Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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/-
Copyright (c) 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 | 275 | 276 | theorem preimage_sub_const_Iio : (fun x => x - a) ⁻¹' Iio b = Iio (b + a) := by |
simp [sub_eq_add_neg]
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc429200506... | Mathlib/Data/Set/Image.lean | 1,300 | 1,304 | theorem Surjective.preimage_subset_preimage_iff {s t : Set β} (hf : Surjective f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by |
apply Set.preimage_subset_preimage_iff
rw [hf.range_eq]
apply subset_univ
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes, Floris van Doorn, Yaël Dillies
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.Common
import Mathlib.Tactic.Monoto... | Mathlib/Data/Nat/Factorial/Basic.lean | 344 | 344 | theorem descFactorial_one (n : ℕ) : n.descFactorial 1 = n := by | simp
|
/-
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.CategoryTheory.Comma.Arrow
import Mathlib.CategoryTheory.Pi.Basic
import Mathlib.Order.CompleteBooleanAlgebra
#align_import category_theory.morphism_propert... | Mathlib/CategoryTheory/MorphismProperty/Basic.lean | 338 | 343 | theorem RespectsIso.monomorphisms : RespectsIso (monomorphisms C) := by |
constructor <;>
· intro X Y Z e f
simp only [monomorphisms.iff]
intro
apply mono_comp
|
/-
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.MeasureTheory.Integral.IntegrableOn
#align_import measure_theory.function.locally_integrable from "leanprover-community/mathlib"@"08a4542bec7242a5... | Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | 218 | 230 | theorem locallyIntegrableOn_iff_locallyIntegrable_restrict [OpensMeasurableSpace X]
(hs : IsClosed s) : LocallyIntegrableOn f s μ ↔ LocallyIntegrable f (μ.restrict s) := by |
refine ⟨fun hf x => ?_, locallyIntegrableOn_of_locallyIntegrable_restrict⟩
by_cases h : x ∈ s
· obtain ⟨t, ht_nhds, ht_int⟩ := hf x h
obtain ⟨u, hu_o, hu_x, hu_sub⟩ := mem_nhdsWithin.mp ht_nhds
refine ⟨u, hu_o.mem_nhds hu_x, ?_⟩
rw [IntegrableOn, restrict_restrict hu_o.measurableSet]
exact ht_int... |
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.Array.Lemmas
import Batteries.Tactic.Lint.Misc
namespace Batteries
/-- Union-find node type -/
structure UFNode where
/-- Parent of node -/
... | .lake/packages/batteries/Batteries/Data/UnionFind/Basic.lean | 305 | 317 | theorem parentD_findAux {self : UnionFind} {x : Fin self.size} :
parentD (findAux self x).s i =
if i = x then self.rootD x else parentD (self.findAux ⟨_, self.parent'_lt x⟩).s i := by |
rw [findAux_s]; split <;> [split; skip]
· subst i; rw [rootD_eq_self.2 _] <;> simp [parentD_eq, *, -Array.get_eq_getElem]
· rw [findAux_s]; simp [*, -Array.get_eq_getElem]
· next h =>
rw [parentD]; split <;> rename_i h'
· rw [Array.get_modify (by simpa using h')]
simp [@eq_comm _ i, -Array.get_eq... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.E... | Mathlib/Topology/Instances/ENNReal.lean | 1,464 | 1,480 | theorem cauchySeq_of_edist_le_of_summable [PseudoEMetricSpace α] {f : ℕ → α} (d : ℕ → ℝ≥0)
(hf : ∀ n, edist (f n) (f n.succ) ≤ d n) (hd : Summable d) : CauchySeq f := by |
refine EMetric.cauchySeq_iff_NNReal.2 fun ε εpos ↦ ?_
-- Actually we need partial sums of `d` to be a Cauchy sequence.
replace hd : CauchySeq fun n : ℕ ↦ ∑ x ∈ Finset.range n, d x :=
let ⟨_, H⟩ := hd
H.tendsto_sum_nat.cauchySeq
-- Now we take the same `N` as in one of the definitions of a Cauchy sequen... |
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 778 | 779 | theorem edges_append {u v w : V} (p : G.Walk u v) (p' : G.Walk v w) :
(p.append p').edges = p.edges ++ p'.edges := by | simp [edges]
|
/-
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.LinearAlgebra.Matrix.Adjugate
import Mathlib.RingTheory.PolynomialAlgebra
#align_import linear_algebra.matrix.charpoly.basic from "leanprover-communit... | Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean | 67 | 76 | theorem matPolyEquiv_charmatrix : matPolyEquiv (charmatrix M) = X - C M := by |
ext k i j
simp only [matPolyEquiv_coeff_apply, coeff_sub, Pi.sub_apply]
by_cases h : i = j
· subst h
rw [charmatrix_apply_eq, coeff_sub]
simp only [coeff_X, coeff_C]
split_ifs <;> simp
· rw [charmatrix_apply_ne _ _ _ h, coeff_X, coeff_neg, coeff_C, coeff_C]
split_ifs <;> simp [h]
|
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.nor... | Mathlib/Analysis/Normed/Ring/Seminorm.lean | 116 | 116 | theorem ne_zero_iff {p : RingSeminorm R} : p ≠ 0 ↔ ∃ x, p x ≠ 0 := by | simp [eq_zero_iff]
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.Sub
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_t... | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | 1,044 | 1,055 | theorem rnDeriv_smul_left' (ν μ : Measure α) [SigmaFinite ν] [SigmaFinite μ] (r : ℝ≥0) :
(r • ν).rnDeriv μ =ᵐ[μ] r • ν.rnDeriv μ := by |
rw [← withDensity_eq_iff_of_sigmaFinite]
· simp_rw [ENNReal.smul_def]
rw [withDensity_smul _ (measurable_rnDeriv _ _)]
suffices (r • ν).singularPart μ + withDensity μ (rnDeriv (r • ν) μ)
= (r • ν).singularPart μ + r • withDensity μ (rnDeriv ν μ) by
rwa [Measure.add_right_inj] at this
rw [... |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.LatticeIntervals
import Mathlib.Order.GaloisConnection
#align_import order.modular_lattice from "leanpr... | Mathlib/Order/ModularLattice.lean | 151 | 153 | theorem covBy_sup_of_inf_covBy_right : a ⊓ b ⋖ b → a ⋖ a ⊔ b := by |
rw [sup_comm, inf_comm]
exact covBy_sup_of_inf_covBy_left
|
/-
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.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
#align_import data.... | Mathlib/Data/Sign.lean | 385 | 390 | theorem sign_eq_zero_iff : sign a = 0 ↔ a = 0 := by |
refine ⟨fun h => ?_, fun h => h.symm ▸ sign_zero⟩
rw [sign_apply] at h
split_ifs at h with h_1 h_2
cases' h
exact (le_of_not_lt h_1).eq_of_not_lt h_2
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
im... | Mathlib/Data/List/Basic.lean | 2,566 | 2,568 | theorem mem_pmap {p : α → Prop} {f : ∀ a, p a → β} {l H b} :
b ∈ pmap f l H ↔ ∃ (a : _) (h : a ∈ l), f a (H a h) = b := by |
simp only [pmap_eq_map_attach, mem_map, mem_attach, true_and_iff, Subtype.exists, eq_comm]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yury Kudryashov
-/
import Mathlib.MeasureTheory.OuterMeasure.Basic
/-!
