Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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/-
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.Topology.Order
#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c73... | Mathlib/Topology/Maps.lean | 676 | 678 | theorem closed_iff_preimage_closed (hf : ClosedEmbedding f) {s : Set Y}
(hs : s ⊆ range f) : IsClosed s ↔ IsClosed (f ⁻¹' s) := by |
rw [hf.closed_iff_image_closed, image_preimage_eq_of_subset hs]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Patrick Massot
-/
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.UniformSpace.Separation
import Mathlib.Topology.DenseEmbedding
... | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | 191 | 193 | theorem UniformEmbedding.of_comp_iff {g : β → γ} (hg : UniformEmbedding g) {f : α → β} :
UniformEmbedding (g ∘ f) ↔ UniformEmbedding f := by |
simp_rw [uniformEmbedding_iff, hg.toUniformInducing.of_comp_iff, hg.inj.of_comp_iff f]
|
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.EMetricSpace.Basic
import Mathlib.Topology.Bornology.Constructions
imp... | Mathlib/Topology/MetricSpace/PseudoMetric.lean | 2,078 | 2,085 | theorem secondCountable_of_almost_dense_set
(H : ∀ ε > (0 : ℝ), ∃ s : Set α, s.Countable ∧ ∀ x, ∃ y ∈ s, dist x y ≤ ε) :
SecondCountableTopology α := by |
refine EMetric.secondCountable_of_almost_dense_set fun ε ε0 => ?_
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 ε0 with ⟨ε', ε'0, ε'ε⟩
choose s hsc y hys hyx using H ε' (mod_cast ε'0)
refine ⟨s, hsc, iUnion₂_eq_univ_iff.2 fun x => ⟨y x, hys _, le_trans ?_ ε'ε.le⟩⟩
exact mod_cast hyx x
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Scott Morrison
-/
import Mathlib.CategoryTheory.Comma.Basic
import Mathlib.CategoryTheory.PUnit
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.Category... | Mathlib/CategoryTheory/Comma/StructuredArrow.lean | 756 | 758 | theorem hom_ext (h : IsUniversal f) {c : C} {η η' : c ⟶ f.left}
(w : S.map η ≫ f.hom = S.map η' ≫ f.hom) : η = η' := by |
rw [h.hom_desc η, h.hom_desc η', w]
|
/-
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.NumberTheory.ModularForms.JacobiTheta.TwoVariable
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
#align_import number_theory.modular_forms.jacob... | Mathlib/NumberTheory/ModularForms/JacobiTheta/OneVariable.lean | 96 | 116 | theorem norm_jacobiTheta_sub_one_le {τ : ℂ} (hτ : 0 < im τ) :
‖jacobiTheta τ - 1‖ ≤ 2 / (1 - rexp (-π * τ.im)) * rexp (-π * τ.im) := by |
suffices ‖∑' n : ℕ, cexp (π * I * ((n : ℂ) + 1) ^ 2 * τ)‖ ≤
rexp (-π * τ.im) / (1 - rexp (-π * τ.im)) by
calc
‖jacobiTheta τ - 1‖ = ↑2 * ‖∑' n : ℕ, cexp (π * I * ((n : ℂ) + 1) ^ 2 * τ)‖ := by
rw [sub_eq_iff_eq_add'.mpr (jacobiTheta_eq_tsum_nat hτ), norm_mul, Complex.norm_eq_abs,
Com... |
/-
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.Data.Set.Prod
#align_import data.set.n_ary from "leanprover-community/mathlib"@"5e526d18cea33550268dcbbddcb822d5cde40654"
/-!
# N-ary images of s... | Mathlib/Data/Set/NAry.lean | 214 | 216 | theorem image2_image_left (f : γ → β → δ) (g : α → γ) :
image2 f (g '' s) t = image2 (fun a b => f (g a) b) s t := by |
ext; simp
|
/-
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.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.EuclideanDist
import Mathlib... | Mathlib/Analysis/Fourier/RiemannLebesgueLemma.lean | 201 | 226 | theorem tendsto_integral_exp_inner_smul_cocompact :
Tendsto (fun w : V => ∫ v, 𝐞 (-⟪v, w⟫) • f v) (cocompact V) (𝓝 0) := by |
by_cases hfi : Integrable f; swap
· convert tendsto_const_nhds (x := (0 : E)) with w
apply integral_undef
rwa [Real.fourierIntegral_convergent_iff]
refine Metric.tendsto_nhds.mpr fun ε hε => ?_
obtain ⟨g, hg_supp, hfg, hg_cont, -⟩ :=
hfi.exists_hasCompactSupport_integral_sub_le (div_pos hε two_pos)... |
/-
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.Nat.Defs
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.Co... | Mathlib/Data/Fin/Basic.lean | 329 | 330 | theorem pos_iff_ne_zero' [NeZero n] (a : Fin n) : 0 < a ↔ a ≠ 0 := by |
rw [← val_fin_lt, val_zero', Nat.pos_iff_ne_zero, Ne, Ne, ext_iff, val_zero']
|
/-
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.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.... | Mathlib/Data/Nat/GCD/Basic.lean | 250 | 252 | theorem coprime_pow_right_iff {n : ℕ} (hn : 0 < n) (a b : ℕ) :
Nat.Coprime a (b ^ n) ↔ Nat.Coprime a b := by |
rw [Nat.coprime_comm, coprime_pow_left_iff hn, Nat.coprime_comm]
|
/-
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.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Ma... | Mathlib/Computability/TuringMachine.lean | 836 | 854 | theorem mem_eval {σ} {f : σ → Option σ} {a b} : b ∈ eval f a ↔ Reaches f a b ∧ f b = none := by |
refine ⟨fun h ↦ ?_, fun ⟨h₁, h₂⟩ ↦ ?_⟩
· -- Porting note: Explicitly specify `c`.
refine @evalInduction _ _ _ (fun a ↦ Reaches f a b ∧ f b = none) _ h fun a h IH ↦ ?_
cases' e : f a with a'
· rw [Part.mem_unique h
(PFun.mem_fix_iff.2 <| Or.inl <| Part.mem_some_iff.2 <| by rw [e] <;> rfl)]
... |
/-
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.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Ma... | Mathlib/Computability/TuringMachine.lean | 461 | 466 | theorem ListBlank.append_assoc {Γ} [Inhabited Γ] (l₁ l₂ : List Γ) (l₃ : ListBlank Γ) :
ListBlank.append (l₁ ++ l₂) l₃ = ListBlank.append l₁ (ListBlank.append l₂ l₃) := by |
refine l₃.inductionOn fun l ↦ ?_
-- Porting note: Added `suffices` to get `simp` to work.
