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.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75... | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | 959 | 965 | theorem sup_direction_le (s1 s2 : AffineSubspace k P) :
s1.direction ⊔ s2.direction ≤ (s1 ⊔ s2).direction := by |
simp only [direction_eq_vectorSpan, vectorSpan_def]
exact
sup_le
(sInf_le_sInf fun p hp => Set.Subset.trans (vsub_self_mono (le_sup_left : s1 ≤ s1 ⊔ s2)) hp)
(sInf_le_sInf fun p hp => Set.Subset.trans (vsub_self_mono (le_sup_right : s2 ≤ s1 ⊔ s2)) hp)
|
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee
-/
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
/-!
# Reflections, inversions, and inversion sequences
Throughout this file, `B` is a type and... | Mathlib/GroupTheory/Coxeter/Inversion.lean | 334 | 344 | theorem getD_leftInvSeq_mul_wordProd (ω : List B) (j : ℕ) :
((lis ω).getD j 1) * π ω = π (ω.eraseIdx j) := by |
rw [getD_leftInvSeq, eraseIdx_eq_take_drop_succ]
nth_rw 4 [← take_append_drop (j + 1) ω]
rw [take_succ]
obtain lt | le := lt_or_le j ω.length
· rw [get?_eq_get lt]
simp only [wordProd_append, wordProd_cons, mul_assoc]
simp
· rw [get?_eq_none.mpr le]
simp
|
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.ProperSpace
import Mathlib.Topology.MetricSpace.Cauchy
/-... | Mathlib/Topology/MetricSpace/Bounded.lean | 515 | 522 | theorem diam_union {t : Set α} (xs : x ∈ s) (yt : y ∈ t) :
diam (s ∪ t) ≤ diam s + dist x y + diam t := by |
simp only [diam, dist_edist]
refine (ENNReal.toReal_le_add' (EMetric.diam_union xs yt) ?_ ?_).trans
(add_le_add_right ENNReal.toReal_add_le _)
· simp only [ENNReal.add_eq_top, edist_ne_top, or_false]
exact fun h ↦ top_unique <| h ▸ EMetric.diam_mono subset_union_left
· exact fun h ↦ top_unique <| h ▸ E... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Aurélien Saue, Anne Baanen
-/
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
/-!
#... | Mathlib/Tactic/Ring/Basic.lean | 890 | 893 | theorem inv_mul {R} [DivisionRing R] {a₁ a₂ a₃ b₁ b₃ c}
(_ : (a₁⁻¹ : R) = b₁) (_ : (a₃⁻¹ : R) = b₃)
(_ : b₃ * (b₁ ^ a₂ * (nat_lit 1).rawCast) = c) :
(a₁ ^ a₂ * a₃ : R)⁻¹ = c := by | subst_vars; simp
|
/-
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 | 1,417 | 1,419 | theorem eval₂_map [CommSemiring S₂] (f : R →+* S₁) (g : σ → S₂) (φ : S₁ →+* S₂)
(p : MvPolynomial σ R) : eval₂ φ g (map f p) = eval₂ (φ.comp f) g p := by |
rw [← eval_map, ← eval_map, map_map]
|
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral
#align_import measure_theory.integral.layercake from "leanprover-community/mathlib"@"08a4542bec7242a5c60f179e4e49d... | Mathlib/MeasureTheory/Integral/Layercake.lean | 447 | 459 | theorem lintegral_eq_lintegral_meas_le (μ : Measure α) (f_nn : 0 ≤ᵐ[μ] f)
(f_mble : AEMeasurable f μ) :
∫⁻ ω, ENNReal.ofReal (f ω) ∂μ = ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} := by |
set cst := fun _ : ℝ => (1 : ℝ)
have cst_intble : ∀ t > 0, IntervalIntegrable cst volume 0 t := fun _ _ =>
intervalIntegrable_const
have key :=
lintegral_comp_eq_lintegral_meas_le_mul μ f_nn f_mble cst_intble
(eventually_of_forall fun _ => zero_le_one)