# The “almost everywhere” filter of co-null sets.
If `μ` is an outer measure or a measure on `α... | Mathlib/MeasureTheory/OuterMeasure/AE.lean | 169 | 171 | theorem diff_ae_eq_self : (s \ t : Set α) =ᵐ[μ] s ↔ μ (s ∩ t) = 0 := by |
simp [eventuallyLE_antisymm_iff, ae_le_set, diff_diff_right, diff_diff,
diff_eq_empty.2 Set.subset_union_right]
|
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 172 | 173 | theorem memℒp_zero_iff_aestronglyMeasurable {f : α → E} :
Memℒp f 0 μ ↔ AEStronglyMeasurable f μ := by | simp [Memℒp, snorm_exponent_zero]
|
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Joachim Breitner
-/
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.GroupTh... | Mathlib/GroupTheory/CoprodI.lean | 828 | 837 | theorem mulHead_prod {i j : ι} (w : NeWord M i j) (x : M i) (hnotone : x * w.head ≠ 1) :
(mulHead w x hnotone).prod = of x * w.prod := by |
unfold mulHead
induction' w with _ _ _ _ _ _ _ _ _ _ w_ih_w₁ w_ih_w₂
· simp [mulHead, replaceHead]
· specialize w_ih_w₁ _ hnotone
clear w_ih_w₂
simp? [replaceHead, ← mul_assoc] at * says
simp only [replaceHead, head, append_prod, ← mul_assoc] at *
congr 1
|
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov, Hunter Monroe
-/
import Mathlib.Combinatorics.SimpleGraph.Init
import Mathlib.Data.Rel
import Mathlib... | Mathlib/Combinatorics/SimpleGraph/Basic.lean | 739 | 743 | theorem incidenceSet_inter_incidenceSet_of_adj (h : G.Adj a b) :
G.incidenceSet a ∩ G.incidenceSet b = {s(a, b)} := by |
refine (G.incidenceSet_inter_incidenceSet_subset <| h.ne).antisymm ?_
rintro _ (rfl : _ = s(a, b))
exact ⟨G.mk'_mem_incidenceSet_left_iff.2 h, G.mk'_mem_incidenceSet_right_iff.2 h⟩
|
/-
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.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Anal... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 384 | 387 | theorem integral_sub {f g : ℂ → E} {c : ℂ} {R : ℝ} (hf : CircleIntegrable f c R)
(hg : CircleIntegrable g c R) :
(∮ z in C(c, R), f z - g z) = (∮ z in C(c, R), f z) - ∮ z in C(c, R), g z := by |
simp only [circleIntegral, smul_sub, intervalIntegral.integral_sub hf.out hg.out]
|
/-
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.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dea... | Mathlib/Data/Multiset/Bind.lean | 82 | 86 | theorem map_join (f : α → β) (S : Multiset (Multiset α)) :
map f (join S) = join (map (map f) S) := by |
induction S using Multiset.induction with
| empty => simp
| cons _ _ ih => simp [ih]
|
/-
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.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mat... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | 916 | 919 | theorem le_iff_mem_closure (x y : PrimeSpectrum R) :
x ≤ y ↔ y ∈ closure ({x} : Set (PrimeSpectrum R)) := by |
rw [← asIdeal_le_asIdeal, ← zeroLocus_vanishingIdeal_eq_closure, mem_zeroLocus,
vanishingIdeal_singleton, SetLike.coe_subset_coe]
|
/-
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 | 72 | 73 | theorem logb_mul (hx : x ≠ 0) (hy : y ≠ 0) : logb b (x * y) = logb b x + logb b y := by |
simp_rw [logb, log_mul hx hy, add_div]
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.MeasureTheory.Integral.FundThmCalculus
#align_import analysis.special_functions.non_integrable fr... | Mathlib/Analysis/SpecialFunctions/NonIntegrable.lean | 177 | 189 | theorem not_intervalIntegrable_of_sub_inv_isBigO_punctured {f : ℝ → F} {a b c : ℝ}
(hf : (fun x => (x - c)⁻¹) =O[𝓝[≠] c] f) (hne : a ≠ b) (hc : c ∈ [[a, b]]) :
¬IntervalIntegrable f volume a b := by |
have A : ∀ᶠ x in 𝓝[≠] c, HasDerivAt (fun x => Real.log (x - c)) (x - c)⁻¹ x := by
filter_upwards [self_mem_nhdsWithin] with x hx
simpa using ((hasDerivAt_id x).sub_const c).log (sub_ne_zero.2 hx)
have B : Tendsto (fun x => ‖Real.log (x - c)‖) (𝓝[≠] c) atTop := by
refine tendsto_abs_atBot_atTop.comp (... |
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.Rat.Basic
import Batteries.Tactic.SeqFocus
/-! # Additional lemmas about the Rational Numbers -/
namespace Rat
theorem ext : {p q : Rat} → p.... | .lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean | 259 | 262 | theorem divInt_sub_divInt (n₁ n₂ : Int) {d₁ d₂} (z₁ : d₁ ≠ 0) (z₂ : d₂ ≠ 0) :
n₁ /. d₁ - n₂ /. d₂ = (n₁ * d₂ - n₂ * d₁) /. (d₁ * d₂) := by |
simp only [Rat.sub_eq_add_neg, neg_divInt,
divInt_add_divInt _ _ z₁ z₂, Int.neg_mul, Int.sub_eq_add_neg]
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.C... | Mathlib/FieldTheory/RatFunc/Basic.lean | 625 | 632 | theorem map_apply_div_ne_zero {R F : Type*} [CommRing R] [IsDomain R]
[FunLike F K[X] R[X]] [MonoidHomClass F K[X] R[X]]
(φ : F) (hφ : K[X]⁰ ≤ R[X]⁰.comap φ) (p q : K[X]) (hq : q ≠ 0) :
map φ hφ (algebraMap _ _ p / algebraMap _ _ q) =
algebraMap _ _ (φ p) / algebraMap _ _ (φ q) := by |
have hq' : φ q ≠ 0 := nonZeroDivisors.ne_zero (hφ (mem_nonZeroDivisors_iff_ne_zero.mpr hq))
simp only [← mk_eq_div, mk_eq_localization_mk _ hq, map_apply_ofFractionRing_mk,
mk_eq_localization_mk _ hq']
|
/-
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.Algebra.Group.Commute.Hom
import Mathlib.Data.Fintype.Card
#align_import data.finset.noncomm_prod f... | Mathlib/Data/Finset/NoncommProd.lean | 359 | 362 | theorem noncommProd_map [MonoidHomClass F β γ] (s : Finset α) (f : α → β) (comm) (g : F) :
g (s.noncommProd f comm) =
s.noncommProd (fun i => g (f i)) fun x hx y hy _ => (comm.of_refl hx hy).map g := by |
simp [noncommProd, Multiset.noncommProd_map]
|
/-
Copyright (c) 2020 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.Algebra.Lie.Abelian
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.Algebra.Lie.... | Mathlib/Algebra/Lie/Classical.lean | 264 | 266 | theorem s_as_blocks : S l R = Matrix.fromBlocks 1 0 0 (-1) := by |
rw [← Matrix.diagonal_one, Matrix.diagonal_neg, Matrix.fromBlocks_diagonal]
