Context stringlengths 285 6.98k | file_name stringlengths 21 79 | start int64 14 184 | end int64 18 184 | theorem stringlengths 25 1.34k | proof stringlengths 5 3.43k |
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/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Heather Macbeth, Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Topolo... | Mathlib/Analysis/Normed/Group/InfiniteSum.lean | 40 | 46 | theorem cauchySeq_finset_iff_vanishing_norm {f : ι → E} :
(CauchySeq fun s : Finset ι => ∑ i ∈ s, f i) ↔
∀ ε > (0 : ℝ), ∃ s : Finset ι, ∀ t, Disjoint t s → ‖∑ i ∈ t, f i‖ < ε := by |
rw [cauchySeq_finset_iff_sum_vanishing, nhds_basis_ball.forall_iff]
· simp only [ball_zero_eq, Set.mem_setOf_eq]
· rintro s t hst ⟨s', hs'⟩
exact ⟨s', fun t' ht' => hst <| hs' _ ht'⟩
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Scott Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
#align_import algebra.... | Mathlib/Algebra/Category/ModuleCat/Monoidal/Symmetric.lean | 49 | 52 | theorem braiding_naturality_right (X : ModuleCat R) {Y Z : ModuleCat R} (f : Y ⟶ Z) :
X ◁ f ≫ (braiding X Z).hom = (braiding X Y).hom ≫ f ▷ X := by |
simp_rw [← id_tensorHom]
apply braiding_naturality
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Order.Basic
import Mathlib.Data.Set.Pointwise.Basic
/-!
# Neighborhoods to the left and to the right on an ... | Mathlib/Topology/Order/LeftRightNhds.lean | 99 | 102 | theorem countable_setOf_isolated_right [SecondCountableTopology α] :
{ x : α | 𝓝[>] x = ⊥ }.Countable := by |
simp only [nhdsWithin_Ioi_eq_bot_iff, setOf_or]
exact (subsingleton_isTop α).countable.union countable_setOf_covBy_right
|
/-
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.GDelta
#align_import topology.metric_space.baire from "leanprover-community/mathlib"@"b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a"
/-!
# ... | Mathlib/Topology/Baire/Lemmas.lean | 60 | 63 | theorem dense_biInter_of_isOpen {S : Set α} {f : α → Set X} (ho : ∀ s ∈ S, IsOpen (f s))
(hS : S.Countable) (hd : ∀ s ∈ S, Dense (f s)) : Dense (⋂ s ∈ S, f s) := by |
rw [← sInter_image]
refine dense_sInter_of_isOpen ?_ (hS.image _) ?_ <;> rwa [forall_mem_image]
|
/-
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.Data.Finsupp.Defs
#align_import data.list.to_finsupp from "leanprover-community/mathlib"@"06a655b5fcfbda03502f9158bbf6c0f1400886f9"
/-!
# Lists as... | Mathlib/Data/List/ToFinsupp.lean | 86 | 89 | theorem toFinsupp_nil [DecidablePred fun i => getD ([] : List M) i 0 ≠ 0] :
toFinsupp ([] : List M) = 0 := by |
ext
simp
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Johan Commelin
-/
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
#align_import category_theory.limits.shapes.zero_objects from "leanprover-community/mathlib"@"74... | Mathlib/CategoryTheory/Limits/Shapes/ZeroObjects.lean | 117 | 123 | theorem of_iso (hY : IsZero Y) (e : X ≅ Y) : IsZero X := by |
refine ⟨fun Z => ⟨⟨⟨e.hom ≫ hY.to_ Z⟩, fun f => ?_⟩⟩,
fun Z => ⟨⟨⟨hY.from_ Z ≫ e.inv⟩, fun f => ?_⟩⟩⟩
· rw [← cancel_epi e.inv]
apply hY.eq_of_src
· rw [← cancel_mono e.hom]
apply hY.eq_of_tgt
|
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Measurable
/-! # Measurability of the line derivative
We prove in `me... | Mathlib/Analysis/Calculus/LineDeriv/Measurable.lean | 40 | 45 | theorem measurable_lineDeriv [MeasurableSpace F] [BorelSpace F]
(hf : Continuous f) : Measurable (fun x ↦ lineDeriv 𝕜 f x v) := by |
borelize 𝕜
let g : E → 𝕜 → F := fun x t ↦ f (x + t • v)
have hg : Continuous g.uncurry := by apply hf.comp; continuity
exact (measurable_deriv_with_param hg).comp measurable_prod_mk_right
|
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Scott Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Monoidal.Category
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheo... | Mathlib/CategoryTheory/Monoidal/Functor.lean | 113 | 116 | theorem LaxMonoidalFunctor.μ_natural (F : LaxMonoidalFunctor C D) {X Y X' Y' : C}
(f : X ⟶ Y) (g : X' ⟶ Y') :
(F.map f ⊗ F.map g) ≫ F.μ Y Y' = F.μ X X' ≫ F.map (f ⊗ g) := by |
simp [tensorHom_def]
|
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Jireh Loreaux
-/
import Mathlib.Algebra.Group.Center
#align_import group_theory.subsemigroup.centralizer from "leanprover-community/mathlib"@"cc67cd75b4e54191e13c2e8... | Mathlib/Algebra/Group/Centralizer.lean | 94 | 97 | theorem div_mem_centralizer [Group M] (ha : a ∈ centralizer S) (hb : b ∈ centralizer S) :
a / b ∈ centralizer S := by |
rw [div_eq_mul_inv]
exact mul_mem_centralizer ha (inv_mem_centralizer hb)
|
/-
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.SetTheory.Cardinal.Ordinal
#align_import set_theory.cardinal.continuum from "leanprover-community/mathlib"@"e08a42b2dd544cf11eba72e5fc7bf199d4349925... | Mathlib/SetTheory/Cardinal/Continuum.lean | 83 | 83 | theorem beth_one : beth 1 = 𝔠 := by | simpa using beth_succ 0
|
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Function.LocallyIntegrabl... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 110 | 113 | theorem integral_union_ae (hst : AEDisjoint μ s t) (ht : NullMeasurableSet t μ)
(hfs : IntegrableOn f s μ) (hft : IntegrableOn f t μ) :
∫ x in s ∪ t, f x ∂μ = ∫ x in s, f x ∂μ + ∫ x in t, f x ∂μ := by |
simp only [IntegrableOn, Measure.restrict_union₀ hst ht, integral_add_measure hfs hft]
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Devon Tuma
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.... | Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 42 | 44 | theorem coeff_scaleRoots_natDegree (p : R[X]) (s : R) :
(scaleRoots p s).coeff p.natDegree = p.leadingCoeff := by |
rw [leadingCoeff, coeff_scaleRoots, tsub_self, pow_zero, mul_one]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, James Gallicchio
-/
import Batteries.Data.List.Count
import Batteries.Data.Fin.Lemmas
/-!
