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 |
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/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Simon Hudon
-/
import Batteries.Data.List.Lemmas
import Mathlib.Tactic.TypeStar
/-!
# The Following Are Equivalent
This file allows to state that all propositions in ... | length makes it happier.
Example:
```lean
example (P₁ P₂ P₃ : ℕ → Prop) (H : ∀ n, [P₁ n, P₂ n, P₃ n].TFAE) :
[∃ n, P₁ n, ∃ n, P₂ n, ∃ n, P₃ n].TFAE :=
| Mathlib/Data/List/TFAE.lean | 91 | 96 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Operations
/-!
# Results about division in extended non-negative reals
This file establishes basic properties related t... | norm_cast
rw [inv_pow]
protected theorem mul_inv_cancel (h0 : a ≠ 0) (ht : a ≠ ∞) : a * a⁻¹ = 1 := by
lift a to ℝ≥0 using ht
norm_cast at h0; norm_cast
exact mul_inv_cancel₀ h0
protected theorem inv_mul_cancel (h0 : a ≠ 0) (ht : a ≠ ∞) : a⁻¹ * a = 1 :=
| Mathlib/Data/ENNReal/Inv.lean | 86 | 94 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
import Mathlib.Data.Quot
/-!
# List rotation
This file proves basic results about `List.r... | @[simp]
theorem cyclicPermutations_eq_singleton_iff {l : List α} {x : α} :
| Mathlib/Data/List/Rotate.lean | 583 | 584 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Limits.FunctorCategory.EpiMono
import Mathlib.Tactic.CategoryTheory.Elementwise
import Mathlib.Categ... | /-- A morphism factors through the sheafification of the image presheaf. -/
@[simps!]
def Subpresheaf.toRangeSheafify (f : F' ⟶ F) : F' ⟶ ((Subpresheaf.range f).sheafify J).toPresheaf :=
toRange f ≫ Subpresheaf.homOfLe ((range f).le_sheafify J)
/-- The image sheaf of a morphism between sheaves, defined to be the sh... | Mathlib/CategoryTheory/Sites/Subsheaf.lean | 193 | 199 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap
import Mathlib.Topology.Spectral.Hom
import Mathlib.AlgebraicGeometry.Limits
/-!
# Quasi-compact morphisms
A morp... | lemma QuasiCompact.compactSpace_of_compactSpace {X Y : Scheme.{u}} (f : X ⟶ Y) [QuasiCompact f]
[CompactSpace Y] : CompactSpace X := by
constructor
rw [← Set.preimage_univ (f := f.base)]
exact QuasiCompact.isCompact_preimage _ isOpen_univ CompactSpace.isCompact_univ
instance quasiCompact_isStableUnderComposi... | Mathlib/AlgebraicGeometry/Morphisms/QuasiCompact.lean | 154 | 175 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.SimplicialObject.Split
import Mathlib.AlgebraicTopology.DoldKan.Degeneracies
import Mathlib.AlgebraicTopology.DoldKan.FunctorN
/-!
# Split si... |
@[reassoc (attr := simp)]
theorem cofan_inj_πSummand_eq_zero [HasZeroMorphisms C] {Δ : SimplexCategoryᵒᵖ} (A B : IndexSet Δ)
| Mathlib/AlgebraicTopology/DoldKan/SplitSimplicialObject.lean | 47 | 49 |
/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Subobject.WellPowered
import Mathlib.CategoryTheory.Comma.LocallySmall
import Mathlib.CategoryTheory.Limits.Preserves.Finite
import Mathli... | {A : StructuredArrow S T} :
∀ P : Subobject A, ∃ q, q ≫ T.map (projectSubobject P).arrow = A.hom :=
Subobject.ind _ fun P f hf =>
⟨P.hom ≫ T.map (Subobject.underlyingIso _).inv, by
dsimp
simp [← T.map_comp]⟩
| Mathlib/CategoryTheory/Subobject/Comma.lean | 63 | 68 |
/-
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, Mario Carneiro
-/
import Mathlib.Algebra.Field.IsField
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp.L... | Mathlib/RingTheory/Ideal/Basic.lean | 862 | 868 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Sigma.Basic
import Mathlib.Algebra.Order.Ring.Nat
/-!
# A computable model of ZFA without infinity
In this file we define finite hereditary list... | · rintro _ ⟨⟩
· intro a l IH₁ IH₂
simpa using ⟨IH₁, IH₂⟩
instance instSetoidLists : Setoid (Lists α) :=
| Mathlib/SetTheory/Lists.lean | 295 | 299 |
/-
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.MeasureTheory.MeasurableSpace.CountablyGenerated
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.Measure.Typeclasses... | theorem tsum_indicator_apply_singleton [Countable α] [MeasurableSingletonClass α] (μ : Measure α)
(s : Set α) (hs : MeasurableSet s) : (∑' x : α, s.indicator (fun x => μ {x}) x) = μ s := by
| Mathlib/MeasureTheory/Measure/Dirac.lean | 117 | 118 |
/-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton
-/
import Mathlib.Topology.Hom.ContinuousEval
import Mathlib.Topology.ContinuousMap.Basic
import Mathlib.Topology.Separation.Regular
/-!