rfl
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.Analysis.NormedSpace.IndicatorFunction
#align_import measure_theory.integral.integrable_on... | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 373 | 379 | theorem integrableOn_Lp_of_measure_ne_top {E} [NormedAddCommGroup E] {p : ℝ≥0∞} {s : Set α}
(f : Lp E p μ) (hp : 1 ≤ p) (hμs : μ s ≠ ∞) : IntegrableOn f s μ := by |
refine memℒp_one_iff_integrable.mp ?_
have hμ_restrict_univ : (μ.restrict s) Set.univ < ∞ := by
simpa only [Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply, lt_top_iff_ne_top]
haveI hμ_finite : IsFiniteMeasure (μ.restrict s) := ⟨hμ_restrict_univ⟩
exact ((Lp.memℒp _).restrict s).memℒp_of_exponent... |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Cont... | Mathlib/Data/Real/Cardinality.lean | 82 | 83 | theorem cantorFunctionAux_zero (f : ℕ → Bool) : cantorFunctionAux c f 0 = cond (f 0) 1 0 := by |
cases h : f 0 <;> simp [h]
|
/-
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.Data.Finsupp.Encodable
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Span
import Mathlib.Data.Set.Countable
#align_import linear_algebr... | Mathlib/LinearAlgebra/Finsupp.lean | 692 | 694 | theorem apply_total (f : M →ₗ[R] M') (v) (l : α →₀ R) :
f (Finsupp.total α M R v l) = Finsupp.total α M' R (f ∘ v) l := by |
apply Finsupp.induction_linear l <;> simp (config := { contextual := true })
|
/-
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.Integrals
import Mathlib.MeasureTheory.Integral.PeakFunction
#align_import analysis.special_functions.trigonometric.euler_si... | Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean | 151 | 161 | theorem integral_cos_mul_cos_pow (hn : 2 ≤ n) (hz : z ≠ 0) :
(((1 : ℂ) - (4 : ℂ) * z ^ 2 / (n : ℂ) ^ 2) *
∫ x in (0 : ℝ)..π / 2, Complex.cos (2 * z * x) * (cos x : ℂ) ^ n) =
(n - 1 : ℂ) / n *
∫ x in (0 : ℝ)..π / 2, Complex.cos (2 * z * x) * (cos x : ℂ) ^ (n - 2) := by |
have nne : (n : ℂ) ≠ 0 := by
contrapose! hn; rw [Nat.cast_eq_zero] at hn; rw [hn]; exact zero_lt_two
have := integral_cos_mul_cos_pow_aux hn hz
rw [integral_sin_mul_sin_mul_cos_pow_eq hn hz, sub_eq_neg_add, mul_add, ← sub_eq_iff_eq_add]
at this
convert congr_arg (fun u : ℂ => -u * (2 * z) ^ 2 / n ^ 2) ... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingul... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,479 | 1,482 | theorem MeasurePreserving.set_lintegral_comp_preimage_emb {mb : MeasurableSpace β} {ν : Measure β}
{g : α → β} (hg : MeasurePreserving g μ ν) (hge : MeasurableEmbedding g) (f : β → ℝ≥0∞)
(s : Set β) : ∫⁻ a in g ⁻¹' s, f (g a) ∂μ = ∫⁻ b in s, f b ∂ν := by |
rw [← hg.map_eq, hge.restrict_map, hge.lintegral_map]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique
import Mathlib.MeasureTheory.Function.L2Space
#a... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean | 282 | 300 | theorem condexpL2_comp_continuousLinearMap (hm : m ≤ m0) (T : E' →L[ℝ] E'') (f : α →₂[μ] E') :
(condexpL2 E'' 𝕜' hm (T.compLp f) : α →₂[μ] E'') =ᵐ[μ]
T.compLp (condexpL2 E' 𝕜 hm f : α →₂[μ] E') := by |
refine Lp.ae_eq_of_forall_setIntegral_eq' 𝕜' hm _ _ two_ne_zero ENNReal.coe_ne_top
(fun s _ hμs => integrableOn_condexpL2_of_measure_ne_top hm hμs.ne _) (fun s _ hμs =>
integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs.ne) ?_ ?_ ?_
· intro s hs hμs
rw [T.setIntegral_compLp _ (hm s... |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.Normed.Order.Lattice
import Mathlib.MeasureTheory.Function.LpSpace
#align_import measure_theory.function.lp_order from "leanprover-community/math... | Mathlib/MeasureTheory/Function/LpOrder.lean | 45 | 50 | theorem coeFn_nonneg (f : Lp E p μ) : 0 ≤ᵐ[μ] f ↔ 0 ≤ f := by |
rw [← coeFn_le]
have h0 := Lp.coeFn_zero E p μ
constructor <;> intro h <;> filter_upwards [h, h0] with _ _ h2
· rwa [h2]
· rwa [← h2]
|
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Idempotents.Basic
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Equivalence
#align_import category_theo... | Mathlib/CategoryTheory/Idempotents/Karoubi.lean | 85 | 85 | theorem p_comp {P Q : Karoubi C} (f : Hom P Q) : P.p ≫ f.f = f.f := by | rw [f.comm, ← assoc, P.idem]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanp... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 67 | 68 | theorem prod_val [CommMonoid α] (s : Finset α) : s.1.prod = s.prod id := by |
rw [Finset.prod, Multiset.map_id]
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.GeneralLinearGroup
#align_import linear_al... | Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean | 649 | 650 | theorem homothety_neg_one_apply (c p : P₁) : homothety c (-1 : R') p = pointReflection R' c p := by |
simp [homothety_apply, pointReflection_apply]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker
-/
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.Algebra.Gr... | Mathlib/Algebra/Associated.lean | 294 | 302 | theorem irreducible_mul_units (a : αˣ) (b : α) : Irreducible (b * ↑a) ↔ Irreducible b := by |
simp only [irreducible_iff, Units.isUnit_mul_units, and_congr_right_iff]
refine fun _ => ⟨fun h A B HAB => ?_, fun h A B HAB => ?_⟩
· rw [← Units.isUnit_mul_units B a]
apply h
rw [← mul_assoc, ← HAB]
· rw [← Units.isUnit_mul_units B a⁻¹]
apply h
rw [← mul_assoc, ← HAB, Units.mul_inv_cancel_righ... |
/-
Copyright (c) 2022 Pim Otte. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Pim Otte
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Factorial.BigOperators
import Mathlib.Data.Fin.VecNotation
import ... | Mathlib/Data/Nat/Choose/Multinomial.lean | 177 | 187 | theorem multinomial_update (a : α) (f : α →₀ ℕ) :
f.multinomial = (f.sum fun _ => id).choose (f a) * (f.update a 0).multinomial := by |
simp only [multinomial_eq]
classical
by_cases h : a ∈ f.support
· rw [← Finset.insert_erase h, Nat.multinomial_insert (Finset.not_mem_erase a _),
