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 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.iterated_deriv from "leanprover-com... | Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 91 | 95 | theorem iteratedFDerivWithin_eq_equiv_comp :
iteratedFDerivWithin 𝕜 n f s =
ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F ∘ iteratedDerivWithin n f s := by |
rw [iteratedDerivWithin_eq_equiv_comp, ← Function.comp.assoc, LinearIsometryEquiv.self_comp_symm,
Function.id_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.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Constructions.Prod.Integral
impor... | Mathlib/Analysis/Convolution.lean | 500 | 503 | theorem ConvolutionExists.distrib_add (hfg : ConvolutionExists f g L μ)
(hfg' : ConvolutionExists f g' L μ) : f ⋆[L, μ] (g + g') = f ⋆[L, μ] g + f ⋆[L, μ] g' := by |
ext x
exact (hfg x).distrib_add (hfg' x)
|
/-
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.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75... | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | 653 | 655 | theorem lt_iff_le_and_exists (s1 s2 : AffineSubspace k P) :
s1 < s2 ↔ s1 ≤ s2 ∧ ∃ p ∈ s2, p ∉ s1 := by |
rw [lt_iff_le_not_le, not_le_iff_exists]
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Sébastien Gouëzel, Zhouhang Zhou, Reid Barton
-/
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.DenseEmbedding
import Mathlib.Topology.Support
impo... | Mathlib/Topology/Homeomorph.lean | 425 | 426 | theorem image_frontier (h : X ≃ₜ Y) (s : Set X) : h '' frontier s = frontier (h '' s) := by |
rw [← preimage_symm, preimage_frontier]
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kevin Buzzard, Scott Morrison, Johan Commelin, Chris Hughes,
Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Data.FunLike.Basi... | Mathlib/Algebra/Group/Hom/Defs.lean | 1,172 | 1,175 | theorem comp_one [MulOneClass M] [MulOneClass N] [MulOneClass P] (f : N →* P) :
f.comp (1 : M →* N) = 1 := by |
ext
simp only [map_one, coe_comp, Function.comp_apply, one_apply]
|
/-
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.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib... | Mathlib/Algebra/Polynomial/RingDivision.lean | 124 | 126 | theorem natDegree_mul (hp : p ≠ 0) (hq : q ≠ 0) : (p*q).natDegree = p.natDegree + q.natDegree := by |
rw [← Nat.cast_inj (R := WithBot ℕ), ← degree_eq_natDegree (mul_ne_zero hp hq),
Nat.cast_add, ← degree_eq_natDegree hp, ← degree_eq_natDegree hq, degree_mul]
|
/-
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 | 676 | 678 | theorem intCast_mod (a : ℤ) (b : ℕ) : ((a % b : ℤ) : ZMod b) = (a : ZMod b) := by |
rw [ZMod.intCast_eq_intCast_iff]
apply Int.mod_modEq
|
/-
Copyright (c) 2021 Paul Lezeau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Paul Lezeau
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Order.Hom.Bounded
import Mathlib.Algebra.GCDMonoid.Basic
#align_impor... | Mathlib/RingTheory/ChainOfDivisors.lean | 224 | 231 | theorem factor_orderIso_map_one_eq_bot {m : Associates M} {n : Associates N}
(d : { l : Associates M // l ≤ m } ≃o { l : Associates N // l ≤ n }) :
(d ⟨1, one_dvd m⟩ : Associates N) = 1 := by |
letI : OrderBot { l : Associates M // l ≤ m } := Subtype.orderBot bot_le
letI : OrderBot { l : Associates N // l ≤ n } := Subtype.orderBot bot_le
simp only [← Associates.bot_eq_one, Subtype.mk_bot, bot_le, Subtype.coe_eq_bot_iff]
letI : BotHomClass ({ l // l ≤ m } ≃o { l // l ≤ n }) _ _ := OrderIsoClass.toBotH... |
/-
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, Mario Carneiro, Johan Commelin
-/
import Mathlib.NumberTheory.Padics.PadicNumbers
import Mathlib.RingTheory.DiscreteValuationRing.Basic
#align_import number_theory.p... | Mathlib/NumberTheory/Padics/PadicIntegers.lean | 577 | 581 | theorem norm_lt_one_iff_dvd (x : ℤ_[p]) : ‖x‖ < 1 ↔ ↑p ∣ x := by |
have := norm_le_pow_iff_mem_span_pow x 1
rw [Ideal.mem_span_singleton, pow_one] at this
rw [← this, norm_le_pow_iff_norm_lt_pow_add_one]
simp only [zpow_zero, Int.ofNat_zero, Int.ofNat_succ, add_left_neg, zero_add]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
import Mathlib.Ord... | Mathlib/Order/Filter/Bases.lean | 672 | 678 | theorem _root_.Pairwise.exists_mem_filter_basis_of_disjoint {I} [Finite I] {l : I → Filter α}
{ι : I → Sort*} {p : ∀ i, ι i → Prop} {s : ∀ i, ι i → Set α} (hd : Pairwise (Disjoint on l))
(h : ∀ i, (l i).HasBasis (p i) (s i)) :
∃ ind : ∀ i, ι i, (∀ i, p i (ind i)) ∧ Pairwise (Disjoint on fun i => s i (ind i)... |
rcases hd.exists_mem_filter_of_disjoint with ⟨t, htl, hd⟩
choose ind hp ht using fun i => (h i).mem_iff.1 (htl i)
exact ⟨ind, hp, hd.mono fun i j hij => hij.mono (ht _) (ht _)⟩
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Na... | Mathlib/Data/Complex/Exponential.lean | 1,744 | 1,745 | theorem abs_exp_ofReal_mul_I (x : ℝ) : abs (exp (x * I)) = 1 := by |
rw [exp_mul_I, abs_cos_add_sin_mul_I]
|
/-
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.Equivalence
#align_import algebraic_topology.dold_kan.compatibility from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504... | Mathlib/AlgebraicTopology/DoldKan/Compatibility.lean | 103 | 105 | theorem equivalence₁UnitIso_eq : (equivalence₁ hF).unitIso = equivalence₁UnitIso hF := by |
ext X
simp [equivalence₁]
|
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.PFunctor.Univariate.M
#align_import data.qpf.univariate.basic from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d561dee7"
/-!
