
(p ⊔ p').toAddSubgroup = p.toAddSubgroup ⊔ p'.toAddSubgroup := by... |
5,203 | lean_file_chunk | 347921c163c058ff6d72f71bd40ee82e4a6d61f0fa6a985b8381112b320b52d7 | 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.LinearAlgebra.QuadraticForm.IsometryEquiv
public import Mathlib.Algebra.Category.ModuleCat.Basic
/-!
# The category of quadratic modules
-/
... |
7,403 | lean_file_chunk | 3ec4d0f8046c13fb596e2350c7180b1d46ea4febc20c22fc9caf572f728c3df3 | 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.LinearAlgebra.BilinearForm.TensorProduct
public import Mathlib.LinearAlgebra.QuadraticForm.Basic
public import Mathlib.Tactic.LinearCombinatio... |
4,009 | lean_file_chunk | 5103dd8bfaca843125b603916ba30faf18ab83ebe2500ecc922a72f88263a675 | 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.LinearAlgebra.QuadraticForm.Basic
/-!
# Isometric linear maps
## Main definitions
* `QuadraticMap.Isometry`: `LinearMap`s which map between... |
3,091 | lean_file_chunk | be4acca3789726997a194d0504fda3aa7f4460f41f1604bc2a4bb78ff22e40c7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Eric Wieser
-/
module
public import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
public import Mathlib.FieldTheory.IsAlgClosed.Basic
/-!
# Quadratic forms... |
14,215 | lean_file_chunk | bff089a52f44268f7cc481dffe1746e1bda4360829c029fd17c4d4c4acaa4e7c | 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.LinearAlgebra.QuadraticForm.IsometryEquiv
/-! # Quadratic form on product and pi types
## Main definitions
* `QuadraticForm.prod Q₁ Q₂`: th... |
9,975 | lean_file_chunk | b9aa88fbc7e322edb343ded6b102c52df6c1b8c91a4fbed2f131fd3d0ee56511 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
module
public import Mathlib.LinearAlgebra.QuadraticForm.Radical
/-!
# Signature of a quadratic form
We define the signature of a quadratic form over a linearly o... |
23,994 | lean_file_chunk | 5dbc006b155303b5d44b4af113302147901b853a6c169654cffa505ba2cf6505 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Eric Wieser
-/
module
public import Mathlib.Data.Finset.Sym
public import Mathlib.LinearAlgebra.BilinearMap
public import Mathlib.LinearAlgebra.FiniteDimensiona... |
23,990 | lean_file_chunk | 1178119e75e75a681b38c21fa568a5158bcb2048e8959e8ab7eae86fff050efa | lean_source_file | pkg_mathlib_source_file_chunks | rfl
variable {n : Type*}
/-- `sq` is the quadratic map sending the vector `x : A` to `x * x` -/
@[simps!]
def sq : QuadraticMap R A A :=
linMulLin LinearMap.id LinearMap.id
/-- `proj i j` is the quadratic map sending the vector `x : n → R` to `x i * x j` -/
def proj (i j : n) : QuadraticMap R (n → A) A :=
linMul... |
3,246 | lean_file_chunk | a50ca5418336d0a93e1c5787af46c738fbda245bef7fc32dbf535b394c87d5a2 | lean_source_file | pkg_mathlib_source_file_chunks | rintro c y hy hc
rw [add_eq_zero_iff_neg_eq] at hc
rw [← hc, Submodule.neg_mem_iff] at hy
have := (isCompl_span_singleton_orthogonal hx).disjoint
rw [Submodule.disjoint_def] at this
have := this (c • x) (Submodule.smul_mem _ _ <| Submodule.mem_span_singleton_self _) hy
ex... |
1,268 | lean_file_chunk | 4fc0dbd6471af9da27bd5ed8e40438267a311337021776cbcf19ec019777d6a9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Eric Wieser
-/
module
public import Mathlib.Data.Complex.Basic
public import Mathlib.LinearAlgebra.QuadraticForm.AlgClosed
public import Mathlib.Algebra.CharP.I... |
7,456 | lean_file_chunk | ec1987330cae08370fe15b54a07fab40ae1adc139aba5e2c66576971294f5fb5 | 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.LinearAlgebra.Dual.Lemmas
public import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
public import Mathlib.LinearAlgebra.QuadraticForm.Pr... |
6,360 | lean_file_chunk | b87dfac0fbc92b649c6d80e10e3f02ddf0cdb1b62f447bea939f6214d2a5191e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 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.Sym
public import Mathlib.Data.Finsupp.Pointwise
public import Mathlib.Data.Sym.Sym2.Finsupp
public import Mathlib.Linear... |
7,445 | lean_file_chunk | 54f7c8bd86718552f0fa031ff01467ac4dce6cabe2bacae21971f3726fa9abb4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Eric Wieser
-/
module
public import Mathlib.LinearAlgebra.QuadraticForm.Basic
public import Mathlib.LinearAlgebra.QuadraticForm.Isometry
/-!
