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, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanp... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 42 | 50 | theorem ediam_smul₀ (c : 𝕜) (s : Set E) : EMetric.diam (c • s) = ‖c‖₊ • EMetric.diam s := by |
refine le_antisymm (ediam_smul_le c s) ?_
obtain rfl | hc := eq_or_ne c 0
· obtain rfl | hs := s.eq_empty_or_nonempty
· simp
simp [zero_smul_set hs, ← Set.singleton_zero]
· have := (lipschitzWith_smul c⁻¹).ediam_image_le (c • s)
rwa [← smul_eq_mul, ← ENNReal.smul_def, Set.image_smul, inv_smul_smul₀... |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Interval.Multiset
#align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
/-!
# Finite inter... | Mathlib/Order/Interval/Finset/Nat.lean | 120 | 121 | theorem card_fintypeIco : Fintype.card (Set.Ico a b) = b - a := by |
rw [Fintype.card_ofFinset, card_Ico]
|
/-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler, Yaël Dillies
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
#align_import analysis.special_functions.trigonometric.bounds from "leanprover-commu... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean | 39 | 49 | theorem sin_lt (h : 0 < x) : sin x < x := by |
cases' lt_or_le 1 x with h' h'
· exact (sin_le_one x).trans_lt h'
have hx : |x| = x := abs_of_nonneg h.le
have := le_of_abs_le (sin_bound <| show |x| ≤ 1 by rwa [hx])
rw [sub_le_iff_le_add', hx] at this
apply this.trans_lt
rw [sub_add, sub_lt_self_iff, sub_pos, div_eq_mul_inv (x ^ 3)]
refine mul_lt_mul... |
/-
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 | 87 | 90 | theorem disjoint_refl_iff : Disjoint f f ↔ f = 1 := by |
refine ⟨fun h => ?_, fun h => h.symm ▸ disjoint_one_left 1⟩
ext x
cases' h x with hx hx <;> simp [hx]
|
/-
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 | 86 | 89 | theorem Ico_mul_Ioc_subset' (a b c d : α) : Ico a b * Ioc c d ⊆ Ioo (a * c) (b * d) := by |
haveI := covariantClass_le_of_lt
rintro x ⟨y, ⟨hya, hyb⟩, z, ⟨hzc, hzd⟩, rfl⟩
exact ⟨mul_lt_mul_of_le_of_lt hya hzc, mul_lt_mul_of_lt_of_le hyb hzd⟩
|
/-
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.Data.Set.Function
import Mathlib.Order.Interval.Set.OrdConnected
#align_import data.set.intervals.proj_Icc from "leanprover-co... | Mathlib/Order/Interval/Set/ProjIcc.lean | 102 | 102 | theorem projIic_eq_self : projIic b x = ⟨b, le_rfl⟩ ↔ b ≤ x := by | simp [projIic, Subtype.ext_iff]
|
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.Category.GroupCat.EquivalenceGroupAddGroup
import Mathlib.GroupTheory.QuotientGroup
#align_import algebra.category.Group.epi_mono from "leanprover... | Mathlib/Algebra/Category/GroupCat/EpiMono.lean | 145 | 152 | theorem fromCoset_eq_of_mem_range {b : B} (hb : b ∈ f.range) :
fromCoset ⟨b • ↑f.range, b, rfl⟩ = fromCoset ⟨f.range, 1, one_leftCoset _⟩ := by |
congr
let b : B.α := b
change b • (f.range : Set B) = f.range
nth_rw 2 [show (f.range : Set B.α) = (1 : B) • f.range from (one_leftCoset _).symm]
rw [leftCoset_eq_iff, mul_one]
exact Subgroup.inv_mem _ hb
|
/-
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, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.... | Mathlib/Order/Interval/Set/OrderIso.lean | 103 | 104 | theorem image_Icc (e : α ≃o β) (a b : α) : e '' Icc a b = Icc (e a) (e b) := by |
rw [e.image_eq_preimage, e.symm.preimage_Icc, e.symm_symm]
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Topology.Order.Basic
/-!
# Set neighborhoods of intervals
In this file we prove basic theorems about `𝓝ˢ s`,
where `s` is one of the intervals
`Se... | Mathlib/Topology/Order/NhdsSet.lean | 44 | 45 | theorem nhdsSet_Ioc (h : a < b) : 𝓝ˢ (Ioc a b) = 𝓝 b ⊔ 𝓟 (Ioo a b) := by |
rw [← Ioo_insert_right h, nhdsSet_insert, nhdsSet_Ioo]
|
/-
Copyright (c) 2022 Antoine Labelle, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle, Rémi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
#align_import combinatorics.quiver.cast from "... | Mathlib/Combinatorics/Quiver/Cast.lean | 50 | 54 | theorem Hom.cast_cast {u v u' v' u'' v'' : U} (e : u ⟶ v) (hu : u = u') (hv : v = v')
(hu' : u' = u'') (hv' : v' = v'') :
(e.cast hu hv).cast hu' hv' = e.cast (hu.trans hu') (hv.trans hv') := by |
subst_vars
rfl
|
/-
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 | 54 | 68 | theorem cauchySeq_finset_of_norm_bounded_eventually {f : ι → E} {g : ι → ℝ} (hg : Summable g)
(h : ∀ᶠ i in cofinite, ‖f i‖ ≤ g i) : CauchySeq fun s => ∑ i ∈ s, f i := by |
refine cauchySeq_finset_iff_vanishing_norm.2 fun ε hε => ?_
rcases summable_iff_vanishing_norm.1 hg ε hε with ⟨s, hs⟩
classical
refine ⟨s ∪ h.toFinset, fun t ht => ?_⟩
have : ∀ i ∈ t, ‖f i‖ ≤ g i := by
intro i hi
simp only [disjoint_left, mem_union, not_or, h.mem_toFinset, Set.mem_compl_iff,
Cl... |
/-
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.Topology.Bornology.Basic
#align_import topology.bornology.constructions from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621... | Mathlib/Topology/Bornology/Constructions.lean | 94 | 96 | theorem isBounded_prod_self : IsBounded (s ×ˢ s) ↔ IsBounded s := by |
rcases s.eq_empty_or_nonempty with (rfl | hs); · simp
exact (isBounded_prod_of_nonempty (hs.prod hs)).trans and_self_iff
|
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Matrix.Basic
/-!
