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 |
|---|---|---|---|---|---|
10,353 | lean_file_chunk | dbd36f24ef4c2a31f8f53136cda57bcd578c3fc1a56ea43c2efd435c1fa38c50 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jiedong Jiang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu, Jiedong Jiang
-/
module
public import Mathlib.FieldTheory.Normal.Closure
public import Mathlib.FieldTheory.IsAlgClosed.Basic
public import Mathlib.FieldTheory.IntermediateFi... |
17,371 | lean_file_chunk | ae9a8760dad2645442da69b7b6010995f5670ab8deede902159f4d35f26b61ae | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Junyan Xu
-/
module
public import Mathlib.Data.Fintype.Order
public import Mathlib.FieldTheory.IntermediateField.Adjoin.Basic
/-!
# Extension of field embeddings
`... |
2,838 | lean_file_chunk | bd391e905b78e2f7f3dc680dd11b10289ee91e2df5b74c053235c1240359c0ec | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
module
public import Mathlib.Algebra.Field.TransferInstance
public import Mathlib.Algebra.Field.ULift
public import Mathlib.Algebra.MvPolynomial.Cardinal
public impo... |
9,049 | lean_file_chunk | e91c63bc62d264f2b8dbe60d54bfe0d8224cc90bbff198a4ca1311f8d08b59e7 | 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.FieldTheory.Finite.Basic
/-!
# The Chevalley–Warning theorem
This file contains a proof of the Chevalley–Warning theorem.
Throughout m... |
13,097 | lean_file_chunk | 7722806801e5d58b3555fd531845c477e4632f3730fbee0f0322b55706376b20 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
module
public import Mathlib.FieldTheory.Galois.Basic
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pro... |
23,970 | lean_file_chunk | 2be946864abc7f561a95b3142bbab5b9987f4f4812bca3eeb2bb330a046f12ba | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Andrew Yang, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots
public import Mathlib.FieldTheory.Galois.Basic
public import Mathlib.FieldTheory.KummerPo... |
1,496 | lean_file_chunk | 6787fddc9ae823345eab565a39c20375ce1e8084bc017fe00a67ba628f89ccd1 | lean_source_file | pkg_mathlib_source_file_chunks | · rw [eq_top_iff, ← IntermediateField.top_toSubalgebra, ← hα,
IntermediateField.adjoin_simple_toSubalgebra_of_isAlgebraic (IsAlgebraic.of_finite K α)]
apply Algebra.adjoin_mono
rw [Set.singleton_subset_iff, mem_rootSet_of_ne (X_pow_sub_C_ne_zero finrank_pos a),
aeval_def, eval₂_sub, eval₂_X_pow, eva... |
11,931 | lean_file_chunk | 71578310e119ed941a432062b8161cd332ceb971ca2120b684fb94f151b42057 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Michal Staromiejski. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michal Staromiejski
-/
module
public import Mathlib.RingTheory.Trace.Basic
/-!
# Normalized trace
This file defines the *normalized trace* map; that is, an `F`-linear map from... |
23,944 | lean_file_chunk | b4aee80d91aba46ca4dc64362f54f72e8e3a890b4f9b15af34b46d9cec487aff | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
module
public import Mathlib.FieldTheory.SeparableDegree
public import Mathlib.FieldTheory.IsSepClosed
public import Mathlib.RingTheory.AlgebraicIndependent.AlgebraicClosure
/-!
#... |
166 | lean_file_chunk | 94abedda76812b47342098b73260c58f612a263cbbc354437d3ef8a2c201fb9d | lean_source_file | pkg_mathlib_source_file_chunks | theorem Algebra.IsSeparable.finInsepDegree_eq [Algebra.IsSeparable F E] : finInsepDegree F E = 1 :=
Cardinal.one_toNat ▸ congr(Cardinal.toNat $(insepDegree_eq F E)) |
22,220 | lean_file_chunk | 05251daa60b0b2633492462e967de9bcea30316a0b78735638aabe32c88f1de6 | 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.FieldTheory.IntermediateField.Adjoin.Basic
/-!
# Relative rank of subfields and intermediate fields
This file contains basics about the relative rank... |
5,704 | lean_file_chunk | 558a48b77ccc2b2cfb911cfb4e5c154079bd8466cc83a7a3e41b1808cc89e761 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
module
public import Mathlib.FieldTheory.Normal.Basic
public import Mathlib.FieldTheory.PrimitiveElement
public import Mathlib.GroupTheory.CosetCover
/-!
# Algebraic extensio... |
16,289 | lean_file_chunk | a9160e9a08df1a9b97c8c7d48b5b1ea052678f71bcc650f15a7de6034f7034d0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
module
public import Mathlib.FieldTheory.SeparableClosure
public import Mathlib.FieldTheory.PurelyInseparable.Basic
public import Mathlib.LinearAlgebra.FreeAlgebra
public impo... |
18,527 | lean_file_chunk | 33dce184006ccfed7cbf703075c2481c8a178f7bc6ff050d6b8a62456e637edf | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
module
public import Mathlib.Data.Fintype.Pigeonhole
public import Mathlib.FieldTheory.IsAlgClosed.Basic
public import Mathlib.FieldThe... |
23,975 | lean_file_chunk | becf2743e98a09fc9d9a3c4216429fee1836236eb58a599c442156371efa5a74 | 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.FieldTheory.Galois.Basic
public import Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality
public import Mathlib.RingTheory.LinearDisjoint
/-!
