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 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Init.Core
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.affine_space.finite_d... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 296 | 307 | theorem AffineIndependent.affineSpan_image_finset_eq_of_le_of_card_eq_finrank_add_one
[DecidableEq P] {p : ι → P}
(hi : AffineIndependent k p) {s : Finset ι} {sp : AffineSubspace k P}
[FiniteDimensional k sp.direction] (hle : affineSpan k (s.image p : Set P) ≤ sp)
(hc : Finset.card s = finrank k sp.dire... |
have hn : s.Nonempty := by
rw [← Finset.card_pos, hc]
apply Nat.succ_pos
refine eq_of_direction_eq_of_nonempty_of_le ?_ ((hn.image p).to_set.affineSpan k) hle
have hd := direction_le hle
rw [direction_affineSpan] at hd ⊢
exact hi.vectorSpan_image_finset_eq_of_le_of_card_eq_finrank_add_one hd hc
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Scott Morrison, Ainsley Pahljina
-/
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Basic
import Mathli... | Mathlib/NumberTheory/LucasLehmer.lean | 457 | 461 | theorem two_lt_q (p' : ℕ) : 2 < q (p' + 2) := by |
refine (minFac_prime (one_lt_mersenne.2 ?_).ne').two_le.lt_of_ne' ?_
· exact le_add_left _ _
· rw [Ne, minFac_eq_two_iff, mersenne, Nat.pow_succ']
exact Nat.two_not_dvd_two_mul_sub_one Nat.one_le_two_pow
|
/-
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 | 1,038 | 1,039 | theorem realize_sentence (g : M ≃[L] N) (φ : L.Sentence) : M ⊨ φ ↔ N ⊨ φ := by |
rw [Sentence.Realize, Sentence.Realize, ← g.realize_formula, Unique.eq_default (g ∘ default)]
|
/-
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 | 338 | 341 | theorem tendsto_prod_top_iff {c : β} :
Tendsto (↿F) (p ×ˢ ⊤) (𝓝 c) ↔ TendstoUniformly F (fun _ => c) p := by |
rw [tendstoUniformly_iff_tendstoUniformlyOnFilter]
exact tendsto_prod_filter_iff
|
/-
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.Order.SuccPred.Basic
import Mathlib.Topology.Order.Basic
import Mathlib.Topology.Metrizable.Uniformity
#align_import topology.instances.discrete from "lea... | Mathlib/Topology/Instances/Discrete.lean | 111 | 117 | theorem discreteTopology_iff_orderTopology_of_pred_succ [LinearOrder α] [PredOrder α]
[SuccOrder α] : DiscreteTopology α ↔ OrderTopology α := by |
refine ⟨fun h => ⟨?_⟩, fun h => ⟨?_⟩⟩
· rw [h.eq_bot]
exact LinearOrder.bot_topologicalSpace_eq_generateFrom
· rw [h.topology_eq_generate_intervals]
exact LinearOrder.bot_topologicalSpace_eq_generateFrom.symm
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.Tactic.SuppressCo... | Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 844 | 855 | theorem map_range_eq_span_tmul (f : M →ₗ[R] P) (g : N →ₗ[R] Q) :
range (map f g) = Submodule.span R { t | ∃ m n, f m ⊗ₜ g n = t } := by |
simp only [← Submodule.map_top, ← span_tmul_eq_top, Submodule.map_span, Set.mem_image,
Set.mem_setOf_eq]
congr; ext t
constructor
· rintro ⟨_, ⟨⟨m, n, rfl⟩, rfl⟩⟩
use m, n
simp only [map_tmul]
· rintro ⟨m, n, rfl⟩
refine ⟨_, ⟨⟨m, n, rfl⟩, ?_⟩⟩
simp only [map_tmul]
|
/-
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 | 1,173 | 1,176 | theorem ofReal_integral_norm_eq_lintegral_nnnorm {P : Type*} [NormedAddCommGroup P] {f : α → P}
(hf : Integrable f μ) : ENNReal.ofReal (∫ x, ‖f x‖ ∂μ) = ∫⁻ x, ‖f x‖₊ ∂μ := by |
rw [integral_norm_eq_lintegral_nnnorm hf.aestronglyMeasurable,
ENNReal.ofReal_toReal (lt_top_iff_ne_top.mp hf.2)]
|
/-
Copyright (c) 2021 Roberto Alvarez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Roberto Alvarez
-/
import Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Logic.Equiv.TransferInstance
import Ma... | Mathlib/Topology/Homotopy/HomotopyGroup.lean | 74 | 78 | theorem insertAt_boundary (i : N) {t₀ : I} {t}
(H : (t₀ = 0 ∨ t₀ = 1) ∨ t ∈ boundary { j // j ≠ i }) : insertAt i ⟨t₀, t⟩ ∈ boundary N := by |
obtain H | ⟨j, H⟩ := H
· use i; rwa [funSplitAt_symm_apply, dif_pos rfl]
· use j; rwa [funSplitAt_symm_apply, dif_neg j.prop, Subtype.coe_eta]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Regular.Pow
import Mathl... | Mathlib/Algebra/MvPolynomial/Basic.lean | 786 | 792 | theorem support_symmDiff_support_subset_support_add [DecidableEq σ] (p q : MvPolynomial σ R) :
p.support ∆ q.support ⊆ (p + q).support := by |
rw [symmDiff_def, Finset.sup_eq_union]
apply Finset.union_subset
· exact support_sdiff_support_subset_support_add p q
· rw [add_comm]
exact support_sdiff_support_subset_support_add q p
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn, Heather Macbeth
-/
import Mathlib.Topology.FiberBundle.Trivialization
import Mathlib.Topology.Order.LeftRightNhds
#align_import topology.fiber_... | Mathlib/Topology/FiberBundle/Basic.lean | 786 | 789 | theorem continuous_symm_of_mem_pretrivializationAtlas (he : e ∈ a.pretrivializationAtlas) :
@ContinuousOn _ _ _ a.totalSpaceTopology e.toPartialEquiv.symm e.target := by |
refine fun z H U h => preimage_nhdsWithin_coinduced' H (le_def.1 (nhds_mono ?_) U h)
exact le_iSup₂ (α := TopologicalSpace (TotalSpace F E)) e he
|
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.Infix
#align_import data.list.rdrop from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
/-!
