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) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Separation
/-!
# Order-closed topologies
In this file we introduce 3 typeclass mixins that relate topology ... | Mathlib/Topology/Order/OrderClosed.lean | 610 | 612 | theorem continuousWithinAt_Icc_iff_Ici (h : a < b) :
ContinuousWithinAt f (Icc a b) a ↔ ContinuousWithinAt f (Ici a) a := by |
simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Ici h]
|
/-
Copyright (c) 2023 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Topology.Connected.Basic
import Mathlib.Topology.Separation
/-!
# Separated maps and locally injective maps out of a topological space.
This module introduces a... | Mathlib/Topology/SeparatedMap.lean | 155 | 160 | theorem discreteTopology_iff_locallyInjective (y : Y) :
DiscreteTopology X ↔ IsLocallyInjective fun _ : X ↦ y := by |
rw [discreteTopology_iff_singleton_mem_nhds, isLocallyInjective_iff_nhds]
refine forall_congr' fun x ↦ ⟨fun h ↦ ⟨{x}, h, Set.injOn_singleton _ _⟩, fun ⟨U, hU, inj⟩ ↦ ?_⟩
convert hU; ext x'; refine ⟨?_, fun h ↦ inj h (mem_of_mem_nhds hU) rfl⟩
rintro rfl; exact mem_of_mem_nhds 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
-/
import Mathlib.MeasureTheory.Measure.Restrict
/-!
# Classes of measures
We introduce the following typeclasses for measures:
* `IsProbabilityMeasur... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 983 | 989 | theorem iSup_restrict_spanningSets [SigmaFinite μ] (s : Set α) :
⨆ i, μ.restrict (spanningSets μ i) s = μ s := by |
rw [← measure_toMeasurable s,
← iSup_restrict_spanningSets_of_measurableSet (measurableSet_toMeasurable _ _)]
simp_rw [restrict_apply' (measurable_spanningSets μ _), Set.inter_comm s,
← restrict_apply (measurable_spanningSets μ _), ← restrict_toMeasurable_of_sFinite s,
restrict_apply (measurable_spanni... |
/-
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 | 721 | 722 | theorem disjoint_principal_principal {s t : Set α} : Disjoint (𝓟 s) (𝓟 t) ↔ Disjoint s t := by |
rw [← subset_compl_iff_disjoint_left, disjoint_principal_left, mem_principal]
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Data.PNat.Defs
#align_import data.pnat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecf... | Mathlib/Data/PNat/Interval.lean | 118 | 119 | theorem card_fintype_Ioc : Fintype.card (Set.Ioc a b) = b - a := by |
rw [← card_Ioc, Fintype.card_ofFinset]
|
/-
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.Init.Core
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.affine_space.finite_d... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 395 | 399 | theorem collinear_iff_finrank_le_one {s : Set P} [FiniteDimensional k (vectorSpan k s)] :
Collinear k s ↔ finrank k (vectorSpan k s) ≤ 1 := by |
have h := collinear_iff_rank_le_one k s
rw [← finrank_eq_rank] at h
exact mod_cast h
|
/-
Copyright (c) 2022 Jiale Miao. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jiale Miao, Kevin Buzzard, Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.Block
#align_import analysis.inner_product_space.gram_s... | Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | 395 | 399 | theorem gramSchmidtOrthonormalBasis_det [DecidableEq ι] :
(gramSchmidtOrthonormalBasis h f).toBasis.det f =
∏ i, ⟪gramSchmidtOrthonormalBasis h f i, f i⟫ := by |
convert Matrix.det_of_upperTriangular (gramSchmidtOrthonormalBasis_inv_blockTriangular h f)
exact ((gramSchmidtOrthonormalBasis h f).repr_apply_apply (f _) _).symm
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.PrimitiveElement
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Matrix.Char... | Mathlib/RingTheory/Norm.lean | 313 | 322 | theorem isIntegral_norm [Algebra R L] [Algebra R K] [IsScalarTower R K L] [IsSeparable K L]
[FiniteDimensional K L] {x : L} (hx : IsIntegral R x) : IsIntegral R (norm K x) := by |
have hx' : IsIntegral K x := hx.tower_top
rw [← isIntegral_algebraMap_iff (algebraMap K (AlgebraicClosure L)).injective, norm_eq_prod_roots]
· refine (IsIntegral.multiset_prod fun y hy => ?_).pow _
rw [mem_roots_map (minpoly.ne_zero hx')] at hy
use minpoly R x, minpoly.monic hx
rw [← aeval_def] at hy... |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Data.Finset.Image
#align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb8... | Mathlib/Data/Finset/Card.lean | 697 | 699 | theorem card_le_one_iff : s.card ≤ 1 ↔ ∀ {a b}, a ∈ s → b ∈ s → a = b := by |
rw [card_le_one]
tauto
|
/-
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 | 146 | 148 | theorem mulRight_mul (a b : A) : mulRight R (a * b) = (mulRight R b).comp (mulRight R a) := by |
ext
simp only [mulRight_apply, comp_apply, mul_assoc]
|
/-
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.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.r... | Mathlib/Data/ENNReal/Operations.lean | 71 | 77 | theorem mul_left_strictMono (h0 : a ≠ 0) (hinf : a ≠ ∞) : StrictMono (a * ·) := by |
lift a to ℝ≥0 using hinf
rw [coe_ne_zero] at h0
intro x y h
contrapose! h
simpa only [← mul_assoc, ← coe_mul, inv_mul_cancel h0, coe_one, one_mul]
using mul_le_mul_left' h (↑a⁻¹)
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Mario Carneiro, Sean Leather
-/
import Mathlib.Data.Finset.Card
#align_import data.finset.option from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0... | Mathlib/Data/Finset/Option.lean | 105 | 108 | theorem eraseNone_eq_biUnion [DecidableEq α] (s : Finset (Option α)) :
eraseNone s = s.biUnion Option.toFinset := by |
ext
simp
|
/-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton
-/
import Mathlib.Topology.Bases
import Mathlib.Topology.DenseEmbedding
#align_import topology.stone_cech from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44... | Mathlib/Topology/StoneCech.lean | 110 | 117 | theorem ultrafilter_comap_pure_nhds (b : Ultrafilter α) : comap pure (𝓝 b) ≤ b := by |
rw [TopologicalSpace.nhds_generateFrom]
simp only [comap_iInf, comap_principal]
intro s hs
rw [← le_principal_iff]
refine iInf_le_of_le { u | s ∈ u } ?_
refine iInf_le_of_le ⟨hs, ⟨s, rfl⟩⟩ ?_
exact principal_mono.2 fun a => id
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot
-/
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Dual
import Mathlib.Data.Fin.FlagRange
/-!
# Flag of submodules defined by a basis
... | Mathlib/LinearAlgebra/Basis/Flag.lean | 35 | 36 | theorem flag_last (b : Basis (Fin n) R M) : b.flag (.last n) = ⊤ := by |
simp [flag, Fin.castSucc_lt_last]
|
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez, Eric Wieser
-/
import Mathlib.Data.List.Chain
#align_import data.list.destutter from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
/-!
