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 |
|---|---|---|---|---|---|
18,659 | lean_file_chunk | 343fb63715b50892067a900b43323d6665cbbc05fa4e6593b1d42d7eea79be7e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow, Kexing Ying
-/
module
public import Mathlib.LinearAlgebra.BilinearForm.Hom
public import Mathlib.LinearAlgebra.Dual.Lemmas
/-!
# Bilinear form
This file defines ... |
9,333 | lean_file_chunk | 10fb629309e8d048919b782df4f1e923929a07600995c4cdfdd89a82ddf25a98 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow, Kexing Ying
-/
module
public import Mathlib.Algebra.Algebra.Bilinear
public import Mathlib.LinearAlgebra.Basis.Defs
public import Mathlib.LinearAlgebra.BilinearFor... |
19,542 | lean_file_chunk | 0dc546a327e153ee4921f124086f4e5e340863d76b8e7c7c39680fa85ef708da | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhangir Azerbayev, Adam Topaz, Eric Wieser
-/
module
public import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
public import Mathlib.LinearAlgebra.Alternating.Curry
public import Mathlib.... |
4,195 | lean_file_chunk | c8a970df1b7699ad85e17e5529b9b2c5b2bdc214aeabd8636fcb93e43a78f0dc | 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.ExteriorAlgebra.Basic
public import Mathlib.RingTheory.GradedAlgebra.Basic
/-!
# Results about the grading structure of the ext... |
2,364 | lean_file_chunk | c932c58d6bf4e11bc385076064081f04873a53ed0430d2007bdffd676d0b65f9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Daniel Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Daniel Morrison
-/
module
public import Mathlib.LinearAlgebra.ExteriorAlgebra.Grading
public import Mathlib.LinearAlgebra.ExteriorPower.Basis
/-!
# Basis for `ExteriorAlgebra`
-/
@... |
6,039 | lean_file_chunk | 72475d68c2a52ea5ca57514e3548fa515256f13e28311d6d8957c3c21bcc02ad | 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.ExteriorAlgebra.Basic
/-!
# Extending an alternating map to the exteri... |
4,593 | lean_file_chunk | a21a3acfaab0a6397c30600e233403cdc2e8c56fc1bc87908ca45b225208ca93 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
module
public import Mathlib.LinearAlgebra.Projectivization.Basic
/-!
# Independence in Projective Space
In this file we define independence and dependence of famili... |
9,087 | lean_file_chunk | d034d37139a2c16454217752a8108006660f2d23df12c463d323a69627f73167 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
module
public import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
public import Mathlib.LinearAlgebra.FiniteDimensional.Basic
/-!
# Projective Spaces
This f... |
10,786 | lean_file_chunk | 23b5dc5ec1fc27524dfceb12d6c5408deec7bbd4b2e3ea87959106c616cc354b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
module
public import Mathlib.LinearAlgebra.Projectivization.Basic
/-!
# Subspaces of Projective Space
In this file we define subspaces of a projective space, and sho... |
5,066 | lean_file_chunk | 87e044a18715bc705ac3e65c7bd57cf5ba63e1488630bfa3616aab9e5b4d9d73 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Judith Ludwig, Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Judith Ludwig, Christian Merten
-/
module
public import Mathlib.Algebra.Ring.GeomSum
public import Mathlib.Data.Finite.Sum
public import Mathlib.Data.Fintype.Units
pu... |
4,952 | lean_file_chunk | b5c5ffad29ddb9d613d00e0964f522191e4a12660b3fe54dc4f18ff2ab52a146 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
module
public import Mathlib.LinearAlgebra.CrossProduct
public import Mathlib.LinearAlgebra.Matrix.DotProduct
public import Mathlib.LinearAlgebra.Projectivization.... |
6,310 | lean_file_chunk | e45df9e5b83c0a2db938e3a8bd640d6d3388dd171d1060647911eaf0ce270ee6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler, Antoine Chambert-Loir
-/
module
public import Mathlib.GroupTheory.GroupAction.MultipleTransitivity
public import Mathlib.GroupTheory.GroupAction.Ring
public import Mat... |
12,470 | lean_file_chunk | e83dfc5ac49038d97bda4ba7b25f0c8e1c61fec86298e634e7a8060def81233a | 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
-/
module
public import Mathlib.Algebra.BigOperators.GroupWithZero.Action
public import Mathlib.Algebra.Module.Equiv.Basic
public import Mathlib.Algebra.Module.Submodul... |
8,888 | lean_file_chunk | 7d14325d4f6ba84641b774f06b1e00b6ea57c955439a4e84abf5fd1899f5f721 | 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
-/
module
public import Mathlib.Algebra.FreeAbelianGroup.Finsupp
public import Mathlib.Algebra.MonoidAlgebra.Defs
public import Mathlib.Algebra.Polynomial.Basic
public ... |
9,758 | lean_file_chunk | 0dc5b9dccc697b29b724520a760013799b43d369da566d05fe128e4bab6056ae | 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
-/
module
public import Mathlib.LinearAlgebra.Finsupp.LSum
public import Mathlib.LinearAlgebra.Pi
public import Mathlib.Algebra.Order.Group.Nat
/-!
