Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot
-/
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
i... | Mathlib/Data/Set/Pointwise/Interval.lean | 207 | 208 | theorem preimage_add_const_Ioo : (fun x => x + a) ⁻¹' Ioo b c = Ioo (b - a) (c - a) := by |
simp [← Ioi_inter_Iio]
|
/-
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 | 443 | 446 | theorem LinearMap.toMatrix₂_comp (B : M₁ →ₗ[R] M₂ →ₗ[R] R) (f : M₁' →ₗ[R] M₁) :
LinearMap.toMatrix₂ b₁' b₂ (B.comp f) = (toMatrix b₁' b₁ f)ᵀ * LinearMap.toMatrix₂ b₁ b₂ B := by |
rw [← LinearMap.compl₂_id (B.comp f), ← LinearMap.compl₁₂, LinearMap.toMatrix₂_compl₁₂ b₁ b₂]
simp
|
/-
Copyright (c) 2023 Moritz Firsching. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Firsching, Ashvni Narayanan, Michael Stoll
-/
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Data.ZMod.Basic
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Rin... | Mathlib/Data/ZMod/Units.lean | 38 | 63 | theorem unitsMap_surjective [hm : NeZero m] (h : n ∣ m) :
Function.Surjective (unitsMap h) := by |
suffices ∀ x : ℕ, x.Coprime n → ∃ k : ℕ, (x + k * n).Coprime m by
intro x
have ⟨k, hk⟩ := this x.val.val (val_coe_unit_coprime x)
refine ⟨unitOfCoprime _ hk, Units.ext ?_⟩
have : NeZero n := ⟨fun hn ↦ hm.out (eq_zero_of_zero_dvd (hn ▸ h))⟩
simp [unitsMap_def]
intro x hx
let ps := m.primeFacto... |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Adjunction.Evaluation
import Mathlib.Tactic.CategoryTheory.Elementwise
import Mathlib.CategoryTheory... | Mathlib/CategoryTheory/Sites/Subsheaf.lean | 122 | 130 | theorem Subpresheaf.eq_top_iff_isIso : G = ⊤ ↔ IsIso G.ι := by |
constructor
· rintro rfl
infer_instance
· intro H
ext U x
apply iff_true_iff.mpr
rw [← IsIso.inv_hom_id_apply (G.ι.app U) x]
exact ((inv (G.ι.app U)) x).2
|
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
#align_import ring_... | Mathlib/RingTheory/Derivation/Basic.lean | 431 | 435 | theorem leibniz_inv {K : Type*} [Field K] [Module K M] [Algebra R K] (D : Derivation R K M)
(a : K) : D a⁻¹ = -a⁻¹ ^ 2 • D a := by |
rcases eq_or_ne a 0 with (rfl | ha)
· simp
· exact D.leibniz_of_mul_eq_one (inv_mul_cancel ha)
|
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 283 | 294 | theorem snorm'_const' [IsFiniteMeasure μ] (c : F) (hc_ne_zero : c ≠ 0) (hq_ne_zero : q ≠ 0) :
snorm' (fun _ : α => c) q μ = (‖c‖₊ : ℝ≥0∞) * μ Set.univ ^ (1 / q) := by |
rw [snorm', lintegral_const, ENNReal.mul_rpow_of_ne_top _ (measure_ne_top μ Set.univ)]
· congr
rw [← ENNReal.rpow_mul]
suffices hp_cancel : q * (1 / q) = 1 by rw [hp_cancel, ENNReal.rpow_one]
rw [one_div, mul_inv_cancel hq_ne_zero]
· rw [Ne, ENNReal.rpow_eq_top_iff, not_or, not_and_or, not_and_or]
... |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
/-!
# Noncomputable... | Mathlib/Data/Set/Card.lean | 78 | 80 | theorem encard_eq_coe_toFinset_card (s : Set α) [Fintype s] : encard s = s.toFinset.card := by |
have h := toFinite s
rw [h.encard_eq_coe_toFinset_card, toFinite_toFinset]
|
/-
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 | 470 | 472 | theorem mul_sub (h : 0 < c → c < b → a ≠ ∞) : a * (b - c) = a * b - a * c := by |
simp only [mul_comm a]
exact sub_mul h
|
/-
Copyright (c) 2024 Emilie Burgun. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emilie Burgun
-/
import Mathlib.Dynamics.PeriodicPts
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.GroupAction.Basic
/-!
# Period of a group action
This module defin... | Mathlib/GroupTheory/GroupAction/Period.lean | 117 | 120 | theorem period_bounded_of_exponent_pos (exp_pos : 0 < Monoid.exponent M) (m : M) :
BddAbove (Set.range (fun a : α => period m a)) := by |
use Monoid.exponent M
simpa [upperBounds] using period_le_exponent exp_pos _
|
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Star.Basic
import Mathlib.Data.Set.Finite
import Mathlib.Data.Set.Pointwise.Basic
#align_import algebra.star.pointwise from "leanprover-communit... | Mathlib/Algebra/Star/Pointwise.lean | 115 | 117 | theorem star_singleton {β : Type*} [InvolutiveStar β] (x : β) : ({x} : Set β)⋆ = {x⋆} := by |
ext1 y
rw [mem_star, mem_singleton_iff, mem_singleton_iff, star_eq_iff_star_eq, eq_comm]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLim... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 1,232 | 1,233 | theorem map_zero (f : α → β) : (0 : Measure α).map f = 0 := by |
by_cases hf : AEMeasurable f (0 : Measure α) <;> simp [map, hf]
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.Fixed
import Mathlib.FieldTheory.NormalClosure
import Mathlib.FieldTheory.PrimitiveElement
import Mathlib.Gro... | Mathlib/FieldTheory/Galois.lean | 332 | 335 | theorem of_fixedField_eq_bot [FiniteDimensional F E]
(h : IntermediateField.fixedField (⊤ : Subgroup (E ≃ₐ[F] E)) = ⊥) : IsGalois F E := by |
rw [← isGalois_iff_isGalois_bot, ← h]
classical exact IsGalois.of_fixed_field E (⊤ : Subgroup (E ≃ₐ[F] E))
|
/-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Measure.MeasureSpace
/-!
