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) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
#align_import ring_theory.ideal.operations from "leanpro... | Mathlib/RingTheory/Ideal/Operations.lean | 283 | 289 | theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M)
(H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by |
suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by
rw [top_smul] at this
exact this (subset_span (Set.mem_singleton x))
rw [← hs, span_smul_span, span_le]
simpa using H
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
/-!
... | Mathlib/Data/ENNReal/Real.lean | 654 | 655 | theorem mul_iInf {ι} [Nonempty ι] {f : ι → ℝ≥0∞} {x : ℝ≥0∞} (h : x ≠ ∞) :
x * iInf f = ⨅ i, x * f i := by | simpa only [mul_comm] using iInf_mul h
|
/-
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 | 1,107 | 1,109 | theorem prod_rightUnitor_inv_naturality [HasBinaryProducts C] (f : X ⟶ Y) :
(prod.rightUnitor X).inv ≫ prod.map f (𝟙 _) = f ≫ (prod.rightUnitor Y).inv := by |
rw [Iso.inv_comp_eq, ← Category.assoc, Iso.eq_comp_inv, prod.rightUnitor_hom_naturality]
|
/-
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.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | Mathlib/CategoryTheory/Quotient.lean | 148 | 151 | theorem compClosure_eq_self [h : Congruence r] :
CompClosure r = r := by |
ext
simp only [compClosure_iff_self]
|
/-
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.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@... | Mathlib/Data/List/Intervals.lean | 220 | 224 | theorem filter_le_of_bot {n m : ℕ} (hnm : n < m) : ((Ico n m).filter fun x => x ≤ n) = [n] := by |
rw [← filter_lt_of_succ_bot hnm]
exact filter_congr' fun _ _ => by
rw [decide_eq_true_eq, decide_eq_true_eq]
exact Nat.lt_succ_iff.symm
|
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedd... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 312 | 314 | theorem convexBodySumFun_smul (c : ℝ) (x : E K) :
convexBodySumFun (c • x) = |c| * convexBodySumFun x := by |
simp_rw [convexBodySumFun, normAtPlace_smul, ← mul_assoc, mul_comm, Finset.mul_sum, mul_assoc]
|
/-
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.Preserves.Shapes.Zero
#align_import category_theory.limits.shapes.kernels from "leanprover-community/mathlib"@"95... | Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean | 381 | 383 | theorem kernelIsoOfEq_refl {h : f = f} : kernelIsoOfEq h = Iso.refl (kernel f) := by |
ext
simp [kernelIsoOfEq]
|
/-
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.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Star.Unitary
import Mathlib.Data.Nat.ModEq
import Mathlib.Numb... | Mathlib/NumberTheory/PellMatiyasevic.lean | 657 | 665 | theorem eq_of_xn_modEq_lem2 {n} (h : 2 * xn a1 n = xn a1 (n + 1)) : a = 2 ∧ n = 0 := by |
rw [xn_succ, mul_comm] at h
have : n = 0 :=
n.eq_zero_or_pos.resolve_right fun np =>
_root_.ne_of_lt
(lt_of_le_of_lt (Nat.mul_le_mul_left _ a1)
(Nat.lt_add_of_pos_right <| mul_pos (d_pos a1) (strictMono_y a1 np)))
h
cases this; simp at h; exact ⟨h.symm, rfl⟩
|
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.Algebraic... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 471 | 504 | theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ ⊥) :
I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by |
-- We'll show `1 ≤ J⁻¹ = (I * I⁻¹)⁻¹ ≤ 1`.
apply mul_inv_cancel_of_le_one hI0
by_cases hJ0 : I * (I : FractionalIdeal A⁰ K)⁻¹ = 0
· rw [hJ0, inv_zero']; exact zero_le _
intro x hx
-- In particular, we'll show all `x ∈ J⁻¹` are integral.
suffices x ∈ integralClosure A K by
rwa [IsIntegrallyClosed.inte... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#alig... | Mathlib/Algebra/Polynomial/Roots.lean | 76 | 79 | theorem card_roots' (p : R[X]) : Multiset.card p.roots ≤ natDegree p := by |
by_cases hp0 : p = 0
· simp [hp0]
exact WithBot.coe_le_coe.1 (le_trans (card_roots hp0) (le_of_eq <| degree_eq_natDegree hp0))
|
/-
Copyright (c) 2021 Martin Zinkevich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Martin Zinkevich, Rémy Degenne
-/
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
#align_import measure_theory.pi_system fro... | Mathlib/MeasureTheory/PiSystem.lean | 580 | 583 | theorem has_union {s₁ s₂ : Set α} (h₁ : d.Has s₁) (h₂ : d.Has s₂) (h : Disjoint s₁ s₂) :
d.Has (s₁ ∪ s₂) := by |
rw [union_eq_iUnion]
exact d.has_iUnion (pairwise_disjoint_on_bool.2 h) (Bool.forall_bool.2 ⟨h₂, h₁⟩)
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Finsupp.Multiset
import Math... | Mathlib/RingTheory/UniqueFactorizationDomain.lean | 1,530 | 1,533 | theorem factors_le {a b : Associates α} : a.factors ≤ b.factors ↔ a ≤ b := by |
refine ⟨fun h ↦ ?_, factors_mono⟩
have : a.factors.prod ≤ b.factors.prod := prod_mono h
rwa [factors_prod, factors_prod] at this
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Logic.Relation
import Mathlib.Data.Option.Basic
import Mathlib.Data.Seq.Seq
#align_import data.seq.wseq from "leanprover-community/mathlib"@"a7... | Mathlib/Data/Seq/WSeq.lean | 909 | 920 | theorem mem_rec_on {C : WSeq α → Prop} {a s} (M : a ∈ s) (h1 : ∀ b s', a = b ∨ C s' → C (cons b s'))
(h2 : ∀ s, C s → C (think s)) : C s := by |
apply Seq.mem_rec_on M
intro o s' h; cases' o with b
· apply h2
cases h
· contradiction
· assumption
· apply h1
apply Or.imp_left _ h
intro h
injection h
|
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.denoms_clearable from "leanprover-community/mathlib"@"... | Mathlib/Algebra/Polynomial/DenomsClearable.lean | 47 | 54 | theorem denomsClearable_C_mul_X_pow {N : ℕ} (a : R) (bu : bi * i b = 1) {n : ℕ} (r : R)
(nN : n ≤ N) : DenomsClearable a b N (C r * X ^ n) i := by |
refine ⟨r * a ^ n * b ^ (N - n), bi, bu, ?_⟩
rw [C_mul_X_pow_eq_monomial, map_monomial, ← C_mul_X_pow_eq_monomial, eval_mul, eval_pow, eval_C]
rw [RingHom.map_mul, RingHom.map_mul, RingHom.map_pow, RingHom.map_pow, eval_X, mul_comm]
rw [← tsub_add_cancel_of_le nN]
conv_lhs => rw [← mul_one (i a), ← bu]
sim... |
/-
Copyright (c) 2021 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Yaël Dillies
-/
import Mathlib.Computability.NFA
#align_import computability.epsilon_NFA from "leanprover-community/mathlib"@"28aa996fc6fb4317f0083c4e6daf79878d81be33"
/-!