# Isometric equivalences with r... |
8,406 | lean_file_chunk | f8df1d0ef729de4965fbe287573ef6a4178962dbdb9f5f6cf62e4f19f87b0f00 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
module
public import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
public import Mathlib.LinearAlgebra.Quotient.Bilinear
/-!
# The radical of a quadratic form
... |
4,287 | lean_file_chunk | 0eeef67e612615b4d9150629999295ef72d3dc90758e0397f3717d03c521db05 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Eric Wieser
-/
module
public import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
public import Mathlib.Data.Sign.Basic
public import Mathlib.Algebra.CharP.... |
2,009 | lean_file_chunk | 1c24a815788bc980718ced800f0eda828c17b8e1f4a970cf39eefbdb735a54c5 | 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.LinearAlgebra.CliffordAlgebra.Conjugation
/-!
# Star structure on `CliffordAlgebra`
This file defines the "clifford conjugation", equal to `... |
7,497 | lean_file_chunk | 1f27d68afb550c6d0ec26fe243788ae83300ead4b88956c241b763057f8e39e0 | 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.LinearAlgebra.CliffordAlgebra.Grading
public import Mathlib.LinearAlgebra.TensorProduct.Graded.Internal
public import Mathlib.LinearAlgebra.Qu... |
16,220 | lean_file_chunk | 57d2b9dd0856772c70363bcf217a5f3ee337746085d35a082f253b8a1133919c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Jiale Miao. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jiale Miao, Utensil Song, Eric Wieser
-/
module
public import Mathlib.Algebra.Ring.Action.ConjAct
public import Mathlib.GroupTheory.GroupAction.ConjAct
public import Mathlib.Algebra.Star.... |
17,327 | lean_file_chunk | c80b6c896062acd853b9fcedc676dc4e0ca14cff78a6b66ab01763b818713350 | 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, Utensil Song
-/
module
public import Mathlib.Algebra.RingQuot
public import Mathlib.LinearAlgebra.TensorAlgebra.Basic
public import Mathlib.LinearAlgebra.QuadraticForm.Isome... |
9,410 | lean_file_chunk | 82521355360f83d94b9afe92e15b848333c0f371c703cf838a156a057d784c77 | 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.LinearAlgebra.CliffordAlgebra.Basic
public import Mathlib.RingTheory.GradedAlgebra.Basic
/-!
# Results about the grading structure of the cli... |
15,480 | lean_file_chunk | 59f40e1cac6cdd2e2f5c030a328bf0be5ddb400f39edddc0d7c3743158ec5693 | 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.LinearAlgebra.CliffordAlgebra.Conjugation
public import Mathlib.LinearAlgebra.CliffordAlgebra.Fold
public import Mathlib.LinearAlgebra.Exterio... |
9,813 | lean_file_chunk | 9b12b075667b15eb5a2643c6915853cc6d8b56e9d6ed3b40d83e6eceab1dc42a | 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.LinearAlgebra.CliffordAlgebra.Fold
public import Mathlib.LinearAlgebra.CliffordAlgebra.Grading
/-!