# D... | Mathlib/Data/List/Destutter.lean | 48 | 49 | theorem destutter'_cons_pos (h : R b a) : (a :: l).destutter' R b = b :: l.destutter' R a := by |
rw [destutter', if_pos h]
|
/-
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 | 164 | 169 | theorem coe_vanishingIdeal (t : Set (PrimeSpectrum R)) :
(vanishingIdeal t : Set R) = { f : R | ∀ x : PrimeSpectrum R, x ∈ t → f ∈ x.asIdeal } := by |
ext f
rw [vanishingIdeal, SetLike.mem_coe, Submodule.mem_iInf]
apply forall_congr'; intro x
rw [Submodule.mem_iInf]
|
/-
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, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 814 | 814 | theorem div_mul_eq_div_mul_one_div : a / (b * c) = a / b * (1 / c) := by | simp
|
/-
Copyright (c) 2020 James Arthur. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: James Arthur, Chris Hughes, Shing Tak Lam
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.speci... | Mathlib/Analysis/SpecialFunctions/Arsinh.lean | 181 | 184 | theorem hasStrictDerivAt_arsinh (x : ℝ) : HasStrictDerivAt arsinh (√(1 + x ^ 2))⁻¹ x := by |
convert sinhHomeomorph.toPartialHomeomorph.hasStrictDerivAt_symm (mem_univ x) (cosh_pos _).ne'
(hasStrictDerivAt_sinh _) using 2
exact (cosh_arsinh _).symm
|
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Data.ZMod.Quotient
#align_import group_theory.complement from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
/-!
# Compl... | Mathlib/GroupTheory/Complement.lean | 713 | 717 | theorem isComplement'_of_disjoint_and_mul_eq_univ (h1 : Disjoint H K)
(h2 : ↑H * ↑K = (Set.univ : Set G)) : IsComplement' H K := by |
refine ⟨mul_injective_of_disjoint h1, fun g => ?_⟩
obtain ⟨h, hh, k, hk, hg⟩ := Set.eq_univ_iff_forall.mp h2 g
exact ⟨(⟨h, hh⟩, ⟨k, hk⟩), hg⟩
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
#align_import analysis.ca... | Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 701 | 707 | theorem fderivWithin_univ : fderivWithin 𝕜 f univ = fderiv 𝕜 f := by |
ext1 x
nontriviality E
have H : 𝓝[univ \ {x}] x ≠ ⊥ := by
rw [← compl_eq_univ_diff, ← neBot_iff]
exact Module.punctured_nhds_neBot 𝕜 E x
simp [fderivWithin, fderiv, H]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.Tactic.SuppressCo... | Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 1,625 | 1,627 | theorem rTensor_neg (f : N →ₗ[R] P) : (-f).rTensor M = -f.rTensor M := by |
simp only [← coe_rTensorHom]
exact (rTensorHom (R := R) (N := N) (P := P) M).map_neg f
|
/-
Copyright (c) 2018 Sean Leather. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sean Leather, Mario Carneiro
-/
import Mathlib.Data.List.AList
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Part
#align_import data.finmap from "leanprover-community/mathlib"@"c... | Mathlib/Data/Finmap.lean | 134 | 136 | theorem liftOn₂_toFinmap {γ} (s₁ s₂ : AList β) (f : AList β → AList β → γ) (H) :
liftOn₂ ⟦s₁⟧ ⟦s₂⟧ f H = f s₁ s₂ := by |
cases s₁; cases s₂; rfl
|
/-
Copyright (c) 2023 Michael Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Lee
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Data.Finset.Card
impo... | Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean | 245 | 269 | theorem psum_eq_mul_esymm_sub_sum (k : ℕ) (h : 0 < k) : psum σ R k =
(-1) ^ (k + 1) * k * esymm σ R k -
∑ a ∈ (antidiagonal k).filter (fun a ↦ a.fst ∈ Set.Ioo 0 k),
(-1) ^ a.fst * esymm σ R a.fst * psum σ R a.snd := by |
simp only [Set.Ioo, Set.mem_setOf_eq, and_comm]
have hesymm := mul_esymm_eq_sum σ R k
rw [← (sum_filter_add_sum_filter_not ((antidiagonal k).filter (fun a ↦ a.fst < k))
(fun a ↦ 0 < a.fst) (fun a ↦ (-1) ^ a.fst * esymm σ R a.fst * psum σ R a.snd))] at hesymm
have sub_both_sides := congrArg (· - (-1 : MvPol... |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Mario Carneiro, Yury Kudryashov, Heather Macbeth
-/
import Mathlib.Algebra.Module.MinimalAxioms
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.... | Mathlib/Topology/ContinuousFunction/Bounded.lean | 1,655 | 1,659 | theorem abs_self_eq_nnrealPart_add_nnrealPart_neg (f : α →ᵇ ℝ) :
abs ∘ ⇑f = (↑) ∘ f.nnrealPart + (↑) ∘ (-f).nnrealPart := by |
funext x
dsimp
simp only [max_zero_add_max_neg_zero_eq_abs_self]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.ChartedSpace
#align_import geometry.manifold.local_invariant_properties from "leanprover-community/mathlib"@... | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 283 | 289 | theorem liftPropWithinAt_indep_chart_target_aux {g : X → M'} {e : PartialHomeomorph X H} {x : X}
{s : Set X} (xe : x ∈ e.source) (hf : f ∈ G'.maximalAtlas M') (xf : g x ∈ f.source)
(hf' : f' ∈ G'.maximalAtlas M') (xf' : g x ∈ f'.source) (hgs : ContinuousWithinAt g s x) :
P (f ∘ g ∘ e.symm) (e.symm ⁻¹' s) (e... |
rw [← e.left_inv xe] at xf xf' hgs
refine hG.liftPropWithinAt_indep_chart_target_aux2 (g ∘ e.symm) hf xf hf' xf' ?_
exact hgs.comp (e.symm.continuousAt <| e.mapsTo xe).continuousWithinAt Subset.rfl
|
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.NormedSpace.BallAction
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.... | Mathlib/Geometry/Manifold/Instances/Sphere.lean | 368 | 369 | theorem stereographic'_target {n : ℕ} [Fact (finrank ℝ E = n + 1)] (v : sphere (0 : E) 1) :
(stereographic' n v).target = Set.univ := by | simp [stereographic']
|
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Anatole Dedecker
-/
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
import Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | 394 | 397 | theorem toLinearEquiv_toContinuousLinearEquiv_symm (e : E ≃ₗ[𝕜] F) :
e.toContinuousLinearEquiv.symm.toLinearEquiv = e.symm := by |
ext x
rfl