suffices append (l₁ ++ l₂) (mk l) = append l₁ (append l₂ (mk l)) by exact this
simp only [ListBlank.append_mk, List.append_assoc]
|
/-
Copyright (c) 2022 Vincent Beffara. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vincent Beffara
-/
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Calculus.Dslope
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Anal... | Mathlib/Analysis/Analytic/IsolatedZeros.lean | 44 | 45 | theorem hasSum_at_zero (a : ℕ → E) : HasSum (fun n => (0 : 𝕜) ^ n • a n) (a 0) := by |
convert hasSum_single (α := E) 0 fun b h ↦ _ <;> simp [*]
|
/-
Copyright (c) 2020 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.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTh... | Mathlib/MeasureTheory/Function/LpSpace.lean | 1,311 | 1,313 | theorem continuous_negPart [Fact (1 ≤ p)] : Continuous fun f : Lp ℝ p μ => negPart f := by |
unfold negPart
exact continuous_posPart.comp continuous_neg
|
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 369 | 370 | theorem zero_testAgainstNN_apply (f : Ω →ᵇ ℝ≥0) : (0 : FiniteMeasure Ω).testAgainstNN f = 0 := by |
simp only [testAgainstNN, toMeasure_zero, lintegral_zero_measure, ENNReal.zero_toNNReal]
|
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.EMetricSpace.Basic
import Mathlib.Topology.Bornology.Constructions
imp... | Mathlib/Topology/MetricSpace/PseudoMetric.lean | 595 | 596 | theorem closedBall_eq_bInter_ball : closedBall x ε = ⋂ δ > ε, ball x δ := by |
ext y; rw [mem_closedBall, ← forall_lt_iff_le', mem_iInter₂]; rfl
|
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou, Scott Morrison, Adam Topaz
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathlib.C... | Mathlib/CategoryTheory/Limits/Shapes/ConcreteCategory.lean | 324 | 333 | theorem widePushout_exists_rep {B : C} {α : Type _} {X : α → C} (f : ∀ j : α, B ⟶ X j)
[HasWidePushout.{v} B X f] [PreservesColimit (wideSpan B X f) (forget C)]
(x : ↑(widePushout B X f)) : (∃ y : B, head f y = x) ∨ ∃ (i : α) (y : X i), ι f i y = x := by |
obtain ⟨_ | j, y, rfl⟩ := Concrete.colimit_exists_rep _ x
· left
use y
rfl
· right
use j, y
rfl
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34... | Mathlib/Data/Finsupp/Multiset.lean | 122 | 125 | theorem toMultiset_inf [DecidableEq α] (f g : α →₀ ℕ) :
toMultiset (f ⊓ g) = toMultiset f ∩ toMultiset g := by |
ext
simp_rw [Multiset.count_inter, Finsupp.count_toMultiset, Finsupp.inf_apply, inf_eq_min]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Regular.Pow
import Mathl... | Mathlib/Algebra/MvPolynomial/Basic.lean | 817 | 821 | theorem coeff_mul_X' [DecidableEq σ] (m) (s : σ) (p : MvPolynomial σ R) :
coeff m (p * X s) = if s ∈ m.support then coeff (m - Finsupp.single s 1) p else 0 := by |
refine (coeff_mul_monomial' _ _ _ _).trans ?_
simp_rw [Finsupp.single_le_iff, Finsupp.mem_support_iff, Nat.succ_le_iff, pos_iff_ne_zero,
mul_one]
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-communit... | Mathlib/MeasureTheory/Integral/Bochner.lean | 432 | 439 | theorem norm_setToSimpleFunc_le_integral_norm (T : Set α → E →L[ℝ] F) {C : ℝ}
(hT_norm : ∀ s, MeasurableSet s → μ s < ∞ → ‖T s‖ ≤ C * (μ s).toReal) {f : α →ₛ E}
(hf : Integrable f μ) : ‖f.setToSimpleFunc T‖ ≤ C * (f.map norm).integral μ :=
calc
‖f.setToSimpleFunc T‖ ≤ C * ∑ x ∈ f.range, ENNReal.toReal (μ ... |
rw [map_integral f norm hf norm_zero]; simp_rw [smul_eq_mul]
|
/-
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.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.QuotientGrou... | Mathlib/Topology/Algebra/Group/Basic.lean | 1,379 | 1,381 | theorem singleton_mul_mem_nhds (a : α) {b : α} (h : s ∈ 𝓝 b) : {a} * s ∈ 𝓝 (a * b) := by |
have := smul_mem_nhds a h
rwa [← singleton_smul] at this
|
/-
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
#align_import topology.metric_space.hausdorff_distance from "lea... | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 215 | 216 | theorem infEdist_image (hΦ : Isometry Φ) : infEdist (Φ x) (Φ '' t) = infEdist x t := by |
simp only [infEdist, iInf_image, hΦ.edist_eq]
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Ma... | Mathlib/LinearAlgebra/LinearIndependent.lean | 126 | 128 | theorem linearIndependent_iff :
LinearIndependent R v ↔ ∀ l, Finsupp.total ι M R v l = 0 → l = 0 := by |
simp [LinearIndependent, LinearMap.ker_eq_bot']
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-communit... | Mathlib/MeasureTheory/Integral/Bochner.lean | 1,743 | 1,747 | theorem integral_subtype_comap {α} [MeasurableSpace α] {μ : Measure α} {s : Set α}
(hs : MeasurableSet s) (f : α → G) :
∫ x : s, f (x : α) ∂(Measure.comap Subtype.val μ) = ∫ x in s, f x ∂μ := by |
rw [← map_comap_subtype_coe hs]
exact ((MeasurableEmbedding.subtype_coe hs).integral_map _).symm
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Antoine Chambert-Loir
-/
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Ring.Action.Basic
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Algebra.Group.Hom.CompType... | Mathlib/GroupTheory/GroupAction/Hom.lean | 150 | 154 | theorem _root_.IsScalarTower.smulHomClass [MulOneClass X] [SMul X Y] [IsScalarTower M' X Y]
[MulActionHomClass F X X Y] : MulActionHomClass F M' X Y where
map_smulₛₗ f m x := by |
rw [← mul_one (m • x), ← smul_eq_mul, map_smul, smul_assoc, ← map_smul,
smul_eq_mul, mul_one, id_eq]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
i... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 561 | 573 | theorem differentiable_set_subset_D :
{ x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K } ⊆ D f K := by |
intro x hx
rw [D, mem_iInter]
intro e
have : (0 : ℝ) < (1 / 2) ^ e := pow_pos (by norm_num) _
rcases mem_A_of_differentiable this hx.1 with ⟨R, R_pos, hR⟩
obtain ⟨n, hn⟩ : ∃ n : ℕ, (1 / 2) ^ n < R :=
exists_pow_lt_of_lt_one R_pos (by norm_num : (1 : ℝ) / 2 < 1)
simp only [mem_iUnion, mem_iInter, B, m... |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Ma... | Mathlib/LinearAlgebra/LinearIndependent.lean | 1,054 | 1,072 | theorem surjective_of_linearIndependent_of_span [Nontrivial R] (hv : LinearIndependent R v)
(f : ι' ↪ ι) (hss : range v ⊆ span R (range (v ∘ f))) : Surjective f := by |
intro i
let repr : (span R (range (v ∘ f)) : Type _) → ι' →₀ R := (hv.comp f f.injective).repr
let l := (repr ⟨v i, hss (mem_range_self i)⟩).mapDomain f
have h_total_l : Finsupp.total ι M R v l = v i := by
dsimp only [l]
rw [Finsupp.total_mapDomain]
rw [(hv.comp f f.injective).total_repr]
-- Po... |
/-
Copyright (c) 2021 Jakob Scholbach. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob Scholbach
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.FieldTheory.Separable
#align_import field_theory.separable_degree from "lea... | Mathlib/RingTheory/Polynomial/SeparableDegree.lean | 111 | 116 | theorem _root_.Irreducible.hasSeparableContraction (q : ℕ) [hF : ExpChar F q] {f : F[X]}
(irred : Irreducible f) : HasSeparableContraction q f := by |
cases hF
· exact ⟨f, irred.separable, ⟨0, by rw [pow_zero, expand_one]⟩⟩
· rcases exists_separable_of_irreducible q irred ‹q.Prime›.ne_zero with ⟨n, g, hgs, hge⟩
exact ⟨g, hgs, n, hge⟩
|
/-
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, Floris van Doorn
-/
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheor... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 1,082 | 1,083 | theorem mk_embedding_eq_zero_iff_lt : #(α ↪ β) = 0 ↔ #β < #α := by |
rw [mk_embedding_eq_zero_iff_lift_lt, lift_lt]
|
/-
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
-/
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.LinearAlgebra.Span
#align_import linear_algebra.quotient from... | Mathlib/LinearAlgebra/Quotient.lean | 588 | 590 | theorem ker_le_range_iff {f : M →ₛₗ[τ₁₂] M₂} {g : M₂ →ₛₗ[τ₂₃] M₃} :
ker g ≤ range f ↔ f.range.mkQ.comp g.ker.subtype = 0 := by |
rw [← range_le_ker_iff, Submodule.ker_mkQ, Submodule.range_subtype]
|
/-
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.Set.Card
import Mathlib.Order.Minimal
import Mathlib.Data.Matroid.Init
/-!