# The universal property of the even subal... |
9,116 | lean_file_chunk | 68badcc91cf334127b7e1d883e17f1d6eb6af3b9acf8c8d3f76ac1a4d8079c98 | 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.LinearAlgebra.CliffordAlgebra.Conjugation
/-!
# Recursive computation rules for the Clifford algebra
This file provides API for a special ca... |
1,057 | lean_file_chunk | 1d594de3798efb3ac1c935ac10b659a133dfa569fdf443792a39c9f3d1785484 | 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.LinearAlgebra.CliffordAlgebra.Basic
public import Mathlib.LinearAlgebra.QuadraticForm.QuadraticModuleCat
public import Mathlib.Algebra.Categor... |
8,950 | lean_file_chunk | 914cf370f3587c3c7f6d476769668e7b19260bc2a16f1abe1e1dc35904f4a542 | 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.LinearAlgebra.QuadraticForm.TensorProduct
public import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
public import Mathlib.LinearAlgebra.... |
11,377 | lean_file_chunk | e07e163126434c90c6cdaa24bf8003fb1361e97093de5b1b4643b8467010f64e | 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.LinearAlgebra.CliffordAlgebra.Grading
public import Mathlib.Algebra.Module.Opposite
/-!
# Conjugations
This file defines the grade reversal ... |
2,904 | lean_file_chunk | 5bae2887a4a4a94fb5e64b836d11c8597bd8a3579442383cddb80e51ddb7812e | 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.LinearAlgebra.CliffordAlgebra.Contraction
/-! # Results about inverses in Clifford algebras
This contains some basic results about the inver... |
11,131 | lean_file_chunk | 7ec21cece39b8d2fa68698b348561b40ec0059007215cdb4d6b038b76cbc181a | 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.LinearAlgebra.CliffordAlgebra.Conjugation
public import Mathlib.LinearAlgebra.CliffordAlgebra.Even
public import Mathlib.LinearAlgebra.Quadrat... |
13,658 | lean_file_chunk | 51f45bd9adb0664df12f2513806faa7e08ea0ce925ceaece06b104c32b68e994 | 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.DualNumber
public import Mathlib.Algebra.QuaternionBasis
public import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
public import... |
11,535 | lean_file_chunk | d50b2d63f18bc07d59d882291bacafb8ff90c5316ac5f361db6209daa469a604 | 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.Finite
public import Mathlib.GroupTheory.Coset.Card
public import Mathlib.GroupTheory.GroupAction.Quotient
public impor... |
3,045 | lean_file_chunk | 23f94cf0ff0c87d20360d91baf363748d2c72c2b3006aa5451424fc3a3d4b3f8 | 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.LinearAlgebra.Alternating.Basic
public import Mathlib.LinearAlgebra.Multilinear.Curry
/-!
# Currying alternating forms
In this file we defin... |
23,931 | lean_file_chunk | 8bc64c3e6b7bb4d28a3c56ecafc143b2a3a444b1f2d37e8ab6e09564606b1549 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Zhangir Azerbayev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Zhangir Azerbayev
-/
module
public import Mathlib.GroupTheory.Perm.Sign
public import Mathlib.LinearAlgebra.LinearIndependent.Defs
public import Mathlib.LinearAlgebra.... |
9,968 | lean_file_chunk | 156abdc43f40bf6b0d43a67b358e7eff46527b30aef876e2d6e77fbb13faa714 | lean_source_file | pkg_mathlib_source_file_chunks | /-- Transfer the arguments to a map along an equivalence between argument indices.