... | Mathlib/Data/QPF/Univariate/Basic.lean | 377 | 379 | theorem corecF_eq {α : Type _} (g : α → F α) (x : α) :
PFunctor.M.dest (corecF g x) = q.P.map (corecF g) (repr (g x)) := by |
rw [corecF, PFunctor.M.dest_corec]
|
/-
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 | 1,710 | 1,717 | theorem IsBigOWith.inv_rev {f : α → 𝕜} {g : α → 𝕜'} (h : IsBigOWith c l f g)
(h₀ : ∀ᶠ x in l, f x = 0 → g x = 0) : IsBigOWith c l (fun x => (g x)⁻¹) fun x => (f x)⁻¹ := by |
refine IsBigOWith.of_bound (h.bound.mp (h₀.mono fun x h₀ hle => ?_))
rcases eq_or_ne (f x) 0 with hx | hx
· simp only [hx, h₀ hx, inv_zero, norm_zero, mul_zero, le_rfl]
· have hc : 0 < c := pos_of_mul_pos_left ((norm_pos_iff.2 hx).trans_le hle) (norm_nonneg _)
replace hle := inv_le_inv_of_le (norm_pos_iff.... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Data.Vector.Defs
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.I... | Mathlib/Data/Vector/Basic.lean | 280 | 281 | theorem get_cons_succ (a : α) (v : Vector α n) (i : Fin n) : get (a ::ᵥ v) i.succ = get v i := by |
rw [← get_tail_succ, tail_cons]
|
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, David Loeffler, Heather Macbeth, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Ana... | Mathlib/Analysis/Fourier/FourierTransformDeriv.lean | 769 | 779 | theorem fourierIntegral_deriv
{f : ℝ → E} (hf : Integrable f) (h'f : Differentiable ℝ f) (hf' : Integrable (deriv f)) :
𝓕 (deriv f) = fun (x : ℝ) ↦ (2 * π * I * x) • (𝓕 f x) := by |
ext x
have I : Integrable (fun x ↦ fderiv ℝ f x) := by
simpa only [← deriv_fderiv] using (ContinuousLinearMap.smulRightL ℝ ℝ E 1).integrable_comp hf'
have : 𝓕 (deriv f) x = 𝓕 (fderiv ℝ f) x 1 := by
simp only [fourierIntegral_continuousLinearMap_apply I, fderiv_deriv]
rw [this, fourierIntegral_fderiv ... |
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import line... | Mathlib/LinearAlgebra/Charpoly/Basic.lean | 71 | 75 | theorem aeval_self_charpoly : aeval f f.charpoly = 0 := by |
apply (LinearEquiv.map_eq_zero_iff (algEquivMatrix (chooseBasis R M)).toLinearEquiv).1
rw [AlgEquiv.toLinearEquiv_apply, ← AlgEquiv.coe_algHom, ← Polynomial.aeval_algHom_apply _ _ _,
charpoly_def]
exact Matrix.aeval_self_charpoly _
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison, Adam Topaz
-/
import Mathlib.AlgebraicTopology.SimplexCategory
import Mathlib.CategoryTheory.Comma.Arrow
import Mathlib.CategoryTheory.Limits.FunctorCat... | Mathlib/AlgebraicTopology/SimplicialObject.lean | 400 | 401 | theorem augment_hom_zero (X : SimplicialObject C) (X₀ : C) (f : X _[0] ⟶ X₀) (w) :
(X.augment X₀ f w).hom.app (op [0]) = f := by | simp
|
/-
Copyright (c) 2021 Lu-Ming Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lu-Ming Zhang
-/
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.LinearAlgebra.Matrix.Orthogonal
import Mathlib.Data.Matrix.Kronecker
#align_import linear_algebra.matrix.i... | Mathlib/LinearAlgebra/Matrix/IsDiag.lean | 184 | 188 | theorem IsDiag.fromBlocks_of_isSymm [Zero α] {A : Matrix m m α} {C : Matrix n m α}
{D : Matrix n n α} (h : (A.fromBlocks 0 C D).IsSymm) (ha : A.IsDiag) (hd : D.IsDiag) :
(A.fromBlocks 0 C D).IsDiag := by |
rw [← (isSymm_fromBlocks_iff.1 h).2.1]
exact ha.fromBlocks hd
|
/-
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.Data.Sigma.Basic
import Mathlib.Algebra.Order.Ring.Nat
#align_import set_theory.lists from "leanprover-community/mathlib"@"497d1e06409995dd8ec95301fa8... | Mathlib/SetTheory/Lists.lean | 313 | 349 | theorem Equiv.trans : ∀ {l₁ l₂ l₃ : Lists α}, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ := by |
let trans := fun l₁ : Lists α => ∀ ⦃l₂ l₃⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃
suffices PProd (∀ l₁, trans l₁) (∀ (l : Lists' α true), ∀ l' ∈ l.toList, trans l') by exact this.1
apply inductionMut
· intro a l₂ l₃ h₁ h₂
rwa [← equiv_atom.1 h₁] at h₂
· intro l₁ IH l₂ l₃ h₁ h₂
-- Porting note: Two 'have's are ... |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Set.Function
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Says
#align_import logic.equiv.set from "leanprover-... | Mathlib/Logic/Equiv/Set.lean | 642 | 648 | theorem ofLeftInverse_eq_ofInjective {α β : Type*} (f : α → β) (f_inv : Nonempty α → β → α)
(hf : ∀ h : Nonempty α, LeftInverse (f_inv h) f) :
ofLeftInverse f f_inv hf =
ofInjective f ((isEmpty_or_nonempty α).elim (fun h _ _ _ => Subsingleton.elim _ _)
(fun h => (hf h).injective)) := by |
ext
simp
|
/-
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 | 364 | 365 | theorem whisker_assoc_symm (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d) :
f ◁ η ▷ h = (α_ f g h).inv ≫ (f ◁ η) ▷ h ≫ (α_ f g' h).hom := by | simp
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Bhavik Mehta, Stuart Presnell
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Basic
#align_import data.nat.choose.basic from "leanprover-community... | Mathlib/Data/Nat/Choose/Basic.lean | 235 | 240 | theorem ascFactorial_eq_factorial_mul_choose (n k : ℕ) :
(n + 1).ascFactorial k = k ! * (n + k).choose k := by |
rw [Nat.mul_comm]
apply Nat.mul_right_cancel (n + k - k).factorial_pos
rw [choose_mul_factorial_mul_factorial <| Nat.le_add_left k n, Nat.add_sub_cancel_right,
← factorial_mul_ascFactorial, Nat.mul_comm]
|
/-
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.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Poly... | Mathlib/FieldTheory/RatFunc/Degree.lean | 65 | 68 | theorem intDegree_polynomial {p : K[X]} :
intDegree (algebraMap K[X] (RatFunc K) p) = natDegree p := by |
rw [intDegree, RatFunc.num_algebraMap, RatFunc.denom_algebraMap, Polynomial.natDegree_one,
Int.ofNat_zero, sub_zero]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Truncated