simp_rw [cst, ENNReal.ofReal_one, mul_one] a... |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.... | Mathlib/Data/Nat/GCD/Basic.lean | 206 | 207 | theorem coprime_add_mul_right_left (m n k : ℕ) : Coprime (m + k * n) n ↔ Coprime m n := by |
rw [Coprime, Coprime, gcd_add_mul_right_left]
|
/-
Copyright (c) 2021 Alex J. Best. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best
-/
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.GroupTheory.GroupAction.Pi
#align_import algebra.module.pointwise_pi from "leanprover-community/mathlib"@"9003f2879... | Mathlib/Algebra/Module/PointwisePi.lean | 29 | 32 | theorem smul_pi_subset [∀ i, SMul K (R i)] (r : K) (s : Set ι) (t : ∀ i, Set (R i)) :
r • pi s t ⊆ pi s (r • t) := by |
rintro x ⟨y, h, rfl⟩ i hi
exact smul_mem_smul_set (h i hi)
|
/-
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
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingul... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 786 | 790 | theorem lintegral_indicator₀ (f : α → ℝ≥0∞) {s : Set α} (hs : NullMeasurableSet s μ) :
∫⁻ a, s.indicator f a ∂μ = ∫⁻ a in s, f a ∂μ := by |
rw [← lintegral_congr_ae (indicator_ae_eq_of_ae_eq_set hs.toMeasurable_ae_eq),
lintegral_indicator _ (measurableSet_toMeasurable _ _),
Measure.restrict_congr_set hs.toMeasurable_ae_eq]
|
/-
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.LinearAlgebra.TensorProduct.Basic
import Mathlib.RingTheory.Finiteness
/-!
# Some finiteness results of tensor product
This file contains some finiteness results of... | Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean | 98 | 116 | theorem exists_finite_submodule_of_finite (s : Set (M ⊗[R] N)) (hs : s.Finite) :
∃ (M' : Submodule R M) (N' : Submodule R N), Module.Finite R M' ∧ Module.Finite R N' ∧
s ⊆ LinearMap.range (mapIncl M' N') := by |
simp_rw [Module.Finite.iff_fg]
refine hs.induction_on ⟨_, _, fg_bot, fg_bot, Set.empty_subset _⟩ ?_
rintro a s - - ⟨M', N', hM', hN', h⟩
refine TensorProduct.induction_on a ?_ (fun x y ↦ ?_) fun x y hx hy ↦ ?_
· exact ⟨M', N', hM', hN', Set.insert_subset (zero_mem _) h⟩
· refine ⟨_, _, hM'.sup (fg_span_sin... |
/-
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 | 436 | 436 | theorem gcd_one_right' [GCDMonoid α] (a : α) : Associated (gcd a 1) 1 := by | simp
|
/-
Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.Convex.Extreme
import Mathlib.Analysis.Convex.Function
import Mathlib.Topology.Algebra.Module.Basic
import Mathlib.Top... | Mathlib/Analysis/Convex/Exposed.lean | 156 | 166 | theorem sInter [ContinuousAdd 𝕜] {F : Finset (Set E)} (hF : F.Nonempty)
(hAF : ∀ B ∈ F, IsExposed 𝕜 A B) : IsExposed 𝕜 A (⋂₀ F) := by |
induction F using Finset.induction with
| empty => exfalso; exact Finset.not_nonempty_empty hF
| @insert C F _ hF' =>
rw [Finset.coe_insert, sInter_insert]
obtain rfl | hFnemp := F.eq_empty_or_nonempty
· rw [Finset.coe_empty, sInter_empty, inter_univ]
exact hAF C (Finset.mem_singleton_self C)
... |
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Adam Topaz
-/
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.Limits.Preserves.Basic
import Mathlib.CategoryTheory.Limits.TypesFilter... | Mathlib/CategoryTheory/Limits/ConcreteCategory.lean | 41 | 54 | theorem Concrete.to_product_injective_of_isLimit {D : Cone F} (hD : IsLimit D) :