# Dropping or taki... | Mathlib/Data/List/DropRight.lean | 198 | 201 | theorem rtakeWhile_concat (x : α) :
rtakeWhile p (l ++ [x]) = if p x then rtakeWhile p l ++ [x] else [] := by |
simp only [rtakeWhile, takeWhile, reverse_append, reverse_singleton, singleton_append]
split_ifs with h <;> simp [h]
|
/-
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.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.... | Mathlib/LinearAlgebra/Determinant.lean | 601 | 606 | theorem Basis.det_comp_basis [Module A M'] (b : Basis ι A M) (b' : Basis ι A M') (f : M →ₗ[A] M') :
b'.det (f ∘ b) = LinearMap.det (f ∘ₗ (b'.equiv b (Equiv.refl ι) : M' →ₗ[A] M)) := by |
rw [Basis.det_apply, ← LinearMap.det_toMatrix b', LinearMap.toMatrix_comp _ b, Matrix.det_mul,
LinearMap.toMatrix_basis_equiv, Matrix.det_one, mul_one]
congr 1; ext i j
rw [Basis.toMatrix_apply, LinearMap.toMatrix_apply, Function.comp_apply]
|
/-
Copyright (c) 2021 Ashwin Iyengar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Ashwin Iyengar, Patrick Massot
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebr... | Mathlib/Topology/Algebra/Nonarchimedean/Basic.lean | 140 | 148 | theorem mul_subset (U : OpenAddSubgroup R) : ∃ V : OpenAddSubgroup R, (V : Set R) * V ⊆ U := by |
let ⟨V, H⟩ := prod_self_subset <| (U.isOpen.preimage continuous_mul).mem_nhds <| by
simpa only [Set.mem_preimage, Prod.snd_zero, mul_zero] using U.zero_mem
use V
rintro v ⟨a, ha, b, hb, hv⟩
have hy := H (Set.mk_mem_prod ha hb)
simp only [Set.mem_preimage, SetLike.mem_coe, hv] at hy
rw [SetLike.mem_coe]... |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc429200506... | Mathlib/Data/Set/Image.lean | 1,460 | 1,462 | theorem preimage_coe_nonempty {s t : Set α} :
(((↑) : s → α) ⁻¹' t).Nonempty ↔ (s ∩ t).Nonempty := by |
rw [← image_preimage_coe, image_nonempty]
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Sphere.Basic
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tactic.DeriveFintype
#align_import geometry.eucl... | Mathlib/Geometry/Euclidean/Circumcenter.lean | 771 | 776 | theorem circumradius_eq_of_cospherical_subset {s : AffineSubspace ℝ P} {ps : Set P} (h : ps ⊆ s)
[Nonempty s] {n : ℕ} [FiniteDimensional ℝ s.direction] (hd : finrank ℝ s.direction = n)
(hc : Cospherical ps) {sx₁ sx₂ : Simplex ℝ P n} (hsx₁ : Set.range sx₁.points ⊆ ps)
(hsx₂ : Set.range sx₂.points ⊆ ps) : sx₁... |
rcases exists_circumradius_eq_of_cospherical_subset h hd hc with ⟨r, hr⟩
rw [hr sx₁ hsx₁, hr sx₂ hsx₂]
|
/-
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.Algebra.RestrictScalars
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.LinearAlgebra.Quotient
import Mathlib.LinearAlgebra.StdB... | Mathlib/RingTheory/Finiteness.lean | 82 | 134 | theorem exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul {R : Type*} [CommRing R] {M : Type*}
[AddCommGroup M] [Module R M] (I : Ideal R) (N : Submodule R M) (hn : N.FG) (hin : N ≤ I • N) :
∃ r : R, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = (0 : M) := by |
rw [fg_def] at hn
rcases hn with ⟨s, hfs, hs⟩
have : ∃ r : R, r - 1 ∈ I ∧ N ≤ (I • span R s).comap (LinearMap.lsmul R M r) ∧ s ⊆ N := by
refine ⟨1, ?_, ?_, ?_⟩
· rw [sub_self]
exact I.zero_mem
· rw [hs]
intro n hn
rw [mem_comap]
change (1 : R) • n ∈ I • N
rw [one_smul]
... |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | Mathlib/MeasureTheory/Measure/Portmanteau.lean | 105 | 123 | theorem le_measure_compl_liminf_of_limsup_measure_le {ι : Type*} {L : Filter ι} {μ : Measure Ω}
{μs : ι → Measure Ω} [IsProbabilityMeasure μ] [∀ i, IsProbabilityMeasure (μs i)] {E : Set Ω}
(E_mble : MeasurableSet E) (h : (L.limsup fun i => μs i E) ≤ μ E) :
μ Eᶜ ≤ L.liminf fun i => μs i Eᶜ := by |
rcases L.eq_or_neBot with rfl | hne
· simp only [liminf_bot, le_top]
have meas_Ec : μ Eᶜ = 1 - μ E := by
simpa only [measure_univ] using measure_compl E_mble (measure_lt_top μ E).ne
have meas_i_Ec : ∀ i, μs i Eᶜ = 1 - μs i E := by
intro i
simpa only [measure_univ] using measure_compl E_mble (measur... |
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.Rat.Basic
import Batteries.Tactic.SeqFocus
/-! # Additional lemmas about the Rational Numbers -/
namespace Rat
theorem ext : {p q : Rat} → p.... | .lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean | 76 | 82 | theorem maybeNormalize_eq_normalize {num : Int} {den g : Nat} (den_nz reduced)
(hn : ↑g ∣ num) (hd : g ∣ den) :
maybeNormalize num den g den_nz reduced = normalize num den (mt (by simp [·]) den_nz) := by |
simp only [maybeNormalize_eq, mk_eq_normalize, Int.div_eq_ediv_of_dvd hn]
have : g ≠ 0 := mt (by simp [·]) den_nz
rw [← normalize_mul_right _ this, Int.ediv_mul_cancel hn]
congr 1; exact Nat.div_mul_cancel hd
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.shapes.kernels from "leanprover-community/mathlib"@"95... | Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean | 868 | 870 | theorem cokernelZeroIsoTarget_hom :
cokernelZeroIsoTarget.hom = cokernel.desc (0 : X ⟶ Y) (𝟙 Y) (by simp) := by |
ext; simp [cokernelZeroIsoTarget]
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.BoxIntegral.Partition.Basic
#align_import analysis.box_integral.partition.split from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf973... | Mathlib/Analysis/BoxIntegral/Partition/Split.lean | 65 | 70 | theorem coe_splitLower : (splitLower I i x : Set (ι → ℝ)) = ↑I ∩ { y | y i ≤ x } := by |
rw [splitLower, coe_mk']
ext y
simp only [mem_univ_pi, mem_Ioc, mem_inter_iff, mem_coe, mem_setOf_eq, forall_and, ← Pi.le_def,
le_update_iff, le_min_iff, and_assoc, and_forall_ne (p := fun j => y j ≤ upper I j) i, mem_def]
rw [and_comm (a := y i ≤ x)]
|
/-
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 | 179 | 181 | theorem sigma_insert {a : ∀ i, α i} :
s.sigma (fun i ↦ insert (a i) (t i)) = (fun i ↦ ⟨i, a i⟩) '' s ∪ s.sigma t := by |
simp_rw [insert_eq, sigma_union, sigma_singleton]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Set.Finite
import Mathlib.Data.Countable.Basic
import Mathlib.Logic.Equiv.List
import Mathlib.Data.Set.Subsingleton
#align_import data.set.counta... | Mathlib/Data/Set/Countable.lean | 309 | 311 | theorem countable_pi {π : α → Type*} [Finite α] {s : ∀ a, Set (π a)} (hs : ∀ a, (s a).Countable) :
{ f : ∀ a, π a | ∀ a, f a ∈ s a }.Countable := by |
simpa only [← mem_univ_pi] using countable_univ_pi hs