# Row and column matrices
This file provides results about row and column matrices
## Main definitions
* `Matrix.row r... | Mathlib/Data/Matrix/RowCol.lean | 154 | 158 | theorem vecMulVec_eq [Mul α] [AddCommMonoid α] (w : m → α) (v : n → α) :
vecMulVec w v = col w * row v := by |
ext
simp only [vecMulVec, mul_apply, Fintype.univ_punit, Finset.sum_singleton]
rfl
|
/-
Copyright (c) 2021 Jakob Scholbach. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob Scholbach
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.FieldTheory.Separable
#align_import field_theory.separable_degree from "lea... | Mathlib/RingTheory/Polynomial/SeparableDegree.lean | 121 | 131 | theorem contraction_degree_eq_or_insep [hq : NeZero q] [CharP F q] (g g' : F[X]) (m m' : ℕ)
(h_expand : expand F (q ^ m) g = expand F (q ^ m') g') (hg : g.Separable) (hg' : g'.Separable) :
g.natDegree = g'.natDegree := by |
wlog hm : m ≤ m'
· exact (this q hf g' g m' m h_expand.symm hg' hg (le_of_not_le hm)).symm
obtain ⟨s, rfl⟩ := exists_add_of_le hm
rw [pow_add, expand_mul, expand_inj (pow_pos (NeZero.pos q) m)] at h_expand
subst h_expand
rcases isUnit_or_eq_zero_of_separable_expand q s (NeZero.pos q) hg with (h | rfl)
· ... |
/-
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.Module.LinearMap.Basic
import Mathlib.RingTheory.HahnSeries.Basic
#align_import ring_theory.hahn_series from "leanprover-community/mathlib"@"a... | Mathlib/RingTheory/HahnSeries/Addition.lean | 171 | 172 | theorem sub_coeff {x y : HahnSeries Γ R} {a : Γ} : (x - y).coeff a = x.coeff a - y.coeff a := by |
simp
|
/-
Copyright (c) 2019 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
/-!
# Operator norm as an `NNNorm`
Operator norm as an `NNNorm`, i.e. takin... | Mathlib/Analysis/NormedSpace/OperatorNorm/NNNorm.lean | 100 | 101 | theorem isLeast_opNNNorm : IsLeast {C : ℝ≥0 | ∀ x, ‖f x‖₊ ≤ C * ‖x‖₊} ‖f‖₊ := by |
simpa only [← opNNNorm_le_iff] using isLeast_Ici
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.LinearAlgebra.Tenso... | Mathlib/Algebra/Algebra/Bilinear.lean | 140 | 142 | theorem mulLeft_mul (a b : A) : mulLeft R (a * b) = (mulLeft R a).comp (mulLeft R b) := by |
ext
simp only [mulLeft_apply, comp_apply, mul_assoc]
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Yury Kudryashov
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.LinearAlg... | Mathlib/Analysis/NormedSpace/AddTorsor.lean | 68 | 72 | theorem dist_lineMap_lineMap (p₁ p₂ : P) (c₁ c₂ : 𝕜) :
dist (lineMap p₁ p₂ c₁) (lineMap p₁ p₂ c₂) = dist c₁ c₂ * dist p₁ p₂ := by |
rw [dist_comm p₁ p₂]
simp only [lineMap_apply, dist_eq_norm_vsub, vadd_vsub_vadd_cancel_right,
← sub_smul, norm_smul, vsub_eq_sub]
|
/-
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.Sites.Sheaf
#align_import category_theory.sites.plus from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# The... | Mathlib/CategoryTheory/Sites/Plus.lean | 71 | 77 | theorem diagramNatTrans_id (X : C) (P : Cᵒᵖ ⥤ D) :
J.diagramNatTrans (𝟙 P) X = 𝟙 (J.diagram P X) := by |
ext : 2
refine Multiequalizer.hom_ext _ _ _ (fun i => ?_)
dsimp
simp only [limit.lift_π, Multifork.ofι_pt, Multifork.ofι_π_app, Category.id_comp]
erw [Category.comp_id]
|
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "lean... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 113 | 115 | theorem χ₄_int_three_mod_four {n : ℤ} (hn : n % 4 = 3) : χ₄ n = -1 := by |
rw [χ₄_int_mod_four, hn]
rfl
|
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Devon Tuma
-/
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Asymptotics.SpecificAsymptoti... | Mathlib/Analysis/SpecialFunctions/Polynomials.lean | 73 | 76 | theorem tendsto_atBot_iff_leadingCoeff_nonpos :
Tendsto (fun x => eval x P) atTop atBot ↔ 0 < P.degree ∧ P.leadingCoeff ≤ 0 := by |
simp only [← tendsto_neg_atTop_iff, ← eval_neg, tendsto_atTop_iff_leadingCoeff_nonneg,
degree_neg, leadingCoeff_neg, neg_nonneg]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34... | Mathlib/Data/Finsupp/Multiset.lean | 67 | 68 | theorem card_toMultiset (f : α →₀ ℕ) : Multiset.card (toMultiset f) = f.sum fun _ => id := by |
simp [toMultiset_apply, map_finsupp_sum, Function.id_def]
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.RelClasses
#align_import data.sigma.lex from "leanprover-community/mathlib"@"41cf0cc2f528dd40a8f2db167ea4fb37b8fde7f3"
/-!
# Lexicographic order on... | Mathlib/Data/Sigma/Lex.lean | 80 | 83 | theorem lex_swap : Lex (Function.swap r) s a b ↔ Lex r (fun i => Function.swap (s i)) b a := by |
constructor <;>
· rintro (⟨a, b, h⟩ | ⟨a, b, h⟩)
exacts [Lex.left _ _ h, Lex.right _ _ h]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a... | Mathlib/Order/Interval/Finset/Basic.lean | 62 | 63 | theorem nonempty_Ico : (Ico a b).Nonempty ↔ a < b := by |
rw [← coe_nonempty, coe_Ico, Set.nonempty_Ico]
|
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Analysis.Normed.Fi... | Mathlib/Topology/Algebra/Polynomial.lean | 105 | 120 | theorem tendsto_abv_eval₂_atTop {R S k α : Type*} [Semiring R] [Ring S] [LinearOrderedField k]
(f : R →+* S) (abv : S → k) [IsAbsoluteValue abv] (p : R[X]) (hd : 0 < degree p)
(hf : f p.leadingCoeff ≠ 0) {l : Filter α} {z : α → S} (hz : Tendsto (abv ∘ z) l atTop) :
Tendsto (fun x => abv (p.eval₂ f (z x))) l... |
revert hf; refine degree_pos_induction_on p hd ?_ ?_ ?_ <;> clear hd p
· rintro _ - hc
rw [leadingCoeff_mul_X, leadingCoeff_C] at hc
simpa [abv_mul abv] using hz.const_mul_atTop ((abv_pos abv).2 hc)
· intro _ _ ihp hf
rw [leadingCoeff_mul_X] at hf
simpa [abv_mul abv] using (ihp hf).atTop_mul_atTo... |
/-
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.SetTheory.Cardinal.Ordinal
import Mathlib.RingTheory.Artinian
#align_import ring_theory.localization.cardinality from "leanprover-community/mathlib"@"... | Mathlib/RingTheory/Localization/Cardinality.lean | 38 | 48 | theorem card_le : #L ≤ #R := by |
classical
cases fintypeOrInfinite R
· exact Cardinal.mk_le_of_surjective (IsArtinianRing.localization_surjective S _)
erw [← Cardinal.mul_eq_self <| Cardinal.aleph0_le_mk R]
set f : R × R → L := fun aa => IsLocalization.mk' _ aa.1 (if h : aa.2 ∈ S then ⟨aa.2, h⟩ else 1)
refine @Cardinal.mk_le_of_... |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Alex Meiburg
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
#align_import data.polynomial.erase_lead from "leanprover-communi... | Mathlib/Algebra/Polynomial/EraseLead.lean | 83 | 85 | theorem self_sub_C_mul_X_pow {R : Type*} [Ring R] (f : R[X]) :
f - C f.leadingCoeff * X ^ f.natDegree = f.eraseLead := by |
rw [C_mul_X_pow_eq_monomial, self_sub_monomial_natDegree_leadingCoeff]
|
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Anatole Dedecker
-/
import Mathlib.Topology.Separation
#align_import topology.extend_from from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b2... | Mathlib/Topology/ExtendFrom.lean | 86 | 89 | theorem continuous_extendFrom [RegularSpace Y] {f : X → Y} {A : Set X} (hA : Dense A)
(hf : ∀ x, ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)) : Continuous (extendFrom A f) := by |
rw [continuous_iff_continuousOn_univ]
exact continuousOn_extendFrom (fun x _ ↦ hA x) (by simpa using hf)
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Group.Invertible.Basic
import Mathlib.Algebra.GroupWithZero.Units.Basic
#align_import algebra.invertible from "leanprover-community/mathlib"@"722b3b... | Mathlib/Algebra/GroupWithZero/Invertible.lean | 23 | 28 | theorem nonzero_of_invertible [MulZeroOneClass α] (a : α) [Nontrivial α] [Invertible a] : a ≠ 0 :=
fun ha =>
zero_ne_one <|
calc
0 = ⅟ a * a := by | simp [ha]
_ = 1 := invOf_mul_self a
|
/-
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.Restrict
/-!