# The compact-open topology
In this ... | continuous_compactOpen.2 fun _K hK _U hU ↦ isOpen_setOf_mapsTo hK (hU.preimage g.2)
/-- If `g : C(Y, Z)` is a topology inducing map,
then the composition `ContinuousMap.comp g : C(X, Y) → C(X, Z)` is a topology inducing map too. -/
| Mathlib/Topology/CompactOpen.lean | 97 | 100 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 1,952 | 1,957 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Group.Subgroup.Ker
import Mathlib.Algebra.Module.Submodule.... | rw [← Module.End.mul_eq_comp, h.eq, Module.End.mul_eq_comp]
exact ker_le_ker_comp f g
| Mathlib/Algebra/Module/Submodule/Ker.lean | 92 | 93 |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Kim Morrison
-/
import Mathlib.CategoryTheory.Limits.ExactFunctor
import Mathlib.CategoryTheory.Limits.Preserves.Finite
import Mathlib.CategoryTheory.Preadditive.Biproducts
imp... |
namespace Equivalence
variable {C D : Type*} [Category C] [Category D] [Preadditive C] [Preadditive D]
instance inverse_additive (e : C ≌ D) [e.functor.Additive] : e.inverse.Additive where
| Mathlib/CategoryTheory/Preadditive/AdditiveFunctor.lean | 184 | 189 |
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Grading
import Mathlib.Algebra.Module.Opposite
/-!
# Conjugations
This file defines the grade reversal and grade involution f... | | algebraMap r => exact AlgHom.commutes _ _
| add x y _hx _hy ihx ihy =>
rw [map_add, ihx, ihy]
| Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean | 302 | 304 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Violeta Hernández Palacios, Grayson Burton, Floris van Doorn
-/
import Mathlib.Order.Antisymmetrization
import Mathlib.Order.Hom.WithTopBot
import Mathlib.Order.Interval.Se... |
variable [Preorder α]
lemma exists_covBy_of_wellFoundedLT [wf : WellFoundedLT α] ⦃a : α⦄ (h : ¬ IsMax a) :
| Mathlib/Order/Cover.lean | 612 | 615 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Measure.Content
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Topology.Algebra.Group.Compact
/-!
# Haar measure
In this fi... | index K V ≤ index K' V := by
rcases index_elim hK' hV with ⟨s, h1s, h2s⟩
apply Nat.sInf_le; rw [mem_image]; exact ⟨s, Subset.trans h h1s, h2s⟩
@[to_additive addIndex_union_le]
theorem index_union_le (K₁ K₂ : Compacts G) {V : Set G} (hV : (interior V).Nonempty) :
index (K₁.1 ∪ K₂.1) V ≤ index K₁.1 V + index... | Mathlib/MeasureTheory/Measure/Haar/Basic.lean | 185 | 196 |
/-
Copyright (c) 2020 Heather Macbeth, Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Patrick Massot
-/
import Mathlib.Algebra.Group.Subgroup.Order
import Mathlib.Algebra.Order.Archimedean.Basic
/-!
# Archimedean groups
This file prov... | exists a minimal element `a` of `H ∩ G_{>1}` then `H` is generated by `a`. -/
@[to_additive AddSubgroup.cyclic_of_min "Given a subgroup `H` of a decidable linearly ordered
archimedean abelian group `G`, if there exists a minimal element `a` of `H ∩ G_{>0}` then `H` is
generated by `a`. "]
theorem Subgroup.cyclic_of_min... | Mathlib/GroupTheory/Archimedean.lean | 40 | 54 |
/-
Copyright (c) 2023 Martin Dvorak. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Martin Dvorak
-/
import Mathlib.Computability.Language
/-!
# Context-Free Grammars
This file contains the definition of a context-free grammar, which is a grammar that has a single
no... | obtain ⟨w, l, r⟩ := hguv
exact ⟨w, l, r.nonterminal_input_mem⟩
lemma derives_nonterminal {t : g.NT} (hgt : ∀ r ∈ g.rules, r.input ≠ t)
| Mathlib/Computability/ContextFreeGrammar.lean | 213 | 216 |
/-
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.... | Mathlib/Algebra/Order/UpperLower.lean | 314 | 316 | |
/-
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.Group.Nat.Units
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.GroupWi... | Mathlib/Data/Nat/GCD/Basic.lean | 331 | 344 | |
/-
Copyright (c) 2022 Jake Levinson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jake Levinson
-/
import Mathlib.Combinatorics.Young.YoungDiagram
/-!
# Semistandard Young tableaux
A semistandard Young tableau is a filling of a Young diagram by natural numbers, suc... | (h : entry' = T) : SemistandardYoungTableau μ where
entry := entry'
row_weak' := h.symm ▸ T.row_weak'
col_strict' := h.symm ▸ T.col_strict'
zeros' := h.symm ▸ T.zeros'
| Mathlib/Combinatorics/Young/SemistandardTableau.lean | 86 | 90 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Find
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
/-!
# Coinductive formalization of unbounded computations.
This fil... | Mathlib/Data/Seq/Computation.lean | 1,121 | 1,133 | |
/-
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, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... | end Disjoint
| Mathlib/Data/Set/Lattice.lean | 1,245 | 1,246 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.WSeq.Relation
/-!
# Parallel computation
Parallel computation of a computable sequence of computations by
a diagonal enumeration.