# Matroids
A `Matroid` is a structure that combinatorially abstracts
the ... | Mathlib/Data/Matroid/Basic.lean | 932 | 933 | theorem Basis.basis_union (hIX : M.Basis I X) (hIY : M.Basis I Y) : M.Basis I (X ∪ Y) := by |
convert hIX.union_basis_union hIY _ <;> rw [union_self]; exact hIX.indep
|
/-
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.Module.Submodule.Lattice
import Mathlib.Algebra.Module.Sub... | Mathlib/Algebra/Module/Submodule/Map.lean | 585 | 591 | theorem inf_comap_le_comap_add (f₁ f₂ : M →ₛₗ[τ₁₂] M₂) :
comap f₁ q ⊓ comap f₂ q ≤ comap (f₁ + f₂) q := by |
rw [SetLike.le_def]
intro m h
change f₁ m + f₂ m ∈ q
change f₁ m ∈ q ∧ f₂ m ∈ q at h
apply q.add_mem h.1 h.2
|
/-
Copyright (c) 2024 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.Data.Setoid.Partition
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.Group... | Mathlib/GroupTheory/GroupAction/Blocks.lean | 280 | 296 | theorem IsBlock.iInter {ι : Type*} {B : ι → Set X} (hB : ∀ i : ι, IsBlock G (B i)) :
IsBlock G (⋂ i, B i) := by |
by_cases hι : (IsEmpty ι)
· -- ι = ∅, block = ⊤
suffices (⋂ i : ι, B i) = Set.univ by simpa only [this] using isBlock_top X
simpa only [Set.top_eq_univ, Set.iInter_eq_univ] using (hι.elim' ·)
rw [IsBlock.def_one]
intro g
rw [Set.smul_set_iInter]
by_cases h : ∃ i : ι, Disjoint (g • B i) (B i)
· ri... |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma, Oliver Nash
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Ring.Opposite
import Mathlib.GroupTheory.Gro... | Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean | 244 | 250 | theorem map_le_nonZeroDivisors_of_injective [NoZeroDivisors M'] [MonoidWithZeroHomClass F M M']
(f : F) (hf : Function.Injective f) {S : Submonoid M} (hS : S ≤ M⁰) : S.map f ≤ M'⁰ := by |
cases subsingleton_or_nontrivial M
· simp [Subsingleton.elim S ⊥]
· exact le_nonZeroDivisors_of_noZeroDivisors fun h ↦
let ⟨x, hx, hx0⟩ := h
zero_ne_one (hS (hf (hx0.trans (map_zero f).symm) ▸ hx : 0 ∈ S) 1 (mul_zero 1)).symm
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.Sub
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_t... | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | 1,068 | 1,084 | theorem rnDeriv_smul_right' (ν μ : Measure α) [SigmaFinite ν] [SigmaFinite μ]
{r : ℝ≥0} (hr : r ≠ 0) :
ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ := by |
refine (absolutelyContinuous_smul <| ENNReal.coe_ne_zero.2 hr).ae_le
(?_ : ν.rnDeriv (r • μ) =ᵐ[r • μ] r⁻¹ • ν.rnDeriv μ)
rw [← withDensity_eq_iff_of_sigmaFinite]
· simp_rw [ENNReal.smul_def]
rw [withDensity_smul _ (measurable_rnDeriv _ _)]
suffices ν.singularPart (r • μ) + withDensity (r • μ) (rnDer... |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Interval.Set.OrderIso
import Mathlib.Data.... | Mathlib/Order/UpperLower/Basic.lean | 623 | 623 | theorem coe_iSup (f : ι → UpperSet α) : (↑(⨆ i, f i) : Set α) = ⋂ i, f i := by | simp [iSup]
|
/-
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.Analysis.Normed.Group.Quotient
import Mathlib.Topology.Instances.AddCircle
#align_import analysis.normed.group.add_circle from "leanprover-community/mathlib... | Mathlib/Analysis/Normed/Group/AddCircle.lean | 278 | 278 | theorem norm_eq {x : ℝ} : ‖(x : UnitAddCircle)‖ = |x - round x| := by | simp [AddCircle.norm_eq]
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Sébastien Gouëzel, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.LinearAlgebra.FiniteDimen... | Mathlib/Analysis/InnerProductSpace/PiL2.lean | 200 | 211 | theorem DirectSum.IsInternal.isometryL2OfOrthogonalFamily_symm_apply [DecidableEq ι]
{V : ι → Submodule 𝕜 E} (hV : DirectSum.IsInternal V)
(hV' : OrthogonalFamily 𝕜 (fun i => V i) fun i => (V i).subtypeₗᵢ) (w : PiLp 2 fun i => V i) :
(hV.isometryL2OfOrthogonalFamily hV').symm w = ∑ i, (w i : E) := by |
classical
let e₁ := DirectSum.linearEquivFunOnFintype 𝕜 ι fun i => V i
let e₂ := LinearEquiv.ofBijective (DirectSum.coeLinearMap V) hV
suffices ∀ v : ⨁ i, V i, e₂ v = ∑ i, e₁ v i by exact this (e₁.symm w)
intro v
-- Porting note: added `DFinsupp.lsum`
simp [e₁, e₂, DirectSum.coeLinearMap, Di... |
/-
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
#align_import measure_theo... | Mathlib/MeasureTheory/Covering/DensityTheorem.lean | 112 | 132 | theorem tendsto_closedBall_filterAt {K : ℝ} {x : α} {ι : Type*} {l : Filter ι} (w : ι → α)
(δ : ι → ℝ) (δlim : Tendsto δ l (𝓝[>] 0)) (xmem : ∀ᶠ j in l, x ∈ closedBall (w j) (K * δ j)) :
Tendsto (fun j => closedBall (w j) (δ j)) l ((vitaliFamily μ K).filterAt x) := by |
refine (vitaliFamily μ K).tendsto_filterAt_iff.mpr ⟨?_, fun ε hε => ?_⟩
· filter_upwards [xmem, δlim self_mem_nhdsWithin] with j hj h'j
exact closedBall_mem_vitaliFamily_of_dist_le_mul μ hj h'j
· rcases l.eq_or_neBot with rfl | h
· simp
have hK : 0 ≤ K := by
rcases (xmem.and (δlim self_mem_nhds... |
/-
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.Algebra.Group.Nat
import Mathlib.Algebra.Order.Sub.Canonical
import Mathlib.Data.List.Perm
import Mathlib.Data.Set.List
import Mathlib.Init.Quot... | Mathlib/Data/Multiset/Basic.lean | 2,679 | 2,681 | theorem inter_replicate (s : Multiset α) (n : ℕ) (x : α) :
s ∩ replicate n x = replicate (min (s.count x) n) x := by |
rw [inter_comm, replicate_inter, min_comm]
|
/-
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.Deriv.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Covering.Bes... | Mathlib/MeasureTheory/Function/Jacobian.lean | 1,109 | 1,118 | theorem map_withDensity_abs_det_fderiv_eq_addHaar (hs : MeasurableSet s)
(hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) (h'f : Measurable f) :
Measure.map f ((μ.restrict s).withDensity fun x => ENNReal.ofReal |(f' x).det|) =
μ.restrict (f '' s) := by |
apply Measure.ext fun t ht => ?_
rw [map_apply h'f ht, withDensity_apply _ (h'f ht), Measure.restrict_apply ht,