# Properties of the... |
13,050 | lean_file_chunk | f34533bae0eca18c7843149ebe3407e1cd8c3916b70d859739b742ece63fb9bb | 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
-/
module
public import Mathlib.Algebra.Module.Equiv.Defs
public import Mathlib.Algebra.Module.LinearMap.End
public import Mathlib.Algebra.Module.Pi
public import Mathl... |
5,889 | lean_file_chunk | 92b10f9db1ba81d6591a6310349ee2bc621bbcd1aee2b896c7609ad363f4f992 | 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
-/
module
public import Mathlib.LinearAlgebra.Finsupp.LSum
public import Mathlib.LinearAlgebra.Span.Basic
/-!
# Finitely supported functions and spans
## Tags
functi... |
10,993 | lean_file_chunk | 82bd036d89889d0d86fcfb04e6fc62790047a7f35e1c0c73c79a7c43feabe65d | 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
-/
module
public import Mathlib.Algebra.Module.Submodule.Range
public import Mathlib.LinearAlgebra.Finsupp.LSum
public import Mathlib.LinearAlgebra.Span.Defs
/-!
# `Fi... |
3,085 | lean_file_chunk | 85c3595603d0505f164f047a8de4c7a8d8973d3b2f1b3981102a72a3fd45f066 | 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
-/
module
public import Mathlib.Algebra.Module.Equiv.Defs
public import Mathlib.Algebra.Module.Pi
public import Mathlib.Algebra.Module.Prod
public import Mathlib.Data.F... |
22,379 | lean_file_chunk | be20be82f1ed65e642bd5b73d2487e200917052c231129099b3ae97929518b0c | 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
-/
module
public import Mathlib.Algebra.Module.Submodule.Equiv
public import Mathlib.Data.Finsupp.Option
public import Mathlib.LinearAlgebra.Finsupp.Supported
/-!
# `F... |
1,456 | lean_file_chunk | 671b02706c473283920b9f9e877a09521416c06d7e04bde968767d568611e139 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 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.CharP.Lemmas
public import Mathlib.LinearAlgebra.SModEq.Basic
public import Mathlib.RingTheory.Ideal.Operations
/-! # Lemmas about SModE... |
858 | lean_file_chunk | 13fa1fdb8a03e50d07ac3c2536a7117036dc895332078bc5ac0433534717a21b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Jiedong Jiang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jiedong Jiang
-/
module
public import Mathlib.Algebra.Algebra.Operations
public import Mathlib.LinearAlgebra.SModEq.Basic
/-!
# Pointwise lemmas for modular equivalence
In this file,... |
6,631 | lean_file_chunk | 9e6a546555a174d3213277bec742919f4cc8305680228c3705c2c883865c33bb | 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
-/
module
public import Mathlib.Algebra.Module.Submodule.Map
public import Mathlib.Algebra.Polynomial.Eval.Defs
public import Mathlib.RingTheory.Ideal.Quotient.Defs
public import... |
880 | lean_file_chunk | 8c2e929c8c938c1362930d390a32182e04f8ea566bba7f98dc341d10a91e5e9c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel
-/
module
public import Mathlib.Algebra.DirectSum.Module
public import Mathlib.RingTheory.Finiteness.Basic
/-!
# A finite direct sum of finite modules is finite
This file... |
6,798 | lean_file_chunk | e336a607da1ec8cb3d44ccd4a7ee98de43cdaf119b342cea1fa1f2cb0359f531 | 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, Mario Carneiro, Eric Wieser
-/
module
public import Mathlib.LinearAlgebra.TensorProduct.Tower
public import Mathlib.Algebra.DirectSum.Module
/-!
# Tensor products of direct sums... |
13,780 | lean_file_chunk | 513140009a602f6189a7e2a51509f81eb6f17d1f5d99879b04cf056971a3a07f | 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, Antoine Chambert-Loir
-/
module
public import Mathlib.Algebra.DirectSum.Finsupp
public import Mathlib.LinearAlgebra.DirectSum.TensorProduct
public import Mathlib.Linea... |
855 | lean_file_chunk | 360d9b285fa1704aacc23a4a75f9d91b8048b4a3fc12bfb13273f771e07d9320 | 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.DirectSum.Module
public import Mathlib.LinearAlgebra.Finsupp.VectorSpace
/-!