# The multiplicative and additive convolution of measures
In... | Mathlib/MeasureTheory/Group/Convolution.lean | 65 | 67 | theorem zero_mconv (μ : Measure M) : μ ∗ (0 : Measure M) = (0 : Measure M) := by |
unfold mconv
simp
|
/-
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.Order.Ring.WithTop
import Mathlib.Algebra.Order.Sub.WithTop
import Mathlib.Data.Real.NNReal
import Mathlib.Order.Interval.Set.... | Mathlib/Data/ENNReal/Basic.lean | 718 | 719 | theorem iInter_Ici_coe_nat : ⋂ n : ℕ, Ici (n : ℝ≥0∞) = {∞} := by |
simp only [← compl_Iio, ← compl_iUnion, iUnion_Iio_coe_nat, compl_compl]
|
/-
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, Winston Yin
-/
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.Topology.MetricSpace.Contracting
#align_import analysis.ODE.picard_lindelof fr... | Mathlib/Analysis/ODE/PicardLindelof.lean | 329 | 333 | theorem dist_iterate_next_le (f₁ f₂ : FunSpace v) (n : ℕ) :
dist (next^[n] f₁) (next^[n] f₂) ≤ (v.L * v.tDist) ^ n / n ! * dist f₁ f₂ := by |
refine dist_le_of_forall fun t => (dist_iterate_next_apply_le _ _ _ _).trans ?_
have : |(t - v.t₀ : ℝ)| ≤ v.tDist := v.dist_t₀_le t
gcongr
|
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Integral.FundThmCalcu... | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 545 | 552 | theorem integrable_of_intervalIntegral_norm_bounded (I : ℝ)
(hfi : ∀ i, IntegrableOn f (Ioc (a i) (b i)) μ) (ha : Tendsto a l atBot)
(hb : Tendsto b l atTop) (h : ∀ᶠ i in l, (∫ x in a i..b i, ‖f x‖ ∂μ) ≤ I) : Integrable f μ := by |
have hφ : AECover μ l _ := aecover_Ioc ha hb
refine hφ.integrable_of_integral_norm_bounded I hfi (h.mp ?_)
filter_upwards [ha.eventually (eventually_le_atBot 0),
hb.eventually (eventually_ge_atTop 0)] with i hai hbi ht
rwa [← intervalIntegral.integral_of_le (hai.trans hbi)]
|
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Bryan Gin-ge Chen, Patrick Massot, Wen Yang, Johan Commelin
-/
import Mathlib.Data.Set.Finite
import Mathlib.Order.Partition.Finpartition
#align_import data.seto... | Mathlib/Data/Setoid/Partition.lean | 67 | 71 | theorem classes_ker_subset_fiber_set {β : Type*} (f : α → β) :
(Setoid.ker f).classes ⊆ Set.range fun y => { x | f x = y } := by |
rintro s ⟨x, rfl⟩
rw [Set.mem_range]
exact ⟨f x, rfl⟩
|
/-
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 | 1,114 | 1,117 | theorem coequalizer.isoTargetOfSelf_hom :
(coequalizer.isoTargetOfSelf f).hom = coequalizer.desc (𝟙 Y) (by simp) := by |
ext
simp [coequalizer.isoTargetOfSelf]
|
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Variance
#align_import probability.moments from "leanprover-community/mathlib"@"85453a2a14be8da64caf15ca50930cf4c6e5d8de"
/-!
# Moments and m... | Mathlib/Probability/Moments.lean | 118 | 118 | theorem cgf_zero_fun : cgf 0 μ t = log (μ Set.univ).toReal := by | simp only [cgf, mgf_zero_fun]
|
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.Calculus.BumpFunction.Normed
import Mathlib.MeasureTheory.Integral.Average
import Mathlib.MeasureTheor... | Mathlib/Analysis/Calculus/BumpFunction/Convolution.lean | 110 | 139 | theorem ae_convolution_tendsto_right_of_locallyIntegrable
{ι} {φ : ι → ContDiffBump (0 : G)} {l : Filter ι} {K : ℝ}
(hφ : Tendsto (fun i ↦ (φ i).rOut) l (𝓝 0))
(h'φ : ∀ᶠ i in l, (φ i).rOut ≤ K * (φ i).rIn) (hg : LocallyIntegrable g μ) : ∀ᵐ x₀ ∂μ,
Tendsto (fun i ↦ ((φ i).normed μ ⋆[lsmul ℝ ℝ, μ] g) x₀) ... |
have : IsAddHaarMeasure μ := ⟨⟩
-- By Lebesgue differentiation theorem, the average of `g` on a small ball converges
-- almost everywhere to the value of `g` as the radius shrinks to zero.
-- We will see that this set of points satisfies the desired conclusion.
filter_upwards [(Besicovitch.vitaliFamily μ).ae... |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Set.Card
import Mathlib.Order.Minimal
import Mathlib.Data.Matroid.Init
/-!
# Matroids
A `Matroid` is a structure that combinatorially abstracts
the ... | Mathlib/Data/Matroid/Basic.lean | 962 | 965 | theorem Basis'.inter_eq_of_subset_indep (hI : M.Basis' I X) (hIJ : I ⊆ J) (hJ : M.Indep J) :
J ∩ X = I := by |
rw [← hI.basis_inter_ground.inter_eq_of_subset_indep hIJ hJ, inter_comm X, ← inter_assoc,
inter_eq_self_of_subset_left hJ.subset_ground]
|
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 624 | 629 | theorem mem_support_append_iff {t u v w : V} (p : G.Walk u v) (p' : G.Walk v w) :
t ∈ (p.append p').support ↔ t ∈ p.support ∨ t ∈ p'.support := by |
simp only [mem_support_iff, mem_tail_support_append_iff]
obtain rfl | h := eq_or_ne t v <;> obtain rfl | h' := eq_or_ne t u <;>
-- this `have` triggers the unusedHavesSuffices linter:
(try have := h'.symm) <;> simp [*]
|
/-
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.Order.Atoms
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.RelIso.Set
import Mathlib.Order.SupClosed
import Mathlib.Order.SupIndep
import Mathlib.Orde... | Mathlib/Order/CompactlyGenerated/Basic.lean | 215 | 224 | theorem IsSupFiniteCompact.isSupClosedCompact (h : IsSupFiniteCompact α) :
IsSupClosedCompact α := by |
intro s hne hsc; obtain ⟨t, ht₁, ht₂⟩ := h s; clear h
rcases t.eq_empty_or_nonempty with h | h
· subst h
rw [Finset.sup_empty] at ht₂
rw [ht₂]
simp [eq_singleton_bot_of_sSup_eq_bot_of_nonempty ht₂ hne]
· rw [ht₂]
exact hsc.finsetSup_mem h ht₁
|
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Ma... | Mathlib/Data/Nat/Factorization/Basic.lean | 934 | 943 | theorem eq_iff_prime_padicValNat_eq (a b : ℕ) (ha : a ≠ 0) (hb : b ≠ 0) :
a = b ↔ ∀ p : ℕ, p.Prime → padicValNat p a = padicValNat p b := by |
constructor
· rintro rfl
simp
· intro h
refine eq_of_factorization_eq ha hb fun p => ?_
by_cases pp : p.Prime
· simp [factorization_def, pp, h p pp]
· simp [factorization_eq_zero_of_non_prime, pp]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Regular.SMul
import Mathlib.Data.Finset.Preimag... | Mathlib/Data/Finsupp/Basic.lean | 597 | 602 | theorem embDomain_eq_mapDomain (f : α ↪ β) (v : α →₀ M) : embDomain f v = mapDomain f v := by |
ext a
by_cases h : a ∈ Set.range f
· rcases h with ⟨a, rfl⟩
rw [mapDomain_apply f.injective, embDomain_apply]
· rw [mapDomain_notin_range, embDomain_notin_range] <;> assumption
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Measure.MutuallySingular
#align_import measure_theory.decomposition.jordan from "leanpro... | Mathlib/MeasureTheory/Decomposition/Jordan.lean | 505 | 506 | theorem totalVariation_neg (s : SignedMeasure α) : (-s).totalVariation = s.totalVariation := by |