Finset.add_sum_erase _ f h, support_update_zero]
congr 1
exact Nat.multinomial_congr fun _ h ↦ (Function.update_noteq (mem_erase.1 h).1 0 f).... |
/-
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.Computability.PartrecCode
import Mathlib.Data.Set.Subsingleton
#align_import computability.halting from "leanprover-community/mathlib"@"a50170a88a4757... | Mathlib/Computability/Halting.lean | 202 | 216 | theorem rice (C : Set (ℕ →. ℕ)) (h : ComputablePred fun c => eval c ∈ C) {f g} (hf : Nat.Partrec f)
(hg : Nat.Partrec g) (fC : f ∈ C) : g ∈ C := by |
cases' h with _ h
obtain ⟨c, e⟩ :=
fixed_point₂
(Partrec.cond (h.comp fst) ((Partrec.nat_iff.2 hg).comp snd).to₂
((Partrec.nat_iff.2 hf).comp snd).to₂).to₂
simp only [Bool.cond_decide] at e
by_cases H : eval c ∈ C
· simp only [H, if_true] at e
change (fun b => g b) ∈ C
rwa [← e]
... |
/-
Copyright (c) 2022 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Order.Basic
import Mathlib.Tactic.NoncommRing
#align_import... | Mathlib/Analysis/NormedSpace/MStructure.lean | 264 | 265 | theorem mul_compl_self {P : { P : M // IsLprojection X P }} : (↑P : M) * ↑Pᶜ = 0 := by |
rw [coe_compl, mul_sub, mul_one, P.prop.proj.eq, sub_self]
|
/-
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.Option.NAry
import Mathlib.Data.Seq.Computation
#align_import data.seq.seq from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b34... | Mathlib/Data/Seq/Seq.lean | 811 | 812 | theorem ofList_append (l l' : List α) : ofList (l ++ l') = append (ofList l) (ofList l') := by |
induction l <;> simp [*]
|
/-
Copyright (c) 2020 Google LLC. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Wong
-/
import Mathlib.Data.List.Basic
#align_import data.list.palindrome from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
/-!
# Palindromes
This mod... | Mathlib/Data/List/Palindrome.lean | 50 | 52 | theorem reverse_eq {l : List α} (p : Palindrome l) : reverse l = l := by |
induction p <;> try (exact rfl)
simpa
|
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Order.ToIntervalMod
import Mathlib.Algebra.Ring.AddAut
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.Divisible
import Mathlib.Topology.C... | Mathlib/Topology/Instances/AddCircle.lean | 435 | 441 | theorem addOrderOf_div_of_gcd_eq_one' {m : ℤ} {n : ℕ} (hn : 0 < n) (h : m.natAbs.gcd n = 1) :
addOrderOf (↑(↑m / ↑n * p) : AddCircle p) = n := by |
induction m
· simp only [Int.ofNat_eq_coe, Int.cast_natCast, Int.natAbs_ofNat] at h ⊢
exact addOrderOf_div_of_gcd_eq_one hn h
· simp only [Int.cast_negSucc, neg_div, neg_mul, coe_neg, addOrderOf_neg]
exact addOrderOf_div_of_gcd_eq_one hn h
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Data.List.Prime
#align_import data.polynomial.splits from "leanpro... | Mathlib/Algebra/Polynomial/Splits.lean | 246 | 249 | theorem splits_of_splits_of_dvd {f g : K[X]} (hf0 : f ≠ 0) (hf : Splits i f) (hgf : g ∣ f) :
Splits i g := by |
obtain ⟨f, rfl⟩ := hgf
exact (splits_of_splits_mul i hf0 hf).1
|
/-
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
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.Tactic.SuppressCo... | Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 1,388 | 1,390 | theorem rTensor_pow (f : M →ₗ[R] M) (n : ℕ) : f.rTensor N ^ n = (f ^ n).rTensor N := by |
have h := TensorProduct.map_pow f (id : N →ₗ[R] N) n
rwa [id_pow] at h
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Shing Tak Lam, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathli... | Mathlib/Data/Nat/Digits.lean | 203 | 204 | theorem ofDigits_one_cons {α : Type*} [Semiring α] (h : ℕ) (L : List ℕ) :
ofDigits (1 : α) (h :: L) = h + ofDigits 1 L := by | simp [ofDigits]
|
/-
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 | 674 | 676 | theorem preimage_mul_const_Ico_of_neg (a b : α) {c : α} (h : c < 0) :
(fun x => x * c) ⁻¹' Ico a b = Ioc (b / c) (a / c) := by |
simp [← Ici_inter_Iio, ← Ioi_inter_Iic, h, inter_comm]
|
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne, Adam Topaz
-/
import Mathlib.Data.Setoid.Partition
import Mathlib.Topology.Separation
import Mathlib.Topology.LocallyConstant.Basic
#align_import topology.discrete_quotient f... | Mathlib/Topology/DiscreteQuotient.lean | 326 | 326 | theorem map_id : map _ (leComap_id A) = id := by | ext ⟨⟩; rfl
|
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Scott Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Limits.IsLimit
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory... | Mathlib/CategoryTheory/Limits/HasLimits.lean | 474 | 481 | theorem limit.pre_post {D : Type u'} [Category.{v'} D] (E : K ⥤ J) (F : J ⥤ C) (G : C ⥤ D)
[HasLimit F] [HasLimit (E ⋙ F)] [HasLimit (F ⋙ G)]
[h : HasLimit ((E ⋙ F) ⋙ G)] :-- G (limit F) ⟶ G (limit (E ⋙ F)) ⟶ limit ((E ⋙ F) ⋙ G) vs
-- G (limit F) ⟶ limit F ⋙ G ⟶ limit (E ⋙ (F ⋙ G)) or
haveI : Ha... |
haveI : HasLimit (E ⋙ F ⋙ G) := h
ext; erw [assoc, limit.post_π, ← G.map_comp, limit.pre_π, assoc, limit.pre_π, limit.post_π]
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.ContDiff.RCLike
import Mathlib.MeasureTheory.Measure.Hausdorff
#align_import topology.metric_space.hausdorff_dimension from "leanp... | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | 622 | 626 | theorem ContDiff.dimH_range_le {f : E → F} (h : ContDiff ℝ 1 f) : dimH (range f) ≤ finrank ℝ E :=
calc
dimH (range f) = dimH (f '' univ) := by | rw [image_univ]
_ ≤ dimH (univ : Set E) := h.contDiffOn.dimH_image_le convex_univ Subset.rfl
_ = finrank ℝ E := Real.dimH_univ_eq_finrank E
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Analysis.Calculus.FDe... | Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean | 1,178 | 1,182 | theorem extend_preimage_mem_nhds {x : M} (h : x ∈ f.source) (ht : t ∈ 𝓝 x) :
(f.extend I).symm ⁻¹' t ∈ 𝓝 (f.extend I x) := by |