# Bases for direct sum of modules
This file def... |
3,364 | lean_file_chunk | 535053d92a0ae95c0925629c786689246e3a85c0ab6f3562fd9c2fb0ef878dd3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel
-/
module
public import Mathlib.Algebra.DirectSum.Module
public import Mathlib.LinearAlgebra.Multilinear.DFinsupp
/-!
# Multilinear maps from direct sums
This file descri... |
15,830 | lean_file_chunk | a3994dc26dcbe4972f19b864a49637c861adefad97d37b6506cc5eed1877a47d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Data.Fintype.Sort
public import Mathlib.LinearAlgebra.Multilinear.Basic
/-!
# Currying of multilinear maps
We register isomorphis... |
4,141 | lean_file_chunk | 3eb4ba25441a875661574c94282a6093ef6e0cfaa0ef081b710199a83484e717 | 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.TensorProduct.Basic
public import Mathlib.LinearAlgebra.Multilinear.Basic
/-!
# Constructions relating multilinear maps and ten... |
5,809 | lean_file_chunk | 8819421e76672e5dbcd523d0cb235e810e628a4834aee9f59f0221241e69a47a | 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.LinearAlgebra.Pi
public import Mathlib.LinearAlgebra.Multilinear.Basic
/-!
# Interactions between (dependent) functions and multilinear maps
... |
23,986 | lean_file_chunk | bd77f8914f7e05b91067f9c7a780d29f78d6b293eaa02425b003c02be60c3f7f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Algebra.BigOperators.Group.Finset.Powerset
public import Mathlib.Data.Finset.Sort
public import Mathlib.Data.Fintype.BigOperators
... |
23,977 | lean_file_chunk | 724eec5e5508ed2d9d474175c8b2aadb29117fd62a3f3a1ecbb553e70cdaec96 | lean_source_file | pkg_mathlib_source_file_chunks | Function.update (fun i => ∑ j ∈ A i, g i j) i₀
((∑ j ∈ B i₀, g i₀ j) + ∑ j ∈ C i₀, g i₀ j) := by
ext i
by_cases hi : i = i₀
· rw [hi, update_self]
have : A i₀ = B i₀ ∪ C i₀ := by
simp only [B, C, Function.update_self, Finset.sdiff_union_self_eq_union]
symm
simp only [hj... |
13,518 | lean_file_chunk | 4904ce1abb0dc6e6929271fdde6f6ae3d5318bdb4052b593d5edf053e3b50597 | lean_source_file | pkg_mathlib_source_file_chunks | convert g.map_update_add (fun j ↦ f j (x j)) i (f₁ (x i)) (f₂ (x i)) with j j j
· exact Function.apply_update c f i (f₁ + f₂) j
· exact Function.apply_update c f i f₁ j
· exact Function.apply_update c f i f₂ j
map_update_smul' := by
intro _ f i a f₀
ext g x
change (g fun j ↦ update f i (a • f₀... |
2,423 | lean_file_chunk | 2efd473c2c46c52ebad928130bdbc35e4739a5bb144c5dc45c48a71a933f8772 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Daniel Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Daniel Morrison
-/
module
public import Mathlib.LinearAlgebra.Multilinear.DFinsupp
/-!
# Interactions between finitely-supported functions and multilinear maps
## Main definitions
... |
3,162 | lean_file_chunk | 47884b9fdaaf1f3ed7485798294f8f4226533a386e427fc022a3b39180f65176 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Eric Wieser, Daniel Morrison
-/
module
public import Mathlib.LinearAlgebra.Basis.Defs
public import Mathlib.LinearAlgebra.Multilinear.Finsupp
/-!
# Multilinear maps in re... |
11,762 | lean_file_chunk | a1c2393067d1d8124b5109e19f8268db7aa9c7ef070f402fc6b151661dac4917 | 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, Sophie Morel
-/
module
public import Mathlib.Data.Fintype.Quotient
public import Mathlib.LinearAlgebra.DFinsupp
public import Mathlib.LinearAlgebra.Multilinear.Basic
/-!
# ... |
2,655 | lean_file_chunk | 17e7a93f5ef6e7f19327f5ee060850fbf702db090238517f15ba3b5229a6f402 | 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.LinearAlgebra.Multilinear.Curry
public import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
/-! # Multilinear maps over finite-dimensional s... |
23,917 | lean_file_chunk | 0686638aa42ea0c57ebb13d6d8dc3f981001a97eb0375858a68e699142dc2926 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Alexander Bentkamp, Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Sébastien Gouëzel, Eric Wieser
-/
module
public import Mathlib.Algebra.Algebra.RestrictScalars
public import Mathlib.Algebra.CharP.Invertibl... |
631 | lean_file_chunk | 96bfd982b109b16faddf3057d47ec5ee8278e901cab95e4f3d2a55751e11b3ac | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
module
public import Mathlib.LinearAlgebra.Complex.Module
public import Mathlib.LinearAlgebra.Orientation
/-!
