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 |
|---|---|---|---|---|---|
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Ring.Parity
import Mathlib.Combinatorics.SimpleGraph.Connectivity
#align_import combinatorics.simple_graph.trails from "leanprover-community/mathlib... | Mathlib/Combinatorics/SimpleGraph/Trails.lean | 93 | 98 | theorem IsEulerian.isTrail {u v : V} {p : G.Walk u v} (h : p.IsEulerian) : p.IsTrail := by |
rw [isTrail_def, List.nodup_iff_count_le_one]
intro e
by_cases he : e ∈ p.edges
· exact (h e (edges_subset_edgeSet _ he)).le
· simp [he]
|
/-
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.Order.Ring.Abs
#align_import data.int.order.units from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
/-!
# Lemmas a... | Mathlib/Data/Int/Order/Units.lean | 17 | 18 | theorem isUnit_iff_abs_eq {x : ℤ} : IsUnit x ↔ abs x = 1 := by |
rw [isUnit_iff_natAbs_eq, abs_eq_natAbs, ← Int.ofNat_one, natCast_inj]
|
/-
Copyright (c) 2021 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.Quasispectrum
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
import Mathlib.Analysis.Complex.Liouville
import Mathlib.Analysis.Complex.P... | Mathlib/Analysis/NormedSpace/Spectrum.lean | 158 | 161 | theorem _root_.quasispectrum.isCompact (a : B) : IsCompact (quasispectrum 𝕜 a) := by |
rw [Unitization.quasispectrum_eq_spectrum_inr' 𝕜 𝕜,
← AlgEquiv.spectrum_eq (WithLp.unitizationAlgEquiv 𝕜).symm (a : Unitization 𝕜 B)]
exact spectrum.isCompact _
|
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Yury Kudryashov
-/
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.Equicontinuity
import Mathlib.Topology.Separation
import... | Mathlib/Topology/UniformSpace/Compact.lean | 169 | 174 | theorem IsCompact.uniformContinuousOn_of_continuous {s : Set α} {f : α → β} (hs : IsCompact s)
(hf : ContinuousOn f s) : UniformContinuousOn f s := by |
rw [uniformContinuousOn_iff_restrict]
rw [isCompact_iff_compactSpace] at hs
rw [continuousOn_iff_continuous_restrict] at hf
exact CompactSpace.uniformContinuous_of_continuous hf
|
/-
Copyright (c) 2022 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.Algebra.Associated
import Mathlib.NumberTheory.Divisors
#align_import algebra.is_prime_pow from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0... | Mathlib/Algebra/IsPrimePow.lean | 107 | 111 | theorem Nat.disjoint_divisors_filter_isPrimePow {a b : ℕ} (hab : a.Coprime b) :
Disjoint (a.divisors.filter IsPrimePow) (b.divisors.filter IsPrimePow) := by |
simp only [Finset.disjoint_left, Finset.mem_filter, and_imp, Nat.mem_divisors, not_and]
rintro n han _ha hn hbn _hb -
exact hn.ne_one (Nat.eq_one_of_dvd_coprimes hab han hbn)
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.Set.Lattice
#align_import data.set.sigma from "leanprover-community/mathlib"@"2258b40dacd2942571c8ce136215350c702dc78f"... | Mathlib/Data/Set/Sigma.lean | 117 | 119 | theorem singleton_sigma_singleton {a : ∀ i, α i} :
(({i} : Set ι).sigma fun i ↦ ({a i} : Set (α i))) = {⟨i, a i⟩} := by |
rw [sigma_singleton, image_singleton]
|
/-
Copyright (c) 2020 Paul van Wamelen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul van Wamelen
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Int.Basic
import Mat... | Mathlib/NumberTheory/PythagoreanTriples.lean | 210 | 215 | theorem isClassified_of_isPrimitiveClassified (hp : h.IsPrimitiveClassified) : h.IsClassified := by |
obtain ⟨m, n, H⟩ := hp
use 1, m, n
rcases H with ⟨t, co, _⟩
rw [one_mul, one_mul]
exact ⟨t, co⟩
|
/-
Copyright (c) 2020 Hanting Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hanting Zhang, Johan Commelin
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.MvPolynomial.CommRing
#align_import r... | Mathlib/RingTheory/MvPolynomial/Symmetric.lean | 217 | 218 | theorem map_esymm (n : ℕ) (f : R →+* S) : map f (esymm σ R n) = esymm σ S n := by |
simp_rw [esymm, map_sum, map_prod, map_X]
|
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.ZPow
#align_import linear_algebra.matrix.hermitian from "leanprover-commun... | Mathlib/LinearAlgebra/Matrix/Hermitian.lean | 262 | 263 | theorem IsHermitian.adjugate [Fintype m] [DecidableEq m] {A : Matrix m m α} (hA : A.IsHermitian) :
A.adjugate.IsHermitian := by | simp [IsHermitian, adjugate_conjTranspose, hA.eq]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Ordering.Basic
import Mathlib.Order.Synonym
#align_import order.compare from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb... | Mathlib/Order/Compare.lean | 161 | 164 | theorem ofDual_compares_ofDual [LT α] {a b : αᵒᵈ} {o : Ordering} :
Compares o (ofDual a) (ofDual b) ↔ Compares o b a := by |
cases o
exacts [Iff.rfl, eq_comm, Iff.rfl]
|
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.JapaneseBracket
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Group.Integral
import Mathlib... | Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean | 32 | 38 | theorem integrableOn_exp_Iic (c : ℝ) : IntegrableOn exp (Iic c) := by |
refine
integrableOn_Iic_of_intervalIntegral_norm_bounded (exp c) c
(fun y => intervalIntegrable_exp.1) tendsto_id
(eventually_of_mem (Iic_mem_atBot 0) fun y _ => ?_)
simp_rw [norm_of_nonneg (exp_pos _).le, integral_exp, sub_le_self_iff]
exact (exp_pos _).le
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Segment
import Mathlib.LinearAlgebra.AffineSpace.Fi... | Mathlib/Analysis/Convex/Between.lean | 486 | 490 | theorem Wbtw.trans_sbtw_right [NoZeroSMulDivisors R V] {w x y z : P} (h₁ : Wbtw R w x z)
(h₂ : Sbtw R x y z) : Sbtw R w y z := by |
rw [wbtw_comm] at *
rw [sbtw_comm] at *
exact h₁.trans_sbtw_left h₂
|
/-
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, Bhavik Mehta, Andrew Yang, Emily Riehl
-/
import Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryPro... | Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean | 1,970 | 1,973 | theorem pullbackSymmetry_hom_of_mono_eq [Mono f] : (pullbackSymmetry f f).hom = 𝟙 _ := by |
ext
· simp [fst_eq_snd_of_mono_eq]
· simp [fst_eq_snd_of_mono_eq]
|
/-