... |
14,511 | lean_file_chunk | fbb365a29fa8aa50e40197c86c64e8ea6ec9c15c8d6666ba372dbf0e665b143f | lean_source_file | pkg_mathlib_source_file_chunks | [Algebra.IsAlgebraic F L] : Module.rank L (adjoin L (A : Set E)) = Module.rank F A :=
H.adjoin_rank_eq_rank_left_of_isAlgebraic (.inr ‹_›)
/-- If `A` and `L` are linearly disjoint over `F`, one of them is algebraic,
then `[L(A) : A] = [L : F]`. Note that in Lean `L(A)` is not naturally an `A`-algebra,
so this result... |
15,439 | lean_file_chunk | 4c0158de0cdd2bc05f0c9bcdad9f6b464bab8f73d986cce741b01d7e7327d23b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Sebastian Monnet. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sebastian Monnet
-/
module
public import Mathlib.FieldTheory.Galois.Basic
public import Mathlib.Topology.Algebra.FilterBasis
public import Mathlib.Topology.Algebra.OpenSubgroup
/-!... |
23,946 | lean_file_chunk | 0b5f316da0722b91f9d4275e46949d171dae1b44c7025a006e5e54b6edfdd472 | 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.Polynomial.Expand
public import Mathlib.Algebra.Polynomial.Splits
public import Mathlib.Algebra.Squarefree.Basic
public import Mathlib.Fie... |
7,716 | lean_file_chunk | e8975794bf831b44f8a6737423ac57d9fedfcfa5117a1a865dcbfecac08cc0ae | lean_source_file | pkg_mathlib_source_file_chunks | theorem Algebra.IsSeparable.isIntegral [Algebra.IsSeparable F K] : ∀ x : K, IsIntegral F x :=
fun x ↦ _root_.IsSeparable.isIntegral (Algebra.IsSeparable.isSeparable F x)
variable (K) in
instance Algebra.IsSeparable.isAlgebraic [Nontrivial F] [Algebra.IsSeparable F K] :
Algebra.IsAlgebraic F K :=
⟨fun x ↦ (Alge... |
16,664 | lean_file_chunk | 2119bd9ba3596b2e4ee06fa5b53950b34edef80daa0661d310d4930abc9f4359 | 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.Polynomial.GroupRingAction
public import Mathlib.Algebra.Ring.Action.Field
public import Mathlib.Algebra.Ring.Action.Invariant
public impo... |
18,907 | lean_file_chunk | f10ab4e920009e5ac424973fd2c38960c052a1731518f555e07a5b23ef108442 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov
-/
module
public import Mathlib.Algebra.CharP.Lemmas
public import Mathlib.FieldTheory.Perfect
/-!
# The perfect closure of a characteristic `p` ring
## Main ... |
4,077 | lean_file_chunk | 1097d5abbc914be44088b572df8cf6690e7d2c8a083cfdefbc13bddedc3703bb | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
module
public import Mathlib.LinearAlgebra.Basis.VectorSpace
public import Mathlib.LinearAlgebra.Dimension.Constructions
public import Mathlib.LinearAlgebra.Dimension.Fi... |
23,942 | lean_file_chunk | 29bd8853a464beb73c88457b6c37341ea01ba5f6918c27f44611f5d4b51e06eb | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.AlgebraicGeometry.Cover.Open
public import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
public import Mathlib.AlgebraicGeometry.Restrict
publ... |
23,982 | lean_file_chunk | 1d435b227dd7243d45c6c20ae6e851bcdbd7c5c66aaca0e57303e4db8518aa37 | lean_source_file | pkg_mathlib_source_file_chunks | rw [fromSpec_preimage_basicOpen', ← basicOpen_eq_of_affine]
theorem fromSpec_image_basicOpen :
hU.fromSpec ''ᵁ PrimeSpectrum.basicOpen f = X.basicOpen f := by
rw [← hU.fromSpec_preimage_basicOpen]
ext1
change hU.fromSpec '' (hU.fromSpec ⁻¹' (X.basicOpen f : Set X)) = _
rw [Set.image_preimage_eq_inter_range... |
10,658 | lean_file_chunk | 612d3617efd7a39d9f41362642c8c756a39741d964b17df9b1b50f83f45253e5 | lean_source_file | pkg_mathlib_source_file_chunks | is the zero locus in terms of the prime spectrum of `Γ(X, ⊤)`. -/
lemma isoSpec_image_zeroLocus [IsAffine X]
(s : Set Γ(X, ⊤)) :
X.isoSpec.hom '' X.zeroLocus s = PrimeSpectrum.zeroLocus s := by
rw [← X.toSpecΓ_preimage_zeroLocus]
simp [Scheme.isoSpec, Set.image_preimage_eq (h := (bijective_of_isIso _).surje... |
4,593 | lean_file_chunk | 2de788148753c02807f0024ed99c42237188452aa92d626b267af66ea0f0dbd8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
module
public import Mathlib.AlgebraicGeometry.Morphisms.Basic
public import Mathlib.CategoryTheory.MorphismProperty.Comma
/-!