# The standard orientation on `ℂ`.
This had previou... |
1,040 | lean_file_chunk | c5c053c6e05fe88e88082b3db24b13bfcad1b9c0509e41b39f2993a3708ae557 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 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.Complex.Module
public import Mathlib.LinearAlgebra.Determinant
/-!
# Determinants of maps in the complex numbers as a vector ... |
2,893 | lean_file_chunk | 0a104796dca24f556775f21a23f6673282793685b55c5c9314b656c45697cbee | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Alexander Bentkamp, Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Sébastien Gouëzel, Eric Wieser
-/
module
public import Mathlib.Algebra.Algebra.Rat
public import Mathlib.Analysis.Complex.Cardinality
public... |
9,386 | lean_file_chunk | 5a15c0738e2f44b5aa682f1cb4e53be678253a03eaf8fb9bce4faabca72f6a9c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Raphael Douglas Giles. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Raphael Douglas Giles, Zhixuan Dai, Zhenyan Fu, Yiming Fu, Jingting Wang, Eric Wieser
-/
module
public import Mathlib.LinearAlgebra.TensorAlgebra.Basic
/-!
# Symmetric Algebra... |
4,138 | lean_file_chunk | 92627e60c338abc19d340ac6c51024b3f0a294d98b6f7370dffa297413766601 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Raphael Douglas Giles. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Raphael Douglas Giles, Zhixuan Dai, Zhenyan Fu, Yiming Fu, Jingting Wang, Eric Wieser
-/
module
public import Mathlib.LinearAlgebra.SymmetricAlgebra.Basic
public import Mathlib... |
11,648 | lean_file_chunk | 0f9726cfa5e01339e17540c5e83971bf653515f8c635cc4b40218ac9771c286d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
module
public import Mathlib.LinearAlgebra.Eigenspace.Basic
public import Mathlib.FieldTheory.IsAlgClosed.Spectrum
public import Mathlib.LinearAlgebra.FreeMo... |
1,065 | lean_file_chunk | aaa7d651aeb684bddec7355a301c97aa8df02888890176ff84a3d1426c5935a3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
module
public import Mathlib.LinearAlgebra.Eigenspace.Basic
public import Mathlib.Topology.Algebra.Module.LinearMap
/-!
# Eigenspaces of continuous linear maps
T... |
5,417 | lean_file_chunk | b2673c27006d12daf4ce1ba63e58c0540ce6f5a60e5bfd07f39d8daa6c9059b1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jon Bannon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jon Bannon, Jireh Loreaux
-/
module
public import Mathlib.LinearAlgebra.Eigenspace.Basic
/-!
# Eigenvalues, Eigenvectors and Spectrum for Matrices
This file collects results about eige... |
2,078 | lean_file_chunk | ed71c62b7a67a7be95c58aba789e86d09866d95fd7ff07ee89d6c74ce6a53dec | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Lawrence Wu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lawrence Wu
-/
module
public import Mathlib.LinearAlgebra.Charpoly.BaseChange
public import Mathlib.LinearAlgebra.Charpoly.ToMatrix
public import Mathlib.LinearAlgebra.Eigenspace.Basic
p... |
10,274 | lean_file_chunk | d676ebd11f82fee4e25013428aa0149b3300b3296bbf65c06b47c1235fbbaaaf | 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.LinearAlgebra.Eigenspace.Triangularizable
/-!
# Simultaneous eigenvectors and eigenvalues for families of endomorphisms
In finite dimensions... |
23,987 | lean_file_chunk | 3f621017159f164bb571582b408c936a48383a5e3f824be9de3b8079889bad8a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
module
public import Mathlib.Algebra.Algebra.Spectrum.Basic
public import Mathlib.Algebra.Module.LinearMap.Basic
public import Mathlib.LinearAlgebra.FiniteDi... |
17,293 | lean_file_chunk | 2b0a2dcccc0593dab8037adffbf9ac0f07ffda50bfa08d5499e7b827d9d76aea | lean_source_file | pkg_mathlib_source_file_chunks | theorem maxGenEigenspace_eq [IsNoetherian R M] (f : End R M) (μ : R) :
maxGenEigenspace f μ = f.genEigenspace μ (maxGenEigenspaceIndex f μ) :=
genEigenspace_top_eq_maxUnifEigenspaceIndex _ _
theorem maxGenEigenspace_eq_maxGenEigenspace_zero (f : End R M) (μ : R) :
maxGenEigenspace f μ = maxGenEigenspace (f -... |
4,273 | lean_file_chunk | 6e2949f902cf5286a24604108acd5b0a58d069c260108cd80c09bcdad11b59c2 | 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.LinearAlgebra.Eigenspace.Basic
public import Mathlib.LinearAlgebra.Eigenspace.Triangularizable
public import Mathlib.LinearAlgebra.Semisimple
... |
8,362 | lean_file_chunk | 79a1d345f7df77fb1061dbbd1a609323178efd30ad76d13914b77c71f0f6dfff | 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.Order.Archimedean.Basic
public import Mathlib.LinearAlgebra.Charpoly.ToMatrix
public import Mathlib.LinearAlgebra.Determinant
pu... |
5,089 | lean_file_chunk | 7e07709551fb4f04fb2c16f85260586633d3c1dfa1ea95fa923425b62920ed75 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
module
public import Mathlib.Algebra.Polynomial.Roots
public import Mathlib.Algebra.Module.Torsion.Field
public import Mathlib.FieldTheory.Minpoly.Basic
publ... |
2,280 | lean_file_chunk | 3be637ed2bc4a8ba3adddfb6f538bf90ac141f6537d194bdcb9c54eac35343c9 | 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.RingTheory.TensorProduct.Maps
public import Mathlib.Algebra.Algebra.Opposite
/-! # `MulOpposite` distributes over `⊗`
The main result in thi... |
10,091 | lean_file_chunk | 86eaa03276771faf68af33c1f645e13a8dfccc6b3bde597f05ed8fe7dc71e998 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
module
public import Mathlib.LinearAlgebra.TensorProduct.Submodule
public import Mathlib.RingTheory.TensorProduct.Maps
/-!