Function.Injective fun (x : D.pt) (j : J) => D.π.app j x := by |
let E := (forget C).mapCone D
let hE : IsLimit E := isLimitOfPreserves _ hD
let G := Types.limitCone.{w, v} (F ⋙ forget C)
let hG := Types.limitConeIsLimit.{w, v} (F ⋙ forget C)
let T : E.pt ≅ G.pt := hE.conePointUniqueUpToIso hG
change Function.Injective (T.hom ≫ fun x j => G.π.app j x)
have h : Functio... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Int.Lemm... | Mathlib/Algebra/Order/Floor.lean | 779 | 781 | theorem floor_add_one (a : α) : ⌊a + 1⌋ = ⌊a⌋ + 1 := by |
-- Porting note: broken `convert floor_add_int a 1`
rw [← cast_one, floor_add_int]
|
/-
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, Scott Morrison
-/
import Mathlib.Data.List.Chain
import Mathlib.Data.List.Enum
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Pairwise
import Mathli... | Mathlib/Data/List/Range.lean | 159 | 160 | theorem length_finRange (n : ℕ) : (finRange n).length = n := by |
rw [finRange, length_pmap, length_range]
|
/-
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.Circumcenter
#align_import geometry.euclidean.monge_point from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0... | Mathlib/Geometry/Euclidean/MongePoint.lean | 264 | 273 | theorem mongePlane_comm {n : ℕ} (s : Simplex ℝ P (n + 2)) (i₁ i₂ : Fin (n + 3)) :
s.mongePlane i₁ i₂ = s.mongePlane i₂ i₁ := by |
simp_rw [mongePlane_def]
congr 3
· congr 1
exact pair_comm _ _
· ext
simp_rw [Submodule.mem_span_singleton]
constructor
all_goals rintro ⟨r, rfl⟩; use -r; rw [neg_smul, ← smul_neg, neg_vsub_eq_vsub_rev]
|
/-
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 | 906 | 910 | theorem strongDownwardInductionOn_eq {p : Multiset α → Sort*} (s : Multiset α) {n : ℕ}
(H : ∀ t₁, (∀ {t₂ : Multiset α}, card t₂ ≤ n → t₁ < t₂ → p t₂) → card t₁ ≤ n → p t₁) :
s.strongDownwardInductionOn H = H s fun {t} ht _h => t.strongDownwardInductionOn H ht := by |
dsimp only [strongDownwardInductionOn]
rw [strongDownwardInduction]
|
/-
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 | 1,269 | 1,272 | theorem Injective.preimage_surjective (hf : Injective f) : Surjective (preimage f) := by |
intro s
use f '' s
rw [hf.preimage_image]
|
/-
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.Algebra.Category.GroupCat.Abelian
import Mathlib.CategoryTheory.Limits.Shapes.Images
#align_import algebra.category.Group.images from "leanprover-comm... | Mathlib/Algebra/Category/GroupCat/Images.lean | 56 | 58 | theorem image.fac : factorThruImage f ≫ image.ι f = f := by |
ext
rfl
|
/-
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 | 246 | 252 | theorem sSup_eq_eqvGen (S : Set (Setoid α)) :
sSup S = EqvGen.Setoid fun x y => ∃ r : Setoid α, r ∈ S ∧ r.Rel x y := by |
rw [eqvGen_eq]
apply congr_arg sInf
simp only [upperBounds, le_def, and_imp, exists_imp]
ext
exact ⟨fun H x y r hr => H hr, fun H r hr x y => H r hr⟩
|
/-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.Setoid.Partition
imp... | Mathlib/Combinatorics/SimpleGraph/Coloring.lean | 114 | 119 | theorem Coloring.card_colorClasses_le [Fintype α] [Fintype C.colorClasses] :
Fintype.card C.colorClasses ≤ Fintype.card α := by |
simp [colorClasses]