import Mathlib.RingTheory.WittVector.Identities
import Mathlib.NumberTheory.Padics.RingHoms
#align_im... | Mathlib/RingTheory/WittVector/Compare.lean | 183 | 189 | theorem toPadicInt_comp_fromPadicInt : (toPadicInt p).comp (fromPadicInt p) = RingHom.id ℤ_[p] := by |
rw [← PadicInt.toZModPow_eq_iff_ext]
intro n
rw [← RingHom.comp_assoc, toPadicInt, PadicInt.lift_spec]
simp only [fromPadicInt, toZModPow, RingHom.comp_id]
rw [RingHom.comp_assoc, truncate_comp_lift, ← RingHom.comp_assoc]
simp only [RingEquiv.symm_toRingHom_comp_toRingHom, RingHom.id_comp]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Sébastien Gouëzel, Yury G. Kudryashov, Dylan MacKenzie, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Basic
impor... | Mathlib/Analysis/SpecificLimits/Normed.lean | 111 | 114 | theorem isLittleO_pow_pow_of_abs_lt_left {r₁ r₂ : ℝ} (h : |r₁| < |r₂|) :
(fun n : ℕ ↦ r₁ ^ n) =o[atTop] fun n ↦ r₂ ^ n := by |
refine (IsLittleO.of_norm_left ?_).of_norm_right
exact (isLittleO_pow_pow_of_lt_left (abs_nonneg r₁) h).congr (pow_abs r₁) (pow_abs r₂)
|
/-
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.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SumOverResidueClass
#alig... | Mathlib/Analysis/PSeries.lean | 161 | 171 | theorem tsum_schlomilch_le {C : ℕ} (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
(h_nonneg : ∀ n, 0 ≤ f n) (hu : Monotone u) (h_succ_diff : SuccDiffBounded C u) :
∑' k : ℕ, (u (k + 1) - u k) * f (u k) ≤ (u 1 - u 0) * f (u 0) + C * ∑' k, f k := by |
rw [ENNReal.tsum_eq_iSup_nat' (tendsto_atTop_mono Nat.le_succ tendsto_id)]
refine
iSup_le fun n =>
le_trans ?_
(add_le_add_left
(mul_le_mul_of_nonneg_left (ENNReal.sum_le_tsum <| Finset.Ico (u 0 + 1) (u n + 1)) ?_) _)
simpa using Finset.sum_schlomilch_le hf h_pos h_nonneg hu h_succ_di... |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.RingTheory.Ideal.Maps
#align_import ring_theory.ideal.prod from "leanprover-community/mathlib"@"052f6013363326d50cb99c6939814a4b8eb7b301"
/-!
# Ideals ... | Mathlib/RingTheory/Ideal/Prod.lean | 157 | 173 | theorem ideal_prod_prime (I : Ideal (R × S)) :
I.IsPrime ↔
(∃ p : Ideal R, p.IsPrime ∧ I = Ideal.prod p ⊤) ∨
∃ p : Ideal S, p.IsPrime ∧ I = Ideal.prod ⊤ p := by |
constructor
· rw [ideal_prod_eq I]
intro hI
rcases ideal_prod_prime_aux hI with (h | h)
· right
rw [h] at hI ⊢
exact ⟨_, ⟨isPrime_of_isPrime_prod_top' hI, rfl⟩⟩
· left
rw [h] at hI ⊢
exact ⟨_, ⟨isPrime_of_isPrime_prod_top hI, rfl⟩⟩
· rintro (⟨p, ⟨h, rfl⟩⟩ | ⟨p, ⟨h, rfl⟩⟩)
... |
/-
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, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 100 | 112 | theorem rpow_def_of_neg {x : ℝ} (hx : x < 0) (y : ℝ) : x ^ y = exp (log x * y) * cos (y * π) := by |
rw [rpow_def, Complex.cpow_def, if_neg]
· have : Complex.log x * y = ↑(log (-x) * y) + ↑(y * π) * Complex.I := by
simp only [Complex.log, abs_of_neg hx, Complex.arg_ofReal_of_neg hx, Complex.abs_ofReal,
Complex.ofReal_mul]
ring
rw [this, Complex.exp_add_mul_I, ← Complex.ofReal_exp, ← Comple... |
/-
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 | 569 | 571 | theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) :
I * span {x} = J * span {x} ↔ I = J := by |
simp only [le_antisymm_iff, span_singleton_mul_left_mono hx]
|
/-
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 | 262 | 282 | theorem lift_cof (o) : Cardinal.lift.{u, v} (cof o) = cof (Ordinal.lift.{u, v} o) := by |
refine inductionOn o ?_
intro α r _
apply le_antisymm
· refine le_cof_type.2 fun S H => ?_
have : Cardinal.lift.{u, v} #(ULift.up ⁻¹' S) ≤ #(S : Type (max u v)) := by
rw [← Cardinal.lift_umax.{v, u}, ← Cardinal.lift_id'.{v, u} #S]
exact mk_preimage_of_injective_lift.{v, max u v} ULift.up S (ULi... |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Ma... | Mathlib/LinearAlgebra/LinearIndependent.lean | 319 | 324 | theorem LinearIndependent.of_comp (f : M →ₗ[R] M') (hfv : LinearIndependent R (f ∘ v)) :
LinearIndependent R v :=
linearIndependent_iff'.2 fun s g hg i his =>
have : (∑ i ∈ s, g i • f (v i)) = 0 := by |
simp_rw [← map_smul, ← map_sum, hg, f.map_zero]
linearIndependent_iff'.1 hfv s g this i his
|
/-
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,
Scott Morrison
-/
import Mathlib.Data.List.Basic
#align_import data.list.lattice from "leanprov... | Mathlib/Data/List/Lattice.lean | 211 | 214 | theorem cons_bagInter_of_neg (l₁ : List α) (h : a ∉ l₂) :
(a :: l₁).bagInter l₂ = l₁.bagInter l₂ := by |
cases l₂; · simp only [bagInter_nil]
simp only [erase_of_not_mem h, List.bagInter, if_neg (mt mem_of_elem_eq_true h)]
|
/-
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 | 68 | 69 | theorem gcd_mul_left_add_left (m n k : ℕ) : gcd (n * k + m) n = gcd m n := by |
rw [gcd_comm, gcd_mul_left_add_right, gcd_comm]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Nat.Dist
import Mathlib.Data.Ordmap.Ordnode
import Mathlib.Tactic.Abel
imp... | Mathlib/Data/Ordmap/Ordset.lean | 850 | 854 | theorem size_balanceL {l x r} (hl : Balanced l) (hr : Balanced r) (sl : Sized l) (sr : Sized r)
(H : (∃ l', Raised l' (size l) ∧ BalancedSz l' (size r)) ∨
∃ r', Raised (size r) r' ∧ BalancedSz (size l) r') :
size (@balanceL α l x r) = size l + size r + 1 := by |
rw [balanceL_eq_balance' hl hr sl sr H, size_balance' sl sr]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Data.Option.Defs
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Sigma.Basic
import Mathlib... | Mathlib/Logic/Equiv/Basic.lean | 1,624 | 1,628 | theorem swapCore_comm (r a b : α) : swapCore a b r = swapCore b a r := by |
unfold swapCore
-- Porting note: whatever solution works for `swapCore_swapCore` will work here too.
