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) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Ideal
import Mathlib.... | refine ⟨y' • y, Submodule.smul_mem _ _ hy, z' * z, ?_⟩
rw [← IsScalarTower.algebraMap_smul S y', smul_smul, ← mk'_mul, smul_smul,
mul_comm (mk' S _ _), mul_mk'_eq_mk'_of_mul]
· rintro ⟨y, hy, z, rfl⟩
exact Submodule.smul_mem _ _ (Submodule.span_subset_span R S _ hy)
theorem mem_span_map {x : ... | Mathlib/RingTheory/Localization/Submodule.lean | 138 | 162 |
/-
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.Order.Atoms
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.RelIso.Set
import Mathlib.Order.SupClosed
import Mathlib.Order.SupIndep
import Mathlib.Orde... | @[deprecated (since := "2024-11-24")]
alias CompleteLattice.setIndependent_iff_finite := sSupIndep_iff_finite
lemma iSupIndep_iff_supIndep_of_injOn {ι : Type*} {f : ι → α}
(hf : InjOn f {i | f i ≠ ⊥}) :
iSupIndep f ↔ ∀ (s : Finset ι), s.SupIndep f := by
refine ⟨fun h ↦ h.supIndep', fun h ↦ iSupIndep_def'.mpr... | Mathlib/Order/CompactlyGenerated/Basic.lean | 450 | 471 |
/-
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.Order.Bounds.Defs
import Mathlib.Order.Directed
import Mathlib.Order.BoundedOrder.Monotone
import Mathlib.Order.Interval.Set.Basic
/-... |
theorem IsGLB.exists_between (h : IsGLB s a) (hb : a < b) : ∃ c ∈ s, a ≤ c ∧ c < b :=
let ⟨c, hcs, hbc⟩ := (isGLB_lt_iff h).1 hb
⟨c, hcs, h.1 hcs, hbc⟩
| Mathlib/Order/Bounds/Basic.lean | 908 | 911 |
/-
Copyright (c) 2024 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel
-/
import Mathlib.Analysis.NormedSpace.PiTensorProduct.ProjectiveSeminorm
import Mathlib.LinearAlgebra.Isomorphisms
/-!
# Injective seminorm on the tensor of a finite famil... | ext
simp only [mapL_coe, ContinuousLinearMap.coe_id, map_id, LinearMap.compMultilinearMap_apply,
LinearMap.id_coe, id_eq]
@[simp]
| Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean | 375 | 379 |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | theorem toReal_coe_eq_self_sub_two_mul_int_mul_pi_iff {θ : ℝ} {k : ℤ} :
(θ : Angle).toReal = θ - 2 * k * π ↔ θ ∈ Set.Ioc ((2 * k - 1 : ℝ) * π) ((2 * k + 1) * π) := by
rw [← sub_zero (θ : Angle), ← zsmul_zero k, ← coe_two_pi, ← coe_zsmul, ← coe_sub, zsmul_eq_mul, ←
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 559 | 561 |
/-
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.FullyFaithful
import Mathlib.CategoryTheory.ObjectProperty.FullSubcategory
import Mat... | instance Equivalence.isEquivalence_functor (F : C ≌ D) : IsEquivalence F.functor where
| Mathlib/CategoryTheory/Equivalence.lean | 520 | 521 |
/-
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.Data.Bundle
import Mathlib.Data.Set.Image
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.O... | cases e; cases e'; congr
-- TODO: move `ext` here?
lemma ext {e e' : Pretrivialization F proj} (h₁ : ∀ x, e x = e' x)
| Mathlib/Topology/FiberBundle/Trivialization.lean | 86 | 89 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Simon Hudon
-/
import Mathlib.Data.Fin.Fin2
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Common
/-!
# Tuples of types, and their categorica... | end Liftp'
@[simp]
theorem dropFun_diag {α} : dropFun (@prod.diag (n + 1) α) = prod.diag := by
| Mathlib/Data/TypeVec.lean | 570 | 573 |
/-
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.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Function.AEMeasurableOrder
import Mathlib.MeasureTheory.Integral.Average
import ... | obtain ⟨U, sU, U_open, νU⟩ : ∃ (U : Set α), s ⊆ U ∧ IsOpen U ∧ ν U ≤ ν s + ε :=
exists_isOpen_le_add s ν (ENNReal.coe_pos.2 εpos).ne'
let f : α → Set (Set α) := fun _ => {a | ρ a ≤ ν a ∧ a ⊆ U}
have h : v.FineSubfamilyOn f s := by
apply v.fineSubfamilyOn_of_frequently f s fun x hx => ?_
have :=
... | Mathlib/MeasureTheory/Covering/Differentiation.lean | 125 | 149 |
/-
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.Data.ENNReal.Real
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Topology... | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 1,274 | 1,277 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.MonoidAlgebra.Defs
import Mathlib.Algebra.Order.Mon... | simp
end Semiring
| Mathlib/Algebra/Polynomial/Basic.lean | 1,055 | 1,057 |
/-
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.Analysis.MellinTransform
/-!