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.Complex.Log
#align_import ana... | Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean | 86 | 88 | theorem one_cpow (x : ℂ) : (1 : ℂ) ^ x = 1 := by |
rw [cpow_def]
split_ifs <;> simp_all [one_ne_zero]
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.Multilinear.Curry
#align_import analysis.calculus.formal_multilinear_series from "leanprover-community/mathlib"@"f2ce608671... | Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean | 243 | 243 | theorem order_zero : (0 : FormalMultilinearSeries 𝕜 E F).order = 0 := by | simp [order]
|
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Algebra.Polynomial.Splits
import... | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 465 | 479 | theorem cyclotomic_eq_prod_X_pow_sub_one_pow_moebius {n : ℕ} (R : Type*) [CommRing R]
[IsDomain R] : algebraMap _ (RatFunc R) (cyclotomic n R) =
∏ i ∈ n.divisorsAntidiagonal, algebraMap R[X] _ (X ^ i.snd - 1) ^ μ i.fst := by |
rcases n.eq_zero_or_pos with (rfl | hpos)
· simp
have h : ∀ n : ℕ, 0 < n → (∏ i ∈ Nat.divisors n, algebraMap _ (RatFunc R) (cyclotomic i R)) =
algebraMap _ _ (X ^ n - 1 : R[X]) := by
intro n hn
rw [← prod_cyclotomic_eq_X_pow_sub_one hn R, map_prod]
rw [(prod_eq_iff_prod_pow_moebius_eq_of_nonzero ... |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Combinatorics.SimpleGraph.Density
import Mathlib.Data.Nat.Cast.Field
import Mathlib.Or... | Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean | 136 | 139 | theorem left_nonuniformWitnesses_subset (h : ¬G.IsUniform ε s t) :
(G.nonuniformWitnesses ε s t).1 ⊆ s := by |
rw [nonuniformWitnesses, dif_pos h]
exact (not_isUniform_iff.1 h).choose_spec.1
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import Mathlib.Data.List.Forall2
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Init.Data.Fin.Basic
#align_import data.list.nodup from "leanprover-... | Mathlib/Data/List/Nodup.lean | 238 | 248 | theorem inj_on_of_nodup_map {f : α → β} {l : List α} (d : Nodup (map f l)) :
∀ ⦃x⦄, x ∈ l → ∀ ⦃y⦄, y ∈ l → f x = f y → x = y := by |
induction' l with hd tl ih
· simp
· simp only [map, nodup_cons, mem_map, not_exists, not_and, ← Ne.eq_def] at d
simp only [mem_cons]
rintro _ (rfl | h₁) _ (rfl | h₂) h₃
· rfl
· apply (d.1 _ h₂ h₃.symm).elim
· apply (d.1 _ h₁ h₃).elim
· apply ih d.2 h₁ h₂ h₃
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic fro... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,271 | 1,272 | theorem lt_sup {ι : Type u} {f : ι → Ordinal.{max u v}} {a} : a < sup.{_, v} f ↔ ∃ i, a < f i := by |
simpa only [not_forall, not_le] using not_congr (@sup_le_iff.{_, v} _ f a)
|
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.MeasureTheory.Function.... | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | 133 | 138 | theorem ae_eq_zero_of_lintegral_rpow_eq_zero {p : ℝ} (hp0 : 0 ≤ p) {f : α → ℝ≥0∞}
(hf : AEMeasurable f μ) (hf_zero : ∫⁻ a, f a ^ p ∂μ = 0) : f =ᵐ[μ] 0 := by |
rw [lintegral_eq_zero_iff' (hf.pow_const p)] at hf_zero
filter_upwards [hf_zero] with x
rw [Pi.zero_apply, ← not_imp_not]
exact fun hx => (rpow_pos_of_nonneg (pos_iff_ne_zero.2 hx) hp0).ne'
|
/-
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, Jeremy Avigad
-/
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Defs.Filter
#align_import topology.basic from "leanprover-community/mathlib"@... | Mathlib/Topology/Basic.lean | 1,207 | 1,209 | theorem isOpen_singleton_iff_punctured_nhds (x : X) : IsOpen ({x} : Set X) ↔ 𝓝[≠] x = ⊥ := by |
rw [isOpen_singleton_iff_nhds_eq_pure, nhdsWithin, ← mem_iff_inf_principal_compl, ← le_pure_iff,
nhds_neBot.le_pure_iff]
|
/-
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 | 208 | 209 | theorem mul_self_mem_of_index_two (h : H.index = 2) (a : G) : a * a ∈ H := by |
rw [mul_mem_iff_of_index_two h]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Embedding.Set
#align_import logic.equiv.fin from "leanprover-community/mathlib"@"bd835ef554f37ef9b804f0903089211f89cb3... | Mathlib/Logic/Equiv/Fin.lean | 206 | 210 | theorem finSuccEquiv'_ne_last_apply {i j : Fin (n + 1)} (hi : i ≠ Fin.last n) (hj : j ≠ i) :
finSuccEquiv' i j = (i.castLT (Fin.val_lt_last hi)).predAbove j := by |
rcases Fin.exists_succAbove_eq hj with ⟨j, rfl⟩
rcases Fin.exists_castSucc_eq.2 hi with ⟨i, rfl⟩
simp
|
/-
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, Junyan Xu
-/
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimen... | Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 332 | 336 | theorem lift_rank_lt_rank_dual {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V]
(h : ℵ₀ ≤ Module.rank K V) :
Cardinal.lift.{u} (Module.rank K V) < Module.rank K (V →ₗ[K] K) := by |
rw [rank_dual_eq_card_dual_of_aleph0_le_rank h, ← rank_dual_eq_card_dual_of_aleph0_le_rank' h]
exact lift_rank_lt_rank_dual' h
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Data.Stream.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Init.Data.List.Basic
import Mathlib.Data.List.Basic
#align_import data.s... | Mathlib/Data/Stream/Init.lean | 240 | 242 | theorem const_eq (a : α) : const a = a::const a := by |
apply Stream'.ext; intro n
cases n <;> rfl
|
/-
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.Algebra.UniformConvergence
#align_import topology.algebra.module.strong_topology from "leanprover-community/mathlib"@"8905e5ed90859939681... | Mathlib/Topology/Algebra/Module/StrongTopology.lean | 180 | 189 | theorem hasBasis_nhds_zero_of_basis [TopologicalSpace F] [TopologicalAddGroup F]
{ι : Type*} (𝔖 : Set (Set E)) (h𝔖₁ : 𝔖.Nonempty) (h𝔖₂ : DirectedOn (· ⊆ ·) 𝔖) {p : ι → Prop}
{b : ι → Set F} (h : (𝓝 0 : Filter F).HasBasis p b) :
(𝓝 (0 : UniformConvergenceCLM σ F 𝔖)).HasBasis
(fun Si : Set E × ι... |
letI : UniformSpace F := TopologicalAddGroup.toUniformSpace F
haveI : UniformAddGroup F := comm_topologicalAddGroup_is_uniform
rw [(embedding_coeFn σ F 𝔖).toInducing.nhds_eq_comap]
exact (UniformOnFun.hasBasis_nhds_zero_of_basis 𝔖 h𝔖₁ h𝔖₂ h).comap DFunLike.coe
