Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yury Kudryashov
-/
import Mathlib.Analysis.NormedSpace.Real
import Mathlib.Analysis.Seminorm
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Applications of the Hausdorf... | classical
obtain ⟨x, hx⟩ : ∃ x : E, x ∉ F := hF
let d := Metric.infDist x F
have hFn : (F : Set E).Nonempty := ⟨_, F.zero_mem⟩
have hdp : 0 < d :=
lt_of_le_of_ne Metric.infDist_nonneg fun heq =>
hx ((hFc.mem_iff_infDist_zero hFn).2 heq.symm)
let r' := max r 2⁻¹
have hr' : r' < 1 ... | Mathlib/Analysis/NormedSpace/RieszLemma.lean | 41 | 70 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Comma.StructuredArrow.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Equivalence
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Terminal... | @[simps!]
noncomputable def rightKanExtensionUnique
(F'' : D ⥤ H) (α' : L ⋙ F'' ⟶ F) [F''.IsRightKanExtension α'] : F' ≅ F'' :=
rightKanExtensionUniqueOfIso F' α (Iso.refl _) F'' α'
lemma isRightKanExtension_iff_isIso {F' : D ⥤ H} {F'' : D ⥤ H} (φ : F'' ⟶ F')
{L : C ⥤ D} {F : C ⥤ H} (α : L ⋙ F' ⟶ F) (α' : L... | Mathlib/CategoryTheory/Functor/KanExtension/Basic.lean | 134 | 145 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Peter Pfaffelhuber, Yaël Dillies, Kin Yau James Wong
-/
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Topo... |
theorem union_cylinder (s₁ s₂ : Finset ι) (S₁ : Set (∀ i : s₁, α i)) (S₂ : Set (∀ i : s₂, α i))
[DecidableEq ι] :
cylinder s₁ S₁ ∪ cylinder s₂ S₂ =
cylinder (s₁ ∪ s₂)
(Finset.restrict₂ Finset.subset_union_left ⁻¹' S₁ ∪
Finset.restrict₂ Finset.subset_union_right ⁻¹' S₂) := by
| Mathlib/MeasureTheory/Constructions/Cylinders.lean | 197 | 203 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Enum
import Mathlib.Tactic.TFAE
import Mathlib.Topology.Order.Monotone
/-!
### Topology of ordinals
We prov... | simp only [frontier_eq_closure_inter_closure, Set.mem_inter_iff, mem_closure_iff] at ha
by_contra h
rw [← isOpen_singleton_iff] at h
rcases ha.1 _ h rfl with ⟨b, hb, hb'⟩
rcases ha.2 _ h rfl with ⟨c, hc, hc'⟩
rw [Set.mem_singleton_iff] at *
subst hb; subst hc
exact hc' hb'
| Mathlib/SetTheory/Ordinal/Topology.lean | 174 | 182 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.CoreAttrs
/-!
# (Semi-)lattices
Semilatti... | variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Max (α' i)] : Max (∀ i, α' i) :=
| Mathlib/Order/Lattice.lean | 793 | 795 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.LinearAlgebra.SModEq
import Mathlib.RingTheory.Ideal.BigOperators
/-!
# Power basis
This file defines a structure ... | @[simp]
theorem natDegree_minpoly [Nontrivial A] (pb : PowerBasis A S) :
| Mathlib/RingTheory/PowerBasis.lean | 211 | 212 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.MeasureTheory.Function.SimpleFuncDense
/-!
# Strongly measurable and finitely strongly meas... |
@[measurability]
theorem _root_.MeasurableEmbedding.stronglyMeasurable_extend {f : α → β} {g : α → γ} {g' : γ → β}
{mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [TopologicalSpace β]
(hg : MeasurableEmbedding g) (hf : StronglyMeasurable f) (hg' : StronglyMeasurable g') :
StronglyMeasurable (Function.ext... | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 741 | 753 |
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.Reverse
/-!
# "Mirror" of a univariate polynomial
In this file we define `Po... | mul_one]
theorem mirror_C (a : R) : (C a).mirror = C a :=
mirror_monomial 0 a
theorem mirror_X : X.mirror = (X : R[X]) :=
mirror_monomial 1 (1 : R)
| Mathlib/Algebra/Polynomial/Mirror.lean | 47 | 53 |
/-
Copyright (c) 2021 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Adjunction.Reflective
/-!