This is the alternating version of `MultilinearMap.domDomCongr`. -/
@[simps]
def domDomCongr (σ : ι ≃ ι') (f : M [⋀^ι]→ₗ[R] N) : M [⋀^ι']→ₗ[R] N :=
{ f.toMultilinearMap.domDomCongr σ with
toFun := fun v => f (v ∘ σ)
map_eq_zer... |
9,808 | lean_file_chunk | 1bd15f90627b2958c1912a1a478148a21ba5d2a7bb1935cee36319aecc9e048f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Robert Maxton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Maxton
-/
module
public import Mathlib.Algebra.DirectSum.Basic
public import Mathlib.LinearAlgebra.TensorAlgebra.ToTensorPower
/-!
# The free product of $R$-algebras
We define... |
5,997 | lean_file_chunk | 812199fafd9386918fc1ae8fa738957ca5b3f8779cba692e3d2c9bbdf037a905 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Lean FRO, LLC. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.LinearAlgebra.ConvexSpace
public import Mathlib.LinearAlgebra.AffineSpace.Combination
public import Mathlib.LinearAlgebra.AffineSpace.Affin... |
23,945 | lean_file_chunk | 6e808d5abbd8cfd07f3f28ce5e9ebbbf5609e0dd44404e66c1fb4ba8e81f875a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Fabian Glöckle, Kyle Miller
-/
module
public import Mathlib.Algebra.Module.LinearMap.DivisionRing
public import Mathlib.LinearAlgebra.Basis.Basic
public import Mathlib... |
23,954 | lean_file_chunk | 4522b3e531523534ce93b83baf92fcd0201157c9a28d130ffc7cc1bd2bbc63d7 | lean_source_file | pkg_mathlib_source_file_chunks | This is `Submodule.dualRestrict` factored through the quotient by its kernel (which
is `W.dualAnnihilator` by definition).
See `Subspace.dualPairing_nondegenerate`. -/
def dualPairing (W : Submodule R M) : Module.Dual R M ⧸ W.dualAnnihilator →ₗ[R] W →ₗ[R] R :=
W.dualAnnihilator.liftQ W.dualRestrict le_rfl
@[simp]
t... |
1,084 | lean_file_chunk | d222a8547a784f971dc7a80cfeeef3dbb7b03f389aa2d9a63c24ea133e851f06 | lean_source_file | pkg_mathlib_source_file_chunks | /-- A linear equivalence between `Dual M ⊗ Dual N` and `Dual (M ⊗ N)` given bases for `M` and `N`.
It sends `f ⊗ g` to the composition of `TensorProduct.map f g` with the natural
isomorphism `R ⊗ R ≃ R`.
-/
@[simps!]
noncomputable def dualDistribEquivOfBasis (b : Basis ι R M) (c : Basis κ R N) :
Dual R M ⊗[R] Dual ... |
18,965 | lean_file_chunk | 0f0c8d640a0ffeec93c742bf9bf2d01c14776de5a86666f8af81152e4e182d1f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Fabian Glöckle, Kyle Miller
-/
module
public import Mathlib.LinearAlgebra.BilinearMap
public import Mathlib.LinearAlgebra.Span.Defs
/-!
# Dual vector spaces
The dual... |
10,770 | lean_file_chunk | 5b9efe0a77c504cdab9528b9e003f3f266da9a326643f1f9dae4b5508ae639c2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Fabian Glöckle, Kyle Miller
-/
module
public import Mathlib.LinearAlgebra.Basis.Defs
public import Mathlib.LinearAlgebra.Dual.Defs
/-!
# Bases of dual vector spaces
... |
4,907 | lean_file_chunk | cd94cd3c46bfdd44e1116889f703edd253151750f421a716c5db7ef15172e4be | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
module
public import Mathlib.LinearAlgebra.Dual.Defs
public import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
public import Mathlib.RingTheory.Tenso... |
781 | lean_file_chunk | 4db79bfff04000781ebb2c216f13750c8d124e5c48016621e1c82daa6c1d46d3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov
-/
module
public import Mathlib.LinearAlgebra.Quotient.Defs
public import Mathlib.SetTheory.Cardinal.Finite
public impor... |
2,114 | lean_file_chunk | f44ebbc13f4720636f60646f1e952890c4165cced8f97b4098751988b2ea2c49 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
module
public import Mathlib.LinearAlgebra.Quotient.Basic
public import Mathlib.LinearAlgebra.SesquilinearForm.Basic
/-!