# (Co)limits in over categories
... |
23,975 | lean_file_chunk | ae1d8745afd7feb6254ebb5dcb5b50fa3b3f97e330dcdd79d968162a06e138ed | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski
public import Mathlib.Tactic.DepRewrite
public import Mathlib.AlgebraicGeometry.Morphisms.Integral
... |
11,194 | lean_file_chunk | 5e30176eac9dcd324e8d19d36869b54d61ec4f9fe6aaeeb9fab4d31edf42928a | lean_source_file | pkg_mathlib_source_file_chunks | rw [this, f.toNormalization_app_preimage U]
simp [appIso_hom', IsAffineOpen.SpecMap_appLE_fromSpec_assoc _ _ (isAffineOpen_top (Spec _)),
IsAffineOpen.fromSpec_top, normalizationObjIso, normalizationDiagram]
#adaptation_note /-- Before #36613, the following simp call was not needed. -/
simp [←... |
7,176 | lean_file_chunk | edffa507eae249e4b90b80016768b2b5645b233090387715742cb0ab6d169e0d | 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.AlgebraicGeometry.PullbackCarrier
public import Mathlib.RingTheory.LocalRing.ResidueField.Fiber
public import Mathlib.RingTheory.Spectrum.Prim... |
17,042 | lean_file_chunk | 78cef1cd7def745870d4a3f467b985ccd28a3be715720756ef6e57bed3a5b67f | 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.AlgebraicGeometry.Morphisms.FiniteType
public import Mathlib.AlgebraicGeometry.Noetherian
public import Mathlib.AlgebraicGeometry.Stalk
public... |
22,904 | lean_file_chunk | 0167894e1191de80d60678510ce9dbea9a1316786b29db9b8668a379af83412e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.AlgebraicGeometry.Morphisms.Etale
public import Mathlib.AlgebraicGeometry.Morphisms.FlatDescent
public import Mathlib.AlgebraicGeometry.Morphi... |
16,991 | lean_file_chunk | d6c8d3d41c3436205ecccdd25862986448b0ba911374a5c2b842453e3693913d | 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, Fangming Li
-/
module
public import Mathlib.AlgebraicGeometry.AffineScheme
public import Mathlib.AlgebraicGeometry.Morphisms.Preimmersion
/-!
# Stalks of a Scheme
## Main ... |
23,931 | lean_file_chunk | a454af97a47996f0f0bbe82349712664f4a4f368fa9583f2f3a3782448f5e938 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.Geometry.RingedSpace.OpenImmersion
public import Mathlib.AlgebraicGeometry.Scheme
public import Mathlib.CategoryTheory.MorphismProperty.Limits... |
10,869 | lean_file_chunk | 782b85cf1d6fea6bfae28a7dfdfa399893acd03985ef87ac0ded3fff874d6910 | lean_source_file | pkg_mathlib_source_file_chunks | PreservesPullback.iso_hom_fst Scheme.forgetToTop g f, TopCat.coe_comp, Set.range_comp,
Set.range_eq_univ.mpr, ← @Set.preimage_univ _ _ (pullback.snd g.base f.base)]
-- Porting note (https://github.com/leanprover-community/mathlib4/issues/11224): was `rw`
· erw [TopCat.pullback_fst_image_snd_preimage]
rw [Se... |
6,524 | lean_file_chunk | 43dbe1c64c008bf626318e7fd9a7daa9a3ae9c850ebed2edf8f109505a50bf35 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yong-Gyu Choi. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yong-Gyu Choi
-/
module
public import Mathlib.Algebra.Category.Ring.EqualizerPushout
public import Mathlib.AlgebraicGeometry.Morphisms.Flat
public import Mathlib.Topology.Category.TopC... |
1,989 | lean_file_chunk | 7a091ff95049b08ad89e823047e10b26b639e3006fad86803e2000286aa10d4f | 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.AlgebraicGeometry.Scheme
public import Mathlib.CategoryTheory.Comma.Over.OverClass
/-!
# Typeclasses for `S`-schemes and `S`-morphisms
We de... |
6,518 | lean_file_chunk | e6f49e8ec5cf04f1cc718c824a14ca68c5b5f03179c7199abffd59d5948207a7 | 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.AlgebraicGeometry.Morphisms.Immersion
/-!
# `Π Rᵢ`-Points of Schemes
We show that the canonical map `X(Π Rᵢ) ⟶ Π X(Rᵢ)` (`AlgebraicGeometry... |
23,947 | lean_file_chunk | 541301cf62609c9e78d43e2313a7c1667a0d3a6de566c48f504269b4bc887f25 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.AlgebraicGeometry.Cover.Open
public import Mathlib.AlgebraicGeometry.Over
/-!