apply (continuousAt_extend_symm f I h).preimage_mem_nhds
rwa [(f.extend I).left_inv]
rwa [f.extend_source]
|
/-
Copyright (c) 2021 Vladimir Goryachev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Goryachev, Kyle Miller, Scott Morrison, Eric Rodriguez
-/
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.Ring
#align_import data.nat.count fr... | Mathlib/Data/Nat/Count.lean | 74 | 83 | theorem count_add (a b : ℕ) : count p (a + b) = count p a + count (fun k ↦ p (a + k)) b := by |
have : Disjoint ((range a).filter p) (((range b).map <| addLeftEmbedding a).filter p) := by
apply disjoint_filter_filter
rw [Finset.disjoint_left]
simp_rw [mem_map, mem_range, addLeftEmbedding_apply]
rintro x hx ⟨c, _, rfl⟩
exact (self_le_add_right _ _).not_lt hx
simp_rw [count_eq_card_filter_r... |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Da... | Mathlib/GroupTheory/OrderOfElement.lean | 460 | 462 | theorem orderOf_mul_dvd_lcm : orderOf (x * y) ∣ Nat.lcm (orderOf x) (orderOf y) := by |
rw [orderOf, ← comp_mul_left]
exact Function.Commute.minimalPeriod_of_comp_dvd_lcm h.function_commute_mul_left
|
/-
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.SpecificLimits.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.specific_limits.floor_pow from "leanprove... | Mathlib/Analysis/SpecificLimits/FloorPow.lean | 28 | 182 | theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (l : ℝ)
(hmono : Monotone u)
(hlim : ∀ a : ℝ, 1 < a → ∃ c : ℕ → ℕ, (∀ᶠ n in atTop, (c (n + 1) : ℝ) ≤ a * c n) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / c n) atTop (𝓝 l)) :
Tendsto (fun n => u n / n) atTop (𝓝 l) := b... |
/- To check the result up to some `ε > 0`, we use a sequence `c` for which the ratio
`c (N+1) / c N` is bounded by `1 + ε`. Sandwiching a given `n` between two consecutive values of
`c`, say `c N` and `c (N+1)`, one can then bound `u n / n` from above by `u (c N) / c (N - 1)`
and from below by `u (c (N -... |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Algebra.Order.Group.OrderIso
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Order.UpperLower.Basic
#al... | Mathlib/Algebra/Order/UpperLower.lean | 293 | 295 | theorem upperClosure_mul : ↑(upperClosure s) * t = upperClosure (s * t) := by |
simp_rw [mul_comm _ t]
exact mul_upperClosure _ _
|
/-
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.Data.Int.Interval
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Nat.Count
import Mathlib.Data.Rat.Floor
import Mathlib.Order.Interval.Finset.Nat
/-!
# Cou... | Mathlib/Data/Int/CardIntervalMod.lean | 99 | 102 | theorem Ico_filter_modEq_card (v : ℕ) : ((Ico a b).filter (· ≡ v [MOD r])).card =
max (⌈(b - v) / (r : ℚ)⌉ - ⌈(a - v) / (r : ℚ)⌉) 0 := by |
simp_rw [← Ico_filter_modEq_cast _ _ ▸ card_map _,
Int.Ico_filter_modEq_card _ _ (cast_lt.mpr hr), Int.cast_natCast]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Ordinal.FixedPoint
#align_import set_theory.cardinal... | Mathlib/SetTheory/Cardinal/Cofinality.lean | 614 | 644 | theorem exists_fundamental_sequence (a : Ordinal.{u}) :
∃ f, IsFundamentalSequence a a.cof.ord f := by |
suffices h : ∃ o f, IsFundamentalSequence a o f by
rcases h with ⟨o, f, hf⟩
exact ⟨_, hf.ord_cof⟩
rcases exists_lsub_cof a with ⟨ι, f, hf, hι⟩
rcases ord_eq ι with ⟨r, wo, hr⟩
haveI := wo
let r' := Subrel r { i | ∀ j, r j i → f j < f i }
let hrr' : r' ↪r r := Subrel.relEmbedding _ _
haveI := hrr'... |
/-
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.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.... | Mathlib/Topology/Compactness/Compact.lean | 1,107 | 1,113 | theorem Filter.coprod_cocompact :
(Filter.cocompact X).coprod (Filter.cocompact Y) = Filter.cocompact (X × Y) := by |
apply le_antisymm
· exact sup_le (comap_cocompact_le continuous_fst) (comap_cocompact_le continuous_snd)
· refine (hasBasis_cocompact.coprod hasBasis_cocompact).ge_iff.2 fun K hK ↦ ?_
rw [← univ_prod, ← prod_univ, ← compl_prod_eq_union]
exact (hK.1.prod hK.2).compl_mem_cocompact
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Comma.Over
import Mathlib.CategoryTheory.DiscreteCategory
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryThe... | Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | 878 | 881 | theorem coprod.map_desc {S T U V W : C} [HasBinaryCoproduct U W] [HasBinaryCoproduct T V]
(f : U ⟶ S) (g : W ⟶ S) (h : T ⟶ U) (k : V ⟶ W) :
coprod.map h k ≫ coprod.desc f g = coprod.desc (h ≫ f) (k ≫ g) := by |
ext <;> simp
|
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin
-/
import Mathlib.Data.Matrix.Basic
#align_import data.matrix.block from "leanprover-community/mathlib"@"c060baa79af5ca... | Mathlib/Data/Matrix/Block.lean | 505 | 509 | theorem blockDiagonal_smul {R : Type*} [Monoid R] [AddMonoid α] [DistribMulAction R α] (x : R)
(M : o → Matrix m n α) : blockDiagonal (x • M) = x • blockDiagonal M := by |
ext
simp only [blockDiagonal_apply, Pi.smul_apply, smul_apply]
split_ifs <;> simp
|
/-
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.GroupTheory.QuotientGroup
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.class_group from "leanprover-community/mathlib"@"565eb991... | Mathlib/RingTheory/ClassGroup.lean | 347 | 361 | theorem ClassGroup.mk_eq_one_iff {I : (FractionalIdeal R⁰ K)ˣ} :
ClassGroup.mk I = 1 ↔ (I : Submodule R K).IsPrincipal := by |
rw [← (ClassGroup.equiv K).injective.eq_iff]
simp only [equiv_mk, canonicalEquiv_self, RingEquiv.coe_mulEquiv_refl, QuotientGroup.mk'_apply,
_root_.map_one, QuotientGroup.eq_one_iff, MonoidHom.mem_range, ext_iff, coe_toPrincipalIdeal,
coe_mapEquiv, MulEquiv.refl_apply]
refine ⟨fun ⟨x, hx⟩ => ⟨⟨x, by rw [... |
/-
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 | 84 | 85 | theorem i_mul_k : q.i * q.k = c₁ • q.j := by |
rw [← i_mul_j, ← mul_assoc, i_mul_i, smul_mul_assoc, one_mul]
