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 |
|---|---|---|---|---|---|
/-
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.PiTensorProduct
import Mathlib.Logic.Equiv.Fin
import Mathlib.Algebra.DirectSum.Algebra
#align_import linear_algebra.tensor_power from "leanpr... | Mathlib/LinearAlgebra/TensorPower.lean | 138 | 142 | theorem gradedMonoid_eq_of_cast {a b : GradedMonoid fun n => ⨂[R] _ : Fin n, M} (h : a.fst = b.fst)
(h2 : cast R M h a.snd = b.snd) : a = b := by |
refine gradedMonoid_eq_of_reindex_cast h ?_
rw [cast] at h2
rw [← finCongr_eq_equivCast, ← h2]
|
/-
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, Patrick Massot
-/
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
i... | Mathlib/Data/Set/Pointwise/Interval.lean | 529 | 530 | theorem preimage_sub_const_uIcc : (fun x => x - a) ⁻¹' [[b, c]] = [[b + a, c + a]] := by |
simp [sub_eq_add_neg]
|
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Notation
import Mathlib.Init.Order.Defs
set_option autoImplicit true
structure UFModel (n) where
parent : Fin n → Fin n
rank : Nat ... | Mathlib/Data/UnionFind.lean | 205 | 207 | theorem parent_lt (self : UnionFind α) (i : Nat) (h) : self.arr[i].parent < self.size := by |
let ⟨m, hm⟩ := self.model'
simp [hm.parent_eq, size, (m.parent ⟨i, h⟩).2, h]
|
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhangir Azerbayev, Adam Topaz, Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
import Mathlib.LinearAlgebra.Alternating.Basic
#align_import linear_algebra.exterior_algeb... | Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean | 440 | 449 | theorem ιInv_comp_map (f : M →ₗ[R] N) :
ιInv.comp (map f).toLinearMap = f.comp ιInv := by |
letI : Module Rᵐᵒᵖ M := Module.compHom _ ((RingHom.id R).fromOpposite mul_comm)
haveI : IsCentralScalar R M := ⟨fun r m => rfl⟩
letI : Module Rᵐᵒᵖ N := Module.compHom _ ((RingHom.id R).fromOpposite mul_comm)
haveI : IsCentralScalar R N := ⟨fun r m => rfl⟩
unfold ιInv
conv_lhs => rw [LinearMap.comp_assoc, ←... |
/-
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.Init.Data.List.Basic
import Mathlib.Data.List.Basic
#align_import data.s... | Mathlib/Data/Stream/Init.lean | 377 | 378 | theorem corec_eq (f : α → β) (g : α → α) (a : α) : corec f g a = f a::corec f g (g a) := by |
rw [corec_def, map_eq, head_iterate, tail_iterate]; rfl
|
/-
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.SmoothManifoldWithCorners
import Mathlib.Geometry.Manifold.LocalInvariantProperties
#align_import geometry.m... | Mathlib/Geometry/Manifold/ContMDiff/Defs.lean | 399 | 407 | theorem contMDiffWithinAt_iff_image {x : M} (he : e ∈ maximalAtlas I M)
(he' : e' ∈ maximalAtlas I' M') (hs : s ⊆ e.source) (hx : x ∈ e.source) (hy : f x ∈ e'.source) :
ContMDiffWithinAt I I' n f s x ↔
ContinuousWithinAt f s x ∧
ContDiffWithinAt 𝕜 n (e'.extend I' ∘ f ∘ (e.extend I).symm) (e.exten... |
rw [contMDiffWithinAt_iff_of_mem_maximalAtlas he he' hx hy, and_congr_right_iff]
refine fun _ => contDiffWithinAt_congr_nhds ?_
simp_rw [nhdsWithin_eq_iff_eventuallyEq, e.extend_symm_preimage_inter_range_eventuallyEq I hs hx]
|
/-
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.BoxIntegral.Partition.Basic
#align_import analysis.box_integral.partition.split from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf973... | Mathlib/Analysis/BoxIntegral/Partition/Split.lean | 147 | 153 | theorem splitLower_ne_splitUpper (I : Box ι) (i : ι) (x : ℝ) :
I.splitLower i x ≠ I.splitUpper i x := by |
cases' le_or_lt x (I.lower i) with h
· rw [splitUpper_eq_self.2 h, splitLower_eq_bot.2 h]
exact WithBot.bot_ne_coe
· refine (disjoint_splitLower_splitUpper I i x).ne ?_
rwa [Ne, splitLower_eq_bot, not_le]
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Baire.Lemmas
import Mathlib.Topology.Baire.CompleteMetrizable
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathli... | Mathlib/Analysis/NormedSpace/Banach.lean | 320 | 325 | theorem continuous_symm (e : E ≃ₛₗ[σ] F) (h : Continuous e) : Continuous e.symm := by |
rw [continuous_def]
intro s hs
rw [← e.image_eq_preimage]
rw [← e.coe_coe] at h ⊢
exact ContinuousLinearMap.isOpenMap (σ := σ) ⟨↑e, h⟩ e.surjective s hs
|
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne, Adam Topaz
-/
import Mathlib.Data.Setoid.Partition
import Mathlib.Topology.Separation
import Mathlib.Topology.LocallyConstant.Basic
#align_import topology.discrete_quotient f... | Mathlib/Topology/DiscreteQuotient.lean | 150 | 152 | theorem isClopen_setOf_rel (x : X) : IsClopen (setOf (S.Rel x)) := by |
rw [← fiber_eq]
apply isClopen_preimage
|
/-
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 | 191 | 207 | theorem exists_eventuallyEq_pow_smul_nonzero_iff (hf : AnalyticAt 𝕜 f z₀) :
(∃ (n : ℕ), ∃ (g : 𝕜 → E), AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧
∀ᶠ z in 𝓝 z₀, f z = (z - z₀) ^ n • g z) ↔ (¬∀ᶠ z in 𝓝 z₀, f z = 0) := by |
constructor
· rintro ⟨n, g, hg_an, hg_ne, hg_eq⟩
contrapose! hg_ne
apply EventuallyEq.eq_of_nhds
rw [EventuallyEq, ← AnalyticAt.frequently_eq_iff_eventually_eq hg_an analyticAt_const]
refine (eventually_nhdsWithin_iff.mpr ?_).frequently
filter_upwards [hg_eq, hg_ne] with z hf_eq hf0 hz
rwa ... |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Data.Real.Sqrt
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedS... | Mathlib/Analysis/RCLike/Basic.lean | 532 | 537 | theorem inv_def (z : K) : z⁻¹ = conj z * ((‖z‖ ^ 2)⁻¹ : ℝ) := by |
rcases eq_or_ne z 0 with (rfl | h₀)
· simp
· apply inv_eq_of_mul_eq_one_right
rw [← mul_assoc, mul_conj, ofReal_inv, ofReal_pow, mul_inv_cancel]
simpa
|
/-
Copyright (c) 2022 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Data.Set.L... | Mathlib/Topology/Algebra/Module/CharacterSpace.lean | 166 | 176 | theorem eq_set_map_one_map_mul [Nontrivial 𝕜] :
characterSpace 𝕜 A = {φ : WeakDual 𝕜 A | φ 1 = 1 ∧ ∀ x y : A, φ (x * y) = φ x * φ y} := by |