-- Porting note: brute force instance declaration `[Fintype (Setoid.classes (Setoid.ker C))]`
haveI : Fintype (Setoid.classes (Setoid.ker C)) := by assumption
convert Setoid.card_classes_ker_le C
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 526 | 529 | theorem Finsupp.inner_sum {ι : Type*} (l : ι →₀ 𝕜) (v : ι → E) (x : E) :
⟪x, l.sum fun (i : ι) (a : 𝕜) => a • v i⟫ = l.sum fun (i : ι) (a : 𝕜) => a • ⟪x, v i⟫ := by |
convert _root_.inner_sum (𝕜 := 𝕜) l.support (fun a => l a • v a) x
simp only [inner_smul_right, Finsupp.sum, smul_eq_mul]
|
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.LineDeriv.Measurable
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
i... | Mathlib/Analysis/Calculus/Rademacher.lean | 368 | 372 | theorem ae_differentiableWithinAt {f : E → F} (hf : LipschitzOnWith C f s)
(hs : MeasurableSet s) :
∀ᵐ x ∂(μ.restrict s), DifferentiableWithinAt ℝ f s x := by |
rw [ae_restrict_iff' hs]
exact hf.ae_differentiableWithinAt_of_mem
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Group.Int
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.... | Mathlib/Data/Rat/Lemmas.lean | 192 | 196 | theorem div_int_inj {a b c d : ℤ} (hb0 : 0 < b) (hd0 : 0 < d) (h1 : Nat.Coprime a.natAbs b.natAbs)
(h2 : Nat.Coprime c.natAbs d.natAbs) (h : (a : ℚ) / b = (c : ℚ) / d) : a = c ∧ b = d := by |
apply And.intro
· rw [← num_div_eq_of_coprime hb0 h1, h, num_div_eq_of_coprime hd0 h2]
· rw [← den_div_eq_of_coprime hb0 h1, h, den_div_eq_of_coprime hd0 h2]
|
/-
Copyright (c) 2022 Filippo A. E. Nuccio Mortarino Majno di Capriglio. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Filippo A. E. Nuccio, Junyan Xu
-/
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Topology.Homoto... | Mathlib/Topology/Homotopy/HSpaces.lean | 246 | 248 | theorem delayReflRight_one (γ : Path x y) : delayReflRight 1 γ = γ := by |
ext t
exact congr_arg γ (qRight_one_right t)
|
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Functor.Currying
import Mathlib.CategoryTheory.Limits.Preserves.Limits
#align_import category_theory.limits.functor_category from "lean... | Mathlib/CategoryTheory/Limits/FunctorCategory.lean | 207 | 212 | theorem limitObjIsoLimitCompEvaluation_hom_π [HasLimitsOfShape J C] (F : J ⥤ K ⥤ C) (j : J)
(k : K) :
(limitObjIsoLimitCompEvaluation F k).hom ≫ limit.π (F ⋙ (evaluation K C).obj k) j =
(limit.π F j).app k := by |
dsimp [limitObjIsoLimitCompEvaluation]
simp
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Top... | Mathlib/Analysis/Normed/Group/Basic.lean | 1,281 | 1,297 | theorem Filter.Tendsto.op_one_isBoundedUnder_le' {f : α → E} {g : α → F} {l : Filter α}
(hf : Tendsto f l (𝓝 1)) (hg : IsBoundedUnder (· ≤ ·) l (norm ∘ g)) (op : E → F → G)
(h_op : ∃ A, ∀ x y, ‖op x y‖ ≤ A * ‖x‖ * ‖y‖) : Tendsto (fun x => op (f x) (g x)) l (𝓝 1) := by |
cases' h_op with A h_op
rcases hg with ⟨C, hC⟩; rw [eventually_map] at hC
rw [NormedCommGroup.tendsto_nhds_one] at hf ⊢
intro ε ε₀
rcases exists_pos_mul_lt ε₀ (A * C) with ⟨δ, δ₀, hδ⟩
filter_upwards [hf δ δ₀, hC] with i hf hg
refine (h_op _ _).trans_lt ?_
rcases le_total A 0 with hA | hA
· exact (mul... |
/-
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.PurelyInseparable
import Mathlib.FieldTheory.PerfectClosure
/-!