... | Mathlib/Computability/EpsilonNFA.lean | 110 | 112 | theorem evalFrom_append_singleton (S : Set σ) (x : List α) (a : α) :
M.evalFrom S (x ++ [a]) = M.stepSet (M.evalFrom S x) a := by |
rw [evalFrom, List.foldl_append, List.foldl_cons, List.foldl_nil]
|
/-
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.Probability.Independence.Basic
import Mathlib.Probability.Independence.Conditional
#align_import probability.independence.zero_one from "leanprover-commun... | Mathlib/Probability/Independence/ZeroOne.lean | 294 | 306 | theorem kernel.indep_limsup_atBot_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα) :
Indep (limsup s atBot) (limsup s atBot) κ μα := by |
let ns : ι → Set ι := Set.Ici
have hnsp : ∀ i, BddBelow (ns i) := fun i => bddBelow_Ici
refine indep_limsup_self h_le h_indep ?_ ?_ hnsp ?_
· simp only [mem_atBot_sets, ge_iff_le, Set.mem_compl_iff, BddBelow, lowerBounds, Set.Nonempty]
rintro t ⟨a, ha⟩
obtain ⟨b, hb⟩ : ∃ b, b < a := exists_lt a
ref... |
/-
Copyright (c) 2023 Andrew Yang, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.RootsOfUnity.Basic
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.FieldTheory.Galois
import Mathlib.LinearAlgebra.Eigenspace.Mi... | Mathlib/FieldTheory/KummerExtension.lean | 151 | 179 | theorem X_pow_sub_C_irreducible_of_prime {p : ℕ} (hp : p.Prime) {a : K} (ha : ∀ b : K, b ^ p ≠ a) :
Irreducible (X ^ p - C a) := by |
-- First of all, We may find an irreducible factor `g` of `X ^ p - C a`.
have : ¬ IsUnit (X ^ p - C a) := by
rw [Polynomial.isUnit_iff_degree_eq_zero, degree_X_pow_sub_C hp.pos, Nat.cast_eq_zero]
exact hp.ne_zero
have ⟨g, hg, hg'⟩ := WfDvdMonoid.exists_irreducible_factor this (X_pow_sub_C_ne_zero hp.pos ... |
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral
#align_import measure_theory.integral.layercake from "leanprover-community/mathlib"@"08a4542bec7242a5c60f179e4e49d... | Mathlib/MeasureTheory/Integral/Layercake.lean | 395 | 437 | theorem lintegral_comp_eq_lintegral_meas_le_mul (μ : Measure α) (f_nn : 0 ≤ᵐ[μ] f)
(f_mble : AEMeasurable f μ) (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t)
(g_nn : ∀ᵐ t ∂volume.restrict (Ioi 0), 0 ≤ g t) :
∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ =
∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * E... |
obtain ⟨G, G_mble, G_nn, g_eq_G⟩ : ∃ G : ℝ → ℝ, Measurable G ∧ 0 ≤ G
∧ g =ᵐ[volume.restrict (Ioi 0)] G := by
refine AEMeasurable.exists_measurable_nonneg ?_ g_nn
exact aemeasurable_Ioi_of_forall_Ioc fun t ht => (g_intble t ht).1.1.aemeasurable
have g_eq_G_on : ∀ t, g =ᵐ[volume.restrict (Ioc 0 t)] G :... |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.... | Mathlib/Data/Nat/GCD/Basic.lean | 299 | 305 | theorem dvd_mul {x m n : ℕ} : x ∣ m * n ↔ ∃ y z, y ∣ m ∧ z ∣ n ∧ y * z = x := by |
constructor
· intro h
obtain ⟨⟨⟨y, hy⟩, ⟨z, hz⟩⟩, rfl⟩ := prod_dvd_and_dvd_of_dvd_prod h
exact ⟨y, z, hy, hz, rfl⟩
· rintro ⟨y, z, hy, hz, rfl⟩
exact mul_dvd_mul hy hz
|
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 363 | 365 | theorem testAgainstNN_one (μ : FiniteMeasure Ω) : μ.testAgainstNN 1 = μ.mass := by |
simp only [testAgainstNN, coe_one, Pi.one_apply, ENNReal.coe_one, lintegral_one]
rfl
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, E. W. Ayers
-/
import Mathlib.CategoryTheory.Comma.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Yoneda
import Mathlib.Data.Set.L... | Mathlib/CategoryTheory/Sites/Sieves.lean | 630 | 640 | theorem pullbackArrows_comm [HasPullbacks C] {X Y : C} (f : Y ⟶ X) (R : Presieve X) :
Sieve.generate (R.pullbackArrows f) = (Sieve.generate R).pullback f := by |
ext W g
constructor
· rintro ⟨_, h, k, hk, rfl⟩
cases' hk with W g hg
change (Sieve.generate R).pullback f (h ≫ pullback.snd)
rw [Sieve.pullback_apply, assoc, ← pullback.condition, ← assoc]
exact Sieve.downward_closed _ (by exact Sieve.le_generate R W hg) (h ≫ pullback.fst)
· rintro ⟨W, h, k, h... |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Yury Kudryashov
-/
import Mathlib.Topology.Order.LocalExtr
import Mathlib.Topology.Order.IntermediateValue
import Mathlib.Topology.Support
import Mathlib.Topology.Order... | Mathlib/Topology/Algebra/Order/Compact.lean | 357 | 360 | theorem Continuous.exists_forall_le [ClosedIicTopology α] [Nonempty β] {f : β → α}
(hf : Continuous f) (hlim : Tendsto f (cocompact β) atTop) : ∃ x, ∀ y, f x ≤ f y := by |
inhabit β
exact hf.exists_forall_le' default (hlim.eventually <| eventually_ge_atTop _)
|
/-
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 | 134 | 136 | theorem commute_mulLeft_right (a b : A) : Commute (mulLeft R a) (mulRight R b) := by |
ext c
exact (mul_assoc a c b).symm
|
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
impo... | Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean | 322 | 357 | theorem determinant_aux (hyp : n = 0 ∨ ¬(of v).TerminatedAt (n - 1)) :
((of v).continuantsAux n).a * ((of v).continuantsAux (n + 1)).b -
((of v).continuantsAux n).b * ((of v).continuantsAux (n + 1)).a = (-1) ^ n := by |
induction n with
| zero => simp [continuantsAux]
| succ n IH =>
-- set up some shorthand notation
let g := of v
let conts := continuantsAux g (n + 2)
set pred_conts := continuantsAux g (n + 1) with pred_conts_eq
set ppred_conts := continuantsAux g n with ppred_conts_eq
let pA := pred_cont... |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.LinearMap.Basic