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Ta... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 392 | 393 | theorem get?_set_ne (a : α) {m n} (l : List α) (h : m ≠ n) : (set l m a).get? n = l.get? n := by |
simp only [set_eq_modifyNth, get?_modifyNth_ne _ _ h]
|
/-
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 | 187 | 191 | theorem abs_rpow_le_exp_log_mul (x y : ℝ) : |x ^ y| ≤ exp (log x * y) := by |
refine (abs_rpow_le_abs_rpow x y).trans ?_
by_cases hx : x = 0
· by_cases hy : y = 0 <;> simp [hx, hy, zero_le_one]
· rw [rpow_def_of_pos (abs_pos.2 hx), log_abs]
|
/-
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 | 1,010 | 1,013 | theorem isInt_rpow_neg {a b : ℝ} {nb ne : ℕ}
(pb : IsInt b (Int.negOfNat nb)) (pe' : IsInt (a ^ (Int.negOfNat nb)) (Int.negOfNat ne)) :
IsInt (a ^ b) (Int.negOfNat ne) := by |
rwa [pb.out, Real.rpow_intCast]
|
/-
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 | 352 | 359 | theorem _root_.HasCompactSupport.convolutionExists_left_of_continuous_right
(hcf : HasCompactSupport f) (hf : LocallyIntegrable f μ) (hg : Continuous g) :
ConvolutionExists f g L μ := by |
intro x₀
refine HasCompactSupport.convolutionExistsAt L ?_ hf hg
refine hcf.mono ?_
refine fun t => mt fun ht : f t = 0 => ?_
simp_rw [ht, L.map_zero₂]
|
/-
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.Set.Countable
import Mathlib.Order.Disjointed
import Mathlib.Tactic.Measurability
#align_import measure_theory.measurable_space_d... | Mathlib/MeasureTheory/MeasurableSpace/Defs.lean | 530 | 532 | theorem measurableSet_iSup {ι} {m : ι → MeasurableSpace α} {s : Set α} :
MeasurableSet[iSup m] s ↔ GenerateMeasurable { s : Set α | ∃ i, MeasurableSet[m i] s } s := by |
simp only [iSup, measurableSet_sSup, exists_range_iff]
|
/-
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.BigOperators.Group.Finset
import Mathlib.Algebra.Module.LinearMap.Basic
#align_import algebra.algebra.basic from "leanprover-community/... | Mathlib/Algebra/Algebra/Defs.lean | 388 | 390 | theorem _root_.smul_algebraMap {α : Type*} [Monoid α] [MulDistribMulAction α A]
[SMulCommClass α R A] (a : α) (r : R) : a • algebraMap R A r = algebraMap R A r := by |
rw [algebraMap_eq_smul_one, smul_comm a r (1 : A), smul_one]
|
/-
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 Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Gr... | Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 302 | 307 | theorem measure_eq_tsum_of_ac (h : IsFundamentalDomain G s μ) (hν : ν ≪ μ) (t : Set α) :
ν t = ∑' g : G, ν (t ∩ g • s) := by |
have H : ν.restrict t ≪ μ := Measure.restrict_le_self.absolutelyContinuous.trans hν
simpa only [set_lintegral_one, Pi.one_def,
Measure.restrict_apply₀ ((h.nullMeasurableSet_smul _).mono_ac H), inter_comm] using
h.lintegral_eq_tsum_of_ac H 1
|
/-
Copyright (c) 2021 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Thomas Browning
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import dat... | Mathlib/Data/Nat/Choose/Central.lean | 124 | 126 | theorem two_dvd_centralBinom_of_one_le {n : ℕ} (h : 0 < n) : 2 ∣ centralBinom n := by |
rw [← Nat.succ_pred_eq_of_pos h]
exact two_dvd_centralBinom_succ n.pred
|
/-
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.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.G... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 240 | 248 | theorem map_le_lineMap_iff_slope_le_slope_right (h : 0 < (1 - r) * (b - a)) :
f c ≤ lineMap (f a) (f b) r ↔ slope f a b ≤ slope f c b := by |
rw [← lineMap_apply_one_sub, ← lineMap_apply_one_sub _ _ r]
revert h; generalize 1 - r = r'; clear! r; intro h
simp_rw [lineMap_apply, slope, vsub_eq_sub, vadd_eq_add, smul_eq_mul]
rw [sub_add_eq_sub_sub_swap, sub_self, zero_sub, neg_mul_eq_mul_neg, neg_sub,
le_inv_smul_iff_of_pos h, smul_smul, mul_inv_can... |
/-
Copyright (c) 2019 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Data.Bracket
import Mathlib.LinearAlgebra.Basic
#align_import algebra.lie.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd298... | Mathlib/Algebra/Lie/Basic.lean | 802 | 804 | theorem coe_injective : @Function.Injective (M →ₗ⁅R,L⁆ N) (M → N) (↑) := by |
rintro ⟨⟨⟨f, _⟩⟩⟩ ⟨⟨⟨g, _⟩⟩⟩ h
congr
|
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Topology.Basic
import Mathlib.Order.UpperLower.Basic
import Mathlib.Order.OmegaCompletePartialOrder
#align_import topology.omega_complete_partial_order from... | Mathlib/Topology/OmegaCompletePartialOrder.lean | 116 | 129 | theorem scottContinuous_of_continuous {α β} [OmegaCompletePartialOrder α]
[OmegaCompletePartialOrder β] (f : Scott α → Scott β) (hf : Continuous f) :
OmegaCompletePartialOrder.Continuous' f := by |
have h : Monotone f := fun x y h ↦ by
have hf : IsUpperSet {x | ¬f x ≤ f y} := ((notBelow_isOpen (f y)).preimage hf).isUpperSet
simpa only [mem_setOf_eq, le_refl, not_true, imp_false, not_not] using hf h
refine ⟨h, fun c ↦ eq_of_forall_ge_iff fun z ↦ ?_⟩
rcases (notBelow_isOpen z).preimage hf with ⟨hf, h... |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Grou... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 93 | 94 | theorem mem_rootsOfUnity' (k : ℕ+) (ζ : Mˣ) : ζ ∈ rootsOfUnity k M ↔ (ζ : M) ^ (k : ℕ) = 1 := by |
rw [mem_rootsOfUnity]; norm_cast
|
/-
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 | 896 | 899 | theorem coprod.map_map {A₁ A₂ A₃ B₁ B₂ B₃ : C} [HasBinaryCoproduct A₁ B₁] [HasBinaryCoproduct A₂ B₂]
[HasBinaryCoproduct A₃ B₃] (f : A₁ ⟶ A₂) (g : B₁ ⟶ B₂) (h : A₂ ⟶ A₃) (k : B₂ ⟶ B₃) :
coprod.map f g ≫ coprod.map h k = coprod.map (f ≫ h) (g ≫ k) := by |
ext <;> simp
|
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.Tactic.WLOG
#align_import set_theory.cardinal.divisibility from "leanprove... | Mathlib/SetTheory/Cardinal/Divisibility.lean | 100 | 108 | theorem nat_coe_dvd_iff : (n : Cardinal) ∣ m ↔ n ∣ m := by |