# Pairwise relations on a list
This file provides basic results about `List.... | .lake/packages/batteries/Batteries/Data/List/Pairwise.lean | 57 | 63 | theorem Pairwise.and (hR : Pairwise R l) (hS : Pairwise S l) :
l.Pairwise fun a b => R a b ∧ S a b := by |
induction hR with
| nil => simp only [Pairwise.nil]
| cons R1 _ IH =>
simp only [Pairwise.nil, pairwise_cons] at hS ⊢
exact ⟨fun b bl => ⟨R1 b bl, hS.1 b bl⟩, IH hS.2⟩
|
/-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.LinearAlgebra.Matrix.Spec... | Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean | 125 | 125 | theorem Gamma0_det (N : ℕ) (A : Gamma0 N) : (A.1.1.det : ZMod N) = 1 := by | simp [A.1.property]
|
/-
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.Algebra.Polynomial.Module.Basic
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import... | Mathlib/RingTheory/Filtration.lean | 74 | 76 | theorem pow_smul_le_pow_smul (i j k : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • F.N (i + j) := by |
rw [add_comm, pow_add, mul_smul]
exact smul_mono_right _ (F.pow_smul_le i j)
|
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.Algebra.Field.ULift
import Mathlib.Algebra.MvPolynomial.Cardinal
import Mathlib.Data.Nat.Factorization.PrimePow
import Mathlib.Data.Rat.Denumerable
imp... | Mathlib/FieldTheory/Cardinality.lean | 80 | 85 | theorem Field.nonempty_iff {α : Type u} : Nonempty (Field α) ↔ IsPrimePow #α := by |
rw [Cardinal.isPrimePow_iff]
cases' fintypeOrInfinite α with h h
· simpa only [Cardinal.mk_fintype, Nat.cast_inj, exists_eq_left',
(Cardinal.nat_lt_aleph0 _).not_le, false_or_iff] using Fintype.nonempty_field_iff
· simpa only [← Cardinal.infinite_iff, h, true_or_iff, iff_true_iff] using Infinite.nonempty... |
/-
Copyright (c) 2021 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
#align_import data.bool.set from "leanprover-community/mathlib"@"ed60ee25ed00d7a62a0d1e5808092e1324cee451"
/-!
# Booleans and ... | Mathlib/Data/Bool/Set.lean | 27 | 28 | theorem range_eq {α : Type*} (f : Bool → α) : range f = {f false, f true} := by |
rw [← image_univ, univ_eq, image_pair]
|
/-
Copyright (c) 2024 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Order.Filter.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
/-!
#... | Mathlib/Order/Filter/CardinalInter.lean | 90 | 94 | theorem cardinal_iInter_mem {s : ι → Set α} (hic : #ι < c) :
(⋂ i, s i) ∈ l ↔ ∀ i, s i ∈ l := by |
rw [← sInter_range _]
apply (cardinal_sInter_mem (lt_of_le_of_lt Cardinal.mk_range_le hic)).trans
exact forall_mem_range
|
/-
Copyright (c) 2022 George Peter Banyard, Yaël Dillies, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: George Peter Banyard, Yaël Dillies, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Connectivity
#align_import combinatorics.simple_graph.prod... | Mathlib/Combinatorics/SimpleGraph/Prod.lean | 65 | 66 | theorem boxProd_adj_right : (G □ H).Adj (a, b₁) (a, b₂) ↔ H.Adj b₁ b₂ := by |
simp only [boxProd_adj, SimpleGraph.irrefl, false_and, and_true, false_or]
|
/-
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.Analysis.Convex.Exposed
import Mathlib.Analysis.NormedSpace.HahnBanach.Separation
import Mathlib.Topology.Algebra.ContinuousAffineMap
#align_import analys... | Mathlib/Analysis/Convex/KreinMilman.lean | 63 | 90 | theorem IsCompact.extremePoints_nonempty (hscomp : IsCompact s) (hsnemp : s.Nonempty) :
(s.extremePoints ℝ).Nonempty := by |
let S : Set (Set E) := { t | t.Nonempty ∧ IsClosed t ∧ IsExtreme ℝ s t }
rsuffices ⟨t, ⟨⟨x, hxt⟩, htclos, hst⟩, hBmin⟩ : ∃ t ∈ S, ∀ u ∈ S, u ⊆ t → u = t
· refine ⟨x, IsExtreme.mem_extremePoints ?_⟩
rwa [← eq_singleton_iff_unique_mem.2 ⟨hxt, fun y hyB => ?_⟩]
by_contra hyx
obtain ⟨l, hl⟩ := geometric_... |
/-
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 | 100 | 104 | theorem iteratedFDerivWithin_apply_eq_iteratedDerivWithin_mul_prod {m : Fin n → 𝕜} :
(iteratedFDerivWithin 𝕜 n f s x : (Fin n → 𝕜) → F) m =
(∏ i, m i) • iteratedDerivWithin n f s x := by |
rw [iteratedDerivWithin_eq_iteratedFDerivWithin, ← ContinuousMultilinearMap.map_smul_univ]
simp
|
/-
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.Algebra.Spectrum
import Mathlib.FieldTheory.IsAlgClosed.Basic
#align_import field_theory.is_alg_closed.spectrum from "leanprover-community/mathl... | Mathlib/FieldTheory/IsAlgClosed/Spectrum.lean | 81 | 91 | theorem subset_polynomial_aeval (a : A) (p : 𝕜[X]) : (eval · p) '' σ a ⊆ σ (aeval a p) := by |
rintro _ ⟨k, hk, rfl⟩
let q := C (eval k p) - p
have hroot : IsRoot q k := by simp only [q, eval_C, eval_sub, sub_self, IsRoot.def]
rw [← mul_div_eq_iff_isRoot, ← neg_mul_neg, neg_sub] at hroot
have aeval_q_eq : ↑ₐ (eval k p) - aeval a p = aeval a q := by
simp only [q, aeval_C, AlgHom.map_sub, sub_left_i... |
/-
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.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 95 | 99 | theorem ascPochhammer_eval_comp {R : Type*} [CommSemiring R] (n : ℕ) (p : R[X]) [Algebra R S]
(x : S) : ((ascPochhammer S n).comp (p.map (algebraMap R S))).eval x =
(ascPochhammer S n).eval (p.eval₂ (algebraMap R S) x) := by |
rw [ascPochhammer_eval₂ (algebraMap R S), ← eval₂_comp', ← ascPochhammer_map (algebraMap R S),
← map_comp, eval_map]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Cone.Extension
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Analysis.NormedSpace.Extend
import Mathlib.... | Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean | 73 | 109 | theorem exists_extension_norm_eq (p : Subspace 𝕜 E) (f : p →L[𝕜] 𝕜) :
∃ g : E →L[𝕜] 𝕜, (∀ x : p, g x = f x) ∧ ‖g‖ = ‖f‖ := by |
letI : Module ℝ E := RestrictScalars.module ℝ 𝕜 E
letI : IsScalarTower ℝ 𝕜 E := RestrictScalars.isScalarTower _ _ _
letI : NormedSpace ℝ E := NormedSpace.restrictScalars _ 𝕜 _
-- Let `fr: p →L[ℝ] ℝ` be the real part of `f`.
let fr := reCLM.comp (f.restrictScalars ℝ)
-- Use the real version to get a norm... |
/-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Prod
#align_import data.nat.cast.prod from "leanprover-community/mathlib"@"ee0c179cd3c8a45aa5bffbf1b41d8dbede452865"
/-!