|
/-
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, Minchao Wu, Mario Carneiro
-/
import Mathlib.Algebra.Group.Embedding
import Mathlib.Data.Fin.Basic
import Mathlib.Data.Finset.Union
#align_imp... | Mathlib/Data/Finset/Image.lean | 374 | 377 | theorem mem_image_const : c ∈ s.image (const α b) ↔ s.Nonempty ∧ b = c := by |
rw [mem_image]
simp only [exists_prop, const_apply, exists_and_right]
rfl
|
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.Algebra.Field.Opposite
import Mathlib.Algebra.Group.Subgroup.ZPowers
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Rin... | Mathlib/Algebra/Periodic.lean | 77 | 82 | theorem _root_.List.periodic_prod [Add α] [Monoid β] (l : List (α → β))
(hl : ∀ f ∈ l, Periodic f c) : Periodic l.prod c := by |
induction' l with g l ih hl
· simp
· rw [List.forall_mem_cons] at hl
simpa only [List.prod_cons] using hl.1.mul (ih hl.2)
|
/-
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.Basis
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
#align_import linear_algebra.affine_space.matrix from "leanprover-com... | Mathlib/LinearAlgebra/AffineSpace/Matrix.lean | 48 | 50 | theorem toMatrix_self [DecidableEq ι] : b.toMatrix b = (1 : Matrix ι ι k) := by |
ext i j
rw [toMatrix_apply, coord_apply, Matrix.one_eq_pi_single, Pi.single_apply]
|
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Algebra.Category.ModuleCat.Free
import Mathlib.Topology.Category.Profinite.CofilteredLimit
import Mathlib.Topology.Category.Profinite.Product
impor... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 1,550 | 1,554 | theorem max_eq_eval_unapply :
(Linear_CC' C hsC ho) ∘ (fun (l : MaxProducts C ho) ↦ Products.eval C l.val) =
(fun l ↦ l.val.Tail.eval (C' C ho)) := by |
ext1 l
exact max_eq_eval _ _ _ _
|
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Fintype.Card
#align_import data.multiset.fintype from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f... | Mathlib/Data/Multiset/Fintype.lean | 130 | 141 | theorem Multiset.toEnumFinset_subset_iff {m₁ m₂ : Multiset α} :
m₁.toEnumFinset ⊆ m₂.toEnumFinset ↔ m₁ ≤ m₂ := by |
refine ⟨fun h ↦ ?_, Multiset.toEnumFinset_mono⟩
rw [Multiset.le_iff_count]
intro x
by_cases hx : x ∈ m₁
· apply Nat.le_of_pred_lt
have : (x, m₁.count x - 1) ∈ m₁.toEnumFinset := by
rw [Multiset.mem_toEnumFinset]
exact Nat.pred_lt (ne_of_gt (Multiset.count_pos.mpr hx))
simpa only [Multiset... |
/-
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, Kyle Miller
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finite.Basic
import Mathlib.Data.Set.Functor
import Mathlib.Data.Set.Lattice
#align... | Mathlib/Data/Set/Finite.lean | 1,526 | 1,531 | theorem exists_lower_bound_image [Nonempty α] [LinearOrder β] (s : Set α) (f : α → β)
(h : s.Finite) : ∃ a : α, ∀ b ∈ s, f a ≤ f b := by |
rcases s.eq_empty_or_nonempty with rfl | hs
· exact ‹Nonempty α›.elim fun a => ⟨a, fun _ => False.elim⟩
· rcases Set.exists_min_image s f h hs with ⟨x₀, _, hx₀⟩
exact ⟨x₀, fun x hx => hx₀ x hx⟩
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
import Mat... | Mathlib/NumberTheory/ArithmeticFunction.lean | 216 | 219 | theorem intCoe_one [AddGroupWithOne R] : ((1 : ArithmeticFunction ℤ) :
ArithmeticFunction R) = 1 := by |
ext n
simp [one_apply]
|
/-
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.AEval
#align_import data.polynomial.module from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
/-!
# P... | Mathlib/Algebra/Polynomial/Module/Basic.lean | 157 | 169 | theorem smul_single_apply (i : ℕ) (f : R[X]) (m : M) (n : ℕ) :
(f • single R i m) n = ite (i ≤ n) (f.coeff (n - i) • m) 0 := by |
induction' f using Polynomial.induction_on' with p q hp hq
· rw [add_smul, Finsupp.add_apply, hp, hq, coeff_add, add_smul]
split_ifs
exacts [rfl, zero_add 0]
· rw [monomial_smul_single, single_apply, coeff_monomial, ite_smul, zero_smul]
by_cases h : i ≤ n
· simp_rw [eq_tsub_iff_add_eq_of_le h, if... |
/-
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.Algebra.Hom
import Mathlib.RingTheory.Ideal.Quotient
#align_import algebra.ring_quot from "leanprover-community/mathlib"@"e5820f6c8fcf1b75bcd7... | Mathlib/Algebra/RingQuot.lean | 491 | 495 | theorem eq_lift_comp_mkRingHom {r : R → R → Prop} (f : RingQuot r →+* T) :
f = lift ⟨f.comp (mkRingHom r), fun x y h ↦ congr_arg f (mkRingHom_rel h)⟩ := by |
conv_lhs => rw [← lift.apply_symm_apply f]
rw [lift_def]
rfl
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Analysis.Filter
import Mathlib.Topology.Bases
import Mathlib.Topology.LocallyFinite
#align_import data.analysis.topology from "leanprover-communi... | Mathlib/Data/Analysis/Topology.lean | 79 | 80 | theorem ofEquiv_val (E : σ ≃ τ) (F : Ctop α σ) (a : τ) : F.ofEquiv E a = F (E.symm a) := by |
cases F; rfl
|
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community... | Mathlib/Logic/Equiv/Fintype.lean | 72 | 75 | theorem Equiv.Perm.viaFintypeEmbedding_apply_image (a : α) :
e.viaFintypeEmbedding f (f a) = f (e a) := by |
rw [Equiv.Perm.viaFintypeEmbedding]
convert Equiv.Perm.extendDomain_apply_image e (Function.Embedding.toEquivRange f) a
|
/-
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, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde9... | Mathlib/MeasureTheory/Integral/Average.lean | 361 | 362 | theorem average_congr {f g : α → E} (h : f =ᵐ[μ] g) : ⨍ x, f x ∂μ = ⨍ x, g x ∂μ := by |
simp only [average_eq, integral_congr_ae h]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.AdjoinRoot
#align_import ring_theory.adjoin.field from "leanpro... | Mathlib/RingTheory/Adjoin/Field.lean | 56 | 81 | theorem Polynomial.lift_of_splits {F K L : Type*} [Field F] [Field K] [Field L] [Algebra F K]
[Algebra F L] (s : Finset K) : (∀ x ∈ s, IsIntegral F x ∧
Splits (algebraMap F L) (minpoly F x)) → Nonempty (Algebra.adjoin F (s : Set K) →ₐ[F] L) := by |