The imp... | induction' l with c l IH <;> simp
rw [IH]
cases List.foldr _ _ _
· simp
· cases destruct c <;> simp
simp only [BisimO, destruct_map, lmap, rmap, corec_eq, parallel.aux1.eq_1]
rw [this]
rcases parallel.aux2 l with a | l' <;> simp
induction' S using WSeq.rec... | Mathlib/Data/Seq/Parallel.lean | 269 | 293 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Group.Pointwise.Finset.Basic
import Mathlib.Algebra.Group.Pointwise.Set.B... | theorem span_mul_span : span R S * span R T = span R (S * T) := by
rw [mul_eq_map₂]; apply map₂_span_span
lemma mul_def : M * N = span R (M * N : Set A) := by simp [← span_mul_span]
| Mathlib/Algebra/Algebra/Operations.lean | 399 | 402 |
/-
Copyright (c) 2023 Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémi Bottinelli
-/
import Mathlib.Data.Set.Function
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.EMetricSpace.BoundedVariation
/-!
# Constant speed
This file defines... | (y - x) * l = l * (y - x) := by rw [mul_comm]
_ = variationOnFromTo (f ∘ φ) s x y := (hfφ.2 xs ys).symm
_ = variationOnFromTo f (φ '' s) (φ x) (φ y) :=
(variationOnFromTo.comp_eq_of_monotoneOn f φ φm xs ys)
_ = l' * (φ y - φ x) := (hf.2 ⟨x, xs, rfl⟩ ⟨y, ys, rfl⟩)
_ = (φ y - φ x) * l' := by rw ... | Mathlib/Analysis/ConstantSpeed.lean | 176 | 190 |
/-
Copyright (c) 2022 Daniel Roca González. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Daniel Roca González
-/
import Mathlib.Analysis.InnerProductSpace.Dual
/-!
# The Lax-Milgram Theorem
We consider a Hilbert space `V` over `ℝ`
equipped with a bounded bilinear f... |
theorem isClosed_range (coercive : IsCoercive B) : IsClosed (range B♯ : Set V) := by
rcases coercive.antilipschitz with ⟨_, _, antilipschitz⟩
exact antilipschitz.isClosed_range B♯.uniformContinuous
| Mathlib/Analysis/InnerProductSpace/LaxMilgram.lean | 74 | 77 |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | Mathlib/Data/Fin/Basic.lean | 1,444 | 1,444 | |
/-
Copyright (c) 2022 Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémi Bottinelli, Junyan Xu
-/
import Mathlib.Algebra.Group.Subgroup.Defs
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathlib... | mul := fun hp _ hq ↦ by
rw [mem_iInter₂] at hp hq ⊢
exact fun S hS => S.mul (hp S hS) (hq S hS) }⟩
| Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 207 | 209 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Orientation
/-!
# Orientations of real inner product spaces.
Th... | simp [e.toBasis.orientation_eq_iff_det_pos,
(e.det_to_matrix_orthonormalBasis_real f).resolve_right h]
variable {e f}
| Mathlib/Analysis/InnerProductSpace/Orientation.lean | 65 | 69 |
/-
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.Topology.ContinuousMap.Bounded.ArzelaAscoli
import Mathlib.Topology.ContinuousMap.Bounded.Normed
import Mathlib.Topology.MetricSpace.Gluing
impor... | /-- To apply Arzela-Ascoli, we need to check that the set of candidates is closed and
equicontinuous. Equicontinuity follows from the Lipschitz control, we check closedness. -/
private theorem closed_candidatesB : IsClosed (candidatesB X Y) := by
have I1 : ∀ x y, IsClosed { f : Cb X Y | f (inl x, inl y) = dist x y } ... | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | 198 | 205 |
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Group.Hom.End
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Group.Subsemigroup.Membership
import Mathlib.Algebra.Group... | variable {F : Type*} [FunLike F R S]
theorem eq_of_eqOn_stop {f g : F}
| Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | 622 | 624 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | rw [← inner_conj_symm, mul_comm]
exact re_eq_norm_of_mul_conj (inner y x)
| Mathlib/Analysis/InnerProductSpace/Basic.lean | 225 | 227 |
/-
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.Polynomial.FieldDivision
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Data.List.Prime
import Mathlib.RingTheory.Polynomial.Tower
/-!
# S... | absurd hg.1 <|
Classical.not_not.2 <|
isUnit_iff_degree_eq_zero.2 <| by
have := congr_arg degree hp
rw [h, degree_C ha, degree_mul, @eq_comm (WithBot ℕ) 0,
Nat.WithBot.add_eq_zero_iff] at this
exact this.1
@[simp]
theorem splits_C (a : K) : Sp... | Mathlib/Algebra/Polynomial/Splits.lean | 56 | 67 |
/-
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 : ∀ ⦃f g : α → ℝ≥0∞⦄, Disjoint (support f) (support g) →
Measurable f → Measurable g → motive f → motive g → motive (f + g))
(iSup : ∀ ⦃f : ℕ → α → ℝ≥0∞⦄, (∀ n, Measurable (f n)) → Monotone f →
(∀ n, motive (f n)) → motive fun x => ⨆ n, f n x)
⦃f : α → ℝ≥0∞⦄ (hf : Measurable f) : motive f :... | Mathlib/MeasureTheory/Function/SimpleFunc.lean | 1,318 | 1,339 |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Convex.Jensen
import M... | push_cast at this
refine le_trans (sum_le_sum fun i _ => ?_) this
simp only [← abs_mul, le_abs_self]
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_... | Mathlib/Analysis/MeanInequalities.lean | 709 | 718 |
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SimpleGraph.Triangle.Basic
/-!