split_ifs with h₁ h₂ h₃ <;> try simp
· cases h₁; cases h₂; rfl
|
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Combinatorics.Enumerative.Partition
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Cycle.F... | Mathlib/GroupTheory/Perm/Cycle/Type.lean | 593 | 605 | theorem _root_.card_support_eq_three_iff : σ.support.card = 3 ↔ σ.IsThreeCycle := by |
refine ⟨fun h => ?_, IsThreeCycle.card_support⟩
by_cases h0 : σ.cycleType = 0
· rw [← sum_cycleType, h0, sum_zero] at h
exact (ne_of_lt zero_lt_three h).elim
obtain ⟨n, hn⟩ := exists_mem_of_ne_zero h0
by_cases h1 : σ.cycleType.erase n = 0
· rw [← sum_cycleType, ← cons_erase hn, h1, cons_zero, Multiset.... |
/-
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 | 135 | 149 | theorem isBigO_iff'' {g : α → E'''} :
f =O[l] g ↔ ∃ c > 0, ∀ᶠ x in l, c * ‖f x‖ ≤ ‖g x‖ := by |
refine ⟨fun h => ?mp, fun h => ?mpr⟩
case mp =>
rw [isBigO_iff'] at h
obtain ⟨c, ⟨hc_pos, hc⟩⟩ := h
refine ⟨c⁻¹, ⟨by positivity, ?_⟩⟩
filter_upwards [hc] with x hx
rwa [inv_mul_le_iff (by positivity)]
case mpr =>
rw [isBigO_iff']
obtain ⟨c, ⟨hc_pos, hc⟩⟩ := h
refine ⟨c⁻¹, ⟨by posi... |
/-
Copyright (c) 2024 Raghuram Sundararajan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Raghuram Sundararajan
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
/-!
# Extensionality lemmas for rings and similar structures
In this file we prove e... | Mathlib/Algebra/Ring/Ext.lean | 231 | 234 | theorem toNonUnitalNonAssocring_injective :
Function.Injective (@toNonUnitalNonAssocRing R) := by |
intro _ _ _
ext <;> congr
|
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
#align_import group_theory.perm.list from "leanprove... | Mathlib/GroupTheory/Perm/List.lean | 379 | 382 | theorem formPerm_apply_mem_ne_self_iff (hl : Nodup l) (x : α) (hx : x ∈ l) :
formPerm l x ≠ x ↔ 2 ≤ l.length := by |
rw [Ne, formPerm_apply_mem_eq_self_iff _ hl x hx, not_le]
exact ⟨Nat.succ_le_of_lt, Nat.lt_of_succ_le⟩
|
/-
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 | 110 | 111 | theorem mk_eq_mkRat (num den nz c) : ⟨num, den, nz, c⟩ = mkRat num den := by |
simp [mk_eq_normalize, normalize_eq_mkRat]
|
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, François Dupuis
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Filter.Extr
import Mathlib.Tactic.GCongr
#align_import analysis.convex.function fr... | Mathlib/Analysis/Convex/Function.lean | 745 | 748 | theorem ConvexOn.le_left_of_right_le (hf : ConvexOn 𝕜 s f) {x y z : E} (hx : x ∈ s) (hy : y ∈ s)
(hz : z ∈ openSegment 𝕜 x y) (hyz : f y ≤ f z) : f z ≤ f x := by |
obtain ⟨a, b, ha, hb, hab, rfl⟩ := hz
exact hf.le_left_of_right_le' hx hy ha hb.le hab hyz
|
/-
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 | 430 | 432 | theorem IsCompact.inf_nhdsSet_eq_biSup {K : Set X} (hK : IsCompact K) (l : Filter X) :
l ⊓ (𝓝ˢ K) = ⨆ x ∈ K, l ⊓ 𝓝 x := by |
simp only [inf_comm l, hK.nhdsSet_inf_eq_biSup]
|
/-
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 | 341 | 344 | theorem pb_inv [Invertible (2 : R)] : PB l R * Matrix.fromBlocks 1 0 0 (⅟ (PD l R)) = 1 := by |
rw [PB, Matrix.fromBlocks_multiply, mul_invOf_self]
simp only [Matrix.mul_zero, Matrix.mul_one, Matrix.zero_mul, zero_add, add_zero,
Matrix.fromBlocks_one]
|
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.EMetricSpace.Basic
import Mathlib.Topology.Bornology.Constructions
imp... | Mathlib/Topology/MetricSpace/PseudoMetric.lean | 392 | 393 | theorem dist_edist (x y : α) : dist x y = (edist x y).toReal := by |
rw [edist_dist, ENNReal.toReal_ofReal dist_nonneg]
|
/-
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.Topology.Algebra.InfiniteSum.Order
import Mathlib.Topology.Algebra.InfiniteSum.Ring
import Mathlib.Topology.Instances.Real
import Mathlib.Topology.Metr... | Mathlib/Topology/Instances/NNReal.lean | 274 | 277 | theorem _root_.Real.tendsto_of_bddAbove_monotone {f : ℕ → ℝ} (h_bdd : BddAbove (Set.range f))
(h_mon : Monotone f) : ∃ r : ℝ, Tendsto f atTop (𝓝 r) := by |
obtain ⟨B, hB⟩ := Real.exists_isLUB (Set.range_nonempty f) h_bdd
exact ⟨B, tendsto_atTop_isLUB h_mon hB⟩
|
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Trace
import Mathlib.RingTheory.Norm
#align_import ring_theory.discriminant from "leanprover-c... | Mathlib/RingTheory/Discriminant.lean | 113 | 116 | theorem discr_of_matrix_vecMul (b : ι → B) (P : Matrix ι ι A) :
discr A (b ᵥ* P.map (algebraMap A B)) = P.det ^ 2 * discr A b := by |
rw [discr_def, traceMatrix_of_matrix_vecMul, det_mul, det_mul, det_transpose, mul_comm, ←
mul_assoc, discr_def, pow_two]
|
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.Dimension.LinearMap
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
#align_import linear_algebra.free_module.finite.matrix... | Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean | 85 | 89 | theorem cardinal_mk_algHom_le_rank : #(M →ₐ[K] L) ≤ lift.{v} (Module.rank K M) := by |
convert (linearIndependent_algHom_toLinearMap K M L).cardinal_lift_le_rank
· rw [lift_id]
· have := Module.nontrivial K L
rw [lift_id, FiniteDimensional.rank_linearMap_self]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, James Gallicchio
-/
import Batteries.Data.List.Count
import Batteries.Data.Fin.Lemmas
/-!