# Abstract functional equations for Mellin transforms
This file formalises a general version of an argument used to pro... | /-- Reformulated functional equation with `f` and `g` interchanged. -/
lemma WeakFEPair.h_feq' (P : WeakFEPair E) (x : ℝ) (hx : 0 < x) :
P.g (1 / x) = (P.ε⁻¹ * ↑(x ^ P.k)) • P.f x := by
rw [(div_div_cancel₀ (one_ne_zero' ℝ) ▸ P.h_feq (1 / x) (one_div_pos.mpr hx):), ← mul_smul]
convert (one_smul ℂ (P.g (1 / x)))... | Mathlib/NumberTheory/LSeries/AbstractFuncEq.lean | 103 | 109 |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calcul... |
end Zero
| Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 477 | 479 |
/-
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, Kim Morrison, Alex Keizer
-/
import Mathlib.Data.List.OfFn
import Batteries.Data.List.Perm
import Mathlib.Data.List.Nodup
/-!
# Lists of elements of `Fin n`... | simp [count_eq_of_nodup (nodup_finRange n)]
| Mathlib/Data/List/FinRange.lean | 54 | 55 |
/-
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.LinearAlgebra.Quotient.Basic
import Mathlib.Algebra.Category.ModuleCat.Basic
import Mathlib.CategoryTheory.ConcreteCategory.EpiMono
/-!
# Monomorphisms in... | theorem mono_iff_injective : Mono f ↔ Function.Injective f := by
rw [mono_iff_ker_eq_bot, LinearMap.ker_eq_bot]
| Mathlib/Algebra/Category/ModuleCat/EpiMono.lean | 39 | 41 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Yuyang Zhao
-/
import Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic
import Mathlib.Algebra.Order.GroupWithZero.Unbundled.Defs
import Mathlib.Tactic.Linter.Deprecate... | Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean | 598 | 605 | |
/-
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... | theorem cyclicPermutations_inj {l l' : List α} :
cyclicPermutations l = cyclicPermutations l' ↔ l = l' :=
cyclicPermutations_injective.eq_iff
theorem length_mem_cyclicPermutations (l : List α) (h : l' ∈ cyclicPermutations l) :
| Mathlib/Data/List/Rotate.lean | 546 | 550 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.RingTheory.Ideal.Maps
/-!
# Ideals in product rings
For commutative rings `R` and `S` and ideals `I ≤ R`, `J ≤ S`, we define `Ideal.prod I J` as the
pr... | theorem ideal_prod_prime_aux {I : Ideal R} {J : Ideal S} :
(Ideal.prod I J).IsPrime → I = ⊤ ∨ J = ⊤ := by
contrapose!
simp only [ne_top_iff_one, isPrime_iff, not_and, not_forall, not_or]
exact fun ⟨hI, hJ⟩ _ => ⟨⟨0, 1⟩, ⟨1, 0⟩, by simp, by simp [hJ], by simp [hI]⟩
/-- Classification of prime ideals in produc... | Mathlib/RingTheory/Ideal/Prod.lean | 129 | 138 |
/-
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.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlg... | refine ⟨fun h => ?_, @congrArg _ _ fs₁ fs₂ (fun z => Finset.centroid k z s.points)⟩
rw [Finset.centroid_eq_affineCombination_fintype,
Finset.centroid_eq_affineCombination_fintype] at h
| Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 919 | 921 |
/-
Copyright (c) 2023 Yaël Dillies, Chenyi Li. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chenyi Li, Ziyu Wang, Yaël Dillies
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.InnerProductSpace.Basic
/-!
# Uniformly and strongly convex functions
I... |
lemma strongConcaveOn_iff_convex :
StrongConcaveOn s m f ↔ ConcaveOn ℝ s fun x ↦ f x + m / (2 : ℝ) * ‖x‖ ^ 2 := by
refine and_congr_right fun _ ↦ forall₄_congr fun x _ y _ ↦ forall₅_congr fun a b ha hb hab ↦ ?_
| Mathlib/Analysis/Convex/Strong.lean | 170 | 173 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Matroid.Init
import Mathlib.Data.Set.Card
import Mathlib.Data.Set.Finite.Powerset
import Mathlib.Order.UpperLower.Clos... | M.isBase_exchange.encard_diff_eq hB₁ hB₂
theorem IsBase.ncard_diff_comm (hB₁ : M.IsBase B₁) (hB₂ : M.IsBase B₂) :
(B₁ \ B₂).ncard = (B₂ \ B₁).ncard := by
rw [ncard_def, hB₁.encard_diff_comm hB₂, ← ncard_def]
| Mathlib/Data/Matroid/Basic.lean | 415 | 419 |
/-
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.Fin.VecNotation
import Mathlib.Logic.Small.Basic
import Mathlib.SetTheory.ZFC.PSet
/-!
# A model of ZFC
In this file, we model Zermelo-Fraenkel ... | Mathlib/SetTheory/ZFC/Basic.lean | 1,034 | 1,034 | |
/-
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... | Mathlib/Algebra/Ring/Ext.lean | 451 | 453 | |
/-
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.Data.Fintype.List
import Mathlib.Data.Fintype.OfMap
/-!