classical
refine Finset.induction_on s (fun _ ↦ ?_) fun a s _ ih H ↦ ?_
· rw [coe_empty, Algebra.adjoin_empty]
exact ⟨(Algebra.ofId F L).comp (Algebra.botEquiv F K)⟩
rw [forall_mem_insert] at H
rcases H with ⟨⟨H1, H2⟩, H3⟩
cases' ih H3 with f
choose H3 _ using H3
rw [coe_insert, Set... |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mat... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 118 | 132 | theorem isBigO_iff' {g : α → E'''} :
f =O[l] g ↔ ∃ c > 0, ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ := by |
refine ⟨fun h => ?mp, fun h => ?mpr⟩
case mp =>
rw [isBigO_iff] at h
obtain ⟨c, hc⟩ := h
refine ⟨max c 1, zero_lt_one.trans_le (le_max_right _ _), ?_⟩
filter_upwards [hc] with x hx
apply hx.trans
gcongr
exact le_max_left _ _
case mpr =>
rw [isBigO_iff]
obtain ⟨c, ⟨_, hc⟩⟩ := h... |
/-
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.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set... | Mathlib/Analysis/NormedSpace/lpSpace.lean | 90 | 92 | theorem memℓp_infty_iff {f : ∀ i, E i} : Memℓp f ∞ ↔ BddAbove (Set.range fun i => ‖f i‖) := by |
dsimp [Memℓp]
rw [if_neg ENNReal.top_ne_zero, if_pos rfl]
|
/-
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.SpecialFunctions.Gamma.Basic
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.Convex.Complex
#align_import a... | Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean | 135 | 144 | theorem integrableOn_Ioi_exp_neg_mul_sq_iff {b : ℝ} :
IntegrableOn (fun x : ℝ => exp (-b * x ^ 2)) (Ioi 0) ↔ 0 < b := by |
refine ⟨fun h => ?_, fun h => (integrable_exp_neg_mul_sq h).integrableOn⟩
by_contra! hb
have : ∫⁻ _ : ℝ in Ioi 0, 1 ≤ ∫⁻ x : ℝ in Ioi 0, ‖exp (-b * x ^ 2)‖₊ := by
apply lintegral_mono (fun x ↦ _)
simp only [neg_mul, ENNReal.one_le_coe_iff, ← toNNReal_one, toNNReal_le_iff_le_coe,
Real.norm_of_nonneg... |
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Notation
import Mathlib.Init.Order.Defs
set_option autoImplicit true
structure UFModel (n) where
parent : Fin n → Fin n
rank : Nat ... | Mathlib/Data/UnionFind.lean | 91 | 101 | theorem push {arr : Array α} {n} {m : Fin n → β} (H : Agrees arr f m)
(k) (hk : k = n + 1) (x) (m' : Fin k → β)
(hm₁ : ∀ (i : Fin k) (h : i < n), m' i = m ⟨i, h⟩)
(hm₂ : ∀ (h : n < k), f x = m' ⟨n, h⟩) : Agrees (arr.push x) f m' := by |
cases H
have : k = (arr.push x).size := by simp [hk]
refine mk' this fun i h₁ h₂ ↦ ?_
simp [Array.get_push]; split <;> (rename_i h; simp at hm₁ ⊢)
· rw [← hm₁ ⟨i, h₂⟩]; assumption
· cases show i = arr.size by apply Nat.le_antisymm <;> simp_all [Nat.lt_succ]
rw [hm₂]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Prod
import Mathlib.Order.Cover
#align_import algebra.support from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62... | Mathlib/Algebra/Group/Support.lean | 93 | 95 | theorem mulSupport_update_one [DecidableEq α] (f : α → M) (x : α) :
mulSupport (update f x 1) = mulSupport f \ {x} := by |
ext a; rcases eq_or_ne a x with rfl | hne <;> simp [*]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Yury G. Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Iterate
import Math... | Mathlib/Analysis/SpecificLimits/Basic.lean | 552 | 560 | theorem Set.Countable.exists_pos_hasSum_le {ι : Type*} {s : Set ι} (hs : s.Countable) {ε : ℝ}
(hε : 0 < ε) : ∃ ε' : ι → ℝ, (∀ i, 0 < ε' i) ∧ ∃ c, HasSum (fun i : s ↦ ε' i) c ∧ c ≤ ε := by |
haveI := hs.toEncodable
rcases posSumOfEncodable hε s with ⟨f, hf0, ⟨c, hfc, hcε⟩⟩
refine ⟨fun i ↦ if h : i ∈ s then f ⟨i, h⟩ else 1, fun i ↦ ?_, ⟨c, ?_, hcε⟩⟩
· conv_rhs => simp
split_ifs
exacts [hf0 _, zero_lt_one]
· simpa only [Subtype.coe_prop, dif_pos, Subtype.coe_eta]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Regular.SMul
import Mathlib.Data.Finset.Preimag... | Mathlib/Data/Finsupp/Basic.lean | 68 | 74 | theorem mk_mem_graph_iff {a : α} {m : M} {f : α →₀ M} : (a, m) ∈ f.graph ↔ f a = m ∧ m ≠ 0 := by |
simp_rw [graph, mem_map, mem_support_iff]
constructor
· rintro ⟨b, ha, rfl, -⟩
exact ⟨rfl, ha⟩
· rintro ⟨rfl, ha⟩
exact ⟨a, ha, rfl⟩
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.rig... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 284 | 291 | theorem oangle_sub_right_eq_arcsin_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle y (y - x) = Real.arcsin (‖x‖ / ‖y - x‖) := by |
have hs : (o.oangle y (y - x)).sign = 1 := by
rw [oangle_sign_sub_right_swap, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs,
InnerProductGeometry.angle_sub_eq_arcsin_of_inner_eq_zero
(o.inner_rev_eq_zero_of_oangle_eq_pi_div_two h)
(Or.inl (o.right_ne_zero_of_oangle_... |
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.Algebra.FreeAlgebra
import Mathlib.Algebra.RingQuot
import Mathlib.Algebra.TrivSqZeroExt
import Mathlib.Algebra.Algebra.Operations
import Mathlib.LinearAlgebra... | Mathlib/LinearAlgebra/TensorAlgebra/Basic.lean | 352 | 353 | theorem toTensor_ι (m : M) : FreeAlgebra.toTensor (FreeAlgebra.ι R m) = TensorAlgebra.ι R m := by |
simp [toTensor]
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Oliver Nash
-/
import Mathlib.Data.Finset.Card
#align_import data.finset.prod from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267... | Mathlib/Data/Finset/Prod.lean | 226 | 229 | theorem singleton_product {a : α} :
({a} : Finset α) ×ˢ t = t.map ⟨Prod.mk a, Prod.mk.inj_left _⟩ := by |
ext ⟨x, y⟩
simp [and_left_comm, eq_comm]
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
#align_import topology.metric_space.hausdorff_distance from "lea... | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 251 | 255 | theorem _root_.IsCompact.exists_infEdist_eq_edist (hs : IsCompact s) (hne : s.Nonempty) (x : α) :
∃ y ∈ s, infEdist x s = edist x y := by |
have A : Continuous fun y => edist x y := continuous_const.edist continuous_id