simp [totalVariation, toJordanDecomposition_neg, add_comm]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Anne Baanen
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.BigOperators.Finsupp
#align_import algebra.big_operators.associated from "leanp... | Mathlib/Algebra/BigOperators/Associated.lean | 145 | 150 | theorem prod_le_prod {p q : Multiset (Associates α)} (h : p ≤ q) : p.prod ≤ q.prod := by |
haveI := Classical.decEq (Associates α)
haveI := Classical.decEq α
suffices p.prod ≤ (p + (q - p)).prod by rwa [add_tsub_cancel_of_le h] at this
suffices p.prod * 1 ≤ p.prod * (q - p).prod by simpa
exact mul_mono (le_refl p.prod) one_le
|
/-
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.AffineScheme
import Mathlib.AlgebraicGeometry.Pullbacks
import Mathlib.CategoryTheory.MorphismProperty.Limits
import Mathlib.Data.List.TFAE... | Mathlib/AlgebraicGeometry/Morphisms/Basic.lean | 490 | 504 | theorem AffineTargetMorphismProperty.diagonal_respectsIso (P : AffineTargetMorphismProperty)
(hP : P.toProperty.RespectsIso) : P.diagonal.toProperty.RespectsIso := by |
delta AffineTargetMorphismProperty.diagonal
apply AffineTargetMorphismProperty.respectsIso_mk
· introv H _ _
rw [pullback.mapDesc_comp, affine_cancel_left_isIso hP, affine_cancel_right_isIso hP]
-- Porting note: add the following two instances
have i1 : IsOpenImmersion (f₁ ≫ e.hom) := PresheafedSpace... |
/-
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 | 245 | 247 | theorem Ico_subset_Icc_self : Ico a b ⊆ Icc a b := by |
rw [← coe_subset, coe_Ico, coe_Icc]
exact Set.Ico_subset_Icc_self
|
/-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.BernoulliPolynomials
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathl... | Mathlib/NumberTheory/ZetaValues.lean | 126 | 129 | theorem bernoulliFourierCoeff_zero {k : ℕ} (hk : k ≠ 0) : bernoulliFourierCoeff k 0 = 0 := by |
simp_rw [bernoulliFourierCoeff, fourierCoeffOn_eq_integral, neg_zero, fourier_zero, sub_zero,
div_one, one_smul, intervalIntegral.integral_ofReal, integral_bernoulliFun_eq_zero hk,
ofReal_zero]
|
/-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Category.ModuleCat.Projective
import Mathlib.AlgebraicTopology.ExtraDegeneracy
import Mathlib.CategoryTheory.Abelian.Ext
import Mathlib.R... | Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean | 398 | 407 | theorem diagonalHomEquiv_symm_partialProd_succ (f : (Fin n → G) → A) (g : Fin (n + 1) → G)
(a : Fin (n + 1)) :
((diagonalHomEquiv n A).symm f).hom (Finsupp.single (Fin.partialProd g ∘ a.succ.succAbove) 1)
= f (Fin.contractNth a (· * ·) g) := by |
simp only [diagonalHomEquiv_symm_apply, Function.comp_apply, Fin.succ_succAbove_zero,
Fin.partialProd_zero, map_one, Fin.succ_succAbove_succ, LinearMap.one_apply,
Fin.partialProd_succ]
congr
ext
rw [← Fin.partialProd_succ, Fin.inv_partialProd_mul_eq_contractNth]
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Wrenna Robson
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebr... | Mathlib/LinearAlgebra/Lagrange.lean | 403 | 406 | theorem eq_interpolate_of_eval_eq {f : F[X]} (hvs : Set.InjOn v s) (degree_f_lt : f.degree < s.card)
(eval_f : ∀ i ∈ s, f.eval (v i) = r i) : f = interpolate s v r := by |
rw [eq_interpolate hvs degree_f_lt]
exact interpolate_eq_of_values_eq_on _ _ eval_f
|
/-
Copyright (c) 2020 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Johan Commelin
-/
import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Sets.Opens
import Mathlib.D... | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | 264 | 266 | theorem zeroLocus_bUnion (s : Set (Set A)) :
zeroLocus 𝒜 (⋃ s' ∈ s, s' : Set A) = ⋂ s' ∈ s, zeroLocus 𝒜 s' := by |
simp only [zeroLocus_iUnion]
|
/-
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 | 751 | 755 | theorem coe_lieSpan_submodule_eq_iff {p : Submodule R M} :
(lieSpan R L (p : Set M) : Submodule R M) = p ↔ ∃ N : LieSubmodule R L M, ↑N = p := by |
rw [p.exists_lieSubmodule_coe_eq_iff L]; constructor <;> intro h
· intro x m hm; rw [← h, mem_coeSubmodule]; exact lie_mem _ (subset_lieSpan hm)
· rw [← coe_toSubmodule_mk p @h, coe_toSubmodule, coe_toSubmodule_eq_iff, lieSpan_eq]
|
/-
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, Eric Wieser
-/
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.LinearAlgebra.Matrix.Sym... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 968 | 968 | theorem isOrtho_comm {x y : M} : IsOrtho Q x y ↔ IsOrtho Q y x := by | simp_rw [isOrtho_def, add_comm]
|
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
#align_import algebra.continued_fractions.translations from "leanprove... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 116 | 117 | theorem exists_conts_b_of_denom {B : K} (nth_denom_eq : g.denominators n = B) :
∃ conts, g.continuants n = conts ∧ conts.b = B := by | simpa
|
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.LocalExtr.Rolle
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Topology.Algebra.Polynomial
... | Mathlib/Analysis/Calculus/LocalExtr/Polynomial.lean | 91 | 94 | theorem card_rootSet_le_derivative {F : Type*} [CommRing F] [Algebra F ℝ] (p : F[X]) :
Fintype.card (p.rootSet ℝ) ≤ Fintype.card (p.derivative.rootSet ℝ) + 1 := by |
simpa only [rootSet_def, Finset.coe_sort_coe, Fintype.card_coe, derivative_map] using
card_roots_toFinset_le_derivative (p.map (algebraMap F ℝ))
|
/-
Copyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Fin
import Mathlib.GroupTheo... | Mathlib/Algebra/BigOperators/Fin.lean | 200 | 203 | theorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :
(∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by |
subst h
congr
|
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
#align_import analysis.calculus.deriv.bas... | Mathlib/Analysis/Calculus/Deriv/Basic.lean | 540 | 543 | theorem deriv_mem_iff {f : 𝕜 → F} {s : Set F} {x : 𝕜} :
deriv f x ∈ s ↔
DifferentiableAt 𝕜 f x ∧ deriv f x ∈ s ∨ ¬DifferentiableAt 𝕜 f x ∧ (0 : F) ∈ s := by |
by_cases hx : DifferentiableAt 𝕜 f x <;> simp [deriv_zero_of_not_differentiableAt, *]
|
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedd... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 491 | 494 | theorem minkowskiBound_pos : 0 < minkowskiBound K I := by |
refine zero_lt_iff.mpr (mul_ne_zero ?_ ?_)
· exact Zspan.measure_fundamentalDomain_ne_zero _
· exact ENNReal.pow_ne_zero two_ne_zero _
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yury Kudryashov
-/
import Mathlib.MeasureTheory.OuterMeasure.Basic
/-!