# Cycles of a list
Lists have an equivalence relation of whether they are rotational permut... | Mathlib/Data/List/Cycle.lean | 959 | 961 | |
/-
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.Conj
import Mathlib.Algebra.Group.Subgroup.Lattice
import Mathlib.Algebra.Group.Submonoid.BigOperators
import Mathlib.Data.Finset.Fin
import ... | ∀ a ∈ finPairsLT n, ∃ b ∈ finPairsLT n, signBijAux f b = a :=
fun ⟨a₁, a₂⟩ ha =>
if hxa : f⁻¹ a₂ < f⁻¹ a₁ then
⟨⟨f⁻¹ a₁, f⁻¹ a₂⟩, mem_finPairsLT.2 hxa, by
dsimp [signBijAux]
rw [apply_inv_self, apply_inv_self, if_pos (mem_finPairsLT.1 ha)]⟩
else
⟨⟨f⁻¹ a₂, f⁻¹ a₁⟩,
mem_f... | Mathlib/GroupTheory/Perm/Sign.lean | 195 | 204 |
/-
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.Data.Nat.ModEq
/-!
# Congruences modulo an integer
This file defines the equivalence relation `a ≡ b [ZMOD n]` on the integers, similarly to how
`Data.N... | Mathlib/Data/Int/ModEq.lean | 303 | 308 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Order.Group.Pointwise.Bounds
import Mathlib.Data.Real.Basic
import Mathlib.Ord... | contrapose! ha
refine le_of_forall_lt_rat_imp_le ?_
| Mathlib/Data/Real/Archimedean.lean | 368 | 369 |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Anatole Dedecker
-/
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.LinearAlgebra.FreeM... | t = hnorm.toUniformSpace.toTopologicalSpace := by
-- Let `𝓣₀` denote the topology on `𝕜` induced by the norm, and `𝓣` be any T2 vector
-- topology on `𝕜`. To show that `𝓣₀ = 𝓣`, it suffices to show that they have the same
-- neighborhoods of 0.
refine IsTopologicalAddGroup.ext h₁ inferInstance (le_ant... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | 77 | 127 |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Group.Unbundled.Int
import Mathlib.Algebra.Module.NatInt
import Mathlib.GroupTheory.QuotientGroup.Defs
impor... | n • ψ = n • θ ↔ ∃ k : Fin n, ψ = θ + (k : ℕ) • (p / n : R) := by
rw [← natCast_zsmul ψ, ← natCast_zsmul θ,
zmultiples_zsmul_eq_zsmul_iff (Int.natCast_ne_zero.mpr hz), Int.cast_natCast]
rfl
| Mathlib/Algebra/CharZero/Quotient.lean | 67 | 71 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Interval.Finset.Basic
/-!
# Intervals as multisets
This file defines intervals as multisets.
## Main declarations
In a `LocallyFiniteOrder`,
* `M... | @[simp]
theorem Ico_inter_Ico_of_le [DecidableEq α] {a b c d : α} (h : b ≤ c) : Ico a b ∩ Ico c d = 0 :=
Multiset.inter_eq_zero_iff_disjoint.2 <| Ico_disjoint_Ico h
| Mathlib/Order/Interval/Multiset.lean | 243 | 246 |
/-
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.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Covering.Besicovitch
import Mathlib.Tactic.AdaptationNote
import Mathlib.Algeb... | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | 507 | 516 | |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Group.Submonoid.Pointwise
/-!
# Submonoid of inverses
Given a submonoid `N` of a monoid `M`, we define the submonoid `N.leftInv` as the submonoid ... |
section CommGroup
| Mathlib/GroupTheory/Submonoid/Inverses.lean | 196 | 197 |
/-
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.Data.ENNReal.Real
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Topology... | simp only [@nhds_eq_comap_uniformity α, Metric.uniformity_eq_comap_nhds_zero, comap_comap,
Function.comp_def, dist_comm]
| Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 1,117 | 1,119 |
/-
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... | theorem inv_top : (⊤ : Rel α β).inv = (⊤ : Rel β α) := by
simp [Top.top, inv, Function.flip_def]
| Mathlib/Data/Rel.lean | 141 | 143 |
/-
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... | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 828 | 832 | |
/-
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.Algebra.Order.Group.Indicator
import Mathlib.MeasureTheory.OuterMeasure.Basic
/-!
# Operations on outer measures
In this file we defi... |
theorem le_comap_map {β} (f : α → β) (m : OuterMeasure α) : m ≤ comap f (map f m) := fun _ =>
m.mono <| subset_preimage_image _ _
| Mathlib/MeasureTheory/OuterMeasure/Operations.lean | 332 | 334 |
/-
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.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Fi... | Mathlib/RingTheory/Polynomial/Content.lean | 106 | 106 | |
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.FiniteDimensional
import Mathlib.Da... | apply mt o.eq_zero_or_eq_zero_of_kahler_eq_zero
tauto
theorem kahler_ne_zero_iff (x y : E) : o.kahler x y ≠ 0 ↔ x ≠ 0 ∧ y ≠ 0 := by
refine ⟨?_, fun h => o.kahler_ne_zero h.1 h.2⟩
| Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 488 | 492 |
/-
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.List.TakeDrop
import Mathlib.Data.List.Induction
/-!