# Lifting bilinear forms to quotients
-/
@[... |
5,011 | lean_file_chunk | 271ac6215539756bbbe500013ea60a8280ecc9b644d54d9dded53fdb1bd233d9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Alex J. Best
-/
module
public import Mathlib.LinearAlgebra.Pi
public import Mathlib.LinearAlgebra.Quotient.Basic
/-!
# Submodule quotients and direct sums
This file contai... |
16,432 | lean_file_chunk | 915e66d47bc248eecd77927c865b2fb85ee420e3198c352ab1e3718130701078 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov
-/
module
public import Mathlib.Algebra.Module.Equiv.Basic
public import Mathlib.GroupTheory.QuotientGroup.Basic
public ... |
8,812 | lean_file_chunk | e16a3937b3e97b9b46da0b545af8c0d688acf955ba21a516e74c2360dd0eec85 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov
-/
module
public import Mathlib.Algebra.Module.Equiv.Defs
public import Mathlib.Algebra.Module.Submodule.Defs
public imp... |
23,958 | lean_file_chunk | 3211ad232ed08d90c4f10951d1596d9fd8f8be6947febdb9afbea9ff09bd1481 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Anne Baanen
-/
module
public import Mathlib.Data.Fin.Tuple.Reflection
public import Mathlib.LinearAlgebra.Dual.Defs
public import Math... |
14,283 | lean_file_chunk | 3c7ac88a2d49ccf5b6b07387eae9c45b788149ffc6164954df90c0840100e434 | lean_source_file | pkg_mathlib_source_file_chunks | @range_unique _ PUnit, @linearIndependent_unique_iff PUnit, disjoint_span_singleton]
dsimp [(· ∘ ·)]
refine ⟨fun h => ⟨h.1, fun hx => h.2.1 <| h.2.2 hx⟩, fun h => ⟨h.1, ?_, fun hx => (h.2 hx).elim⟩⟩
rintro rfl
exact h.2 (zero_mem _)
theorem LinearIndependent.option (hv : LinearIndependent K v)
(hx : x ∉ Su... |
23,964 | lean_file_chunk | 79c5267a12838f490ff9176943e41325fa7112740b0360840c0edbae72597845 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Anne Baanen
-/
module
public import Mathlib.Algebra.BigOperators.Fin
public import Mathlib.LinearAlgebra.LinearIndependent.Defs
/-!
#... |
2,669 | lean_file_chunk | af2baba70b7aee885e24181016e169c54914297d8129922f61849a61d896e66c | lean_source_file | pkg_mathlib_source_file_chunks | ∑ i ∈ insert a s, a x * (g i * i y) := by
congrm ∑ i ∈ insert a s, ?_ - ∑ i ∈ insert a s, ?_
· rw [mul_assoc]
· rw [mul_assoc, mul_left_comm]
_ = (∑ i ∈ insert a s, g i • i : G → L) (x * y) -
a x * (∑ i ∈ insert a s, (g i • (i : G → L))) y := by simp [Finset.mul_sum]
_ = ... |
23,926 | lean_file_chunk | db06288c712c78970dba79cb2168fb96c85fb9133d1feca31ec8af290b3a04f6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Anne Baanen
-/
module
public import Mathlib.Algebra.Order.Sub.Basic
public import Mathlib.LinearAlgebra.Finsupp.LinearCombination
publ... |
17,296 | lean_file_chunk | 3a9e79f229020957f7bfec4bc06575f83841be61eabd984f8197896bff5eb0d0 | lean_source_file | pkg_mathlib_source_file_chunks | A linearly independent family is maximal if there is no strictly larger linearly independent family.