# The product ... | Mathlib/Data/Nat/Cast/Prod.lean | 29 | 29 | theorem fst_natCast (n : ℕ) : (n : α × β).fst = n := by | induction n <;> simp [*]
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.RowCol
#align_import linear_algebra.matrix.trace from "leanprov... | Mathlib/LinearAlgebra/Matrix/Trace.lean | 154 | 155 | theorem trace_one : trace (1 : Matrix n n R) = Fintype.card n := by |
simp_rw [trace, diag_one, Pi.one_def, Finset.sum_const, nsmul_one, Finset.card_univ]
|
/-
Copyright (c) 2023 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.List.Sym
/-! # Unordered tuples of elements of a multiset
Defines `Multiset.sym` and the specialized `Multiset.sym2` for computing multisets of all
un... | Mathlib/Data/Multiset/Sym.lean | 70 | 73 | theorem card_sym2 {m : Multiset α} :
Multiset.card m.sym2 = Nat.choose (Multiset.card m + 1) 2 := by |
refine m.inductionOn fun xs => ?_
simp [List.length_sym2]
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability... | Mathlib/Probability/Variance.lean | 75 | 89 | theorem evariance_eq_top [IsFiniteMeasure μ] (hXm : AEStronglyMeasurable X μ) (hX : ¬Memℒp X 2 μ) :
evariance X μ = ∞ := by |
by_contra h
rw [← Ne, ← lt_top_iff_ne_top] at h
have : Memℒp (fun ω => X ω - μ[X]) 2 μ := by
refine ⟨hXm.sub aestronglyMeasurable_const, ?_⟩
rw [snorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top]
simp only [coe_two, ENNReal.one_toReal, ENNReal.rpow_two, Ne]
exact ENNReal.rpow_lt_top_o... |
/-
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.Associated
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.Ring.Int
import Ma... | Mathlib/Data/Nat/Prime.lean | 99 | 109 | theorem prime_def_lt'' {p : ℕ} : Prime p ↔ 2 ≤ p ∧ ∀ m, m ∣ p → m = 1 ∨ m = p := by |
refine ⟨fun h => ⟨h.two_le, h.eq_one_or_self_of_dvd⟩, fun h => ?_⟩
-- Porting note: needed to make ℕ explicit
have h1 := (@one_lt_two ℕ ..).trans_le h.1
refine ⟨mt Nat.isUnit_iff.mp h1.ne', fun a b hab => ?_⟩
simp only [Nat.isUnit_iff]
apply Or.imp_right _ (h.2 a _)
· rintro rfl
rw [← mul_right_inj' ... |
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
#align_import control.traversable.lemmas from "leanprover-community/mathlib"@"3342d1b2178381196... | Mathlib/Control/Traversable/Lemmas.lean | 70 | 73 | theorem map_traverse (x : t α) : map f <$> traverse g x = traverse (map f ∘ g) x := by |
rw [map_eq_traverse_id f]
refine (comp_traverse (pure ∘ f) g x).symm.trans ?_
congr; apply Comp.applicative_comp_id
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Order.IsLUB
/-!
# Order topology on a densely ordered set
-/
open Set Filter TopologicalSpace Topology Func... | Mathlib/Topology/Order/DenselyOrdered.lean | 52 | 61 | theorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by |
apply Subset.antisymm
· exact closure_minimal Ioo_subset_Icc_self isClosed_Icc
· cases' hab.lt_or_lt with hab hab
· rw [← diff_subset_closure_iff, Icc_diff_Ioo_same hab.le]
have hab' : (Ioo a b).Nonempty := nonempty_Ioo.2 hab
simp only [insert_subset_iff, singleton_subset_iff]
exact ⟨(isGLB... |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Range
#align_import data.list.nat_antidiagonal from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601... | Mathlib/Data/List/NatAntidiagonal.lean | 38 | 47 | theorem mem_antidiagonal {n : ℕ} {x : ℕ × ℕ} : x ∈ antidiagonal n ↔ x.1 + x.2 = n := by |
rw [antidiagonal, mem_map]; constructor
· rintro ⟨i, hi, rfl⟩
rw [mem_range, Nat.lt_succ_iff] at hi
exact Nat.add_sub_cancel' hi
· rintro rfl
refine ⟨x.fst, ?_, ?_⟩
· rw [mem_range]
omega
· exact Prod.ext rfl (by simp only [Nat.add_sub_cancel_left])
|
/-
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
-/
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction... | Mathlib/Algebra/Polynomial/Derivative.lean | 103 | 104 | theorem derivative_C_mul_X_sq (a : R) : derivative (C a * X ^ 2) = C (a * 2) * X := by |
rw [derivative_C_mul_X_pow, Nat.cast_two, pow_one]
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Order.UpperLower.Basic
#align_import combinatorics.set_family.intersecting from "leanprover-community/mathlib"@"d90e4e186... | Mathlib/Combinatorics/SetFamily/Intersecting.lean | 81 | 92 | theorem intersecting_iff_pairwise_not_disjoint :
s.Intersecting ↔ (s.Pairwise fun a b => ¬Disjoint a b) ∧ s ≠ {⊥} := by |
refine ⟨fun h => ⟨fun a ha b hb _ => h ha hb, ?_⟩, fun h a ha b hb hab => ?_⟩
· rintro rfl
exact intersecting_singleton.1 h rfl
have := h.1.eq ha hb (Classical.not_not.2 hab)
rw [this, disjoint_self] at hab
rw [hab] at hb
exact
h.2
(eq_singleton_iff_unique_mem.2
⟨hb, fun c hc => not_n... |
/-
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.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.Set.Function
import Mathlib.Order.In... | Mathlib/Algebra/Order/Interval/Set/Monoid.lean | 123 | 124 | theorem image_const_add_Icc : (fun x => a + x) '' Icc b c = Icc (a + b) (a + c) := by |
simp only [add_comm a, image_add_const_Icc]
|
/-
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.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
#align_im... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 101 | 105 | theorem dist_eq_iff_eq_sq_sinh (hr : 0 ≤ r) :
dist z w = r ↔ dist (z : ℂ) w ^ 2 / (4 * z.im * w.im) = sinh (r / 2) ^ 2 := by |
rw [dist_eq_iff_eq_sinh, ← sq_eq_sq, div_pow, mul_pow, sq_sqrt, mul_assoc]
· norm_num
all_goals positivity
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yury G. Kudryashov
-/
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.MkIffOfInductiveProp
#align_import data.sum.basic from "leanprover-community/mathlib"@"... | Mathlib/Data/Sum/Basic.lean | 66 | 67 | theorem getRight_eq_getRight? (h₁ : x.isRight) (h₂ : x.getRight?.isSome) :
x.getRight h₁ = x.getRight?.get h₂ := by | simp [← getRight?_eq_some_iff]
|
/-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.pow fr... | Mathlib/Algebra/Regular/Pow.lean | 54 | 58 | theorem IsRightRegular.pow_iff {n : ℕ} (n0 : 0 < n) :
IsRightRegular (a ^ n) ↔ IsRightRegular a := by |
refine ⟨?_, IsRightRegular.pow n⟩
rw [← Nat.succ_pred_eq_of_pos n0, pow_succ']
exact IsRightRegular.of_mul
|
/-
Copyright (c) 2024 Emilie Burgun. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emilie Burgun
-/
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.Dynamics.PeriodicPts
import Mathlib.Data.Set.Pointwise.SMul
/-!