# Prefixes, suffixes, infixes
This file proves properties about
* `List.isPrefix`: `l₁` is ... | Mathlib/Data/List/Infix.lean | 307 | 313 | |
/-
Copyright (c) 2022 Antoine Labelle, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle, Rémi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
/-!
# Rewriting arrows and paths along vertex... |
theorem Path.cast_cons {u v w u' w' : U} (p : Path u v) (e : v ⟶ w) (hu : u = u') (hw : w = w') :
(p.cons e).cast hu hw = (p.cast hu rfl).cons (e.cast rfl hw) := by
subst_vars
| Mathlib/Combinatorics/Quiver/Cast.lean | 112 | 115 |
/-
Copyright (c) 2024 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.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Principal
/-!
# Ordinal arithmetic with cardinals
This file co... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 1,137 | 1,137 | |
/-
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.ConditionalProbability
import Mathlib.Probability.Kernel.Basic
import Mathlib.Probability.Kernel.Composition.MeasureComp
import Mathlib.Tactic.... |
@[nontriviality, simp]
lemma iIndep.of_subsingleton [Subsingleton ι] {m : ι → MeasurableSpace Ω} {κ : Kernel α Ω}
[IsMarkovKernel κ] : iIndep m κ μ := by simp [iIndep]
@[nontriviality, simp]
lemma iIndepFun.of_subsingleton [Subsingleton ι] {β : ι → Type*} {m : ∀ i, MeasurableSpace (β i)}
| Mathlib/Probability/Independence/Kernel.lean | 242 | 248 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
import Mathlib.Analysis.As... | @[simp]
theorem fderivWithin_univ : fderivWithin 𝕜 f univ = fderiv 𝕜 f := by
| Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 216 | 217 |
/-
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.Interval.Set.Basic
import Mathlib.Data.Set.Lattice.Image
import Mathlib.Data.SetLike.Basic
/-!
# Order intervals
This file defines (nonempty) close... | · apply iInf₂_le
exact ha
· exact le_iSup₂_of_le _ ha le_rfl
sSup_le := fun s s ha => by
| Mathlib/Order/Interval/Basic.lean | 601 | 604 |
/-
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.LinearAlgebra.ExteriorAlgebra.Basic
import Mathlib.RingTheory.GradedAlgebra.Basic
/-!
# Results about the grading structure of the exterior algebra
Many of... | lift R ⟨by apply GradedAlgebra.ι R M, GradedAlgebra.ι_sq_zero R M⟩
theorem GradedAlgebra.liftι_eq (i : ℕ) (x : ⋀[R]^i M) :
| Mathlib/LinearAlgebra/ExteriorAlgebra/Grading.lean | 52 | 54 |
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kim Morrison
-/
import Mathlib.Algebra.Order.Interval.Set.Instances
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Algebra.Ring.Real
/-!
# The unit ... | theorem symm_lt_comm {i j : I} : σ i < j ↔ σ j < i := by
| Mathlib/Topology/UnitInterval.lean | 167 | 167 |
/-
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... | by_cases h : f = 0
· exact C_eq_zero.mp (eq_zero_of_zero_dvd (by rwa [h] at hp))
· contrapose! hf with h_contra
rw [← degree_C h_contra]
apply le_antisymm (degree_le_of_dvd hp (by rwa [Ne, C_eq_zero])) _
rwa [degree_C h_contra, zero_le_degree_iff]
variable [CommRing S]
| Mathlib/RingTheory/AdjoinRoot.lean | 225 | 233 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Ultraproducts
import Mathlib.ModelTheory.Bundled
import Mathlib.ModelTheory.Skolem
import Mathlib.Order.Filter.AtTopBot.Basic
/-!
# First-... | Mathlib/ModelTheory/Satisfiability.lean | 553 | 554 | |
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Derivation.Killing
import Mathlib.Algebra.Lie.Killing
import Mathlib.Algebra.Lie.Sl2
import Mathlib.Algebra.Lie.Weights.Chain
import Mathlib.Line... | replace hy' : y ∈ H := by
suffices y ∈ H.normalizer by rwa [LieSubalgebra.IsCartanSubalgebra.self_normalizing] at this
exact (H.mem_normalizer_iff y).mpr fun z hz ↦ hy' z hz ▸ LieSubalgebra.zero_mem H
/- It suffices to show `x = y` since `S = ad K L y` is semisimple. -/
suffices x = y by rwa [this, ← LieD... | Mathlib/Algebra/Lie/Weights/Killing.lean | 303 | 307 |
/-
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.Algebra.Order.Archimedean.IndicatorCard
import Mathlib.Probability.Martingale.Centering
import Mathlib.Probability.Martingale.Convergence
import Mathlib.Prob... |
theorem adapted_process (hs : ∀ n, MeasurableSet[ℱ n] (s n)) : Adapted ℱ (process s) := fun _ =>
Finset.stronglyMeasurable_sum' _ fun _ hk =>
stronglyMeasurable_one.indicator <| ℱ.mono (Finset.mem_range.1 hk) _ <| hs _
| Mathlib/Probability/Martingale/BorelCantelli.lean | 270 | 274 |
/-
Copyright (c) 2019 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
/-!