# Restriction of Schemes and Morphisms
## Main definition
- `A... |
10,526 | lean_file_chunk | ca88342d99335fb00630eea645b92bf54fb545fba3827523a12deb83c37644f9 | lean_source_file | pkg_mathlib_source_file_chunks | congr(Y.presheaf.map (eqToHom (congr_arg (U.ι ''ᵁ ·) (U.ι.preimage_image_eq V).symm)).op ≫
$(Scheme.Hom.congr_app (morphismRestrict_ι f U) (U.ι ''ᵁ V)))
theorem morphismRestrict_appTop {X Y : Scheme.{u}} (f : X ⟶ Y) (U : Y.Opens) :
(f ∣_ U).appTop = f.app (U.ι ''ᵁ ⊤) ≫
X.presheaf.map (eqToHom (image_... |
23,980 | lean_file_chunk | a1dfb30e3c360814c8b59209f944e50be2f63297c2179bb2a964216d2955639d | 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: Johan Commelin, Kim Morrison, Andrew Yang
-/
module
public import Mathlib.Algebra.Category.ModuleCat.Stalk
public import Mathlib.Algebra.Category.Ring.Limits
public import Mathlib.RingT... |
23,914 | lean_file_chunk | 688a1abf84a4588b954b675342556de676cc258d5ede86bc1b6b830e506e6685 | lean_source_file | pkg_mathlib_source_file_chunks | /-- The canonical ring homomorphism interpreting an element of `R` as an element of
the stalk of `structureSheaf R` at `x`. -/
@[expose] public def toStalk (x : PrimeSpectrum.Top R) :
CommRingCat.of R ⟶ (structurePresheafInCommRingCat R).stalk x :=
CommRingCat.ofHom (algebraMap _ _) ≫ (structurePresheafInCommRing... |
6,430 | lean_file_chunk | d82cf887f82d8582bae1198d0d6cb5d413e2854985f517f6828e7c437f2fe201 | lean_source_file | pkg_mathlib_source_file_chunks | comapₗ f U V hUV (const a b U hb) = const (f a) (σ b) V (hUV.trans (Set.preimage_mono hb)) :=
Subtype.ext <| funext fun _ ↦ by simp [comapₗ, comapFun]
section Ring
open Spec (structureSheaf)
variable {S : Type u} [CommRing S] {P : Type u} [CommRing P]
set_option backward.isDefEq.respectTransparency false in
@[sim... |
18,079 | lean_file_chunk | 596e395fcf060dc5f362dcdfc7409ad12f0f9498bf718539c128bc320a543a8f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Qi Ge, Christian Merten, Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Qi Ge, Christian Merten, Andrew Yang
-/
module
public import Mathlib.Algebra.Category.Ring.LinearAlgebra
public import Mathlib.AlgebraicGeometry.ResidueField
/-... |
14,965 | lean_file_chunk | 7dd3671ac52c97e18b1a9ab63e3a920cd668041e16f1cf67e143b0592ffd4e6b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Andrew Yang, Qi Ge, Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Qi Ge, Christian Merten
-/
module
public import Mathlib.AlgebraicGeometry.Morphisms.Immersion
public import Mathlib.AlgebraicGeometry.Morphisms.Prop... |
23,978 | lean_file_chunk | 944e53fabdda606b19b6a0ddae87122573e09c4bfdb74410c5f4ba98b147981a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.AlgebraicGeometry.Restrict
public import Mathlib.CategoryTheory.LocallyDirected
public import Mathlib.CategoryTheory.MorphismProperty.Local
pu... |
13,167 | lean_file_chunk | a70c5dd245510dfcb032d8327a9357bc610f650d02f3efcc6714ea6fc2e960ac | lean_source_file | pkg_mathlib_source_file_chunks | let α : F.obj l ⟶ pullback ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) :=
pullback.lift
(F.map hli ≫ (F.map k₁.2.1).isoOpensRange.hom)
(F.map hlj ≫ (F.map k₂.2.1).isoOpensRange.hom)
(by simp [← cancel_mono (Scheme.Opens.ι _), ← Functor.map_comp,
Subsingleton.elim (hli ≫ k₁.2.1) (hlj ≫ k₂... |
17,440 | lean_file_chunk | f5b0dbd55d38f13a0b5ebc23c63521a0556f025b8b0faf6a09f0461e8840a835 | 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, Justus Springer
-/
module
public import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
public import Mathlib.AlgebraicGeometry.StructureSheaf
public import Mathlib.RingTh... |
12,639 | lean_file_chunk | b783e7c58b25148f5d3abc03011baf5762f0713dd1477981b5176312019c6462 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
module
public import Mathlib.AlgebraicGeometry.RelativeGluing
public import Mathlib.CategoryTheory.MorphismProperty.OverAdjunction
/-!
# Colimits in `P.Over ⊤ ... |
7,125 | lean_file_chunk | 01e1884020c3042e46c0617c3c0b9a60d9155c7aa37743c4020cffdff9e4d5d4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
module
public import Mathlib.AlgebraicGeometry.Cover.Directed
public import Mathlib.AlgebraicGeometry.Morphisms.UniversallyInjective
/-!
# Relative gluing
In t... |
23,947 | lean_file_chunk | c4696edb885a14572a80827bcfa37766715f40547533b16b883dbe9e446ea24e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.AlgebraicGeometry.Spec
public import Mathlib.Algebra.Category.Ring.Constructions
public import Mathlib.CategoryTheory.Elementwise
/-!