refine ⟨?_, fun ⟨h, ht⟩ => ⟨h, mod_cast ht⟩⟩
rintro ⟨k, hk⟩
have : ↑m < ℵ₀ := nat_lt_aleph0 m
rw [hk, mul_lt_aleph0_iff] at this
rcases this with (h | h | ⟨-, hk'⟩)
iterate 2 simp only [h, mul_zero, zero_mul, Nat.cast_eq_zero] at hk; simp [hk]
lift k to ℕ using hk'
exact ⟨k, mod_cast hk⟩
|
/-
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.Init.ZeroOne
import Mathlib.Data.Set.Defs
import Mathlib.Order.Basic
import Mathlib.Order.SymmDiff
import Mathlib.Tactic.Tauto
import ... | Mathlib/Data/Set/Basic.lean | 1,976 | 1,977 | theorem insert_inter_of_mem (h : a ∈ t) : insert a s ∩ t = insert a (s ∩ t) := by |
rw [insert_inter_distrib, insert_eq_of_mem h]
|
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Set.Pointwise.Basic
#align_import data.set.pointwise.big_operators from "leanprover-community/mathlib"... | Mathlib/Data/Set/Pointwise/BigOperators.lean | 125 | 129 | theorem multiset_prod_mem_multiset_prod (t : Multiset ι) (f : ι → Set α) (g : ι → α)
(hg : ∀ i ∈ t, g i ∈ f i) : (t.map g).prod ∈ (t.map f).prod := by |
induction t using Quotient.inductionOn
simp_rw [Multiset.quot_mk_to_coe, Multiset.map_coe, Multiset.prod_coe]
exact list_prod_mem_list_prod _ _ _ hg
|
/-
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 | 210 | 213 | theorem midpoint_add_self (x y : V) : midpoint R x y + midpoint R x y = x + y :=
calc
midpoint R x y +ᵥ midpoint R x y = midpoint R x y +ᵥ midpoint R y x := by | rw [midpoint_comm]
_ = x + y := by rw [midpoint_vadd_midpoint, vadd_eq_add, vadd_eq_add, add_comm, midpoint_self]
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Data.List.Join
#align_import data.list.permutation from "leanprover-community/... | Mathlib/Data/List/Permutation.lean | 104 | 108 | theorem map_permutationsAux2 (t : α) (ts : List α) (ys : List α) (f : List α → β) :
(permutationsAux2 t ts [] ys id).2.map f = (permutationsAux2 t ts [] ys f).2 := by |
rw [map_permutationsAux2' id, map_id, map_id]
· rfl
simp
|
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import comp... | Mathlib/Computability/TMToPartrec.lean | 211 | 212 | theorem pred_eval (v) : pred.eval v = pure [v.headI.pred] := by |
simp [pred]; cases v.headI <;> simp
|
/-
Copyright (c) 2023 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Ev... | Mathlib/Algebra/Polynomial/Smeval.lean | 88 | 90 | theorem smeval_X :
(X : R[X]).smeval x = x ^ 1 := by |
simp only [smeval_eq_sum, smul_pow, zero_smul, sum_X_index, one_smul]
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Antichain
import Mathlib.Order.UpperLower.Basic
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.RelIso.Set
#align_import order.minimal ... | Mathlib/Order/Minimal.lean | 411 | 415 | theorem inter_preimage_minimals_eq_of_rel_iff_rel_on_of_subset
(hf : ∀ ⦃a a'⦄, a ∈ x → a' ∈ x → (r a a' ↔ s (f a) (f a'))) (hy : y ⊆ f '' x) :
x ∩ f ⁻¹' (minimals s y) = minimals r (x ∩ f ⁻¹' y) := by |
rw [← inter_eq_self_of_subset_right hy, inter_minimals_preimage_inter_eq_of_rel_iff_rel_on hf,
preimage_inter, ← inter_assoc, inter_eq_self_of_subset_left (subset_preimage_image f x)]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Joey van Langen, Casper Putz
-/
import Mathlib.FieldTheory.Separable
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Tactic.App... | Mathlib/FieldTheory/Finite/Basic.lean | 424 | 432 | theorem Nat.sq_add_sq_zmodEq (p : ℕ) [Fact p.Prime] (x : ℤ) :
∃ a b : ℕ, a ≤ p / 2 ∧ b ≤ p / 2 ∧ (a : ℤ) ^ 2 + (b : ℤ) ^ 2 ≡ x [ZMOD p] := by |
rcases ZMod.sq_add_sq p x with ⟨a, b, hx⟩
refine ⟨a.valMinAbs.natAbs, b.valMinAbs.natAbs, ZMod.natAbs_valMinAbs_le _,
ZMod.natAbs_valMinAbs_le _, ?_⟩
rw [← a.coe_valMinAbs, ← b.coe_valMinAbs] at hx
push_cast
rw [sq_abs, sq_abs, ← ZMod.intCast_eq_intCast_iff]
exact mod_cast hx
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Grou... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 478 | 486 | theorem pow_mul_pow_lcm {ζ' : M} {k' : ℕ} (hζ : IsPrimitiveRoot ζ k) (hζ' : IsPrimitiveRoot ζ' k')
(hk : k ≠ 0) (hk' : k' ≠ 0) :
IsPrimitiveRoot
(ζ ^ (k / Nat.factorizationLCMLeft k k') * ζ' ^ (k' / Nat.factorizationLCMRight k k'))
(Nat.lcm k k') := by |
convert IsPrimitiveRoot.orderOf _
convert ((Commute.all ζ ζ').orderOf_mul_pow_eq_lcm
(by simpa [← hζ.eq_orderOf]) (by simpa [← hζ'.eq_orderOf])).symm using 2
all_goals simp [hζ.eq_orderOf, hζ'.eq_orderOf]
|
/-
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 | 2,027 | 2,029 | theorem div (hf : LipschitzWith Kf f) (hg : LipschitzWith Kg g) :
LipschitzWith (Kf + Kg) fun x => f x / g x := by |
simpa only [div_eq_mul_inv] using hf.mul' hg.inv
|
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Topology.UniformSpace.Equiv
#align_import topology... | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | 1,087 | 1,095 | theorem isClosed_setOf_continuous_of_le [t : TopologicalSpace α]
(h : t ≤ ⨆ s ∈ 𝔖, .coinduced (Subtype.val : s → α) inferInstance) :
IsClosed {f : α →ᵤ[𝔖] β | Continuous (toFun 𝔖 f)} := by |
refine isClosed_iff_forall_filter.2 fun f u _ hu huf ↦ ?_
rw [← tendsto_id', UniformOnFun.tendsto_iff_tendstoUniformlyOn] at huf
have hcont : ∀ s ∈ 𝔖, ContinuousOn f s := fun s hs ↦
(huf s hs).continuousOn <| hu fun _ ↦ Continuous.continuousOn
refine continuous_le_dom h ?_
simpa only [continuous_iSup_do... |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
#align_import data.int.bitwise from "leanprover-community/mathlib"@"0743cc... | Mathlib/Data/Int/Bitwise.lean | 257 | 259 | theorem bit_negSucc (b) (n : ℕ) : bit b -[n+1] = -[Nat.bit (not b) n+1] := by |
rw [bit_val, Nat.bit_val]
cases b <;> rfl
|
/-
Copyright (c) 2023 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Bicategory.Coherence
/-!