# Classes of measures
We introduce the following typeclasses for measures:
* `IsProbabilityMeasur... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 65 | 72 | theorem measure_compl_le_add_of_le_add [IsFiniteMeasure μ] (hs : MeasurableSet s)
(ht : MeasurableSet t) {ε : ℝ≥0∞} (h : μ s ≤ μ t + ε) : μ tᶜ ≤ μ sᶜ + ε := by |
rw [measure_compl ht (measure_ne_top μ _), measure_compl hs (measure_ne_top μ _),
tsub_le_iff_right]
calc
μ univ = μ univ - μ s + μ s := (tsub_add_cancel_of_le <| measure_mono s.subset_univ).symm
_ ≤ μ univ - μ s + (μ t + ε) := add_le_add_left h _
_ = _ := by rw [add_right_comm, add_assoc]
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Module.Defs
import Mathlib.Alge... | Mathlib/RingTheory/Nilpotent/Basic.lean | 100 | 102 | theorem isReduced_iff_pow_one_lt [MonoidWithZero R] (k : ℕ) (hk : 1 < k) :
IsReduced R ↔ ∀ x : R, x ^ k = 0 → x = 0 := by |
simp_rw [← zero_isRadical_iff, isRadical_iff_pow_one_lt k hk, zero_dvd_iff]
|
/-
Copyright (c) 2022 Pierre-Alexandre Bazin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pierre-Alexandre Bazin
-/
import Mathlib.Algebra.Module.Torsion
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import algebra.module.dedekind_domain from "leanprover-co... | Mathlib/Algebra/Module/DedekindDomain.lean | 37 | 59 | theorem isInternal_prime_power_torsion_of_is_torsion_by_ideal {I : Ideal R} (hI : I ≠ ⊥)
(hM : Module.IsTorsionBySet R M I) :
DirectSum.IsInternal fun p : (factors I).toFinset =>
torsionBySet R M (p ^ (factors I).count ↑p : Ideal R) := by |
let P := factors I
have prime_of_mem := fun p (hp : p ∈ P.toFinset) =>
prime_of_factor p (Multiset.mem_toFinset.mp hp)
apply torsionBySet_isInternal (p := fun p => p ^ P.count p) _
· convert hM
rw [← Finset.inf_eq_iInf, IsDedekindDomain.inf_prime_pow_eq_prod, ← Finset.prod_multiset_count,
← assoc... |
/-
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.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimension.Constructions
import Mathlib.LinearAlgebra.Dimension.Finite
#align_import field_the... | Mathlib/FieldTheory/Finiteness.lean | 32 | 43 | theorem iff_rank_lt_aleph0 : IsNoetherian K V ↔ Module.rank K V < ℵ₀ := by |
let b := Basis.ofVectorSpace K V
rw [← b.mk_eq_rank'', lt_aleph0_iff_set_finite]
constructor
· intro
exact (Basis.ofVectorSpaceIndex.linearIndependent K V).set_finite_of_isNoetherian
· intro hbfinite
refine
@isNoetherian_of_linearEquiv K (⊤ : Submodule K V) V _ _ _ _ _ (LinearEquiv.ofTop _ rfl)... |
/-
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 | 40 | 49 | theorem Fintype.isPrimePow_card_of_field {α} [Fintype α] [Field α] : IsPrimePow ‖α‖ := by |
-- TODO: `Algebra` version of `CharP.exists`, of type `∀ p, Algebra (ZMod p) α`
cases' CharP.exists α with p _
haveI hp := Fact.mk (CharP.char_is_prime α p)
letI : Algebra (ZMod p) α := ZMod.algebra _ _
let b := IsNoetherian.finsetBasis (ZMod p) α
rw [Module.card_fintype b, ZMod.card, isPrimePow_pow_iff]
... |
/-
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.MvPolynomial.Expand
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.RingTheory.MvPolynomial.Basic
#align_import field_theory.finite.pol... | Mathlib/FieldTheory/Finite/Polynomial.lean | 33 | 38 | theorem frobenius_zmod (f : MvPolynomial σ (ZMod p)) : frobenius _ p f = expand p f := by |
apply induction_on f
· intro a; rw [expand_C, frobenius_def, ← C_pow, ZMod.pow_card]
· simp only [AlgHom.map_add, RingHom.map_add]; intro _ _ hf hg; rw [hf, hg]
· simp only [expand_X, RingHom.map_mul, AlgHom.map_mul]
intro _ _ hf; rw [hf, frobenius_def]
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison, Adam Topaz
-/
import Mathlib.AlgebraicTopology.SimplexCategory
import Mathlib.CategoryTheory.Comma.Arrow
import Mathlib.CategoryTheory.Limits.FunctorCat... | Mathlib/AlgebraicTopology/SimplicialObject.lean | 107 | 110 | theorem δ_comp_δ {n} {i j : Fin (n + 2)} (H : i ≤ j) :
X.δ j.succ ≫ X.δ i = X.δ (Fin.castSucc i) ≫ X.δ j := by |
dsimp [δ]
simp only [← X.map_comp, ← op_comp, SimplexCategory.δ_comp_δ H]
|
/-
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.Algebra.Order.Ring.Nat
import Mathlib.Data.List.Chain
#align_import data.bool.count from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d8... | Mathlib/Data/Bool/Count.lean | 33 | 34 | theorem count_add_count_not (l : List Bool) (b : Bool) : count b l + count (!b) l = length l := by |
rw [add_comm, count_not_add_count]
|
/-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Init.Order.Defs