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Ideal
import Mathlib.... | Mathlib/RingTheory/Localization/Submodule.lean | 48 | 49 | theorem coeSubmodule_bot : coeSubmodule S (⊥ : Ideal R) = ⊥ := by |
rw [coeSubmodule, Submodule.map_bot]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.LinearAlgebra.AffineSpace.Independent
import Mathlib.Tactic.FieldSimp
#align_import analysi... | Mathlib/Analysis/Convex/Caratheodory.lean | 151 | 161 | theorem convexHull_eq_union : convexHull 𝕜 s =
⋃ (t : Finset E) (hss : ↑t ⊆ s) (hai : AffineIndependent 𝕜 ((↑) : t → E)), convexHull 𝕜 ↑t := by |
apply Set.Subset.antisymm
· intro x hx
simp only [exists_prop, Set.mem_iUnion]
exact ⟨Caratheodory.minCardFinsetOfMemConvexHull hx,
Caratheodory.minCardFinsetOfMemConvexHull_subseteq hx,
Caratheodory.affineIndependent_minCardFinsetOfMemConvexHull hx,
Caratheodory.mem_minCardFinsetOfMemCon... |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.RingTheory.HahnSeries.Basic
#align_import ring_theory.hahn_series from "leanprover-community/mathlib"@"a... | Mathlib/RingTheory/HahnSeries/Addition.lean | 166 | 168 | theorem sub_coeff' {x y : HahnSeries Γ R} : (x - y).coeff = x.coeff - y.coeff := by |
ext
simp [sub_eq_add_neg]
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Prod
import Mathlib.Data.Seq.WSeq
#align_import data.seq.parallel from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7... | Mathlib/Data/Seq/Parallel.lean | 122 | 186 | theorem terminates_parallel {S : WSeq (Computation α)} {c} (h : c ∈ S) [T : Terminates c] :
Terminates (parallel S) := by |
suffices
∀ (n) (l : List (Computation α)) (S c),
c ∈ l ∨ some (some c) = Seq.get? S n → Terminates c → Terminates (corec parallel.aux1 (l, S))
from
let ⟨n, h⟩ := h
this n [] S c (Or.inr h) T
intro n; induction' n with n IH <;> intro l S c o T
· cases' o with a a
· exact terminates_paral... |
/-
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.Ring.Regular
import Mathlib.Data.Int.GCD
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Tactic.NormNum.Basic
#align_import data.nat.modeq... | Mathlib/Data/Nat/ModEq.lean | 380 | 385 | theorem chineseRemainder'_lt_lcm (h : a ≡ b [MOD gcd n m]) (hn : n ≠ 0) (hm : m ≠ 0) :
↑(chineseRemainder' h) < lcm n m := by |
dsimp only [chineseRemainder']
rw [dif_neg hn, dif_neg hm, Subtype.coe_mk, xgcd_val, ← Int.toNat_natCast (lcm n m)]
have lcm_pos := Int.natCast_pos.mpr (Nat.pos_of_ne_zero (lcm_ne_zero hn hm))
exact (Int.toNat_lt_toNat lcm_pos).mpr (Int.emod_lt_of_pos _ lcm_pos)
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kevin Buzzard
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Fintype.BigOperators
import Mathlib.RingTheory.P... | Mathlib/NumberTheory/Bernoulli.lean | 104 | 106 | theorem bernoulli'_zero : bernoulli' 0 = 1 := by |
rw [bernoulli'_def]
norm_num
|
/-
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
-/
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Tactic.AdaptationNote
#align_import geometry.eu... | Mathlib/Geometry/Euclidean/Inversion/Basic.lean | 69 | 69 | theorem inversion_self (c : P) (R : ℝ) : inversion c R c = c := by | simp [inversion]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.To... | Mathlib/Topology/UniformSpace/Basic.lean | 1,397 | 1,401 | theorem uniformContinuous_iInf_dom {f : α → β} {u₁ : ι → UniformSpace α} {u₂ : UniformSpace β}
{i : ι} (hf : UniformContinuous[u₁ i, u₂] f) : UniformContinuous[iInf u₁, u₂] f := by |
delta UniformContinuous
rw [iInf_uniformity]
exact tendsto_iInf' i hf
|
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set... | Mathlib/Analysis/NormedSpace/lpSpace.lean | 968 | 970 | theorem _root_.algebraMap_memℓp_infty (k : 𝕜) : Memℓp (algebraMap 𝕜 (∀ i, B i) k) ∞ := by |
rw [Algebra.algebraMap_eq_smul_one]
exact (one_memℓp_infty.const_smul k : Memℓp (k • (1 : ∀ i, B i)) ∞)
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Nat
import Mathlib.Algebra.Order.Sub.Canonical
import Mathlib.Data.List.Perm
import Mathlib.Data.Set.List
import Mathlib.Init.Quot... | Mathlib/Data/Multiset/Basic.lean | 2,530 | 2,532 | theorem count_replicate (a b : α) (n : ℕ) : count a (replicate n b) = if a = b then n else 0 := by |
convert List.count_replicate a b n
rw [← coe_count, coe_replicate]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5... | Mathlib/Order/Filter/Basic.lean | 678 | 679 | theorem principal_mono {s t : Set α} : 𝓟 s ≤ 𝓟 t ↔ s ⊆ t := by |
simp only [le_principal_iff, iff_self_iff, mem_principal]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov, Yakov Pechersky
-/
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Da... | Mathlib/Algebra/Group/Subsemigroup/Basic.lean | 395 | 399 | theorem dense_induction {p : M → Prop} (x : M) {s : Set M} (hs : closure s = ⊤)
(mem : ∀ x ∈ s, p x)
(mul : ∀ x y, p x → p y → p (x * y)) : p x := by |
have : ∀ x ∈ closure s, p x := fun x hx => closure_induction hx mem mul
simpa [hs] using this x
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.AlgebraicGeometry.Scheme
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
#align_import algebraic_geomet... | Mathlib/AlgebraicGeometry/OpenImmersion.lean | 367 | 370 | theorem scheme_toScheme {X Y : Scheme.{u}} (f : X ⟶ Y) [AlgebraicGeometry.IsOpenImmersion f] :
toScheme Y f.1 = X := by |
apply scheme_eq_of_locallyRingedSpace_eq
exact locallyRingedSpace_toLocallyRingedSpace f
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
import Mathlib.Tactic.AdaptationNote
... | Mathlib/Probability/Martingale/Upcrossing.lean | 201 | 203 | theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by |
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.Data.Set.Pairwise.Lattice
#align_... | Mathlib/MeasureTheory/Covering/Vitali.lean | 205 | 388 | theorem exists_disjoint_covering_ae [MetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α]
[SecondCountableTopology α] (μ : Measure α) [IsLocallyFiniteMeasure μ] (s : Set α) (t : Set ι)
(C : ℝ≥0) (r : ι → ℝ) (c : ι → α) (B : ι → Set α) (hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a))
(μB : ∀ a ∈ t, μ (clo... |
/- The idea of the proof is the following. Assume for simplicity that `μ` is finite. Applying the
abstract Vitali covering theorem with `δ = r` given by `hf`, one obtains a disjoint subfamily `u`,
such that any element of `t` intersects an element of `u` with comparable radius. Fix `ε > 0`.