restrict_restrict (h'f ht),
lintegral_abs_det_fderiv_eq_addHaar_image μ ((h'f ht).inter hs)
(fun x hx => (hf' x hx.2).mono inter_subset_right) (hf.mono inter_subset_right),
image_preima... |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedd... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 472 | 484 | theorem volume_fundamentalDomain_fractionalIdealLatticeBasis :
volume (fundamentalDomain (fractionalIdealLatticeBasis K I)) =
.ofReal (FractionalIdeal.absNorm I.1) * volume (fundamentalDomain (latticeBasis K)) := by |
let e : (Module.Free.ChooseBasisIndex ℤ I) ≃ (Module.Free.ChooseBasisIndex ℤ (𝓞 K)) := by
refine Fintype.equivOfCardEq ?_
rw [← finrank_eq_card_chooseBasisIndex, ← finrank_eq_card_chooseBasisIndex,
fractionalIdeal_rank]
rw [← fundamentalDomain_reindex (fractionalIdealLatticeBasis K I) e,
measure... |
/-
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.Algebra.GroupWithZero.Indicator
import Mathlib.Topology.ContinuousOn
import Mathlib.Topology.Instances.ENNReal
#align_import topology.semicontin... | Mathlib/Topology/Semicontinuous.lean | 353 | 366 | theorem lowerSemicontinuous_iff_isClosed_epigraph {f : α → γ} :
LowerSemicontinuous f ↔ IsClosed {p : α × γ | f p.1 ≤ p.2} := by |
constructor
· rw [lowerSemicontinuous_iff_le_liminf, isClosed_iff_forall_filter]
rintro hf ⟨x, y⟩ F F_ne h h'
rw [nhds_prod_eq, le_prod] at h'
calc f x ≤ liminf f (𝓝 x) := hf x
_ ≤ liminf f (map Prod.fst F) := liminf_le_liminf_of_le h'.1
_ = liminf (f ∘ Prod.fst) F := (Filter.liminf_comp _ _ _... |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.Data.List.Chain
import Mathlib.CategoryTheory.PUnit
import Mathlib.CategoryTheory.Groupoid
import Mathlib.CategoryTheory.Category.ULift
... | Mathlib/CategoryTheory/IsConnected.lean | 115 | 117 | theorem any_functor_const_on_obj [IsPreconnected J] {α : Type u₂} (F : J ⥤ Discrete α) (j j' : J) :
F.obj j = F.obj j' := by |
ext; exact ((isoConstant F j').hom.app j).down.1
|
/-
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.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingul... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 223 | 239 | theorem exists_simpleFunc_forall_lintegral_sub_lt_of_pos {f : α → ℝ≥0∞} (h : ∫⁻ x, f x ∂μ ≠ ∞)
{ε : ℝ≥0∞} (hε : ε ≠ 0) :
∃ φ : α →ₛ ℝ≥0,
(∀ x, ↑(φ x) ≤ f x) ∧
∀ ψ : α →ₛ ℝ≥0, (∀ x, ↑(ψ x) ≤ f x) → (map (↑) (ψ - φ)).lintegral μ < ε := by |
rw [lintegral_eq_nnreal] at h
have := ENNReal.lt_add_right h hε
erw [ENNReal.biSup_add] at this <;> [skip; exact ⟨0, fun x => zero_le _⟩]
simp_rw [lt_iSup_iff, iSup_lt_iff, iSup_le_iff] at this
rcases this with ⟨φ, hle : ∀ x, ↑(φ x) ≤ f x, b, hbφ, hb⟩
refine ⟨φ, hle, fun ψ hψ => ?_⟩
have : (map (↑) φ).li... |
/-
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.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.To... | Mathlib/Topology/UniformSpace/Basic.lean | 199 | 202 | theorem subset_iterate_compRel {s t : Set (α × α)} (h : idRel ⊆ s) (n : ℕ) :
t ⊆ (s ○ ·)^[n] t := by |
induction' n with n ihn generalizing t
exacts [Subset.rfl, (right_subset_compRel h).trans ihn]
|
/-
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.Init.Control.Combinators
import Mathlib.Init.Function
import Mathlib.Tactic.CasesM
import Mathlib.Tactic.Attr.Core
#align_import control.basic from "l... | Mathlib/Control/Basic.lean | 189 | 190 | theorem guard_false {h : Decidable False} : @guard F _ False h = failure := by |
simp [guard, if_neg not_false]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
import Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable
#align_import measure_theory.function.cond... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Unique.lean | 76 | 93 | theorem Lp.ae_eq_zero_of_forall_setIntegral_eq_zero' (hm : m ≤ m0) (f : Lp E' p μ)
(hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
(hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn f s μ)
(hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
(hf_meas : AEStronglyMeasurabl... |
let f_meas : lpMeas E' 𝕜 m p μ := ⟨f, hf_meas⟩
-- Porting note: `simp only` does not call `rfl` to try to close the goal. See https://github.com/leanprover-community/mathlib4/issues/5025
have hf_f_meas : f =ᵐ[μ] f_meas := by simp only [Subtype.coe_mk]; rfl
refine hf_f_meas.trans ?_
refine lpMeas.ae_eq_zero_... |
/-
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, Scott Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#alig... | Mathlib/Algebra/Polynomial/Roots.lean | 309 | 311 | theorem mem_nthRoots {n : ℕ} (hn : 0 < n) {a x : R} : x ∈ nthRoots n a ↔ x ^ n = a := by |
rw [nthRoots, mem_roots (X_pow_sub_C_ne_zero hn a), IsRoot.def, eval_sub, eval_C, eval_pow,
eval_X, sub_eq_zero]
|
/-
Copyright (c) 2022 Rémy Degenne, Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.MeasureTheory.Function.Egorov
import Mathlib.MeasureTheory.Function.LpSpace
#a... | Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean | 128 | 138 | theorem tendstoInMeasure_of_tendsto_ae [IsFiniteMeasure μ] (hf : ∀ n, AEStronglyMeasurable (f n) μ)
(hfg : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))) : TendstoInMeasure μ f atTop g := by |
have hg : AEStronglyMeasurable g μ := aestronglyMeasurable_of_tendsto_ae _ hf hfg
refine TendstoInMeasure.congr (fun i => (hf i).ae_eq_mk.symm) hg.ae_eq_mk.symm ?_
refine tendstoInMeasure_of_tendsto_ae_of_stronglyMeasurable
(fun i => (hf i).stronglyMeasurable_mk) hg.stronglyMeasurable_mk ?_
have hf_eq_ae :... |
/-
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
#align_... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 149 | 150 | theorem eq_zero_rpow_iff {x : ℝ} {a : ℝ} : a = 0 ^ x ↔ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1 := by |
rw [← zero_rpow_eq_iff, eq_comm]