#align_import init.algebra.functions from "leanprover-community/lean"@"c2bcdbcbe741ed37c361a30d38e179182b989f... | Mathlib/Init/Order/LinearOrder.lean | 117 | 122 | theorem max_assoc (a b c : α) : max (max a b) c = max a (max b c) := by |
apply eq_max
· apply le_trans; apply le_max_left a b; apply le_max_left
· apply max_le; apply le_trans; apply le_max_right a b; apply le_max_left; apply le_max_right
· intro d h₁ h₂; apply max_le; apply max_le h₁; apply le_trans (le_max_left _ _) h₂
apply le_trans (le_max_right _ _) h₂
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Order.Monoid.Canoni... | Mathlib/Algebra/Order/Monoid/WithTop.lean | 132 | 136 | theorem add_eq_top : a + b = ⊤ ↔ a = ⊤ ∨ b = ⊤ := by |
match a, b with
| ⊤, _ => simp
| _, ⊤ => simp
| (a : α), (b : α) => simp only [← coe_add, coe_ne_top, or_false]
|
/-
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 | 140 | 141 | theorem Sized.size_eq_zero {t : Ordnode α} (ht : Sized t) : size t = 0 ↔ t = nil := by |
cases t <;> [simp;simp [ht.1]]
|
/-
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 | 100 | 101 | theorem perpBisector_comm (p₁ p₂ : P) : perpBisector p₁ p₂ = perpBisector p₂ p₁ := by |
ext c; simp only [mem_perpBisector_iff_dist_eq, eq_comm]
|
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 84 | 84 | theorem inv_logb (a b : ℝ) : (logb a b)⁻¹ = logb b a := by | simp_rw [logb, inv_div]
|
/-
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.Homotopies
import Mathlib.Tactic.Ring
#align_import algebraic_topology.dold_kan.faces from "leanprover-community/mathlib"@"32a7e535287... | Mathlib/AlgebraicTopology/DoldKan/Faces.lean | 53 | 58 | theorem comp_δ_eq_zero {Y : C} {n : ℕ} {q : ℕ} {φ : Y ⟶ X _[n + 1]} (v : HigherFacesVanish q φ)
(j : Fin (n + 2)) (hj₁ : j ≠ 0) (hj₂ : n + 2 ≤ (j : ℕ) + q) : φ ≫ X.δ j = 0 := by |
obtain ⟨i, rfl⟩ := Fin.eq_succ_of_ne_zero hj₁
apply v i
simp only [Fin.val_succ] at hj₂
omega
|
/-
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 | 115 | 124 | theorem finiteType_of_monoid_fg [Monoid.FG M] : Algebra.FiniteType R S := by |
have := Monoid.fg_of_surjective _ (toInvSubmonoid_surjective M S)
rw [Monoid.fg_iff_submonoid_fg] at this
rcases this with ⟨s, hs⟩
refine ⟨⟨s, ?_⟩⟩
rw [eq_top_iff]
rintro x -
change x ∈ (Subalgebra.toSubmodule (Algebra.adjoin R _ : Subalgebra R S) : Set S)
rw [Algebra.adjoin_eq_span, hs, span_invSubmon... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.UniformGroup
/-!
# Infinite sums and products in topological groups
Lemmas on topo... | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 50 | 53 | theorem HasProd.div (hf : HasProd f a₁) (hg : HasProd g a₂) :
HasProd (fun b ↦ f b / g b) (a₁ / a₂) := by |
simp only [div_eq_mul_inv]
exact hf.mul hg.inv
|
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
#align_import group_theory.perm.list from "leanprove... | Mathlib/GroupTheory/Perm/List.lean | 131 | 133 | theorem mem_of_formPerm_apply_mem (h : l.formPerm x ∈ l) : x ∈ l := by |
contrapose h
rwa [formPerm_apply_of_not_mem h]
|
/-
Copyright (c) 2022 Hans Parshall. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hans Parshall
-/
import Mathlib.Analysis.InnerProductSpace.Adjoint
import Mathlib.Analysis.Matrix
import Mathlib.Analysis.RCLike.Basic
import Mathlib.LinearAlgebra.UnitaryGroup
import M... | Mathlib/Analysis/NormedSpace/Star/Matrix.lean | 83 | 90 | theorem entrywise_sup_norm_bound_of_unitary {U : Matrix n n 𝕜} (hU : U ∈ Matrix.unitaryGroup n 𝕜) :
‖U‖ ≤ 1 := by |
conv => -- Porting note: was `simp_rw [pi_norm_le_iff_of_nonneg zero_le_one]`
rw [pi_norm_le_iff_of_nonneg zero_le_one]
intro
rw [pi_norm_le_iff_of_nonneg zero_le_one]
intros
exact entry_norm_bound_of_unitary hU _ _
|
/-
Copyright (c) 2022 Niels Voss. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Niels Voss
-/
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.Order.Filter.Cofinite
#align_import number_theory.fermat_psp from "leanprover-community/mathlib"@"c0439b4877c24a117bfd... | Mathlib/NumberTheory/FermatPsp.lean | 75 | 99 | theorem coprime_of_probablePrime {n b : ℕ} (h : ProbablePrime n b) (h₁ : 1 ≤ n) (h₂ : 1 ≤ b) :
Nat.Coprime n b := by |
by_cases h₃ : 2 ≤ n
· -- To prove that `n` is coprime with `b`, we need to show that for all prime factors of `n`,
-- we can derive a contradiction if `n` divides `b`.
apply Nat.coprime_of_dvd
-- If `k` is a prime number that divides both `n` and `b`, then we know that `n = m * k` and
-- `b = j * k... |
/-
Copyright (c) 2024 Thomas Browning, Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Junyan Xu
-/
import Mathlib.Data.Set.Finite
import Mathlib.GroupTheory.GroupAction.FixedPoints
import Mathlib.GroupTheory.Perm.Support
/-!