import ... | Mathlib/Data/DFinsupp/Basic.lean | 1,269 | 1,271 | theorem subtypeDomain_def (f : Π₀ i, β i) :
f.subtypeDomain p = mk (f.support.subtype p) fun i => f i := by |
ext i; by_cases h2 : f i ≠ 0 <;> try simp at h2; dsimp; simp [h2]
|
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Mul
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
imp... | Mathlib/Analysis/MeanInequalitiesPow.lean | 150 | 160 | theorem rpow_add_le_mul_rpow_add_rpow (z₁ z₂ : ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(z₁ + z₂) ^ p ≤ (2 : ℝ≥0) ^ (p - 1) * (z₁ ^ p + z₂ ^ p) := by |
rcases eq_or_lt_of_le hp with (rfl | h'p)
· simp only [rpow_one, sub_self, rpow_zero, one_mul]; rfl
convert rpow_arith_mean_le_arith_mean2_rpow (1 / 2) (1 / 2) (2 * z₁) (2 * z₂) (add_halves 1) hp
using 1
· simp only [one_div, inv_mul_cancel_left₀, Ne, mul_eq_zero, two_ne_zero, one_ne_zero,
not_false_... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
import Mathlib.Analysis.SpecialFunctions... | Mathlib/Analysis/SpecialFunctions/Trigonometric/InverseDeriv.lean | 82 | 90 | theorem differentiableWithinAt_arcsin_Ici {x : ℝ} :
DifferentiableWithinAt ℝ arcsin (Ici x) x ↔ x ≠ -1 := by |
refine ⟨?_, fun h => (hasDerivWithinAt_arcsin_Ici h).differentiableWithinAt⟩
rintro h rfl
have : sin ∘ arcsin =ᶠ[𝓝[≥] (-1 : ℝ)] id := by
filter_upwards [Icc_mem_nhdsWithin_Ici ⟨le_rfl, neg_lt_self (zero_lt_one' ℝ)⟩] with x using
sin_arcsin'
have := h.hasDerivWithinAt.sin.congr_of_eventuallyEq this.s... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algeb... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 544 | 546 | theorem sin_eq_sqrt_one_sub_cos_sq {x : ℝ} (hl : 0 ≤ x) (hu : x ≤ π) :
sin x = √(1 - cos x ^ 2) := by |
rw [← abs_sin_eq_sqrt_one_sub_cos_sq, abs_of_nonneg (sin_nonneg_of_nonneg_of_le_pi hl hu)]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import Mathlib.GroupTheory.GroupAction.BigOperators
import Mathlib.Logic.Equiv.Fin
import Mathlib.Algebra... | Mathlib/LinearAlgebra/Pi.lean | 458 | 459 | theorem piRing_symm_apply (f : ι → M) (g : ι → R) : (piRing R M ι S).symm f g = ∑ i, g i • f i := by |
simp [piRing, LinearMap.lsum_apply]
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Limits.Products
#align_import topology.category.Top.limits.pullbacks from "leanp... | Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | 323 | 329 | theorem snd_embedding_of_left_embedding {X Y S : TopCat} {f : X ⟶ S} (H : Embedding f) (g : Y ⟶ S) :
Embedding <| ⇑(pullback.snd : pullback f g ⟶ Y) := by |
convert (homeoOfIso (asIso (pullback.snd : pullback (𝟙 S) g ⟶ _))).embedding.comp
(pullback_map_embedding_of_embeddings (i₂ := 𝟙 Y)
f g (𝟙 S) g H (homeoOfIso (Iso.refl _)).embedding (𝟙 _) rfl (by simp))
erw [← coe_comp]
simp
|
/-
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 | 733 | 738 | theorem exists_not_mem_null_laverage_le (hμ : μ ≠ 0) (hint : ∫⁻ a : α, f a ∂μ ≠ ∞) (hN : μ N = 0) :
∃ x, x ∉ N ∧ ⨍⁻ a, f a ∂μ ≤ f x := by |
have := measure_laverage_le_pos hμ hint
rw [← measure_diff_null hN] at this
obtain ⟨x, hx, hxN⟩ := nonempty_of_measure_ne_zero this.ne'
exact ⟨x, hxN, 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.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.Set.Function
import Mathlib.Order.In... | Mathlib/Algebra/Order/Interval/Set/Monoid.lean | 27 | 32 | theorem Ici_add_bij : BijOn (· + d) (Ici a) (Ici (a + d)) := by |
refine
⟨fun x h => add_le_add_right (mem_Ici.mp h) _, (add_left_injective d).injOn, fun _ h => ?_⟩
obtain ⟨c, rfl⟩ := exists_add_of_le (mem_Ici.mp h)
rw [mem_Ici, add_right_comm, add_le_add_iff_right] at h
exact ⟨a + c, h, by rw [add_right_comm]⟩
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Topology.Maps
import Mathlib.Topology.NhdsSet
#align_import topology.constructions from "leanprover-community/mathlib"... | Mathlib/Topology/Constructions.lean | 1,137 | 1,138 | theorem map_nhds_subtype_val {s : Set X} (x : s) : map ((↑) : s → X) (𝓝 x) = 𝓝[s] ↑x := by |
rw [inducing_subtype_val.map_nhds_eq, Subtype.range_val]
|
/-
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 | 262 | 264 | theorem eq_iff_eq_of_cmp_eq_cmp (h : cmp x y = cmp x' y') : x = y ↔ x' = y' := by |
rw [le_antisymm_iff, le_antisymm_iff, le_iff_le_of_cmp_eq_cmp h,
le_iff_le_of_cmp_eq_cmp (cmp_eq_cmp_symm.1 h)]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.To... | Mathlib/Topology/UniformSpace/Basic.lean | 1,487 | 1,490 | theorem uniformContinuousOn_iff_restrict [UniformSpace α] [UniformSpace β] {f : α → β} {s : Set α} :
UniformContinuousOn f s ↔ UniformContinuous (s.restrict f) := by |
delta UniformContinuousOn UniformContinuous
rw [← map_uniformity_set_coe, tendsto_map'_iff]; rfl
|
/-
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 | 299 | 304 | theorem divInt_mul_divInt (n₁ n₂ : Int) {d₁ d₂} (z₁ : d₁ ≠ 0) (z₂ : d₂ ≠ 0) :
(n₁ /. d₁) * (n₂ /. d₂) = (n₁ * n₂) /. (d₁ * d₂) := by |
rcases Int.eq_nat_or_neg d₁ with ⟨_, rfl | rfl⟩ <;>
rcases Int.eq_nat_or_neg d₂ with ⟨_, rfl | rfl⟩ <;>
simp_all [-Int.natCast_mul, divInt_neg', Int.mul_neg, Int.ofNat_mul_ofNat, Int.neg_mul,
mkRat_mul_mkRat]
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriente... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 540 | 541 | theorem oangle_sub_right {x y z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0) :
o.oangle x z - o.oangle y z = o.oangle x y := by | rw [sub_eq_iff_eq_add, o.oangle_add hx hy hz]
|
/-