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Scott Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Data.Set.Finite
#align_import data.finsupp.defs fr... | Mathlib/Data/Finsupp/Defs.lean | 361 | 362 | theorem single_apply_eq_zero {a x : α} {b : M} : single a b x = 0 ↔ x = a → b = 0 := by |
simp [single_eq_set_indicator]
|
/-
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.CharZero.Lemmas
import Mathlib.Algebra.Module.Submodule.Ker
import Mathlib.Algebra.Module.Submodule.... | Mathlib/Algebra/Algebra/Basic.lean | 353 | 355 | theorem algebraMap_injective [CommRing R] [Ring A] [Nontrivial A] [Algebra R A]
[NoZeroSMulDivisors R A] : Function.Injective (algebraMap R A) := by |
simpa only [algebraMap_eq_smul_one'] using smul_left_injective R one_ne_zero
|
/-
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.FreeAlgebra
import Mathlib.GroupTheory.Finiteness
import Mathlib.RingTheory.Adjoin.Tower
import Mathlib.RingTheory.Finiteness
import Mathlib.Ri... | Mathlib/RingTheory/FiniteType.lean | 267 | 277 | theorem comp {g : B →+* C} {f : A →+* B} (hg : g.FiniteType) (hf : f.FiniteType) :
(g.comp f).FiniteType := by |
let _ : Algebra A B := f.toAlgebra
let _ : Algebra A C := (g.comp f).toAlgebra
let _ : Algebra B C := g.toAlgebra
exact @Algebra.FiniteType.trans A B C _ _ _ f.toAlgebra (g.comp f).toAlgebra g.toAlgebra
⟨by
intro a b c
simp [Algebra.smul_def, RingHom.map_mul, mul_assoc]
rfl⟩
hf hg
|
/-
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 | 1,054 | 1,058 | theorem _root_.HasCompactSupport.hasFDerivAt_convolution_left [IsNegInvariant μ]
(hcf : HasCompactSupport f) (hf : ContDiff 𝕜 1 f) (hg : LocallyIntegrable g μ) (x₀ : G) :
HasFDerivAt (f ⋆[L, μ] g) ((fderiv 𝕜 f ⋆[L.precompL G, μ] g) x₀) x₀ := by |
simp (config := { singlePass := true }) only [← convolution_flip]
exact hcf.hasFDerivAt_convolution_right L.flip hg hf x₀
|
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.EMetricSpace.Basic
import Mathlib.Topology.Bornology.Constructions
imp... | Mathlib/Topology/MetricSpace/PseudoMetric.lean | 360 | 361 | theorem edist_lt_ofReal {x y : α} {r : ℝ} : edist x y < ENNReal.ofReal r ↔ dist x y < r := by |
rw [edist_dist, ENNReal.ofReal_lt_ofReal_iff_of_nonneg dist_nonneg]
|
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Wieser
-/
import Mathlib.LinearAlgebra.Matrix.DotProduct
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
#align_import data.... | Mathlib/Data/Matrix/Rank.lean | 244 | 246 | theorem rank_self_mul_conjTranspose (A : Matrix m n R) : (A * Aᴴ).rank = A.rank := by |
simpa only [rank_conjTranspose, conjTranspose_conjTranspose] using
rank_conjTranspose_mul_self Aᴴ
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Int.LeastGreatest
import Mathlib.Data.Rat.Floor
import Mathlib.Data.NNRat.Defs
#align_import algebra.ord... | Mathlib/Algebra/Order/Archimedean.lean | 254 | 260 | theorem exists_nat_pow_near_of_lt_one (xpos : 0 < x) (hx : x ≤ 1) (ypos : 0 < y) (hy : y < 1) :
∃ n : ℕ, y ^ (n + 1) < x ∧ x ≤ y ^ n := by |
rcases exists_nat_pow_near (one_le_inv_iff.2 ⟨xpos, hx⟩) (one_lt_inv_iff.2 ⟨ypos, hy⟩) with
⟨n, hn, h'n⟩
refine ⟨n, ?_, ?_⟩
· rwa [inv_pow, inv_lt_inv xpos (pow_pos ypos _)] at h'n
· rwa [inv_pow, inv_le_inv (pow_pos ypos _) xpos] at hn
|
/-
Copyright (c) 2021 Paul Lezeau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Paul Lezeau
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Order.Hom.Bounded
import Mathlib.Algebra.GCDMonoid.Basic
#align_impor... | Mathlib/RingTheory/ChainOfDivisors.lean | 427 | 452 | theorem multiplicity_factor_dvd_iso_eq_multiplicity_of_mem_normalizedFactors {m p : M} {n : N}
(hm : m ≠ 0) (hn : n ≠ 0) (hp : p ∈ normalizedFactors m)
{d : { l : M // l ∣ m } ≃ { l : N // l ∣ n }} (hd : ∀ l l', (d l : N) ∣ d l' ↔ (l : M) ∣ l') :
multiplicity (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : N) n =... |
apply Eq.symm
suffices multiplicity (Associates.mk p) (Associates.mk m) = multiplicity (Associates.mk
↑(d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), by
simp [dvd_of_mem_normalizedFactors hp]⟩)) (Associates.mk n) by
simpa only [multiplicity_mk_eq_multiplicity, associatesEquiv... |
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.Charpoly.Basic
import Mathlib.LinearAlgebra.Matrix.Basis
#align_import linear_algebra.charpoly.to_matrix from "leanprover-community/ma... | Mathlib/LinearAlgebra/Charpoly/ToMatrix.lean | 48 | 87 | theorem charpoly_toMatrix {ι : Type w} [DecidableEq ι] [Fintype ι] (b : Basis ι R M) :
(toMatrix b b f).charpoly = f.charpoly := by |
let A := toMatrix b b f
let b' := chooseBasis R M
let ι' := ChooseBasisIndex R M
let A' := toMatrix b' b' f
let e := Basis.indexEquiv b b'
let φ := reindexLinearEquiv R R e e
let φ₁ := reindexLinearEquiv R R e (Equiv.refl ι')
let φ₂ := reindexLinearEquiv R R (Equiv.refl ι') (Equiv.refl ι')
let φ₃ := ... |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker
-/
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.Algebra.Gr... | Mathlib/Algebra/Associated.lean | 1,275 | 1,280 | theorem DvdNotUnit.ne [CancelCommMonoidWithZero α] {p q : α} (h : DvdNotUnit p q) : p ≠ q := by |
by_contra hcontra
obtain ⟨hp, x, hx', hx''⟩ := h
conv_lhs at hx'' => rw [← hcontra, ← mul_one p]
rw [(mul_left_cancel₀ hp hx'').symm] at hx'
exact hx' isUnit_one
|
/-
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, Huỳnh Trần Khanh, Stuart Presnell
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Fintype... | Mathlib/Data/Sym/Card.lean | 167 | 170 | theorem card_image_offDiag (s : Finset α) :
(s.offDiag.image Sym2.mk).card = s.card.choose 2 := by |
rw [Nat.choose_two_right, Nat.mul_sub_left_distrib, mul_one, ← offDiag_card,
Nat.div_eq_of_eq_mul_right Nat.zero_lt_two (two_mul_card_image_offDiag s).symm]
|
/-
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 | 891 | 895 | theorem exists_finset_of_mem_supr' {ι : Type*} {f : ι → IntermediateField F E} {x : E}
(hx : x ∈ ⨆ i, f i) : ∃ s : Finset (Σ i, f i), x ∈ ⨆ i ∈ s, F⟮(i.2 : E)⟯ := by |
-- Porting note: writing `fun i x h => ...` does not work.