# Adjunctions in bicategories
For 1-morphisms `f : a ⟶ b` and `g : b... | Mathlib/CategoryTheory/Bicategory/Adjunction.lean | 79 | 91 | theorem rightZigzag_idempotent_of_left_triangle
(η : 𝟙 a ⟶ f ≫ g) (ε : g ≫ f ⟶ 𝟙 b) (h : leftZigzag η ε = (λ_ _).hom ≫ (ρ_ _).inv) :
rightZigzag η ε ⊗≫ rightZigzag η ε = rightZigzag η ε := by |
dsimp only [rightZigzag]
calc
_ = g ◁ η ⊗≫ ((ε ▷ g ▷ 𝟙 a) ≫ (𝟙 b ≫ g) ◁ η) ⊗≫ ε ▷ g := by
simp [bicategoricalComp]; coherence
_ = 𝟙 _ ⊗≫ g ◁ (η ▷ 𝟙 a ≫ (f ≫ g) ◁ η) ⊗≫ (ε ▷ (g ≫ f) ≫ 𝟙 b ◁ ε) ▷ g ⊗≫ 𝟙 _ := by
rw [← whisker_exchange]; simp [bicategoricalComp]; coherence
_ = g ◁ η ⊗≫ g ... |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Sheaf
#align_import category_theory.sites.plus from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# The... | Mathlib/CategoryTheory/Sites/Plus.lean | 316 | 320 | theorem plusLift_unique {P Q : Cᵒᵖ ⥤ D} (η : P ⟶ Q) (hQ : Presheaf.IsSheaf J Q)
(γ : J.plusObj P ⟶ Q) (hγ : J.toPlus P ≫ γ = η) : γ = J.plusLift η hQ := by |
dsimp only [plusLift]
rw [Iso.eq_comp_inv, ← hγ, plusMap_comp]
simp
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Gluing
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.CategoryTheory.Limits.Sh... | Mathlib/AlgebraicGeometry/Pullbacks.lean | 249 | 251 | theorem p_comm : p1 𝒰 f g ≫ f = p2 𝒰 f g ≫ g := by |
apply Multicoequalizer.hom_ext
simp [p1, p2, pullback.condition]
|
/-
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.Set.Lattice
#align_import data.semiquot from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-! # Semiquotients
A da... | Mathlib/Data/Semiquot.lean | 47 | 50 | theorem ext_s {q₁ q₂ : Semiquot α} : q₁ = q₂ ↔ q₁.s = q₂.s := by |
refine ⟨congr_arg _, fun h => ?_⟩
cases' q₁ with _ v₁; cases' q₂ with _ v₂; congr
exact Subsingleton.helim (congrArg Trunc (congrArg Set.Elem h)) v₁ v₂
|
/-
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 | 1,156 | 1,157 | theorem neg_smul_set : -a • t = -(a • t) := by |
simp_rw [← image_smul, ← image_neg, image_image, neg_smul]
|
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll, Ralf Stephan
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Squarefree
/-!
# Smooth numbers
For `s : Finset ℕ` we define the set `Nat.factoredNu... | Mathlib/NumberTheory/SmoothNumbers.lean | 334 | 337 | theorem mul_mem_smoothNumbers {m₁ m₂ n : ℕ}
(hm1 : m₁ ∈ n.smoothNumbers) (hm2 : m₂ ∈ n.smoothNumbers) : m₁ * m₂ ∈ n.smoothNumbers := by |
rw [smoothNumbers_eq_factoredNumbers] at hm1 hm2 ⊢
exact mul_mem_factoredNumbers hm1 hm2
|
/-
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 | 636 | 647 | theorem cast_sub_one {R : Type*} [Ring R] {n : ℕ} (k : ZMod n) :
(cast (k - 1 : ZMod n) : R) = (if k = 0 then (n : R) else cast k) - 1 := by |
split_ifs with hk
· rw [hk, zero_sub, ZMod.cast_neg_one]
· cases n
· dsimp [ZMod, ZMod.cast]
rw [Int.cast_sub, Int.cast_one]
· dsimp [ZMod, ZMod.cast, ZMod.val]
rw [Fin.coe_sub_one, if_neg]
· rw [Nat.cast_sub, Nat.cast_one]
rwa [Fin.ext_iff, Fin.val_zero, ← Ne, ← Nat.one_le_iff_... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | Mathlib/GroupTheory/Sylow.lean | 428 | 444 | theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by |
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q :... |
/-
Copyright (c) 2022 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer, Kevin Klinge, Andrew Yang
-/
import Mathlib.RingTheory.OreLocalization.OreSet
import Mathlib.Algebra.Group.Submonoid.Operations
#align_import ring_theory.ore_local... | Mathlib/RingTheory/OreLocalization/Basic.lean | 465 | 467 | theorem oreDiv_one_smul (r : R) (x : X[S⁻¹]) : (r /ₒ (1 : S)) • x = r • x := by |
induction' x using OreLocalization.ind with r s
rw [smul_oreDiv, oreDiv_smul_oreDiv, mul_one]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Yury G. Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Iterate
import Math... | Mathlib/Analysis/SpecificLimits/Basic.lean | 575 | 583 | theorem exists_pos_sum_of_countable {ε : ℝ≥0} (hε : ε ≠ 0) (ι) [Countable ι] :
∃ ε' : ι → ℝ≥0, (∀ i, 0 < ε' i) ∧ ∃ c, HasSum ε' c ∧ c < ε := by |
cases nonempty_encodable ι
obtain ⟨a, a0, aε⟩ := exists_between (pos_iff_ne_zero.2 hε)
obtain ⟨ε', hε', c, hc, hcε⟩ := posSumOfEncodable a0 ι
exact
⟨fun i ↦ ⟨ε' i, (hε' i).le⟩, fun i ↦ NNReal.coe_lt_coe.1 <| hε' i,
⟨c, hasSum_le (fun i ↦ (hε' i).le) hasSum_zero hc⟩, NNReal.hasSum_coe.1 hc,
aε.t... |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Moritz Doll
-/
import Mathlib.LinearAlgebra.FinsuppVectorSpace
import Mathlib.LinearAlgebra.Matrix.Basis
import Mathlib.LinearAlgebra.Matrix.Nondegenerate
import... | Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean | 449 | 452 | theorem LinearMap.toMatrix₂_compl₂ (B : M₁ →ₗ[R] M₂ →ₗ[R] R) (f : M₂' →ₗ[R] M₂) :
LinearMap.toMatrix₂ b₁ b₂' (B.compl₂ f) = LinearMap.toMatrix₂ b₁ b₂ B * toMatrix b₂' b₂ f := by |
rw [← LinearMap.comp_id B, ← LinearMap.compl₁₂, LinearMap.toMatrix₂_compl₁₂ b₁ b₂]
simp
|
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhangir Azerbayev, Adam Topaz, Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
import Mathlib.LinearAlgebra.Alternating.Basic
#align_import linear_algebra.exterior_algeb... | Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean | 241 | 250 | theorem ι_eq_algebraMap_iff (x : M) (r : R) : ι R x = algebraMap R _ r ↔ x = 0 ∧ r = 0 := by |
refine ⟨fun h => ?_, ?_⟩
· letI : Module Rᵐᵒᵖ M := Module.compHom _ ((RingHom.id R).fromOpposite mul_comm)
haveI : IsCentralScalar R M := ⟨fun r m => rfl⟩
have hf0 : toTrivSqZeroExt (ι R x) = (0, x) := toTrivSqZeroExt_ι _
rw [h, AlgHom.commutes] at hf0
have : r = 0 ∧ 0 = x := Prod.ext_iff.1 hf0
... |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.ContMDiff.Defs
/-!