# Construct a tripartite graph from its triangles
This file contains the constru... | lemma in₂₀_iff' :
(graph t).Adj (in₂ c) (in₀ a) ↔ ∃ x : α × β × γ, x ∈ t ∧ x.2.2 = c ∧ x.1 = a where
mp := by rintro ⟨⟩; exact ⟨_, ‹_›, by simp⟩
mpr := by rintro ⟨⟨a, b, c⟩, h, rfl, rfl⟩; constructor; assumption
| Mathlib/Combinatorics/SimpleGraph/Triangle/Tripartite.lean | 97 | 100 |
/-
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.Data.ENNReal.Action
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.MeasureTheory.OuterMeasure.Caratheodory
... | theorem trim_eq_iInf' (s : Set α) : m.trim s = ⨅ t : { t // s ⊆ t ∧ MeasurableSet t }, m t := by
simp [iInf_subtype, iInf_and, trim_eq_iInf]
theorem trim_trim (m : OuterMeasure α) : m.trim.trim = m.trim :=
trim_eq_trim_iff.2 fun _s => m.trim_eq
| Mathlib/MeasureTheory/OuterMeasure/Induced.lean | 347 | 351 |
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.LinearAlgebra.Matrix.Gershgorin
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
import Mathlib.NumberTheory.NumberField.Units.Basic... | rw [← Units.finrank_eq_rank, ZLattice.rank ℝ]
/-- The map obtained by quotienting by the kernel of `logEmbedding`. -/
| Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean | 373 | 375 |
/-
Copyright (c) 2024 Raghuram Sundararajan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Raghuram Sundararajan
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
/-!
# Extensionality lemmas for rings and similar structures
In this file we prove e... | · exact congrArg (·.toAdd.add x y) h
· exact congrArg (·.toMul.mul x y) h
end Semiring
| Mathlib/Algebra/Ring/Ext.lean | 301 | 304 |
/-
Copyright (c) 2019 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.Submodule.Equiv
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.Algebra.Module.Rat
im... |
@[simp]
theorem lie_smul : ⁅x, t • m⁆ = t • ⁅x, m⁆ :=
| Mathlib/Algebra/Lie/Basic.lean | 152 | 154 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.CharP.Lemmas
import Mathlib.GroupTheory.OrderOfElement
/-!
# Lemmas about rings of characteristic two
This file contains results about `CharP R 2`,... |
theorem multiset_sum_sq (l : Multiset R) : l.sum ^ 2 = (l.map (· ^ 2)).sum :=
| Mathlib/Algebra/CharP/Two.lean | 99 | 100 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Independence.Kernel
import Mathlib.Probability.Kernel.Condexp
/-!
# Conditional Independence
We define conditional independence of sets/σ-alg... | μ⟦⋂ i ∈ s, f i | m'⟧ =ᵐ[μ] ∏ i ∈ s, (μ⟦f i | m'⟧) := by
simp only [iCondIndepSets, Kernel.iIndepSets]
have h_eq' : ∀ (s : Finset ι) (f : ι → Set Ω) (_H : ∀ i, i ∈ s → f i ∈ π i) i (_hi : i ∈ s),
(fun ω ↦ ENNReal.toReal (condExpKernel μ m' ω (f i))) =ᵐ[μ] μ⟦f i | m'⟧ :=
fun s f H i hi ↦ condExpKernel... | Mathlib/Probability/Independence/Conditional.lean | 154 | 191 |
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheo... |
theorem whiskerRight_iff {X Y : C} (f g : X ⟶ Y) : f ▷ 𝟙_ C = g ▷ 𝟙_ C ↔ f = g := by simp
| Mathlib/CategoryTheory/Monoidal/Category.lean | 490 | 491 |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.PurelyInseparable.Basic
import Mathlib.FieldTheory.PerfectClosure
/-!
# `IsPerfectClosure` predicate
This file contains `IsPerfectClosure` which asserts... | @[simp]
theorem lift_lift : lift g (lift i j p) p = lift (g.comp i) j p := by
refine IsPRadical.injective_comp_of_perfect _ (g.comp i) p ?_
| Mathlib/FieldTheory/IsPerfectClosure.lean | 413 | 415 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 1,832 | 1,836 | |
/-
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, Patrick Massot
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Topology.UniformSpace.Defs
import Mathlib.Topology.ContinuousOn
/-!
# Basic resu... |
theorem Filter.HasBasis.uniformity_closure {p : ι → Prop} {U : ι → Set (α × α)}
| Mathlib/Topology/UniformSpace/Basic.lean | 183 | 184 |
/-
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.Data.Finset.Card
import Mathlib.Data.Finset.Lattice.Fold
/-!
# Down-compressions
This file defines down-compression.
Down-compressing `𝒜 : Finset (Fins... |
@[simp] lemma memberSubfamily_image_erase : (𝒜.image (erase · a)).memberSubfamily a = ∅ := by
simp [eq_empty_iff_forall_not_mem,
(ne_of_mem_of_not_mem' (mem_insert_self _ _) (not_mem_erase _ _)).symm]
| Mathlib/Combinatorics/SetFamily/Compression/Down.lean | 133 | 136 |
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.MeasureTheory.Measure.Doubling
import Mathlib.MeasureTheory.Covering.Vitali
import Mathlib.MeasureTheory.Covering.Differentiation
/-!