refine exists_finset_of_mem_iSup (SetLike.le_def.mp (iSup_le fun i ↦ ?_) hx)
exact fun x h ↦ SetLike.le_def.mp (le_iSup_of_le ⟨i, x, h⟩ (by simp)) (mem_adjoin_simple_self F x)
|
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov, Hunter Monroe
-/
import Mathlib.Combinatorics.SimpleGraph.Init
import Mathlib.Data.Rel
import Mathlib... | Mathlib/Combinatorics/SimpleGraph/Basic.lean | 934 | 935 | theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) :
(G.deleteEdges s).deleteEdges s' = G.deleteEdges (s ∪ s') := by | simp [deleteEdges, sdiff_sdiff]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Exponential
#align_import set_theory.ordinal.cantor_normal_form from "leanprover-communi... | Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | 62 | 64 | theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)
(H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :
@CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _) := by | rw [CNFRec, dif_neg ho]
|
/-
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.Finset.Lattice
import Mathlib.Data.Multiset.Powerset
#align_import data.finset.powerset from "leanprover-community/mathlib"@"9003f28797c0664a49e4... | Mathlib/Data/Finset/Powerset.lean | 312 | 325 | theorem powersetCard_sup [DecidableEq α] (u : Finset α) (n : ℕ) (hn : n < u.card) :
(powersetCard n.succ u).sup id = u := by |
apply le_antisymm
· simp_rw [Finset.sup_le_iff, mem_powersetCard]
rintro x ⟨h, -⟩
exact h
· rw [sup_eq_biUnion, le_iff_subset, subset_iff]
intro x hx
simp only [mem_biUnion, exists_prop, id]
obtain ⟨t, ht⟩ : ∃ t, t ∈ powersetCard n (u.erase x) := powersetCard_nonempty.2
(le_trans (Nat.l... |
/-
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.Topology.Sets.Opens
#align_import topology.local_at_target from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# Properties ... | Mathlib/Topology/LocalAtTarget.lean | 148 | 156 | theorem embedding_iff_embedding_of_iSup_eq_top (h : Continuous f) :
Embedding f ↔ ∀ i, Embedding ((U i).1.restrictPreimage f) := by |
simp_rw [embedding_iff]
rw [forall_and]
apply and_congr
· apply inducing_iff_inducing_of_iSup_eq_top <;> assumption
· apply Set.injective_iff_injective_of_iUnion_eq_univ
convert congr_arg SetLike.coe hU
simp
|
/-
Copyright (c) 2022 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde9... | Mathlib/MeasureTheory/Integral/Average.lean | 332 | 333 | theorem average_eq (f : α → E) : ⨍ x, f x ∂μ = (μ univ).toReal⁻¹ • ∫ x, f x ∂μ := by |
rw [average_eq', integral_smul_measure, ENNReal.toReal_inv]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Independence.Basic
import Mathlib.Probability.Independence.Conditional
#align_import probability.independence.zero_one from "leanprover-commun... | Mathlib/Probability/Independence/ZeroOne.lean | 156 | 167 | theorem kernel.indep_iSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα)
(hf : ∀ t, p t → tᶜ ∈ f)
(hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
Indep (⨆ n, s n) (limsup s f) κ μα := by |
suffices (⨆ a, ⨆ n ∈ ns a, s n) = ⨆ n, s n by
rw [← this]
exact indep_iSup_directed_limsup h_le h_indep hf hns hnsp
rw [iSup_comm]
refine iSup_congr fun n => ?_
have h : ⨆ (i : α) (_ : n ∈ ns i), s n = ⨆ _ : ∃ i, n ∈ ns i, s n := by rw [iSup_exists]
haveI : Nonempty (∃ i : α, n ∈ ns i) := ⟨hns_univ n... |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Integral.IntegrableOn
#align_import measure_theory.function.locally_integrable from "leanprover-community/mathlib"@"08a4542bec7242a5... | Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | 301 | 305 | theorem LocallyIntegrable.indicator (hf : LocallyIntegrable f μ) {s : Set X}
(hs : MeasurableSet s) : LocallyIntegrable (s.indicator f) μ := by |
intro x
rcases hf x with ⟨U, hU, h'U⟩
exact ⟨U, hU, h'U.indicator hs⟩
|
/-
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.Finset.Lattice
import Mathlib.Data.Multiset.Powerset
#align_import data.finset.powerset from "leanprover-community/mathlib"@"9003f28797c0664a49e4... | Mathlib/Data/Finset/Powerset.lean | 233 | 235 | theorem map_val_val_powersetCard (s : Finset α) (i : ℕ) :
(s.powersetCard i).val.map Finset.val = s.1.powersetCard i := by |
simp [Finset.powersetCard, map_pmap, pmap_eq_map, map_id']
|
/-
Copyright (c) 2023 Ziyu Wang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ziyu Wang, Chenyi Li, Sébastien Gouëzel, Penghao Yu, Zhipeng Cao
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.Calc... | Mathlib/Analysis/Calculus/Gradient/Basic.lean | 173 | 176 | theorem HasDerivAt.hasGradientAt (h : HasDerivAt g g' u) :
HasGradientAt g (starRingEnd 𝕜 g') u := by |
rw [hasGradientAt_iff_hasFDerivAt, hasFDerivAt_iff_hasDerivAt]
simpa
|
/-
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 | 73 | 74 | theorem midpoint_comm (x y : P) : midpoint R x y = midpoint R y x := by |
rw [midpoint, ← lineMap_apply_one_sub, one_sub_invOf_two, midpoint]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Patrick Massot
-/
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.UniformSpace.Separation
import Mathlib.Topology.DenseEmbedding
... | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | 456 | 470 | theorem uniform_extend_subtype [CompleteSpace γ] {p : α → Prop} {e : α → β} {f : α → γ} {b : β}
{s : Set α} (hf : UniformContinuous fun x : Subtype p => f x.val) (he : UniformEmbedding e)
(hd : ∀ x : β, x ∈ closure (range e)) (hb : closure (e '' s) ∈ 𝓝 b) (hs : IsClosed s)
(hp : ∀ x ∈ s, p x) : ∃ c, Tendst... |
have de : DenseEmbedding e := he.denseEmbedding hd
have de' : DenseEmbedding (DenseEmbedding.subtypeEmb p e) := de.subtype p
have ue' : UniformEmbedding (DenseEmbedding.subtypeEmb p e) := uniformEmbedding_subtypeEmb _ he de
have : b ∈ closure (e '' { x | p x }) :=
(closure_mono <| monotone_image <| hp) (me... |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | Mathlib/SetTheory/Game/Nim.lean | 272 | 308 | theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impartial.forall_leftMoves_fuzzy_iff_equiv_zero]
intro i
apply leftMoves_add_cases i
· intro i₁
rw [add_moveLeft_inl]
apply
(fuzzy_congr_left... |
revert i₂
rw [grundyValue_eq_mex_left]
intro i₂
have hnotin : _ ∉ _ := fun hin =>
(le_not_le_of_lt (Ordinal.typein_lt_self i₂)).2 (csInf_le' hin)
simpa using hnotin
cases' h' with i hi
use toLeftMovesAdd (Sum.inl i)
rw [add_moveLeft_inl, moveLeft_mk]
... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic fro... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 2,257 | 2,272 | theorem enumOrd_surjective (hS : Unbounded (· < ·) S) : ∀ s ∈ S, ∃ a, enumOrd S a = s := fun s hs =>
⟨sSup { a | enumOrd S a ≤ s }, by
apply le_antisymm
· rw [enumOrd_def]
refine csInf_le' ⟨hs, fun a ha => ?_⟩
have : enumOrd S 0 ≤ s := by |
rw [enumOrd_zero]
exact csInf_le' hs
-- Porting note: `flip` is required to infer a metavariable.