ext φ
refine ⟨?_, ?_⟩
· rintro hφ
lift φ to characterSpace 𝕜 A using hφ
exact ⟨map_one φ, map_mul φ⟩
· rintro ⟨hφ₁, hφ₂⟩
refine ⟨?_, hφ₂⟩
rintro rfl
exact zero_ne_one hφ₁
|
/-
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
-/
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Na... | Mathlib/Data/Complex/Exponential.lean | 742 | 744 | theorem inv_one_add_tan_sq {x : ℂ} (hx : cos x ≠ 0) : (1 + tan x ^ 2)⁻¹ = cos x ^ 2 := by |
rw [tan_eq_sin_div_cos, div_pow]
field_simp
|
/-
Copyright (c) 2019 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.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import ... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 117 | 118 | theorem contDiffWithinAt_of_subsingleton [Subsingleton F] : ContDiffWithinAt 𝕜 n f s x := by |
rw [Subsingleton.elim f fun _ => 0]; exact contDiffWithinAt_const
|
/-
Copyright (c) 2020 Hanting Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hanting Zhang, Johan Commelin
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.MvPolynomial.CommRing
#align_import r... | Mathlib/RingTheory/MvPolynomial/Symmetric.lean | 279 | 296 | theorem degrees_esymm [Nontrivial R] (n : ℕ) (hpos : 0 < n) (hn : n ≤ Fintype.card σ) :
(esymm σ R n).degrees = (univ : Finset σ).val := by |
classical
have :
(Finsupp.toMultiset ∘ fun t : Finset σ => ∑ i ∈ t, Finsupp.single i 1) = val := by
funext
simp [Finsupp.toMultiset_sum_single]
rw [degrees_def, support_esymm, sup_image, this]
have : ((powersetCard n univ).sup (fun (x : Finset σ) => x)).val
= sup (powersetCard n... |
/-
Copyright (c) 2023 Bulhwi Cha. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bulhwi Cha, Mario Carneiro
-/
import Batteries.Data.Char
import Batteries.Data.List.Lemmas
import Batteries.Data.String.Basic
import Batteries.Tactic.Lint.Misc
import Batteries.Tactic.SeqF... | .lake/packages/batteries/Batteries/Data/String/Lemmas.lean | 233 | 237 | theorem prev_of_valid' (cs cs' : List Char) :
prev ⟨cs ++ cs'⟩ ⟨utf8Len cs⟩ = ⟨utf8Len cs.dropLast⟩ := by |
match cs, cs.eq_nil_or_concat with
| _, .inl rfl => rfl
| _, .inr ⟨cs, c, rfl⟩ => simp [prev_of_valid]
|
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.Algebra.UniformConvergence
#align_import topology.algebra.module.strong_topology from "leanprover-community/mathlib"@"8905e5ed90859939681... | Mathlib/Topology/Algebra/Module/StrongTopology.lean | 168 | 177 | theorem continuousSMul [RingHomSurjective σ] [RingHomIsometric σ]
[TopologicalSpace F] [TopologicalAddGroup F] [ContinuousSMul 𝕜₂ F] (𝔖 : Set (Set E))
(h𝔖₃ : ∀ S ∈ 𝔖, Bornology.IsVonNBounded 𝕜₁ S) :
ContinuousSMul 𝕜₂ (UniformConvergenceCLM σ F 𝔖) := by |
letI : UniformSpace F := TopologicalAddGroup.toUniformSpace F
haveI : UniformAddGroup F := comm_topologicalAddGroup_is_uniform
let φ : (UniformConvergenceCLM σ F 𝔖) →ₗ[𝕜₂] E → F :=
⟨⟨DFunLike.coe, fun _ _ => rfl⟩, fun _ _ => rfl⟩
exact UniformOnFun.continuousSMul_induced_of_image_bounded 𝕜₂ E F (Uniform... |
/-
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.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.Option
import Mathlib.Logic.Equiv.Fin
import Mathlib.Logic.Equiv.Fintype
#align_import group_the... | Mathlib/GroupTheory/Perm/Fin.lean | 110 | 114 | theorem sign_finRotate (n : ℕ) : Perm.sign (finRotate (n + 1)) = (-1) ^ n := by |
induction' n with n ih
· simp
· rw [finRotate_succ_eq_decomposeFin]
simp [ih, pow_succ]
|
/-
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 | 458 | 462 | theorem toSignedMeasure_smul (μ : Measure α) [IsFiniteMeasure μ] (r : ℝ≥0) :
(r • μ).toSignedMeasure = r • μ.toSignedMeasure := by |
ext i hi
rw [toSignedMeasure_apply_measurable hi, VectorMeasure.smul_apply,
toSignedMeasure_apply_measurable hi, coe_smul, Pi.smul_apply, ENNReal.toReal_smul]
|
/-
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.Trim
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
#align_import measure_theory.measure.ae_measurable fr... | Mathlib/MeasureTheory/Measure/AEMeasurable.lean | 220 | 223 | theorem exists_measurable_nonneg {β} [Preorder β] [Zero β] {mβ : MeasurableSpace β} {f : α → β}
(hf : AEMeasurable f μ) (f_nn : ∀ᵐ t ∂μ, 0 ≤ f t) : ∃ g, Measurable g ∧ 0 ≤ g ∧ f =ᵐ[μ] g := by |
obtain ⟨G, hG_meas, hG_mem, hG_ae_eq⟩ := hf.exists_ae_eq_range_subset f_nn ⟨0, le_rfl⟩
exact ⟨G, hG_meas, fun x => hG_mem (mem_range_self x), hG_ae_eq⟩
|
/-
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.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive... | Mathlib/Data/Option/Basic.lean | 108 | 110 | theorem bind_congr {f g : α → Option β} {x : Option α}
(h : ∀ a ∈ x, f a = g a) : x.bind f = x.bind g := by |
cases x <;> simp only [some_bind, none_bind, mem_def, h]
|
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Louis Carlin, Mario Carneiro
-/
import Mathlib.Algebra.EuclideanDomain.Defs
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.Grou... | Mathlib/Algebra/EuclideanDomain/Basic.lean | 225 | 229 | theorem gcd_eq_gcd_ab (a b : R) : (gcd a b : R) = a * gcdA a b + b * gcdB a b := by |
have :=
@xgcdAux_P _ _ _ a b a b 1 0 0 1 (by dsimp [P]; rw [mul_one, mul_zero, add_zero])
(by dsimp [P]; rw [mul_one, mul_zero, zero_add])
rwa [xgcdAux_val, xgcd_val] at this
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.Minpoly.Field
import Mathli... | Mathlib/FieldTheory/Separable.lean | 138 | 149 | theorem _root_.Associated.separable {f g : R[X]}
(ha : Associated f g) (h : f.Separable) : g.Separable := by |
obtain ⟨⟨u, v, h1, h2⟩, ha⟩ := ha
obtain ⟨a, b, h⟩ := h
refine ⟨a * v + b * derivative v, b * v, ?_⟩
replace h := congr($h * $(h1))
have h3 := congr(derivative $(h1))
simp only [← ha, derivative_mul, derivative_one] at h3 ⊢
calc
_ = (a * f + b * derivative f) * (u * v)
+ (b * f) * (derivative u... |
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SeparableDegree
import Mathlib.FieldTheory.IsSepClosed
/-!