... | Mathlib/GroupTheory/GroupAction/FixedPoints.lean | 102 | 105 | theorem smul_fixedBy (g h: G) :
h • fixedBy α g = fixedBy α (h * g * h⁻¹) := by |
ext a
simp_rw [Set.mem_smul_set_iff_inv_smul_mem, mem_fixedBy, mul_smul, smul_eq_iff_eq_inv_smul h]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.GroupTheory.Submonoid.Inverses
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.Loc... | Mathlib/RingTheory/Localization/InvSubmonoid.lean | 108 | 112 | theorem span_invSubmonoid : Submodule.span R (invSubmonoid M S : Set S) = ⊤ := by |
rw [eq_top_iff]
rintro x -
rcases IsLocalization.surj'' M x with ⟨r, m, rfl⟩
exact Submodule.smul_mem _ _ (Submodule.subset_span (toInvSubmonoid M S m).prop)
|
/-
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.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.LinearAlgebra.Basis
impo... | Mathlib/RingTheory/Adjoin/Basic.lean | 79 | 80 | theorem adjoin_attach_biUnion [DecidableEq A] {α : Type*} {s : Finset α} (f : s → Finset A) :
adjoin R (s.attach.biUnion f : Set A) = ⨆ x, adjoin R (f x) := by | simp [adjoin_iUnion]
|
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Data.List.OfFn
import Mathlib.Data.Set.Pointwise.Basic
#align_import data.set.pointwise.list_of_fn from "lean... | Mathlib/Data/Set/Pointwise/ListOfFn.lean | 52 | 54 | theorem mem_pow {a : α} {n : ℕ} :
a ∈ s ^ n ↔ ∃ f : Fin n → s, (List.ofFn fun i ↦ (f i : α)).prod = a := by |
rw [← mem_prod_list_ofFn, List.ofFn_const, List.prod_replicate]
|
/-
Copyright (c) 2023 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Prod
import Mathlib.Tactic.Common
/-!
# Lemmas about the divisibility relation in product (sem... | Mathlib/Algebra/Divisibility/Prod.lean | 35 | 36 | theorem pi_dvd_iff {x y : ∀ i, G i} : x ∣ y ↔ ∀ i, x i ∣ y i := by |
simp_rw [dvd_def, Function.funext_iff, Classical.skolem]; rfl
|
/-
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 | 71 | 73 | theorem vars_def [DecidableEq σ] (p : MvPolynomial σ R) : p.vars = p.degrees.toFinset := by |
rw [vars]
convert rfl
|
/-
Copyright (c) 2023 Koundinya Vajjha. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Koundinya Vajjha, Thomas Browning
-/
import Mathlib.NumberTheory.Harmonic.Defs
import Mathlib.NumberTheory.Padics.PadicNumbers
/-!
The nth Harmonic number is not an integer. We for... | Mathlib/NumberTheory/Harmonic/Int.lean | 42 | 46 | theorem harmonic_not_int {n : ℕ} (h : 2 ≤ n) : ¬ (harmonic n).isInt := by |
apply padicNorm.not_int_of_not_padic_int 2
rw [padicNorm.eq_zpow_of_nonzero (harmonic_pos (ne_zero_of_lt h)).ne',
padicValRat_two_harmonic, neg_neg, zpow_natCast]
exact one_lt_pow one_lt_two (Nat.log_pos one_lt_two h).ne'
|
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.Order.Group.WithTop
import Mathlib.RingTheory.HahnSeries.Multiplication
import Mathlib.RingTheory.V... | Mathlib/RingTheory/HahnSeries/Summable.lean | 96 | 106 | theorem isPWO_iUnion_support_powers [LinearOrderedCancelAddCommMonoid Γ] [Ring R] [IsDomain R]
{x : HahnSeries Γ R} (hx : 0 < addVal Γ R x) : (⋃ n : ℕ, (x ^ n).support).IsPWO := by |
apply (x.isWF_support.isPWO.addSubmonoid_closure _).mono _
· exact fun g hg => WithTop.coe_le_coe.1 (le_trans (le_of_lt hx) (addVal_le_of_coeff_ne_zero hg))
refine Set.iUnion_subset fun n => ?_
induction' n with n ih <;> intro g hn
· simp only [Nat.zero_eq, pow_zero, support_one, Set.mem_singleton_iff] at hn... |
/-
Copyright (c) 2021 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Monoidal.Free.Basic
import Mathlib.CategoryTheory.Groupoid
import Mathlib.CategoryTheory.DiscreteCategory
#align_import category_theory.m... | Mathlib/CategoryTheory/Monoidal/Free/Coherence.lean | 91 | 95 | theorem inclusion_map {X Y : N C} (f : X ⟶ Y) :
inclusion.map f = eqToHom (congr_arg _ (Discrete.ext _ _ (Discrete.eq_of_hom f))) := by |
rcases f with ⟨⟨⟩⟩
cases Discrete.ext _ _ (by assumption)
apply inclusion.map_id
|
/-
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.Complex.RemovableSingularity
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Analysis.NormedSpace.FunctionSeries
#align_... | Mathlib/Analysis/Complex/LocallyUniformLimit.lean | 79 | 86 | theorem norm_cderiv_lt (hr : 0 < r) (hfM : ∀ w ∈ sphere z r, ‖f w‖ < M)
(hf : ContinuousOn f (sphere z r)) : ‖cderiv r f z‖ < M / r := by |
obtain ⟨L, hL1, hL2⟩ : ∃ L < M, ∀ w ∈ sphere z r, ‖f w‖ ≤ L := by
have e1 : (sphere z r).Nonempty := NormedSpace.sphere_nonempty.mpr hr.le
have e2 : ContinuousOn (fun w => ‖f w‖) (sphere z r) := continuous_norm.comp_continuousOn hf
obtain ⟨x, hx, hx'⟩ := (isCompact_sphere z r).exists_isMaxOn e1 e2
ex... |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Data.Multiset.Nodup
import Mathlib.Data.List.NatAntidiagonal
#align_import data.multiset.nat_antidiagonal from "leanprover-community/mathlib"@"9003f28... | Mathlib/Data/Multiset/NatAntidiagonal.lean | 42 | 43 | theorem card_antidiagonal (n : ℕ) : card (antidiagonal n) = n + 1 := by |
rw [antidiagonal, coe_card, List.Nat.length_antidiagonal]
|
/-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
#align_import category_theory.limits.shapes.strict_initia... | Mathlib/CategoryTheory/Limits/Shapes/StrictInitial.lean | 74 | 77 | theorem IsInitial.strict_hom_ext (hI : IsInitial I) {A : C} (f g : A ⟶ I) : f = g := by |
haveI := hI.isIso_to f
haveI := hI.isIso_to g
exact eq_of_inv_eq_inv (hI.hom_ext (inv f) (inv g))
|
/-
Copyright (c) 2022 Moritz Firsching. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Firsching, Fabian Kruse, Nikolas Kuhn
-/
import Mathlib.Analysis.PSeries
import Mathlib.Data.Real.Pi.Wallis
import Mathlib.Tactic.AdaptationNote
#align_import analysis.specia... | Mathlib/Analysis/SpecialFunctions/Stirling.lean | 56 | 57 | theorem stirlingSeq_zero : stirlingSeq 0 = 0 := by |
rw [stirlingSeq, cast_zero, mul_zero, Real.sqrt_zero, zero_mul, div_zero]
|
/-
Copyright (c) 2020 James Arthur. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: James Arthur, Chris Hughes, Shing Tak Lam