# The... |
13,847 | lean_file_chunk | 6a05e00345391d563807eaa8110862e2ec2f70f2577bb9053be63f543e82db09 | lean_source_file | pkg_mathlib_source_file_chunks | theorem basicOpen_res_eq (i : op U ⟶ op V) [IsIso i] :
X.basicOpen (X.presheaf.map i f) = X.basicOpen f :=
RingedSpace.basicOpen_res_eq _ i f
@[sheaf_restrict]
theorem basicOpen_le : X.basicOpen f ≤ U :=
RingedSpace.basicOpen_le _ _
@[sheaf_restrict]
lemma basicOpen_restrict (i : V ⟶ U) (f : Γ(X, U)) :
X.... |
23,946 | lean_file_chunk | 536ed23388e6458a4774e67dfd2332db113b920e542bcc2ed6a9bb5aba47ccd4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Christian Merten
-/
module
public import Mathlib.Algebra.Category.Ring.FinitePresentation
public import Mathlib.AlgebraicGeometry.IdealSheaf.Functorial
public import Mathlib... |
23,994 | lean_file_chunk | 33471c046cbbe273799fdc62982a9a5c8ade0aaff3ae5050b721110851914321 | lean_source_file | pkg_mathlib_source_file_chunks | ((D.map hii' ≫ pullback.lift A.a A.b (A.ha.symm.trans A.hb)) ⁻¹'
((Scheme.Pullback.diagonalCoverDiagonalRange f A.𝒰S A.𝒰X : Set <|
↑(pullback f f))ᶜ)) = ∅ := by
let W := Scheme.Pullback.diagonalCoverDiagonalRange f A.𝒰S A.𝒰X
by_contra! h
let Z (i' : I) (hii' : i' ⟶ A.i) :=
(D.map hii' ≫ pull... |
19,455 | lean_file_chunk | 1e06a4f1a2c5079ee011125fe3343b6604ac3a0b2016235e8904c1b6841626d9 | lean_source_file | pkg_mathlib_source_file_chunks | rw [← ConcreteCategory.comp_apply, ← ConcreteCategory.comp_apply,
← ConcreteCategory.comp_apply]
congr 2
simp [Scheme.Hom.app_eq_appLE]
include hc in
lemma nonempty_isColimit_Γ_mapCocone [∀ {i j} (f : i ⟶ j), IsAffineHom (D.map f)]
[∀ i, CompactSpace (D.obj i)] [∀ i, QuasiSeparatedSpace (D.obj i)] :
... |
23,907 | lean_file_chunk | 87009bbe162ee19bfbd9e93a3e293af2c817340377d17bede455df8993a98c64 | 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.AlgebraicGeometry.SpreadingOut
public import Mathlib.AlgebraicGeometry.FunctionField
public import Mathlib.AlgebraicGeometry.Morphisms.Separat... |
1,535 | lean_file_chunk | 95ad1fe733e80eacbbee11f435e3b1affeeb8daa45665ec72aa526fab99cfbbf | lean_source_file | pkg_mathlib_source_file_chunks | convert (PartialMap.equiv_iff_of_isSeparated (S := ⊤_ _) (f := g x) (g := g y)).mp ?_ using 1
· dsimp; congr 1; simp [g, ← cancel_mono (Opens.ι _)]
· dsimp; congr 1; simp [g, ← cancel_mono (Opens.ι _)]
· rw [← PartialMap.toRationalMap_eq_iff, hg₁, hg₁]
lemma PartialMap.toPartialMap_toRationalMap_restrict [IsRedu... |
8,725 | lean_file_chunk | 000f90c5c6fd6856d0f28836d0b688d8cf91dc558eb1512596d323e6478cf1ad | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Brian Nugent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Brian Nugent
-/
module
public import Mathlib.AlgebraicGeometry.Noetherian
public import Mathlib.AlgebraicGeometry.Morphisms.Immersion
public import Mathlib.RingTheory.HopkinsLevitzki
/... |
2,746 | lean_file_chunk | 817b398c2ac9181aeb8a877b6e49c2e3f4e7752060c93c6f3cd1cbf5f612f06e | 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: Calle Sönne, Joël Riou, Ravi Vakil
-/
module
public import Mathlib.AlgebraicGeometry.Gluing
public import Mathlib.AlgebraicGeometry.Sites.BigZariski
public import Mathlib.CategoryTheory.Li... |
7,035 | lean_file_chunk | e87afa4d7336ae7f85341bc9e6a6bbecc0e8405903e94b939fb71bccc4fdc13e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.AlgebraicGeometry.Properties
/-!
# Function field of integral schemes
We define the function field of an irreducible scheme as the stalk of ... |
12,768 | lean_file_chunk | d52a7da7c2f5c5c7a2ba0dfc8f09877c1d43a43ead7b43e0af14e24a963d46a1 | 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.AlgebraicGeometry.Stalk
public import Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField
/-!