rcases flip exists_lt_of_lt_csSup ha ⟨0, this⟩ with ⟨b, hb, hab⟩
exact (enumOrd_strictMono hS hab).trans_le hb
· by_contra! h
exact
(le_csSup ⟨s, fun a => (lt_wf.self_le... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Topology.Maps
import Mathlib.Topology.NhdsSet
#align_import topology.constructions from "leanprover-community/mathlib"... | Mathlib/Topology/Constructions.lean | 1,654 | 1,658 | theorem continuous_sigma_iff {f : Sigma σ → X} :
Continuous f ↔ ∀ i, Continuous fun a => f ⟨i, a⟩ := by |
delta instTopologicalSpaceSigma
rw [continuous_iSup_dom]
exact forall_congr' fun _ => continuous_coinduced_dom
|
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Topology.MetricSpace.Basic
#align_import topology.metric_space.infsep from "leanprover-community/mathlib"@"5316314b553dcf8c6716541851517c1a9715e22b"
/-... | Mathlib/Topology/MetricSpace/Infsep.lean | 209 | 212 | theorem einfsep_pair (hxy : x ≠ y) : ({x, y} : Set α).einfsep = edist x y := by |
nth_rw 1 [← min_self (edist x y)]
convert einfsep_pair_eq_inf hxy using 2
rw [edist_comm]
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "... | Mathlib/Data/Set/Function.lean | 1,822 | 1,826 | theorem injOn_image (h : Semiconj f fa fb) (ha : InjOn fa s) (hf : InjOn f (fa '' s)) :
InjOn fb (f '' s) := by |
rintro _ ⟨x, hx, rfl⟩ _ ⟨y, hy, rfl⟩ H
simp only [← h.eq] at H
exact congr_arg f (ha hx hy <| hf (mem_image_of_mem fa hx) (mem_image_of_mem fa hy) 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.List
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Data.List.FinRange
import Mathlib.Algebra.Group.Action.Defs
import Mathli... | Mathlib/Algebra/GradedMonoid.lean | 598 | 605 | theorem pow_mem_graded (n : ℕ) {r : R} {i : ι} (h : r ∈ A i) : r ^ n ∈ A (n • i) := by |
match n with
| 0 =>
rw [pow_zero, zero_nsmul]
exact one_mem_graded _
| n+1 =>
rw [pow_succ', succ_nsmul']
exact mul_mem_graded h (pow_mem_graded n h)
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | Mathlib/Topology/Sheaves/Stalks.lean | 590 | 600 | theorem app_isIso_of_stalkFunctor_map_iso {F G : Sheaf C X} (f : F ⟶ G) (U : Opens X)
[∀ x : U, IsIso ((stalkFunctor C x.val).map f.1)] : IsIso (f.1.app (op U)) := by |
-- Since the forgetful functor of `C` reflects isomorphisms, it suffices to see that the
-- underlying map between types is an isomorphism, i.e. bijective.
suffices IsIso ((forget C).map (f.1.app (op U))) by
exact isIso_of_reflects_iso (f.1.app (op U)) (forget C)
rw [isIso_iff_bijective]
apply app_biject... |
/-
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 | 179 | 180 | theorem midpoint_eq_right_iff {x y : P} : midpoint R x y = y ↔ x = y := by |
rw [midpoint_comm, midpoint_eq_left_iff, eq_comm]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Star.Unitary
import Mathlib.RingTheory.Int.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathli... | Mathlib/NumberTheory/Zsqrtd/Basic.lean | 589 | 590 | theorem isUnit_iff_norm_isUnit {d : ℤ} (z : ℤ√d) : IsUnit z ↔ IsUnit z.norm := by |
rw [Int.isUnit_iff_natAbs_eq, norm_eq_one_iff]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Scott Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Data.Set.Finite
#align_import data.finsupp.defs fr... | Mathlib/Data/Finsupp/Defs.lean | 369 | 370 | theorem mem_support_single (a a' : α) (b : M) : a ∈ (single a' b).support ↔ a = a' ∧ b ≠ 0 := by |
simp [single_apply_eq_zero, not_or]
|
/-
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.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.pi_nat from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5... | Mathlib/Topology/MetricSpace/PiNat.lean | 557 | 563 | theorem disjoint_cylinder_of_longestPrefix_lt {s : Set (∀ n, E n)} (hs : IsClosed s) {x : ∀ n, E n}
(hx : x ∉ s) {n : ℕ} (hn : longestPrefix x s < n) : Disjoint s (cylinder x n) := by |
contrapose! hn
rcases not_disjoint_iff_nonempty_inter.1 hn with ⟨y, ys, hy⟩
apply le_trans _ (firstDiff_le_longestPrefix hs hx ys)
apply (mem_cylinder_iff_le_firstDiff (ne_of_mem_of_not_mem ys hx).symm _).1
rwa [mem_cylinder_comm]
|
/-
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
-/
import Mathlib.Logic.Relation
import Mathlib.Order.GaloisConnection
#align_import data.setoid.basic from "leanprover-community/mathlib"@"bbe... | Mathlib/Data/Setoid/Basic.lean | 476 | 482 | theorem Quotient.subsingleton_iff {s : Setoid α} : Subsingleton (Quotient s) ↔ s = ⊤ := by |
simp only [_root_.subsingleton_iff, eq_top_iff, Setoid.le_def, Setoid.top_def, Pi.top_apply,
forall_const]
refine (surjective_quotient_mk' _).forall.trans (forall_congr' fun a => ?_)
refine (surjective_quotient_mk' _).forall.trans (forall_congr' fun b => ?_)
simp_rw [Prop.top_eq_true, true_implies, Quotien... |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Algebra.Group.Sub... | Mathlib/Algebra/Group/Submonoid/Membership.lean | 198 | 207 | theorem mem_iSup_of_directed {ι} [hι : Nonempty ι] {S : ι → Submonoid M} (hS : Directed (· ≤ ·) S)
{x : M} : (x ∈ ⨆ i, S i) ↔ ∃ i, x ∈ S i := by |
refine ⟨?_, fun ⟨i, hi⟩ ↦ le_iSup S i hi⟩
suffices x ∈ closure (⋃ i, (S i : Set M)) → ∃ i, x ∈ S i by
simpa only [closure_iUnion, closure_eq (S _)] using this
refine fun hx ↦ closure_induction hx (fun _ ↦ mem_iUnion.1) ?_ ?_
· exact hι.elim fun i ↦ ⟨i, (S i).one_mem⟩
· rintro x y ⟨i, hi⟩ ⟨j, hj⟩
rcas... |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Topology.EMetricSpace.Paracompact
import Mathlib.Topology.Instances.ENNReal
import Mathlib.Analysis.Convex.PartitionOfUnity
#align_import topology.m... | Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | 87 | 93 | theorem exists_continuous_real_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
(hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
∃ δ : C(X, ℝ), (∀ x, 0 < δ x) ∧