# Separable closure
This file contains basics about the (relative) separable closure of a fie... | Mathlib/FieldTheory/SeparableClosure.lean | 330 | 332 | theorem insepDegree_bot : insepDegree F (⊥ : IntermediateField F E) = 1 := by |
have := lift_insepDegree_eq_of_equiv _ _ _ (botEquiv F E)
rwa [insepDegree_self, Cardinal.lift_one, ← Cardinal.lift_one.{u, v}, Cardinal.lift_inj] at this
|
/-
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.Data.Set.Equitable
import Mathlib.Logic.Equiv.Fin
import Mathlib.Order.Partition.Finpartition
#align_import order.partition.eq... | Mathlib/Order/Partition/Equipartition.lean | 61 | 66 | theorem IsEquipartition.card_part_eq_average_iff (hP : P.IsEquipartition) (ht : t ∈ P.parts) :
t.card = s.card / P.parts.card ↔ t.card ≠ s.card / P.parts.card + 1 := by |
have a := hP.card_parts_eq_average ht
have b : ¬(t.card = s.card / P.parts.card ∧ t.card = s.card / P.parts.card + 1) := by
by_contra h; exact absurd (h.1 ▸ h.2) (lt_add_one _).ne
tauto
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Set.Subsingleton
#align_import combinatorics.double_counting from "leanprover-community/mathli... | Mathlib/Combinatorics/Enumerative/DoubleCounting.lean | 110 | 120 | theorem card_le_card_of_forall_subsingleton (hs : ∀ a ∈ s, ∃ b, b ∈ t ∧ r a b)
(ht : ∀ b ∈ t, ({ a ∈ s | r a b } : Set α).Subsingleton) : s.card ≤ t.card := by |
classical
rw [← mul_one s.card, ← mul_one t.card]
exact card_mul_le_card_mul r
(fun a h ↦ card_pos.2 (by
rw [← coe_nonempty, coe_bipartiteAbove]
exact hs _ h : (t.bipartiteAbove r a).Nonempty))
(fun b h ↦ card_le_one.2 (by
simp_rw [mem_bipartiteBelow]
exact ht _ h)... |
/-
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.Order.Interval.Finset.Nat
import Mathlib.Topol... | Mathlib/Topology/EMetricSpace/Basic.lean | 357 | 364 | theorem tendstoLocallyUniformlyOn_iff {ι : Type*} [TopologicalSpace β] {F : ι → β → α} {f : β → α}
{p : Filter ι} {s : Set β} :
TendstoLocallyUniformlyOn F f p s ↔
∀ ε > 0, ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, ∀ᶠ n in p, ∀ y ∈ t, edist (f y) (F n y) < ε := by |
refine ⟨fun H ε hε => H _ (edist_mem_uniformity hε), fun H u hu x hx => ?_⟩
rcases mem_uniformity_edist.1 hu with ⟨ε, εpos, hε⟩
rcases H ε εpos x hx with ⟨t, ht, Ht⟩
exact ⟨t, ht, Ht.mono fun n hs x hx => hε (hs x hx)⟩
|
/-
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
-/
import Mathlib.Logic.Function.Basic
#align_import logic.is_empty from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
/-!
# Types that... | Mathlib/Logic/IsEmpty.lean | 159 | 160 | theorem isEmpty_pi {π : α → Sort*} : IsEmpty (∀ a, π a) ↔ ∃ a, IsEmpty (π a) := by |
simp only [← not_nonempty_iff, Classical.nonempty_pi, not_forall]
|
/-
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.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.Thin
#align_import category_theory.limits.shapes.wide_pullbacks from "lean... | Mathlib/CategoryTheory/Limits/Shapes/WidePullbacks.lean | 379 | 383 | theorem hom_eq_lift (g : X ⟶ widePullback _ _ arrows) :
g = lift (g ≫ base arrows) (fun j => g ≫ π arrows j) (by aesop_cat) := by |
apply eq_lift_of_comp_eq
· aesop_cat
· rfl -- Porting note: quite a few missing refl's in aesop_cat now
|
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Dart
import Mathlib.Data.FunLike.Fintype
/-!
# Maps between graphs
This file defines two functions and three str... | Mathlib/Combinatorics/SimpleGraph/Maps.lean | 319 | 323 | theorem mapEdgeSet.injective (hinj : Function.Injective f) : Function.Injective f.mapEdgeSet := by |
rintro ⟨e₁, h₁⟩ ⟨e₂, h₂⟩
dsimp [Hom.mapEdgeSet]
repeat rw [Subtype.mk_eq_mk]
apply Sym2.map.injective hinj
|
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee
-/
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
/-!
# Reflections, inversions, and inversion sequences
Throughout this file, `B` is a type and... | Mathlib/GroupTheory/Coxeter/Inversion.lean | 307 | 315 | theorem isReflection_of_mem_rightInvSeq (ω : List B) {t : W} (ht : t ∈ ris ω) :
cs.IsReflection t := by |
induction' ω with i ω ih
· simp at ht
· dsimp [rightInvSeq] at ht
rcases ht with _ | ⟨_, mem⟩
· use (π ω)⁻¹, i
group
· exact ih mem
|
/-
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.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a5... | Mathlib/Algebra/MvPolynomial/Variables.lean | 335 | 338 | theorem mem_vars_rename (f : σ → τ) (φ : MvPolynomial σ R) {j : τ} (h : j ∈ (rename f φ).vars) :
∃ i : σ, i ∈ φ.vars ∧ f i = j := by |
classical
simpa only [exists_prop, Finset.mem_image] using vars_rename f φ h
|
/-
Copyright (c) 2021 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Tactic.NoncommRing
#align_import algebra.algebra.spectrum from "leanprover-c... | Mathlib/Algebra/Algebra/Spectrum.lean | 230 | 232 | theorem smul_mem_smul_iff {a : A} {s : R} {r : Rˣ} : r • s ∈ σ (r • a) ↔ s ∈ σ a := by |
simp only [mem_iff, not_iff_not, Algebra.algebraMap_eq_smul_one, smul_assoc, ← smul_sub,
isUnit_smul_iff]
|
/-
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 | 1,555 | 1,566 | theorem _root_.NNReal.count_const_le_le_of_tsum_le [MeasurableSingletonClass α] {a : α → ℝ≥0}
(a_mble : Measurable a) (a_summable : Summable a) {c : ℝ≥0} (tsum_le_c : ∑' i, a i ≤ c)
{ε : ℝ≥0} (ε_ne_zero : ε ≠ 0) : Measure.count { i : α | ε ≤ a i } ≤ c / ε := by |
rw [show (fun i => ε ≤ a i) = fun i => (ε : ℝ≥0∞) ≤ ((↑) ∘ a) i by
funext i
simp only [ENNReal.coe_le_coe, Function.comp]]
apply
ENNReal.count_const_le_le_of_tsum_le (measurable_coe_nnreal_ennreal.comp a_mble) _
(mod_cast ε_ne_zero) (@ENNReal.coe_ne_top ε)
convert ENNReal.coe_le_coe.mpr tsu... |
/-
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.EqLocus
import Mathlib.Algebra.Module.Subm... | Mathlib/LinearAlgebra/Span.lean | 666 | 667 | theorem map_subtype_span_singleton {p : Submodule R M} (x : p) :