# Adjunction between `Γ` and ... |
/-- `toSpecΓ _` is an isomorphism so these are mutually two-sided inverses. -/
theorem Γ_Spec_left_triangle : toSpecΓ (Γ.obj (op X)) ≫ X.toΓSpec.c.app (op ⊤) = 𝟙 _ := by
unfold toSpecΓ
rw [← toOpen_res _ (basicOpen (1 : Γ.obj (op X))) ⊤ (eqToHom basicOpen_one.symm),
Category.assoc, NatTrans.naturality, ← Cate... | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | 258 | 274 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Yaël Dillies, Yuyang Zhao
-/
import Mathlib.Algebra.Order.Ring.Unbundled.Basic
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Alge... | Mathlib/Algebra/Order/Ring/Defs.lean | 1,198 | 1,204 | |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.RingTheory.Nilpotent.Basic
import Mathlib.RingTheory.UniqueFactorizationDomain.GCDMonoid
import Mathlib.RingTheory.UniqueFactorizationDomain.Multiplici... | rw [mul_assoc, mul_ne_zero_iff] at h0; exact h0.2
namespace Squarefree
| Mathlib/Algebra/Squarefree/Basic.lean | 171 | 174 |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Sébastien Gouëzel, Yury Kudryashov, Yuyang Zhao
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calcu... |
/-- The chain rule.
Note that the function `h₂` is a function on an algebra. If you are looking for the chain rule
| Mathlib/Analysis/Calculus/Deriv/Comp.lean | 232 | 235 |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.SetNotation
/-!
# Properties of unbundled ... | section NoMinOrder
variable [NoMinOrder α]
| Mathlib/Order/UpperLower/Basic.lean | 298 | 301 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Field.Rat
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.GroupWithZero.Un... | simp [cast_def]
| Mathlib/Data/Rat/Cast/Defs.lean | 120 | 121 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Image
/-!
# Cardinality of a finite set
This defines the cardinality of a `Fins... | · rintro ⟨y, rfl⟩
simp
· exact fun h => ⟨x, eq_singleton_iff_unique_mem.2 ⟨hx, fun y hy => h _ hy _ hx⟩⟩
theorem card_le_one_iff : #s ≤ 1 ↔ ∀ {a b}, a ∈ s → b ∈ s → a = b := by
| Mathlib/Data/Finset/Card.lean | 614 | 618 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
/-! # P... | theorem rpow_eq_zero_iff_of_nonneg (hx : 0 ≤ x) : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 := by
simp only [rpow_def_of_nonneg hx]
split_ifs <;> simp [*, exp_ne_zero]
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 64 | 66 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yaël Dillies
-/
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.GroupWithZero.Action.Defs
import Mathlib.Algebra.Order.Group.OrderIso
impor... | Mathlib/Order/Filter/Pointwise.lean | 1,407 | 1,411 | |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Composition.MapComap
import Mathlib.Probability.Martingale.Convergence
import Mathlib.Probability.Process.PartitionFiltration
/-!
# Ker... | /-- Auxiliary lemma for `setIntegral_density`. -/
lemma setIntegral_density_of_measurableSet (hκν : fst κ ≤ ν)
[IsFiniteKernel ν] (n : ℕ) (a : α) {s : Set β} (hs : MeasurableSet s) {A : Set γ}
(hA : MeasurableSet[countableFiltration γ n] A) :
∫ x in A, density κ ν a x s ∂(ν a) = (κ a).real (A ×ˢ s) := by
... | Mathlib/Probability/Kernel/Disintegration/Density.lean | 511 | 516 |
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Riccardo Brasca, Eric Rodriguez
-/
import Mathlib.Data.PNat.Prime
import Mathlib.NumberTheory.Cyclotomic.Basic
import Mathlib.RingTheory.Adjoin.PowerBasis
import Mathlib... | by_cases h1 : n = 1
· rw [h1, one_coe, one_right_iff] at hζ
rw [hζ, show 1 = algebraMap K L 1 by simp, Algebra.norm_algebraMap, one_pow]
| Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean | 289 | 291 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.FunLike.Basic
import Mathlib.Logic.Embedding.Basic
import Mathlib.Order.RelClasses
/-!
# Relation homomorphisms, embeddings, isomorphisms
This f... | def ofUniqueOfIrrefl (r : α → α → Prop) (s : β → β → Prop) [IsIrrefl α r]
[IsIrrefl β s] [Unique α] [Unique β] : r ≃r s :=
⟨Equiv.ofUnique α β, iff_of_false (not_rel_of_subsingleton s _ _)
(not_rel_of_subsingleton r _ _) ⟩
| Mathlib/Order/RelIso/Basic.lean | 750 | 753 |
/-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Lie.Weights.Killing
import Mathlib.LinearAlgebra.RootSystem.Basic
import Mathlib.LinearAlgebra.RootSystem.Reduced
import Mathlib.LinearAlgebra.RootSy... | (chainLength α β : K) • x = ⁅coroot α, x⁆ := by
rw [Nat.cast_smul_eq_nsmul, chainLength_nsmul _ _ hx]
lemma apply_coroot_eq_cast' :
β (coroot α) = ↑(chainLength α β - 2 * chainTopCoeff α β : ℤ) := by
by_cases hα : α.IsZero
· rw [coroot_eq_zero_iff.mpr hα, chainLength, dif_pos hα, hα.eq, chainTopCoeff_zer... | Mathlib/Algebra/Lie/Weights/RootSystem.lean | 82 | 93 |
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical
import Mathlib.Algebra.O... | Mathlib/Data/NNReal/Basic.lean | 1,249 | 1,253 | |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Category.Grp.Basic
/-! The cohomology of a sheaf of groups in degree 1
In this file, we shall define the cohomology in degree 1 of a sheaf
of groups (T... | exact ⟨_, h.symm⟩
trans := by
rintro γ₁ γ₂ γ₂ ⟨α, h⟩ ⟨β, h'⟩
exact ⟨_, h.trans h'⟩
end OneCocycle
variable (G U) in
/-- The cohomology in degree 1 of a presheaf of groups
| Mathlib/CategoryTheory/Sites/NonabelianCohomology/H1.lean | 183 | 191 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Option
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Data.Set.Pairwise.Lattice
/-!