∀ (i), ∀ x ∈ K i, closedBall x (ENNReal.ofReal <| δ x) ⊆ U i := by |
simpa only [mem_inter_iff, forall_and, mem_preimage, mem_iInter, @forall_swap ι X] using
exists_continuous_forall_mem_convex_of_local_const exists_forall_closedBall_subset_aux₂
(exists_forall_closedBall_subset_aux₁ hK hU hKU hfin)
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attr
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Logic.Equiv.Set
import Math... | Mathlib/Data/Finset/Basic.lean | 3,303 | 3,306 | theorem toFinset_replicate_of_ne_zero {n : ℕ} (hn : n ≠ 0) :
(List.replicate n a).toFinset = {a} := by |
ext x
simp [hn, List.mem_replicate]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Data.Finset.So... | Mathlib/Algebra/Polynomial/Basic.lean | 430 | 430 | theorem card_support_eq_zero : p.support.card = 0 ↔ p = 0 := by | simp
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.RingTheory.EuclideanDo... | Mathlib/Algebra/Polynomial/FieldDivision.lean | 564 | 567 | theorem coe_normUnit_of_ne_zero [DecidableEq R] (hp : p ≠ 0) :
(normUnit p : R[X]) = C p.leadingCoeff⁻¹ := by |
have : p.leadingCoeff ≠ 0 := mt leadingCoeff_eq_zero.mp hp
simp [CommGroupWithZero.coe_normUnit _ this]
|
/-
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.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.... | Mathlib/Analysis/Calculus/FDeriv/Add.lean | 561 | 563 | theorem HasFDerivAtFilter.sub_const (hf : HasFDerivAtFilter f f' x L) (c : F) :
HasFDerivAtFilter (fun x => f x - c) f' x L := by |
simpa only [sub_eq_add_neg] using hf.add_const (-c)
|
/-
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 | 108 | 116 | theorem mk_eq_mk_iff' (v w : V) (hv : v ≠ 0) (hw : w ≠ 0) :
mk K v hv = mk K w hw ↔ ∃ a : K, a • w = v := by |
rw [mk_eq_mk_iff K v w hv hw]
constructor
· rintro ⟨a, ha⟩
exact ⟨a, ha⟩
· rintro ⟨a, ha⟩
refine ⟨Units.mk0 a fun c => hv.symm ?_, ha⟩
rwa [c, zero_smul] at ha
|
/-
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.IsAlgClosed.AlgebraicClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_e... | Mathlib/FieldTheory/PrimitiveElement.lean | 366 | 371 | theorem AlgHom.card_of_splits (L : Type*) [Field L] [Algebra F L]
(hL : ∀ x : E, (minpoly F x).Splits (algebraMap F L)) :
Fintype.card (E →ₐ[F] L) = finrank F E := by |
rw [← Fintype.ofEquiv_card <| Algebra.IsAlgebraic.algHomEquivAlgHomOfSplits
(AlgebraicClosure L) _ hL]
convert AlgHom.card F E (AlgebraicClosure L)
|
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SeparableClosure
import Mathlib.Algebra.CharP.IntermediateField
/-!
# Purely inseparable extension and relative perfect closure
This file contains basic... | Mathlib/FieldTheory/PurelyInseparable.lean | 843 | 851 | theorem finSepDegree_mul_finInsepDegree : finSepDegree F E * finInsepDegree F E = finrank F E := by |
by_cases halg : Algebra.IsAlgebraic F E
· have := congr_arg Cardinal.toNat (sepDegree_mul_insepDegree F E)
rwa [Cardinal.toNat_mul, ← finSepDegree_eq F E] at this
rw [finInsepDegree, finrank_of_infinite_dimensional (K := F) (V := E) fun _ ↦
halg (Algebra.IsAlgebraic.of_finite F E),
finrank_of_infin... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic fro... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,288 | 1,294 | theorem sup_eq_zero_iff {ι : Type u} {f : ι → Ordinal.{max u v}} :
sup.{_, v} f = 0 ↔ ∀ i, f i = 0 := by |
refine
⟨fun h i => ?_, fun h =>
le_antisymm (sup_le fun i => Ordinal.le_zero.2 (h i)) (Ordinal.zero_le _)⟩
rw [← Ordinal.le_zero, ← h]
exact le_sup f i
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic fro... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 296 | 297 | theorem IsLimit.one_lt {o : Ordinal} (h : IsLimit o) : 1 < o := by |
simpa only [succ_zero] using h.2 _ h.pos
|
/-
Copyright (c) 2019 Zhouhang Zhou. 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.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-communit... | Mathlib/MeasureTheory/Integral/Bochner.lean | 1,977 | 2,006 | theorem integral_trim (hm : m ≤ m0) {f : β → G} (hf : StronglyMeasurable[m] f) :
∫ x, f x ∂μ = ∫ x, f x ∂μ.trim hm := by |
by_cases hG : CompleteSpace G; swap
· simp [integral, hG]
borelize G
by_cases hf_int : Integrable f μ
swap
· have hf_int_m : ¬Integrable f (μ.trim hm) := fun hf_int_m =>
hf_int (integrable_of_integrable_trim hm hf_int_m)
rw [integral_undef hf_int, integral_undef hf_int_m]
haveI : SeparableSpace... |
/-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
#align_import category_theory.limits.shapes.wide_equalizers from "l... | Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean | 271 | 272 | theorem Trident.condition (j₁ j₂ : J) (t : Trident f) : t.ι ≫ f j₁ = t.ι ≫ f j₂ := by |
rw [t.app_zero, t.app_zero]
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | Mathlib/Topology/Inseparable.lean | 457 | 458 | theorem map_mk_nhdsSet : map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s) := by |
rw [← comap_mk_nhdsSet_image, map_comap_of_surjective surjective_mk]
|
/-
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.Order.Interval.Set.Basic
import Mathlib.Data.Set.NAry
import Mathlib.Order.Directed
#align_import order.bounds.basic from "leanprover... | Mathlib/Order/Bounds/Basic.lean | 1,645 | 1,654 | theorem isLUB_pi {s : Set (∀ a, π a)} {f : ∀ a, π a} :
IsLUB s f ↔ ∀ a, IsLUB (Function.eval a '' s) (f a) := by |
classical
refine
⟨fun H a => ⟨(Function.monotone_eval a).mem_upperBounds_image H.1, fun b hb => ?_⟩, fun H =>
⟨?_, ?_⟩⟩
· suffices h : Function.update f a b ∈ upperBounds s from Function.update_same a b f ▸ H.2 h a
exact fun g hg => le_update_iff.2 ⟨hb <| mem_image_of_mem _ hg, fun i _ =>... |
/-
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.Filter.Prod
#align_import order.filter.n_ary from "leanprover-community/mathlib"@"78f647f8517f021d839a7553d5dc97e79b508dea"
/-!