# Some results on tensor product of subalgebras
## Line... |
2,383 | lean_file_chunk | c126fa3d8b8a839f3bea5734e03357b0499d730cdd71e6ee8e6c076babbbd47b | 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.Algebra.DirectSum.Decomposition
public import Mathlib.LinearAlgebra.DirectSum.TensorProduct
/-! # Decomposition of tensor product
In this file w... |
3,599 | lean_file_chunk | 4ae3a64be10d29491bff4f49da3222eeffee5d5443cbd733033c4239f002b8e2 | 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.Matrix.Kronecker
public import Mathlib.LinearAlgebra.Matrix.ToLin
public import Mathlib.LinearAlgebra.TensorProduct.Basis
/-!
#... |
13,807 | lean_file_chunk | aa08667a5f417c2df8c772aeb3d3f00b62ddbd42570bceed54095c756384b554 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee, Junyan Xu
-/
module
public import Mathlib.LinearAlgebra.TensorProduct.RightExactness
public import Mathlib.LinearAlgebra.TensorProduct.Finiteness
public import Mathlib.Lin... |
2,539 | lean_file_chunk | da3bb212e23f94512c9af8162228688bda61bb97cea95e0a8e7e842fdf638db5 | 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.Prod
public import Mathlib.LinearAlgebra.TensorProduct.Tower
/-!
# Tensor products of products
This file shows that taking `Te... |
23,998 | lean_file_chunk | 8d41938dfb8b34019897961d1d7dd828167d5089c1e5b394cc57f3b520fbb29e | 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, Johan Commelin, Eric Wieser
-/
module
public import Mathlib.Algebra.Algebra.Tower
public import Mathlib.LinearAlgebra.TensorProduct.Associator
/-!
# The `A`-module struct... |
5,495 | lean_file_chunk | fa27acade5b041f2a952c2868e7b5fa103042c78d0f4f4c7d1c62e876455fa9c | lean_source_file | pkg_mathlib_source_file_chunks | @[simp] lemma _root_.LinearEquiv.baseChange_tmul {e : M ≃ₗ[R] N} (a : A) (m : M) :
e.baseChange R A M N (a ⊗ₜ m) = a ⊗ₜ e m :=
rfl
@[simp] lemma _root_.LinearEquiv.baseChange_symm_tmul {e : M ≃ₗ[R] N} (a : A) (n : N) :
(e.baseChange R A).symm (a ⊗ₜ n) = a ⊗ₜ e.symm n :=
rfl
@[simp]
theorem _root_.LinearEq... |
23,989 | lean_file_chunk | 9624e09e144d9cc12b929e15f1276166ddb9239547ae8a0e3685581d841b9a01 | 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, Mario Carneiro
-/
module
public import Mathlib.LinearAlgebra.TensorProduct.Basic
/-!
# Tensor products and linear maps
This file defines `TensorProduct.map`, the `R`-linear ma... |
3,784 | lean_file_chunk | 7951bbcdbfa6c98c5e4a44faf8a952a7abce12e3c37def456862c2f278a3d706 | lean_source_file | pkg_mathlib_source_file_chunks | theorem rTensor_refl_apply : (refl R N).rTensor M y = y := by rw [rTensor_refl, refl_apply]
variable {N}
@[simp] theorem rTensor_trans_lTensor (f : M ≃ₗ[R] P) (g : N ≃ₗ[R] Q) :
f.rTensor N ≪≫ₗ g.lTensor P = TensorProduct.congr f g :=
toLinearMap_injective <| LinearMap.lTensor_comp_rTensor M _ _
@[simp] theorem... |
4,896 | lean_file_chunk | f7e5414c5de5f5d8620acd704899ce65c01b8bf5d5678e08bad7e4fb8cf32726 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
module
public import Mathlib.Algebra.Colimit.Module
public import Mathlib.LinearAlgebra.TensorProduct.Map
/-!