# Residue fields of points
## Main d... |
23,983 | lean_file_chunk | 6714d347e072e6bffd940494e0b08ad2fea953e907f9722420c66795a42ec06a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products
public import Mathlib.AlgebraicGeometry.Pullbacks
public import Mathlib.AlgebraicGeometry.Affi... |
8,392 | lean_file_chunk | 5f9035ff7c10424da323844e1f0b5815eeb680d198d2c7984445ad8a316c3b7b | lean_source_file | pkg_mathlib_source_file_chunks | have : IsOpenImmersion (Spec.map (CommRingCat.ofHom (RingHom.snd R S))) :=
IsOpenImmersion.of_isLocalization (0, 1)
infer_instance
instance : IsIso (coprodSpec R S) := by
rw [isIso_iff_isIso_stalkMap]
refine ⟨?_, isIso_stalkMap_coprodSpec R S⟩
convert_to IsIso (TopCat.isoOfHomeo (X := Spec (.of <| R × ... |
18,223 | lean_file_chunk | 7098f4d8edc089ee7e0d7e896a1608b4a9dcf1d413398fd136ad5ca5ec988089 | 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.Algebra.MvPolynomial.Monad
public import Mathlib.AlgebraicGeometry.Morphisms.Finite
public import Mathlib.AlgebraicGeometry.Morphisms.FinitePr... |
7,237 | lean_file_chunk | 52ce247e6afc74c7377f0c51929495ee44b964ee88e07a55e4d3f0f7ef32975d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.AlgebraicGeometry.Morphisms.Immersion
/-!
# Quasi-affine schemes
## Main results
- `IsQuasiAffine`:
A scheme `X` is quasi-affine if it is... |
23,984 | lean_file_chunk | c9b2886e3ff54472a42cece0fc8ad274cef8ae79ea2f8fc8b89f9677d27f2b8f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
module
public import Mathlib.AlgebraicGeometry.Restrict
public import Mathlib.CategoryTheory.Adjunction.Limits
public import Mathlib.CategoryTheory.Adjunction.Opposites
public... |
676 | lean_file_chunk | f364c8c7a3a7d17ba9a6385427c9b275ab5b7207711a2a51f540eb8e33bde397 | lean_source_file | pkg_mathlib_source_file_chunks | Spec.map_injective (by simp)
@[simp, reassoc]
lemma Spec.preimage_comp {R S T : CommRingCat} (f : Spec R ⟶ Spec S) (g : Spec S ⟶ Spec T) :
Spec.preimage (f ≫ g) = Spec.preimage g ≫ Spec.preimage f :=
Spec.map_injective (by simp)
end
instance : Reflective Spec.toLocallyRingedSpace where
L := Γ.rightOp
adj :... |
14,927 | lean_file_chunk | 25b49f2cfb497e93fe134e6804bb845cbfa31b88b7a97187e9447c9e7982f793 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.AlgebraicGeometry.AffineScheme
public import Mathlib.AlgebraicGeometry.Limits
public import Mathlib.RingTheory.KrullDimension.Zero
public impo... |
23,993 | lean_file_chunk | 18281798da11a6b57dfe278f203d5965d8aaf9227ece8ee0a07504eb683f1502 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.AlgebraicGeometry.AffineScheme
public import Mathlib.AlgebraicGeometry.Gluing
public import Mathlib.CategoryTheory.Limits.Opposites
public imp... |
13,174 | lean_file_chunk | 0ec63066cbbc86b1d90023180409b7557348d87527728ae7a0fe8d7436f54d28 | lean_source_file | pkg_mathlib_source_file_chunks | @[simps! I₀ X f]
def openCoverOfLeftRight (𝒰X : OpenCover.{v} X) (𝒰Y : OpenCover.{w} Y) (f : X ⟶ Z) (g : Y ⟶ Z) :
OpenCover.{max v w} (pullback f g) := by
fapply
Cover.copy ((openCoverOfLeft 𝒰X f g).bind fun x => openCoverOfRight 𝒰Y (𝒰X.f x ≫ f) g)
(𝒰X.I₀ × 𝒰Y.I₀) (fun ij => pullback (𝒰X.f ij.1 ... |
15,734 | lean_file_chunk | ec912fdb2100410edb489873f82b550647e34c6fe6b6b6cd9883d65198088f99 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Geno Racklin Asher. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Geno Racklin Asher
-/
module
public import Mathlib.AlgebraicGeometry.Morphisms.Immersion
public import Mathlib.AlgebraicGeometry.Morphisms.FinitePresentation
public import Mathlib... |
6,670 | lean_file_chunk | e3bf6b0d58532f1909d479030869f54a74ad89dbbf0eea79402a3de8a209c14e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.Algebra.Group.TypeTags.Finite
public import Mathlib.Algebra.MonoidAlgebra.Basic
public import Mathlib.LinearAlgebra.Basis.VectorSpace
public... |
13,634 | lean_file_chunk | ae510d60158e0a3af2b24d4421140d8296b91117454d5405b99cb8d22af12303 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
module
public import Mathlib.CategoryTheory.Preadditive.Projective.Preserves
public import Mathlib.RepresentationTheory.Intertwining
public import Mathlib.Repr... |
8,594 | lean_file_chunk | 141aa9336cd1da22f98dcc756259f39aa1ef94204ae4b3a99edc3f34f3faceb4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.Algebra.Category.FGModuleCat.Limits
public import Mathlib.Algebra.Category.FGModuleCat.Colimits
public import Mathlib.CategoryTheory.Monoida... |
21,980 | lean_file_chunk | 60ba24101720423f47d12c02ea2318d2ca463f49787542d7fe8bb2eff51a3bd0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
module
public import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
public import Mathlib.RepresentationTheory.Rep.Res
public import Mathlib.RepresentationThe... |
8,258 | lean_file_chunk | dfaa45ad8623784429af74ae58ffc9f2448d4ad22a0bb6da9096828f73d7b435 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yacine Benmeuraiem. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yacine Benmeuraiem
-/
module
public import Mathlib.RepresentationTheory.FDRep
/-!