# N-ary maps of fil... | Mathlib/Order/Filter/NAry.lean | 53 | 55 | theorem map_prod_eq_map₂ (m : α → β → γ) (f : Filter α) (g : Filter β) :
Filter.map (fun p : α × β => m p.1 p.2) (f ×ˢ g) = map₂ m f g := by |
rw [map₂, copy_eq, uncurry_def]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.Products.Basic
import Mathlib.CategoryTheory.Functor.Currying
import Mathlib.CategoryTheo... | Mathlib/CategoryTheory/Limits/Fubini.lean | 497 | 501 | theorem colimitIsoColimitCurryCompColim_ι_hom {j} {k} :
colimit.ι _ (j, k) ≫ (colimitIsoColimitCurryCompColim G).hom =
(colimit.ι (_) k ≫ colimit.ι (curry.obj G ⋙ colim) j : _ ⟶ colimit (_ ⋙ colim)) := by |
rw [← cancel_mono (colimitIsoColimitCurryCompColim G).inv]
simp
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Eric Wieser, Yaël Dillies
-/
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.normed_spa... | Mathlib/Analysis/NormedSpace/Real.lean | 81 | 98 | theorem interior_closedBall (x : E) {r : ℝ} (hr : r ≠ 0) :
interior (closedBall x r) = ball x r := by |
cases' hr.lt_or_lt with hr hr
· rw [closedBall_eq_empty.2 hr, ball_eq_empty.2 hr.le, interior_empty]
refine Subset.antisymm ?_ ball_subset_interior_closedBall
intro y hy
rcases (mem_closedBall.1 <| interior_subset hy).lt_or_eq with (hr | rfl)
· exact hr
set f : ℝ → E := fun c : ℝ => c • (y - x) + x
suf... |
/-
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.Init.Function
import Mathlib.Logic.Function.Basic
#align_import data.sigma.basic from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f13... | Mathlib/Data/Sigma/Basic.lean | 120 | 122 | theorem fst_injective [h : ∀ a, Subsingleton (β a)] : Injective (fst : (Σ a, β a) → α) := by |
rintro ⟨a₁, b₁⟩ ⟨a₂, b₂⟩ (rfl : a₁ = a₂)
exact congr_arg (mk a₁) <| Subsingleton.elim _ _
|
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.CartanSubalgebra
import Mathlib.Algebra.Lie.Weights.Basic
/-!
# Weights and roots of Lie modules and Lie algebras with respect to Cartan subalge... | Mathlib/Algebra/Lie/Weights/Cartan.lean | 154 | 157 | theorem rootSpaceProduct_tmul
(χ₁ χ₂ χ₃ : H → R) (hχ : χ₁ + χ₂ = χ₃) (x : rootSpace H χ₁) (y : rootSpace H χ₂) :
(rootSpaceProduct R L H χ₁ χ₂ χ₃ hχ (x ⊗ₜ y) : L) = ⁅(x : L), (y : L)⁆ := by |
simp only [rootSpaceProduct_def, coe_rootSpaceWeightSpaceProduct_tmul]
|
/-
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.Basic
import Mathlib.RingTheory.Noetherian
#align_import algebra.lie.subalgebra from "leanprover-community/mathlib"@"6d584f1709bedbed9175bd9350d... | Mathlib/Algebra/Lie/Subalgebra.lean | 352 | 358 | theorem Submodule.exists_lieSubalgebra_coe_eq_iff (p : Submodule R L) :
(∃ K : LieSubalgebra R L, ↑K = p) ↔ ∀ x y : L, x ∈ p → y ∈ p → ⁅x, y⁆ ∈ p := by |
constructor
· rintro ⟨K, rfl⟩ _ _
exact K.lie_mem'
· intro h
use { p with lie_mem' := h _ _ }
|
/-
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.Associated
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Module.Defs
import Mathlib.Alge... | Mathlib/RingTheory/Nilpotent/Basic.lean | 174 | 178 | theorem isNilpotent_sub (hx : IsNilpotent x) (hy : IsNilpotent y) : IsNilpotent (x - y) := by |
rw [← neg_right_iff] at h_comm
rw [← isNilpotent_neg_iff] at hy
rw [sub_eq_add_neg]
exact h_comm.isNilpotent_add hx hy
|
/-
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.Data.ENat.Lattice
import Mathlib.Order.OrderIsoNat
import Mathlib.Tactic.TFAE
#align_import order.height from "leanprover-community/mathlib"@"bf27744463e962... | Mathlib/Order/Height.lean | 309 | 312 | theorem chainHeight_insert_of_forall_lt (a : α) (ha : ∀ b ∈ s, b < a) :
(insert a s).chainHeight = s.chainHeight + 1 := by |
rw [← chainHeight_dual, ← chainHeight_dual s]
exact chainHeight_insert_of_forall_gt _ ha
|
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.LinearAlgebra.Basis
impo... | Mathlib/RingTheory/Adjoin/Basic.lean | 343 | 351 | theorem adjoin_adjoin_of_tower (s : Set A) : adjoin S (adjoin R s : Set A) = adjoin S s := by |
apply le_antisymm (adjoin_le _)
· exact adjoin_mono subset_adjoin
· change adjoin R s ≤ (adjoin S s).restrictScalars R
refine adjoin_le ?_
-- Porting note: unclear why this was broken
have : (Subalgebra.restrictScalars R (adjoin S s) : Set A) = adjoin S s := rfl
rw [this]
exact subset_adjoin
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Ring.Regular
import Mathlib.Tactic.Common
#align_import algebra.gcd_monoid.basic from "leanp... | Mathlib/Algebra/GCDMonoid/Basic.lean | 468 | 479 | theorem gcd_mul_left' [GCDMonoid α] (a b c : α) :
Associated (gcd (a * b) (a * c)) (a * gcd b c) := by |
obtain rfl | ha := eq_or_ne a 0
· simp only [zero_mul, gcd_zero_left']
obtain ⟨d, eq⟩ := dvd_gcd (dvd_mul_right a b) (dvd_mul_right a c)
apply associated_of_dvd_dvd
· rw [eq]
apply mul_dvd_mul_left
exact
dvd_gcd ((mul_dvd_mul_iff_left ha).1 <| eq ▸ gcd_dvd_left _ _)
((mul_dvd_mul_iff_le... |
/-
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 | 372 | 381 | theorem pow_inj (h : IsPrimitiveRoot ζ k) ⦃i j : ℕ⦄ (hi : i < k) (hj : j < k) (H : ζ ^ i = ζ ^ j) :
i = j := by |
wlog hij : i ≤ j generalizing i j
· exact (this hj hi H.symm (le_of_not_le hij)).symm
apply le_antisymm hij
rw [← tsub_eq_zero_iff_le]
apply Nat.eq_zero_of_dvd_of_lt _ (lt_of_le_of_lt tsub_le_self hj)
apply h.dvd_of_pow_eq_one
rw [← ((h.isUnit (lt_of_le_of_lt (Nat.zero_le _) hi)).pow i).mul_left_inj, ← p... |
/-
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 | 161 | 162 | theorem mgf_zero [IsProbabilityMeasure μ] : mgf X μ 0 = 1 := by |