# Operator norm as an `NNNorm`
Operator norm as an `NNNorm`, i.e. takin... | theorem nndist_le_opNNNorm (x y : E) : nndist (f x) (f y) ≤ ‖f‖₊ * nndist x y :=
dist_le_opNorm f x y
| Mathlib/Analysis/NormedSpace/OperatorNorm/NNNorm.lean | 100 | 101 |
/-
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.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
imp... | end Pdiv
section ProdPrimeFactors
| Mathlib/NumberTheory/ArithmeticFunction.lean | 509 | 512 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
/-!
# Differentiability of models with corners and (extended) charts
In this file... | lemma mfderiv_extChartAt_comp_mfderivWithin_extChartAt_symm {x : M}
{y : E} (hy : y ∈ (extChartAt I x).target) :
(mfderiv I 𝓘(𝕜, E) (extChartAt I x) ((extChartAt I x).symm y)) ∘L
(mfderivWithin 𝓘(𝕜, E) I (extChartAt I x).symm (range I) y) = ContinuousLinearMap.id _ _ := by
have U : UniqueMDiffWithin... | Mathlib/Geometry/Manifold/MFDeriv/Atlas.lean | 263 | 273 |
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Order.Group.Multiset
import Mathlib.Data.Setoid.Basic
import Mathlib.Data.Vector.Basic
import Mathlib.Logic.Nontrivial.Basic
import Mathlib.Tactic.Ap... | @[simps]
def equivCongr (e : α ≃ β) : Sym α n ≃ Sym β n where
| Mathlib/Data/Sym/Basic.lean | 388 | 389 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Topology.Category.Profinite.Nobeling.Basic
import Mathlib.Topology.Category.Profinite.Nobeling.Induction
import Mathlib.Topology.Category.Profinite... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 370 | 373 | |
/-
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... | section Prime
variable {f}
| Mathlib/RingTheory/AdjoinRoot.lean | 315 | 318 |
/-
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.MeasurableSpace.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... | /-! ### The complete lattice of measures -/
| Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 941 | 942 |
/-
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
-/
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Lemmas
import Mathlib.Tactic.Peel
import... | exact mod_cast hp.1.one_lt⟩ }
protected theorem image {q : ℚ_[p]} : q ≠ 0 → ∃ n : ℤ, ‖q‖ = ↑((p : ℚ) ^ (-n)) :=
Quotient.inductionOn q fun f hf ↦
| Mathlib/NumberTheory/Padics/PadicNumbers.lean | 796 | 799 |
/-
Copyright (c) 2022 Tian Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tian Chen, Mantas Bakšys
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.D... | rw [Int.ofNat_sub (Nat.pow_le_pow_left hyx n)]
rw [← Int.natCast_dvd_natCast] at hxy hx
rw [Int.natCast_sub hyx] at *
push_cast at *
exact Int.emultiplicity_pow_sub_pow hp hp1 hxy hx n
· simp only [Nat.sub_eq_zero_iff_le.mpr (Nat.pow_le_pow_left hyx n), emultiplicity_zero,
| Mathlib/NumberTheory/Multiplicity.lean | 239 | 244 |
/-
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
-/
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Module.Rat
import Mathlib.GroupThe... | Commute (a₁ ⊗ₜ[R] b₁) (a₂ ⊗ₜ[R] b₂) :=
ha.tmul hb
| Mathlib/RingTheory/TensorProduct/Basic.lean | 300 | 301 |
/-
Copyright (c) 2024 Johan Commelin. 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.Module.Equiv.Opposite
import Mathlib.Algebra.... | to the space of linear maps `M → M₃`. -/
def compRight (f : M₂ →ₗ[R] M₃) : (M →ₗ[R] M₂) →ₗ[R] M →ₗ[R] M₃ where
| Mathlib/Algebra/Module/LinearMap/End.lean | 368 | 369 |
/-
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.Finset.Lattice.Fold
/-!
# Irreducible and prime elements in an order
This file defines irreducible and prime elements in an order and shows that in ... | Mathlib/Order/Irreducible.lean | 307 | 309 | |
/-
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.Order.IsLUB
/-!
# Order topology on a densely ordered set
-/
open Set Filter TopologicalSpace Topology Func... | theorem interior_Iic' {a : α} (ha : (Ioi a).Nonempty) : interior (Iic a) = Iio a :=
interior_Ici' (α := αᵒᵈ) ha
| Mathlib/Topology/Order/DenselyOrdered.lean | 83 | 84 |
/-
Copyright (c) 2022 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Devon Tuma
-/
import Mathlib.Data.Vector.Basic
/-!