# Uniformly locally do... |
end Applications
end IsUnifLocDoublingMeasure
| Mathlib/MeasureTheory/Covering/DensityTheorem.lean | 166 | 172 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.List.Sublists
import Mathlib.Data.List.Zip
import Mathlib.Data.Multiset.Bind
import Mathlib.Data.Multiset.Range
/-!
# The powerset of a multiset
... | | nil => simp
| cons _ _ IH =>
simp only [powersetAux'_cons]
| Mathlib/Data/Multiset/Powerset.lean | 55 | 57 |
/-
Copyright (c) 2021 Roberto Alvarez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Roberto Alvarez
-/
import Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Algebra.Equiv.TransferInstance
import ... | def transAt (i : N) (f g : Ω^ N X x) : Ω^ N X x :=
copy (fromLoop i <| (toLoop i f).trans <| toLoop i g)
(fun t => if (t i : ℝ) ≤ 1 / 2
then f (Function.update t i <| Set.projIcc 0 1 zero_le_one (2 * t i))
else g (Function.update t i <| Set.projIcc 0 1 zero_le_one (2 * t i - 1)))
(by
ext1; s... | Mathlib/Topology/Homotopy/HomotopyGroup.lean | 319 | 341 |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.Maps
import Mathlib.Data.Finset.Max
import Mathlib.Data.Sy... | ∃ v, G.maxDegree = G.degree v := by
obtain ⟨t, ht⟩ := max_of_nonempty (univ_nonempty.image fun v => G.degree v)
have ht₂ := mem_of_max ht
| Mathlib/Combinatorics/SimpleGraph/Finite.lean | 341 | 343 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Group.MeasurableEquiv
import Mathlib... | /-- If `μ` is inner regular for finite measure sets with respect to compact sets,
then any measurable set of finite measure can be approximated by a
compact subset. See also `MeasurableSet.exists_lt_isCompact_of_ne_top`. -/
theorem _root_.MeasurableSet.exists_isCompact_lt_add [InnerRegularCompactLTTop μ]
⦃A : Set α... | Mathlib/MeasureTheory/Measure/Regular.lean | 704 | 710 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Kim Morrison
-/
import Mathlib.CategoryTheory.Subobject.Lattice
/-!
# Specific subobjects
We define `equalizerSubobject`, `kernelSubobject` and `imageSubobject`, which ar... |
theorem kernelSubobject_factors_iff {W : C} (h : W ⟶ X) :
(kernelSubobject f).Factors h ↔ h ≫ f = 0 :=
| Mathlib/CategoryTheory/Subobject/Limits.lean | 104 | 106 |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
import Mathlib.Comp... | Mathlib/Computability/TMToPartrec.lean | 146 | 146 | |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Matrix.ConjTranspose
/-!
# Row and column matrices
This file provides results about row and column matrices.
## Main definitions
* `Mat... | theorem updateCol_ne [DecidableEq n] {j' : n} (j_ne : j' ≠ j) :
updateCol M j c i j' = M i j' :=
Function.update_of_ne (β := fun _ => α) j_ne (c i) (M i)
| Mathlib/Data/Matrix/RowCol.lean | 255 | 258 |
/-
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, Patrick Massot
-/
import Mathlib.Algebra.Group.TypeTags.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Piecewise
import Mathlib.Or... | (hf : Tendsto f l (𝓝 x)) (i : ι) {g : Y → π i} {xi : π i} (hg : Tendsto g l (𝓝 xi)) :
Tendsto (fun a => update (f a) i (g a)) l (𝓝 <| update x i xi) :=
tendsto_pi_nhds.2 fun j => by rcases eq_or_ne j i with (rfl | hj) <;> simp [*, hf.apply_nhds]
theorem ContinuousAt.update [DecidableEq ι] {x : X} (hf : Co... | Mathlib/Topology/Constructions.lean | 725 | 733 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Batteries.WF
import Mathlib.Data.Part
import Mathlib.Data.Rel
import Mathlib.Tactic.GeneralizeProofs
/-!
# Partial functions
This... | rfl
@[simp]
theorem id_comp (f : α →. β) : (PFun.id β).comp f = f :=
| Mathlib/Data/PFun.lean | 488 | 491 |
/-
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... | theorem prod_eq_prod_of_nonempty {eX eX' : PartialHomeomorph X X'} {eY eY' : PartialHomeomorph Y Y'}
(h : (eX.prod eY).source.Nonempty) : eX.prod eY = eX'.prod eY' ↔ eX = eX' ∧ eY = eY' := by
| Mathlib/Topology/PartialHomeomorph.lean | 924 | 925 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Defs
import Mathlib.A... | Mathlib/Algebra/Group/Submonoid/Membership.lean | 735 | 744 | |
/-
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.Algebra.BigOperators.Group.Finset.Sigma
import Mathlib.Algebra.Order.Interval.Finset.Basic
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Tact... | ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a (j + 1), f i j := by
rw [Finset.sum_sigma', Finset.sum_sigma']
refine sum_nbij' (fun x ↦ ⟨x.2, x.1⟩) (fun x ↦ ⟨x.2, x.1⟩) ?_ ?_ (fun _ _ ↦ rfl) (fun _ _ ↦ rfl)
(fun _ _ ↦ rfl) <;>
simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>
rintro a b ⟨⟨h₁, h... | Mathlib/Algebra/BigOperators/Intervals.lean | 176 | 183 |
/-
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.Finset.Basic
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
/-!