# The “almost everywhere” filter of co-null sets.
If `μ` is an outer measure or a measure on `α... | Mathlib/MeasureTheory/OuterMeasure/AE.lean | 178 | 179 | theorem ae_eq_set {s t : Set α} : s =ᵐ[μ] t ↔ μ (s \ t) = 0 ∧ μ (t \ s) = 0 := by |
simp [eventuallyLE_antisymm_iff, ae_le_set]
|
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Ma... | Mathlib/Data/Nat/Factorization/Basic.lean | 262 | 266 | theorem eq_pow_of_factorization_eq_single {n p k : ℕ} (hn : n ≠ 0)
(h : n.factorization = Finsupp.single p k) : n = p ^ k := by |
-- Porting note: explicitly added `Finsupp.prod_single_index`
rw [← Nat.factorization_prod_pow_eq_self hn, h, Finsupp.prod_single_index]
simp
|
/-
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 | 627 | 631 | theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι]
{I : ι → ℕ} (hI : Pairwise fun i j => (I i).Coprime (I j)) :
⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by |
rw [iInf_span_singleton, Nat.cast_prod]
exact fun i j h ↦ (hI h).cast
|
/-
Copyright (c) 2019 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.Order.Archimedean
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Tactic.GCongr
#align_import order.filter.archimedean fr... | Mathlib/Order/Filter/Archimedean.lean | 302 | 303 | theorem Tendsto.atBot_zsmul_neg_const {f : α → ℤ} (hr : r < 0) (hf : Tendsto f l atBot) :
Tendsto (fun x => f x • r) l atTop := by | simpa using hf.atBot_zsmul_const (neg_pos.2 hr)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp
-/
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
i... | Mathlib/LinearAlgebra/Basis.lean | 582 | 587 | theorem mem_submodule_iff {P : Submodule R M} (b : Basis ι R P) {x : M} :
x ∈ P ↔ ∃ c : ι →₀ R, x = Finsupp.sum c fun i x => x • (b i : M) := by |
conv_lhs =>
rw [← P.range_subtype, ← Submodule.map_top, ← b.span_eq, Submodule.map_span, ← Set.range_comp,
← Finsupp.range_total]
simp [@eq_comm _ x, Function.comp, Finsupp.total_apply]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algeb... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 1,315 | 1,316 | theorem tan_periodic : Function.Periodic tan π := by |
simpa only [tan_eq_sin_div_cos] using sin_antiperiodic.div cos_antiperiodic
|
/-
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.Topology.GDelta
#align_import topology.metric_space.baire from "leanprover-community/mathlib"@"b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a"
/-!
# ... | Mathlib/Topology/Baire/Lemmas.lean | 151 | 156 | theorem IsGδ.dense_biUnion_interior_of_closed {t : Set α} {s : Set X} (hs : IsGδ s) (hd : Dense s)
(ht : t.Countable) {f : α → Set X} (hc : ∀ i ∈ t, IsClosed (f i)) (hU : s ⊆ ⋃ i ∈ t, f i) :
Dense (⋃ i ∈ t, interior (f i)) := by |
haveI := ht.to_subtype
simp only [biUnion_eq_iUnion, SetCoe.forall'] at *
exact hs.dense_iUnion_interior_of_closed hd hc hU
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import data.polynomial.expand from "leanprover-community/mathlib"@"bbeb185db4ccee8e... | Mathlib/Algebra/Polynomial/Expand.lean | 48 | 49 | theorem expand_eq_sum {f : R[X]} : expand R p f = f.sum fun e a => C a * (X ^ p) ^ e := by |
simp [expand, eval₂]
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.Algebra.Algebra.Subalgebra.Directed
import Mathlib.FieldTheory.IntermediateField
import Mathlib.FieldTheory.Separable
imp... | Mathlib/FieldTheory/Adjoin.lean | 1,018 | 1,020 | theorem finrank_adjoin_simple_eq_one_iff :
finrank F F⟮α⟯ = 1 ↔ α ∈ (⊥ : IntermediateField F E) := by |
rw [finrank_adjoin_eq_one_iff]; exact Set.singleton_subset_iff
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Topology.Order.ProjIcc
#al... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 406 | 407 | theorem arccos_eq_pi {x} : arccos x = π ↔ x ≤ -1 := by |
rw [arccos, sub_eq_iff_eq_add, ← sub_eq_iff_eq_add', div_two_sub_self, neg_pi_div_two_eq_arcsin]
|
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | Mathlib/Analysis/MeanInequalities.lean | 409 | 426 | theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by |
have H₁ : ∀ s : Finset ι,
∑ i ∈ s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul ?_ ?_ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _... |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat... | Mathlib/Data/Num/Lemmas.lean | 860 | 861 | theorem cast_lt [LinearOrderedSemiring α] {m n : Num} : (m : α) < n ↔ m < n := by |
rw [← cast_to_nat m, ← cast_to_nat n, Nat.cast_lt (α := α), lt_to_nat]
|
/-
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.Basic
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.Topology.MetricSpace.IsometricSMul
#alig... | Mathlib/Analysis/Normed/Group/Pointwise.lean | 101 | 103 | theorem inv_cthickening : (cthickening δ s)⁻¹ = cthickening δ s⁻¹ := by |
simp_rw [cthickening, ← infEdist_inv]
rfl
|
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Eric Wieser
-/
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Multilinear.TensorProduct
import Mathlib.Ta... | Mathlib/LinearAlgebra/PiTensorProduct.lean | 165 | 171 | theorem smul_tprodCoeff [DecidableEq ι] (z : R) (f : Π i, s i) (i : ι) (r : R₁) [SMul R₁ R]
[IsScalarTower R₁ R R] [SMul R₁ (s i)] [IsScalarTower R₁ R (s i)] :
tprodCoeff R z (update f i (r • f i)) = tprodCoeff R (r • z) f := by |
have h₁ : r • z = r • (1 : R) * z := by rw [smul_mul_assoc, one_mul]
have h₂ : r • f i = (r • (1 : R)) • f i := (smul_one_smul _ _ _).symm
rw [h₁, h₂]
exact smul_tprodCoeff_aux z f i _
|
/-
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.Real
#align_import analys... | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 639 | 646 | theorem prod_rpow_of_ne_top {ι} {s : Finset ι} {f : ι → ℝ≥0∞} (hf : ∀ i ∈ s, f i ≠ ∞) (r : ℝ) :
∏ i ∈ s, f i ^ r = (∏ i ∈ s, f i) ^ r := by |
induction s using Finset.induction with
| empty => simp
| @insert i s hi ih =>
have h2f : ∀ i ∈ s, f i ≠ ∞ := fun i hi ↦ hf i <| mem_insert_of_mem hi
rw [prod_insert hi, prod_insert hi, ih h2f, ← mul_rpow_of_ne_top <| hf i <| mem_insert_self ..]