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.Pi.Basic
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Data.Set.Lattice
#... | Mathlib/Order/Interval/Set/Pi.lean | 112 | 118 | theorem disjoint_pi_univ_Ioc_update_left_right {x y : ∀ i, α i} {i₀ : ι} {m : α i₀} :
Disjoint (pi univ fun i ↦ Ioc (x i) (update y i₀ m i))
(pi univ fun i ↦ Ioc (update x i₀ m i) (y i)) := by |
rw [disjoint_left]
rintro z h₁ h₂
refine (h₁ i₀ (mem_univ _)).2.not_lt ?_
simpa only [Function.update_same] using (h₂ i₀ (mem_univ _)).1
|
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.Module.Card
import Mathlib.SetTheory.Cardinal.CountableCover
import Mathlib.SetTheory.Cardinal.Continuum
import Mathlib.Analysis.Specific... | Mathlib/Topology/Algebra/Module/Cardinality.lean | 119 | 123 | theorem continuum_le_cardinal_of_isOpen
{E : Type*} (𝕜 : Type*) [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] [AddCommGroup E]
[Module 𝕜 E] [Nontrivial E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul 𝕜 E]
{s : Set E} (hs : IsOpen s) (h's : s.Nonempty) : 𝔠 ≤ #s := by |
simpa [cardinal_eq_of_isOpen 𝕜 hs h's] using continuum_le_cardinal_of_module 𝕜 E
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprov... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,391 | 1,393 | theorem setToFun_sub (hT : DominatedFinMeasAdditive μ T C) (hf : Integrable f μ)
(hg : Integrable g μ) : setToFun μ T hT (f - g) = setToFun μ T hT f - setToFun μ T hT g := by |
rw [sub_eq_add_neg, sub_eq_add_neg, setToFun_add hT hf hg.neg, setToFun_neg hT g]
|
/-
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.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.midpoint from "leanprover-community/mathlib"@"2196ab363eb097c008d449... | Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 68 | 71 | theorem Equiv.pointReflection_midpoint_left (x y : P) :
(Equiv.pointReflection (midpoint R x y)) x = y := by |
rw [midpoint, pointReflection_apply, lineMap_apply, vadd_vsub, vadd_vadd, ← add_smul, ← two_mul,
mul_invOf_self, one_smul, vsub_vadd]
|
/-
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
-/
import Mathlib.Data.Sum.Order
import Mathlib.Order.InitialSeg
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
#align_impor... | Mathlib/SetTheory/Ordinal/Basic.lean | 760 | 761 | theorem lift_lt {a b : Ordinal} : lift.{u,v} a < lift.{u,v} b ↔ a < b := by |
simp only [lt_iff_le_not_le, lift_le]
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Floris van Doorn
-/
import Mathlib.Algebra.Module.Defs
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.GroupTheory.GroupAc... | Mathlib/Data/Set/Pointwise/SMul.lean | 986 | 994 | theorem smul_inter_ne_empty_iff {s t : Set α} {x : α} :
x • s ∩ t ≠ ∅ ↔ ∃ a b, (a ∈ t ∧ b ∈ s) ∧ a * b⁻¹ = x := by |
rw [← nonempty_iff_ne_empty]
constructor
· rintro ⟨a, h, ha⟩
obtain ⟨b, hb, rfl⟩ := mem_smul_set.mp h
exact ⟨x • b, b, ⟨ha, hb⟩, by simp⟩
· rintro ⟨a, b, ⟨ha, hb⟩, rfl⟩
exact ⟨a, mem_inter (mem_smul_set.mpr ⟨b, hb, by simp⟩) ha⟩
|
/-
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.Data.Finset.Update
import Mathlib.Data.Prod.TProd
import Mathlib.GroupTheory.Coset
import Mathlib.Logic.Equiv.Fin
import Mathlib.Measur... | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 1,017 | 1,019 | theorem measurable_eq_mp {i i' : δ} (h : i = i') : Measurable (congr_arg π h).mp := by |
cases h
exact measurable_id
|
/-
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.Logic.Equiv.Option
import Mathlib.Order.RelIso.Basic
import Mathlib.Order.Disjoint
import Mathlib.Order.WithBot
import Mathlib.Tactic.Monotonicity.Attr... | Mathlib/Order/Hom/Basic.lean | 1,278 | 1,281 | theorem Disjoint.map_orderIso [SemilatticeInf α] [OrderBot α] [SemilatticeInf β] [OrderBot β]
{a b : α} (f : α ≃o β) (ha : Disjoint a b) : Disjoint (f a) (f b) := by |
rw [disjoint_iff_inf_le, ← f.map_inf, ← f.map_bot]
exact f.monotone ha.le_bot
|
/-
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.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | Mathlib/Analysis/Fourier/PoissonSummation.lean | 183 | 201 | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : f =O[cocompact ℝ] fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
(fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) =O[cocompact ℝ] (|·| ^ (-b)) := by |
obtain ⟨r, hr⟩ := K.isCompact.isBounded.subset_closedBall 0
rw [closedBall_eq_Icc, zero_add, zero_sub] at hr
have : ∀ x : ℝ,
‖(f.comp (ContinuousMap.addRight x)).restrict K‖ ≤ ‖f.restrict (Icc (x - r) (x + r))‖ := by
intro x
rw [ContinuousMap.norm_le _ (norm_nonneg _)]
rintro ⟨y, hy⟩
refine... |
/-
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 | 268 | 268 | theorem proj_bot_inj [DiscreteTopology X] {x y : X} : proj ⊥ x = proj ⊥ y ↔ x = y := by | simp
|
/-
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.Data.Set.Finite
import Mathlib.Data.Countable.Basic
import Mathlib.Logic.Equiv.List
import Mathlib.Data.Set.Subsingleton
#align_import data.set.counta... | Mathlib/Data/Set/Countable.lean | 157 | 161 | theorem Countable.exists_eq_range {s : Set α} (hc : s.Countable) (hs : s.Nonempty) :
∃ f : ℕ → α, s = range f := by |
rcases hc.exists_surjective hs with ⟨f, hf⟩
refine ⟨(↑) ∘ f, ?_⟩
rw [hf.range_comp, Subtype.range_coe]
|
/-
Copyright (c) 2024 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel
-/
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.LinearAlgebra.PiTensorProduct
/-!