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.speci... | Mathlib/Analysis/SpecialFunctions/Arsinh.lean | 65 | 65 | theorem arsinh_zero : arsinh 0 = 0 := by | simp [arsinh]
|
/-
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.SmoothSeries
import Mathlib.Analysis.Calculus.BumpFunction.InnerProduct
import Mathlib.Analysis.Convolution
import Mathlib.Anal... | Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean | 43 | 73 | theorem exists_smooth_tsupport_subset {s : Set E} {x : E} (hs : s ∈ 𝓝 x) :
∃ f : E → ℝ,
tsupport f ⊆ s ∧ HasCompactSupport f ∧ ContDiff ℝ ⊤ f ∧ range f ⊆ Icc 0 1 ∧ f x = 1 := by |
obtain ⟨d : ℝ, d_pos : 0 < d, hd : Euclidean.closedBall x d ⊆ s⟩ :=
Euclidean.nhds_basis_closedBall.mem_iff.1 hs
let c : ContDiffBump (toEuclidean x) :=
{ rIn := d / 2
rOut := d
rIn_pos := half_pos d_pos
rIn_lt_rOut := half_lt_self d_pos }
let f : E → ℝ := c ∘ toEuclidean
have f_supp ... |
/-
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 | 82 | 84 | theorem IsDiag.neg [AddGroup α] {A : Matrix n n α} (ha : A.IsDiag) : (-A).IsDiag := by |
intro i j h
simp [ha h]
|
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.LocallyConvex.Basic
#align_import analysis.locally_convex.balanced_core_hull from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547... | Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean | 85 | 90 | theorem smul_balancedCore_subset (s : Set E) {a : 𝕜} (ha : ‖a‖ ≤ 1) :
a • balancedCore 𝕜 s ⊆ balancedCore 𝕜 s := by |
rintro x ⟨y, hy, rfl⟩
rw [mem_balancedCore_iff] at hy
rcases hy with ⟨t, ht1, ht2, hy⟩
exact ⟨t, ⟨ht1, ht2⟩, ht1 a ha (smul_mem_smul_set hy)⟩
|
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Order.Zorn
import Mathlib.Order.Atoms
#align_import order.zorn_atoms from "leanprover-community/mathlib"@"9aba7801eeecebb61f58a5763c2b6dd1b47dc6ef"
... | Mathlib/Order/ZornAtoms.lean | 24 | 36 | theorem IsCoatomic.of_isChain_bounded {α : Type*} [PartialOrder α] [OrderTop α]
(h :
∀ c : Set α,
IsChain (· ≤ ·) c → c.Nonempty → ⊤ ∉ c → ∃ x ≠ ⊤, x ∈ upperBounds c) :
IsCoatomic α := by |
refine ⟨fun x => le_top.eq_or_lt.imp_right fun hx => ?_⟩
have : ∃ y ∈ Ico x ⊤, x ≤ y ∧ ∀ z ∈ Ico x ⊤, y ≤ z → z = y := by
refine zorn_nonempty_partialOrder₀ (Ico x ⊤) (fun c hxc hc y hy => ?_) x (left_mem_Ico.2 hx)
rcases h c hc ⟨y, hy⟩ fun h => (hxc h).2.ne rfl with ⟨z, hz, hcz⟩
exact ⟨z, ⟨le_trans (h... |
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.LinearAlgebra.Determinant
/-!
# Gershgorin's circle theorem
Th... | Mathlib/LinearAlgebra/Matrix/Gershgorin.lean | 59 | 66 | theorem det_ne_zero_of_sum_row_lt_diag (h : ∀ k, ∑ j ∈ Finset.univ.erase k, ‖A k j‖ < ‖A k k‖) :
A.det ≠ 0 := by |
contrapose! h
suffices ∃ k, 0 ∈ Metric.closedBall (A k k) (∑ j ∈ Finset.univ.erase k, ‖A k j‖) by
exact this.imp (fun a h ↦ by rwa [mem_closedBall_iff_norm', sub_zero] at h)
refine eigenvalue_mem_ball ?_
rw [Module.End.HasEigenvalue, Module.End.eigenspace_zero, ne_comm]
exact ne_of_lt (LinearMap.bot_lt_... |
/-
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 | 62 | 65 | theorem hasStrictDerivAt_of_bilinear (hu : HasStrictDerivAt u u' x) (hv : HasStrictDerivAt v v' x) :
HasStrictDerivAt (fun x ↦ B (u x) (v x)) (B (u x) v' + B u' (v x)) x := by |
simpa using
(B.hasStrictFDerivAt_of_bilinear hu.hasStrictFDerivAt hv.hasStrictFDerivAt).hasStrictDerivAt
|
/-
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.CategoryTheory.Subobject.Limits
#align_import algebra.homology.image_to_kernel from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a14... | Mathlib/Algebra/Homology/ImageToKernel.lean | 147 | 152 | theorem imageToKernel_comp_hom_inv_comp [HasEqualizers V] [HasImages V] {Z : V} {i : B ≅ Z} (w) :
imageToKernel (f ≫ i.hom) (i.inv ≫ g) w =
(imageSubobjectCompIso _ _).hom ≫
imageToKernel f g (by simpa using w) ≫ (kernelSubobjectIsoComp i.inv g).inv := by |
ext
simp
|
/-
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.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
#align_im... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 71 | 73 | theorem cosh_dist (z w : ℍ) : cosh (dist z w) = 1 + dist (z : ℂ) w ^ 2 / (2 * z.im * w.im) := by |
rw [dist_eq, cosh_two_mul, cosh_sq', add_assoc, ← two_mul, sinh_arsinh, div_pow, mul_pow,
sq_sqrt, sq (2 : ℝ), mul_assoc, ← mul_div_assoc, mul_assoc, mul_div_mul_left] <;> positivity
|
/-
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.Data.Nat.Prime
import Mathlib.Tactic.NormNum.Basic
#align_import data.nat.prime_norm_num from "leanprover-com... | Mathlib/Tactic/NormNum/Prime.lean | 58 | 65 | theorem minFacHelper_0 (n : ℕ)
(h1 : Nat.ble (nat_lit 2) n = true) (h2 : nat_lit 1 = n % (nat_lit 2)) :
MinFacHelper n (nat_lit 3) := by |
refine ⟨by norm_num, by norm_num, ?_⟩
refine (le_minFac'.mpr λ p hp hpn ↦ ?_).resolve_left (Nat.ne_of_gt (Nat.le_of_ble_eq_true h1))
rcases hp.eq_or_lt with rfl|h
· simp [(Nat.dvd_iff_mod_eq_zero ..).1 hpn] at h2
· exact h
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Devon Tuma
-/
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.JacobsonIdeal
#align_import ring_theory.jacobson from "leanp... | Mathlib/RingTheory/Jacobson.lean | 70 | 78 | theorem isJacobson_iff_prime_eq : IsJacobson R ↔ ∀ P : Ideal R, IsPrime P → P.jacobson = P := by |
refine isJacobson_iff.trans ⟨fun h I hI => h I hI.isRadical, ?_⟩
refine fun h I hI ↦ le_antisymm (fun x hx ↦ ?_) (fun x hx ↦ mem_sInf.mpr fun _ hJ ↦ hJ.left hx)
rw [← hI.radical, radical_eq_sInf I, mem_sInf]
intro P hP
rw [Set.mem_setOf_eq] at hP
erw [mem_sInf] at hx
erw [← h P hP.right, mem_sInf]
exac... |
/-
Copyright (c) 2021 Shing Tak Lam. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam
-/
import Mathlib.CategoryTheory.Category.Grpd
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Homotopy.Path
i... | Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean | 131 | 135 | theorem continuous_transReflReparamAux : Continuous transReflReparamAux := by |
refine continuous_if_le ?_ ?_ (Continuous.continuousOn ?_) (Continuous.continuousOn ?_) ?_ <;>