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Functor.Category
import Mathlib.CategoryTheory.Iso
#align_import category_theory.natural... | Mathlib/CategoryTheory/NatIso.lean | 254 | 262 | theorem isIso_map_iff {F₁ F₂ : C ⥤ D} (e : F₁ ≅ F₂) {X Y : C} (f : X ⟶ Y) :
IsIso (F₁.map f) ↔ IsIso (F₂.map f) := by |
revert F₁ F₂
suffices ∀ {F₁ F₂ : C ⥤ D} (_ : F₁ ≅ F₂) (_ : IsIso (F₁.map f)), IsIso (F₂.map f) by
exact fun F₁ F₂ e => ⟨this e, this e.symm⟩
intro F₁ F₂ e hf
refine IsIso.mk ⟨e.inv.app Y ≫ inv (F₁.map f) ≫ e.hom.app X, ?_, ?_⟩
· simp only [NatTrans.naturality_assoc, IsIso.hom_inv_id_assoc, Iso.inv_hom_id... |
/-
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.GroupPower.IterateHom
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Da... | Mathlib/GroupTheory/OrderOfElement.lean | 381 | 383 | theorem Function.Injective.isOfFinOrder_iff [Monoid H] {f : G →* H} (hf : Injective f) :
IsOfFinOrder (f x) ↔ IsOfFinOrder x := by |
rw [← orderOf_pos_iff, orderOf_injective f hf x, ← orderOf_pos_iff]
|
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Measure.GiryMonad
#align_import probability.kernel.basic from "leanprover-community/mathlib"@"... | Mathlib/Probability/Kernel/Basic.lean | 580 | 583 | theorem setIntegral_restrict {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
{f : β → E} {a : α} (hs : MeasurableSet s) (t : Set β) :
∫ x in t, f x ∂kernel.restrict κ hs a = ∫ x in t ∩ s, f x ∂κ a := by |
rw [restrict_apply, Measure.restrict_restrict' hs]
|
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Topology.MetricSpace.Isometry
import Mathlib.Topology.MetricSpace.Lipschitz
#align_import topology.metric_spa... | Mathlib/Topology/MetricSpace/IsometricSMul.lean | 310 | 313 | theorem preimage_mul_right_ball [IsometricSMul Gᵐᵒᵖ G] (a b : G) (r : ℝ≥0∞) :
(fun x => x * a) ⁻¹' ball b r = ball (b / a) r := by |
rw [div_eq_mul_inv]
exact preimage_smul_ball (MulOpposite.op a) b r
|
/-
Copyright (c) 2020 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.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.iterated_deriv from "leanprover-com... | Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 236 | 238 | theorem iteratedFDeriv_apply_eq_iteratedDeriv_mul_prod {m : Fin n → 𝕜} :
(iteratedFDeriv 𝕜 n f x : (Fin n → 𝕜) → F) m = (∏ i, m i) • iteratedDeriv n f x := by |
rw [iteratedDeriv_eq_iteratedFDeriv, ← ContinuousMultilinearMap.map_smul_univ]; simp
|
/-
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
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
import Mat... | Mathlib/NumberTheory/ArithmeticFunction.lean | 300 | 303 | theorem natCoe_mul [Semiring R] {f g : ArithmeticFunction ℕ} :
(↑(f * g) : ArithmeticFunction R) = f * g := by |
ext n
simp
|
/-
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.Group.Int
import Mathlib.Algebra.Order.Group.Abs
#align_import data.int.order.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f3... | Mathlib/Algebra/Order/Group/Int.lean | 115 | 116 | theorem emod_lt (a : ℤ) {b : ℤ} (H : b ≠ 0) : a % b < |b| := by |
rw [← emod_abs]; exact emod_lt_of_pos _ (abs_pos.2 H)
|
/-
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.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanp... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 2,133 | 2,135 | theorem prod_sdiff_eq_div (h : s₁ ⊆ s₂) :
∏ x ∈ s₂ \ s₁, f x = (∏ x ∈ s₂, f x) / ∏ x ∈ s₁, f x := by |
rw [eq_div_iff_mul_eq', prod_sdiff h]
|
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Sébastien Gouëzel
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 375 | 381 | theorem addHaar_preimage_smul {r : ℝ} (hr : r ≠ 0) (s : Set E) :
μ ((r • ·) ⁻¹' s) = ENNReal.ofReal (abs (r ^ finrank ℝ E)⁻¹) * μ s :=
calc
μ ((r • ·) ⁻¹' s) = Measure.map (r • ·) μ s :=
((Homeomorph.smul (isUnit_iff_ne_zero.2 hr).unit).toMeasurableEquiv.map_apply s).symm
_ = ENNReal.ofReal (abs (r ... |
rw [map_addHaar_smul μ hr, coe_smul, Pi.smul_apply, smul_eq_mul]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Nat.Dist
import Mathlib.Data.Ordmap.Ordnode
import Mathlib.Tactic.Abel
imp... | Mathlib/Data/Ordmap/Ordset.lean | 1,391 | 1,401 | theorem Valid'.eraseMax_aux {s l x r o₁ o₂} (H : Valid' o₁ (.node s l x r) o₂) :
Valid' o₁ (@eraseMax α (.node' l x r)) ↑(findMax' x r) ∧
size (.node' l x r) = size (eraseMax (.node' l x r)) + 1 := by |
have := H.2.eq_node'; rw [this] at H; clear this
induction' r with rs rl rx rr _ IHrr generalizing l x o₁
· exact ⟨H.left, rfl⟩
have := H.2.2.2.eq_node'; rw [this] at H ⊢
rcases IHrr H.right with ⟨h, e⟩
refine ⟨Valid'.balanceL H.left h (Or.inr ⟨_, Or.inr e, H.3.1⟩), ?_⟩
rw [eraseMax, size_balanceL H.3.2.... |
/-
Copyright (c) 2021 Martin Zinkevich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Martin Zinkevich, Rémy Degenne
-/
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
#align_import measure_theory.pi_system fro... | Mathlib/MeasureTheory/PiSystem.lean | 280 | 308 | theorem mem_generatePiSystem_iUnion_elim {α β} {g : β → Set (Set α)} (h_pi : ∀ b, IsPiSystem (g b))
(t : Set α) (h_t : t ∈ generatePiSystem (⋃ b, g b)) :
∃ (T : Finset β) (f : β → Set α), (t = ⋂ b ∈ T, f b) ∧ ∀ b ∈ T, f b ∈ g b := by |
induction' h_t with s h_s s t' h_gen_s h_gen_t' h_nonempty h_s h_t'
· rcases h_s with ⟨t', ⟨⟨b, rfl⟩, h_s_in_t'⟩⟩
refine ⟨{b}, fun _ => s, ?_⟩
simpa using h_s_in_t'
· rcases h_t' with ⟨T_t', ⟨f_t', ⟨rfl, h_t'⟩⟩⟩
rcases h_s with ⟨T_s, ⟨f_s, ⟨rfl, h_s⟩⟩⟩
use T_s ∪ T_t', fun b : β =>
if b ∈ T_... |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprov... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,502 | 1,524 | theorem tendsto_setToFun_of_L1 (hT : DominatedFinMeasAdditive μ T C) {ι} (f : α → E)
(hfi : Integrable f μ) {fs : ι → α → E} {l : Filter ι} (hfsi : ∀ᶠ i in l, Integrable (fs i) μ)
(hfs : Tendsto (fun i => ∫⁻ x, ‖fs i x - f x‖₊ ∂μ) l (𝓝 0)) :