# The fold operation for a commutative associative operation ... |
theorem fold_const [hd : Decidable (s = ∅)] (c : β) (h : op c (op b c) = op b c) :
Finset.fold op b (fun _ => c) s = if s = ∅ then b else op b c := by
| Mathlib/Data/Finset/Fold.lean | 73 | 75 |
/-
Copyright (c) 2023 Ruben Van de Velde. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ruben Van de Velde
-/
import Mathlib.Algebra.MvPolynomial.Eval
import Mathlib.Topology.Algebra.Ring.Basic
/-!
# Multivariate polynomials and continuity
In this file we prove the... | theorem MvPolynomial.continuous_eval : Continuous fun x ↦ eval x p := by
continuity
| Mathlib/Topology/Algebra/MvPolynomial.lean | 25 | 26 |
/-
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.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.LocalExtr.Basic
/-!
# Darboux's theorem
In ... | (hmb : m < f' b) : m ∈ f' '' Ioo a b := by
rcases hab.eq_or_lt with (rfl | hab')
· exact (lt_asymm hma hmb).elim
set g : ℝ → ℝ := fun x => f x - m * x
have hg : ∀ x ∈ Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x := by
intro x hx
simpa using (hf x hx).sub ((hasDerivWithinAt_id x _).const_mul m)... | Mathlib/Analysis/Calculus/Darboux.lean | 28 | 60 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... |
/-- The closure of `K` is the full space iff `Kᗮ` is trivial. -/
theorem topologicalClosure_eq_top_iff [CompleteSpace E] :
K.topologicalClosure = ⊤ ↔ Kᗮ = ⊥ := by
rw [← K.orthogonal_orthogonal_eq_closure]
constructor <;> intro h
· rw [← Submodule.triorthogonal_eq_orthogonal, h, Submodule.top_orthogonal_eq_bo... | Mathlib/Analysis/InnerProductSpace/Projection.lean | 909 | 915 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Order.Group.Int
import Mathlib.Algebra.Order.Group.Unbundled.Int
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Ring.Int.Parity
im... | Mathlib/Algebra/Order/Ring/Int.lean | 59 | 60 | |
/-
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.FieldTheory.Finite.Polynomial
import Mathlib.NumberTheory.Basic
import Mathlib.RingTheory.WittVector.WittPolynomial
/-!
# Witt struct... | replace := congr_arg (bind₁ fun k : ℕ => bind₁ (fun i => rename (Prod.mk i) (W_ ℚ k)) Φ) this
rw [map_mul, bind₁_C_right] at this
rw [wittStructureRat, this]; clear this
conv_lhs => simp only [map_sub, bind₁_X_right]
rw [sub_right_inj]
simp only [map_sum, map_mul, bind₁_C_right, map_pow]
rfl
/-- Write `w... | Mathlib/RingTheory/WittVector/StructurePolynomial.lean | 165 | 176 |
/-
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.Analytic.Within
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
/-!
# Higher d... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 1,679 | 1,683 | |
/-
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.Dynamics.Ergodic.MeasurePreserving
import Mathlib.Dynamics.Minimal
import Mathlib.GroupTheory.GroupAction.Hom
import Mathlib.MeasureTheory.Group.Meas... | @[to_additive]
theorem MeasurePreserving.smulInvariantMeasure_iterateMulAct
{f : α → α} {_ : MeasurableSpace α} {μ : Measure α} (hf : MeasurePreserving f μ μ) :
SMulInvariantMeasure (IterateMulAct f) α μ :=
⟨fun n _s hs ↦ (hf.iterate n.val).measure_preimage hs.nullMeasurableSet⟩
@[to_additive]
theorem smulIn... | Mathlib/MeasureTheory/Group/Action.lean | 156 | 184 |
/-
Copyright (c) 2022 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.Nat.Cast.Field
import Mathlib.NumberTheory.ArithmeticFunction
/-!