apply prod_lt_top h2f |>.ne
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Aesop
import Mathlib.Order.BoundedOrder
#align_import order.disjoint from "leanprover-community/mathlib"@"22c4d2ff43714b6ff724b2745ccfdc0f236a4a76"
/-!
# Dis... | Mathlib/Order/Disjoint.lean | 809 | 810 | theorem disjoint_coe : Disjoint (a : α) b ↔ Disjoint a b := by |
rw [disjoint_iff, disjoint_iff, ← coe_inf, ← coe_bot, coe_inj]
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Ri... | Mathlib/LinearAlgebra/Matrix/ToLin.lean | 632 | 633 | theorem Matrix.toLin_one : Matrix.toLin v₁ v₁ 1 = LinearMap.id := by |
rw [← LinearMap.toMatrix_id v₁, Matrix.toLin_toMatrix]
|
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.MonoidAlgebra.Basic
#align_import algebra.monoid_algebra.division from "leanprover-community/mathlib"@"72c366d0475675f1309d3027d3d7d47ee4423951"
/-... | Mathlib/Algebra/MonoidAlgebra/Division.lean | 77 | 79 | theorem divOf_zero (x : k[G]) : x /ᵒᶠ 0 = x := by |
refine Finsupp.ext fun _ => ?_ -- Porting note: `ext` doesn't work
simp only [AddMonoidAlgebra.divOf_apply, zero_add]
|
/-
Copyright (c) 2023 Alex Keizer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Keizer
-/
import Mathlib.Data.Vector.Basic
/-!
This file establishes a `snoc : Vector α n → α → Vector α (n+1)` operation, that appends a single
element to the back of a vector.... | Mathlib/Data/Vector/Snoc.lean | 154 | 159 | theorem mapAccumr₂_snoc (f : α → β → σ → σ × φ) (x : α) (y : β) :
mapAccumr₂ f (xs.snoc x) (ys.snoc y) c
= let q := f x y c
let r := mapAccumr₂ f xs ys q.1
(r.1, r.2.snoc q.2) := by |
induction xs, ys using Vector.inductionOn₂ <;> simp_all
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanp... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 417 | 418 | theorem prod_congr (h : s₁ = s₂) : (∀ x ∈ s₂, f x = g x) → s₁.prod f = s₂.prod g := by |
rw [h]; exact fold_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.Data.List.Sigma
import Mathlib.Data.Int.Range
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Finsupp.ToDFinsupp
import Mathlib.Testing.SlimCheck.Sample... | Mathlib/Testing/SlimCheck/Functions.lean | 369 | 378 | theorem List.applyId_eq_self [DecidableEq α] {xs ys : List α} (x : α) :
x ∉ xs → List.applyId.{u} (xs.zip ys) x = x := by |
intro h
dsimp [List.applyId]
rw [List.dlookup_eq_none.2]
· rfl
simp only [List.keys, not_exists, Prod.toSigma, exists_and_right, exists_eq_right, List.mem_map,
Function.comp_apply, List.map_map, Prod.exists]
intro y hy
exact h (List.mem_zip hy).1
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | Mathlib/LinearAlgebra/AffineSpace/Slope.lean | 62 | 63 | theorem sub_smul_slope_vadd (f : k → PE) (a b : k) : (b - a) • slope f a b +ᵥ f a = f b := by |
rw [sub_smul_slope, vsub_vadd]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.LinearAlgebra.Span
#align_import linear_algebra.quotient from... | Mathlib/LinearAlgebra/Quotient.lean | 557 | 566 | theorem Quotient.equiv_trans {N O : Type*} [AddCommGroup N] [Module R N] [AddCommGroup O]
[Module R O] (P : Submodule R M) (Q : Submodule R N) (S : Submodule R O) (e : M ≃ₗ[R] N)
(f : N ≃ₗ[R] O) (he : P.map e = Q) (hf : Q.map f = S) (hef : P.map (e.trans f) = S) :
Quotient.equiv P S (e.trans f) hef =
... |
ext
-- `simp` can deal with `hef` depending on `e` and `f`
simp only [Quotient.equiv_apply, LinearEquiv.trans_apply, LinearEquiv.coe_trans]
-- `rw` can deal with `mapQ_comp` needing extra hypotheses coming from the RHS
rw [mapQ_comp, LinearMap.comp_apply]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.... | Mathlib/Order/Interval/Set/OrderIso.lean | 58 | 59 | theorem preimage_Ioc (e : α ≃o β) (a b : β) : e ⁻¹' Ioc a b = Ioc (e.symm a) (e.symm b) := by |
simp [← Ioi_inter_Iic]
|
/-
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 | 379 | 381 | theorem product_sdiff_diag : s ×ˢ s \ s.diag = s.offDiag := by |
rw [← diag_union_offDiag, union_comm, union_sdiff_self,
sdiff_eq_self_of_disjoint (disjoint_diag_offDiag _).symm]
|
/-
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 | 205 | 206 | theorem rightZigzagIso_inv : (rightZigzagIso η ε).inv = leftZigzag ε.inv η.inv := by |
simp [bicategoricalComp, bicategoricalIsoComp]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Order.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_imp... | Mathlib/Order/Interval/Set/Basic.lean | 1,034 | 1,034 | theorem Icc_top : Icc a ⊤ = Ici a := by | simp [← Ici_inter_Iic]
|
/-
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.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Compl... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 217 | 231 | theorem isBoundedLinearMap_prod_multilinear {E : ι → Type*} [∀ i, NormedAddCommGroup (E i)]