[continuity; continuity; continuity; continuity; skip]
intro x hx
simp [hx]
|
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Scott Carnahan
-/
import Mathlib.RingTheory.HahnSeries.Addition
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Data.Finset.MulAntidiagonal
#align_impor... | Mathlib/RingTheory/HahnSeries/Multiplication.lean | 65 | 68 | theorem order_one [MulZeroOneClass R] : order (1 : HahnSeries Γ R) = 0 := by |
cases subsingleton_or_nontrivial R
· rw [Subsingleton.elim (1 : HahnSeries Γ R) 0, order_zero]
· exact order_single one_ne_zero
|
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Alex Keizer
-/
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.Bits
import Mathlib.Algebra.Ring.Nat
import Mathlib.Order.Basic
import Mathlib.Tactic.AdaptationNote
... | Mathlib/Data/Nat/Bitwise.lean | 75 | 81 | theorem binaryRec_of_ne_zero {C : Nat → Sort*} (z : C 0) (f : ∀ b n, C n → C (bit b n)) {n}
(h : n ≠ 0) :
binaryRec z f n = bit_decomp n ▸ f (bodd n) (div2 n) (binaryRec z f (div2 n)) := by |
rw [Eq.rec_eq_cast]
rw [binaryRec]
dsimp only
rw [dif_neg h, eq_mpr_eq_cast]
|
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.FunctorGamma
import Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject
import Mathlib.CategoryTheory.Idempotents.HomologicalComple... | Mathlib/AlgebraicTopology/DoldKan/GammaCompN.lean | 76 | 82 | theorem N₁Γ₀_app (K : ChainComplex C ℕ) :
N₁Γ₀.app K = (Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.symm ≪≫
(toKaroubi _).mapIso (Γ₀NondegComplexIso K) := by |
ext1
dsimp [N₁Γ₀]
erw [id_comp, comp_id, comp_id]
rfl
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.PrimitiveElement
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Matrix.Char... | Mathlib/RingTheory/Norm.lean | 91 | 97 | theorem norm_algebraMap_of_basis [Fintype ι] (b : Basis ι R S) (x : R) :
norm R (algebraMap R S x) = x ^ Fintype.card ι := by |
haveI := Classical.decEq ι
rw [norm_apply, ← det_toMatrix b, lmul_algebraMap]
convert @det_diagonal _ _ _ _ _ fun _ : ι => x
· ext (i j); rw [toMatrix_lsmul]
· rw [Finset.prod_const, Finset.card_univ]
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johan Commelin
-/
import Mathlib.RingTheory.IntegralClosure
#align_import field_theory.minpoly.basic from "leanprover-community/mathlib"@"df0098f0db291900600f32070f6abb3e1... | Mathlib/FieldTheory/Minpoly/Basic.lean | 96 | 97 | theorem aeval_algHom (f : B →ₐ[A] B') (x : B) : (Polynomial.aeval (f x)) (minpoly A x) = 0 := by |
rw [Polynomial.aeval_algHom, AlgHom.coe_comp, comp_apply, aeval, map_zero]
|
/-
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.Topology.Algebra.Module.WeakDual
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.Analysis.LocallyConvex.WithSeminorms
#align_import analysis.local... | Mathlib/Analysis/LocallyConvex/WeakDual.lean | 68 | 70 | theorem toSeminorm_ball_zero {f : E →ₗ[𝕜] 𝕜} {r : ℝ} :
Seminorm.ball f.toSeminorm 0 r = { x : E | ‖f x‖ < r } := by |
simp only [Seminorm.ball_zero_eq, toSeminorm_apply]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Basic
import Mathlib.RingT... | Mathlib/RingTheory/Localization/LocalizationLocalization.lean | 73 | 89 | theorem localization_localization_surj [IsLocalization N T] (x : T) :
∃ y : R × localizationLocalizationSubmodule M N,
x * algebraMap R T y.2 = algebraMap R T y.1 := by |
rcases IsLocalization.surj N x with ⟨⟨y, s⟩, eq₁⟩
-- x = y / s
rcases IsLocalization.surj M y with ⟨⟨z, t⟩, eq₂⟩
-- y = z / t
rcases IsLocalization.surj M (s : S) with ⟨⟨z', t'⟩, eq₃⟩
-- s = z' / t'
dsimp only at eq₁ eq₂ eq₃
refine ⟨⟨z * t', z' * t, ?_⟩, ?_⟩ -- x = y / s = (z * t') / (z' * t)
· rw [m... |
/-
Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheo... | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 104 | 105 | theorem mem_eigenspace_iff {f : End R M} {μ : R} {x : M} : x ∈ eigenspace f μ ↔ f x = μ • x := by |
rw [eigenspace, LinearMap.mem_ker, LinearMap.sub_apply, algebraMap_end_apply, sub_eq_zero]
|
/-
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.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.alg... | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | 53 | 57 | theorem to_affineMap_injective {f g : P →ᴬ[R] Q} (h : (f : P →ᵃ[R] Q) = (g : P →ᵃ[R] Q)) :
f = g := by |
cases f
cases g
congr
|
/-
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.RingTheory.Polynomial.Pochhammer
#align_import data.nat.factorial.cast from "leanprover-community/mathlib"@"d50b12ae8e2bd910d08a94823976adae9825718b"
/-!... | Mathlib/Data/Nat/Factorial/Cast.lean | 51 | 52 | theorem cast_factorial : (a ! : S) = (ascPochhammer S a).eval 1 := by |
rw [← one_ascFactorial, cast_ascFactorial, cast_one]
|
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.NormedSpace.BallAction
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.... | Mathlib/Geometry/Manifold/Instances/Sphere.lean | 98 | 104 | theorem contDiffOn_stereoToFun :
ContDiffOn ℝ ⊤ (stereoToFun v) {x : E | innerSL _ v x ≠ (1 : ℝ)} := by |
refine ContDiffOn.smul ?_ (orthogonalProjection (ℝ ∙ v)ᗮ).contDiff.contDiffOn
refine contDiff_const.contDiffOn.div ?_ ?_
· exact (contDiff_const.sub (innerSL ℝ v).contDiff).contDiffOn
· intro x h h'
exact h (sub_eq_zero.mp h').symm
|
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Data.ENNReal.Basic
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.MetricSpace.Thickening
#align_import topology.metric_space.t... | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 58 | 66 | theorem continuous_thickenedIndicatorAux {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) :
Continuous (thickenedIndicatorAux δ E) := by |
unfold thickenedIndicatorAux
let f := fun x : α => (⟨1, infEdist x E / ENNReal.ofReal δ⟩ : ℝ≥0 × ℝ≥0∞)
let sub := fun p : ℝ≥0 × ℝ≥0∞ => (p.1 : ℝ≥0∞) - p.2
rw [show (fun x : α => (1 : ℝ≥0∞) - infEdist x E / ENNReal.ofReal δ) = sub ∘ f by rfl]
apply (@ENNReal.continuous_nnreal_sub 1).comp
apply (ENNReal.cont... |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
/-!