# Tensor product and direct limits commute with each other... |
6,278 | lean_file_chunk | 8b8e0f32b9055f7d282f298b71438e8d0a9fbbe73877d0742a13da3825054a77 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Judith Ludwig, Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Judith Ludwig, Christian Merten
-/
module
public import Mathlib.LinearAlgebra.TensorProduct.Tower
public import Mathlib.LinearAlgebra.Pi
/-!
# Tensor product and pr... |
16,423 | lean_file_chunk | 4645c7e50966122f00ac02ac99db356792fd3d7a0b528b3e8fc8bf6c8c6a7ab2 | 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, Mario Carneiro
-/
module
public import Mathlib.LinearAlgebra.TensorProduct.Defs
/-!
# Universal property of the tensor product
Given any bilinear map `f : M →ₛₗ[σ₁₂] N →ₛₗ[σ₁₂... |
13,927 | lean_file_chunk | 36b305087973c482c8c4a82e58327c86220049694f352cdac4d3a0cdb195fc5f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, Jujian Zhang
-/
module
public import Mathlib.LinearAlgebra.Quotient.Basic
public import Mathlib.LinearAlgebra.TensorProduct.Tower
public import Mathlib.R... |
23,962 | lean_file_chunk | 1e27bdb16eedb42c61c2a1872dd37ca2aa537483164e82b9a804368187ef1bac | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 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.Algebra.Exact
public import Mathlib.RingTheory.Ideal.Maps
public import Mathlib.RingTheory.Ideal.Quotient.Defs
public impo... |
2,285 | lean_file_chunk | 95e49b134f78f9f6857bcde2b9bf9d106265506f8df41e5765f39bc400b46b92 | lean_source_file | pkg_mathlib_source_file_chunks | rw [(lTensor_exact A g.toLinearMap.exact_subtype_ker_map hg).linearMap_ker_eq]
rfl
/-- If `f` is surjective, then the kernel of `f ⊗ (id B)` is generated by the kernel of `f` -/
lemma Algebra.TensorProduct.rTensor_ker (hf : Function.Surjective f) :
RingHom.ker (map f (AlgHom.id R C)) =
(RingHom.ker f).map ... |
17,719 | lean_file_chunk | 89910a9e03856cac6e2441ec5228edab9634eebae96916c06c53798e9de574c6 | 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, Mario Carneiro
-/
module
public import Mathlib.Algebra.Module.Equiv.Basic
public import Mathlib.Algebra.Module.Shrink
public import Mathlib.Algebra.Module.Submodule.Bilinear
pub... |
11,713 | lean_file_chunk | 9119426aa15e61d4754c41cc16379e41897e7fde10f3b2bb5f95526354292c78 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
module
public import Mathlib.Algebra.Algebra.Operations
public import Mathlib.Algebra.Algebra.Subalgebra.Lattice
public import Mathlib.LinearAlgebra.DirectSum.Finsupp
/-!
# Some r... |
14,145 | lean_file_chunk | b3cd333541317a2c1632f560fe0937c2553727ae9b0d7c9189e78fe077823d15 | 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, Mario Carneiro
-/
module
public import Mathlib.Algebra.Algebra.Hom
public import Mathlib.LinearAlgebra.TensorProduct.Map
/-!
# Associators and unitors for tensor products of mo... |
7,850 | lean_file_chunk | b0d60a43f4c6b0ec445006039283ec08005b37c2cb88f270e02a9ca6825bf35b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
module
public import Mathlib.LinearAlgebra.Basis.Basic
public import Mathlib.LinearAlgebra.DirectSum.Finsupp
public import Mathlib.LinearAlgebra.Finsupp.VectorSp... |
1,329 | lean_file_chunk | ee2d866773b645417ed63476be42ee3746762b7ef84ea6db16f9b8eb6c65dbfe | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Bryan Wang
-/
module
public import Mathlib.LinearAlgebra.TensorProduct.Pi
public import Mathlib.LinearAlgebra.DirectSum.Finsupp
/-!
# Tensor product with free modules.
... |
8,860 | lean_file_chunk | 07913ddf471a51a251621b88c9da2996f0b5cff7acf93ec7b7a2a7794639e5be | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
module
public import Mathlib.LinearAlgebra.DFinsupp
public import Mathlib.RingTheory.Finiteness.Basic
public import Mathlib.LinearAlgebra.TensorProduct.Map
/-!