# Partitions of rectangular b... | theorem filter_le (π : Prepartition I) (p : Box ι → Prop) : π.filter p ≤ π := fun J hJ =>
let ⟨hπ, _⟩ := π.mem_filter.1 hJ
⟨J, hπ, le_rfl⟩
theorem filter_of_true {p : Box ι → Prop} (hp : ∀ J ∈ π, p J) : π.filter p = π := by
ext J
simpa using hp J
@[simp]
| Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | 533 | 541 |
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.GroupWithZero.Subgroup
import Mathlib.Data.Finite.Card
import Mathlib.Data.Finite.Pr... | @[to_additive (attr := simp)]
theorem relindex_sup_left [K.Normal] : K.relindex (K ⊔ H) = K.relindex H := by
| Mathlib/GroupTheory/Index.lean | 134 | 135 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Lattice
import Mathlib.Algebra.Order.Group.Defs
import Mathlib.Algebra.... | exact isUpperSet_iUnion₂ fun x _ ↦ ht.smul
@[to_additive]
| Mathlib/Algebra/Order/UpperLower.lean | 56 | 58 |
/-
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, Sébastien Gouëzel
-/
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
import Mathlib.MeasureTheory... | exact hr (pow_eq_zero h)
simp only [f, map_linearMap_addHaar_eq_smul_addHaar μ hf, mul_one, LinearMap.det_smul, map_one]
theorem quasiMeasurePreserving_smul {r : ℝ} (hr : r ≠ 0) :
QuasiMeasurePreserving (r • ·) μ μ := by
| Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 342 | 346 |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Order.Antichain
import Mathlib.Topology.ContinuousOn
/-!
# Left and right continuity
In this file we prove a few lemmas about left and right cont... |
@[deprecated (since := "2024-12-21")] alias nhds_left'_sup_nhds_right' := nhdsLT_sup_nhdsGT
| Mathlib/Topology/Order/LeftRight.lean | 123 | 124 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes, Daniel Weber
-/
import Batteries.Data.Nat.Gcd
import Mathlib.Algebra.GroupWithZero.Associated
import Mathlib.Algebra.Ring.Divisibility.Basic
import Math... | constructor
· exact fun h n hab ↦ pow_dvd_of_le_emultiplicity (le_trans (le_emultiplicity_of_pow_dvd hab) h)
· intro h
unfold emultiplicity
-- aesop? says
split
next h_1 =>
| Mathlib/RingTheory/Multiplicity.lean | 426 | 432 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Logic.Equiv.PartialEquiv
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.Sets.Opens
/-!
# Partial homeomorphisms
This file de... | show _ = U.partialHomeomorphSubtypeCoe hU _
rw [(U.partialHomeomorphSubtypeCoe hU).right_inv hy.2]
end subtypeRestr
| Mathlib/Topology/PartialHomeomorph.lean | 1,356 | 1,359 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Multiset.Fintype
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.FieldTheory.SplittingField.Construction
/... | [IsScalarTower R S k] : IsScalarTower R S (AlgebraicClosure k) :=
Ideal.Quotient.isScalarTower _ _ _
instance instGroupWithZero : GroupWithZero (AlgebraicClosure k) where
__ := Ideal.Quotient.field _
instance instField : Field (AlgebraicClosure k) where
__ := instCommRing _
__ := instGroupWithZero _
nnq... | Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean | 137 | 147 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Simple functions
A function `f` from a measurable ... | _ < _ := add_lt_top.2 ⟨hf, hg⟩
protected theorem map₂ [Zero δ] (hf : f.FinMeasSupp μ) {g : α →ₛ γ} (hg : g.FinMeasSupp μ)
| Mathlib/MeasureTheory/Function/SimpleFunc.lean | 1,093 | 1,095 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.OuterMeasure.Operations
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Outer measures from functions
Given an arbit... |
theorem map_iInf_le {ι β} (f : α → β) (m : ι → OuterMeasure α) :
map f (⨅ i, m i) ≤ ⨅ i, map f (m i) :=
(map_mono f).map_iInf_le
| Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean | 402 | 405 |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subgroup.Ker
/-!
# Basic results on subgroups
We prove basic results... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 3,038 | 3,039 | |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Homology.Single
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
/-!