# Subgroups gen... | Mathlib/GroupTheory/Perm/ClosureSwap.lean | 59 | 70 | theorem exists_smul_not_mem_of_subset_orbit_closure (S : Set G) (T : Set α) {a : α}
(hS : ∀ g ∈ S, g⁻¹ ∈ S) (subset : T ⊆ orbit (closure S) a) (not_mem : a ∉ T)
(nonempty : T.Nonempty) : ∃ σ ∈ S, ∃ a ∈ T, σ • a ∉ T := by |
have key0 : ¬ closure S ≤ stabilizer G T := by
have ⟨b, hb⟩ := nonempty
obtain ⟨σ, rfl⟩ := subset hb
contrapose! not_mem with h
exact smul_mem_smul_set_iff.mp ((h σ.2).symm ▸ hb)
contrapose! key0
refine (closure_le _).mpr fun σ hσ ↦ ?_
simp_rw [SetLike.mem_coe, mem_stabilizer_iff, Set.ext_iff, ... |
/-
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.Data.Ordering.Basic
import Mathlib.Order.Synonym
#align_import order.compare from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb... | Mathlib/Order/Compare.lean | 67 | 71 | theorem compares_swap [LT α] {a b : α} {o : Ordering} : o.swap.Compares a b ↔ o.Compares b a := by |
cases o
· exact Iff.rfl
· exact eq_comm
· exact Iff.rfl
|
/-
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.MeasureSpace
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.Topology.Sets.Compacts
#align_import measure_theory... | Mathlib/MeasureTheory/Measure/Content.lean | 108 | 111 | theorem sup_le (K₁ K₂ : Compacts G) : μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂ := by |
simp only [apply_eq_coe_toFun]
norm_cast
exact μ.sup_le' _ _
|
/-
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.InnerProductSpace.Orientation
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.measure.haar.inner_prod... | Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean | 102 | 108 | theorem EuclideanSpace.volume_preserving_measurableEquiv :
MeasurePreserving (EuclideanSpace.measurableEquiv ι) := by |
suffices volume = map (EuclideanSpace.measurableEquiv ι).symm volume by
convert ((EuclideanSpace.measurableEquiv ι).symm.measurable.measurePreserving _).symm
rw [← addHaarMeasure_eq_volume_pi, ← Basis.parallelepiped_basisFun, ← Basis.addHaar_def,
coe_measurableEquiv_symm, ← PiLp.continuousLinearEquiv_symm_... |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Michael Howes
-/
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Commutator
import Mathlib.GroupTheory.Finiteness
#align_import group_theory.abelianization from "lean... | Mathlib/GroupTheory/Abelianization.lean | 53 | 54 | theorem commutator_eq_normalClosure : commutator G = Subgroup.normalClosure (commutatorSet G) := by |
simp [commutator, Subgroup.commutator_def', commutatorSet]
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
/-!
# Neig... | Mathlib/Topology/ContinuousOn.lean | 63 | 67 | theorem eventually_nhdsWithin_nhdsWithin {a : α} {s : Set α} {p : α → Prop} :
(∀ᶠ y in 𝓝[s] a, ∀ᶠ x in 𝓝[s] y, p x) ↔ ∀ᶠ x in 𝓝[s] a, p x := by |
refine ⟨fun h => ?_, fun h => (eventually_nhds_nhdsWithin.2 h).filter_mono inf_le_left⟩
simp only [eventually_nhdsWithin_iff] at h ⊢
exact h.mono fun x hx hxs => (hx hxs).self_of_nhds hxs
|
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Data.Finsupp.Defs
#align_import data.finsupp.ne_locus from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
/-!
# Locus of une... | Mathlib/Data/Finsupp/NeLocus.lean | 52 | 54 | theorem coe_neLocus : ↑(f.neLocus g) = { x | f x ≠ g x } := by |
ext
exact mem_neLocus
|
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Analysis.Quaternion
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Series
#align_import analysis.nor... | Mathlib/Analysis/NormedSpace/QuaternionExponential.lean | 82 | 94 | theorem hasSum_expSeries_of_imaginary {q : Quaternion ℝ} (hq : q.re = 0) {c s : ℝ}
(hc : HasSum (fun n => (-1 : ℝ) ^ n * ‖q‖ ^ (2 * n) / (2 * n)!) c)
(hs : HasSum (fun n => (-1 : ℝ) ^ n * ‖q‖ ^ (2 * n + 1) / (2 * n + 1)!) s) :
HasSum (fun n => expSeries ℝ (Quaternion ℝ) n fun _ => q) (↑c + (s / ‖q‖) • q) :=... |
replace hc := hasSum_coe.mpr hc
replace hs := (hs.div_const ‖q‖).smul_const q
refine HasSum.even_add_odd ?_ ?_
· convert hc using 1
ext n : 1
rw [expSeries_even_of_imaginary hq]
· convert hs using 1
ext n : 1
rw [expSeries_odd_of_imaginary hq]
|
/-
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.Logic.Basic
import Mathlib.Tactic.Positivity.Basic
#align_import algebra.order.hom.basic from "leanprover-community/mathlib"@"28aa996fc6fb4317f0083c4e6daf... | Mathlib/Algebra/Order/Hom/Basic.lean | 124 | 126 | theorem le_map_mul_map_div [Group α] [CommSemigroup β] [LE β] [SubmultiplicativeHomClass F α β]
(f : F) (a b : α) : f a ≤ f b * f (a / b) := by |
simpa only [mul_comm, div_mul_cancel] using map_mul_le_mul f (a / b) b
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Rémy Degenne
-/
import Mathlib.Probability.Process.Stopping
import Mathlib.Tactic.AdaptationNote
#align_import probability.process.hitting_time from "leanprover-community/ma... | Mathlib/Probability/Process/HittingTime.lean | 102 | 109 | theorem hitting_of_le {m : ι} (hmn : m ≤ n) : hitting u s n m ω = m := by |
obtain rfl | h := le_iff_eq_or_lt.1 hmn
· rw [hitting, ite_eq_right_iff, forall_exists_index]
conv => intro; rw [Set.mem_Icc, Set.Icc_self, and_imp, and_imp]
intro i hi₁ hi₂ hi
rw [Set.inter_eq_left.2, csInf_singleton]
exact Set.singleton_subset_iff.2 (le_antisymm hi₂ hi₁ ▸ hi)
· exact hitting_of... |
/-
Copyright (c) 2024 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.SpecificLimits.Normed
/-!