# Pairwise relations on a list
This file provides basic results about `List.... | .lake/packages/batteries/Batteries/Data/List/Pairwise.lean | 108 | 112 | theorem pairwise_append_comm {R : α → α → Prop} (s : ∀ {x y}, R x y → R y x) {l₁ l₂ : List α} :
Pairwise R (l₁ ++ l₂) ↔ Pairwise R (l₂ ++ l₁) := by |
have (l₁ l₂ : List α) (H : ∀ x : α, x ∈ l₁ → ∀ y : α, y ∈ l₂ → R x y)
(x : α) (xm : x ∈ l₂) (y : α) (ym : y ∈ l₁) : R x y := s (H y ym x xm)
simp only [pairwise_append, and_left_comm]; rw [Iff.intro (this l₁ l₂) (this l₂ l₁)]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.... | Mathlib/LinearAlgebra/Multilinear/Basic.lean | 1,811 | 1,823 | theorem curryFinFinset_symm_apply_piecewise_const {k l n : ℕ} {s : Finset (Fin n)} (hk : s.card = k)
(hl : sᶜ.card = l)
(f : MultilinearMap R (fun _ : Fin k => M') (MultilinearMap R (fun _ : Fin l => M') M₂))
(x y : M') :
(curryFinFinset R M₂ M' hk hl).symm f (s.piecewise (fun _ => x) fun _ => y) =
... |
rw [curryFinFinset_symm_apply]; congr
· ext
rw [finSumEquivOfFinset_inl, Finset.piecewise_eq_of_mem]
apply Finset.orderEmbOfFin_mem
· ext
rw [finSumEquivOfFinset_inr, Finset.piecewise_eq_of_not_mem]
exact Finset.mem_compl.1 (Finset.orderEmbOfFin_mem _ _ _)
|
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.ProperSpace
import Mathlib.Topology.MetricSpace.Cauchy
/-... | Mathlib/Topology/MetricSpace/Bounded.lean | 456 | 460 | theorem dist_le_diam_of_mem' (h : EMetric.diam s ≠ ⊤) (hx : x ∈ s) (hy : y ∈ s) :
dist x y ≤ diam s := by |
rw [diam, dist_edist]
rw [ENNReal.toReal_le_toReal (edist_ne_top _ _) h]
exact EMetric.edist_le_diam_of_mem hx hy
|
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Pigeonhole
imp... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 323 | 327 | theorem two_div_one_sub_two_div_e_le_eight : 2 / (1 - 2 / exp 1) ≤ 8 := by |
rw [div_le_iff, mul_sub, mul_one, mul_div_assoc', le_sub_comm, div_le_iff (exp_pos _)]
· have : 16 < 6 * (2.7182818283 : ℝ) := by norm_num
linarith [exp_one_gt_d9]
rw [sub_pos, div_lt_one] <;> exact exp_one_gt_d9.trans' (by norm_num)
|
/-
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 | 571 | 574 | theorem dist_le_infDist_add_diam (hs : IsBounded s) (hy : y ∈ s) :
dist x y ≤ infDist x s + diam s := by |
rw [infDist, diam, dist_edist]
exact toReal_le_add (edist_le_infEdist_add_ediam hy) (infEdist_ne_top ⟨y, hy⟩) hs.ediam_ne_top
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Na... | Mathlib/Data/Complex/Exponential.lean | 761 | 767 | theorem sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 := by |
have h1 : x + 2 * x = 3 * x := by ring
rw [← h1, sin_add x (2 * x)]
simp only [cos_two_mul, sin_two_mul, cos_sq']
have h2 : cos x * (2 * sin x * cos x) = 2 * sin x * cos x ^ 2 := by ring
rw [h2, cos_sq']
ring
|
/-
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 | 522 | 535 | theorem tendsto_zero_testAgainstNN_of_tendsto_zero_mass {γ : Type*} {F : Filter γ}
{μs : γ → FiniteMeasure Ω} (mass_lim : Tendsto (fun i => (μs i).mass) F (𝓝 0)) (f : Ω →ᵇ ℝ≥0) :
Tendsto (fun i => (μs i).testAgainstNN f) F (𝓝 0) := by |
apply tendsto_iff_dist_tendsto_zero.mpr
have obs := fun i => (μs i).testAgainstNN_lipschitz_estimate f 0
simp_rw [testAgainstNN_zero, zero_add] at obs
simp_rw [show ∀ i, dist ((μs i).testAgainstNN f) 0 = (μs i).testAgainstNN f by
simp only [dist_nndist, NNReal.nndist_zero_eq_val', eq_self_iff_true, imp_t... |
/-
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 | 653 | 660 | theorem localization_comap_injective [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
Function.Injective (comap (algebraMap R S)) := by |
intro p q h
replace h := congr_arg (fun x : PrimeSpectrum R => Ideal.map (algebraMap R S) x.asIdeal) h
dsimp only [comap, ContinuousMap.coe_mk] at h
rw [IsLocalization.map_comap M S, IsLocalization.map_comap M S] at h
ext1
exact h
|
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
#align_import combinatorics.simple_graph.subg... | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 686 | 695 | theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by |
intro H H' h
constructor
· intro
simp only [comap_verts, Set.mem_preimage]
apply h.1
· intro v w
simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff]
intro
apply h.2
|
/-
Copyright (c) 2022 Kevin H. Wilson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin H. Wilson
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Data.Set.Function
#align_import analysis.sum_integral_comparisons from "leanprover-community/... | Mathlib/Analysis/SumIntegralComparisons.lean | 156 | 159 | theorem MonotoneOn.sum_le_integral_Ico (hab : a ≤ b) (hf : MonotoneOn f (Set.Icc a b)) :
∑ x ∈ Finset.Ico a b, f x ≤ ∫ x in a..b, f x := by |
rw [← neg_le_neg_iff, ← Finset.sum_neg_distrib, ← intervalIntegral.integral_neg]
exact hf.neg.integral_le_sum_Ico hab
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Topology.UniformSpace.CompleteSeparated
import Mathlib.Topology.EMetricSpace.Lipschitz
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Topol... | Mathlib/Topology/MetricSpace/Antilipschitz.lean | 77 | 79 | theorem mul_le_nndist (hf : AntilipschitzWith K f) (x y : α) :
K⁻¹ * nndist x y ≤ nndist (f x) (f y) := by |
simpa only [div_eq_inv_mul] using NNReal.div_le_of_le_mul' (hf.le_mul_nndist x y)
|
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Ines Wright, Joachim Breitner
-/
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Solvable
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory... | Mathlib/GroupTheory/Nilpotent.lean | 638 | 645 | theorem nilpotencyClass_eq_quotient_center_plus_one [hH : IsNilpotent G] [Nontrivial G] :
Group.nilpotencyClass G = Group.nilpotencyClass (G ⧸ center G) + 1 := by |
rw [nilpotencyClass_quotient_center]
rcases h : Group.nilpotencyClass G with ⟨⟩
· exfalso
rw [nilpotencyClass_zero_iff_subsingleton] at h
apply false_of_nontrivial_of_subsingleton G
· simp
|
/-
Copyright (c) 2021 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Devon Tuma
-/
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Topology.Algebra.Order.Liminf... | Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean | 287 | 289 | theorem superpolynomialDecay_mul_param_iff (hk : Tendsto k l atTop) :