-/
@[nolint unusedArguments]
def LinearIndependent.Maximal {ι : Type w} {R : Type u} [Semiring R] {M : Type v} [AddCommMonoid M]
[Module R M] {v : ι → M} (_i : LinearIndependent R v) : Prop :=
∀ (s : Set M) (_i' :... |
3,012 | lean_file_chunk | 5a7aafe4f3aacec2e035acc356a30a5de078ae3f77652c289dbf6ffb207c2c63 | 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.LinearAlgebra.Basis.VectorSpace
public import Mathlib.LinearAlgebra.TensorProduct.Basis
public import Mathlib.RingTheory.TensorProduct.IsBaseC... |
9,110 | lean_file_chunk | fbeddd0f2a252989d603d3d0b74c42dabcf64ae7b82c6962cfdcb28293a99c3b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
module
public import Mathlib.Algebra.EuclideanDomain.Int
public import Mathlib.Data.ZMod.QuotientGroup
public import Mathlib.GroupTheory.Index
public import Mathlib.Line... |
2,677 | lean_file_chunk | c69d9a882f1e8d930c162dca43b50b8e3a1c2289ec097c7b0823d38f714f9216 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
module
public import Mathlib.LinearAlgebra.FreeModule.Finite.Quotient
/-! # Ideals in free modules over PIDs
## Main results
- `Ideal.quotientEquivPiSpan`: `S ⧸ I`, if ... |
1,587 | lean_file_chunk | c6bb349c2c8396beca7378744eaf9bd62aadc30cb192b444a5dcfafafd4795b7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
module
public import Mathlib.RingTheory.FiniteType
public import Mathlib.LinearAlgebra.InvariantBasisNumber
/-!
# Strong rank condition for commutative rings
We... |
4,545 | lean_file_chunk | abec39f6cf2a58728f1e20ef67c5bb92d8398d8be9ee1cd5a6e7aa564bc6f776 | 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.EuclideanDomain.Int
public import Mathlib.Algebra.Module.ZMod
public import Mathlib.LinearAlgebra.Dimension.Free
/-!
# Quotienting ... |
3,070 | lean_file_chunk | 6312b128daa5a13e8eabcfb34faee78cf31a85ec74841e922003172ded4cfde5 | 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.LinearAlgebra.FreeModule.IdealQuotient
public import Mathlib.RingTheory.AdjoinRoot
public import Mathlib.RingTheory.Norm.Defs
/-!
# Norms on free... |
23,971 | lean_file_chunk | 62e6eda5af15222704b34cd3daeec1af932f033faa36d514723c4d7e3e331dc8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
module
public import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
public import Mathlib.LinearAlgebra.FreeModule.Basic
public import Mathlib.LinearAlgebra.Matrix.To... |
10,132 | lean_file_chunk | c0d9e9c09d41fa3eb5c9dda9d245893f39ce212741650b5f19b68424e94496c0 | lean_source_file | pkg_mathlib_source_file_chunks | This is a strengthening of `Submodule.basisOfPid`.
See also `Ideal.smithNormalForm`, which moreover proves that the dimension of
an ideal is the same as the dimension of the whole ring.
-/
noncomputable def Submodule.smithNormalForm [Finite ι] (b : Basis ι R M) (N : Submodule R M) :
Σ n : ℕ, Basis.SmithNormalForm ... |
8,049 | lean_file_chunk | 2529caa035bed591934d160a1c52918823aca1c96c18e20a9965a4e4efdca362 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
module
public import Mathlib.Algebra.Algebra.Defs
public import Mathlib.Algebra.Module.Shrink
public import Mathlib.Algebra.Module.ULift
public import Mathlib.Data... |
1,000 | lean_file_chunk | bed3f16ef311a87f91136ecea5ebcffaf9fd43a52e4173c2e4ed3d1425af9a6e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Alex J. Best
-/
module
public import Mathlib.LinearAlgebra.Determinant
public import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
/-!
# Determinants in free (finite) modul... |
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