## Basic properties of smooth functions between manifolds
In this file, we show that sta... | Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | 380 | 387 | theorem contMDiff_inclusion {n : ℕ∞} {U V : Opens M} (h : U ≤ V) :
ContMDiff I I n (Set.inclusion h : U → V) := by |
rintro ⟨x, hx : x ∈ U⟩
apply (contDiffWithinAt_localInvariantProp I I n).liftProp_inclusion
intro y
dsimp only [ContDiffWithinAtProp, id_comp, preimage_univ]
rw [Set.univ_inter]
exact contDiffWithinAt_id.congr I.rightInvOn (congr_arg I (I.left_inv y))
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.finset_ops from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
/-!
# Prepa... | Mathlib/Data/Multiset/FinsetOps.lean | 79 | 80 | theorem length_ndinsert_of_not_mem {a : α} {s : Multiset α} (h : a ∉ s) :
card (ndinsert a s) = card s + 1 := by | simp [h]
|
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.factorization from "leanprover-community/mat... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 522 | 529 | theorem finprod_heightOneSpectrum_factorization' {I : FractionalIdeal R⁰ K} (hI : I ≠ 0) :
∏ᶠ v : HeightOneSpectrum R, (v.asIdeal : FractionalIdeal R⁰ K) ^ (count K v I) = I := by |
have h := (choose_spec (choose_spec (exists_eq_spanSingleton_mul I))).2
conv_rhs => rw [← finprod_heightOneSpectrum_factorization hI h]
apply finprod_congr
intro w
apply congr_arg
rw [count_ne_zero K w hI]
|
/-
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 | 687 | 693 | theorem IsNormal.cof_le {f} (hf : IsNormal f) (a) : cof a ≤ cof (f a) := by |
rcases zero_or_succ_or_limit a with (rfl | ⟨b, rfl⟩ | ha)
· rw [cof_zero]
exact zero_le _
· rw [cof_succ, Cardinal.one_le_iff_ne_zero, cof_ne_zero, ← Ordinal.pos_iff_ne_zero]
exact (Ordinal.zero_le (f b)).trans_lt (hf.1 b)
· rw [hf.cof_eq ha]
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.MeasureTheory.Integral.Pi
impor... | Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | 274 | 277 | theorem integrable_cexp_neg_mul_sum_add {ι : Type*} [Fintype ι] (hb : 0 < b.re) (c : ι → ℂ) :
Integrable (fun (v : ι → ℝ) ↦ cexp (- b * ∑ i, (v i : ℂ) ^ 2 + ∑ i, c i * v i)) := by |
simp_rw [neg_mul, Finset.mul_sum]
exact integrable_cexp_neg_sum_mul_add (fun _ ↦ hb) c
|
/-
Copyright (c) 2021 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.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis... | Mathlib/Analysis/Convex/Gauge.lean | 119 | 121 | theorem gauge_of_subset_zero (h : s ⊆ 0) : gauge s = 0 := by |
obtain rfl | rfl := subset_singleton_iff_eq.1 h
exacts [gauge_empty, gauge_zero']
|
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.SimplicialObject
import Mathlib.CategoryTheory.Limits.Shapes.Products
#align_import algebraic_topology.split_simplicial_object from "leanprove... | Mathlib/AlgebraicTopology/SplitSimplicialObject.lean | 291 | 293 | theorem ι_desc {Z : C} (Δ : SimplexCategoryᵒᵖ) (F : ∀ A : IndexSet Δ, s.N A.1.unop.len ⟶ Z)
(A : IndexSet Δ) : (s.cofan Δ).inj A ≫ s.desc Δ F = F A := by |
apply Cofan.IsColimit.fac
|
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Data.ZMod.Quotient
#align_import group_theory.complement from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
/-!
# Compl... | Mathlib/GroupTheory/Complement.lean | 232 | 243 | theorem mem_rightTransversals_iff_existsUnique_mul_inv_mem :
S ∈ rightTransversals T ↔ ∀ g : G, ∃! s : S, g * (s : G)⁻¹ ∈ T := by |
rw [rightTransversals, Set.mem_setOf_eq, isComplement_iff_existsUnique]
refine ⟨fun h g => ?_, fun h g => ?_⟩
· obtain ⟨x, h1, h2⟩ := h g
exact
⟨x.2, (congr_arg (· ∈ T) (eq_mul_inv_of_mul_eq h1)).mp x.1.2, fun y hy =>
(Prod.ext_iff.mp (h2 ⟨⟨g * (↑y)⁻¹, hy⟩, y⟩ (inv_mul_cancel_right g y))).2⟩
... |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
#align_import analysis.ca... | Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 724 | 726 | theorem fderiv_mem_iff {f : E → F} {s : Set (E →L[𝕜] F)} {x : E} : fderiv 𝕜 f x ∈ s ↔
DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ s ∨ ¬DifferentiableAt 𝕜 f x ∧ (0 : E →L[𝕜] F) ∈ s := by |
by_cases hx : DifferentiableAt 𝕜 f x <;> simp [fderiv_zero_of_not_differentiableAt, *]
|
/-
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 | 518 | 524 | theorem convolution_mono_right {f g g' : G → ℝ} (hfg : ConvolutionExistsAt f g x (lsmul ℝ ℝ) μ)
(hfg' : ConvolutionExistsAt f g' x (lsmul ℝ ℝ) μ) (hf : ∀ x, 0 ≤ f x) (hg : ∀ x, g x ≤ g' x) :
(f ⋆[lsmul ℝ ℝ, μ] g) x ≤ (f ⋆[lsmul ℝ ℝ, μ] g') x := by |
apply integral_mono hfg hfg'
simp only [lsmul_apply, Algebra.id.smul_eq_mul]
intro t
apply mul_le_mul_of_nonneg_left (hg _) (hf _)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Separation
/-!