[∀ i, NormedSpace 𝕜 (E i)] :
IsBoundedLinearMap 𝕜 fun p : ContinuousMultilinearMap 𝕜 E F × ContinuousMultilinearMap 𝕜 E G =>
p.1.prod p.2 where
map_add p₁ p₂ := by | ext : 1; rfl
map_smul c p := by ext : 1; rfl
bound := by
refine ⟨1, zero_lt_one, fun p ↦ ?_⟩
rw [one_mul]
apply ContinuousMultilinearMap.opNorm_le_bound _ (norm_nonneg _) _
intro m
rw [ContinuousMultilinearMap.prod_apply, norm_prod_le_iff]
constructor
· exact (p.1.le_opNorm m).trans (mu... |
/-
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.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | Mathlib/ModelTheory/FinitelyGenerated.lean | 284 | 291 | theorem Substructure.cg_iff_structure_cg (S : L.Substructure M) : S.CG ↔ Structure.CG L S := by |
rw [Structure.cg_def]
refine ⟨fun h => CG.of_map_embedding S.subtype ?_, fun h => ?_⟩
· rw [← Hom.range_eq_map, range_subtype]
exact h
· have h := h.map S.subtype.toHom
rw [← Hom.range_eq_map, range_subtype] at h
exact h
|
/-
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.Algebra.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_seri... | Mathlib/RingTheory/PowerSeries/Order.lean | 250 | 259 | theorem coeff_mul_prod_one_sub_of_lt_order {R ι : Type*} [CommRing R] (k : ℕ) (s : Finset ι)
(φ : R⟦X⟧) (f : ι → R⟦X⟧) :
(∀ i ∈ s, ↑k < (f i).order) → coeff R k (φ * ∏ i ∈ s, (1 - f i)) = coeff R k φ := by |
classical
induction' s using Finset.induction_on with a s ha ih t
· simp
· intro t
simp only [Finset.mem_insert, forall_eq_or_imp] at t
rw [Finset.prod_insert ha, ← mul_assoc, mul_right_comm, coeff_mul_one_sub_of_lt_order _ t.1]
exact ih t.2
|
/-
Copyright (c) 2023 Jonas van der Schaaf. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Christian Merten, Jonas van der Schaaf
-/
import Mathlib.AlgebraicGeometry.OpenImmersion
import Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact
import Mathlib... | Mathlib/AlgebraicGeometry/Morphisms/ClosedImmersion.lean | 79 | 89 | theorem spec_of_surjective {R S : CommRingCat} (f : R ⟶ S) (h : Function.Surjective f) :
IsClosedImmersion (Scheme.specMap f) where
base_closed := PrimeSpectrum.closedEmbedding_comap_of_surjective _ _ h
surj_on_stalks x := by |
erw [← localRingHom_comp_stalkIso, CommRingCat.coe_comp, CommRingCat.coe_comp]
apply Function.Surjective.comp (Function.Surjective.comp _ _) _
· exact (ConcreteCategory.bijective_of_isIso (StructureSheaf.stalkIso S x).inv).2
· exact surjective_localRingHom_of_surjective f h x.asIdeal
· let g := (St... |
/-
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 | 118 | 121 | theorem PowerBasis.norm_gen_eq_coeff_zero_minpoly (pb : PowerBasis R S) :
norm R pb.gen = (-1) ^ pb.dim * coeff (minpoly R pb.gen) 0 := by |
rw [norm_eq_matrix_det pb.basis, det_eq_sign_charpoly_coeff, charpoly_leftMulMatrix,
Fintype.card_fin]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Finset.Update
import Mathlib.Data.Prod.TProd
import Mathlib.GroupTheory.Coset
import Mathlib.Logic.Equiv.Fin
import Mathlib.Measur... | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 1,908 | 1,917 | theorem iff_comap_eq :
MeasurableEmbedding f ↔
Injective f ∧ MeasurableSpace.comap f ‹_› = ‹_› ∧ MeasurableSet (range f) :=
⟨fun hf ↦ ⟨hf.injective, hf.comap_eq, hf.measurableSet_range⟩, fun hf ↦
{ injective := hf.1
measurable := by | rw [← hf.2.1]; exact comap_measurable f
measurableSet_image' := by
rw [← hf.2.1]
rintro _ ⟨s, hs, rfl⟩
simpa only [image_preimage_eq_inter_range] using hs.inter hf.2.2 }⟩
|
/-
Copyright (c) 2019 mathlib community. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Wojciech Nawrocki
-/
import Batteries.Data.RBMap.Basic
import Mathlib.Init.Data.Nat.Notation
import Mathlib.Mathport.Rename
import Mathlib.Tactic.TypeStar
import Mat... | Mathlib/Data/Tree/Basic.lean | 94 | 96 | theorem numLeaves_pos (x : Tree α) : 0 < x.numLeaves := by |
rw [numLeaves_eq_numNodes_succ]
exact x.numNodes.zero_lt_succ
|
/-
Copyright (c) 2022 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.projective_space.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccb... | Mathlib/LinearAlgebra/Projectivization/Basic.lean | 228 | 233 | theorem map_comp {F U : Type*} [Field F] [AddCommGroup U] [Module F U] {σ : K →+* L} {τ : L →+* F}
{γ : K →+* F} [RingHomCompTriple σ τ γ] (f : V →ₛₗ[σ] W) (hf : Function.Injective f)
(g : W →ₛₗ[τ] U) (hg : Function.Injective g) :
map (g.comp f) (hg.comp hf) = map g hg ∘ map f hf := by |
ext ⟨v⟩
rfl
|
/-
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.Submodule
#align_import algebra.lie.ideal_operations from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
/-!