SuperpolynomialDecay l k (f * k) ↔ SuperpolynomialDecay l k f := by |
simpa [mul_comm k] using superpolynomialDecay_param_mul_iff f hk
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Ta... | Mathlib/RingTheory/Valuation/Basic.lean | 337 | 339 | theorem map_one_add_of_lt (h : v x < 1) : v (1 + x) = 1 := by |
rw [← v.map_one] at h
simpa only [v.map_one] using v.map_add_eq_of_lt_left h
|
/-
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.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Ma... | Mathlib/Computability/TuringMachine.lean | 587 | 590 | theorem Tape.move_left_mk' {Γ} [Inhabited Γ] (L R : ListBlank Γ) :
(Tape.mk' L R).move Dir.left = Tape.mk' L.tail (R.cons L.head) := by |
simp only [Tape.move, Tape.mk', ListBlank.head_cons, eq_self_iff_true, ListBlank.cons_head_tail,
and_self_iff, ListBlank.tail_cons]
|
/-
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.RepresentationTheory.Action.Limits
import Mathlib.RepresentationTheory.Action.Concrete
import Mathlib.CategoryTheory.Monoidal.FunctorCategory
import Ma... | Mathlib/RepresentationTheory/Action/Monoidal.lean | 98 | 100 | theorem leftUnitor_hom_hom {X : Action V G} : Hom.hom (λ_ X).hom = (λ_ X.V).hom := by |
dsimp
simp
|
/-
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.Gamma.Beta
import Mathlib.NumberTheory.LSeries.HurwitzZeta
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathli... | Mathlib/NumberTheory/LSeries/RiemannZeta.lean | 203 | 208 | theorem zeta_eq_tsum_one_div_nat_add_one_cpow {s : ℂ} (hs : 1 < re s) :
riemannZeta s = ∑' n : ℕ, 1 / (n + 1 : ℂ) ^ s := by |
have := zeta_eq_tsum_one_div_nat_cpow hs
rw [tsum_eq_zero_add] at this
· simpa [zero_cpow (Complex.ne_zero_of_one_lt_re hs)]
· rwa [Complex.summable_one_div_nat_cpow]
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
Coinductive formalization of unbounded computations.
-/
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
#align_import data.seq.computation from "le... | Mathlib/Data/Seq/Computation.lean | 759 | 773 | theorem bind_assoc (s : Computation α) (f : α → Computation β) (g : β → Computation γ) :
bind (bind s f) g = bind s fun x : α => bind (f x) g := by |
apply
eq_of_bisim fun c₁ c₂ =>
c₁ = c₂ ∨ ∃ s, c₁ = bind (bind s f) g ∧ c₂ = bind s fun x : α => bind (f x) g
· intro c₁ c₂ h
match c₁, c₂, h with
| _, c₂, Or.inl (Eq.refl _) => cases' destruct c₂ with b cb <;> simp
| _, _, Or.inr ⟨s, rfl, rfl⟩ =>
apply recOn s <;> intro s <;> simp
... |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanp... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 435 | 439 | theorem smul_sphere [Nontrivial E] (c : 𝕜) (x : E) {r : ℝ} (hr : 0 ≤ r) :
c • sphere x r = sphere (c • x) (‖c‖ * r) := by |
rcases eq_or_ne c 0 with (rfl | hc)
· simp [zero_smul_set, Set.singleton_zero, hr]
· exact smul_sphere' hc x r
|
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.ModelTheory.Substructures
#align_import model_theory.elementary_maps from "leanprover-community/mathlib"@"d11893b411... | Mathlib/ModelTheory/ElementaryMaps.lean | 98 | 100 | theorem map_formula (f : M ↪ₑ[L] N) {α : Type*} (φ : L.Formula α) (x : α → M) :
φ.Realize (f ∘ x) ↔ φ.Realize x := by |
rw [Formula.Realize, Formula.Realize, ← f.map_boundedFormula, Unique.eq_default (f ∘ default)]
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.FullSubcategory
import Mathlib.Catego... | Mathlib/CategoryTheory/Equivalence.lean | 159 | 163 | theorem counitInv_functor_comp (e : C ≌ D) (X : C) :
e.counitInv.app (e.functor.obj X) ≫ e.functor.map (e.unitInv.app X) = 𝟙 (e.functor.obj X) := by |
erw [Iso.inv_eq_inv (e.functor.mapIso (e.unitIso.app X) ≪≫ e.counitIso.app (e.functor.obj X))
(Iso.refl _)]
exact e.functor_unit_comp X
|
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Interval.Set.OrderIso
import Mathlib.Data.... | Mathlib/Order/UpperLower/Basic.lean | 1,417 | 1,419 | theorem coe_lowerClosure (s : Set α) : ↑(lowerClosure s) = ⋃ a ∈ s, Iic a := by |
ext
simp
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a... | Mathlib/Order/Interval/Finset/Basic.lean | 149 | 149 | theorem right_mem_Ioc : b ∈ Ioc a b ↔ a < b := by | simp only [mem_Ioc, and_true_iff, le_rfl]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex
#align_import analysis.special_function... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | 247 | 256 | theorem arctan_add {x y : ℝ} (h : x * y < 1) :
arctan x + arctan y = arctan ((x + y) / (1 - x * y)) := by |
rw [← arctan_tan (x := _ + _)]
· congr
conv_rhs => rw [← tan_arctan x, ← tan_arctan y]
exact tan_add' ⟨arctan_ne_mul_pi_div_two, arctan_ne_mul_pi_div_two⟩
· rw [neg_lt, neg_add, ← arctan_neg, ← arctan_neg]
rw [← neg_mul_neg] at h
exact arctan_add_arctan_lt_pi_div_two h
· exact arctan_add_arctan... |
/-
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.Data.Fin.VecNotation
import Mathlib.Logic.Embedding.Set
#align_import logic.equiv.fin from "leanprover-community/mathlib"@"bd835ef554f37ef9b804f0903089211f89cb3... | Mathlib/Logic/Equiv/Fin.lean | 445 | 449 | theorem coe_finRotate_of_ne_last {i : Fin n.succ} (h : i ≠ Fin.last n) :
(finRotate (n + 1) i : ℕ) = i + 1 := by |
rw [finRotate_succ_apply]
have : (i : ℕ) < n := Fin.val_lt_last h
exact Fin.val_add_one_of_lt this
|
/-
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 | 454 | 462 | theorem integrable_condexpIndSMul (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hs : MeasurableSet s)
(hμs : μ s ≠ ∞) (x : G) : Integrable (condexpIndSMul hm hs hμs x) μ := by |
refine
integrable_of_forall_fin_meas_le' hm (μ s * ‖x‖₊) (ENNReal.mul_lt_top hμs ENNReal.coe_ne_top) ?_
?_
· exact Lp.aestronglyMeasurable _
· refine fun t ht hμt => (set_lintegral_nnnorm_condexpIndSMul_le hm hs hμs x ht hμt).trans ?_
gcongr
apply Set.inter_subset_left
|
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Tactic.Abel
#align_import set_theory.ordinal.natural_ops from "leanprover-communit... | Mathlib/SetTheory/Ordinal/NaturalOps.lean | 799 | 799 | theorem nmul_succ (a b) : a ⨳ succ b = a ⨳ b ♯ a := by | rw [← nadd_one, nmul_nadd_one]
|
/-
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.MeasureSpace
/-!