map p.subtype (R ∙ x) = R ∙ (x : M) := by | simp [← span_image]
|
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Yaël Dillies
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.log from "leanprover-comm... | Mathlib/Data/Nat/Log.lean | 197 | 202 | theorem log_anti_left {b c n : ℕ} (hc : 1 < c) (hb : c ≤ b) : log b n ≤ log c n := by |
rcases eq_or_ne n 0 with (rfl | hn); · rw [log_zero_right, log_zero_right]
apply le_log_of_pow_le hc
calc
c ^ log b n ≤ b ^ log b n := Nat.pow_le_pow_left hb _
_ ≤ n := pow_log_le_self _ hn
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash, Eric Wieser
-/
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.Ring.Basic
import Mathlib.Topology.Algebra.Star
import Mathlib.LinearAlgebra.... | Mathlib/Topology/Instances/Matrix.lean | 332 | 337 | theorem Matrix.conjTranspose_tsum [StarAddMonoid R] [ContinuousStar R] [T2Space R]
{f : X → Matrix m n R} : (∑' x, f x)ᴴ = ∑' x, (f x)ᴴ := by |
by_cases hf : Summable f
· exact hf.hasSum.matrix_conjTranspose.tsum_eq.symm
· have hft := summable_matrix_conjTranspose.not.mpr hf
rw [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable hft, conjTranspose_zero]
|
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Finset.Pairwise
#align_import data.finsupp.big_operators from... | Mathlib/Data/Finsupp/BigOperators.lean | 81 | 96 | theorem List.support_sum_eq [AddMonoid M] (l : List (ι →₀ M))
(hl : l.Pairwise (_root_.Disjoint on Finsupp.support)) :
l.sum.support = l.foldr (Finsupp.support · ⊔ ·) ∅ := by |
induction' l with hd tl IH
· simp
· simp only [List.pairwise_cons] at hl
simp only [List.sum_cons, List.foldr_cons, Function.comp_apply]
rw [Finsupp.support_add_eq, IH hl.right, Finset.sup_eq_union]
suffices _root_.Disjoint hd.support (tl.foldr (fun x y ↦ (Finsupp.support x ⊔ y)) ∅) by
exact Fi... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_th... | Mathlib/GroupTheory/Perm/Support.lean | 415 | 421 | theorem support_prod_of_pairwise_disjoint (l : List (Perm α)) (h : l.Pairwise Disjoint) :
l.prod.support = (l.map support).foldr (· ⊔ ·) ⊥ := by |
induction' l with hd tl hl
· simp
· rw [List.pairwise_cons] at h
have : Disjoint hd tl.prod := disjoint_prod_right _ h.left
simp [this.support_mul, hl h.right]
|
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Anatole Dedecker, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 336 | 339 | theorem HasDerivAt.finset_prod (hf : ∀ i ∈ u, HasDerivAt (f i) (f' i) x) :
HasDerivAt (∏ i ∈ u, f i ·) (∑ i ∈ u, (∏ j ∈ u.erase i, f j x) • f' i) x := by |
simpa [ContinuousLinearMap.sum_apply, ContinuousLinearMap.smul_apply] using
(HasFDerivAt.finset_prod (fun i hi ↦ (hf i hi).hasFDerivAt)).hasDerivAt
|
/-
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, Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Finsupp.Multiset
import Math... | Mathlib/RingTheory/UniqueFactorizationDomain.lean | 613 | 618 | theorem prime_of_normalized_factor {a : α} : ∀ x : α, x ∈ normalizedFactors a → Prime x := by |
rw [normalizedFactors, factors]
split_ifs with ane0; · simp
intro x hx; rcases Multiset.mem_map.1 hx with ⟨y, ⟨hy, rfl⟩⟩
rw [(normalize_associated _).prime_iff]
exact (Classical.choose_spec (UniqueFactorizationMonoid.exists_prime_factors a ane0)).1 y hy
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | Mathlib/Topology/Inseparable.lean | 545 | 547 | theorem continuousOn_lift {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {s : Set (SeparationQuotient X)} :
ContinuousOn (lift f hf) s ↔ ContinuousOn f (mk ⁻¹' s) := by |
simp only [ContinuousOn, surjective_mk.forall, continuousWithinAt_lift, mem_preimage]
|
/-
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
-/
import Mathlib.Analysis.Complex.Asymptotics
import Mathlib.Analysis.SpecificLimits.Normed
#align_import analysis.special_... | Mathlib/Analysis/SpecialFunctions/Exp.lean | 208 | 212 | theorem tendsto_exp_atTop : Tendsto exp atTop atTop := by |
have A : Tendsto (fun x : ℝ => x + 1) atTop atTop :=
tendsto_atTop_add_const_right atTop 1 tendsto_id
have B : ∀ᶠ x in atTop, x + 1 ≤ exp x := eventually_atTop.2 ⟨0, fun x _ => add_one_le_exp x⟩
exact tendsto_atTop_mono' atTop B A
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Combinatorics.SetFamily.Compression.Down
import Mathlib.Order.UpperLower.Basic
import Mathlib.Data.Fintype.Powerset
... | Mathlib/Combinatorics/SetFamily/HarrisKleitman.lean | 122 | 132 | theorem IsUpperSet.le_card_inter_finset (h𝒜 : IsUpperSet (𝒜 : Set (Finset α)))
(hℬ : IsUpperSet (ℬ : Set (Finset α))) :
𝒜.card * ℬ.card ≤ 2 ^ Fintype.card α * (𝒜 ∩ ℬ).card := by |
rw [← isLowerSet_compl, ← coe_compl] at h𝒜
have := h𝒜.card_inter_le_finset hℬ
rwa [card_compl, Fintype.card_finset, tsub_mul, le_tsub_iff_le_tsub, ← mul_tsub, ←
card_sdiff inter_subset_right, sdiff_inter_self_right, sdiff_compl,
_root_.inf_comm] at this
· exact mul_le_mul_left' (card_le_card inter_su... |
/-
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 | 749 | 752 | theorem zmodEquivZPowers_apply_coe_nat (i : ℕ) :
h.zmodEquivZPowers i = Additive.ofMul (⟨ζ ^ i, i, rfl⟩ : Subgroup.zpowers ζ) := by |
have : (i : ZMod k) = (i : ℤ) := by norm_cast
simp only [this, zmodEquivZPowers_apply_coe_int, zpow_natCast]
|
/-
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.ContinuousFunction.Basic
#align_import topology.compact_open from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
/-!
# ... | Mathlib/Topology/CompactOpen.lean | 287 | 290 | theorem nhds_compactOpen_eq_iInf_nhds_induced (f : C(X, Y)) :
𝓝 f = ⨅ (s) (hs : IsCompact s), (𝓝 (f.restrict s)).comap (ContinuousMap.restrict s) := by |
rw [compactOpen_eq_iInf_induced]
simp only [nhds_iInf, nhds_induced]
|
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Restrict
/-!