# Ideal... | Mathlib/Algebra/Lie/IdealOperations.lean | 284 | 286 | theorem comap_bracket_le {J₁ J₂ : LieIdeal R L'} : ⁅comap f J₁, comap f J₂⁆ ≤ comap f ⁅J₁, J₂⁆ := by |
rw [← map_le_iff_le_comap]
exact le_trans (map_bracket_le f) (LieSubmodule.mono_lie _ _ _ _ map_comap_le map_comap_le)
|
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Anatole Dedecker, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 87 | 89 | theorem HasDerivWithinAt.smul (hc : HasDerivWithinAt c c' s x) (hf : HasDerivWithinAt f f' s x) :
HasDerivWithinAt (fun y => c y • f y) (c x • f' + c' • f x) s x := by |
simpa using (HasFDerivWithinAt.smul hc hf).hasDerivWithinAt
|
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.Algebraic... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 1,381 | 1,388 | theorem IsDedekindDomain.exists_representative_mod_finset {ι : Type*} {s : Finset ι}
(P : ι → Ideal R) (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (P i))
(coprime : ∀ᵉ (i ∈ s) (j ∈ s), i ≠ j → P i ≠ P j) (x : ∀ i : s, R ⧸ P i ^ e i) :
∃ y, ∀ (i) (hi : i ∈ s), Ideal.Quotient.mk (P i ^ e i) y = x ⟨i, hi⟩ := by |
let f := IsDedekindDomain.quotientEquivPiOfFinsetProdEq _ P e prime coprime rfl
obtain ⟨y, rfl⟩ := f.surjective x
obtain ⟨z, rfl⟩ := Ideal.Quotient.mk_surjective y
exact ⟨z, fun i _hi => rfl⟩
|
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Data.Set.Lattice
#align_import data.set.accumulate from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-!
# Accumulat... | Mathlib/Data/Set/Accumulate.lean | 56 | 61 | theorem iUnion_accumulate [Preorder α] : ⋃ x, Accumulate s x = ⋃ x, s x := by |
apply Subset.antisymm
· simp only [subset_def, mem_iUnion, exists_imp, mem_accumulate]
intro z x x' ⟨_, hz⟩
exact ⟨x', hz⟩
· exact iUnion_mono fun i => subset_accumulate
|
/-
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 | 888 | 896 | theorem spanSingleton_mul_le_iff {x : P} {I J : FractionalIdeal S P} :
spanSingleton _ x * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by |
simp only [mul_le, mem_singleton_mul, mem_spanSingleton]
constructor
· intro h zI hzI
exact h x ⟨1, one_smul _ _⟩ zI hzI
· rintro h _ ⟨z, rfl⟩ zI hzI
rw [Algebra.smul_mul_assoc]
exact Submodule.smul_mem J.1 _ (h zI hzI)
|
/-
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.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Finiteness
import Mathlib.GroupTheory.GroupActio... | Mathlib/GroupTheory/Index.lean | 326 | 328 | theorem index_map {G' : Type*} [Group G'] (f : G →* G') :
(H.map f).index = (H ⊔ f.ker).index * f.range.index := by |
rw [← comap_map_eq, index_comap, relindex_mul_index (H.map_le_range f)]
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.PUnitInstances
import Mathlib.GroupTheory.Congr... | Mathlib/GroupTheory/Coprod/Basic.lean | 524 | 525 | theorem snd_toProd (x : M ∗ N) : (toProd x).2 = snd x := by |
rw [← snd_comp_toProd]; rfl
|
/-
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 | 239 | 249 | theorem superpolynomialDecay_iff_zpow_tendsto_zero (hk : Tendsto k l atTop) :
SuperpolynomialDecay l k f ↔ ∀ z : ℤ, Tendsto (fun a : α => k a ^ z * f a) l (𝓝 0) := by |
refine ⟨fun h z => ?_, fun h n => by simpa only [zpow_natCast] using h (n : ℤ)⟩
by_cases hz : 0 ≤ z
· unfold Tendsto
lift z to ℕ using hz
simpa using h z
· have : Tendsto (fun a => k a ^ z) l (𝓝 0) :=
Tendsto.comp (tendsto_zpow_atTop_zero (not_le.1 hz)) hk
have h : Tendsto f l (𝓝 0) := by s... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Order.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_imp... | Mathlib/Order/Interval/Set/Basic.lean | 1,799 | 1,800 | theorem Icc_inter_Icc : Icc a₁ b₁ ∩ Icc a₂ b₂ = Icc (a₁ ⊔ a₂) (b₁ ⊓ b₂) := by |
simp only [Ici_inter_Iic.symm, Ici_inter_Ici.symm, Iic_inter_Iic.symm]; ac_rfl
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stephen Morgan, Scott Morrison, Johannes Hölzl
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.Tactic.PPWithUniv
import Math... | Mathlib/CategoryTheory/Types.lean | 170 | 172 | theorem eqToHom_map_comp_apply (p : X = Y) (q : Y = Z) (x : F.obj X) :
F.map (eqToHom q) (F.map (eqToHom p) x) = F.map (eqToHom <| p.trans q) x := by |
aesop_cat
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Logic.Equiv.Option
import Mathlib.Order.RelIso.Basic
import Mathlib.Order.Disjoint
import Mathlib.Order.WithBot
import Mathlib.Tactic.Monotonicity.Attr... | Mathlib/Order/Hom/Basic.lean | 361 | 363 | theorem id_comp (f : α →o β) : comp id f = f := by |
ext
rfl
|
/-
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 | 661 | 673 | theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} :
(H ⊔ H').map f = H.map f ⊔ H'.map f := by |
ext1
· simp only [Set.image_union, map_verts, verts_sup]
· ext
simp only [Relation.Map, map_adj, sup_adj]
constructor
· rintro ⟨a, b, h | h, rfl, rfl⟩
· exact Or.inl ⟨_, _, h, rfl, rfl⟩
· exact Or.inr ⟨_, _, h, rfl, rfl⟩
· rintro (⟨a, b, h, rfl, rfl⟩ | ⟨a, b, h, rfl, rfl⟩)
· exa... |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | Mathlib/ModelTheory/Semantics.lean | 1,011 | 1,028 | theorem realize_boundedFormula (g : M ≃[L] N) (φ : L.BoundedFormula α n) {v : α → M}
{xs : Fin n → M} : φ.Realize (g ∘ v) (g ∘ xs) ↔ φ.Realize v xs := by |
induction' φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3
· rfl
· simp only [BoundedFormula.Realize, ← Sum.comp_elim, Equiv.realize_term, g.injective.eq_iff]
· simp only [BoundedFormula.Realize, ← Sum.comp_elim, Equiv.realize_term]
exact g.map_rel _ _
· rw [BoundedFormula.Realize, ih1, ih2, BoundedFormula.... |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Data.Fintype.Perm
import Mathlib.Data.Int.ModEq