# Theorems about membership of elements in vectors
This file contains theorems for membership in a `v.toList` for a vector `v`.
Having ... | theorem head_mem (v : Vector α (n + 1)) : v.head ∈ v.toList :=
(Vector.mem_iff_get v.head v).2 ⟨0, Vector.get_zero v⟩
| Mathlib/Data/Vector/Mem.lean | 52 | 54 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.LinearAlgebra.AffineSpac... |
variable [Semifield 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E]
{x y z : E}
| Mathlib/Analysis/Convex/Segment.lean | 334 | 336 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Data.Stream.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Data.List.Defs
import Mathlib.Data.Nat.Basic
import Mathlib.Tactic.Common... |
@[deprecated (since := "2025-02-14")] alias get?_take_succ := getElem?_take_succ
@[simp] theorem dropLast_take {n : ℕ} {xs : Stream' α} :
| Mathlib/Data/Stream/Init.lean | 543 | 546 |
/-
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.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Eva... | push_cast
ring
rw [add_pow, map_sum (pderiv true), map_sum (pderiv true), map_sum (MvPolynomial.aeval e),
Finset.sum_mul]
-- Step inside the sum:
refine Finset.sum_congr rfl fun k _ => (w k).trans ?_
simp only [x, y, e, pderiv_true_x, pderiv_true_y, Algebra.id.smul_eq_mul, nsmul_eq... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 338 | 378 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam, Yury Kudryashov
-/
import Mathlib.Algebra.MvPolynomial.Derivation
import Mathlib.Algebra.MvPolynomial.Variables
/-!
# Partial derivatives of polynomials
This file defi... | pderiv i (X j : MvPolynomial σ R) = Pi.single (f := fun _ => _) i 1 j := by
rw [pderiv_def, mkDerivation_X]
| Mathlib/Algebra/MvPolynomial/PDeriv.lean | 89 | 91 |
/-
Copyright (c) 2016 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.Set.Defs
import Mathlib.Logic.Basic
import Mathlib.Logic.ExistsUnique
import Mathlib.Logic.Nonempty
import Mathlib.Logic.Nontrivia... | theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
| Mathlib/Logic/Function/Basic.lean | 568 | 570 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.C... | (hφ : K[X]⁰ ≤ L⁰.comap φ) (f : RatFunc K) :
liftMonoidWithZeroHom φ hφ f = φ f.num / φ f.denom := by
| Mathlib/FieldTheory/RatFunc/Basic.lean | 984 | 985 |
/-
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.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Mul
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
imp... | ENNReal.div_eq_inv_mul, rpow_one, mul_one]
ring
theorem add_rpow_le_rpow_add {p : ℝ} (a b : ℝ≥0∞) (hp1 : 1 ≤ p) : a ^ p + b ^ p ≤ (a + b) ^ p := by
have hp_pos : 0 < p := by positivity
| Mathlib/Analysis/MeanInequalitiesPow.lean | 281 | 285 |
/-
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... | @[to_additive]
lemma QuotientMeasureEqMeasurePreimage.sigmaFiniteQuotient
[i : SigmaFinite ν] [i' : HasFundamentalDomain G α ν]
(μ : Measure (Quotient α_mod_G)) [QuotientMeasureEqMeasurePreimage ν μ] :
SigmaFinite μ := by
rw [sigmaFinite_iff]
obtain ⟨A, hA_meas, hA, hA'⟩ := Measure.toFiniteSpanningSetsI... | Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 828 | 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, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Group.Subgroup.Map
import Mathlib.Algebra.Module.Submodule.... | def submoduleMap (f : M →ₗ[R] M₁) (p : Submodule R M) : p →ₗ[R] p.map f :=
f.restrict fun x hx ↦ Submodule.mem_map.mpr ⟨x, hx, rfl⟩
| Mathlib/Algebra/Module/Submodule/Map.lean | 656 | 658 |
/-
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... | compile_inductive% Code
| Mathlib/Computability/TMToPartrec.lean | 201 | 201 |
/-
Copyright (c) 2022 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.Algebra.Order.Group.Unbundled.Int
import Mathlib.Algebra.Order.Nonneg.Basic
import Mathlib.Algebra.Order.Ring.Unbundled.Rat
imp... | @[simp, norm_cast]
| Mathlib/Data/NNRat/Defs.lean | 142 | 142 |
/-
Copyright (c) 2022 Felix Weilacher. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Felix Weilacher
-/
import Mathlib.Topology.Separation.Regular
/-!