Since the elements ... |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Ordinal.FixedPoint
#align_import set_theory.cardinal... | Mathlib/SetTheory/Cardinal/Cofinality.lean | 451 | 455 | theorem cof_bsup_le_lift {o : Ordinal} {f : ∀ a < o, Ordinal} (H : ∀ i h, f i h < bsup.{u, v} o f) :
cof (bsup.{u, v} o f) ≤ Cardinal.lift.{v, u} o.card := by |
rw [← bsup_eq_blsub_iff_lt_bsup.{u, v}] at H
rw [H]
exact cof_blsub_le_lift.{u, v} f
|
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Tactic.Abel
#align_import set_theory.ordinal.natural_ops from "leanprover-communit... | Mathlib/SetTheory/Ordinal/NaturalOps.lean | 418 | 420 | theorem toNatOrdinal_cast_nat (n : ℕ) : toNatOrdinal n = n := by |
rw [← toOrdinal_cast_nat n]
rfl
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Wieser
-/
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolyn... | Mathlib/RingTheory/MvPolynomial/Homogeneous.lean | 271 | 278 | theorem totalDegree (hφ : IsHomogeneous φ n) (h : φ ≠ 0) : totalDegree φ = n := by |
apply le_antisymm hφ.totalDegree_le
obtain ⟨d, hd⟩ : ∃ d, coeff d φ ≠ 0 := exists_coeff_ne_zero h
simp only [← hφ hd, MvPolynomial.totalDegree, Finsupp.sum]
replace hd := Finsupp.mem_support_iff.mpr hd
simp only [weightedDegree_apply,Pi.one_apply, smul_eq_mul, mul_one, ge_iff_le]
-- Porting note: Original ... |
/-
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.MeasureTheory.MeasurableSpace.Basic
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
#align_import measure_theory.function.ae_measurable_sequence from... | Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean | 81 | 86 | theorem fun_prop_of_mem_aeSeqSet (hf : ∀ i, AEMeasurable (f i) μ) {x : α} (hx : x ∈ aeSeqSet hf p) :
p x fun n => f n x := by |
have h_eq : (fun n => f n x) = fun n => aeSeq hf p n x :=
funext fun n => (aeSeq_eq_fun_of_mem_aeSeqSet hf hx n).symm
rw [h_eq]
exact prop_of_mem_aeSeqSet hf hx
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Felix Weilacher
-/
import Mathlib.Data.Real.Cardinality
import Mathlib.Topology.MetricSpace.Perfect
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
i... | Mathlib/MeasureTheory/Constructions/Polish.lean | 958 | 980 | theorem measurableSet_exists_tendsto [TopologicalSpace γ] [PolishSpace γ] [MeasurableSpace γ]
[hγ : OpensMeasurableSpace γ] [Countable ι] {l : Filter ι}
[l.IsCountablyGenerated] {f : ι → β → γ} (hf : ∀ i, Measurable (f i)) :
MeasurableSet { x | ∃ c, Tendsto (fun n => f n x) l (𝓝 c) } := by |
rcases l.eq_or_neBot with rfl | hl
· simp
letI := upgradePolishSpace γ
rcases l.exists_antitone_basis with ⟨u, hu⟩
simp_rw [← cauchy_map_iff_exists_tendsto]
change MeasurableSet { x | _ ∧ _ }
have : ∀ x, (map (f · x) l ×ˢ map (f · x) l).HasAntitoneBasis fun n =>
((f · x) '' u n) ×ˢ ((f · x) '' u n)... |
/-
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, Johannes Hölzl
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.RelIso.Basic
#align_import order.ord_continuous from "leanprover... | Mathlib/Order/OrdContinuous.lean | 98 | 99 | theorem le_iff (hf : LeftOrdContinuous f) (h : Injective f) {x y} : f x ≤ f y ↔ x ≤ y := by |
simp only [← sup_eq_right, ← hf.map_sup, h.eq_iff]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
impo... | Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | 48 | 54 | theorem Even.strictConvexOn_pow {n : ℕ} (hn : Even n) (h : n ≠ 0) :
StrictConvexOn ℝ Set.univ fun x : ℝ => x ^ n := by |
apply StrictMono.strictConvexOn_univ_of_deriv (continuous_pow n)
rw [deriv_pow']
replace h := Nat.pos_of_ne_zero h
exact StrictMono.const_mul (Odd.strictMono_pow <| Nat.Even.sub_odd h hn <| Nat.odd_iff.2 rfl)
(Nat.cast_pos.2 h)
|
/-
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.Algebra.Order.Ring.Nat
import Mathlib.Combinatorics.SetFamily.Compression.Down
import Mathlib.Order.UpperLower.Basic
import Mathlib.Data.Fintype.Powerset
... | Mathlib/Combinatorics/SetFamily/HarrisKleitman.lean | 114 | 118 | theorem IsLowerSet.card_inter_le_finset (h𝒜 : IsLowerSet (𝒜 : Set (Finset α)))
(hℬ : IsUpperSet (ℬ : Set (Finset α))) :
2 ^ Fintype.card α * (𝒜 ∩ ℬ).card ≤ 𝒜.card * ℬ.card := by |
rw [inter_comm, mul_comm 𝒜.card]
exact hℬ.card_inter_le_finset h𝒜
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
/-!