# Some constructions of matroids
This file defines some very elementary examples of matroids, namely those with at most one bas... | Mathlib/Data/Matroid/Constructions.lean | 57 | 59 | theorem ground_eq_empty_iff : (M.E = ∅) ↔ M = emptyOn α := by |
simp only [emptyOn, eq_iff_indep_iff_indep_forall, iff_self_and]
exact fun h ↦ by simp [h, subset_empty_iff]
|
/-
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.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.... | Mathlib/Data/Finset/Lattice.lean | 1,245 | 1,249 | theorem sup'_inf_distrib_left (f : ι → α) (a : α) :
a ⊓ s.sup' hs f = s.sup' hs fun i ↦ a ⊓ f i := by |
induction hs using Finset.Nonempty.cons_induction with
| singleton => simp
| cons _ _ _ hs ih => simp_rw [sup'_cons hs, inf_sup_left, ih]
|
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow, Kexing Ying
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.LinearAlgebra.BilinearForm.Properties
/-!
# Bilinear form
This file defines ort... | Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean | 224 | 231 | theorem nondegenerate_restrict_of_disjoint_orthogonal (B : BilinForm R₁ M₁) (b : B.IsRefl)
{W : Submodule R₁ M₁} (hW : Disjoint W (B.orthogonal W)) : (B.restrict W).Nondegenerate := by |
rintro ⟨x, hx⟩ b₁
rw [Submodule.mk_eq_zero, ← Submodule.mem_bot R₁]
refine hW.le_bot ⟨hx, fun y hy => ?_⟩
specialize b₁ ⟨y, hy⟩
simp only [restrict_apply, domRestrict_apply] at b₁
exact isOrtho_def.mpr (b x y b₁)
|
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Yaël Dillies
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
#align_import analysis.special_functions.trigonometric.series from "leanprover-community/mathlib"@"ccf84... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Series.lean | 75 | 79 | theorem Complex.hasSum_sin (z : ℂ) :
HasSum (fun n : ℕ => (-1) ^ n * z ^ (2 * n + 1) / ↑(2 * n + 1)!) (Complex.sin z) := by |
convert Complex.hasSum_sin' z using 1
simp_rw [mul_pow, pow_succ, pow_mul, Complex.I_sq, ← mul_assoc, mul_div_assoc, div_right_comm,
div_self Complex.I_ne_zero, mul_comm _ ((-1 : ℂ) ^ _), mul_one_div, mul_div_assoc, mul_assoc]
|
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Yaël Dillies
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.log from "leanprover-comm... | Mathlib/Data/Nat/Log.lean | 251 | 252 | theorem clog_of_left_le_one {b : ℕ} (hb : b ≤ 1) (n : ℕ) : clog b n = 0 := by |
rw [clog, dif_neg fun h : 1 < b ∧ 1 < n => h.1.not_le hb]
|
/-
Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
#align_import linear_algebra.eigenspace.is_alg_closed from "lea... | Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean | 64 | 123 | theorem iSup_genEigenspace_eq_top [IsAlgClosed K] [FiniteDimensional K V] (f : End K V) :
⨆ (μ : K) (k : ℕ), f.genEigenspace μ k = ⊤ := by |
-- We prove the claim by strong induction on the dimension of the vector space.
induction' h_dim : finrank K V using Nat.strong_induction_on with n ih generalizing V
cases' n with n
-- If the vector space is 0-dimensional, the result is trivial.
· rw [← top_le_iff]
simp only [Submodule.finrank_eq_zero.1 ... |
/-
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 | 581 | 592 | theorem realize_onBoundedFormula [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M] {n : ℕ}
(ψ : L.BoundedFormula α n) {v : α → M} {xs : Fin n → M} :
(φ.onBoundedFormula ψ).Realize v xs ↔ ψ.Realize v xs := by |
induction' ψ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3
· rfl
· simp only [onBoundedFormula, realize_bdEqual, realize_onTerm]
rfl
· simp only [onBoundedFormula, realize_rel, LHom.map_onRelation,
Function.comp_apply, realize_onTerm]
rfl
· simp only [onBoundedFormula, ih1, ih2, realize_imp]
· s... |
/-
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.Algebra.Order.Ring.Basic
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align... | Mathlib/Computability/Ackermann.lean | 311 | 376 | theorem exists_lt_ack_of_nat_primrec {f : ℕ → ℕ} (hf : Nat.Primrec f) :
∃ m, ∀ n, f n < ack m n := by |
induction' hf with f g hf hg IHf IHg f g hf hg IHf IHg f g hf hg IHf IHg
-- Zero function:
· exact ⟨0, ack_pos 0⟩
-- Successor function:
· refine ⟨1, fun n => ?_⟩
rw [succ_eq_one_add]
apply add_lt_ack
-- Left projection:
· refine ⟨0, fun n => ?_⟩
rw [ack_zero, Nat.lt_succ_iff]
exact unpai... |
/-
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.EqLocus
import Mathlib.Algebra.Module.Subm... | Mathlib/LinearAlgebra/Span.lean | 1,024 | 1,025 | theorem map_le_map_iff' {f : F} (hf : ker f = ⊥) {p p'} : map f p ≤ map f p' ↔ p ≤ p' := by |
rw [LinearMap.map_le_map_iff, hf, sup_bot_eq]
|
/-
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.Complex
#align_import analysis.special_function... | Mathlib/Analysis/SpecialFunctions/Trigonometric/ComplexDeriv.lean | 53 | 56 | theorem continuousAt_tan {x : ℂ} : ContinuousAt tan x ↔ cos x ≠ 0 := by |
refine ⟨fun hc h₀ => ?_, fun h => (hasDerivAt_tan h).continuousAt⟩
exact not_tendsto_nhds_of_tendsto_atTop (tendsto_abs_tan_of_cos_eq_zero h₀) _
(hc.norm.tendsto.mono_left inf_le_left)
|
/-
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.Perfect
import Mathlib.Topology.MetricSpace.Polish
import Mathlib.Topology.MetricSpace.CantorScheme
#align_import topology.perfect from "l... | Mathlib/Topology/MetricSpace/Perfect.lean | 80 | 129 | theorem Perfect.exists_nat_bool_injection [CompleteSpace α] :
∃ f : (ℕ → Bool) → α, range f ⊆ C ∧ Continuous f ∧ Injective f := by |
obtain ⟨u, -, upos', hu⟩ := exists_seq_strictAnti_tendsto' (zero_lt_one' ℝ≥0∞)
have upos := fun n => (upos' n).1
let P := Subtype fun E : Set α => Perfect E ∧ E.Nonempty
choose C0 C1 h0 h1 hdisj using
fun {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε) =>
hC.small_diam_spl... |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.ContDiff.Bounds
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysi... | Mathlib/Analysis/Distribution/SchwartzSpace.lean | 210 | 214 | theorem decay_smul_aux (k n : ℕ) (f : 𝓢(E, F)) (c : 𝕜) (x : E) :
‖x‖ ^ k * ‖iteratedFDeriv ℝ n (c • (f : E → F)) x‖ =
‖c‖ * ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ := by |
rw [mul_comm ‖c‖, mul_assoc, iteratedFDeriv_const_smul_apply (f.smooth _),
norm_smul c (iteratedFDeriv ℝ n (⇑f) x)]
|
/-
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 | 2,736 | 2,739 | theorem map_count_True_eq_filter_card (s : Multiset α) (p : α → Prop) [DecidablePred p] :
(s.map p).count True = card (s.filter p) := by |
simp only [count_eq_card_filter_eq, map_filter, card_map, Function.id_comp,
eq_true_eq_id, Function.comp_apply]
|
/-
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.Subgroup.Finite
import Mathlib.Data.Finset.Fin
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Int.Order.Units
import Mathlib.GroupTheory... | Mathlib/GroupTheory/Perm/Sign.lean | 494 | 497 | theorem sign_eq_sign_of_equiv [DecidableEq β] [Fintype β] (f : Perm α) (g : Perm β) (e : α ≃ β)
(h : ∀ x, e (f x) = g (e x)) : sign f = sign g := by |