# Restricting a measure to a subset or a subtype
Given a measure `μ` on a type `α` and a subse... | Mathlib/MeasureTheory/Measure/Restrict.lean | 243 | 247 | theorem restrict_inter_add_diff₀ (s : Set α) (ht : NullMeasurableSet t μ) :
μ.restrict (s ∩ t) + μ.restrict (s \ t) = μ.restrict s := by |
ext1 u hu
simp only [add_apply, restrict_apply hu, ← inter_assoc, diff_eq]
exact measure_inter_add_diff₀ (u ∩ s) ht
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.UniformEmbedding
import Mathlib.Topology.UniformSpace.Com... | Mathlib/Topology/Algebra/UniformGroup.lean | 679 | 692 | theorem comm_topologicalGroup_is_uniform : UniformGroup G := by |
have :
Tendsto
((fun p : G × G => p.1 / p.2) ∘ fun p : (G × G) × G × G => (p.1.2 / p.1.1, p.2.2 / p.2.1))
(comap (fun p : (G × G) × G × G => (p.1.2 / p.1.1, p.2.2 / p.2.1)) ((𝓝 1).prod (𝓝 1)))
(𝓝 (1 / 1)) :=
(tendsto_fst.div' tendsto_snd).comp tendsto_comap
constructor
rw [UniformCon... |
/-
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 | 826 | 839 | theorem infinite_pigeonhole_set {β α : Type u} {s : Set β} (f : s → α) (θ : Cardinal)
(hθ : θ ≤ #s) (h₁ : ℵ₀ ≤ θ) (h₂ : #α < θ.ord.cof) :
∃ (a : α) (t : Set β) (h : t ⊆ s), θ ≤ #t ∧ ∀ ⦃x⦄ (hx : x ∈ t), f ⟨x, h hx⟩ = a := by |
cases' infinite_pigeonhole_card f θ hθ h₁ h₂ with a ha
refine ⟨a, { x | ∃ h, f ⟨x, h⟩ = a }, ?_, ?_, ?_⟩
· rintro x ⟨hx, _⟩
exact hx
· refine
ha.trans
(ge_of_eq <|
Quotient.sound ⟨Equiv.trans ?_ (Equiv.subtypeSubtypeEquivSubtypeExists _ _).symm⟩)
simp only [coe_eq_subtype, mem_s... |
/-
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 | 668 | 671 | theorem erase_single_ne {a a' : α} {b : M} (h : a ≠ a') : erase a (single a' b) = single a' b := by |
ext s; by_cases hs : s = a
· rw [hs, erase_same, single_eq_of_ne h.symm]
· rw [erase_ne hs]
|
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.ZPow
#align_import linear_algebra.matrix.hermitian from "leanprover-commun... | Mathlib/LinearAlgebra/Matrix/Hermitian.lean | 252 | 253 | theorem IsHermitian.inv [Fintype m] [DecidableEq m] {A : Matrix m m α} (hA : A.IsHermitian) :
A⁻¹.IsHermitian := by | simp [IsHermitian, conjTranspose_nonsing_inv, hA.eq]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Data.Option.Defs
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Sigma.Basic
import Mathlib... | Mathlib/Logic/Equiv/Basic.lean | 2,075 | 2,078 | theorem update_comp_equiv [DecidableEq α'] [DecidableEq α] (f : α → β)
(g : α' ≃ α) (a : α) (v : β) :
update f a v ∘ g = update (f ∘ g) (g.symm a) v := by |
rw [← update_comp_eq_of_injective _ g.injective, g.apply_symm_apply]
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.UniformEmbedding
import Mathlib.Topology.UniformSpace.Com... | Mathlib/Topology/Algebra/UniformGroup.lean | 351 | 355 | theorem Filter.HasBasis.uniformity_of_nhds_one_inv_mul {ι} {p : ι → Prop} {U : ι → Set α}
(h : (𝓝 (1 : α)).HasBasis p U) :
(𝓤 α).HasBasis p fun i => { x : α × α | x.1⁻¹ * x.2 ∈ U i } := by |
rw [uniformity_eq_comap_inv_mul_nhds_one]
exact h.comap _
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#ali... | Mathlib/Algebra/Order/Group/Defs.lean | 165 | 166 | theorem Left.inv_lt_one_iff : a⁻¹ < 1 ↔ 1 < a := by |
rw [← mul_lt_mul_iff_left a, mul_inv_self, mul_one]
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
/-!