# Tannaka duality for finite groups
In this file we prove Tannaka duality for finite groups.
T... |
23,994 | lean_file_chunk | dff3f0eac1f754edd531a869afd75b3f3f3dce54d23de603c7bdad54c01846bd | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Antoine Labelle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle
-/
module
public import Mathlib.LinearAlgebra.Contraction
public import Mathlib.Algebra.Group.Equiv.TypeTags
/-!
# Monoid representations
This file introduces monoi... |
2,147 | lean_file_chunk | ab1fde6bd15b2be3fcf1180e7db1e23d15678e7ad4c8a022b7a9bc368f89f3ca | lean_source_file | pkg_mathlib_source_file_chunks | @[simps -isSimp]
noncomputable def finsupp (α : Type*) :
Representation k G (α →₀ A) where
toFun g := lsum k fun i => (lsingle i).comp (ρ g)
map_one' := lhom_ext (fun _ _ => by simp)
map_mul' _ _ := lhom_ext (fun _ _ => by simp)
@[simp]
lemma finsupp_single (g : G) (x : α) (a : A) :
ρ.finsupp α g (single... |
12,275 | lean_file_chunk | ad244b0202bd6c367a2da4fc75e8361d99fa9e307054541fe25073d4e6e70572 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
module
public import Mathlib.GroupTheory.Index
public import Mathlib.RepresentationTheory.Coinduced
public import Mathlib.RepresentationTheory.Induced
/-!
# (... |
11,755 | lean_file_chunk | 52a1c3b9cef0d7cd0bc44f9eb9610ec150bc213eb75d6040597021c5b2c31960 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Yunzhou Xie. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yunzhou Xie
-/
module
public import Mathlib.RepresentationTheory.Intertwining
public import Mathlib.CategoryTheory.Action.Monoidal
/-!
## Main Purpose
This file is the preliminary for ... |
7,296 | lean_file_chunk | ec1a1cc182271586e9830650e1b69183955a0b0e409b2b0bbe3fc51d6b4a22ae | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Yunzhou Xie. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Edison Xie
-/
module
public import Mathlib.RepresentationTheory.Intertwining
/-!
## Main purpose
This file is a preliminary file for the `Iso`s in `Rep`, we build all the isomorphisms ... |
3,390 | lean_file_chunk | 5fd61aeddd2c7182db0cf415dd93cd817389c6b4990aee792e25a0d4c1fef589 | 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.Module.Submodule.Invariant
public import Mathlib.RepresentationTheory.Basic
/-!
# Invariant submodules of a group representation
-/
... |
6,766 | lean_file_chunk | 9c419ed0762bdb9fb63dde5ea1b16e04fa683ae7595a719909cf7ceb778981a6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 FLT Project. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: FLT Project
-/
module
public import Mathlib.RepresentationTheory.Basic
public import Mathlib.LinearAlgebra.Span.Defs
/-!
# Subrepresentations
This file defines subrepresentations of a ... |
3,776 | lean_file_chunk | 5bab2b88bcdd1b0bfd9afa97c9bf1f04be31dd8c30c41bac5d26ca476f573d6b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Stepan Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stepan Nesterov
-/
module
public import Mathlib.RepresentationTheory.Subrepresentation
public import Mathlib.RepresentationTheory.Intertwining
public import Mathlib.RepresentationThe... |
1,681 | lean_file_chunk | 77aa696cc327224dc82cfee84c62fe9ec57a0b777da256262ffacb537c74d581 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Stepan Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stepan Nesterov
-/
module
public import Mathlib.Data.Nat.Totient
public import Mathlib.Data.Sym.Sym2
public import Mathlib.RepresentationTheory.Subrepresentation
public import Mathli... |
23,931 | lean_file_chunk | 386acbfbeb8838cc229ec30ab492cacd73613de27a438846e200e4c7459c8222 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Stepan Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stepan Nesterov, Edison Xie
-/
module
public import Mathlib.RepresentationTheory.Subrepresentation
/-!