have hg : g = (e.symm.trans f).trans e := Equiv.ext <| by simp [h]
rw [hg, sign_symm_trans_trans]
|
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Measure.Content
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Topology.Algebra.Group.Compact
#align_import measure_theory.m... | Mathlib/MeasureTheory/Measure/Haar/Basic.lean | 575 | 578 | theorem is_left_invariant_haarContent {K₀ : PositiveCompacts G} (g : G) (K : Compacts G) :
haarContent K₀ (K.map _ <| continuous_mul_left g) = haarContent K₀ K := by |
simpa only [ENNReal.coe_inj, ← NNReal.coe_inj, haarContent_apply] using
is_left_invariant_chaar g K
|
/-
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
#align_import measu... | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | 87 | 89 | theorem centerAndRescale_radius {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ) :
a.centerAndRescale.r (last N) = 1 := by |
simp [SatelliteConfig.centerAndRescale, inv_mul_cancel (a.rpos _).ne']
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | Mathlib/FieldTheory/IsAlgClosed/Basic.lean | 104 | 111 | theorem roots_eq_zero_iff [IsAlgClosed k] {p : k[X]} :
p.roots = 0 ↔ p = Polynomial.C (p.coeff 0) := by |
refine ⟨fun h => ?_, fun hp => by rw [hp, roots_C]⟩
rcases le_or_lt (degree p) 0 with hd | hd
· exact eq_C_of_degree_le_zero hd
· obtain ⟨z, hz⟩ := IsAlgClosed.exists_root p hd.ne'
rw [← mem_roots (ne_zero_of_degree_gt hd), h] at hz
simp at hz
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.PEquiv
#align_import data.matrix.pequiv from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa... | Mathlib/Data/Matrix/PEquiv.lean | 62 | 67 | theorem mul_matrix_apply [Fintype m] [DecidableEq m] [Semiring α] (f : l ≃. m) (M : Matrix m n α)
(i j) : (f.toMatrix * M :) i j = Option.casesOn (f i) 0 fun fi => M fi j := by |
dsimp [toMatrix, Matrix.mul_apply]
cases' h : f i with fi
· simp [h]
· rw [Finset.sum_eq_single fi] <;> simp (config := { contextual := true }) [h, eq_comm]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | Mathlib/Order/BooleanAlgebra.lean | 176 | 176 | theorem inf_sdiff_self_left : y \ x ⊓ x = ⊥ := by | rw [inf_comm, inf_sdiff_self_right]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Topology.Instances.RealVectorSpace
import Mathlib.Analysis.NormedSpace.AffineIsometry
#align_import analysis.normed_space.mazur_ulam from "leanprove... | Mathlib/Analysis/NormedSpace/MazurUlam.lean | 158 | 161 | theorem coe_toRealAffineIsometryEquiv (f : PE ≃ᵢ PF) :
f.toRealAffineIsometryEquiv.toIsometryEquiv = f := by |
ext
rfl
|
/-
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 | 1,375 | 1,381 | theorem Valid'.balanceR_aux {l} {x : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hr : Valid' x r o₂)
(H₁ : size r = 0 → size l ≤ 1) (H₂ : 1 ≤ size r → 1 ≤ size l → size l ≤ delta * size r)
(H₃ : 2 * @size α r ≤ 9 * size l + 5 ∨ size r ≤ 3) : Valid' o₁ (@balanceR α l x r) o₂ := by |
rw [Valid'.dual_iff, dual_balanceR]
have := hr.dual.balanceL_aux hl.dual
rw [size_dual, size_dual] at this
exact this H₁ H₂ H₃
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.Algebra.Module.ULift
#align_import ring_theory.is_tensor_product from "leanprover-community/mathlib"@"c4926d76... | Mathlib/RingTheory/IsTensorProduct.lean | 192 | 200 | theorem IsBaseChange.algHom_ext (g₁ g₂ : N →ₗ[S] Q) (e : ∀ x, g₁ (f x) = g₂ (f x)) : g₁ = g₂ := by |
ext x
refine h.inductionOn x ?_ ?_ ?_ ?_
· rw [map_zero, map_zero]
· assumption
· intro s n e'
rw [g₁.map_smul, g₂.map_smul, e']
· intro x y e₁ e₂
rw [map_add, map_add, e₁, e₂]
|
/-
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 | 778 | 786 | theorem UniformSpace.mem_nhds_iff_symm {x : α} {s : Set α} :
s ∈ 𝓝 x ↔ ∃ V ∈ 𝓤 α, SymmetricRel V ∧ ball x V ⊆ s := by |
rw [UniformSpace.mem_nhds_iff]
constructor
· rintro ⟨V, V_in, V_sub⟩
use symmetrizeRel V, symmetrize_mem_uniformity V_in, symmetric_symmetrizeRel V
exact Subset.trans (ball_mono (symmetrizeRel_subset_self V) x) V_sub
· rintro ⟨V, V_in, _, V_sub⟩
exact ⟨V, V_in, V_sub⟩
|
/-
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, Yaël Dillies
-/
import Mathlib.Data.Finset.NAry
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Set.Pointwise.Finite
import Mathlib.Data.Set.Pointwise.SMul
... | Mathlib/Data/Finset/Pointwise.lean | 1,364 | 1,365 | theorem preimage_mul_right_one' : preimage 1 (· * b⁻¹) (mul_left_injective _).injOn = {b} := by |
rw [preimage_mul_right_one, inv_inv]
|
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.MeasureTheory.Integral.FundThmCalculus
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
import Mathlib.Analysis.SpecialFunction... | Mathlib/Analysis/SpecialFunctions/Integrals.lean | 261 | 265 | theorem intervalIntegrable_one_div_one_add_sq :
IntervalIntegrable (fun x : ℝ => 1 / (↑1 + x ^ 2)) μ a b := by |
refine (continuous_const.div ?_ fun x => ?_).intervalIntegrable a b
· continuity
· nlinarith
|
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Monad.Basic
import Mathlib.Data.Part
import Mathlib.Order.Chain
import Mathlib.Order.Hom.Order
import Mathlib.Algebra.Order.Ring.Nat
#align_import o... | Mathlib/Order/OmegaCompletePartialOrder.lean | 214 | 221 | theorem ωSup_total {c : Chain α} {x : α} (h : ∀ i, c i ≤ x ∨ x ≤ c i) : ωSup c ≤ x ∨ x ≤ ωSup c :=
by_cases
(fun (this : ∀ i, c i ≤ x) => Or.inl (ωSup_le _ _ this))
(fun (this : ¬∀ i, c i ≤ x) =>
have : ∃ i, ¬c i ≤ x := by | simp only [not_forall] at this ⊢; assumption
let ⟨i, hx⟩ := this
have : x ≤ c i := (h i).resolve_left hx
Or.inr <| le_ωSup_of_le _ this)
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Wrenna Robson
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebr... | Mathlib/LinearAlgebra/Lagrange.lean | 376 | 379 | theorem degree_interpolate_erase_lt (hvs : Set.InjOn v s) (hi : i ∈ s) :
(interpolate (s.erase i) v r).degree < ↑(s.card - 1) := by |
rw [← Finset.card_erase_of_mem hi]
exact degree_interpolate_lt _ (Set.InjOn.mono (coe_subset.mpr (erase_subset _ _)) hvs)
|
/-
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 | 256 | 258 | theorem weightedVSub_apply (w : ι → k) (p : ι → P) :
s.weightedVSub p w = ∑ i ∈ s, w i • (p i -ᵥ Classical.choice S.nonempty) := by |
simp [weightedVSub, LinearMap.sum_apply]
|
/-
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 Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Ta... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 1,453 | 1,460 | theorem findIdxs_cons_aux (p : α → Bool) :
foldrIdx (fun i a is => if p a = true then (i + 1) :: is else is) [] xs s =
map (· + 1) (foldrIdx (fun i a is => if p a = true then i :: is else is) [] xs s) := by |
induction xs generalizing s with
| nil => rfl
| cons x xs ih =>
simp only [foldrIdx]
split <;> simp [ih]
|
/-
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.Probability.Variance
#align_import probability.moments from "leanprover-community/mathlib"@"85453a2a14be8da64caf15ca50930cf4c6e5d8de"
/-!