# Perfect Sets
In this file we define perfect subsets of a topological space, and prove some basic properties,
inc... | have : {x}ᶜ ∩ C = C := by simp [h]
rw [AccPt, nhdsWithin, inf_assoc, inf_principal, this]
rw [closure_eq_cluster_pts] at hx
exact hx
/-- In a T1 space, being preperfect is equivalent to having perfect closure. -/
theorem preperfect_iff_perfect_closure [T1Space α] : Preperfect C ↔ Perfect (closure C) := by
co... | Mathlib/Topology/Perfect.lean | 120 | 128 |
/-
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.Operator.BoundedLinearMaps
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.Stron... | AEStronglyMeasurable (fun x => (f x : ℝ) • g x) μ := by
constructor
· rintro ⟨g', g'meas, hg'⟩
have A : MeasurableSet { x : α | f x ≠ 0 } := (hf (measurableSet_singleton 0)).compl
refine ⟨fun x => (f x : ℝ) • g' x, hf.coe_nnreal_real.stronglyMeasurable.smul g'meas, ?_⟩
apply @ae_of_ae_restrict_of_... | Mathlib/MeasureTheory/Function/StronglyMeasurable/Lemmas.lean | 85 | 108 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yury Kudryashov
-/
import Mathlib.MeasureTheory.OuterMeasure.Basic
/-!
# The “almost everywhere” filter of co-null sets.
If `μ` is an outer measure or a measure on `α... | eventually_countable_ball hS
lemma ae_eq_refl (f : α → β) : f =ᵐ[μ] f := EventuallyEq.rfl
| Mathlib/MeasureTheory/OuterMeasure/AE.lean | 107 | 109 |
/-
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.Data.Set.SymmDiff
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Irreducible
/-!
# Connected subsets ... | fun hi hj hk => K _ (h1 hi hj hk) (h2 hi hj hk)
refine IsPreconnected.biUnion_of_reflTransGen H fun i hi j hj => ?_
exact reflTransGen_of_succ _ (fun k hk => ⟨h3 hi hj hk, h1 hi hj hk⟩) fun k hk =>
⟨by rw [inter_comm]; exact h3 hj hi hk, h2 hj hi hk⟩
/-- The iUnion of connected sets indexed by a subset o... | Mathlib/Topology/Connected/Basic.lean | 235 | 240 |
/-
Copyright (c) 2023 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Yaël Dillies, Jineon Baek
-/
import Mathlib.Algebra.EuclideanDomain.Int
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Algebr... |
-/
| Mathlib/NumberTheory/FLT/Basic.lean | 42 | 43 |
/-
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 ... | Mathlib/Topology/CompactOpen.lean | 499 | 510 | |
/-
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.Lie.Abelian
import Mathlib.Algebra.Lie.BaseChange
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Order.Hom.Basic
import Mathlib.RingTheory... | rw [LieIdeal.derivedSeries_eq_bot_iff] at hI hJ ⊢
rw [← le_bot_iff]
let D := derivedSeriesOfIdeal R L; change D k I = ⊥ at hI; change D l J = ⊥ at hJ
calc
D (k + l) (I + J) ≤ D k I + D l J := derivedSeriesOfIdeal_add_le_add I J k l
_ ≤ ⊥ := by rw [hI, hJ]; simp
theorem derivedSeries_map_le (k : ℕ) : (d... | Mathlib/Algebra/Lie/Solvable.lean | 169 | 176 |
/-
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 Mathlib.Algebra.Notation.Defs
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
/-!
# Partial values of a type
... | instance [Inter α] : Inter (Part α) where inter a b := (· ∩ ·) <$> a <*> b
| Mathlib/Data/Part.lean | 593 | 594 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker
-/
import Mathlib.Algebra.Ring.Associated
import Mathlib.Algebra.Ring.Regular
/-!
# Monoids with normalization functions, `gcd`, and `lcm`
This file de... | obtain ⟨h0₂, ⟨b', hb'⟩⟩ := pow_dvd_of_mul_eq_pow hb hab h hc hd₂
rw [ha', hb', hc, mul_pow] at h
have h' : a' * b' = 1 := by
apply (mul_right_inj' h0₁).mp
rw [mul_one]
apply (mul_right_inj' h0₂).mp
rw [← h]
rw [mul_assoc, mul_comm a', ← mul_assoc _ b', ← mul_assoc b', mul_comm b']
use Units.... | Mathlib/Algebra/GCDMonoid/Basic.lean | 575 | 589 |
/-
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.Finset.Pi
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Finset.Sum
import Mathlib.Data.Set.Finite.Basic
/-!
# Preimage of ... |
theorem map_subset_iff_subset_preimage {f : α ↪ β} {s : Finset α} {t : Finset β} :
s.map f ⊆ t ↔ s ⊆ t.preimage f f.injective.injOn := by
| Mathlib/Data/Finset/Preimage.lean | 87 | 89 |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Ring.Commute
import Mathlib.Algebra.Ring.Invertible
impo... | theorem same_sub_div {a b : K} (h : b ≠ 0) : (b - a) / b = 1 - a / b := by
simpa only [← @div_self _ _ b h] using (div_sub_div_same b a b).symm
| Mathlib/Algebra/Field/Basic.lean | 96 | 98 |
/-
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.Lie.Subalgebra
import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie submodules, we construct ... | (add : ∀ x y hx hy, p x hx → p y hy → p (x + y) (add_mem ‹_› ‹_›))
(smul : ∀ (a : R) (x hx), p x hx → p (a • x) (SMulMemClass.smul_mem _ hx)) {x}
(lie : ∀ (x : L) (y hy), p y hy → p (⁅x, y⁆) (LieSubmodule.lie_mem _ ‹_›))
| Mathlib/Algebra/Lie/Submodule.lean | 731 | 733 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Algebra.Group.AddChar
import Mathlib.Analysis.Complex.Circle
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.Prod
imp... |
variable [CompleteSpace E] [CompleteSpace F]
/-- The Fourier transform satisfies `∫ 𝓕 f * g = ∫ f * 𝓕 g`, i.e., it is self-adjoint.