# Intertwining maps
This file gives defines intertwining maps of representati... |
2,477 | lean_file_chunk | 6e0b032a30184783426c2b6555ebe31c51cdd0ae8c148ae84a6b5892ae671854 | lean_source_file | pkg_mathlib_source_file_chunks | lemma comm_apply (v : V) (w : W) : comm ρ σ (v ⊗ₜ w) = w ⊗ₜ v := rfl
lemma comm_comp_lTensor (f : IntertwiningMap σ τ) :
(comm ρ τ).comp (f.lTensor ρ) = (f.rTensor ρ).comp (comm ρ σ).toIntertwiningMap := by ext; simp
lemma comm_comp_rTensor (f : IntertwiningMap σ τ) :
(comm τ ρ).comp (f.rTensor ρ) = (f.lTenso... |
11,313 | lean_file_chunk | 94f23907cb969241f873b7193d247ddf6bc69e1363f882ed58ecbd6c9fc8ae10 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Antoine Labelle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle
-/
module
public import Mathlib.RepresentationTheory.Intertwining
public import Mathlib.RepresentationTheory.FDRep
public import Mathlib.RepresentationTheory.Rep.Res
... |
5,639 | lean_file_chunk | 4316a64c41bd151aaa5416327e77db981910cf80ef8fe52e731c6671f8933e05 | 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.Category.FGModuleCat.Abelian
public import Mathlib.Algebra.Category.ModuleCat.Injective
public import Mathlib.RepresentationTheory.C... |
8,931 | lean_file_chunk | 85e5c4bc7fa3175fc4770a09ed38861e2d24637c864690f6d143fe90584dfc8f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Antoine Labelle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle
-/
module
public import Mathlib.RepresentationTheory.FDRep
public import Mathlib.LinearAlgebra.Trace
public import Mathlib.RepresentationTheory.Invariants
public impo... |
11,391 | lean_file_chunk | 30d7d6e590dbbf18b4b40d523639238add5210bc066e8beeb123b32b74b90a9b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
module
public import Mathlib.RepresentationTheory.Coinvariants
/-!
# Induced representations
Given a commutative ring `k`, a group homomorphism `φ : G →* H`,... |
2,842 | lean_file_chunk | 21bff8484ae37bc4f04ba16a67e0491f4b330ffa9177ddf9fc810683df37d987 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.Logic.Basic
public import Mathlib.Tactic.Convert
public import Mathlib.Tactic.SplitIfs
public import Mathlib.Tactic.Tauto
/-!
# More basic ... |
4,081 | lean_file_chunk | 85b0ec1cd3be1ee4ca7e5867753eb2c91f31fdaf7d23f0980d668228a26d1f66 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
module
public import Mathlib.Logic.Function.Basic
public import Mathlib.Data.Set.Defs
public import Mathlib.Data.Nat.Notation
/-!
# Relations holding pairwise
This... |
6,674 | lean_file_chunk | 7427bd8aaf9d020ada5f56663e2f0a1300d8ebb4bbce17108a35b68edbe8f48b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Floris van Doorn
-/
module
public import Mathlib.Init
/-!
# `ExistsUnique`
This file defines the `ExistsUnique` predicate, notated as `∃!`, ... |
385 | lean_file_chunk | 450755f2fed95d4398f4e55daa5d0290afdb2d01df5b45da57963a3a4d44aa86 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
module
public import Mathlib.Logic.Function.Basic
public import Mathlib.Logic.Relator
public import Mathlib.Tactic.Linter.DeprecatedModule
deprecated_module "im... |
3,506 | lean_file_chunk | 24ed6ac76de1c6f0cd47d7cf339a2a1df638dfb644c1f60a04bb98f428e8eda0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.Logic.Small.Defs
/-!
# UnivLE
A proposition expressing a universe inequality. `UnivLE.{u, v}` expresses that `u ≤ v`,
in the form `∀ α : T... |
11,327 | lean_file_chunk | 643bc1386486c990699a024775637838fb6fdf17cc93b75a550b875912d976de | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
module
public import Mathlib.Data.Fintype.EquivFin
public import Mathlib.Data.List.MinMax
public import Mathlib.Data.Nat.Order.Lemmas
public import Mathlib.Logic.Enc... |
4,955 | lean_file_chunk | ea6d54844058478320fe92c80fcd5ea30310c09e923cd3cc478897f476ffe0c7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
module
public import Mathlib.Init
/-!
# Nonempty types
This file proves a few extra facts about `Nonempty`, which is defined in core Lean.
## Main declarations
* ... |
23,976 | lean_file_chunk | 93b17ce51ab01374055ae80a784266518f61520fa50a2bb1519ecac50b343f06 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
module
public import Mathlib.Tactic.AdaptationNote
public import Batteries.Logic
public import Batteries.Util.LibraryNote
import Mathlib.Tactic.Att... |
14,938 | lean_file_chunk | e2d94e18686088e387296d2f828478daf243391f07af9f5aa32eb4891b478fd7 | lean_source_file | pkg_mathlib_source_file_chunks | (fun H ↦ .inl <| by simpa only [H] using h₂), fun h ↦ h.elim (.intro _) (.intro _)⟩
theorem Prop.forall_iff {p : Prop → Prop} : (∀ h, p h) ↔ p False ∧ p True :=
⟨fun H ↦ ⟨H _, H _⟩, fun ⟨h₁, h₂⟩ h ↦ by by_cases H : h <;> simpa only [H]⟩
theorem exists_iff_of_forall {p : Prop} {q : p → Prop} (h : ∀ h, q h) : (∃ h, q... |
8,961 | lean_file_chunk | 80c6028b1e0acd3c79d899cd9a30e4172443affcd0c14dd6d135ff5749c38235 | 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.Logic.Function.Basic
public import Mathlib.Logic.IsEmpty.Defs
public import Mathlib.Tactic.Inhabit
/-!
# Types with a unique term
In t... |
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