# Moments and m... | Mathlib/Probability/Moments.lean | 141 | 147 | theorem cgf_const' [IsFiniteMeasure μ] (hμ : μ ≠ 0) (c : ℝ) :
cgf (fun _ => c) μ t = log (μ Set.univ).toReal + t * c := by |
simp only [cgf, mgf_const']
rw [log_mul _ (exp_pos _).ne']
· rw [log_exp _]
· rw [Ne, ENNReal.toReal_eq_zero_iff, Measure.measure_univ_eq_zero]
simp only [hμ, measure_ne_top μ Set.univ, or_self_iff, not_false_iff]
|
/-
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.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLim... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 1,477 | 1,484 | theorem sum_apply₀ (f : ι → Measure α) {s : Set α} (hs : NullMeasurableSet s (sum f)) :
sum f s = ∑' i, f i s := by |
apply le_antisymm ?_ (le_sum_apply _ _)
rcases hs.exists_measurable_subset_ae_eq with ⟨t, ts, t_meas, ht⟩
calc
sum f s = sum f t := measure_congr ht.symm
_ = ∑' i, f i t := sum_apply _ t_meas
_ ≤ ∑' i, f i s := ENNReal.tsum_le_tsum fun i ↦ measure_mono ts
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Ring.Prod
import Mathlib.Algebra.Ring.Subring.Basic
import Mathlib.Topology.Algebra.... | Mathlib/Topology/Algebra/Ring/Basic.lean | 314 | 316 | theorem toTopologicalSpace_injective :
Injective (toTopologicalSpace : RingTopology α → TopologicalSpace α) := by |
intro f g _; cases f; cases g; congr
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shape... | Mathlib/CategoryTheory/Limits/Shapes/Images.lean | 108 | 115 | theorem ext {F F' : MonoFactorisation f} (hI : F.I = F'.I)
(hm : F.m = eqToHom hI ≫ F'.m) : F = F' := by |
cases' F with _ Fm _ _ Ffac; cases' F' with _ Fm' _ _ Ffac'
cases' hI
simp? at hm says simp only [eqToHom_refl, Category.id_comp] at hm
congr
apply (cancel_mono Fm).1
rw [Ffac, hm, Ffac']
|
/-
Copyright (c) 2022 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012... | Mathlib/Order/Interval/Set/WithBotTop.lean | 59 | 59 | theorem preimage_coe_Icc : (some : α → WithTop α) ⁻¹' Icc a b = Icc a b := by | simp [← Ici_inter_Iic]
|
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.RBMap.Basic
import Batteries.Tactic.SeqFocus
/-!
# Lemmas for Red-black trees
The main theorem in this file is `WF_def`, which shows that the ... | .lake/packages/batteries/Batteries/Data/RBMap/WF.lean | 507 | 511 | theorem WF.out {t : RBNode α} (h : t.WF cmp) : t.Ordered cmp ∧ ∃ c n, t.Balanced c n := by |
induction h with
| mk o h => exact ⟨o, _, _, h⟩
| insert _ ih => have ⟨o, _, _, h⟩ := ih; exact ⟨o.insert, h.insert⟩
| erase _ ih => have ⟨o, _, _, h⟩ := ih; exact ⟨o.erase, _, h.erase⟩
|
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.Analysis.NormedSpace.Star.Basic
#align_import analysis... | Mathlib/Analysis/InnerProductSpace/Dual.lean | 82 | 91 | theorem ext_inner_left_basis {ι : Type*} {x y : E} (b : Basis ι 𝕜 E)
(h : ∀ i : ι, ⟪b i, x⟫ = ⟪b i, y⟫) : x = y := by |
apply (toDualMap 𝕜 E).map_eq_iff.mp
refine (Function.Injective.eq_iff ContinuousLinearMap.coe_injective).mp (Basis.ext b ?_)
intro i
simp only [ContinuousLinearMap.coe_coe]
rw [toDualMap_apply, toDualMap_apply]
rw [← inner_conj_symm]
conv_rhs => rw [← inner_conj_symm]
exact congr_arg conj (h i)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
/-!