# Neig... | Mathlib/Topology/ContinuousOn.lean | 1,280 | 1,283 | theorem continuous_if_const (p : Prop) {f g : α → β} [Decidable p] (hf : p → Continuous f)
(hg : ¬p → Continuous g) : Continuous fun a => if p then f a else g a := by |
split_ifs with h
exacts [hf h, hg h]
|
/-
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 | 824 | 828 | theorem AEEqFun.snorm_compMeasurePreserving {ν : MeasureTheory.Measure β} (g : β →ₘ[ν] E)
(hf : MeasurePreserving f μ ν) :
snorm (g.compMeasurePreserving f hf) p μ = snorm g p ν := by |
rw [snorm_congr_ae (g.coeFn_compMeasurePreserving _)]
exact snorm_comp_measurePreserving g.aestronglyMeasurable hf
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Scott Morrison, Ainsley Pahljina
-/
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Basic
import Mathli... | Mathlib/NumberTheory/LucasLehmer.lean | 93 | 95 | theorem succ_mersenne (k : ℕ) : mersenne k + 1 = 2 ^ k := by |
rw [mersenne, tsub_add_cancel_of_le]
exact one_le_pow_of_one_le (by norm_num) k
|
/-
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.Topology.Constructions
import Mathlib.Topology.ContinuousOn
#align_import topology.bases from "leanprover-community/mathlib"@"bcfa7268... | Mathlib/Topology/Bases.lean | 889 | 896 | theorem countable_cover_nhds [SecondCountableTopology α] {f : α → Set α} (hf : ∀ x, f x ∈ 𝓝 x) :
∃ s : Set α, s.Countable ∧ ⋃ x ∈ s, f x = univ := by |
rcases isOpen_iUnion_countable (fun x => interior (f x)) fun x => isOpen_interior with
⟨s, hsc, hsU⟩
suffices ⋃ x ∈ s, interior (f x) = univ from
⟨s, hsc, flip eq_univ_of_subset this <| iUnion₂_mono fun _ _ => interior_subset⟩
simp only [hsU, eq_univ_iff_forall, mem_iUnion]
exact fun x => ⟨x, mem_inter... |
/-
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.Algebra.Field.Basic
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Tree.Basic
import Mathlib.Logic.Basic
import Mathlib.Tactic.NormNum.C... | Mathlib/Tactic/CancelDenoms/Core.lean | 45 | 47 | theorem div_subst {α} [Field α] {n1 n2 k e1 e2 t1 : α}
(h1 : n1 * e1 = t1) (h2 : n2 / e2 = 1) (h3 : n1 * n2 = k) : k * (e1 / e2) = t1 := by |
rw [← h3, mul_assoc, mul_div_left_comm, h2, ← mul_assoc, h1, mul_comm, one_mul]
|
/-
Copyright (c) 2022 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johanes Hölzl, Patrick Massot, Yury Kudryashov, Kevin Wilson, Heather Macbeth
-/
import Mathlib.Order.Filter.Basic
#align_import order.filter.prod from "leanprover-community/mathlib"@... | Mathlib/Order/Filter/Prod.lean | 412 | 413 | theorem prod_inf {g₁ g₂ : Filter β} : f ×ˢ (g₁ ⊓ g₂) = (f ×ˢ g₁) ⊓ (f ×ˢ g₂) := by |
rw [prod_inf_prod, inf_idem]
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Eric Wieser
-/
import Mathlib.Data.Real.Basic
#align_import data.real.sign from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
/-!
# Real sign fu... | Mathlib/Data/Real/Sign.lean | 39 | 39 | theorem sign_of_pos {r : ℝ} (hr : 0 < r) : sign r = 1 := by | rw [sign, if_pos hr, if_neg hr.not_lt]
|
/-
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 | 757 | 758 | theorem dotProduct_comm [AddCommMonoid α] [CommSemigroup α] (v w : m → α) : v ⬝ᵥ w = w ⬝ᵥ v := by |
simp_rw [dotProduct, mul_comm]
|
/-
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.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
imp... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 570 | 573 | theorem tensor_μ_natural_right (Z₁ Z₂ : C) {X₁ X₂ Y₁ Y₂ : C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) :
(Z₁ ⊗ Z₂) ◁ (f₁ ⊗ f₂) ≫ tensor_μ C (Z₁, Z₂) (Y₁, Y₂) =
tensor_μ C (Z₁, Z₂) (X₁, X₂) ≫ (Z₁ ◁ f₁ ⊗ Z₂ ◁ f₂) := by |
convert tensor_μ_natural C (𝟙 Z₁) (𝟙 Z₂) f₁ f₂ using 1 <;> simp
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.EqLocus
import Mathlib.Algebra.Module.Subm... | Mathlib/LinearAlgebra/Span.lean | 573 | 575 | theorem mem_span_insert {y} :
x ∈ span R (insert y s) ↔ ∃ a : R, ∃ z ∈ span R s, x = a • y + z := by |
simp [span_insert, mem_sup, mem_span_singleton, eq_comm (a := x)]
|
/-
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 | 422 | 423 | theorem diag_insert : (insert a s).diag = insert (a, a) s.diag := by |
rw [insert_eq, insert_eq, diag_union, diag_singleton]
|
/-
Copyright (c) 2021 Jakob Scholbach. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob Scholbach
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-commun... | Mathlib/Algebra/CharP/ExpChar.lean | 320 | 322 | theorem iterateFrobenius_add_apply :
iterateFrobenius R p (m + n) x = iterateFrobenius R p m (iterateFrobenius R p n x) := by |
simp_rw [iterateFrobenius_def, add_comm m n, pow_add, pow_mul]
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import algebra.gcd_monoid.multiset from "leanp... | Mathlib/Algebra/GCDMonoid/Multiset.lean | 185 | 190 | theorem gcd_map_mul (a : α) (s : Multiset α) : (s.map (a * ·)).gcd = normalize a * s.gcd := by |
refine s.induction_on ?_ fun b s ih ↦ ?_
· simp_rw [map_zero, gcd_zero, mul_zero]
· simp_rw [map_cons, gcd_cons, ← gcd_mul_left]
rw [ih]
apply ((normalize_associated a).mul_right _).gcd_eq_right
|
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.Analysis.... | Mathlib/Analysis/LocallyConvex/Bounded.lean | 419 | 428 | theorem isBounded_iff_subset_smul_ball {s : Set E} :
Bornology.IsBounded s ↔ ∃ a : 𝕜, s ⊆ a • Metric.ball (0 : E) 1 := by |
rw [← isVonNBounded_iff 𝕜]
constructor
· intro h
rcases (h (Metric.ball_mem_nhds 0 zero_lt_one)).exists_pos with ⟨ρ, _, hρball⟩
rcases NormedField.exists_lt_norm 𝕜 ρ with ⟨a, ha⟩
exact ⟨a, hρball a ha.le⟩
· rintro ⟨a, ha⟩
exact ((isVonNBounded_ball 𝕜 E 1).image (a • (1 : E →L[𝕜] E))).subset... |
/-
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 | 462 | 465 | theorem HasDerivAt.clm_comp (hc : HasDerivAt c c' x) (hd : HasDerivAt d d' x) :
HasDerivAt (fun y => (c y).comp (d y)) (c'.comp (d x) + (c x).comp d') x := by |
rw [← hasDerivWithinAt_univ] at *
exact hc.clm_comp hd
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
#align_import probability.martingale.basic from "leanprover-community/mathli... | Mathlib/Probability/Martingale/Basic.lean | 347 | 349 | theorem sub_submartingale [Preorder E] [CovariantClass E E (· + ·) (· ≤ ·)]
(hf : Supermartingale f ℱ μ) (hg : Submartingale g ℱ μ) : Supermartingale (f - g) ℱ μ := by |
rw [sub_eq_add_neg]; exact hf.add hg.neg
|
/-
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 | 99 | 100 | theorem T_eq (n : ℤ) : T R n = 2 * X * T R (n - 1) - T R (n - 2) := by |
linear_combination (norm := ring_nf) T_add_two R (n - 2)
|
/-
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 | 325 | 340 | theorem EuclideanSpace.volume_ball (x : EuclideanSpace ℝ ι) (r : ℝ) :
volume (Metric.ball x r) = (.ofReal r) ^ card ι *
.ofReal (Real.sqrt π ^ card ι / Gamma (card ι / 2 + 1)) := by |
obtain hr | hr := le_total r 0
· rw [Metric.ball_eq_empty.mpr hr, measure_empty, ← zero_eq_ofReal.mpr hr, zero_pow card_ne_zero,
zero_mul]
· suffices volume (Metric.ball (0 : EuclideanSpace ℝ ι) 1) =
.ofReal (Real.sqrt π ^ card ι / Gamma (card ι / 2 + 1)) by
rw [Measure.addHaar_ball _ _ hr, t... |
/-
Copyright (c) 2014 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Nat