# Homology is an additive functor
When `V` is preadditive, `Homolog... | Mathlib/Algebra/Homology/Additive.lean | 331 | 334 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.Dedup
/-!
# The fold operation for a commutative associative operation over a multiset.
-/
namespace Multiset
variable {α β : Type*}
... | Mathlib/Data/Multiset/Fold.lean | 102 | 105 | |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Nat.Fib.Basic
/-!
# Zeckendorf's Theorem
This file proves Zeckendorf's theorem: Every natural number can be written uniquely as a sum of
distinct no... | lemma isZeckendorfRep_zeckendorf : ∀ n, (zeckendorf n).IsZeckendorfRep
| 0 => by simp only [zeckendorf_zero, IsZeckendorfRep_nil]
| n + 1 => by
rw [zeckendorf_succ, IsZeckendorfRep, List.cons_append]
refine (isZeckendorfRep_zeckendorf _).cons' (fun a ha ↦ ?_)
obtain h | h := eq_zero_or_pos (n + 1 - fib ... | Mathlib/Data/Nat/Fib/Zeckendorf.lean | 136 | 148 |
/-
Copyright (c) 2023 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Prod
/-!
# Products of finite measures and probability measures
This file i... | @[simp] lemma map_fst_prod : (μ.prod ν).map Prod.fst = ν univ • μ := by ext; simp
| Mathlib/MeasureTheory/Measure/FiniteMeasureProd.lean | 70 | 70 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Batteries.Data.Nat.Gcd
import Mathlib.Algebra.BigOperators.Group.Finset.Defs
/-! # Lemmas about coprimality with big products.
These lemmas ... | Coprime m.prod k ↔ ∀ n ∈ m, Coprime n k := by
induction m using Quotient.inductionOn; simpa using coprime_list_prod_left_iff
| Mathlib/Data/Nat/GCD/BigOperators.lean | 28 | 30 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib... |
theorem pow_eq_zero_of_mem {I : Ideal R} {n m : ℕ} (hnI : I ^ n = 0) (hmn : n ≤ m) {x : R}
(hx : x ∈ I) : x ^ m = 0 := by
simpa [hnI] using pow_le_pow_right hmn <| pow_mem_pow hx m
end Semiring
| Mathlib/RingTheory/Ideal/Operations.lean | 372 | 378 |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Function.Jacobian
imp... | (hnng : ∀ᵐ x ∂μ, 0 ≤ f x) (hbounded : ∀ᶠ i in l, (∫ x in φ i, f x ∂μ) ≤ I) : Integrable f μ :=
hφ.integrable_of_integral_norm_bounded I hfi <| hbounded.mono fun _i hi =>
(integral_congr_ae <| ae_restrict_of_ae <| hnng.mono fun _ => Real.norm_of_nonneg).le.trans hi
theorem AECover.integrable_of_integral_tends... | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 461 | 475 |
/-
Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Algebra.Algebra.Spectrum.Basic
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathl... |
/-- For an endomorphism of a Noetherian module, the maximal eigenspace is always of the form kernel
`(f - μ • id) ^ k` for some `k`. -/
theorem maxGenEigenspace_eq [IsNoetherian R M] (f : End R M) (μ : R) :
maxGenEigenspace f μ = f.genEigenspace μ (maxGenEigenspaceIndex f μ) :=
genEigenspace_top_eq_maxUnifEigens... | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 529 | 550 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
... | (div_le_one (pos_denom x)).2 <| le_add_of_nonneg_right (nonneg _)
| Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean | 167 | 168 |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Analysis.InnerProductSpace.... |
-- simp normal form is `fourier_add'`
theorem fourier_add {m n : ℤ} {x : AddCircle T} : fourier (m+n) x = fourier m x * fourier n x := by
simp_rw [fourier_apply, add_zsmul, toCircle_add, Circle.coe_mul]
@[simp]
| Mathlib/Analysis/Fourier/AddCircle.lean | 154 | 159 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.GroupWithZero.Invertible
import Mathlib.Algebra.Ring.Defs
/-!
# Theorems about invertible elements in rings
-/
universe u
variable {R : Type u}
/... |
@[simp]
theorem one_sub_invOf_two [Ring R] [Invertible (2 : R)] : 1 - (⅟ 2 : R) = ⅟ 2 :=
| Mathlib/Algebra/Ring/Invertible.lean | 25 | 27 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Logic.Function.Basic
/-!
# Semiconjugate and commuting maps
We define the following predicates:
* `Function.Semiconj`: `f : α → β` semiconjugates ... | | none => rfl
| some _ => congr_arg some <| h _
end Semiconj
| Mathlib/Logic/Function/Conjugate.lean | 92 | 96 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.FinMeasAdditive
/-!