# Abel's limit theorem
If a real or complex power series for a function has radius o... | Mathlib/Analysis/Complex/AbelLimit.lean | 56 | 66 | theorem nhdsWithin_lt_le_nhdsWithin_stolzSet {M : ℝ} (hM : 1 < M) :
(𝓝[<] 1).map ofReal' ≤ 𝓝[stolzSet M] 1 := by |
rw [← tendsto_id']
refine tendsto_map' <| tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within ofReal'
(tendsto_nhdsWithin_of_tendsto_nhds <| ofRealCLM.continuous.tendsto' 1 1 rfl) ?_
simp only [eventually_iff, norm_eq_abs, abs_ofReal, abs_lt, mem_nhdsWithin]
refine ⟨Set.Ioo 0 2, isOpen_Ioo, by norm_num... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Floris van Doorn, Sébastien Gouëzel, Alex J. Best
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.Group.Nat
import ... | Mathlib/Algebra/BigOperators/Group/List.lean | 95 | 99 | theorem prod_cons : (a :: l).prod = a * l.prod :=
calc
(a :: l).prod = foldl (· * ·) (a * 1) l := by |
simp only [List.prod, foldl_cons, one_mul, mul_one]
_ = _ := foldl_assoc
|
/-
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.Order.RelIso.Set
import Mathlib.Data.Multiset.Sort
import Mathlib.Data.List.NodupEquivFin
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Fintyp... | Mathlib/Data/Finset/Sort.lean | 79 | 81 | theorem sort_perm_toList (s : Finset α) : sort r s ~ s.toList := by |
rw [← Multiset.coe_eq_coe]
simp only [coe_toList, sort_eq]
|
/-
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.Algebra.Order.Group.Instances
import Mathlib.Analysis.Convex.Segment
import Mathlib.Tactic.GCongr
#align_import analysis.convex.star from "leanprover-comm... | Mathlib/Analysis/Convex/Star.lean | 75 | 80 | theorem starConvex_iff_segment_subset : StarConvex 𝕜 x s ↔ ∀ ⦃y⦄, y ∈ s → [x -[𝕜] y] ⊆ s := by |
constructor
· rintro h y hy z ⟨a, b, ha, hb, hab, rfl⟩
exact h hy ha hb hab
· rintro h y hy a b ha hb hab
exact h hy ⟨a, b, ha, hb, hab, rfl⟩
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#ali... | Mathlib/Algebra/Order/Group/Defs.lean | 82 | 84 | theorem OrderedCommGroup.to_contravariantClass_right_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (swap (· * ·)) (· ≤ ·) where
elim a b c bc := by | simpa using mul_le_mul_right' bc a⁻¹
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Set.Finite
#align_import data.finset.n_ary from "leanprover-community/mathlib"@"eba7871095e834365616b5e43c8c7bb0b3705... | Mathlib/Data/Finset/NAry.lean | 112 | 113 | theorem image₂_subset_iff_right : image₂ f s t ⊆ u ↔ ∀ b ∈ t, (s.image fun a => f a b) ⊆ u := by |
simp_rw [image₂_subset_iff, image_subset_iff, @forall₂_swap α]
|
/-
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.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a... | Mathlib/Logic/Lemmas.lean | 28 | 31 | theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by |
split_ifs <;> rfl
|
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.Calculus.BumpFunction.Normed
import Mathlib.MeasureTheory.Integral.Average
import Mathlib.MeasureTheor... | Mathlib/Analysis/Calculus/BumpFunction/Convolution.lean | 54 | 56 | theorem convolution_eq_right {x₀ : G} (hg : ∀ x ∈ ball x₀ φ.rOut, g x = g x₀) :
(φ ⋆[lsmul ℝ ℝ, μ] g : G → E') x₀ = integral μ φ • g x₀ := by |
simp_rw [convolution_eq_right' _ φ.support_eq.subset hg, lsmul_apply, integral_smul_const]
|
/-
Copyright (c) 2019 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.UniformSpace.UniformEmbedding
#align_import topology.uniform_space.pi from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062... | Mathlib/Topology/UniformSpace/Pi.lean | 46 | 49 | theorem uniformContinuous_pi {β : Type*} [UniformSpace β] {f : β → ∀ i, α i} :
UniformContinuous f ↔ ∀ i, UniformContinuous fun x => f x i := by |
-- Porting note: required `Function.comp` to close
simp only [UniformContinuous, Pi.uniformity, tendsto_iInf, tendsto_comap_iff, Function.comp]
|
/-
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.Algebra.Homology.Homotopy
import Mathlib.Algebra.Category.ModuleCat.Abelian
import Mathlib.Algebra.Category.ModuleCat.Subobject
import Mathlib.Category... | Mathlib/Algebra/Homology/ModuleCat.lean | 91 | 93 | theorem homology'_ext' {M : ModuleCat R} (i : ι) {h k : C.homology' i ⟶ M}
(w : ∀ x : LinearMap.ker (C.dFrom i), h (toHomology' x) = k (toHomology' x)) : h = k := by |
apply homology'_ext _ w
|
/-
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.InnerProductSpace.Rayleigh
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.Line... | Mathlib/Analysis/InnerProductSpace/Spectrum.lean | 68 | 72 | theorem invariant_orthogonalComplement_eigenspace (μ : 𝕜) (v : E) (hv : v ∈ (eigenspace T μ)ᗮ) :
T v ∈ (eigenspace T μ)ᗮ := by |
intro w hw
have : T w = (μ : 𝕜) • w := by rwa [mem_eigenspace_iff] at hw
simp [← hT w, this, inner_smul_left, hv w hw]
|
/-
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.SetTheory.Cardinal.ToNat
import Mathlib.Data.Nat.PartENat
#align_import set_theory.cardinal.basic from "leanprover-community/mathlib"@"3ff3f2d6a3118b8... | Mathlib/SetTheory/Cardinal/PartENat.lean | 39 | 40 | theorem toPartENat_natCast (n : ℕ) : toPartENat n = n := by |
simp only [← partENatOfENat_toENat, toENat_nat, PartENat.ofENat_coe]
|
/-
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.Analysis.NormedSpace.AffineIsometry
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
#align_... | Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean | 118 | 120 | theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by |
ext
rfl
|
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.Indicator
import Mathlib.Algebra.Module.Basic
import Mathlib.Topology.Separation
#align_import topology.supp... | Mathlib/Topology/Support.lean | 124 | 126 | theorem not_mem_mulTSupport_iff_eventuallyEq : x ∉ mulTSupport f ↔ f =ᶠ[𝓝 x] 1 := by |
simp_rw [mulTSupport, mem_closure_iff_nhds, not_forall, not_nonempty_iff_eq_empty, exists_prop,
← disjoint_iff_inter_eq_empty, disjoint_mulSupport_iff, eventuallyEq_iff_exists_mem]
|
/-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Adjoin.Ba... | Mathlib/RingTheory/AlgebraicIndependent.lean | 109 | 118 | theorem linearIndependent : LinearIndependent R x := by |
rw [linearIndependent_iff_injective_total]
have : Finsupp.total ι A R x =
(MvPolynomial.aeval x).toLinearMap.comp (Finsupp.total ι _ R X) := by
ext
simp
rw [this]
refine hx.comp ?_
rw [← linearIndependent_iff_injective_total]
exact linearIndependent_X _ _
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
#align_import data.set.prod from "leanprover-community/mathlib"@"48fb5b5280e7... | Mathlib/Data/Set/Prod.lean | 96 | 98 | theorem univ_prod_univ : @univ α ×ˢ @univ β = univ := by |
ext
exact true_and_iff _
|
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Quaternion
import Mathlib.Tactic.Ring
#align_import algebra.quaternion_basis from "leanprover-community/mathlib"@"3aa5b8a9ed7a7cabd36e6e1d022c9858ab... | Mathlib/Algebra/QuaternionBasis.lean | 114 | 114 | theorem lift_zero : q.lift (0 : ℍ[R,c₁,c₂]) = 0 := by | simp [lift]
|
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
#align_import algebra.continued_fractions.translations from "leanprove... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 58 | 59 | theorem part_num_eq_s_a {gp : Pair α} (s_nth_eq : g.s.get? n = some gp) :
g.partialNumerators.get? n = some gp.a := by | simp [partialNumerators, s_nth_eq]
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
/-!