Version where the multiplication is replaced by a general bilinear form `M`. -/
theorem integral_bilin_fourierIntegral_eq_flip
{f : V → E} {g : W → F} (M : E →L[ℂ] F →L[ℂ] G) (he :... | Mathlib/Analysis/Fourier/FourierTransform.lean | 167 | 175 |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.PUnit
import Mathlib.Algebra.Group.Subgroup.Ker
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.GroupTheory.Congruence... | @[deprecated (since := "2025-03-11")]
alias _root_.AddMonoid.Coprod.toSum_apply_inl := AddMonoid.Coprod.toProd_apply_inl
| Mathlib/GroupTheory/Coprod/Basic.lean | 512 | 513 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,986 | 1,995 | |
/-
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... |
theorem withTopEquiv_zero : withTopEquiv 0 = 0 := by
simp
| Mathlib/Data/Nat/PartENat.lean | 628 | 630 |
/-
Copyright (c) 2020 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.GroupTheory.Complement
/-!
# Semidirect product
This file defines semidirect products of groups, and the canonical maps in and out of the
semidirect prod... | @[simp]
| Mathlib/GroupTheory/SemidirectProduct.lean | 166 | 166 |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Dynamics.BirkhoffSum.NormedSpace
/-!
# Von Neumann Mean Ergodic Theorem in a Hilbert Space
In ... | theorem ContinuousLinearMap.tendsto_birkhoffAverage_orthogonalProjection (f : E →L[𝕜] E)
(hf : ‖f‖ ≤ 1) (x : E) :
Tendsto (birkhoffAverage 𝕜 f _root_.id · x) atTop
(𝓝 <| (LinearMap.eqLocus f 1).orthogonalProjection x) := by
/- Due to the previous theorem, it suffices to verify
that the range of `f ... | Mathlib/Analysis/InnerProductSpace/MeanErgodic.lean | 84 | 103 |
/-
Copyright (c) 2019 Minchao Wu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Minchao Wu, Mario Carneiro
-/
import Mathlib.Computability.Halting
/-!
# Strong reducibility and degrees.
This file defines the notions of computable many-one reduction and one-one
reduc... | Mathlib/Computability/Reduce.lean | 490 | 491 | |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 2,164 | 2,166 | |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Joachim Breitner
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mat... |
-- A group action on α, and the ping-pong sets
variable {α : Type*} [MulAction G α]
variable (X Y : ι → Set α)
| Mathlib/GroupTheory/CoprodI.lean | 963 | 966 |
/-
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.Data.Int.Bitwise
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Data.Set.Function
import Mathlib.Data.Set.Monotone
import Mathlib.Order.Interval.Set... | natAbs a = natAbs b ↔ a = -b := by
simpa only [Int.natAbs_neg] using natAbs_inj_of_nonneg_of_nonneg ha (neg_nonneg_of_nonpos hb)
| Mathlib/Data/Int/Lemmas.lean | 60 | 61 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
/-!
# Preserving images
In this file, we show that if a fu... | end CategoryTheory
| Mathlib/CategoryTheory/Limits/Preserves/Shapes/Images.lean | 62 | 63 |
/-
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... | rw [not_le] at h3
rw [coeff_eq_zero_of_natDegree_lt (lt_tsub_iff_right.mpr (Nat.add_lt_add_left h3 _))]
exact coeff_eq_zero_of_lt_natTrailingDegree (by rwa [mirror_natTrailingDegree])
--TODO: Extract `Finset.sum_range_rev_at` lemma.
theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 := by
simp_rw [eval_eq_su... | Mathlib/Algebra/Polynomial/Mirror.lean | 82 | 97 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jujian Zhang
-/
import Mathlib.Algebra.GroupWithZero.Subgroup
import Mathlib.Algebra.Order.Group.Action
import Mathlib.LinearAlgebra.Finsupp.Supported
import Mathlib.LinearAl... | s • N = ⨆ (a ∈ s), a • N := by
refine Eq.trans (congrArg sInf ?_) csInf_Ici
simp_rw [← Set.Ici_def, iSup_le_iff, @forall_comm M]
exact Set.ext fun _ => forall₂_congr (fun _ _ => Iff.symm map_le_iff_le_comap)
theorem set_smul_span [SMulCommClass S R M] (s : Set S) (t : Set M) :
s • span R t = span R (s • ... | Mathlib/Algebra/Module/Submodule/Pointwise.lean | 366 | 372 |
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