Tendsto (fun i => setToFun μ T hT (fs i)) l (𝓝 <| setToFun μ T h... |
classical
let f_lp := hfi.toL1 f
let F_lp i := if hFi : Integrable (fs i) μ then hFi.toL1 (fs i) else 0
have tendsto_L1 : Tendsto F_lp l (𝓝 f_lp) := by
rw [Lp.tendsto_Lp_iff_tendsto_ℒp']
simp_rw [snorm_one_eq_lintegral_nnnorm, Pi.sub_apply]
refine (tendsto_congr' ?_).mp hfs
filte... |
/-
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
import Mathlib.Algebra.Order.BigOperators.Group.Multiset
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Tactic.Po... | Mathlib/Algebra/Order/BigOperators/Group/Finset.lean | 35 | 44 | theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M → N) (p : M → Prop)
(h_mul : ∀ x y, p x → p y → f (x * y) ≤ f x * f y) (hp_mul : ∀ x y, p x → p y → p (x * y))
(g : ι → M) (s : Finset ι) (hs_nonempty : s.Nonempty) (hs : ∀ i ∈ s, p (g i)) :
f (∏ i ∈ s, g i) ≤ ∏ i ∈ s, f (g i) := by |
refine le_trans
(Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ ?_ ?_) ?_
· simp [hs_nonempty.ne_empty]
· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
|
/-
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.FDeriv.Bilinear
#align_import analysis.calculus.fderiv.mul from "leanprover-community/mathlib"@"d6... | Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 708 | 711 | theorem hasStrictFDerivAt_finset_prod [DecidableEq ι] [Fintype ι] {x : ι → 𝔸'} :
HasStrictFDerivAt (𝕜 := 𝕜) (∏ i ∈ u, · i) (∑ i ∈ u, (∏ j ∈ u.erase i, x j) • proj i) x := by |
simp only [Finset.sum_eq_multiset_sum, Finset.prod_eq_multiset_prod]
exact hasStrictFDerivAt_multiset_prod
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/ma... | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 442 | 443 | theorem rank_self : Module.rank R R = 1 := by |
rw [← Cardinal.lift_inj, ← (Basis.singleton PUnit R).mk_eq_rank, Cardinal.mk_punit]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprov... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 660 | 675 | theorem norm_eq_sum_mul (f : α →₁ₛ[μ] G) :
‖f‖ = ∑ x ∈ (toSimpleFunc f).range, (μ (toSimpleFunc f ⁻¹' {x})).toReal * ‖x‖ := by |
rw [norm_toSimpleFunc, snorm_one_eq_lintegral_nnnorm]
have h_eq := SimpleFunc.map_apply (fun x => (‖x‖₊ : ℝ≥0∞)) (toSimpleFunc f)
simp_rw [← h_eq]
rw [SimpleFunc.lintegral_eq_lintegral, SimpleFunc.map_lintegral, ENNReal.toReal_sum]
· congr
ext1 x
rw [ENNReal.toReal_mul, mul_comm, ← ofReal_norm_eq_coe... |
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Limits.Products
#align_import topology.category.Top.limits.pullbacks from "leanp... | Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | 349 | 355 | theorem snd_openEmbedding_of_left_openEmbedding {X Y S : TopCat} {f : X ⟶ S} (H : OpenEmbedding f)
(g : Y ⟶ S) : OpenEmbedding <| ⇑(pullback.snd : pullback f g ⟶ Y) := by |
convert (homeoOfIso (asIso (pullback.snd : pullback (𝟙 S) g ⟶ _))).openEmbedding.comp
(pullback_map_openEmbedding_of_open_embeddings (i₂ := 𝟙 Y) f g (𝟙 _) g H
(homeoOfIso (Iso.refl _)).openEmbedding (𝟙 _) rfl (by simp))
erw [← coe_comp]
simp
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a... | Mathlib/MeasureTheory/Function/L1Space.lean | 747 | 748 | theorem Integrable.sub {f g : α → β} (hf : Integrable f μ) (hg : Integrable g μ) :
Integrable (f - g) μ := by | simpa only [sub_eq_add_neg] using hf.add hg.neg
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanp... | Mathlib/Data/Int/Cast/Basic.lean | 93 | 98 | theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n := by |
unfold subNatNat
cases e : n - m
· simp only [ofNat_eq_coe]
simp [e, Nat.le_of_sub_eq_zero e]
· rw [cast_negSucc, ← e, Nat.cast_sub <| _root_.le_of_lt <| Nat.lt_of_sub_eq_succ e, neg_sub]
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.Affine... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 406 | 408 | theorem affineCombination_congr {w₁ w₂ : ι → k} (hw : ∀ i ∈ s, w₁ i = w₂ i) {p₁ p₂ : ι → P}
(hp : ∀ i ∈ s, p₁ i = p₂ i) : s.affineCombination k p₁ w₁ = s.affineCombination k p₂ w₂ := by |
simp_rw [affineCombination_apply, s.weightedVSubOfPoint_congr hw hp]
|
/-
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, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Prod
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Span
... | Mathlib/LinearAlgebra/Prod.lean | 756 | 758 | theorem snd_comp_prodComm :
(LinearMap.snd R M₂ M).comp (prodComm R M M₂).toLinearMap = (LinearMap.fst R M M₂) := by |
ext <;> simp
|
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.Algebraic... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 1,095 | 1,103 | theorem idealFactorsFunOfQuotHom_comp {f : R ⧸ I →+* A ⧸ J} {g : A ⧸ J →+* B ⧸ L}
(hf : Function.Surjective f) (hg : Function.Surjective g) :
(idealFactorsFunOfQuotHom hg).comp (idealFactorsFunOfQuotHom hf) =
idealFactorsFunOfQuotHom (show Function.Surjective (g.comp f) from hg.comp hf) := by |
refine OrderHom.ext _ _ (funext fun x => ?_)
rw [idealFactorsFunOfQuotHom, idealFactorsFunOfQuotHom, OrderHom.comp_coe, OrderHom.coe_mk,
OrderHom.coe_mk, Function.comp_apply, idealFactorsFunOfQuotHom, OrderHom.coe_mk,
Subtype.mk_eq_mk, Subtype.coe_mk, map_comap_of_surjective (Ideal.Quotient.mk J)
Quoti... |
/-
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.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
/-!
# Noncomputable... | Mathlib/Data/Set/Card.lean | 1,116 | 1,120 | theorem ncard_eq_three : s.ncard = 3 ↔ ∃ x y z, x ≠ y ∧ x ≠ z ∧ y ≠ z ∧ s = {x, y, z} := by |
rw [← encard_eq_three, ncard_def, ← Nat.cast_inj (R := ℕ∞), Nat.cast_ofNat]
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· rwa [ENat.coe_toNat] at h; rintro h'; simp [h'] at h
rw [h]; rfl
|
/-
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.Set.Card
import Mathlib.Order.Minimal
import Mathlib.Data.Matroid.Init
/-!