# `IsPerfectClosure` predicate
This file contains `IsPerfectClosure` which asserts that ... | Mathlib/FieldTheory/IsPerfectClosure.lean | 96 | 110 | theorem mem_pNilradical {R : Type*} [CommSemiring R] {p : ℕ} {x : R} :
x ∈ pNilradical R p ↔ ∃ n : ℕ, x ^ p ^ n = 0 := by |
by_cases hp : 1 < p
· rw [pNilradical_eq_nilradical hp]
refine ⟨fun ⟨n, h⟩ ↦ ⟨n, ?_⟩, fun ⟨n, h⟩ ↦ ⟨p ^ n, h⟩⟩
rw [← Nat.sub_add_cancel ((Nat.lt_pow_self hp n).le), pow_add, h, mul_zero]
rw [pNilradical_eq_bot hp, Ideal.mem_bot]
refine ⟨fun h ↦ ⟨0, by rw [pow_zero, pow_one, h]⟩, fun ⟨n, h⟩ ↦ ?_⟩
rcas... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Order.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_imp... | Mathlib/Order/Interval/Set/Basic.lean | 1,460 | 1,464 | theorem Iic_union_Ioc (h : min c d < b) : Iic b ∪ Ioc c d = Iic (max b d) := by |
rcases le_total c d with hcd | hcd <;> simp [hcd] at h
· exact Iic_union_Ioc' h
· rw [max_comm]
simp [*, max_eq_right_of_lt h]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Ta... | Mathlib/Logic/Relation.lean | 539 | 542 | theorem TransGen.swap (h : TransGen r b a) : TransGen (swap r) a b := by |
induction' h with b h b c _ hbc ih
· exact TransGen.single h
· exact ih.head hbc
|
/-
Copyright (c) 2021 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.special_functions.log.monotone from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5... | Mathlib/Analysis/SpecialFunctions/Log/Monotone.lean | 56 | 82 | theorem log_div_self_rpow_antitoneOn {a : ℝ} (ha : 0 < a) :
AntitoneOn (fun x : ℝ => log x / x ^ a) { x | exp (1 / a) ≤ x } := by |
simp only [AntitoneOn, mem_setOf_eq]
intro x hex y _ hxy
have x_pos : 0 < x := lt_of_lt_of_le (exp_pos (1 / a)) hex
have y_pos : 0 < y := by linarith
have x_nonneg : 0 ≤ x := le_trans (le_of_lt (exp_pos (1 / a))) hex
have y_nonneg : 0 ≤ y := by linarith
nth_rw 1 [← rpow_one y]
nth_rw 1 [← rpow_one x]
... |
/-
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.Topology.Order.MonotoneContinuity
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.E... | Mathlib/Topology/Instances/ENNReal.lean | 1,568 | 1,569 | theorem diam_Icc {a b : ℝ} (h : a ≤ b) : Metric.diam (Icc a b) = b - a := by |
simp [Metric.diam, ENNReal.toReal_ofReal (sub_nonneg.2 h)]
|
/-
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.Squarefree.Basic
import Mathlib.Data.Nat.Factorization.PrimePow
#align_import data.nat.squarefree from "leanprover-community/mathlib"@"3c1368c... | Mathlib/Data/Nat/Squarefree.lean | 148 | 161 | theorem minSqFacProp_div (n) {k} (pk : Prime k) (dk : k ∣ n) (dkk : ¬k * k ∣ n) {o}
(H : MinSqFacProp (n / k) o) : MinSqFacProp n o := by |
have : ∀ p, Prime p → p * p ∣ n → k * (p * p) ∣ n := fun p pp dp =>
have :=
(coprime_primes pk pp).2 fun e => by
subst e
contradiction