obtain ⟨y, ys, hy⟩ := hs.exists_isMinOn hne A.continuousOn
exact ⟨y, ys, le_antisymm (infEdist_le_edist_of_mem ys) (by rwa [le_infEdist])⟩
|
/-
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.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.GroupTh... | Mathlib/RingTheory/Localization/Basic.lean | 1,137 | 1,138 | theorem algEquiv_symm_mk (x : R) (y : M) : (algEquiv M S).symm (mk' S x y) = mk x y := by |
rw [mk_eq_mk', algEquiv_symm_mk']
|
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro, Anne Baanen,
Frédéric Dupuis, Heather Macbeth, Antoine Chambert-Loir
-/
import Mathlib.Data.Set.Pointwise.SMul
impor... | Mathlib/GroupTheory/GroupAction/Pointwise.lean | 64 | 67 | theorem smul_preimage_set_leₛₗ :
c • h ⁻¹' t ⊆ h ⁻¹' (σ c • t) := by |
rintro x ⟨y, hy, rfl⟩
exact ⟨h y, hy, by rw [map_smulₛₗ]⟩
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.BoxIntegral.Partition.Basic
#align_import analysis.box_integral.partition.split from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf973... | Mathlib/Analysis/BoxIntegral/Partition/Split.lean | 188 | 189 | theorem iUnion_split (I : Box ι) (i : ι) (x : ℝ) : (split I i x).iUnion = I := by |
simp [split, ← inter_union_distrib_left, ← setOf_or, le_or_lt]
|
/-
Copyright (c) 2023 Mark Andrew Gerads. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mark Andrew Gerads, Junyan Xu, Eric Wieser
-/
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Tactic.Ring
#align_import data.nat.hyperoperation from "leanprover-community/mat... | Mathlib/Data/Nat/Hyperoperation.lean | 60 | 65 | theorem hyperoperation_one : hyperoperation 1 = (· + ·) := by |
ext m k
induction' k with bn bih
· rw [Nat.add_zero m, hyperoperation]
· rw [hyperoperation_recursion, bih, hyperoperation_zero]
exact Nat.add_assoc m bn 1
|
/-
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 | 452 | 457 | theorem AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_X_some
(hx : AlgebraicIndependent R x) (i) :
hx.mvPolynomialOptionEquivPolynomialAdjoin (X (some i)) =
Polynomial.C (hx.aevalEquiv (X i)) := by |
rw [AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_apply, aeval_X, Option.elim,
Polynomial.map_C, RingHom.coe_coe]
|
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Group.Subsemigroup.Basic
#align_import group_theory.subsemigroup.membership from "leanprover-community/mathlib"@"6cb77a8eaff0ddd100e87b1591c6d3a... | Mathlib/Algebra/Group/Subsemigroup/Membership.lean | 89 | 91 | theorem mem_sup_right {S T : Subsemigroup M} : ∀ {x : M}, x ∈ T → x ∈ S ⊔ T := by |
have : T ≤ S ⊔ T := le_sup_right
tauto
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic fro... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,087 | 1,090 | theorem mod_mod_of_dvd (a : Ordinal) {b c : Ordinal} (h : c ∣ b) : a % b % c = a % c := by |
nth_rw 2 [← div_add_mod a b]
rcases h with ⟨d, rfl⟩
rw [mul_assoc, mul_add_mod_self]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.constructions.borel_space.basic from "leanprover-community/... | Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | 395 | 399 | theorem Measurable.ennreal_tsum {ι} [Countable ι] {f : ι → α → ℝ≥0∞} (h : ∀ i, Measurable (f i)) :
Measurable fun x => ∑' i, f i x := by |
simp_rw [ENNReal.tsum_eq_iSup_sum]
apply measurable_iSup
exact fun s => s.measurable_sum fun i _ => h i
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
#align_import data.nat.part_enat from "l... | Mathlib/Data/Nat/PartENat.lean | 192 | 194 | theorem coe_add_get {x : ℕ} {y : PartENat} (h : ((x : PartENat) + y).Dom) :
get ((x : PartENat) + y) h = x + get y h.2 := by |
rfl
|
/-
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.Set.Image
import Mathlib.Data.Set.Lattice
#align_import data.set.sigma from "leanprover-community/mathlib"@"2258b40dacd2942571c8ce136215350c702dc78f"... | Mathlib/Data/Set/Sigma.lean | 43 | 50 | theorem image_sigmaMk_preimage_sigmaMap {β : ι' → Type*} {f : ι → ι'} (hf : Function.Injective f)
(g : ∀ i, α i → β (f i)) (i : ι) (s : Set (β (f i))) :
Sigma.mk i '' (g i ⁻¹' s) = Sigma.map f g ⁻¹' (Sigma.mk (f i) '' s) := by |
refine (image_sigmaMk_preimage_sigmaMap_subset f g i s).antisymm ?_
rintro ⟨j, x⟩ ⟨y, hys, hxy⟩
simp only [hf.eq_iff, Sigma.map, Sigma.ext_iff] at hxy
rcases hxy with ⟨rfl, hxy⟩; rw [heq_iff_eq] at hxy; subst y
exact ⟨x, hys, rfl⟩
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shape... | Mathlib/CategoryTheory/Limits/Shapes/Images.lean | 766 | 768 | theorem ImageMap.map_uniq {f g : Arrow C} [HasImage f.hom] [HasImage g.hom]
{sq : f ⟶ g} (F G : ImageMap sq) : F.map = G.map := by |
apply ImageMap.map_uniq_aux _ F.map_ι _ G.map_ι
|
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Basic
#align_import number_theory.padic... | Mathlib/NumberTheory/Padics/PadicNumbers.lean | 1,089 | 1,100 | theorem valuation_map_mul {x y : ℚ_[p]} (hx : x ≠ 0) (hy : y ≠ 0) :
valuation (x * y : ℚ_[p]) = valuation x + valuation y := by |
have h_norm : ‖x * y‖ = ‖x‖ * ‖y‖ := norm_mul x y
have hp_ne_one : (p : ℝ) ≠ 1 := by
rw [← Nat.cast_one, Ne, Nat.cast_inj]
exact Nat.Prime.ne_one hp.elim
have hp_pos : (0 : ℝ) < p := by
rw [← Nat.cast_zero, Nat.cast_lt]
exact Nat.Prime.pos hp.elim
rw [norm_eq_pow_val hx, norm_eq_pow_val hy, nor... |
/-
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 | 177 | 190 | theorem starConvex_iff_forall_ne_pos (hx : x ∈ s) :
StarConvex 𝕜 x s ↔
∀ ⦃y⦄, y ∈ s → x ≠ y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ s := by |
refine ⟨fun h y hy _ a b ha hb hab => h hy ha.le hb.le hab, ?_⟩
intro h y hy a b ha hb hab
obtain rfl | ha' := ha.eq_or_lt
· rw [zero_add] at hab
rwa [hab, zero_smul, one_smul, zero_add]
obtain rfl | hb' := hb.eq_or_lt
· rw [add_zero] at hab
rwa [hab, zero_smul, one_smul, add_zero]
obtain rfl | h... |
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