# Matroids
A `Matroid` is a structure that combinatorially abstracts
the ... | Mathlib/Data/Matroid/Basic.lean | 935 | 939 | theorem Basis.basis_union_of_subset (hI : M.Basis I X) (hJ : M.Indep J) (hIJ : I ⊆ J) :
M.Basis J (J ∪ X) := by |
convert hJ.basis_self.union_basis_union hI _ <;>
rw [union_eq_self_of_subset_right hIJ]
assumption
|
/-
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.Data.Finset.Prod
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
import Mathlib.Data.SetLike.Basic
#align_import data.sym.sym2 from "leanpro... | Mathlib/Data/Sym/Sym2.lean | 304 | 313 | theorem mem_iff' {a b c : α} : Sym2.Mem a s(b, c) ↔ a = b ∨ a = c :=
{ mp := by |
rintro ⟨_, h⟩
rw [eq_iff] at h
aesop
mpr := by
rintro (rfl | rfl)
· exact ⟨_, rfl⟩
rw [eq_swap]
exact ⟨_, rfl⟩ }
|
/-
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.Topology.Order.MonotoneContinuity
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.E... | Mathlib/Topology/Instances/ENNReal.lean | 1,168 | 1,175 | theorem not_summable_iff_tendsto_nat_atTop {f : ℕ → ℝ≥0} :
¬Summable f ↔ Tendsto (fun n : ℕ => ∑ i ∈ Finset.range n, f i) atTop atTop := by |
constructor
· intro h
refine ((tendsto_of_monotone ?_).resolve_right h).comp ?_
exacts [Finset.sum_mono_set _, tendsto_finset_range]
· rintro hnat ⟨r, hr⟩
exact not_tendsto_nhds_of_tendsto_atTop hnat _ (hasSum_iff_tendsto_nat.1 hr)
|
/-
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, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Topology.Order.ProjIcc
#al... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 386 | 386 | theorem arccos_zero : arccos 0 = π / 2 := by | simp [arccos]
|
/-
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 | 427 | 431 | theorem diagonal_transpose [Zero α] (v : n → α) : (diagonal v)ᵀ = diagonal v := by |
ext i j
by_cases h : i = j
· simp [h, transpose]
· simp [h, transpose, diagonal_apply_ne' _ h]
|
/-
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.CategoryTheory.Idempotents.Basic
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Equivalence
#align_import category_theo... | Mathlib/CategoryTheory/Idempotents/Karoubi.lean | 286 | 288 | theorem decomp_p (P : Karoubi C) : (toKaroubi C).map P.p = decompId_p P ≫ decompId_i P := by |
ext
simp only [comp_f, decompId_p_f, decompId_i_f, P.idem, toKaroubi_map_f]
|
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark
-/
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.Data.Matrix.CharP
#align_import l... | Mathlib/LinearAlgebra/Matrix/Charpoly/FiniteField.lean | 47 | 50 | theorem ZMod.charpoly_pow_card {p : ℕ} [Fact p.Prime] (M : Matrix n n (ZMod p)) :
(M ^ p).charpoly = M.charpoly := by |
have h := FiniteField.Matrix.charpoly_pow_card M
rwa [ZMod.card] at h
|
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | Mathlib/ModelTheory/Semantics.lean | 147 | 154 | theorem realize_restrictVarLeft [DecidableEq α] {γ : Type*} {t : L.Term (Sum α γ)} {s : Set α}
(h : ↑t.varFinsetLeft ⊆ s) {v : α → M} {xs : γ → M} :
(t.restrictVarLeft (Set.inclusion h)).realize (Sum.elim (v ∘ (↑)) xs) =
t.realize (Sum.elim v xs) := by |
induction' t with a _ _ _ ih
· cases a <;> rfl
· simp_rw [varFinsetLeft, Finset.coe_biUnion, Set.iUnion_subset_iff] at h
exact congr rfl (funext fun i => ih i (h i (Finset.mem_univ i)))
|
/-
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.PartialHomeomorph
import Mathlib.Topology.Order.Basic
#align_import topology.f... | Mathlib/Topology/FiberBundle/Trivialization.lean | 181 | 185 | theorem preimage_symm_proj_baseSet :
e.toPartialEquiv.symm ⁻¹' (proj ⁻¹' e.baseSet) ∩ e.target = e.target := by |
refine inter_eq_right.mpr fun x hx => ?_
simp only [mem_preimage, PartialEquiv.invFun_as_coe, e.proj_symm_apply hx]
exact e.mem_target.mp hx
|
/-
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.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
#align_import algebraic_geometry.open_immersion.basic from ... | Mathlib/Geometry/RingedSpace/OpenImmersion.lean | 560 | 568 | theorem pullback_snd_isIso_of_range_subset (H : Set.range g.base ⊆ Set.range f.base) :
IsIso (pullback.snd : pullback f g ⟶ _) := by |
haveI := TopCat.snd_iso_of_left_embedding_range_subset hf.base_open.toEmbedding g.base H
have : IsIso (pullback.snd : pullback f g ⟶ _).base := by
delta pullback.snd
rw [← limit.isoLimitCone_hom_π ⟨_, pullbackConeOfLeftIsLimit f g⟩ WalkingCospan.right]
change IsIso (_ ≫ pullback.snd)
infer_instance... |
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SumOverResidueClass
#alig... | Mathlib/Analysis/PSeries.lean | 225 | 235 | theorem summable_schlomilch_iff_of_nonneg {C : ℕ} {u : ℕ → ℕ} {f : ℕ → ℝ} (h_nonneg : ∀ n, 0 ≤ f n)
(hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
(hu_strict : StrictMono u) (hC_nonzero : C ≠ 0) (h_succ_diff : SuccDiffBounded C u) :
(Summable fun k : ℕ => (u (k + 1) - (u k : ℝ)) * f (u k))... |
lift f to ℕ → ℝ≥0 using h_nonneg
simp only [NNReal.coe_le_coe] at *
have (k : ℕ) : (u (k + 1) - (u k : ℝ)) = ((u (k + 1) : ℝ≥0) - (u k : ℝ≥0) : ℝ≥0) := by
have := Nat.cast_le (α := ℝ≥0).mpr <| (hu_strict k.lt_succ_self).le
simp [NNReal.coe_sub this]
simp_rw [this]
exact_mod_cast NNReal.summable_schlo... |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import meas... | Mathlib/MeasureTheory/Function/L2Space.lean | 118 | 121 | theorem snorm_rpow_two_norm_lt_top (f : Lp F 2 μ) : snorm (fun x => ‖f x‖ ^ (2 : ℝ)) 1 μ < ∞ := by |
have h_two : ENNReal.ofReal (2 : ℝ) = 2 := by simp [zero_le_one]
rw [snorm_norm_rpow f zero_lt_two, one_mul, h_two]
exact ENNReal.rpow_lt_top_of_nonneg zero_le_two (Lp.snorm_ne_top f)
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathl... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 1,092 | 1,095 | theorem add [ContinuousAdd M] {v₁ v₂ : VectorMeasure α M} {w : VectorMeasure α N} (hv₁ : v₁ ≪ᵥ w)
(hv₂ : v₂ ≪ᵥ w) : v₁ + v₂ ≪ᵥ w := by |
intro s hs
rw [add_apply, hv₁ hs, hv₂ hs, zero_add]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Nat.Prime
import Mathlib.Data.List.Prime
import Mathlib.Dat... | Mathlib/Data/Nat/Factors.lean | 215 | 221 | theorem perm_factors_mul_of_coprime {a b : ℕ} (hab : Coprime a b) :
(a * b).factors ~ a.factors ++ b.factors := by |
rcases a.eq_zero_or_pos with (rfl | ha)
· simp [(coprime_zero_left _).mp hab]
rcases b.eq_zero_or_pos with (rfl | hb)
· simp [(coprime_zero_right _).mp hab]
exact perm_factors_mul ha.ne' hb.ne'
|
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Topology.MetricSpace.Basic
#align_import topology.metric_space.infsep from "leanprover-community/mathlib"@"5316314b553dcf8c6716541851517c1a9715e22b"
/-... | Mathlib/Topology/MetricSpace/Infsep.lean | 428 | 432 | theorem infsep_triple (hxy : x ≠ y) (hyz : y ≠ z) (hxz : x ≠ z) :
({x, y, z} : Set α).infsep = dist x y ⊓ dist x z ⊓ dist y z := by |
simp only [infsep, einfsep_triple hxy hyz hxz, ENNReal.toReal_inf, edist_ne_top x y,
edist_ne_top x z, edist_ne_top y z, dist_edist, Ne, inf_eq_top_iff, and_self_iff,
not_false_iff]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.Measur... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 609 | 618 | theorem Integrable.ae_eq_of_forall_setIntegral_eq (f g : α → E) (hf : Integrable f μ)
(hg : Integrable g μ)
(hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ) :
f =ᵐ[μ] g := by |
rw [← sub_ae_eq_zero]
have hfg' : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, (f - g) x ∂μ) = 0 := by
intro s hs hμs
rw [integral_sub' hf.integrableOn hg.integrableOn]
exact sub_eq_zero.mpr (hfg s hs hμs)
exact Integrable.ae_eq_zero_of_forall_setIntegral_eq_zero (hf.sub hg) hfg'
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.StructurePolynomial
#align_import ring_theory.witt_vector.defs from "leanprover-community/mathlib"@"f1944b30c97... | Mathlib/RingTheory/WittVector/Defs.lean | 298 | 300 | theorem constantCoeff_wittMul (n : ℕ) : constantCoeff (wittMul p n) = 0 := by |
apply constantCoeff_wittStructureInt p _ _ n
simp only [mul_zero, RingHom.map_mul, constantCoeff_X]
|
/-
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.Analysis.Calculus.ContDiff.RCLike
import Mathlib.MeasureTheory.Measure.Hausdorff
#align_import topology.metric_space.hausdorff_dimension from "leanp... | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | 371 | 376 | theorem dimH_range_le_of_locally_holder_on [SecondCountableTopology X] {r : ℝ≥0} {f : X → Y}
(hr : 0 < r) (hf : ∀ x : X, ∃ C : ℝ≥0, ∃ s ∈ 𝓝 x, HolderOnWith C r f s) :
dimH (range f) ≤ dimH (univ : Set X) / r := by |
rw [← image_univ]
refine dimH_image_le_of_locally_holder_on hr fun x _ => ?_
simpa only [exists_prop, nhdsWithin_univ] using hf x
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Regular.Pow
import Mathl... | Mathlib/Algebra/MvPolynomial/Basic.lean | 460 | 470 | theorem ringHom_ext {A : Type*} [Semiring A] {f g : MvPolynomial σ R →+* A}
(hC : ∀ r, f (C r) = g (C r)) (hX : ∀ i, f (X i) = g (X i)) : f = g := by |
refine AddMonoidAlgebra.ringHom_ext' ?_ ?_
-- Porting note: this has high priority, but Lean still chooses `RingHom.ext`, why?