# Projective seminorm on the tensor of a finite family of normed s... | Mathlib/Analysis/NormedSpace/PiTensorProduct/ProjectiveSeminorm.lean | 134 | 153 | theorem norm_eval_le_projectiveSeminorm (x : ⨂[𝕜] i, E i) (G : Type*) [SeminormedAddCommGroup G]
[NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) :
‖lift f.toMultilinearMap x‖ ≤ projectiveSeminorm x * ‖f‖ := by |
letI := nonempty_subtype.mpr (nonempty_lifts x)
rw [projectiveSeminorm_apply, Real.iInf_mul_of_nonneg (norm_nonneg _), projectiveSeminormAux]
refine le_ciInf ?_
intro ⟨p, hp⟩
rw [mem_lifts_iff] at hp
conv_lhs => rw [← hp, ← List.sum_map_hom, ← Multiset.sum_coe]
refine le_trans (norm_multiset_sum_le _) ?_... |
/-
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.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.... | Mathlib/Algebra/Order/Interval/Set/Group.lean | 151 | 157 | theorem nonempty_Ico_sdiff {x dx y dy : α} (h : dy < dx) (hx : 0 < dx) :
Nonempty ↑(Ico x (x + dx) \ Ico y (y + dy)) := by |
cases' lt_or_le x y with h' h'
· use x
simp [*, not_le.2 h']
· use max x (x + dy)
simp [*, le_refl]
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Alistair Tucker, Wen Yang
-/
import Mathlib.Order.Interval.Set.Image
import Mathlib.Order.CompleteLatticeIntervals
import Mathlib.Topology.Order.DenselyOrdered
... | Mathlib/Topology/Order/IntermediateValue.lean | 352 | 364 | theorem IsClosed.Icc_subset_of_forall_exists_gt {a b : α} {s : Set α} (hs : IsClosed (s ∩ Icc a b))
(ha : a ∈ s) (hgt : ∀ x ∈ s ∩ Ico a b, ∀ y ∈ Ioi x, (s ∩ Ioc x y).Nonempty) : Icc a b ⊆ s := by |
intro y hy
have : IsClosed (s ∩ Icc a y) := by
suffices s ∩ Icc a y = s ∩ Icc a b ∩ Icc a y by
rw [this]
exact IsClosed.inter hs isClosed_Icc
rw [inter_assoc]
congr
exact (inter_eq_self_of_subset_right <| Icc_subset_Icc_right hy.2).symm
exact
IsClosed.mem_of_ge_of_forall_exists_gt... |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Oliver Nash
-/
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.D... | Mathlib/Analysis/NormedSpace/HomeomorphBall.lean | 127 | 128 | theorem univBall_source (c : P) (r : ℝ) : (univBall c r).source = univ := by |
unfold univBall; split_ifs <;> rfl
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Nat
import Mathlib.Algebra.Order.Sub.Canonical
import Mathlib.Data.List.Perm
import Mathlib.Data.Set.List
import Mathlib.Init.Quot... | Mathlib/Data/Multiset/Basic.lean | 1,300 | 1,313 | theorem map_eq_cons [DecidableEq α] (f : α → β) (s : Multiset α) (t : Multiset β) (b : β) :
(∃ a ∈ s, f a = b ∧ (s.erase a).map f = t) ↔ s.map f = b ::ₘ t := by |
constructor
· rintro ⟨a, ha, rfl, rfl⟩
rw [← map_cons, Multiset.cons_erase ha]
· intro h
have : b ∈ s.map f := by
rw [h]
exact mem_cons_self _ _
obtain ⟨a, h1, rfl⟩ := mem_map.mp this
obtain ⟨u, rfl⟩ := exists_cons_of_mem h1
rw [map_cons, cons_inj_right] at h
refine ⟨a, mem_co... |
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 301 | 312 | theorem descPochhammer_succ_right (n : ℕ) :
descPochhammer R (n + 1) = descPochhammer R n * (X - (n : R[X])) := by |
suffices h : descPochhammer ℤ (n + 1) = descPochhammer ℤ n * (X - (n : ℤ[X])) by
apply_fun Polynomial.map (algebraMap ℤ R) at h
simpa [descPochhammer_map, Polynomial.map_mul, Polynomial.map_add, map_X,
Polynomial.map_intCast] using h
induction' n with n ih
· simp [descPochhammer]
· conv_lhs =>
... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLim... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 549 | 571 | theorem measure_iInter_eq_iInf' {α ι : Type*} [MeasurableSpace α] {μ : Measure α}
[Countable ι] [Preorder ι] [IsDirected ι (· ≤ ·)]
{f : ι → Set α} (h : ∀ i, MeasurableSet (f i)) (hfin : ∃ i, μ (f i) ≠ ∞) :
μ (⋂ i, f i) = ⨅ i, μ (⋂ j ≤ i, f j) := by |
let s := fun i ↦ ⋂ j ≤ i, f j
have iInter_eq : ⋂ i, f i = ⋂ i, s i := by
ext x; simp [s]; constructor
· exact fun h _ j _ ↦ h j
· intro h i
rcases directed_of (· ≤ ·) i i with ⟨j, rij, -⟩
exact h j i rij
have ms : ∀ i, MeasurableSet (s i) :=
fun i ↦ MeasurableSet.biInter (countable_un... |
/-
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, Yaël Dillies
-/
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Algebra.Group.Units
import Ma... | Mathlib/Algebra/Order/Ring/Defs.lean | 267 | 271 | theorem add_le_mul_two_add (a2 : 2 ≤ a) (b0 : 0 ≤ b) : a + (2 + b) ≤ a * (2 + b) :=
calc
a + (2 + b) ≤ a + (a + a * b) :=
add_le_add_left (add_le_add a2 <| le_mul_of_one_le_left b0 <| (@one_le_two α).trans a2) a
_ ≤ a * (2 + b) := by | rw [mul_add, mul_two, add_assoc]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Finsupp.Multiset
import Math... | Mathlib/RingTheory/UniqueFactorizationDomain.lean | 1,512 | 1,514 | theorem eq_of_prod_eq_prod [Nontrivial α] {a b : FactorSet α} (h : a.prod = b.prod) : a = b := by |
have : a.prod.factors = b.prod.factors := by rw [h]
rwa [prod_factors, prod_factors] at this
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Covering.Differentiation
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Integral.Lebesgue
import M... | Mathlib/MeasureTheory/Covering/Besicovitch.lean | 488 | 547 | theorem exist_disjoint_covering_families {N : ℕ} {τ : ℝ} (hτ : 1 < τ)
(hN : IsEmpty (SatelliteConfig α N τ)) (q : BallPackage β α) :
∃ s : Fin N → Set β,
(∀ i : Fin N, (s i).PairwiseDisjoint fun j => closedBall (q.c j) (q.r j)) ∧
range q.c ⊆ ⋃ i : Fin N, ⋃ j ∈ s i, ball (q.c j) (q.r j) := by |
-- first exclude the trivial case where `β` is empty (we need non-emptiness for the transfinite
-- induction, to be able to choose garbage when there is no point left).
cases isEmpty_or_nonempty β
· refine ⟨fun _ => ∅, fun _ => pairwiseDisjoint_empty, ?_⟩
rw [← image_univ, eq_empty_of_isEmpty (univ : Set β... |
/-
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 | 1,238 | 1,243 | theorem extend_coord_change_source_mem_nhdsWithin' {x : M} (hxf : x ∈ f.source)
(hxf' : x ∈ f'.source) :
((f.extend I).symm ≫ f'.extend I).source ∈ 𝓝[range I] f.extend I x := by |
apply extend_coord_change_source_mem_nhdsWithin
rw [← extend_image_source_inter]
exact mem_image_of_mem _ ⟨hxf, hxf'⟩
|
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.Analysis.... | Mathlib/Analysis/LocallyConvex/Bounded.lean | 412 | 416 | theorem vonNBornology_eq : Bornology.vonNBornology 𝕜 E = PseudoMetricSpace.toBornology := by |
rw [Bornology.ext_iff_isBounded]
intro s
rw [Bornology.isBounded_iff_isVonNBounded]
exact isVonNBounded_iff _
|
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.LinearAlgebra.Dimension.Constructions
#align_import algebra.linear_recurrence from "leanprover-community/ma... | Mathlib/Algebra/LinearRecurrence.lean | 85 | 88 | theorem is_sol_mkSol (init : Fin E.order → α) : E.IsSolution (E.mkSol init) := by |
intro n
rw [mkSol]
simp
|
/-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Filtered.Final
/-!