# The von Mangoldt Function
In ... | theorem sum_moebius_mul_log_eq {n : ℕ} : (∑ d ∈ n.divisors, (μ d : ℝ) * log d) = -Λ n := by
simp only [← log_mul_moebius_eq_vonMangoldt, mul_comm log, mul_apply, log_apply, intCoe_apply, ←
| Mathlib/NumberTheory/VonMangoldt.lean | 126 | 127 |
/-
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.... | theorem fourier_zero' {x : AddCircle T} : @toCircle T 0 = (1 : ℂ) := by
have : fourier 0 x = @toCircle T 0 := by rw [fourier_apply, zero_smul]
rw [← this]; exact fourier_zero
| Mathlib/Analysis/Fourier/AddCircle.lean | 132 | 135 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Algebra.Basic
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Eval.Algebra
impo... | rw [← descPochhammer_map (algebraMap ℤ R), eval_map, ← eq_intCast (algebraMap ℤ R)]
simp only [algebraMap_int_eq, eq_intCast, eval₂_at_intCast, Nat.cast_id, eq_natCast, Int.cast_id]
theorem descPochhammer_eval_zero {n : ℕ} :
(descPochhammer R n).eval 0 = if n = 0 then 1 else 0 := by
| Mathlib/RingTheory/Polynomial/Pochhammer.lean | 276 | 280 |
/-
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... | (sub_one_lt _).le
/-- The (real) logarithm of a natural number `n` is bounded by a multiple of every power of `n`
with positive exponent. -/
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 883 | 886 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue.Basic
import Mathlib.Algebra.Ring.Opposite
import Mathlib.Algebra.Ring.Prod
import Mathlib.Algebra.Ring.Sub... | section MulOpposite
open MulOpposite
instance [NonUnitalNonAssocSemiring R] [TopologicalSpace R] [ContinuousAdd R] :
ContinuousAdd Rᵐᵒᵖ :=
continuousAdd_induced opAddEquiv.symm
| Mathlib/Topology/Algebra/Ring/Basic.lean | 210 | 216 |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Lie.Engel
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.D... | @[simp]
lemma engel_zero : engel R (0 : L) = ⊤ := by
rw [eq_top_iff]
rintro x -
rw [mem_engel_iff, LieHom.map_zero]
use 1
simp only [pow_one, LinearMap.zero_apply]
| Mathlib/Algebra/Lie/EngelSubalgebra.lean | 70 | 76 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.C... | Mathlib/Analysis/BoundedVariation.lean | 572 | 585 | |
/-
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
-/
import Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots
import Mathlib.FieldTheory.Galois.Basic
import Mathlib.FieldTheory.KummerPolynomial
import Mathlib.Linea... | (X ^ n - C a) = ∏ i ∈ Finset.range n, (X - C (ζ ^ i * α)) := by
let K := FractionRing R
let i := algebraMap R K
have h := FaithfulSMul.algebraMap_injective R K
apply_fun Polynomial.map i using map_injective i h
simpa only [Polynomial.map_sub, Polynomial.map_pow, map_X, map_C, map_mul, map_pow,
Polynom... | Mathlib/FieldTheory/KummerExtension.lean | 74 | 82 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Sigma.Basic
import Mathlib.Algebra.Order.Ring.Nat
/-!
# A computable model of ZFA without infinity
In this file we define finite hereditary list... | Mathlib/SetTheory/Lists.lean | 88 | 88 | |
/-
Copyright (c) 2023 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
import Mathlib.Combinatorics.SimpleGraph.Subgraph
/-!
# Local graph operations
... | contrapose! h
simp_all [incidenceSet]
aesop
theorem card_edgeFinset_replaceVertex_of_not_adj (hn : ¬G.Adj s t) :
#(G.replaceVertex s t).edgeFinset = #G.edgeFinset + G.degree s - G.degree t := by
have inc : G.incidenceFinset t ⊆ G.edgeFinset := by simp [incidenceFinset, incidenceSet_subset]
rw [G.edge... | Mathlib/Combinatorics/SimpleGraph/Operations.lean | 115 | 124 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Symmetric
/-!
# Integer powers of square matrices
In this file, we defi... | Mathlib/LinearAlgebra/Matrix/ZPow.lean | 292 | 297 | |
/-
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.Finite.Defs
import Mathlib.Data.Finset.BooleanAlgebra
import Mathlib.Data.Finset.Image
import Mathlib.Data.Fintype.Defs
import Mathlib.Data.Fintyp... | | n =>
Classical.choose
(h
| Mathlib/Data/Fintype/Basic.lean | 235 | 237 |
/-
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.Algebra.Group.Units.Basic
import Mathlib.Algebra.GroupWithZero.Basic
import Mathlib.Data.Int.Basic
import Mathlib.Lean.Meta.CongrTheorems
import Mathli... | namespace Ring
open Classical in
| Mathlib/Algebra/GroupWithZero/Units/Basic.lean | 73 | 75 |
/-
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.FreeNonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.NonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.UniversalEnveloping
import Mathlib.GroupT... | theorem Rel.neg {a b : lib R X} (h : Rel R X a b) : Rel R X (-a) (-b) := by
simpa only [neg_one_smul] using h.smul (-1)
| Mathlib/Algebra/Lie/Free.lean | 87 | 88 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomotopyCategory
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.CategoryTheory.Shift.Quotient
import Mathlib.CategoryTheory.Linear.LinearF... | ((CategoryTheory.shiftFunctorAdd' (CochainComplex C ℤ) a b ab h).inv.app K).f n =
(K.XIsoOfEq (by dsimp; rw [← h, add_assoc, add_comm a])).hom := by
subst h
rw [shiftFunctorAdd'_eq_shiftFunctorAdd, shiftFunctorAdd_inv_app_f]
| Mathlib/Algebra/Homology/HomotopyCategory/Shift.lean | 143 | 147 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro, Anne Baanen,
Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Algebra.Modul... | ⟨fun a f ↦
{ toFun := a • (f : M → M₂)
map_add' := fun x y ↦ by simp only [Pi.smul_apply, f.map_add, smul_add]
map_smul' := fun c x ↦ by simp [Pi.smul_apply, smul_comm] }⟩
| Mathlib/Algebra/Module/LinearMap/Defs.lean | 734 | 738 |
/-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.BernoulliPolynomials
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import... | theorem bernoulliFun_endpoints_eq_of_ne_one {k : ℕ} (hk : k ≠ 1) :
bernoulliFun k 1 = bernoulliFun k 0 := by
| Mathlib/NumberTheory/ZetaValues.lean | 49 | 50 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau, Yury Kudryashov
-/
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Logic.Function.Iterate
import Mathlib.Logic.Relation
/-!