-- probably because of the type synonym
· ext x
exact hC _
· apply Finsupp.mulHom_ext'; intros x
-- Porting note: `Finsupp.mulHom_ext'` needs to have increased priority
a... |
/-
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.Module.Equiv
import Mathlib.Data.DFinsupp.Basic
import Mathlib.Data.Finsupp.Basic
#align_import data.finsupp.to_dfinsupp from "leanprover-community/... | Mathlib/Data/Finsupp/ToDFinsupp.lean | 323 | 333 | theorem sigmaFinsuppEquivDFinsupp_single [DecidableEq ι] [Zero N] (a : Σi, η i) (n : N) :
sigmaFinsuppEquivDFinsupp (Finsupp.single a n) =
@DFinsupp.single _ (fun i => η i →₀ N) _ _ a.1 (Finsupp.single a.2 n) := by |
obtain ⟨i, a⟩ := a
ext j b
by_cases h : i = j
· subst h
classical simp [split_apply, Finsupp.single_apply]
suffices Finsupp.single (⟨i, a⟩ : Σi, η i) n ⟨j, b⟩ = 0 by simp [split_apply, dif_neg h, this]
have H : (⟨i, a⟩ : Σi, η i) ≠ ⟨j, b⟩ := by simp [h]
classical rw [Finsupp.single_apply, if_neg H]
|
/-
Copyright (c) 2021 Lu-Ming Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lu-Ming Zhang
-/
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.LinearAlgebra.Matrix.Orthogonal
import Mathlib.Data.Matrix.Kronecker
#align_import linear_algebra.matrix.i... | Mathlib/LinearAlgebra/Matrix/IsDiag.lean | 104 | 107 | theorem IsDiag.smul [Monoid R] [AddMonoid α] [DistribMulAction R α] (k : R) {A : Matrix n n α}
(ha : A.IsDiag) : (k • A).IsDiag := by |
intro i j h
simp [ha h]
|
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Game.Ordinal
import Mathlib.SetTheory.Ordinal.NaturalOps
#align_import set_theory.game.birthday from "leanprover-com... | Mathlib/SetTheory/Game/Birthday.lean | 122 | 129 | theorem toPGame_birthday (o : Ordinal) : o.toPGame.birthday = o := by |
induction' o using Ordinal.induction with o IH
rw [toPGame_def, PGame.birthday]
simp only [lsub_empty, max_zero_right]
-- Porting note: was `nth_rw 1 [← lsub_typein o]`
conv_rhs => rw [← lsub_typein o]
congr with x
exact IH _ (typein_lt_self x)
|
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Comp
#align_import analysis.calculus.deriv.inv from "leanpro... | Mathlib/Analysis/Calculus/Deriv/Inv.lean | 126 | 129 | theorem HasDerivWithinAt.inv (hc : HasDerivWithinAt c c' s x) (hx : c x ≠ 0) :
HasDerivWithinAt (fun y => (c y)⁻¹) (-c' / c x ^ 2) s x := by |
convert (hasDerivAt_inv hx).comp_hasDerivWithinAt x hc using 1
field_simp
|
/-
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.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Topology.Order.Basic
#align_import analysis.convex.strict from "leanprove... | Mathlib/Analysis/Convex/Strict.lean | 85 | 92 | theorem Directed.strictConvex_iUnion {ι : Sort*} {s : ι → Set E} (hdir : Directed (· ⊆ ·) s)
(hs : ∀ ⦃i : ι⦄, StrictConvex 𝕜 (s i)) : StrictConvex 𝕜 (⋃ i, s i) := by |
rintro x hx y hy hxy a b ha hb hab
rw [mem_iUnion] at hx hy
obtain ⟨i, hx⟩ := hx
obtain ⟨j, hy⟩ := hy
obtain ⟨k, hik, hjk⟩ := hdir i j
exact interior_mono (subset_iUnion s k) (hs (hik hx) (hjk hy) hxy ha hb hab)
|
/-
Copyright (c) 2022 Antoine Labelle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle
-/
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Algebra.MonoidAlgebra.Ba... | Mathlib/RepresentationTheory/Basic.lean | 238 | 245 | theorem ofModule_asModule_act (g : G) (x : RestrictScalars k (MonoidAlgebra k G) ρ.asModule) :
ofModule (k := k) (G := G) ρ.asModule g x = -- Porting note: more help with implicit
(RestrictScalars.addEquiv _ _ _).symm
(ρ.asModuleEquiv.symm (ρ g (ρ.asModuleEquiv (RestrictScalars.addEquiv _ _ _ x)))) :=... |
apply_fun RestrictScalars.addEquiv _ _ ρ.asModule using
(RestrictScalars.addEquiv _ _ ρ.asModule).injective
dsimp [ofModule, RestrictScalars.lsmul_apply_apply]
simp
|
/-
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.CharP.Two
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Grou... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 1,076 | 1,085 | theorem autToPow_spec (f : S ≃ₐ[R] S) : μ ^ (hμ.autToPow R f : ZMod n).val = f μ := by |
rw [IsPrimitiveRoot.coe_autToPow_apply]
generalize_proofs h
have := h.choose_spec
refine (?_ : ((hμ.toRootsOfUnity : Sˣ) : S) ^ _ = _).trans this.symm
rw [← rootsOfUnity.coe_pow, ← rootsOfUnity.coe_pow]
congr 2
rw [pow_eq_pow_iff_modEq, ZMod.val_natCast, hμ.eq_orderOf, ← Subgroup.orderOf_coe,
← order... |
/-
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... | Mathlib/Order/CompactlyGenerated/Basic.lean | 354 | 356 | theorem sSup_compact_eq_top : sSup { a : α | CompleteLattice.IsCompactElement a } = ⊤ := by |
refine Eq.trans (congr rfl (Set.ext fun x => ?_)) (sSup_compact_le_eq ⊤)
exact (and_iff_left le_top).symm
|
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