# Neig... | Mathlib/Topology/ContinuousOn.lean | 1,318 | 1,324 | theorem IsOpen.ite' {s s' t : Set α} (hs : IsOpen s) (hs' : IsOpen s')
(ht : ∀ x ∈ frontier t, x ∈ s ↔ x ∈ s') : IsOpen (t.ite s s') := by |
classical
simp only [isOpen_iff_continuous_mem, Set.ite] at *
convert continuous_piecewise (fun x hx => propext (ht x hx)) hs.continuousOn hs'.continuousOn
rename_i x
by_cases hx : x ∈ t <;> simp [hx]
|
/-
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, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 2,012 | 2,015 | theorem mulVec_smul_assoc [Fintype n] (A : Matrix m n α) (b : n → α) (a : α) :
A *ᵥ (a • b) = a • A *ᵥ b := by |
ext
apply dotProduct_smul
|
/-
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 | 369 | 372 | theorem linearIndependent_iff_range : LinearIndependent ℤ (GoodProducts.eval C) ↔
LinearIndependent ℤ (fun (p : range C) ↦ p.1) := by |
rw [← @Set.rangeFactorization_eq _ _ (GoodProducts.eval C), ← equiv_toFun_eq_eval C]
exact linearIndependent_equiv (equiv_range C)
|
/-
Copyright (c) 2018 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.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean | 403 | 409 | theorem Fork.equalizer_ext (s : Fork f g) {W : C} {k l : W ⟶ s.pt} (h : k ≫ s.ι = l ≫ s.ι) :
∀ j : WalkingParallelPair, k ≫ s.π.app j = l ≫ s.π.app j
| zero => h
| one => by
have : k ≫ ι s ≫ f = l ≫ ι s ≫ f := by |
simp only [← Category.assoc]; exact congrArg (· ≫ f) h
rw [s.app_one_eq_ι_comp_left, this]
|
/-
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.FieldTheory.IntermediateField
import Mathlib.RingTheory.Adjoin.Field
#align_import field_theory.splitting_field.is_splitting_field from "leanprover-commun... | Mathlib/FieldTheory/SplittingField/IsSplittingField.lean | 136 | 142 | theorem of_algEquiv [Algebra K F] (p : K[X]) (f : F ≃ₐ[K] L) [IsSplittingField K F p] :
IsSplittingField K L p := by |
constructor
· rw [← f.toAlgHom.comp_algebraMap]
exact splits_comp_of_splits _ _ (splits F p)
· rw [← (Algebra.range_top_iff_surjective f.toAlgHom).mpr f.surjective,
adjoin_rootSet_eq_range (splits F p), adjoin_rootSet F p]
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Devon Tuma
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.... | Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 98 | 101 | theorem map_scaleRoots (p : R[X]) (x : R) (f : R →+* S) (h : f p.leadingCoeff ≠ 0) :
(p.scaleRoots x).map f = (p.map f).scaleRoots (f x) := by |
ext
simp [Polynomial.natDegree_map_of_leadingCoeff_ne_zero _ h]
|
/-
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 | 2,321 | 2,324 | theorem isBigO_congr (e : α ≃ₜ β) {b : β} {f : β → E} {g : β → F} :
f =O[𝓝 b] g ↔ (f ∘ e) =O[𝓝 (e.symm b)] (g ∘ e) := by |
simp only [IsBigO_def]
exact exists_congr fun C => e.isBigOWith_congr
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Fabian Glöckle, Kyle Miller
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModu... | Mathlib/LinearAlgebra/Dual.lean | 320 | 326 | theorem toDual_total_right (f : ι →₀ R) (i : ι) :
b.toDual (b i) (Finsupp.total ι M R b f) = f i := by |
rw [Finsupp.total_apply, Finsupp.sum, _root_.map_sum]
simp_rw [LinearMap.map_smul, toDual_apply, smul_eq_mul, mul_boole, Finset.sum_ite_eq]
split_ifs with h
· rfl
· rw [Finsupp.not_mem_support_iff.mp h]
|
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Plus
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
#align_import category_theory.sites.sheafification from "leanprover-com... | Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean | 483 | 486 | theorem sheafifyMap_comp {P Q R : Cᵒᵖ ⥤ D} (η : P ⟶ Q) (γ : Q ⟶ R) :
J.sheafifyMap (η ≫ γ) = J.sheafifyMap η ≫ J.sheafifyMap γ := by |
dsimp [sheafifyMap, sheafify]
simp
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.... | Mathlib/RingTheory/WittVector/Identities.lean | 57 | 61 | theorem coeff_p_pow [CharP R p] (i : ℕ) : ((p : 𝕎 R) ^ i).coeff i = 1 := by |
induction' i with i h
· simp only [Nat.zero_eq, one_coeff_zero, Ne, pow_zero]
· rw [pow_succ, ← frobenius_verschiebung, coeff_frobenius_charP,
verschiebung_coeff_succ, h, one_pow]
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Top... | Mathlib/Analysis/Normed/Group/Basic.lean | 1,901 | 1,902 | theorem ennnorm_eq_ofReal_abs (r : ℝ) : (‖r‖₊ : ℝ≥0∞) = ENNReal.ofReal |r| := by |
rw [← Real.nnnorm_abs r, Real.ennnorm_eq_ofReal (abs_nonneg _)]
|
/-
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.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
... | Mathlib/Algebra/Order/ToIntervalMod.lean | 349 | 350 | theorem toIocDiv_sub (a b : α) : toIocDiv hp a (b - p) = toIocDiv hp a b - 1 := by |
simpa only [one_zsmul] using toIocDiv_sub_zsmul hp a b 1
|
/-
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.Group.Equiv.Basic
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.Opposites
import Mathlib.Algebra.Order.GroupW... | Mathlib/Data/Set/Pointwise/Basic.lean | 1,210 | 1,211 | theorem image_mul_right : (· * b) '' t = (· * b⁻¹) ⁻¹' t := by |
rw [image_eq_preimage_of_inverse] <;> intro c <;> simp
|
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.SetIntegral... | Mathlib/MeasureTheory/Constructions/Prod/Integral.lean | 280 | 285 | theorem integrable_prod_iff' [SigmaFinite μ] ⦃f : α × β → E⦄
(h1f : AEStronglyMeasurable f (μ.prod ν)) :
Integrable f (μ.prod ν) ↔
(∀ᵐ y ∂ν, Integrable (fun x => f (x, y)) μ) ∧ Integrable (fun y => ∫ x, ‖f (x, y)‖ ∂μ) ν := by |
convert integrable_prod_iff h1f.prod_swap using 1
rw [funext fun _ => Function.comp_apply.symm, integrable_swap_iff]
|
/-
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 | 1,170 | 1,174 | theorem support_single_ne_zero {i : ι} {b : β i} (hb : b ≠ 0) : (single i b).support = {i} := by |
ext j; by_cases h : i = j
· subst h
simp [hb]
simp [Ne.symm h, h]
|
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Scott Morrison
-/
import Mathlib.Algebra.Homology.Exact
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.Adjunction.Limits
import Math... | Mathlib/CategoryTheory/Preadditive/Projective.lean | 217 | 223 | theorem projective_of_map_projective (adj : F ⊣ G) [F.Full] [F.Faithful] (P : C)
(hP : Projective (F.obj P)) : Projective P where
factors f g _ := by |
haveI := Adjunction.leftAdjointPreservesColimits.{0, 0} adj
rcases (@hP).1 (F.map f) (F.map g) with ⟨f', hf'⟩
use adj.unit.app _ ≫ G.map f' ≫ (inv <| adj.unit.app _)
exact F.map_injective (by simpa)
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.FractionalIdeal.Basic
#align_import ring_theory.fractional_ideal from "leanprover... | Mathlib/RingTheory/FractionalIdeal/Operations.lean | 128 | 130 | theorem map_symm_map (I : FractionalIdeal S P') (g : P ≃ₐ[R] P') :
(I.map (g.symm : P' →ₐ[R] P)).map (g : P →ₐ[R] P') = I := by |
rw [← map_comp, g.comp_symm, map_id]
|
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