# The fold ope... | Mathlib/Data/Multiset/Fold.lean | 82 | 87 | theorem fold_bind {ι : Type*} (s : Multiset ι) (t : ι → Multiset α) (b : ι → α) (b₀ : α) :
(s.bind t).fold op ((s.map b).fold op b₀) =
(s.map fun i => (t i).fold op (b i)).fold op b₀ := by |
induction' s using Multiset.induction_on with a ha ih
· rw [zero_bind, map_zero, map_zero, fold_zero]
· rw [cons_bind, map_cons, map_cons, fold_cons_left, fold_cons_left, fold_add, ih]
|
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.NormedSpace.Banach
import Mathlib.LinearAlgebra.SesquilinearF... | Mathlib/Analysis/InnerProductSpace/Symmetric.lean | 71 | 72 | theorem IsSymmetric.conj_inner_sym {T : E →ₗ[𝕜] E} (hT : IsSymmetric T) (x y : E) :
conj ⟪T x, y⟫ = ⟪T y, x⟫ := by | rw [hT x y, inner_conj_symm]
|
/-
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.Data.List.Basic
#align_import data.list.forall2 from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
/-!
# ... | Mathlib/Data/List/Forall2.lean | 61 | 69 | theorem forall₂_eq_eq_eq : Forall₂ ((· = ·) : α → α → Prop) = Eq := by |
funext a b; apply propext
constructor
· intro h
induction h
· rfl
simp only [*]
· rintro rfl
exact forall₂_refl _
|
/-
Copyright (c) 2023 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
/-!
# Local graph operations
This file defines some single-graph operations th... | Mathlib/Combinatorics/SimpleGraph/Operations.lean | 115 | 124 | theorem card_edgeFinset_replaceVertex_of_not_adj (hn : ¬G.Adj s t) :
(G.replaceVertex s t).edgeFinset.card = G.edgeFinset.card + G.degree s - G.degree t := by |
have inc : G.incidenceFinset t ⊆ G.edgeFinset := by simp [incidenceFinset, incidenceSet_subset]
rw [G.edgeFinset_replaceVertex_of_not_adj hn,
card_union_of_disjoint G.disjoint_sdiff_neighborFinset_image, card_sdiff inc,
← Nat.sub_add_comm <| card_le_card inc, card_incidenceFinset_eq_degree]
congr 2
rw ... |
/-
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.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
#align_im... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 91 | 93 | theorem dist_le_iff_le_sinh :
dist z w ≤ r ↔ dist (z : ℂ) w / (2 * √(z.im * w.im)) ≤ sinh (r / 2) := by |
rw [← div_le_div_right (zero_lt_two' ℝ), ← sinh_le_sinh, sinh_half_dist]
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import geometry.euclidean.basic from "lean... | Mathlib/Geometry/Euclidean/PerpBisector.lean | 53 | 57 | theorem mem_perpBisector_iff_inner_pointReflection_vsub_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪Equiv.pointReflection c p₁ -ᵥ p₂, p₂ -ᵥ p₁⟫ = 0 := by |
rw [mem_perpBisector_iff_inner_eq_zero, Equiv.pointReflection_apply,
vsub_midpoint, invOf_eq_inv, ← smul_add, real_inner_smul_left, vadd_vsub_assoc]
simp
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Mario Carneiro, Sean Leather
-/
import Mathlib.Data.Finset.Card
#align_import data.finset.option from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0... | Mathlib/Data/Finset/Option.lean | 142 | 144 | theorem eraseNone_empty : eraseNone (∅ : Finset (Option α)) = ∅ := by |
ext
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 | 119 | 122 | theorem rootsOfUnity_le_of_dvd (h : k ∣ l) : rootsOfUnity k M ≤ rootsOfUnity l M := by |
obtain ⟨d, rfl⟩ := h
intro ζ h
simp_all only [mem_rootsOfUnity, PNat.mul_coe, pow_mul, one_pow]
|
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.... | Mathlib/Algebra/GeomSum.lean | 64 | 64 | theorem geom_sum_two {x : α} : ∑ i ∈ range 2, x ^ i = x + 1 := by | simp [geom_sum_succ']
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.RingTheory.DiscreteValuationRing.Basic
import Mathlib.RingTheory.MvPowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.Basic
import M... | Mathlib/RingTheory/PowerSeries/Inverse.lean | 140 | 147 | theorem coeff_inv (n) (φ : k⟦X⟧) :
coeff k n φ⁻¹ =
if n = 0 then (constantCoeff k φ)⁻¹
else
-(constantCoeff k φ)⁻¹ *
∑ x ∈ antidiagonal n,
if x.2 < n then coeff k x.1 φ * coeff k x.2 φ⁻¹ else 0 := by |
rw [inv_eq_inv_aux, coeff_inv_aux n (constantCoeff k φ)⁻¹ φ]
|
/-
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.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c066... | Mathlib/Algebra/GCDMonoid/Finset.lean | 114 | 116 | theorem lcm_image [DecidableEq β] {g : γ → β} (s : Finset γ) :
(s.image g).lcm f = s.lcm (f ∘ g) := by |
classical induction' s using Finset.induction with c s _ ih <;> simp [*]
|
/-
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 | 128 | 136 | theorem ext_iff (v w : VectorMeasure α M) : v = w ↔ ∀ i : Set α, MeasurableSet i → v i = w i := by |
constructor
· rintro rfl _ _
rfl
· rw [ext_iff']
intro h i
by_cases hi : MeasurableSet i
· exact h i hi
· simp_rw [not_measurable _ hi]
|
/-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Splits
#align_import algebra.cubic_discriminant from "leanprover-community/mathlib"@"930133160e24036d5242039fe4... | Mathlib/Algebra/CubicDiscriminant.lean | 121 | 121 | theorem a_of_eq (h : P.toPoly = Q.toPoly) : P.a = Q.a := by | rw [← coeff_eq_a, h, coeff_eq_a]
|
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Ma... | Mathlib/Data/Nat/Factorization/Basic.lean | 133 | 135 | theorem factorization_eq_zero_iff (n p : ℕ) :
n.factorization p = 0 ↔ ¬p.Prime ∨ ¬p ∣ n ∨ n = 0 := by |
simp_rw [← not_mem_support_iff, support_factorization, mem_primeFactors, not_and_or, not_ne_iff]