# Extension of a linear function from indicators to L1
Give... | setToL1S (fun s => c • T s) f = c • setToL1S T f :=
SimpleFunc.setToSimpleFunc_smul_left T c _
theorem setToL1S_smul_left' (T T' : Set α → E →L[ℝ] F) (c : ℝ)
(h_smul : ∀ s, MeasurableSet s → μ s < ∞ → T' s = c • T s) (f : α →₁ₛ[μ] E) :
setToL1S T' f = c • setToL1S T f :=
SimpleFunc.setToSimpleFunc_smul... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 152 | 179 |
/-
Copyright (c) 2023 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.Localization.Module
import Mathlib.RingTheory.Norm.Basic
import Mathlib.RingTheory.Discriminant
/-!
# Field/algebra norm / trace and localizatio... | (algebraMap R Rₘ).mapMatrix (leftMulMatrix b a) =
leftMulMatrix (b.localizationLocalization Rₘ M Sₘ) (algebraMap S Sₘ a) := by
ext i j
simp only [Matrix.map_apply, RingHom.mapMatrix_apply, leftMulMatrix_eq_repr_mul, ← map_mul,
Basis.localizationLocalization_apply, Basis.localizationLocalization_repr_alg... | Mathlib/RingTheory/Localization/NormTrace.lean | 50 | 56 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Joey van Langen, Casper Putz
-/
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.Field.ZMod
import Mathlib.Data.Nat.Prime.In... |
theorem X_pow_card_pow_sub_X_ne_zero (hn : n ≠ 0) (hp : 1 < p) : (X ^ p ^ n - X : K'[X]) ≠ 0 :=
X_pow_card_sub_X_ne_zero K' <| Nat.one_lt_pow hn hp
end
theorem roots_X_pow_card_sub_X : roots (X ^ q - X : K[X]) = Finset.univ.val := by
classical
have aux : (X ^ q - X : K[X]) ≠ 0 := X_pow_card_sub_X_ne_zero K F... | Mathlib/FieldTheory/Finite/Basic.lean | 423 | 431 |
/-
Copyright (c) 2023 Yaël Dillies, Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Data.Finset.Lattice.Prod
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Or... | @[simps! isClosed]
def latticeClosure : ClosureOperator (Set α) :=
.ofCompletePred IsSublattice fun _ ↦ isSublattice_sInter
@[simp] lemma subset_latticeClosure : s ⊆ latticeClosure s := latticeClosure.le_closure _
| Mathlib/Order/SupClosed.lean | 426 | 430 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
/-!
# Natural numbers with infinity
The... | revert h₂
contrapose!
| Mathlib/Data/Nat/PartENat.lean | 712 | 713 |
/-
Copyright (c) 2024 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Category.CoalgebraCat.Basic
import Mathlib.Algebra.Category.AlgebraCat.Basic
import Mathlib.RingTheory.Bialgebra.Equiv
/-!
# The categor... | instance category : Category (BialgebraCat.{v} R) where
Hom X Y := Hom X Y
id X := ⟨BialgHom.id R X⟩
| Mathlib/Algebra/Category/BialgebraCat/Basic.lean | 66 | 68 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.PropInstances
import Mathlib.Order.GaloisConnection.Defs
/-!
# Heyting algebras
This file defines Heyting, co-Heyting and bi-Heyting algebras.
A H... | theorem toDual_compl (a : α) : toDual aᶜ = ￢toDual a :=
rfl
| Mathlib/Order/Heyting/Basic.lean | 765 | 766 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 336 | 337 | |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.InitTail
/-!
# Truncated Witt vectors
The ring of truncated Witt vectors (of length `n`) is a quotient of the... |
/-- We can turn a truncated Witt vector `x` into a Witt vector
| Mathlib/RingTheory/WittVector/Truncated.lean | 90 | 91 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Nat.Totient
import Mathlib.Data.ZMod.Aut
import Mathlib.Data.ZMod.QuotientGroup
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Sub... | theorem isCyclic_of_injective [IsCyclic G'] (f : G →* G') (hf : Function.Injective f) :
IsCyclic G :=
isCyclic_of_surjective (MonoidHom.ofInjective hf).symm (MonoidHom.ofInjective hf).symm.surjective
@[to_additive]
lemma Subgroup.isCyclic_of_le {H H' : Subgroup G} (h : H ≤ H') [IsCyclic H'] : IsCyclic H :=
isC... | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 299 | 330 |
/-
Copyright (c) 2022 Pierre-Alexandre Bazin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pierre-Alexandre Bazin
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.ZMod
import Mathlib.GroupTheory.Torsion
import Mathlib.LinearAlgebra.Isomorphism... | (ht : IsTorsionBySet R M t) : IsTorsionBySet R M s :=
fun m r ↦ @ht m ⟨r, h r.2⟩
@[simp]
| Mathlib/Algebra/Module/Torsion.lean | 304 | 307 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Bases
import Mathlib.Topology.Compactness.LocallyCompact
import Mathlib.Topology.Compactness.LocallyFinite
/... | K.iUnion_eq'
theorem exists_mem (x : X) : ∃ n, x ∈ K n :=
iUnion_eq_univ_iff.1 K.iUnion_eq x
theorem exists_mem_nhds (x : X) : ∃ n, K n ∈ 𝓝 x := by
rcases K.exists_mem x with ⟨n, hn⟩
| Mathlib/Topology/Compactness/SigmaCompact.lean | 380 | 386 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Basic
/-!