simp only [mgf_zero', measure_univ, ENNReal.one_toReal]
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a... | Mathlib/MeasureTheory/Function/L1Space.lean | 511 | 517 | theorem Memℒp.integrable_norm_rpow' [IsFiniteMeasure μ] {f : α → β} {p : ℝ≥0∞} (hf : Memℒp f p μ) :
Integrable (fun x : α => ‖f x‖ ^ p.toReal) μ := by |
by_cases h_zero : p = 0
· simp [h_zero, integrable_const]
by_cases h_top : p = ∞
· simp [h_top, integrable_const]
exact hf.integrable_norm_rpow h_zero h_top
|
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.MeasureTheory.Constructions.HaarToSphere
import Mathlib.MeasureTheory.Integral.Gamma
import Mathlib.MeasureTheory.Integral.Pi
import Mathlib.Analysis.Spe... | Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean | 361 | 370 | theorem InnerProductSpace.volume_ball (x : E) (r : ℝ) :
volume (Metric.ball x r) = (.ofReal r) ^ finrank ℝ E *
.ofReal (sqrt π ^ finrank ℝ E / Gamma (finrank ℝ E / 2 + 1)) := by |
rw [← ((stdOrthonormalBasis ℝ E).measurePreserving_repr_symm).measure_preimage
measurableSet_ball]
have : Nonempty (Fin (finrank ℝ E)) := Fin.pos_iff_nonempty.mp finrank_pos
have := EuclideanSpace.volume_ball (Fin (finrank ℝ E)) ((stdOrthonormalBasis ℝ E).repr x) r
simp_rw [Fintype.card_fin] at this
co... |
/-
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.Data.Countable.Basic
import Mathlib.Logic.Encodable.Basic
import Mathlib.Order.SuccPred.Basic
import Mathlib.Order.Interval.Finset.Defs
#align_import orde... | Mathlib/Order/SuccPred/LinearLocallyFinite.lean | 239 | 247 | theorem toZ_iterate_pred_ge (n : ℕ) : -(n : ℤ) ≤ toZ i0 (pred^[n] i0) := by |
rcases le_or_lt i0 (pred^[n] i0) with h | h
· have h_eq : pred^[n] i0 = i0 := le_antisymm (pred_iterate_le _ _) h
rw [h_eq, toZ_of_eq]
omega
· rw [toZ_of_lt h]
refine Int.neg_le_neg ?_
norm_cast
exact Nat.find_min' _ rfl
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth, Mitchell Lee
-/
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.pol... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 156 | 157 | theorem U_sub_two (n : ℤ) : U R (n - 2) = 2 * X * U R (n - 1) - U R n := by |
linear_combination (norm := ring_nf) U_add_two R (n - 2)
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Separation
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.uniform_space.... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 635 | 639 | theorem tendstoLocallyUniformlyOn_iff_tendstoLocallyUniformly_comp_coe :
TendstoLocallyUniformlyOn F f p s ↔
TendstoLocallyUniformly (fun i (x : s) => F i x) (f ∘ (↑)) p := by |
simp only [tendstoLocallyUniformly_iff_forall_tendsto, Subtype.forall', tendsto_map'_iff,
tendstoLocallyUniformlyOn_iff_forall_tendsto, ← map_nhds_subtype_val, prod_map_right]; rfl
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathl... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 186 | 189 | theorem of_union {A B : Set α} (h : Disjoint A B) (hA : MeasurableSet A) (hB : MeasurableSet B) :
v (A ∪ B) = v A + v B := by |
rw [Set.union_eq_iUnion, of_disjoint_iUnion, tsum_fintype, Fintype.sum_bool, cond, cond]
exacts [fun b => Bool.casesOn b hB hA, pairwise_disjoint_on_bool.2 h]
|
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap
import Mathlib.RingTheory.Adjoin.... | Mathlib/RingTheory/IntegralClosure.lean | 579 | 582 | theorem IsIntegral.det {n : Type*} [Fintype n] [DecidableEq n] {M : Matrix n n A}
(h : ∀ i j, IsIntegral R (M i j)) : IsIntegral R M.det := by |
rw [Matrix.det_apply]
exact IsIntegral.sum _ fun σ _hσ ↦ (IsIntegral.prod _ fun i _hi => h _ _).zsmul _
|
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.UnionFind.Basic
namespace Batteries.UnionFind
@[simp] theorem arr_empty : empty.arr = #[] := rfl
@[simp] theorem parent_empty : empty.parent a... | .lake/packages/batteries/Batteries/Data/UnionFind/Lemmas.lean | 53 | 62 | theorem parent_link {self} {x y : Fin self.size} (yroot) {i} :
(link self x y yroot).parent i =
if x.1 = y then
self.parent i
else
if self.rank y < self.rank x then
if y = i then x else self.parent i
else
if x = i then y else self.parent i := by |
simp [rankD_eq]; exact parentD_linkAux
|
/-
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, Jeremy Avigad
-/
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Defs.Filter
#align_import topology.basic from "leanprover-community/mathlib"@... | Mathlib/Topology/Basic.lean | 728 | 729 | theorem IsOpen.frontier_eq (hs : IsOpen s) : frontier s = closure s \ s := by |
rw [frontier, hs.interior_eq]
|
/-
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 | 552 | 555 | theorem codisjoint_iff_coe_toSubmodule :
Codisjoint N N' ↔ Codisjoint (N : Submodule R M) (N' : Submodule R M) := by |
rw [codisjoint_iff, codisjoint_iff, ← coe_toSubmodule_eq_iff, sup_coe_toSubmodule,
top_coeSubmodule, ← codisjoint_iff]
|
/-
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.Fintype.Perm
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.List
#a... | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 152 | 153 | theorem sameCycle_zpow_right {n : ℤ} : SameCycle f x ((f ^ n) y) ↔ SameCycle f x y := by |
rw [sameCycle_comm, sameCycle_zpow_left, sameCycle_comm]
|
/-
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, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Fiel... | Mathlib/Algebra/Order/Field/Basic.lean | 324 | 325 | theorem div_lt_div_iff (b0 : 0 < b) (d0 : 0 < d) : a / b < c / d ↔ a * d < c * b := by |
rw [lt_div_iff d0, div_mul_eq_mul_div, div_lt_iff b0]
|
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