#align_import data.nat.dist from "leanprover-community/mathlib"@"d50b12ae8e2bd910d08a94823976adae9825718b"
... | Mathlib/Data/Nat/Dist.lean | 103 | 104 | theorem dist_mul_left (k n m : ℕ) : dist (k * n) (k * m) = k * dist n m := by |
rw [mul_comm k n, mul_comm k m, dist_mul_right, mul_comm]
|
/-
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 | 538 | 541 | theorem Algebra.restrictScalars_adjoin (F : Type*) [CommSemiring F] {E : Type*} [CommSemiring E]
[Algebra F E] (K : Subalgebra F E) (S : Set E) :
(Algebra.adjoin K S).restrictScalars F = Algebra.adjoin F (K ∪ S) := by |
conv_lhs => rw [← Algebra.adjoin_eq K, ← Algebra.adjoin_union_eq_adjoin_adjoin]
|
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.ULift
import Mathlib.Data.ZMod.Defs
import Mathlib.SetTheory.Cardinal.PartENat
#align_import set_theory.cardinal.finite from "leanprover-communit... | Mathlib/SetTheory/Cardinal/Finite.lean | 272 | 275 | theorem _root_.Cardinal.natCast_eq_toPartENat_iff {n : ℕ} {c : Cardinal} :
↑n = toPartENat c ↔ ↑n = c := by |
rw [le_antisymm_iff, le_antisymm_iff, Cardinal.toPartENat_le_natCast_iff,
Cardinal.natCast_le_toPartENat_iff]
|
/-
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 | 86 | 96 | theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by |
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [s', s, Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_... |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Floris van Doorn, Mario Carneiro
-/
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
import Batteries.Tactic.Lint.Misc
instance {f :... | .lake/packages/batteries/Batteries/Logic.lean | 42 | 43 | theorem congrArg₂ (f : α → β → γ) {x x' : α} {y y' : β}
(hx : x = x') (hy : y = y') : f x y = f x' y' := by | subst hx hy; rfl
|
/-
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 | 583 | 584 | theorem mem_closedBall_comm : x ∈ closedBall y ε ↔ y ∈ closedBall x ε := by |
rw [mem_closedBall', mem_closedBall]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.... | Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | 1,848 | 1,856 | theorem biprod.conePointUniqueUpToIso_inv (X Y : C) [HasBinaryBiproduct X Y] {b : BinaryBicone X Y}
(hb : b.IsBilimit) :
(hb.isLimit.conePointUniqueUpToIso (BinaryBiproduct.isLimit _ _)).inv =
biprod.desc b.inl b.inr := by |
refine biprod.hom_ext' _ _ (hb.isLimit.hom_ext fun j => ?_) (hb.isLimit.hom_ext fun j => ?_)
all_goals
simp only [Category.assoc, IsLimit.conePointUniqueUpToIso_inv_comp]
rcases j with ⟨⟨⟩⟩
all_goals simp
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Quotient
import Mathlib.Combinatorics.Quiver.Path
#align_import category_theory.path_category fro... | Mathlib/CategoryTheory/PathCategory.lean | 87 | 90 | theorem lift_toPath {C} [Category C] (φ : V ⥤q C) {X Y : V} (f : X ⟶ Y) :
(lift φ).map f.toPath = φ.map f := by |
dsimp [Quiver.Hom.toPath, lift]
simp
|
/-
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 | 789 | 795 | theorem nhds_basis_opens (x : X) :
(𝓝 x).HasBasis (fun s : Set X => x ∈ s ∧ IsOpen s) fun s => s := by |
rw [nhds_def]
exact hasBasis_biInf_principal
(fun s ⟨has, hs⟩ t ⟨hat, ht⟩ =>
⟨s ∩ t, ⟨⟨has, hat⟩, IsOpen.inter hs ht⟩, ⟨inter_subset_left, inter_subset_right⟩⟩)
⟨univ, ⟨mem_univ x, isOpen_univ⟩⟩
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Order.BoundedOrder
import Mathlib.Order.MinMax
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Or... | Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean | 357 | 358 | theorem min_mul_distrib' (a b c : α) : min (a * b) c = min (min a c * min b c) c := by |
simpa [min_comm _ c] using min_mul_distrib c a b
|
/-
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, Kexing Ying
-/
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Function.AEMeasurableSequence
import Mathlib.MeasureTheory.Order.Lat... | Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | 551 | 554 | theorem measurable_of_Ioi {f : δ → α} (hf : ∀ x, MeasurableSet (f ⁻¹' Ioi x)) : Measurable f := by |
convert measurable_generateFrom (α := δ) _
· exact BorelSpace.measurable_eq.trans (borel_eq_generateFrom_Ioi _)
· rintro _ ⟨x, rfl⟩; exact hf x
|
/-
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 | 1,310 | 1,313 | theorem clusterPt_iff_forall_mem_closure {F : Filter X} :
ClusterPt x F ↔ ∀ s ∈ F, x ∈ closure s := by |
simp_rw [ClusterPt, inf_neBot_iff, mem_closure_iff_nhds]
rw [forall₂_swap]
|
/-
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.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Adjugate
#align_import linear_algebra.matrix.non... | Mathlib/LinearAlgebra/Matrix/Nondegenerate.lean | 50 | 63 | theorem nondegenerate_of_det_ne_zero [DecidableEq m] {M : Matrix m m A} (hM : M.det ≠ 0) :
Nondegenerate M := by |
intro v hv
ext i
specialize hv (M.cramer (Pi.single i 1))
refine (mul_eq_zero.mp ?_).resolve_right hM
convert hv
simp only [mulVec_cramer M (Pi.single i 1), dotProduct, Pi.smul_apply, smul_eq_mul]
rw [Finset.sum_eq_single i, Pi.single_eq_same, mul_one]
· intro j _ hj
simp [hj]
· intros
have :... |
/-
Copyright (c) 2021 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.BoundedLinearMaps
import Mathlib.MeasureTheory.Measure.WithDensity
import Mathlib.MeasureTheory.Function.SimpleFunc... | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 1,650 | 1,654 | theorem isSeparable_ae_range (hf : AEStronglyMeasurable f μ) :
∃ t : Set β, IsSeparable t ∧ ∀ᵐ x ∂μ, f x ∈ t := by |
refine ⟨range (hf.mk f), hf.stronglyMeasurable_mk.isSeparable_range, ?_⟩
filter_upwards [hf.ae_eq_mk] with x hx
simp [hx]
|
/-
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 | 1,894 | 1,897 | theorem prod_partition (R : Setoid α) [DecidableRel R.r] :
∏ x ∈ s, f x = ∏ xbar ∈ s.image Quotient.mk'', ∏ y ∈ s.filter (⟦·⟧ = xbar), f y := by |
refine (Finset.prod_image' f fun x _hx => ?_).symm
rfl
|
/-
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.Probability.IdentDistrib
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Analysis.SpecificLimits.FloorPow
import Mathli... | Mathlib/Probability/StrongLaw.lean | 505 | 517 | theorem strong_law_aux2 {c : ℝ} (c_one : 1 < c) :
∀ᵐ ω, (fun n : ℕ => ∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) i ω -
𝔼[∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) i]) =o[atTop] fun n : ℕ => (⌊c ^ n⌋₊ : ℝ) := by |
obtain ⟨v, -, v_pos, v_lim⟩ :
∃ v : ℕ → ℝ, StrictAnti v ∧ (∀ n : ℕ, 0 < v n) ∧ Tendsto v atTop (𝓝 0) :=
exists_seq_strictAnti_tendsto (0 : ℝ)
have := fun i => strong_law_aux1 X hint hindep hident hnonneg c_one (v_pos i)
filter_upwards [ae_all_iff.2 this] with ω hω
apply Asymptotics.isLittleO_iff.2 f... |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Algebra.Star.Order
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.Order.MonotoneContinuity
#align... | Mathlib/Data/Real/Sqrt.lean | 224 | 224 | theorem sqrt_one : √1 = 1 := by | simp [Real.sqrt]
|
/-
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.PropInstances
#align_import order.heyting.basic from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
/-!