# Finally small categories
A category given by `(J : Type u) [Category.{v} J]` is `w`-finally small if there exists a... | Mathlib/CategoryTheory/Limits/FinallySmall.lean | 160 | 164 | theorem InitiallySmall.exists_small_weakly_initial_set [InitiallySmall.{w} J] :
∃ (s : Set J) (_ : Small.{w} s), ∀ i, ∃ j ∈ s, Nonempty (j ⟶ i) := by |
refine ⟨Set.range (fromInitialModel J).obj, inferInstance, fun i => ?_⟩
obtain ⟨f⟩ : Nonempty (CostructuredArrow (fromInitialModel J) i) := IsConnected.is_nonempty
exact ⟨(fromInitialModel J).obj f.left, Set.mem_range_self _, ⟨f.hom⟩⟩
|
/-
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.CategoryTheory.Comma.StructuredArrow
import Mathlib.CategoryTheory.IsConnected
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Terminal
import Ma... | Mathlib/CategoryTheory/Limits/Final.lean | 745 | 761 | theorem final_of_final_comp [hF : Final F] [hFG : Final (F ⋙ G)] : Final G := by |
let s₁ : C ≌ AsSmall.{max u₁ v₁ u₂ v₂ u₃ v₃} C := AsSmall.equiv
let s₂ : D ≌ AsSmall.{max u₁ v₁ u₂ v₂ u₃ v₃} D := AsSmall.equiv
let s₃ : E ≌ AsSmall.{max u₁ v₁ u₂ v₂ u₃ v₃} E := AsSmall.equiv
let _i : s₁.inverse ⋙ (F ⋙ G) ⋙ s₃.functor ≅
(s₁.inverse ⋙ F ⋙ s₂.functor) ⋙ (s₂.inverse ⋙ G ⋙ s₃.functor) :=
... |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov, Yakov Pechersky
-/
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Da... | Mathlib/Algebra/Group/Subsemigroup/Basic.lean | 460 | 462 | theorem iSup_eq_closure {ι : Sort*} (p : ι → Subsemigroup M) :
⨆ i, p i = Subsemigroup.closure (⋃ i, (p i : Set M)) := by |
simp_rw [Subsemigroup.closure_iUnion, Subsemigroup.closure_eq]
|
/-
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.SubboxInduction
import Mathlib.Analysis.BoxIntegral.Partition.Split
#align_import analysis.box_integral.partition.fil... | Mathlib/Analysis/BoxIntegral/Partition/Filter.lean | 485 | 489 | theorem hasBasis_toFilteriUnion (l : IntegrationParams) (I : Box ι) (π₀ : Prepartition I) :
(l.toFilteriUnion I π₀).HasBasis (fun r : ℝ≥0 → (ι → ℝ) → Ioi (0 : ℝ) => ∀ c, l.RCond (r c))
fun r => { π | ∃ c, l.MemBaseSet I c (r c) π ∧ π.iUnion = π₀.iUnion } := by |
have := fun c => l.hasBasis_toFilterDistortioniUnion I c π₀
simpa only [setOf_and, setOf_exists] using hasBasis_iSup this
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.RingTheory.Adjoin.Basic
#align_im... | Mathlib/Algebra/Polynomial/AlgebraMap.lean | 521 | 537 | theorem eval_mul_X_sub_C {p : R[X]} (r : R) : (p * (X - C r)).eval r = 0 := by |
simp only [eval, eval₂_eq_sum, RingHom.id_apply]
have bound :=
calc
(p * (X - C r)).natDegree ≤ p.natDegree + (X - C r).natDegree := natDegree_mul_le
_ ≤ p.natDegree + 1 := add_le_add_left (natDegree_X_sub_C_le _) _
_ < p.natDegree + 2 := lt_add_one _
rw [sum_over_range' _ _ (p.natDegree + ... |
/-
Copyright (c) 2021 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.NatIso
#align_import category_theory.bicategory.basic from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
/-!
# B... | Mathlib/CategoryTheory/Bicategory/Basic.lean | 245 | 248 | theorem inv_whiskerRight {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) [IsIso η] :
inv (η ▷ h) = inv η ▷ h := by |
apply IsIso.inv_eq_of_hom_inv_id
simp only [← comp_whiskerRight, id_whiskerRight, IsIso.hom_inv_id]
|
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Eric Wieser, Jeremy Avigad, Johan Commelin
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Linear... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 374 | 378 | theorem isUnit_fromBlocks_iff_of_invertible₁₁ {A : Matrix m m α} {B : Matrix m n α}
{C : Matrix n m α} {D : Matrix n n α} [Invertible A] :
IsUnit (fromBlocks A B C D) ↔ IsUnit (D - C * ⅟ A * B) := by |
simp only [← nonempty_invertible_iff_isUnit,
(invertibleEquivFromBlocks₁₁Invertible A B C D).nonempty_congr]
|
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Integral.FundThmCalcu... | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 957 | 977 | theorem tendsto_zero_of_hasDerivAt_of_integrableOn_Iic
(hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x)
(f'int : IntegrableOn f' (Iic a)) (fint : IntegrableOn f (Iic a)) :
Tendsto f atBot (𝓝 0) := by |
let F : E →L[ℝ] Completion E := Completion.toComplL
have Fderiv : ∀ x ∈ Iic a, HasDerivAt (F ∘ f) (F (f' x)) x :=
fun x hx ↦ F.hasFDerivAt.comp_hasDerivAt _ (hderiv x hx)
have Fint : IntegrableOn (F ∘ f) (Iic a) := by apply F.integrable_comp fint
have F'int : IntegrableOn (F ∘ f') (Iic a) := by apply F.int... |
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import number_theory.ramification_inertia from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac6... | Mathlib/NumberTheory/RamificationInertia.lean | 389 | 433 | theorem finrank_quotient_map [IsDomain S] [IsDedekindDomain R] [Algebra K L]
[Algebra R L] [IsScalarTower R K L] [IsScalarTower R S L] [IsIntegralClosure S R L]
[hp : p.IsMaximal] [IsNoetherian R S] :
finrank (R ⧸ p) (S ⧸ map (algebraMap R S) p) = finrank K L := by |
-- Choose an arbitrary basis `b` for `[S/pS : R/p]`.
-- We'll use the previous results to turn it into a basis on `[Frac(S) : Frac(R)]`.