# Some finiteness r... |
5,641 | lean_file_chunk | 26b2d99c5652f92eacdc043beb5e01823a587127febc39058f8208718a052549 | 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.LinearAlgebra.FreeModule.Finite.Basic
public import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
public import Mathlib.LinearAlgebra.De... |
3,134 | lean_file_chunk | fbad6a1961fb270427c5a26c31cd06cb5d559bf8e840f195f6be8dc7dbfa9ee1 | 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.LinearAlgebra.Charpoly.Basic
public import Mathlib.LinearAlgebra.Matrix.Basis
public import Mathlib.RingTheory.Finiteness.Prod
/-!
#... |
1,914 | lean_file_chunk | 3a42463014058146365938a89c7d4f86ce74172fbdd236e8a2656a4bddd03e14 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.LinearAlgebra.Charpoly.ToMatrix
public import Mathlib.LinearAlgebra.Determinant
public import Mathlib.RingTheory.TensorProduct.Finite
public i... |
3,236 | lean_file_chunk | ece8012603bfd6431317144b12aef4104b6a697cd92f756f99bba2083fce523d | 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.Equiv
public import Mathlib.Algebra.Ring.Action.ConjAct
public import Mathlib.LinearAlgebra.GeneralLinearGroup.Basic
import M... |
4,641 | lean_file_chunk | b165a6b969767583e6ba4dd73e037dfe9c7901a6dd1e065270a4bdba570b1cd6 | 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
-/
module
public import Mathlib.Algebra.Module.Equiv.Basic
/-!
# The general linear group of linear maps
The general linear group is defined to be the group of invert... |
19,857 | lean_file_chunk | 7df5215fe748b9d708996e090818bafe37e26acb323a60776de6385f774c3105 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel, Joël Riou
-/
module
public import Mathlib.Algebra.Module.Presentation.Basic
public import Mathlib.LinearAlgebra.ExteriorAlgebra.OfAlternating
/-!
# Exterior powers
We st... |
8,671 | lean_file_chunk | ee054975e93da6629c6cf892253cfe9a6edd0b4a9c242b5a28106da5e082762b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Daniel Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel, Daniel Morrison
-/
module
public import Mathlib.LinearAlgebra.ExteriorPower.Basic
public import Mathlib.LinearAlgebra.ExteriorPower.Pairing
public import Mathlib.RingTh... |
4,940 | lean_file_chunk | 59edaf9cad2e46872cbd8120ea4e4430c10c945bd907f235b516845ed9cba1fc | 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, Sophie Morel
-/
module
public import Mathlib.LinearAlgebra.ExteriorPower.Basic
public import Mathlib.LinearAlgebra.TensorPower.Pairing
/-!
# The pairing between the exterior po... |
23,981 | lean_file_chunk | 6d77b084d3ba60af6b6436dc4d8bded4937054b5e51b3ff15bf9aaaa1f0003a5 | 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.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup
public import Mathlib.LinearAlgebra.Charpoly.BaseChange
public impo... |
2,148 | lean_file_chunk | 30c9f2715c5823569920de7cbf221e61ef99ac0cf28d64e0432617936fa73f2b | lean_source_file | pkg_mathlib_source_file_chunks | open IsBaseChange
@[simp] theorem det [Free R V] [Module.Finite R V] (f : Dual R V) (v : V) :
(transvection f v).det = 1 + f v := by
rcases subsingleton_or_nontrivial R with hR | hR
· subsingleton
let b := finBasis R V
set n := finrank R V
let S := MvPolynomial (Fin n ⊕ Fin n) ℤ
let γ : S →+* R :=
... |
2,201 | lean_file_chunk | 7ad5d357eaef60da6949d3443e87bb7e3f6a859a3bc1b1a0c5be4acfb9226590 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel, Eric Wieser
-/
module
public import Mathlib.LinearAlgebra.PiTensorProduct
public import Mathlib.LinearAlgebra.PiTensorProduct.DFinsupp
public import Mathlib.Algebra.Direct... |
2,712 | lean_file_chunk | 74a07ee7f84668e867eb91fdab9344ba251993fb842023c67d788d35fe656a77 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Daniel Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Daniel Morrison
-/
module
public import Mathlib.Data.Finsupp.ToDFinsupp
public import Mathlib.LinearAlgebra.PiTensorProduct.DFinsupp
public import Mathlib.RingTheory.PiTensorProduct
... |
5,271 | lean_file_chunk | 86f53e9315418e26e5c6b1674cad7ff26e890daed8ceb2a4c79f4d52693f3243 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Daniel Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Daniel Morrison, Sophie Morel
-/
module
public import Mathlib.LinearAlgebra.Dual.Basis
public import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
public import Mathlib.LinearAlgebra... |
1,915 | lean_file_chunk | cf870e88dd86005fb680c8a0cf4174ca3180db4c5a07812f3bed515a1063a4da | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Daniel Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Daniel Morrison, Sophie Morel
-/
module
public import Mathlib.LinearAlgebra.Finsupp.VectorSpace
public import Mathlib.LinearAlgebra.PiTensorProduct.Finsupp
/-!