# Heyting algebr... | Mathlib/Order/Heyting/Basic.lean | 474 | 474 | theorem le_sdiff_sup : a ≤ a \ b ⊔ b := by | rw [sup_comm, ← sdiff_le_iff]
|
/-
Copyright (c) 2021 Martin Zinkevich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Martin Zinkevich, Rémy Degenne
-/
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
#align_import measure_theory.pi_system fro... | Mathlib/MeasureTheory/PiSystem.lean | 79 | 82 | theorem IsPiSystem.singleton {α} (S : Set α) : IsPiSystem ({S} : Set (Set α)) := by |
intro s h_s t h_t _
rw [Set.mem_singleton_iff.1 h_s, Set.mem_singleton_iff.1 h_t, Set.inter_self,
Set.mem_singleton_iff]
|
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.adjoint f... | Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 389 | 393 | theorem adjoint_inner_right (A : E →ₗ[𝕜] F) (x : E) (y : F) : ⟪x, adjoint A y⟫ = ⟪A x, y⟫ := by |
haveI := FiniteDimensional.complete 𝕜 E
haveI := FiniteDimensional.complete 𝕜 F
rw [← coe_toContinuousLinearMap A, adjoint_eq_toCLM_adjoint]
exact ContinuousLinearMap.adjoint_inner_right _ x y
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.LinearAlgebra.Matrix.BilinearForm
import Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.... | Mathlib/RingTheory/Trace.lean | 280 | 288 | theorem trace_eq_trace_adjoin [FiniteDimensional K L] (x : L) :
Algebra.trace K L x = finrank K⟮x⟯ L • trace K K⟮x⟯ (AdjoinSimple.gen K x) := by |
-- Porting note: `conv` was
-- `conv in x => rw [← IntermediateField.AdjoinSimple.algebraMap_gen K x]`
-- and it was after the first `rw`.
conv =>
lhs
rw [← IntermediateField.AdjoinSimple.algebraMap_gen K x]
rw [← trace_trace (L := K⟮x⟯), trace_algebraMap, LinearMap.map_smul_of_tower]
|
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Hom.CompleteLattice
#align_import order.liminf_limsup from "leanprover-... | Mathlib/Order/LiminfLimsup.lean | 880 | 882 | theorem blimsup_eq_iInf_biSup_of_nat {p : ℕ → Prop} {u : ℕ → α} :
blimsup u atTop p = ⨅ i, ⨆ (j) (_ : p j ∧ i ≤ j), u j := by |
simp only [atTop_basis.blimsup_eq_iInf_iSup, @and_comm (p _), iSup_and, mem_Ici, iInf_true]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Prod
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Span
... | Mathlib/LinearAlgebra/Prod.lean | 691 | 701 | theorem le_prod_iff {p₁ : Submodule R M} {p₂ : Submodule R M₂} {q : Submodule R (M × M₂)} :
q ≤ p₁.prod p₂ ↔ map (LinearMap.fst R M M₂) q ≤ p₁ ∧ map (LinearMap.snd R M M₂) q ≤ p₂ := by |
constructor
· intro h
constructor
· rintro x ⟨⟨y1, y2⟩, ⟨hy1, rfl⟩⟩
exact (h hy1).1
· rintro x ⟨⟨y1, y2⟩, ⟨hy1, rfl⟩⟩
exact (h hy1).2
· rintro ⟨hH, hK⟩ ⟨x1, x2⟩ h
exact ⟨hH ⟨_, h, rfl⟩, hK ⟨_, h, rfl⟩⟩
|
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Scott Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheory.Products.Basic
#align_import cat... | Mathlib/CategoryTheory/Monoidal/Category.lean | 1,012 | 1,015 | theorem prodMonoidal_leftUnitor_hom_snd (X : C₁ × C₂) :
((λ_ X).hom : 𝟙_ _ ⊗ X ⟶ X).2 = (λ_ X.2).hom := by |
cases X
rfl
|
/-
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.MeasureTheory.MeasurableSpace.Defs
/-!
# σ-algebra of sets invariant under a self-map
In this file we define `MeasurableSpace.invariants (f : α → α)... | Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean | 58 | 60 | theorem measurable_invariants_dom {f : α → α} {g : α → β} :
Measurable[invariants f] g ↔ Measurable g ∧ ∀ s, MeasurableSet s → (g ∘ f) ⁻¹' s = g ⁻¹' s := by |
simp only [Measurable, ← forall_and]; rfl
|
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Eric Rodriguez
-/
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingTheory.Localization.NormTrace
#align_import number_theory.number_field.norm from "... | Mathlib/NumberTheory/NumberField/Norm.lean | 104 | 106 | theorem norm_norm [IsSeparable K L] [Algebra F L] [IsSeparable F L] [FiniteDimensional F L]
[IsScalarTower K F L] (x : 𝓞 L) : norm K (norm F x) = norm K x := by |
rw [RingOfIntegers.ext_iff, coe_norm, coe_norm, coe_norm, Algebra.norm_norm]
|
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical.Basic
import Mathlib.Algebra.Or... | Mathlib/Data/Real/NNReal.lean | 858 | 859 | theorem mul_eq_mul_left {a b c : ℝ≥0} (h : a ≠ 0) : a * b = a * c ↔ b = c := by |
rw [mul_eq_mul_left_iff, or_iff_left h]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 675 | 677 | theorem rpow_le_rpow_of_exponent_ge (hx0 : 0 < x) (hx1 : x ≤ 1) (hyz : z ≤ y) : x ^ y ≤ x ^ z := by |
repeat' rw [rpow_def_of_pos hx0]
rw [exp_le_exp]; exact mul_le_mul_of_nonpos_left hyz (log_nonpos (le_of_lt hx0) hx1)
|
/-
Copyright (c) 2023 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Integral.Lebesgue
/-!