letI : Field (R ⧸ p) := Ideal.Quotient.field _
let ι := Module.Free.ChooseBasisIndex (R ⧸ p) (S ⧸ map (algebraMap R S) p)
let b : Basis ι (R ⧸ p) (S ⧸ map (algebraMap R S) ... |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Eric Wieser
-/
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Multilinear.TensorProduct
import Mathlib.Ta... | Mathlib/LinearAlgebra/PiTensorProduct.lean | 318 | 321 | theorem tprodCoeff_eq_smul_tprod (z : R) (f : Π i, s i) : tprodCoeff R z f = z • tprod R f := by |
have : z = z • (1 : R) := by simp only [mul_one, Algebra.id.smul_eq_mul]
conv_lhs => rw [this]
rfl
|
/-
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.Group.Indicator
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Data.Set.Finite
#align_import data.finsupp.defs fr... | Mathlib/Data/Finsupp/Defs.lean | 504 | 506 | theorem equivFunOnFinite_symm_single [DecidableEq α] [Finite α] (x : α) (m : M) :
Finsupp.equivFunOnFinite.symm (Pi.single x m) = Finsupp.single x m := by |
rw [← equivFunOnFinite_single, Equiv.symm_apply_apply]
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "... | Mathlib/Data/Set/Function.lean | 225 | 226 | theorem EqOn.image_eq_self {f : α → α} (h : Set.EqOn f id s) : f '' s = s := by |
rw [h.image_eq, image_id]
|
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.LinearAlgebra.Basis
impo... | Mathlib/RingTheory/Adjoin/Basic.lean | 261 | 279 | theorem adjoin_inl_union_inr_eq_prod (s) (t) :
adjoin R (LinearMap.inl R A B '' (s ∪ {1}) ∪ LinearMap.inr R A B '' (t ∪ {1})) =
(adjoin R s).prod (adjoin R t) := by |
apply le_antisymm
· simp only [adjoin_le_iff, Set.insert_subset_iff, Subalgebra.zero_mem, Subalgebra.one_mem,
subset_adjoin,-- the rest comes from `squeeze_simp`
Set.union_subset_iff,
LinearMap.coe_inl, Set.mk_preimage_prod_right, Set.image_subset_iff, SetLike.mem_coe,
Set.mk_preimage_prod_... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Exponential
#align_import set_theory.ordinal.cantor_normal_form from "leanprover-communi... | Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | 163 | 174 | theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·) := by |
refine CNFRec b ?_ (fun o ho IH ↦ ?_) o
· simp only [gt_iff_lt, CNF_zero, map_nil, sorted_nil]
· rcases le_or_lt b 1 with hb | hb
· simp only [CNF_of_le_one hb ho, gt_iff_lt, map_cons, map, sorted_singleton]
· cases' lt_or_le o b with hob hbo
· simp only [CNF_of_lt ho hob, gt_iff_lt, map_cons, map,... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 190 | 194 | theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by |
rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero]
conv_lhs =>
rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul,
arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc]
|
/-
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.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.function.simple_func from "leanprover-community/mathlib"@"bf... | Mathlib/MeasureTheory/Function/SimpleFunc.lean | 848 | 852 | theorem approx_comp [TopologicalSpace β] [OrderClosedTopology β] [MeasurableSpace β]
[OpensMeasurableSpace β] [MeasurableSpace γ] {i : ℕ → β} {f : γ → β} {g : α → γ} {n : ℕ} (a : α)
(hf : Measurable f) (hg : Measurable g) :
(approx i (f ∘ g) n : α →ₛ β) a = (approx i f n : γ →ₛ β) (g a) := by |
rw [approx_apply _ hf, approx_apply _ (hf.comp hg), Function.comp_apply]
|
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Eric Wieser
-/
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Multilinear.TensorProduct
import Mathlib.Ta... | Mathlib/LinearAlgebra/PiTensorProduct.lean | 823 | 828 | theorem map_comp_reindex_symm (f : Π i, s i →ₗ[R] t i) (e : ι ≃ ι₂) :
map f ∘ₗ (reindex R s e).symm = (reindex R t e).symm ∘ₗ map (fun i => f (e.symm i)) := by |
ext m
apply LinearEquiv.injective (reindex R t e)
simp only [LinearMap.compMultilinearMap_apply, LinearMap.coe_comp, LinearEquiv.coe_coe,
comp_apply, ← map_reindex, LinearEquiv.apply_symm_apply, map_tprod]
|
/-
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.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | Mathlib/Analysis/Fourier/PoissonSummation.lean | 107 | 121 | theorem Real.tsum_eq_tsum_fourierIntegral {f : C(ℝ, ℂ)}
(h_norm :
∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖)
(h_sum : Summable fun n : ℤ => 𝓕 f n) (x : ℝ) :
∑' n : ℤ, f (x + n) = ∑' n : ℤ, 𝓕 f n * fourier n (x : UnitAddCircle) := by |
let F : C(UnitAddCircle, ℂ) :=
⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
convert (has_pointwise_sum_fourier_series_of_summable this x).tsum_eq.symm usin... |
/-
Copyright (c) 2021 Justus Springer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Justus Springer
-/
import Mathlib.Topology.Sheaves.Forget
import Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections
import Mathlib.CategoryTheory.Limits.Shapes.Types
#alig... | Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean | 112 | 118 | theorem isGluing_iff_pairwise {sf s} : IsGluing F U sf s ↔
∀ i, (F.mapCone (Pairwise.cocone U).op).π.app i s = objPairwiseOfFamily sf i := by |
refine ⟨fun h ↦ ?_, fun h i ↦ h (op <| Pairwise.single i)⟩
rintro (i|⟨i,j⟩)
· exact h i
· rw [← (F.mapCone (Pairwise.cocone U).op).w (op <| Pairwise.Hom.left i j)]
exact congr_arg _ (h i)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5... | Mathlib/Order/Filter/Basic.lean | 3,054 | 3,054 | theorem tendsto_bot {f : α → β} {l : Filter β} : Tendsto f ⊥ l := by | simp [Tendsto]
|
/-
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.Ring.Prod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Tactic.FinCases
#align_import data.zmod.basic from "leanprover-community/mathli... | Mathlib/Data/ZMod/Basic.lean | 580 | 581 | theorem intCast_zmod_eq_zero_iff_dvd (a : ℤ) (b : ℕ) : (a : ZMod b) = 0 ↔ (b : ℤ) ∣ a := by |
rw [← Int.cast_zero, ZMod.intCast_eq_intCast_iff, Int.modEq_zero_iff_dvd]
|
/-
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
-/
import Mathlib.Init.Logic
import Mathlib.Init.Function
import Mathlib.Init.Algebra.Classes
import Batteries.Util.LibraryNote
import Batteries.Tactic.... | Mathlib/Logic/Basic.lean | 1,356 | 1,363 | theorem beq_ext {α : Type*} (inst1 : BEq α) (inst2 : BEq α)
(h : ∀ x y, @BEq.beq _ inst1 x y = @BEq.beq _ inst2 x y) :
inst1 = inst2 := by |
have ⟨beq1⟩ := inst1
have ⟨beq2⟩ := inst2
congr
funext x y
exact h x y
|
/-
Copyright (c) 2022 Yaël Dillies, George Shakan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, George Shakan
-/
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Combinatorics.Enumerative.DoubleCounting
imp... | Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean | 63 | 66 | theorem card_div_mul_le_card_mul_mul_card_mul (A B C : Finset α) :
(A / C).card * B.card ≤ (A * B).card * (B * C).card := by |
rw [← div_inv_eq_mul, ← card_inv B, ← card_inv (B * C), mul_inv, ← div_eq_mul_inv]
exact card_div_mul_le_card_div_mul_card_div _ _ _
|
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical.Basic
import Mathlib.Algebra.Or... | Mathlib/Data/Real/NNReal.lean | 1,133 | 1,135 | theorem le_mul_iInf {a : ℝ≥0} {g : ℝ≥0} {h : ι → ℝ≥0} (H : ∀ j, a ≤ g * h j) : a ≤ g * iInf h := by |
rw [mul_iInf]
exact le_ciInf H
|
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Floris van Doorn
-/
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.NormedSpace.Connected
import Mathlib.LinearAlgebra.AffineSpace.ContinuousAffineEqu... | Mathlib/Analysis/Convex/AmpleSet.lean | 94 | 96 | theorem preimage {s : Set F} (h : AmpleSet s) (L : E ≃ᵃL[ℝ] F) : AmpleSet (L ⁻¹' s) := by |
rw [← L.image_symm_eq_preimage]
exact h.image L.symm
|
/-
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 | 1,327 | 1,328 | theorem infinite_univ_iff : (@univ α).Infinite ↔ Infinite α := by |
rw [Set.Infinite, finite_univ_iff, not_finite_iff_infinite]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Regular.SMul
#align_import data.polynomial.monic from "l... | Mathlib/Algebra/Polynomial/Monic.lean | 214 | 220 | theorem nextCoeff_mul (hp : Monic p) (hq : Monic q) :
nextCoeff (p * q) = nextCoeff p + nextCoeff q := by |
nontriviality
simp only [← coeff_one_reverse]
rw [reverse_mul] <;>
simp [coeff_mul, antidiagonal, hp.leadingCoeff, hq.leadingCoeff, add_comm,
show Nat.succ 0 = 1 from rfl]
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Alex J. Best
-/
import Mathlib.MeasureTheory.Group.Arithmetic
#align_import measure_theory.group.pointwise from "leanprover-community/mathlib"@"66f7114a1d5cba4... | Mathlib/MeasureTheory/Group/Pointwise.lean | 24 | 28 | theorem MeasurableSet.const_smul {G α : Type*} [Group G] [MulAction G α] [MeasurableSpace G]
[MeasurableSpace α] [MeasurableSMul G α] {s : Set α} (hs : MeasurableSet s) (a : G) :
MeasurableSet (a • s) := by |
rw [← preimage_smul_inv]
exact measurable_const_smul _ hs
|
/-
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 | 417 | 424 | theorem eq_top_iff_forall_lt (x : PartENat) : x = ⊤ ↔ ∀ n : ℕ, (n : PartENat) < x := by |
constructor
· rintro rfl n
exact natCast_lt_top _
· contrapose!