# Basis for `PiTensor... |
2,204 | lean_file_chunk | 1eedd0bfbc0768f5ac8629a03632c6088bb5c6551881515a8070b201296148b3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel, Eric Wieser
-/
module
public import Mathlib.LinearAlgebra.PiTensorProduct
public import Mathlib.LinearAlgebra.DFinsupp
public import Mathlib.LinearAlgebra.Multilinear.DFin... |
9,389 | lean_file_chunk | b2aa7f854259a95b92ce68be2a240b3747690393fb365b418a0e70b749edf600 | 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.Module.Basic
public import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
/-!
# Midpoint of a segment
## Main definitions
* ... |
6,265 | lean_file_chunk | 891ec25a5faa94087d0904d1579f1bb9b9f209f241835695988c5112f1148c97 | 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.Basis
public import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
/-!
# Matrix results for barycentric co-ordinat... |
23,950 | lean_file_chunk | f094fa6a326e531b7b145bcc7d759d36d7845f473011efa513d82be454cba978 | 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.BigOperators.Group.Finset.Indicator
public import Mathlib.Algebra.Module.BigOperators
public import Mathlib.LinearAlgebra.AffineSpac... |
23,975 | lean_file_chunk | 60a4d196e4e642ad8fea421d5180c6a81c951b9026c9d72315c7c33315e2d8f0 | lean_source_file | pkg_mathlib_source_file_chunks | theorem affineCombination_eq_of_weightedVSub_eq_zero_of_eq_neg_one {w : ι → k} {p : ι → P}
(hw : s.weightedVSub p w = (0 : V)) {i : ι} [DecidablePred (· ≠ i)] (his : i ∈ s)
(hwi : w i = -1) : {x ∈ s | x ≠ i}.affineCombination k p w = p i := by
classical
rw [← @vsub_eq_zero_iff_eq V, ← hw,
← s.affine... |
362 | lean_file_chunk | e2aa262af20a844360762973170f2d5822ff9b34682f8e2083d12becaa53662f | lean_source_file | pkg_mathlib_source_file_chunks | toFun p := s.weightedVSubOfPoint p.fst p.snd w
linear := ∑ i ∈ s, w i • ((LinearMap.proj i).comp (LinearMap.fst _ _ _) - LinearMap.snd _ _ _)
map_vadd' := by
rintro ⟨p, b⟩ ⟨v, b'⟩
simp [LinearMap.sum_apply, Finset.weightedVSubOfPoint, vsub_vadd_eq_vsub_sub,
vadd_vsub_assoc, ← sub_add_eq_add_sub, smul_a... |
23,915 | lean_file_chunk | b1681d1bd286871cd82d42c862ca505a718f4c4d9b19373425a0ca89cfbde46a | 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.Data.Fin.VecNotation
public import Mathlib.Data.Sign.Basic
public import Mathlib.LinearAlgebra.AffineSpace.Combination
public import Mathlib... |
22,907 | lean_file_chunk | 9417cce92a7b2f49a2516c84c47c9107a4af19dcfb5b41ec98fe2f54f17e8d35 | lean_source_file | pkg_mathlib_source_file_chunks | theorem AffineIndependent.exists_mem_inter_of_exists_mem_inter_affineSpan [Nontrivial k] {p : ι → P}
(ha : AffineIndependent k p) {s1 s2 : Set ι} {p0 : P} (hp0s1 : p0 ∈ affineSpan k (p '' s1))
(hp0s2 : p0 ∈ affineSpan k (p '' s2)) : ∃ i : ι, i ∈ s1 ∩ s2 := by
have hp0' : p0 ∈ affineSpan k (p '' s1) ⊓ affineSp... |
23,984 | lean_file_chunk | 7c27f6e003fee8e2861080405e8b398498badf355959a5201bdc3174d6eb79b6 | 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.Order.Group.Pointwise.Interval
public import Mathlib.Algebra.Order.Module.Defs
public import Mathlib.LinearAlgebra.BilinearMap
publi... |
6,792 | lean_file_chunk | 530e68db9862d0be5559f86090f88438a223354a88cf94c28d82038f10c93a93 | lean_source_file | pkg_mathlib_source_file_chunks | right_inv := by
rintro ⟨v, f⟩
ext <;> simp [const_linear]
map_add' := by simp
map_smul' := by simp
end Module
/-- Interpolating between affine maps with `lineMap` commutes with evaluation. -/
@[simp]
lemma lineMap_apply' [SMulCommClass k k V2] (f g : P1 →ᵃ[k] P2) (c : k)
(p : P1) : lineMap f g c p = l... |
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