# Marginals of multivariate functions
In this ... | Mathlib/MeasureTheory/Integral/Marginal.lean | 214 | 222 | theorem lmarginal_update_of_not_mem {i : δ}
{f : (∀ i, π i) → ℝ≥0∞} (hf : Measurable f) (hi : i ∉ s) (x : ∀ i, π i) (y : π i) :
(∫⋯∫⁻_s, f ∂μ) (Function.update x i y) = (∫⋯∫⁻_s, f ∘ (Function.update · i y) ∂μ) x := by |
induction s using Finset.induction generalizing x with
| empty => simp
| @insert i' s hi' ih =>
rw [lmarginal_insert _ hf hi', lmarginal_insert _ (hf.comp measurable_update_left) hi']
have hii' : i ≠ i' := mt (by rintro rfl; exact mem_insert_self i s) hi
simp_rw [update_comm hii', ih (mt Finset.mem_i... |
/-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.SetLike.Fintype
import Mathlib.Algebra.Divisibility.Prod
import Mathlib.RingTheory.Nakayama
import Mathlib.RingTheory.SimpleModule
import Mathlib.Tact... | Mathlib/RingTheory/Artinian.lean | 490 | 502 | theorem localization_surjective : Function.Surjective (algebraMap R L) := by |
intro r'
obtain ⟨r₁, s, rfl⟩ := IsLocalization.mk'_surjective S r'
-- TODO: can `rsuffices` be used to move the `exact` below before the proof of this `obtain`?
obtain ⟨r₂, h⟩ : ∃ r : R, IsLocalization.mk' L 1 s = algebraMap R L r := by
obtain ⟨n, r, hr⟩ := IsArtinian.exists_pow_succ_smul_dvd (s : R) (1 : ... |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.FieldTheory.Finite.Polynomial
import Mathlib.NumberTheory.Basic
import Mathlib.RingTheory.WittVector.WittPolynomial
#align_import rin... | Mathlib/RingTheory/WittVector/StructurePolynomial.lean | 405 | 409 | theorem wittStructureInt_vars [Fintype idx] (Φ : MvPolynomial idx ℤ) (n : ℕ) :
(wittStructureInt p Φ n).vars ⊆ Finset.univ ×ˢ Finset.range (n + 1) := by |
have : Function.Injective (Int.castRingHom ℚ) := Int.cast_injective
rw [← vars_map_of_injective _ this, map_wittStructureInt]
apply wittStructureRat_vars
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Finsupp.Multiset
import Math... | Mathlib/RingTheory/UniqueFactorizationDomain.lean | 1,286 | 1,292 | theorem FactorSet.prod_eq_zero_iff [Nontrivial α] (p : FactorSet α) : p.prod = 0 ↔ p = ⊤ := by |
unfold FactorSet at p
induction p -- TODO: `induction_eliminator` doesn't work with `abbrev`
· simp only [iff_self_iff, eq_self_iff_true, Associates.prod_top]
· rw [prod_coe, Multiset.prod_eq_zero_iff, Multiset.mem_map, eq_false WithTop.coe_ne_top,
iff_false_iff, not_exists]
exact fun a => not_and_o... |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Interval.Set.OrderIso
import Mathlib.Data.... | Mathlib/Order/UpperLower/Basic.lean | 1,078 | 1,080 | theorem map_refl : map (OrderIso.refl α) = OrderIso.refl _ := by |
ext
simp
|
/-
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 | 1,486 | 1,492 | theorem preimage_invFun_of_not_mem [n : Nonempty α] {f : α → β} (hf : Injective f) {s : Set α}
(h : Classical.choice n ∉ s) : invFun f ⁻¹' s = f '' s := by |
ext x
rcases em (x ∈ range f) with (⟨a, rfl⟩ | hx)
· rw [mem_preimage, leftInverse_invFun hf, hf.mem_set_image]
· have : x ∉ f '' s := fun h' => hx (image_subset_range _ _ h')
simp only [mem_preimage, invFun_neg hx, h, this]
|
/-
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.Matroid.Dual
/-!
# Matroid Restriction
Given `M : Matroid α` and `R : Set α`, the independent sets of `M` that are contained in `R`
are the independ... | Mathlib/Data/Matroid/Restrict.lean | 400 | 405 | theorem Indep.exists_basis_subset_union_basis (hI : M.Indep I) (hIX : I ⊆ X) (hJ : M.Basis J X) :
∃ I', M.Basis I' X ∧ I ⊆ I' ∧ I' ⊆ I ∪ J := by |
obtain ⟨I', hI', hII', hI'IJ⟩ :=
(hI.indep_restrict_of_subset hIX).exists_base_subset_union_base (Basis.base_restrict hJ)
rw [base_restrict_iff] at hI'
exact ⟨I', hI', hII', hI'IJ⟩
|
/-
Copyright (c) 2018 Mitchell Rowett. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Rowett, Scott Morrison
-/
import Mathlib.Algebra.Quotient
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.Algebra.Group.Subgroup.MulOpposite
import Mathlib.Group... | Mathlib/GroupTheory/Coset.lean | 373 | 377 | theorem rightRel_r_eq_rightCosetEquivalence :
@Setoid.r _ (QuotientGroup.rightRel s) = RightCosetEquivalence s := by |
ext
rw [rightRel_eq]
exact (rightCoset_eq_iff s).symm
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.LinearAlgebra.Matrix.AbsoluteValue
import Mathlib.NumberTheory.... | Mathlib/NumberTheory/ClassNumber/Finite.lean | 91 | 114 | theorem norm_lt {T : Type*} [LinearOrderedRing T] (a : S) {y : T}
(hy : ∀ k, (abv (bS.repr a k) : T) < y) :
(abv (Algebra.norm R a) : T) < normBound abv bS * y ^ Fintype.card ι := by |
obtain ⟨i⟩ := bS.index_nonempty
have him : (Finset.univ.image fun k => abv (bS.repr a k)).Nonempty :=
⟨_, Finset.mem_image.mpr ⟨i, Finset.mem_univ _, rfl⟩⟩
set y' : ℤ := Finset.max' _ him with y'_def
have hy' : ∀ k, abv (bS.repr a k) ≤ y' := by
intro k
exact @Finset.le_max' ℤ _ _ _ (Finset.mem_imag... |
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