# Order-closed topologies
In this file we introduce 3 typeclass mixins that relate topology ... | Mathlib/Topology/Order/OrderClosed.lean | 816 | 818 | theorem frontier_lt_subset_eq (hf : Continuous f) (hg : Continuous g) :
frontier { b | f b < g b } ⊆ { b | f b = g b } := by |
simpa only [← not_lt, ← compl_setOf, frontier_compl, eq_comm] using frontier_le_subset_eq hg hf
|
/-
Copyright (c) 2014 Robert Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Lewis, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ring.Commute... | Mathlib/Algebra/Field/Basic.lean | 109 | 114 | theorem div_neg_eq_neg_div (a b : K) : b / -a = -(b / a) :=
calc
b / -a = b * (1 / -a) := by | rw [← inv_eq_one_div, division_def]
_ = b * -(1 / a) := by rw [one_div_neg_eq_neg_one_div]
_ = -(b * (1 / a)) := by rw [neg_mul_eq_mul_neg]
_ = -(b / a) := by rw [mul_one_div]
|
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.factorization from "leanprover-community/mat... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 407 | 409 | theorem count_pow_self (n : ℕ) :
count K v ((v.asIdeal : FractionalIdeal R⁰ K) ^ n) = n := by |
rw [count_pow, count_self, mul_one]
|
/-
Copyright (c) 2023 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Integral.Lebesgue
/-!
# Marginals of multivariate functions
In this ... | Mathlib/MeasureTheory/Integral/Marginal.lean | 110 | 116 | theorem lmarginal_update_of_mem {i : δ} (hi : i ∈ s)
(f : (∀ i, π i) → ℝ≥0∞) (x : ∀ i, π i) (y : π i) :
(∫⋯∫⁻_s, f ∂μ) (Function.update x i y) = (∫⋯∫⁻_s, f ∂μ) x := by |
apply lmarginal_congr
intro j hj
have : j ≠ i := by rintro rfl; exact hj hi
apply update_noteq this
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Anne Baanen
-/
import Mathlib.LinearAlgebra.Dimension.Basic
import Mathlib.SetTheory.Cardinal.ToNat
#align_import linear_algebra.finrank from "leanprover-community/mathlib... | Mathlib/LinearAlgebra/Dimension/Finrank.lean | 92 | 93 | theorem one_lt_rank_of_one_lt_finrank (h : 1 < finrank R M) : 1 < Module.rank R M := by |
simpa using lt_rank_of_lt_finrank h
|
/-
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.Analysis.Calculus.MeanValue
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Order.Filter.Curry
#align_import analysis.calculus.uniform_lim... | Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean | 497 | 523 | theorem hasDerivAt_of_tendstoUniformlyOnFilter [NeBot l]
(hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x))
(hf : ∀ᶠ n : ι × 𝕜 in l ×ˢ 𝓝 x, HasDerivAt (f n.1) (f' n.1 n.2) n.2)
(hfg : ∀ᶠ y in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))) : HasDerivAt g (g' x) x := by |
-- The first part of the proof rewrites `hf` and the goal to be functions so that Lean
-- can recognize them when we apply `hasFDerivAt_of_tendstoUniformlyOnFilter`
let F' n z := (1 : 𝕜 →L[𝕜] 𝕜).smulRight (f' n z)
let G' z := (1 : 𝕜 →L[𝕜] 𝕜).smulRight (g' z)
simp_rw [hasDerivAt_iff_hasFDerivAt] at hf ⊢... |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Callum Sutton, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.Group.Units.Equiv
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mat... | Mathlib/Algebra/Ring/Equiv.lean | 778 | 781 | theorem toRingHom_comp_symm_toRingHom (e : R ≃+* S) :
e.toRingHom.comp e.symm.toRingHom = RingHom.id _ := by |
ext
simp
|
/-
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 | 1,686 | 1,691 | theorem IsBigO.of_pow {f : α → 𝕜} {g : α → R} {n : ℕ} (hn : n ≠ 0) (h : (f ^ n) =O[l] (g ^ n)) :
f =O[l] g := by |
rcases h.exists_pos with ⟨C, _hC₀, hC⟩
obtain ⟨c : ℝ, hc₀ : 0 ≤ c, hc : C ≤ c ^ n⟩ :=
((eventually_ge_atTop _).and <| (tendsto_pow_atTop hn).eventually_ge_atTop C).exists
exact (hC.of_pow hn hc hc₀).isBigO
|
/-
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 | 181 | 182 | theorem factorial_mul_factorial_dvd_factorial {n k : ℕ} (hk : k ≤ n) : k ! * (n - k)! ∣ n ! := by |
rw [← choose_mul_factorial_mul_factorial hk, Nat.mul_assoc]; exact Nat.dvd_mul_left _ _
|
/-
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, Sander Dahmen, Scott Morrison, Chris Hughes, Anne Baanen
-/
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_im... | Mathlib/LinearAlgebra/Dimension/Constructions.lean | 306 | 308 | theorem rank_fun_eq_lift_mul : Module.rank R (η → M) =
(Fintype.card η : Cardinal.{max u₁' v}) * Cardinal.lift.{u₁'} (Module.rank R M) := by |
rw [rank_pi, Cardinal.sum_const, Cardinal.mk_fintype, Cardinal.lift_natCast]
|
/-
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, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.EqLocus
import Mathlib.Algebra.Module.Subm... | Mathlib/LinearAlgebra/Span.lean | 718 | 728 | theorem iSup_induction' {ι : Sort*} (p : ι → Submodule R M) {C : ∀ x, (x ∈ ⨆ i, p i) → Prop}
(mem : ∀ (i) (x) (hx : x ∈ p i), C x (mem_iSup_of_mem i hx)) (zero : C 0 (zero_mem _))
(add : ∀ x y hx hy, C x hx → C y hy → C (x + y) (add_mem ‹_› ‹_›)) {x : M}
(hx : x ∈ ⨆ i, p i) : C x hx := by |
refine Exists.elim ?_ fun (hx : x ∈ ⨆ i, p i) (hc : C x hx) => hc
refine iSup_induction p (C := fun x : M ↦ ∃ (hx : x ∈ ⨆ i, p i), C x hx) hx
(fun i x hx => ?_) ?_ fun x y => ?_
· exact ⟨_, mem _ _ hx⟩
· exact ⟨_, zero⟩
· rintro ⟨_, Cx⟩ ⟨_, Cy⟩
exact ⟨_, add _ _ _ _ Cx Cy⟩
|
/-
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.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.Affine... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 109 | 118 | theorem weightedVSubOfPoint_eq_of_sum_eq_zero (w : ι → k) (p : ι → P) (h : ∑ i ∈ s, w i = 0)
(b₁ b₂ : P) : s.weightedVSubOfPoint p b₁ w = s.weightedVSubOfPoint p b₂ w := by |
apply eq_of_sub_eq_zero
rw [weightedVSubOfPoint_apply, weightedVSubOfPoint_apply, ← sum_sub_distrib]
conv_lhs =>
congr
· skip
· ext
rw [← smul_sub, vsub_sub_vsub_cancel_left]
rw [← sum_smul, h, zero_smul]
|
/-