# R... | protected theorem Chain.sublist [IsTrans α R] (hl : l₂.Chain R a) (h : l₁ <+ l₂) :
l₁.Chain R a := by
rw [chain_iff_pairwise] at hl ⊢
exact hl.sublist (h.cons_cons a)
protected theorem Chain.rel [IsTrans α R] (hl : l.Chain R a) (hb : b ∈ l) : R a b := by
| Mathlib/Data/List/Chain.lean | 109 | 114 |
/-
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... | @[to_additive]
theorem normalizer_eq_top_iff : H.normalizer = ⊤ ↔ H.Normal :=
eq_top_iff.trans
⟨fun h => ⟨fun a ha b => (h (mem_top b) a).mp ha⟩, fun h a _ha b =>
| Mathlib/Algebra/Group/Subgroup/Basic.lean | 341 | 344 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Amelia Livingston, Yury Kudryashov,
Neil Strickland, Aaron Anderson
-/
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathli... | rintro rfl; exact nonzero ⟨zero_dvd_iff.1 hx, zero_dvd_iff.1 hy⟩
| Mathlib/Algebra/GroupWithZero/Divisibility.lean | 92 | 93 |
/-
Copyright (c) 2023 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Topology.Connected.Basic
import Mathlib.Topology.Separation.Hausdorff
import Mathlib.Topology.Connected.Clopen
/-!
# Separated maps and locally injective maps ou... | theorem IsLocallyInjective.comp_left {A} {f : X → Y} (hf : IsLocallyInjective f) {g : Y → A}
(hg : g.Injective) : IsLocallyInjective (g ∘ f) :=
fun x ↦ let ⟨U, hU, hx, inj⟩ := hf x; ⟨U, hU, hx, hg.comp_injOn inj⟩
theorem IsLocallyInjective.comp_right {f : X → Y} (hf : IsLocallyInjective f) {g : A → X}
| Mathlib/Topology/SeparatedMap.lean | 166 | 170 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.GroupWithZero.WithZero
impo... | WithBot.coe_le_iff
@[simp] lemma unzero_le_unzero {a b : WithZero α} (ha hb) :
unzero (x := a) ha ≤ unzero (x := b) hb ↔ a ≤ b := by
| Mathlib/Algebra/Order/GroupWithZero/Canonical.lean | 277 | 280 |
/-
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... | Mathlib/Data/Set/Basic.lean | 1,664 | 1,666 | |
/-
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, Patrick Massot
-/
import Mathlib.Algebra.Group.TypeTags.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Piecewise
import Mathlib.Or... | Mathlib/Topology/Constructions.lean | 1,376 | 1,379 | |
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Yury Kudryashov, Frédéric Dupuis
-/
import Mathlib.Topology.Algebra.InfiniteSum.Constructions
import Mathlib.Topology.Algebra.Module.Equiv
/-! # Infinite sums in top... | by_cases hf : Summable f
· exact
⟨fun h ↦ (e.hasSum.mp ((e.summable.mpr hf).hasSum_iff.mpr h)).tsum_eq, fun h ↦
| Mathlib/Topology/Algebra/InfiniteSum/Module.lean | 157 | 159 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Gluing
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.CategoryTheory.Limits.Sh... | · simp_rw [Category.assoc, cocycle_snd_fst_fst 𝒰 f g i j k]
· simp_rw [Category.assoc, cocycle_snd_fst_snd 𝒰 f g i j k]
· simp_rw [Category.assoc, cocycle_snd_snd 𝒰 f g i j k]
| Mathlib/AlgebraicGeometry/Pullbacks.lean | 190 | 193 |
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Computation.Basic
import Mathlib.Algebra.ContinuedFractions.Translations
import Mathlib.Algebra.Order.Floor.Ring
/-!
# ... | rfl
variable (K)
variable [IsStrictOrderedRing K]
/-- If `a` is an integer, then the coefficient sequence of its continued fraction is empty.
-/
theorem of_s_of_int (a : ℤ) : (of (a : K)).s = Stream'.Seq.nil :=
| Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean | 263 | 270 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Re... | simp only [Ne, not_false_iff, reduceCtorEq]
/-- Multiplying by all the matrices in `listTransvecRow M` does not change the last column. -/
theorem mul_listTransvecRow_last_col (i : Fin r ⊕ Unit) :
(M * (listTransvecRow M).prod) i (inr unit) = M i (inr unit) := by
have A : (listTransvecRow M).length = r := by... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 437 | 451 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Alge... | orderOf_dvd_of_pow_eq_one⟩
| Mathlib/GroupTheory/OrderOfElement.lean | 245 | 246 |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Preserves.Limits
import Mathlib.CategoryTheory.Limits.Yoneda
/-!
# Ind- and ... | noncomputable def colimitCoyonedaHomIsoLimitLeftOp :
(colimit (D ⋙ coyoneda) ⟶ F) ≅ limit (D.leftOp ⋙ F ⋙ uliftFunctor.{u₁}) :=
haveI : HasColimit (D.leftOp.rightOp ⋙ coyoneda) :=
inferInstanceAs <| HasColimit (D ⋙ coyoneda)
colimitCoyonedaHomIsoLimit D.leftOp F
| Mathlib/CategoryTheory/Limits/IndYoneda.lean | 170 | 175 |
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