# Bind operation for multisets
This file defines a few basic operations on `Multiset`, notably the mona... | count_sum
theorem le_bind {α β : Type*} {f : α → Multiset β} (S : Multiset α) {x : α} (hx : x ∈ S) :
| Mathlib/Data/Multiset/Bind.lean | 186 | 188 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
/-!
# Special values of Hurwitz and Riemann zeta functions
This file gives t... | positivity
have h3 : Gammaℂ (2 * k + 1) ≠ 0 := by
refine mul_ne_zero (mul_ne_zero two_ne_zero ?_) (Gamma_ne_zero h1)
simp [pi_ne_zero]
rw [(by simp : -(2 * k : ℂ) = 1 - (2 * k + 1)),
hurwitzZetaOdd_one_sub _ h1, ← Gammaℂ, sinZeta_two_mul_nat_add_one' hk hx, ← mul_assoc,
← mul_div_assoc, mul_asso... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 148 | 169 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
/-!
# Power function... |
theorem one_le_rpow {x : ℝ≥0∞} {z : ℝ} (hx : 1 ≤ x) (hz : 0 < z) : 1 ≤ x ^ z := by
cases x
· simp [top_rpow_of_pos hz]
· simp only [one_le_coe_iff, some_eq_coe] at hx
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 850 | 854 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | Mathlib/Data/Ordmap/Ordset.lean | 991 | 995 | |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Anatole Dedecker
-/
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.Seminorm
import Mathlib.Data.Real.Sqrt
import Mathlib.Topology.Algebra.Equicontinuit... | · rintro ⟨-, ⟨s, r, hr, rfl⟩, hV⟩
exact ⟨s, r, hr, hV⟩
· rintro ⟨s, r, hr, hV⟩
exact ⟨_, ⟨s, r, hr, rfl⟩, hV⟩
| Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | 291 | 294 |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.MonoidLocalization.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matri... | split_ifs with h
· -- i = j: this entry should be `A.det`
| Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 102 | 103 |
/-
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, Andrew Yang, Joël Riou
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
import Mathlib.CategoryTheory.Monoid... | (shiftFunctorComm C m₁ (m₂ + m₃)).hom.app X ≫
((shiftFunctorAdd C m₂ m₃).hom.app X)⟦m₁⟧' =
(shiftFunctorAdd C m₂ m₃).hom.app (X⟦m₁⟧) ≫
((shiftFunctorComm C m₁ m₂).hom.app X)⟦m₃⟧' ≫
(shiftFunctorComm C m₁ m₃).hom.app (X⟦m₂⟧) := by
rw [← cancel_mono ((shiftFunctorComm C m₁ m₃).inv.app (X⟦m₂⟧)),
← ... | Mathlib/CategoryTheory/Shift/Basic.lean | 637 | 660 |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.CartanSubalgebra
import Mathlib.Algebra.Lie.Weights.Basic
/-!
# Weights and roots of Lie modules and Lie algebras with respect to Cartan subalge... | rfl
change (g ^ k).comp (H : Submodule R L).subtype ⟨x, hx⟩ = 0
rw [Module.End.commute_pow_left_of_commute hfg k]
have h := iterate_toEnd_mem_lowerCentralSeries R H H y ⟨x, hx⟩ k
rw [hk, LieSubmodule.mem_bot] at h
simp only [Submodule.subtype_apply, Function.comp_apply, Module.End.pow_apply, LinearMap.coe... | Mathlib/Algebra/Lie/Weights/Cartan.lean | 181 | 202 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.Dedup
import Mathlib.Data.List.Infix
/-!
# Preparations for defining operations on `Finset`.
The operations here ignore multiplicities,... | Mathlib/Data/Multiset/FinsetOps.lean | 279 | 280 | |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Functor.Category
import Mathlib.CategoryTheory.Iso
/-!
# Natural isomorphisms
For the most ... | theorem cancel_natIso_inv_right_assoc {W X X' : D} {Y : C} (f : W ⟶ X) (g : X ⟶ G.obj Y)
(f' : W ⟶ X') (g' : X' ⟶ G.obj Y) :
| Mathlib/CategoryTheory/NatIso.lean | 160 | 161 |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.Order.Compact
import Mathlib.Topology.MetricSpace.ProperSpace
import M... |
open Lean Meta Qq Function
/-- Extension for the `positivity` tactic: the diameter of a set is always nonnegative. -/
@[positivity Metric.diam _]
def evalDiam : PositivityExt where eval {u α} _zα _pα e := do
match u, α, e with
| 0, ~q(ℝ), ~q(@Metric.diam _ $inst $s) =>
| Mathlib/Topology/MetricSpace/Bounded.lean | 515 | 522 |
/-
Copyright (c) 2023 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory.... | v v
A ---- dZero ---> Fun(G, A)
```
where the vertical arrows are `zeroCochainsLequiv` and `oneCochainsLequiv` respectively.