rw [ne_top_iff]
rintro ⟨n, rfl⟩
exact ⟨n, irrefl _⟩
|
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Batteries.Data.List.Lemmas
/-!
# Basic properties of `List.eraseIdx`
`List.eraseIdx l k` erases `k`-th element of `l : List α`.
If `k ≥ length l`, then it ... | .lake/packages/batteries/Batteries/Data/List/EraseIdx.lean | 76 | 80 | theorem mem_eraseIdx_iff_get? {x : α} {l} {k} : x ∈ eraseIdx l k ↔ ∃ i ≠ k, l.get? i = x := by |
simp only [mem_eraseIdx_iff_get, Fin.exists_iff, exists_and_left, get_eq_iff, exists_prop]
refine exists_congr fun i => and_congr_right' <| and_iff_right_of_imp fun h => ?_
obtain ⟨h, -⟩ := get?_eq_some.1 h
exact h
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.NumberTheory.Liouville.Basic
import Mathlib.Topology.Instances.Irrational
#align_impo... | Mathlib/NumberTheory/Liouville/LiouvilleWith.lean | 76 | 85 | theorem exists_pos (h : LiouvilleWith p x) :
∃ (C : ℝ) (_h₀ : 0 < C),
∃ᶠ n : ℕ in atTop, 1 ≤ n ∧ ∃ m : ℤ, x ≠ m / n ∧ |x - m / n| < C / n ^ p := by |
rcases h with ⟨C, hC⟩
refine ⟨max C 1, zero_lt_one.trans_le <| le_max_right _ _, ?_⟩
refine ((eventually_ge_atTop 1).and_frequently hC).mono ?_
rintro n ⟨hle, m, hne, hlt⟩
refine ⟨hle, m, hne, hlt.trans_le ?_⟩
gcongr
apply le_max_left
|
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | Mathlib/ModelTheory/Semantics.lean | 554 | 565 | theorem realize_toPrenex (φ : L.BoundedFormula α n) {v : α → M} :
∀ {xs : Fin n → M}, φ.toPrenex.Realize v xs ↔ φ.Realize v xs := by |
induction' φ with _ _ _ _ _ _ _ _ _ f1 f2 h1 h2 _ _ h
· exact Iff.rfl
· exact Iff.rfl
· exact Iff.rfl
· intros
rw [toPrenex, realize_toPrenexImp f1.toPrenex_isPrenex f2.toPrenex_isPrenex, realize_imp,
realize_imp, h1, h2]
· intros
rw [realize_all, toPrenex, realize_all]
exact forall_congr... |
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Ma... | Mathlib/Data/Nat/Factorization/Basic.lean | 120 | 120 | theorem factorization_one : factorization 1 = 0 := by | ext; simp [factorization]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Topology.Order.ProjIcc
#al... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 153 | 159 | theorem arcsin_le_iff_le_sin' {x y : ℝ} (hy : y ∈ Ico (-(π / 2)) (π / 2)) :
arcsin x ≤ y ↔ x ≤ sin y := by |
rcases le_total x (-1) with hx₁ | hx₁
· simp [arcsin_of_le_neg_one hx₁, hy.1, hx₁.trans (neg_one_le_sin _)]
cases' lt_or_le 1 x with hx₂ hx₂
· simp [arcsin_of_one_le hx₂.le, hy.2.not_le, (sin_le_one y).trans_lt hx₂]
exact arcsin_le_iff_le_sin ⟨hx₁, hx₂⟩ (mem_Icc_of_Ico hy)
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 143 | 152 | theorem ascPochhammer_succ_comp_X_add_one (n : ℕ) :
(ascPochhammer S (n + 1)).comp (X + 1) =
ascPochhammer S (n + 1) + (n + 1) • (ascPochhammer S n).comp (X + 1) := by |
suffices (ascPochhammer ℕ (n + 1)).comp (X + 1) =
ascPochhammer ℕ (n + 1) + (n + 1) * (ascPochhammer ℕ n).comp (X + 1)
by simpa [map_comp] using congr_arg (Polynomial.map (Nat.castRingHom S)) this
nth_rw 2 [ascPochhammer_succ_left]
rw [← add_mul, ascPochhammer_succ_right ℕ n, mul_comp, mul_comm, add_co... |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.LinearAlgebra.AffineSpace.Basic
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.Pi
import ... | Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean | 672 | 674 | theorem decomp' (f : V1 →ᵃ[k] V2) : (f.linear : V1 → V2) = ⇑f - fun _ => f 0 := by |
rw [decomp]
simp only [LinearMap.map_zero, Pi.add_apply, add_sub_cancel_right, zero_add]
|
/-
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, Alexander Bentkamp
-/
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
i... | Mathlib/LinearAlgebra/Basis.lean | 186 | 189 | theorem repr_support_subset_of_mem_span (s : Set ι) {m : M}
(hm : m ∈ span R (b '' s)) : ↑(b.repr m).support ⊆ s := by |
rcases (Finsupp.mem_span_image_iff_total _).1 hm with ⟨l, hl, rfl⟩
rwa [repr_total, ← Finsupp.mem_supported R l]
|
/-
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.Data.Countable.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Order.Disjointed
import Mathlib.MeasureTheory.OuterMeasure.Defs... | Mathlib/MeasureTheory/OuterMeasure/Basic.lean | 125 | 126 | theorem measure_union_null_iff : μ (s ∪ t) = 0 ↔ μ s = 0 ∧ μ t = 0 := by |
simp [union_eq_iUnion, and_comm]
|
/-
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
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.A... | Mathlib/RingTheory/AdjoinRoot.lean | 244 | 245 | theorem eval₂_root (f : R[X]) : f.eval₂ (of f) (root f) = 0 := by |
rw [← algebraMap_eq, ← aeval_def, aeval_eq, mk_self]
|
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