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 | 223 | 228 | theorem smul [NormedField 𝕜] [NormedSpace 𝕜 β] (hT : DominatedFinMeasAdditive μ T C) (c : 𝕜) :
DominatedFinMeasAdditive μ (fun s => c • T s) (‖c‖ * C) := by |
refine ⟨hT.1.smul c, fun s hs hμs => ?_⟩
dsimp only
rw [norm_smul, mul_assoc]
exact mul_le_mul le_rfl (hT.2 s hs hμs) (norm_nonneg _) (norm_nonneg _)
|
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Normalizer
#align_import algebra.lie.engel from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3... | Mathlib/Algebra/Lie/Engel.lean | 165 | 170 | theorem LieAlgebra.isEngelian_of_subsingleton [Subsingleton L] : LieAlgebra.IsEngelian R L := by |
intro M _i1 _i2 _i3 _i4 _h
use 1
suffices (⊤ : LieIdeal R L) = ⊥ by simp [this]
haveI := (LieSubmodule.subsingleton_iff R L L).mpr inferInstance
apply Subsingleton.elim
|
/-
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.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Covering.Besicovitch
import Mathlib.Tactic.AdaptationNote
#align_import measu... | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | 412 | 466 | theorem exists_normalized_aux3 {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ)
(lastc : a.c (last N) = 0) (lastr : a.r (last N) = 1) (hτ : 1 ≤ τ) (δ : ℝ) (hδ1 : τ ≤ 1 + δ / 4)
(i j : Fin N.succ) (inej : i ≠ j) (hi : 2 < ‖a.c i‖) (hij : ‖a.c i‖ ≤ ‖a.c j‖) :
1 - δ ≤ ‖(2 / ‖a.c i‖) • a.c i - (2 / ‖a.c j‖) • a.c j‖... |
have ah :
Pairwise fun i j => a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨
a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j := by
simpa only [dist_eq_norm] using a.h
have δnonneg : 0 ≤ δ := by linarith only [hτ, hδ1]
have hcrj : ‖a.c j‖ ≤ a.r j + 1 := by simpa only [lastc, lastr, dist_zero_right] u... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Yury Kudryashov
-/
import Mathlib.Data.Rat.Sqrt
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Algebraic
import... | Mathlib/Data/Real/Irrational.lean | 601 | 602 | theorem irrational_mul_rat_iff : Irrational (x * q) ↔ q ≠ 0 ∧ Irrational x := by |
rw [mul_comm, irrational_rat_mul_iff]
|
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
import Mathlib.Data.SetLike.Basic
#align_import data.sym.sym2 from "leanpro... | Mathlib/Data/Sym/Sym2.lean | 381 | 383 | theorem other_mem {a : α} {z : Sym2 α} (h : a ∈ z) : Mem.other h ∈ z := by |
convert mem_mk_right a <| Mem.other h
rw [other_spec 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, Yury Kudryashov
-/
import Mathlib.Topology.Separation
/-!
# Order-closed topologies
In this file we introduce 3 typeclass mixins that relate topology ... | Mathlib/Topology/Order/OrderClosed.lean | 314 | 316 | theorem continuousWithinAt_Ioo_iff_Iio (h : a < b) :
ContinuousWithinAt f (Ioo a b) b ↔ ContinuousWithinAt f (Iio b) b := by |
simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Iio h]
|
/-
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.Bool.Set
import Mathlib.Data.Nat.Set
import Mathlib.Data.Set.Prod
import Mathlib.Data.ULift
import Mathlib.Order.Bounds.Basic
import Mathlib.Order... | Mathlib/Order/CompleteLattice.lean | 1,764 | 1,767 | theorem unary_relation_sSup_iff {α : Type*} (s : Set (α → Prop)) {a : α} :
sSup s a ↔ ∃ r : α → Prop, r ∈ s ∧ r a := by |
rw [sSup_apply]
simp [← eq_iff_iff]
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Oliver Nash
-/
import Mathlib.Data.Finset.Card
#align_import data.finset.prod from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267... | Mathlib/Data/Finset/Prod.lean | 220 | 222 | theorem product_eq_empty {s : Finset α} {t : Finset β} : s ×ˢ t = ∅ ↔ s = ∅ ∨ t = ∅ := by |
rw [← not_nonempty_iff_eq_empty, nonempty_product, not_and_or, not_nonempty_iff_eq_empty,
not_nonempty_iff_eq_empty]
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.RingTheory.DiscreteValuationRing.Basic
import Mathlib.RingTheory.MvPowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.Basic
import M... | Mathlib/RingTheory/PowerSeries/Inverse.lean | 54 | 81 | theorem coeff_inv_aux (n : ℕ) (a : R) (φ : R⟦X⟧) :
coeff R n (inv.aux a φ) =
if n = 0 then a
else
-a *
∑ x ∈ antidiagonal n,
if x.2 < n then coeff R x.1 φ * coeff R x.2 (inv.aux a φ) else 0 := by |
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [coeff, inv.aux, MvPowerSeries.coeff_inv_aux]
simp only [Finsupp.single_eq_zero]
split_ifs; · rfl
congr 1
symm
apply Finset.sum_nbij' (fun (a, b) ↦ (single () a, single () b))
fun (f, g) ↦ (f (), g ())
· aesop
· aesop
· ... |
/-
Copyright (c) 2017 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Mario Carneiro
-/
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Real.Basic
import Mathlib.Data.Set.Image
#... | Mathlib/Data/Complex/Basic.lean | 726 | 727 | theorem normSq_add (z w : ℂ) : normSq (z + w) = normSq z + normSq w + 2 * (z * conj w).re := by |
dsimp [normSq]; ring
|
/-
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 | 1,765 | 1,768 | theorem add_mulVec [Fintype n] (A B : Matrix m n α) (x : n → α) :
(A + B) *ᵥ x = A *ᵥ x + B *ᵥ x := by |
ext
apply add_dotProduct
|
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.FiniteSupport
import Mathlib.Algebra... | Mathlib/Algebra/BigOperators/Finprod.lean | 785 | 790 | theorem finprod_mem_union_inter (hs : s.Finite) (ht : t.Finite) :
((∏ᶠ i ∈ s ∪ t, f i) * ∏ᶠ i ∈ s ∩ t, f i) = (∏ᶠ i ∈ s, f i) * ∏ᶠ i ∈ t, f i := by |
lift s to Finset α using hs; lift t to Finset α using ht
classical
rw [← Finset.coe_union, ← Finset.coe_inter]
simp only [finprod_mem_coe_finset, Finset.prod_union_inter]
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Artinian
#align_import algebra.lie.submodule from "leanprover-communit... | Mathlib/Algebra/Lie/Submodule.lean | 1,063 | 1,063 | theorem comap_map_le : I ≤ comap f (map f I) := by | rw [← map_le_iff_le_comap]
|
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