-/
theorem dZero_comp_eq : dZero A ∘ₗ (zeroCochainsLequiv A) =
oneCochainsLequiv A ∘ₗ ((inhomogeneousCochains A).d 0 1).hom := by
| Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 145 | 151 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Topology.ContinuousOn
/-!
### Locally finite families of sets
We say that a family of sets in a topological s... | intersection of the complements to `f i`, `x ∉ f i`, is a neighbourhood of `x`. -/
theorem iInter_compl_mem_nhds (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)) (x : X) :
(⋂ (i) (_ : x ∉ f i), (f i)ᶜ) ∈ 𝓝 x := by
refine IsOpen.mem_nhds ?_ (mem_iInter₂.2 fun i => id)
suffices IsClosed (⋃ i : { i // x ∉ f i },... | Mathlib/Topology/LocallyFinite.lean | 121 | 126 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 2,876 | 2,887 | |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Sophie Morel, Yury Kudryashov
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Logic.Embedding.Basic
import Mathlib.Data.Fintype.Car... | simp [smul_add]
map_smul' c z := by
ext m
simp [smul_smul, mul_comm]
left_inv z := by simp
right_inv f := f.mkPiRing_apply_one_eq_self
norm_map' := norm_mkPiRing
| Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean | 827 | 834 |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
import Mathlib.Tactic.AdaptationNote
... | (hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k ∈ Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_const_mul]
refine integral_mono_of_nonn... | Mathlib/Probability/Martingale/Upcrossing.lean | 601 | 654 |
/-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Topology.IsLocalHomeomorph
import Mathlib.Topology.FiberBundle.Basic
/-!
# Covering Maps
This file defines covering maps.
## Main definitions
* `... |
variable {f}
| Mathlib/Topology/Covering.lean | 140 | 141 |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Set.Lattice
import Mathlib.Order.ConditionallyCompleteLattice.Defs
/-!
# Theory of conditionally complete lattices
A conditionally complet... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 1,148 | 1,150 | |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Functors from a category to a category with a shift
Given a category `C`, and a category `D` equipped with a shift by a mo... | lemma hom_inv_id_hom_app (e : F ≅ G) (n : A) (X : C) :
(e.hom.hom n).app X ≫ (e.inv.hom n).app X = 𝟙 _ := by
rw [← NatTrans.comp_app, hom_inv_id_hom, NatTrans.id_app]
| Mathlib/CategoryTheory/Shift/SingleFunctors.lean | 183 | 186 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
/-!
# Hausdorff distance
The Hausdorff distance on subsets of a... | · exact Or.inl ⟨hs, ht⟩
· rw [hausdorffEdist_comm] at fin
| Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 404 | 405 |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.Pseudo.Basic
import Mathlib.Topology.MetricSpace.Pseudo.Le... | is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
uniformity are defeq in the metric space and the emetric space. -/
| Mathlib/Topology/MetricSpace/Basic.lean | 82 | 83 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.A... | Mathlib/RingTheory/AdjoinRoot.lean | 914 | 923 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.End
import Mathlib.Data.Finset.N... | (h : f.IsSwap) : (ofSubtype f).IsSwap :=
let ⟨⟨x, hx⟩, ⟨y, hy⟩, hxy⟩ := h
⟨x, y, by
simp only [Ne, Subtype.ext_iff] at hxy
exact hxy.1, by
rw [hxy.2, ofSubtype_swap_eq]⟩
theorem ne_and_ne_of_swap_mul_apply_ne_self {f : Perm α} {x y : α} (hy : (swap x (f x) * f) y ≠ y) :
f y ≠ y ∧ y ≠ x := by
... | Mathlib/GroupTheory/Perm/Support.lean | 218 | 236 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
deprecated_module (since := "2025-04-06")
| Mathlib/MeasureTheory/Integral/FundThmCalculus.lean | 607 | 620 | |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
/-!
## The Verschiebung operator
## References
* [Hazewinkel, *Witt Vectors*... | theorem ghostComponent_verschiebungFun [hp : Fact p.Prime] (x : 𝕎 R) (n : ℕ) :
ghostComponent (n + 1) (verschiebungFun x) = p * ghostComponent n x := by
simp only [ghostComponent_apply, aeval_wittPolynomial]
rw [Finset.sum_range_succ', verschiebungFun_coeff, if_pos rfl,
| Mathlib/RingTheory/WittVector/Verschiebung.lean | 58 | 61 |
/-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.MeasureTheory.Measure.MeasureSpace
import Mathlib.MeasureTheory.Measure.Prod
/-!