# The rank of a linear map
## Main Definit... | Mathlib/LinearAlgebra/Dimension/LinearMap.lean | 79 | 81 | theorem rank_comp_le (g : V →ₗ[K] V') (f : V' →ₗ[K] V'₁) :
rank (f.comp g) ≤ min (rank f) (rank g) := by |
simpa only [Cardinal.lift_id] using lift_rank_comp_le g f
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.VectorMeasure
#align_import measure_theory.measure.complex from "leanprover-community/mathlib"@"17b3357baa47f48697ca9c243e300eb8cdd16a... | Mathlib/MeasureTheory/Measure/Complex.lean | 116 | 122 | theorem absolutelyContinuous_ennreal_iff (c : ComplexMeasure α) (μ : VectorMeasure α ℝ≥0∞) :
c ≪ᵥ μ ↔ ComplexMeasure.re c ≪ᵥ μ ∧ ComplexMeasure.im c ≪ᵥ μ := by |
constructor <;> intro h
· constructor <;> · intro i hi; simp [h hi]
· intro i hi
rw [← Complex.re_add_im (c i), (_ : (c i).re = 0), (_ : (c i).im = 0)]
exacts [by simp, h.2 hi, h.1 hi]
|
/-
Copyright (c) 2020 Hanting Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hanting Zhang
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.MvPolynomial.Symmetric
#align_import ring_theory.polynomial.vieta from "leanprover-community/mathlib... | Mathlib/RingTheory/Polynomial/Vieta.lean | 94 | 101 | theorem esymm_neg (s : Multiset R) (k : ℕ) : (map Neg.neg s).esymm k = (-1) ^ k * esymm s k := by |
rw [esymm, esymm, ← Multiset.sum_map_mul_left, Multiset.powersetCard_map, Multiset.map_map,
map_congr rfl]
intro x hx
rw [(mem_powersetCard.mp hx).right.symm, ← prod_replicate, ← Multiset.map_const]
nth_rw 3 [← map_id' x]
rw [← prod_map_mul, map_congr rfl, Function.comp_apply]
exact fun z _ => neg_one_... |
/-
Copyright (c) 2021 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri
-/
import Mathlib.Data.Set.Basic
#align_import data.bundle from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
/-!
# Bundle
Basic data s... | Mathlib/Data/Bundle.lean | 74 | 75 | theorem TotalSpace.mk_inj {b : B} {y y' : E b} : mk' F b y = mk' F b y' ↔ y = y' := by |
simp [TotalSpace.ext_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, Patrick Massot
-/
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.QuotientGrou... | Mathlib/Topology/Algebra/Group/Basic.lean | 71 | 73 | theorem Homeomorph.mulLeft_symm (a : G) : (Homeomorph.mulLeft a).symm = Homeomorph.mulLeft a⁻¹ := by |
ext
rfl
|
/-
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.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
/-!
# The length function, reduced words, and descents
Throughout this file, `B` is a type and `... | Mathlib/GroupTheory/Coxeter/Length.lean | 111 | 113 | theorem length_mul_ge_length_sub_length' (w₁ w₂ : W) :
ℓ w₂ - ℓ w₁ ≤ ℓ (w₁ * w₂) := by |
simpa [Nat.sub_le_of_le_add, add_comm] using cs.length_mul_le w₁⁻¹ (w₁ * w₂)
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Measure.Restrict
#align_import measure_theory.measure.mutually_singular from "leanprover-community/mathlib"@"70a4f2197832bcea... | Mathlib/MeasureTheory/Measure/MutuallySingular.lean | 48 | 52 | theorem mk {s t : Set α} (hs : μ s = 0) (ht : ν t = 0) (hst : univ ⊆ s ∪ t) :
MutuallySingular μ ν := by |
use toMeasurable μ s, measurableSet_toMeasurable _ _, (measure_toMeasurable _).trans hs
refine measure_mono_null (fun x hx => (hst trivial).resolve_left fun hxs => hx ?_) ht
exact subset_toMeasurable _ _ hxs
|
/-
Copyright (c) 2024 Miyahara Kō. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Miyahara Kō
-/
import Mathlib.Data.List.Range
import Mathlib.Algebra.Order.Ring.Nat
/-!
# iterate
Proves various lemmas about `List.iterate`.
-/
variable {α : Type*}
namespace List
... | Mathlib/Data/List/Iterate.lean | 48 | 52 | theorem iterate_add (f : α → α) (a : α) (m n : ℕ) :
iterate f a (m + n) = iterate f a m ++ iterate f (f^[m] a) n := by |
induction m generalizing a with
| zero => simp
| succ n ih => rw [iterate, add_right_comm, iterate, ih, Nat.iterate, cons_append]
|
/-
Copyright (c) 2019 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Bitraversable.Basic
#align_import control.bitraversable.lemmas from "leanprover-community/mathlib"@"58581d0fe523063f5651df0619be2bf65012a94a"
/-!
#... | Mathlib/Control/Bitraversable/Lemmas.lean | 110 | 112 | theorem tfst_eq_fst_id {α α' β} (f : α → α') (x : t α β) :
tfst (F := Id) (pure ∘ f) x = pure (fst f x) := by |
apply bitraverse_eq_bimap_id
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth, Mitchell Lee
-/
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.pol... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 96 | 97 | theorem T_sub_one (n : ℤ) : T R (n - 1) = 2 * X * T R n - T R (n + 1) := by |
linear_combination (norm := ring_nf) T_add_two R (n - 1)
|
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
... | Mathlib/Analysis/MellinTransform.lean | 53 | 56 | theorem MellinConvergent.cpow_smul {f : ℝ → E} {s a : ℂ} :
MellinConvergent (fun t => (t : ℂ) ^ a • f t) s ↔ MellinConvergent f (s + a) := by |
refine integrableOn_congr_fun (fun t ht => ?_) measurableSet_Ioi
simp_rw [← sub_add_eq_add_sub, cpow_add _ _ (ofReal_ne_zero.2 <| ne_of_gt ht), mul_smul]
|
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.List.MinMax
import Mathlib.Algebra.Tropical.Basic
import Mathlib.Order.ConditionallyCompleteLat... | Mathlib/Algebra/Tropical/BigOperators.lean | 70 | 74 | theorem untrop_prod [AddCommMonoid R] (s : Finset S) (f : S → Tropical R) :
untrop (∏ i ∈ s, f i) = ∑ i ∈ s, untrop (f i) := by |
convert Multiset.untrop_prod (s.val.map f)
simp only [Multiset.map_map, Function.comp_apply]
rfl
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Geometry.Euclidean.Inversion.Basic
import Mathlib.Geometry.Euclidean.PerpBisector
/-!