import Mat... | Mathlib/GroupTheory/Perm/Cycle/Factors.lean | 560 | 568 | theorem Disjoint.disjoint_cycleFactorsFinset {f g : Perm α} (h : Disjoint f g) :
_root_.Disjoint (cycleFactorsFinset f) (cycleFactorsFinset g) := by |
rw [disjoint_iff_disjoint_support] at h
rw [Finset.disjoint_left]
intro x hx hy
simp only [mem_cycleFactorsFinset_iff, mem_support] at hx hy
obtain ⟨⟨⟨a, ha, -⟩, hf⟩, -, hg⟩ := hx, hy
have := h.le_bot (by simp [ha, ← hf a ha, ← hg a ha] : a ∈ f.support ∩ g.support)
tauto
|
/-
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 | 922 | 923 | theorem tan_mul_cos {x : ℝ} (hx : cos x ≠ 0) : tan x * cos x = sin x := by |
rw [tan_eq_sin_div_cos, div_mul_cancel₀ _ hx]
|
/-
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.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/... | Mathlib/Algebra/Polynomial/Mirror.lean | 193 | 194 | theorem mirror_neg : (-p).mirror = -p.mirror := by |
rw [mirror, mirror, reverse_neg, natTrailingDegree_neg, neg_mul_eq_neg_mul]
|
/-
Copyright (c) 2023 Antoine Chambert-Loir and María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández, Eric Wieser, Bhavik Mehta
-/
import Mathlib.Data.Finset.Antidiagonal
import Mathlib.Dat... | Mathlib/Data/Finset/PiAntidiagonal.lean | 187 | 209 | theorem mem_finsuppAntidiag_insert [DecidableEq ι] {a : ι} {s : Finset ι}
(h : a ∉ s) (n : μ) {f : ι →₀ μ} :
f ∈ finsuppAntidiag (insert a s) n ↔
∃ m ∈ antidiagonal n, ∃ (g : ι →₀ μ),
f = Finsupp.update g a m.1 ∧ g ∈ finsuppAntidiag s m.2 := by |
simp only [mem_finsuppAntidiag', mem_antidiagonal, Prod.exists, sum_insert h]
constructor
· rintro ⟨hsupp, rfl⟩
refine ⟨_, _, rfl, Finsupp.erase a f, ?_, ?_, ?_⟩
· rw [update_erase_eq_update, update_self]
· rwa [support_erase, ← subset_insert_iff]
· apply sum_congr rfl
intro x hx
rw [... |
/-
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 | 1,810 | 1,814 | theorem diagonal_mulVec_single [Fintype n] [DecidableEq n] [NonUnitalNonAssocSemiring R] (v : n → R)
(j : n) (x : R) : diagonal v *ᵥ Pi.single j x = Pi.single j (v j * x) := by |
ext i
rw [mulVec_diagonal]
exact Pi.apply_single (fun i x => v i * x) (fun i => mul_zero _) j x i
|
/-
Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheo... | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 225 | 227 | theorem mem_maxGenEigenspace (f : End R M) (μ : R) (m : M) :
m ∈ f.maxGenEigenspace μ ↔ ∃ k : ℕ, ((f - μ • (1 : End R M)) ^ k) m = 0 := by |
simp only [maxGenEigenspace, ← mem_genEigenspace, Submodule.mem_iSup_of_chain]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Eric Wieser
-/
import Mathlib.Data.Matrix.Basis
import Mathlib.RingTheory.TensorProduct.Basic
#align_import ring_theory.matrix_algebra from "leanprover-community/mathl... | Mathlib/RingTheory/MatrixAlgebra.lean | 113 | 121 | theorem right_inv (M : Matrix n n A) : (toFunAlgHom R A n) (invFun R A n M) = M := by |
simp only [invFun, AlgHom.map_sum, stdBasisMatrix, apply_ite ↑(algebraMap R A), smul_eq_mul,
mul_boole, toFunAlgHom_apply, RingHom.map_zero, RingHom.map_one, Matrix.map_apply,
Pi.smul_def]
convert Finset.sum_product (β := Matrix n n A)
conv_lhs => rw [matrix_eq_sum_std_basis M]
refine Finset.sum_congr ... |
/-
Copyright (c) 2020 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.Tactic.Linarith.Datatypes
import Mathlib.Tactic.Zify
import Mathlib.Tactic.CancelDenoms.Core
import Batteries.Data.RBMap.Basic
import Mathlib.Data.Ha... | Mathlib/Tactic/Linarith/Preprocessing.lean | 273 | 273 | theorem without_one_mul [MulOneClass M] {a b : M} (h : 1 * a = b) : a = b := by | rwa [one_mul] at h
|
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Algebra.Nonarchimedean.Bases
import Mathlib.Topology.Algebra.UniformFilterBasis
import Mathlib.RingTheory.Valuation.ValuationSubring
#align_i... | Mathlib/Topology/Algebra/Valuation.lean | 37 | 81 | theorem subgroups_basis : RingSubgroupsBasis fun γ : Γ₀ˣ => (v.ltAddSubgroup γ : AddSubgroup R) :=
{ inter := by |
rintro γ₀ γ₁
use min γ₀ γ₁
simp only [ltAddSubgroup, ge_iff_le, Units.min_val, Units.val_le_val, lt_min_iff,
AddSubgroup.mk_le_mk, setOf_subset_setOf, le_inf_iff, and_imp, imp_self, implies_true,
forall_const, and_true]
tauto
mul := by
rintro γ
cases' exists_squa... |
/-
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 | 672 | 676 | theorem derived_le_lower_central (n : ℕ) : derivedSeries G n ≤ lowerCentralSeries G n := by |
induction' n with i ih
· simp
· apply commutator_mono ih
simp
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Jireh Loreaux
-/
import Mathlib.Analysis.MeanInequalities
import Mathlib.Data.Fintype.Order
import Mathlib.LinearAlgebra.Matrix.Basis
import Mathlib.Analysis.Norm... | Mathlib/Analysis/NormedSpace/PiLp.lean | 803 | 823 | theorem nnnorm_equiv_symm_single [hp : Fact (1 ≤ p)] (i : ι) (b : β i) :
‖(WithLp.equiv p (∀ i, β i)).symm (Pi.single i b)‖₊ = ‖b‖₊ := by |
haveI : Nonempty ι := ⟨i⟩
induction p generalizing hp with
| top =>
simp_rw [nnnorm_eq_ciSup, WithLp.equiv_symm_pi_apply]
refine
ciSup_eq_of_forall_le_of_forall_lt_exists_gt (fun j => ?_) fun n hn => ⟨i, hn.trans_eq ?_⟩
· obtain rfl | hij := Decidable.eq_or_ne i j
· rw [Pi.single_eq_same]... |
/-
Copyright (c) 2024 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Filter.CountableInter
import Mathlib.Order.Filter.CardinalInter
import Mathlib.SetTheory.Cardinal.Ordinal
import... | Mathlib/Order/Filter/Cocardinal.lean | 70 | 72 | theorem frequently_cocardinal {p : α → Prop} :
(∃ᶠ x in cocardinal α hreg, p x) ↔ c ≤ # { x | p x } := by |
simp only [Filter.Frequently, eventually_cocardinal, not_not,coe_setOf, not_lt]
|
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