# The multiplicative and additive convolution of measures
In this file w... | · simp only [Function.comp_def, one_mul, map_id']
fun_prop
/-- Convolution of a measure μ with the dirac measure at 1 returns μ. -/
@[to_additive (attr := simp) "Convolution of a measure μ with the dirac measure at 0 returns μ."]
theorem mconv_dirac_one [MeasurableMul₂ M]
(μ : Measure M) [SFinite μ] : μ ∗ (Mea... | Mathlib/MeasureTheory/Group/Convolution.lean | 53 | 65 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 2,462 | 2,463 | |
/-
Copyright (c) 2018 Rohan Mitta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rohan Mitta, Kevin Buzzard, Alistair Tucker, Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topolo... |
end LipschitzOnWith
| Mathlib/Topology/MetricSpace/Lipschitz.lean | 291 | 293 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.Algebra.Ring.Defs
import Mathlib.Data.Nat.Lattice
/-!
# Definition... | IsRadical y ↔ ∀ x, y ∣ x ^ k → y ∣ x :=
⟨(· k), k.pow_imp_self_of_one_lt hk _ fun _ _ h ↦ .inl (dvd_mul_of_dvd_left h _)⟩
namespace Commute
| Mathlib/RingTheory/Nilpotent/Defs.lean | 229 | 232 |
/-
Copyright (c) 2021 Alex Kontorovich and Heather Macbeth and Marc Masdeu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth, Marc Masdeu
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.GeneralLinearG... | Mathlib/NumberTheory/Modular.lean | 337 | 337 | |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Indicator
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
import... | theorem sum_centroidWeightsIndicator_eq_one_of_card_eq_add_one [CharZero k] [Fintype ι] {n : ℕ}
(h : #s = n + 1) : ∑ i, s.centroidWeightsIndicator k i = 1 := by
| Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 847 | 848 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.Set.BooleanAlgebra
import Mathlib.Tactic.AdaptationNote
/-!
# Relations
This file defines bundled relations. A relation between `α` and `β` is a f... |
@[simp]
theorem comp_left_bot (r : Rel α β) : (⊥ : Rel γ α) • r = ⊥ := by
ext x y
| Mathlib/Data/Rel.lean | 112 | 115 |
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 305 | 305 | |
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical
import Mathlib.Algebra.O... | Mathlib/Data/NNReal/Basic.lean | 328 | 332 | |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Sébastien Gouëzel, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.Normed.Lp.PiLp
import Mathlib.LinearAlgebra.FiniteDimensi... | (v.toOrthonormalBasis hv : ι → E) = (v : ι → E) :=
calc
(v.toOrthonormalBasis hv : ι → E) = ((v.toOrthonormalBasis hv).toBasis : ι → E) := by
| Mathlib/Analysis/InnerProductSpace/PiL2.lean | 516 | 518 |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash, Deepro Choudhury, Scott Carnahan
-/
import Mathlib.LinearAlgebra.PerfectPairing.Basic
import Mathlib.LinearAlgebra.Reflection
/-!
# Root data and root systems
This file con... |
@[simp]
lemma reflection_inv :
(P.reflection i)⁻¹ = P.reflection i :=
rfl
| Mathlib/LinearAlgebra/RootSystem/Defs.lean | 251 | 255 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Nat.Even
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Cast.Commute
import Mathlib.Data.Set.Operations
import Mathlib.Logic.Fu... | Mathlib/Algebra/Ring/Parity.lean | 460 | 462 | |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.LocallyConvex.Basic
/-!
# Balanced Core and Balanced Hull
## Main definitions
* `balancedCore`: The largest balanced subset of a set `s`.
* `bala... |
end Topology
| Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean | 271 | 276 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Operations
import Mathlib.Order.Basic
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.B... | simp only [mem_dite, mem_empty_iff_false, imp_false]
exact ⟨fun h => ⟨h.1, h.2 h.1⟩, fun ⟨h₁, h₂⟩ => ⟨fun h => h₁ h, fun _ => h₂⟩⟩
| Mathlib/Data/Set/Basic.lean | 1,309 | 1,310 |
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Group.Submonoid.BigOperators
import Mathlib.GroupTheory.Subsemigroup.Center
import Mathlib.RingTheory... | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | 849 | 851 | |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alex Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Integral.Lebe... | (f : α → E) (hf : Integrable f μ) : ∫ x, f x ∂μ = ∑' g : G, ∫ x in s, f (g • x) ∂μ :=
(integral_eq_tsum' h f hf).trans ((Equiv.inv G).tsum_eq (fun g ↦ ∫ (x : α) in s, f (g • x) ∂μ))
@[to_additive]
theorem setIntegral_eq_tsum (h : IsFundamentalDomain G s μ) {f : α → E} {t : Set α}
(hf : IntegrableOn f t μ) : ... | Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 389 | 411 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.RowCol
import Mathli... | _ = (-1 : R) ^ (i : ℕ) * (Perm.sign i.cycleRange⁻¹) * det A := by simp [-Int.units_mul_self]
rw [this, mul_assoc]
congr
rw [← det_permute, det_succ_row_zero]
refine Finset.sum_congr rfl fun j _ => ?_
rw [mul_assoc, Matrix.submatrix_apply, submatrix_submatrix, id_comp, Function.comp_def, id]
congr
· ... | Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | 774 | 791 |
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