# Image of a hyperplane under inversion
In this file we prove... | Mathlib/Geometry/Euclidean/Inversion/ImageHyperplane.lean | 66 | 71 | theorem preimage_inversion_sphere_dist_center (hR : R ≠ 0) (hy : y ≠ c) :
inversion c R ⁻¹' sphere y (dist y c) =
insert c (perpBisector c (inversion c R y) : Set P) := by |
ext x
rcases eq_or_ne x c with rfl | hx; · simp [dist_comm]
rw [mem_preimage, mem_sphere, ← inversion_mem_perpBisector_inversion_iff hR] <;> simp [*]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau, Scott Morrison
-/
import Mathlib.Data.List.Chain
import Mathlib.Data.List.Enum
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Pairwise
import Mathli... | Mathlib/Data/List/Range.lean | 87 | 90 | theorem take_range (m n : ℕ) : take m (range n) = range (min m n) := by |
apply List.ext_get
· simp
· simp (config := { contextual := true }) [← get_take, Nat.lt_min]
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yourong Zang
-/
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Conformal
import Mat... | Mathlib/Analysis/Complex/RealDeriv.lean | 84 | 89 | theorem ContDiffAt.real_of_complex {n : ℕ∞} (h : ContDiffAt ℂ n e z) :
ContDiffAt ℝ n (fun x : ℝ => (e x).re) z := by |
have A : ContDiffAt ℝ n ((↑) : ℝ → ℂ) z := ofRealCLM.contDiff.contDiffAt
have B : ContDiffAt ℝ n e z := h.restrict_scalars ℝ
have C : ContDiffAt ℝ n re (e z) := reCLM.contDiff.contDiffAt
exact C.comp z (B.comp z A)
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Uni Marx
-/
import Mathlib.CategoryTheory.Iso
import Mathlib.CategoryTheory.EssentialImage
import Mathlib.CategoryTheory.Types
import Mathlib.CategoryTheory.Opposites
i... | Mathlib/CategoryTheory/Category/RelCat.lean | 93 | 122 | theorem rel_iso_iff {X Y : RelCat} (r : X ⟶ Y) :
IsIso (C := RelCat) r ↔ ∃ f : (Iso (C := Type) X Y), graphFunctor.map f.hom = r := by |
constructor
· intro h
have h1 := congr_fun₂ h.hom_inv_id
have h2 := congr_fun₂ h.inv_hom_id
simp only [RelCat.Hom.rel_comp_apply₂, RelCat.Hom.rel_id_apply₂, eq_iff_iff] at h1 h2
obtain ⟨f, hf⟩ := Classical.axiomOfChoice (fun a => (h1 a a).mpr rfl)
obtain ⟨g, hg⟩ := Classical.axiomOfChoice (fun ... |
/-
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.MonoidAlgebra.Basic
#align_import algebra.monoid_algebra.division from "leanprover-community/mathlib"@"72c366d0475675f1309d3027d3d7d47ee4423951"
/-... | Mathlib/Algebra/MonoidAlgebra/Division.lean | 112 | 117 | theorem mul_of'_divOf (x : k[G]) (a : G) : x * of' k G a /ᵒᶠ a = x := by |
refine Finsupp.ext fun _ => ?_ -- Porting note: `ext` doesn't work
rw [AddMonoidAlgebra.divOf_apply, of'_apply, mul_single_apply_aux, mul_one]
intro c
rw [add_comm]
exact add_right_inj _
|
/-
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 | 102 | 105 | theorem partialSum_continuous (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
Continuous (p.partialSum n) := by |
unfold partialSum -- Porting note: added
continuity
|
/-
Copyright (c) 2019 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Analysis.LocallyConvex.WithSeminorms
import Mathlib.Topology.Algebra.Module.Stro... | Mathlib/Analysis/NormedSpace/OperatorNorm/Basic.lean | 54 | 57 | theorem norm_image_of_norm_zero [SemilinearMapClass 𝓕 σ₁₂ E F] (f : 𝓕) (hf : Continuous f) {x : E}
(hx : ‖x‖ = 0) : ‖f x‖ = 0 := by |
rw [← mem_closure_zero_iff_norm, ← specializes_iff_mem_closure, ← map_zero f] at *
exact hx.map hf
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.LinearAlgebra.Quotient
#align_import linear_algebra.isomorphisms from "leanprover-community/mathlib"@"... | Mathlib/LinearAlgebra/Isomorphisms.lean | 88 | 93 | theorem quotientInfEquivSupQuotient_surjective (p p' : Submodule R M) :
Function.Surjective (quotientInfToSupQuotient p p') := by |
rw [← range_eq_top, quotientInfToSupQuotient, range_liftQ, eq_top_iff']
rintro ⟨x, hx⟩; rcases mem_sup.1 hx with ⟨y, hy, z, hz, rfl⟩
use ⟨y, hy⟩; apply (Submodule.Quotient.eq _).2
simp only [mem_comap, map_sub, coeSubtype, coe_inclusion, sub_add_cancel_left, neg_mem_iff, hz]
|
/-
Copyright (c) 2020 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash, Antoine Labelle
-/
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.contraction from "leanprover-community/mathlib"@"... | Mathlib/LinearAlgebra/Contraction.lean | 113 | 118 | theorem map_dualTensorHom (f : Module.Dual R M) (p : P) (g : Module.Dual R N) (q : Q) :
TensorProduct.map (dualTensorHom R M P (f ⊗ₜ[R] p)) (dualTensorHom R N Q (g ⊗ₜ[R] q)) =
dualTensorHom R (M ⊗[R] N) (P ⊗[R] Q) (dualDistrib R M N (f ⊗ₜ g) ⊗ₜ[R] p ⊗ₜ[R] q) := by |
ext m n
simp only [compr₂_apply, mk_apply, map_tmul, dualTensorHom_apply, dualDistrib_apply, ←
smul_tmul_smul]
|
/-
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.Group.Basic
import Mathlib.Algebra.Group.Nat
import Mathlib.Init.Data.Nat.Lemmas
#align_import data.nat.psub from "leanprover-community/mathli... | Mathlib/Data/Nat/PSub.lean | 85 | 93 | theorem psub_eq_none {m n : ℕ} : psub m n = none ↔ m < n := by |
cases s : psub m n <;> simp [eq_comm]
· show m < n
refine lt_of_not_ge fun h => ?_
cases' le.dest h with k e
injection s.symm.trans (psub_eq_some.2 <| (add_comm _ _).trans e)
· show n ≤ m
rw [← psub_eq_some.1 s]
apply Nat.le_add_left
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.SetTheory.Game.State
#align_import set_theory.game.domineering from "leanprover-community/mathlib"@"b134b2f5cf6dd25d4bbfd3c498b6e36c11a17225"
/-!
# D... | Mathlib/SetTheory/Game/Domineering.lean | 101 | 106 | theorem card_of_mem_right {b : Board} {m : ℤ × ℤ} (h : m ∈ right b) : 2 ≤ Finset.card b := by |
have w₁ : m ∈ b := (Finset.mem_inter.1 h).1
have w₂ := fst_pred_mem_erase_of_mem_right h
have i₁ := Finset.card_erase_lt_of_mem w₁
have i₂ := Nat.lt_of_le_of_lt (Nat.zero_le _) (Finset.card_erase_lt_of_mem w₂)
exact Nat.lt_of_le_of_lt i₂ i₁
|
/-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.Elements
import Mathlib.CategoryTheory.IsConnected
import Mathlib.CategoryTheory.SingleObj
import Mathlib.GroupTheory.GroupAction.Quotient
impor... | Mathlib/CategoryTheory/Action.lean | 89 | 89 | theorem back_coe (x : ActionCategory M X) : ↑x.back = x := by | cases x; rfl
|
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