|
/-
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.MeasureTheory.Covering.DensityTheorem
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.covering.one_dim from "l... | Mathlib/MeasureTheory/Covering/OneDim.lean | 33 | 41 | theorem tendsto_Icc_vitaliFamily_right (x : ℝ) :
Tendsto (fun y => Icc x y) (𝓝[>] x) ((vitaliFamily (volume : Measure ℝ) 1).filterAt x) := by |
refine (VitaliFamily.tendsto_filterAt_iff _).2 ⟨?_, ?_⟩
· filter_upwards [self_mem_nhdsWithin] with y hy using Icc_mem_vitaliFamily_at_right hy
· intro ε εpos
have : x ∈ Ico x (x + ε) := ⟨le_refl _, by linarith⟩
filter_upwards [Icc_mem_nhdsWithin_Ioi this] with y hy
rw [closedBall_eq_Icc]
exact I... |
/-
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 | 75 | 76 | theorem preimage_coe_Iio_top : (some : α → WithTop α) ⁻¹' Iio ⊤ = univ := by |
rw [← range_coe, preimage_range]
|
/-
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 | 123 | 125 | theorem comap_monotone (f : V ↪ W) : Monotone (SimpleGraph.comap f) := by |
intro G G' h _ _ ha
exact h ha
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.MeasureTheory.Measure.Dirac
/-!
# Counting measure
In this file we define the counting measure `MeasurTheory.Measure.count`
as `MeasureTheory.Measure... | Mathlib/MeasureTheory/Measure/Count.lean | 73 | 80 | theorem count_apply_infinite (hs : s.Infinite) : count s = ∞ := by |
refine top_unique (le_of_tendsto' ENNReal.tendsto_nat_nhds_top fun n => ?_)
rcases hs.exists_subset_card_eq n with ⟨t, ht, rfl⟩
calc
(t.card : ℝ≥0∞) = ∑ i ∈ t, 1 := by simp
_ = ∑' i : (t : Set α), 1 := (t.tsum_subtype 1).symm
_ ≤ count (t : Set α) := le_count_apply
_ ≤ count s := measure_mono ht
|
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Topology.Algebra.Module.Multilinear.Basic
/-!
# Images of (von Neumann) bounded sets under continuous ... | Mathlib/Topology/Algebra/Module/Multilinear/Bounded.lean | 90 | 96 | theorem image_multilinear [ContinuousSMul 𝕜 F] {s : Set (∀ i, E i)} (hs : IsVonNBounded 𝕜 s)
(f : ContinuousMultilinearMap 𝕜 E F) : IsVonNBounded 𝕜 (f '' s) := by |
cases isEmpty_or_nonempty ι with
| inl h =>
exact (isBounded_iff_isVonNBounded _).1 <|
@Set.Finite.isBounded _ (vonNBornology 𝕜 F) _ (s.toFinite.image _)
| inr h => exact hs.image_multilinear' f
|
/-
Copyright (c) 2021 Gabriel Moise. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Moise, Yaël Dillies, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Matrix.Basic
#align_import combinatorics.... | Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean | 121 | 123 | theorem sum_incMatrix_apply [Fintype (neighborSet G a)] :
∑ e, G.incMatrix R a e = G.degree a := by |
classical simp [incMatrix_apply', sum_boole, Set.filter_mem_univ_eq_toFinset]
|
/-
Copyright (c) 2024 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Filter.CountableInter
import Mathlib.Order.Filter.CardinalInter
import Mathlib.SetTheory.Cardinal.Ordinal
import... | Mathlib/Order/Filter/Cocardinal.lean | 98 | 100 | theorem eventually_cocardinal_ne (x : α) : ∀ᶠ a in cocardinal α hreg, a ≠ x := by |
simp [Set.finite_singleton x]
exact hreg.nat_lt 1
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen, Antoine Labelle
-/
import Mathlib.LinearAlgebra.Contraction
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
#align_import ... | Mathlib/LinearAlgebra/Trace.lean | 55 | 69 | theorem traceAux_eq : traceAux R b = traceAux R c :=
LinearMap.ext fun f =>
calc
Matrix.trace (LinearMap.toMatrix b b f) =
Matrix.trace (LinearMap.toMatrix b b ((LinearMap.id.comp f).comp LinearMap.id)) := by |
rw [LinearMap.id_comp, LinearMap.comp_id]
_ = Matrix.trace (LinearMap.toMatrix c b LinearMap.id * LinearMap.toMatrix c c f *
LinearMap.toMatrix b c LinearMap.id) := by
rw [LinearMap.toMatrix_comp _ c, LinearMap.toMatrix_comp _ c]
_ = Matrix.trace (LinearMap.toMatrix c c f * Linear... |
/-
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.SetTheory.Ordinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Exponential
#align_import set_theory.ordinal.cantor_normal_form from "leanprover-communi... | Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | 121 | 129 | theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :
x ∈ CNF b o → x.1 ≤ log b o := by |
refine CNFRec b ?_ (fun o ho H ↦ ?_) o
· rw [CNF_zero]
intro contra; contradiction
· rw [CNF_ne_zero ho, mem_cons]
rintro (rfl | h)
· exact le_rfl
· exact (H h).trans (log_mono_right _ (mod_opow_log_lt_self b ho).le)
|
/-
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.MeasureTheory.Measure.MeasureSpace
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
#align_import measure_theory.measure.open_pos from "l... | Mathlib/MeasureTheory/Measure/OpenPos.lean | 88 | 90 | theorem _root_.IsOpen.ae_eq_empty_iff_eq (hU : IsOpen U) :
U =ᵐ[μ] (∅ : Set X) ↔ U = ∅ := by |
rw [ae_eq_empty, hU.measure_zero_iff_eq_empty]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Interval
import Mathlib.Order.Interval.Set.Pi
import Mathlib.Tactic.TFAE
import Mathlib.Tactic.NormNum
im... | Mathlib/Topology/Order/Basic.lean | 91 | 93 | theorem isOpen_iff_generate_intervals {s : Set α} :
IsOpen s ↔ GenerateOpen { s | ∃ a, s = Ioi a ∨ s = Iio a } s := by |
rw [t.topology_eq_generate_intervals]; rfl
|
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