... | Mathlib/Data/ENNReal/Real.lean | 335 | 338 | theorem le_ofReal_iff_toReal_le {a : ℝ≥0∞} {b : ℝ} (ha : a ≠ ∞) (hb : 0 ≤ b) :
a ≤ ENNReal.ofReal b ↔ ENNReal.toReal a ≤ b := by |
lift a to ℝ≥0 using ha
simpa [ENNReal.ofReal, ENNReal.toReal] using Real.le_toNNReal_iff_coe_le hb
|
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import... | Mathlib/Algebra/Algebra/Operations.lean | 463 | 467 | theorem le_pow_toAddSubmonoid {n : ℕ} : M.toAddSubmonoid ^ n ≤ (M ^ n).toAddSubmonoid := by |
obtain rfl | hn := Decidable.eq_or_ne n 0
· rw [pow_zero, pow_zero]
exact le_one_toAddSubmonoid
· exact (pow_toAddSubmonoid M hn).ge
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.Topology.MetricSpace.IsometricSMul
#alig... | Mathlib/Analysis/Normed/Group/Pointwise.lean | 246 | 247 | theorem mul_ball : s * ball x δ = x • thickening δ s := by |
rw [← smul_ball_one, mul_smul_comm, mul_ball_one]
|
/-
Copyright (c) 2021 Alex J. Best. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Yaël Dillies
-/
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#a... | Mathlib/Algebra/Order/Pointwise.lean | 211 | 222 | theorem smul_Ico : r • Ico a b = Ico (r • a) (r • b) := by |
ext x
simp only [mem_smul_set, smul_eq_mul, mem_Ico]
constructor
· rintro ⟨a, ⟨a_h_left_left, a_h_left_right⟩, rfl⟩
constructor
· exact (mul_le_mul_left hr).mpr a_h_left_left
· exact (mul_lt_mul_left hr).mpr a_h_left_right
· rintro ⟨a_left, a_right⟩
use x / r
refine ⟨⟨(le_div_iff' hr).mpr... |
/-
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.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Ma... | Mathlib/Analysis/Analytic/Basic.lean | 698 | 709 | theorem HasFPowerSeriesOnBall.uniform_geometric_approx {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1,
∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n, ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n := by |
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine ⟨a, ha, C, hC, fun y hy n => (hp y hy n).trans ?_⟩
have yr' : ‖y‖ < r' := by rwa [ball_zero_eq] at hy
have :=... |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Analysis.Normed.Group.Basic
#align_import analysis.normed.group.hom from "leanprover-community/mathlib"@"3c4225288b55380a90df078ebae0991080b12393"
/-... | Mathlib/Analysis/Normed/Group/Hom.lean | 802 | 804 | theorem incl_range (s : AddSubgroup V₁) : (incl s).range = s := by |
ext x
exact ⟨fun ⟨y, hy⟩ => by rw [← hy]; simp, fun hx => ⟨⟨x, hx⟩, by simp⟩⟩
|
/-
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.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
im... | Mathlib/Data/List/Basic.lean | 1,570 | 1,570 | theorem map_tail (f : α → β) (l) : map f (tail l) = tail (map f l) := by | cases l <;> rfl
|
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
import Mathlib.Data.List.Basic
import Mathlib.Logic.Equiv.D... | Mathlib/Algebra/Free.lean | 565 | 565 | theorem lift_of_mul (x y) : lift f (of x * y) = f x * lift f y := by | rw [map_mul, lift_of]
|
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d... | Mathlib/Order/Interval/Set/Disjoint.lean | 278 | 282 | theorem iInter_Iio_of_not_bddBelow_range (hf : ¬ BddBelow (range f)) : ⋂ i, Iio (f i) = ∅ := by |
refine eq_empty_of_subset_empty ?_
rw [← iInter_Iic_eq_empty_iff.mpr hf]
gcongr
exact Iio_subset_Iic_self
|
/-
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.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathli... | Mathlib/MeasureTheory/Measure/Hausdorff.lean | 796 | 809 | theorem MeasureTheory.Measure.hausdorffMeasure_smul₀ {𝕜 E : Type*} [NormedAddCommGroup E]
[NormedField 𝕜] [NormedSpace 𝕜 E] [MeasurableSpace E] [BorelSpace E] {d : ℝ} (hd : 0 ≤ d)
{r : 𝕜} (hr : r ≠ 0) (s : Set E) : μH[d] (r • s) = ‖r‖₊ ^ d • μH[d] s := by |
suffices ∀ {r : 𝕜}, r ≠ 0 → ∀ s : Set E, μH[d] (r • s) ≤ ‖r‖₊ ^ d • μH[d] s by
refine le_antisymm (this hr s) ?_
rw [← le_inv_smul_iff_of_pos]
· dsimp
rw [← NNReal.inv_rpow, ← nnnorm_inv]
· refine Eq.trans_le ?_ (this (inv_ne_zero hr) (r • s))
rw [inv_smul_smul₀ hr]
· simp [pos_i... |
/-
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 | theorem content_one : content (1 : R[X]) = 1 := by | rw [← C_1, content_C, normalize_one]
|
/-
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.NumberTheory.LegendreSymbol.Basic
import Mathlib.Analysis.Normed.Field.Basic
#align_import number_theory.legendre_symbol.gauss_eisenstein_lemmas from "lea... | Mathlib/NumberTheory/LegendreSymbol/GaussEisensteinLemmas.lean | 193 | 226 | theorem sum_mul_div_add_sum_mul_div_eq_mul (p q : ℕ) [hp : Fact p.Prime] (hq0 : (q : ZMod p) ≠ 0) :
∑ a ∈ Ico 1 (p / 2).succ, a * q / p + ∑ a ∈ Ico 1 (q / 2).succ, a * p / q =
p / 2 * (q / 2) := by |
have hswap :
((Ico 1 (q / 2).succ ×ˢ Ico 1 (p / 2).succ).filter fun x : ℕ × ℕ => x.2 * q ≤ x.1 * p).card =
((Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ).filter fun x : ℕ × ℕ =>
x.1 * q ≤ x.2 * p).card :=
card_equiv (Equiv.prodComm _ _)
(fun ⟨_, _⟩ => by
simp (config := { contextual ... |
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Su... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 93 | 99 | theorem eval_at_1 (n ν : ℕ) : (bernsteinPolynomial R n ν).eval 1 = if ν = n then 1 else 0 := by |
rw [bernsteinPolynomial]
split_ifs with h
· subst h; simp
· obtain hνn | hnν := Ne.lt_or_lt h
· simp [zero_pow $ Nat.sub_ne_zero_of_lt hνn]
· simp [Nat.choose_eq_zero_of_lt hnν]
|
/-
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 | 414 | 415 | theorem arccos_of_one_le {x : ℝ} (hx : 1 ≤ x) : arccos x = 0 := by |
rw [arccos, arcsin_of_one_le hx, sub_self]
|
/-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.Opposites
#align_import category_theory.eq_to_hom from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd29... | Mathlib/CategoryTheory/EqToHom.lean | 287 | 288 | theorem hcongr_hom {F G : C ⥤ D} (h : F = G) {X Y} (f : X ⟶ Y) : HEq (F.map f) (G.map f) := by |
rw [h]
|
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.ConeCategory
#align_import c... | Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean | 305 | 306 | theorem ι_sndSigmaMap (b) : Sigma.ι I.left b ≫ I.sndSigmaMap = I.snd b ≫ Sigma.ι I.right _ := by |
simp [sndSigmaMap]
|
/-
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 Batteries.Data.List.Basic
import Batteries.Data.List.Lemmas
/-!
# Counting in lists
T... | .lake/packages/batteries/Batteries/Data/List/Count.lean | 188 | 190 | theorem count_replicate (a b : α) (n : Nat) : count a (replicate n b) = if a = b then n else 0 := by |
split
exacts [‹a = b› ▸ count_replicate_self .., count_eq_zero.2 <| mt eq_of_mem_replicate ‹a ≠ b›]
|
/-
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.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algeb... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 1,132 | 1,135 | theorem sin_pi_div_two : sin (π / 2) = 1 :=
calc
sin (π / 2) = Real.sin (π / 2) := by | rw [ofReal_sin]; simp
_ = 1 := by simp
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.