(coprime_mul_iff_right.2 ⟨this, this⟩).mul_dvd_of_dvd_of_dvd dk dp
cases' o with d
· rw [MinSqFacProp, squarefree_iff_prime_squarefree] at H ⊢
e... |
/-
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.BigOperators.Finprod
import Mathlib.Algebra.Order.Group.WithTop
import Mathlib.RingTheory.HahnSeries.Multiplication
import Mathlib.RingTheory.V... | Mathlib/RingTheory/HahnSeries/Summable.lean | 89 | 92 | theorem addVal_le_of_coeff_ne_zero {x : HahnSeries Γ R} {g : Γ} (h : x.coeff g ≠ 0) :
addVal Γ R x ≤ g := by |
rw [addVal_apply_of_ne (ne_zero_of_coeff_ne_zero h), WithTop.coe_le_coe]
exact order_le_of_coeff_ne_zero h
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Measure.MutuallySingular
#align_import measure_theory.decomposition.jordan from "leanpro... | Mathlib/MeasureTheory/Decomposition/Jordan.lean | 509 | 516 | theorem null_of_totalVariation_zero (s : SignedMeasure α) {i : Set α}
(hs : s.totalVariation i = 0) : s i = 0 := by |
rw [totalVariation, Measure.coe_add, Pi.add_apply, add_eq_zero_iff] at hs
rw [← toSignedMeasure_toJordanDecomposition s, toSignedMeasure, VectorMeasure.coe_sub,
Pi.sub_apply, Measure.toSignedMeasure_apply, Measure.toSignedMeasure_apply]
by_cases hi : MeasurableSet i
· rw [if_pos hi, if_pos hi]; simp [hs.1,... |
/-
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.Order.SuccPred.Basic
import Mathlib.Order.BoundedOrder
#align_import order.succ_pred.limit from "leanprover-community/mathlib"... | Mathlib/Order/SuccPred/Limit.lean | 283 | 284 | theorem not_isPredLimit_iff_exists_covBy (a : α) : ¬IsPredLimit a ↔ ∃ b, a ⋖ b := by |
simp [IsPredLimit]
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.Algebraic
#align_import field_the... | Mathlib/FieldTheory/Minpoly/Field.lean | 131 | 138 | theorem eq_of_irreducible [Nontrivial B] {p : A[X]} (hp1 : Irreducible p)
(hp2 : Polynomial.aeval x p = 0) : p * C p.leadingCoeff⁻¹ = minpoly A x := by |
have : p.leadingCoeff ≠ 0 := leadingCoeff_ne_zero.mpr hp1.ne_zero
apply eq_of_irreducible_of_monic
· exact Associated.irreducible ⟨⟨C p.leadingCoeff⁻¹, C p.leadingCoeff,
by rwa [← C_mul, inv_mul_cancel, C_1], by rwa [← C_mul, mul_inv_cancel, C_1]⟩, rfl⟩ hp1
· rw [aeval_mul, hp2, zero_mul]
· rwa [Polyno... |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Sébastien Gouëzel, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.LinearAlgebra.FiniteDimen... | Mathlib/Analysis/InnerProductSpace/PiL2.lean | 695 | 698 | theorem Complex.isometryOfOrthonormal_symm_apply (v : OrthonormalBasis (Fin 2) ℝ F) (f : F) :
(Complex.isometryOfOrthonormal v).symm f =
(v.toBasis.coord 0 f : ℂ) + (v.toBasis.coord 1 f : ℂ) * I := by |
simp [Complex.isometryOfOrthonormal]
|
/-
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 | 102 | 103 | theorem lt_iff (hf : LeftOrdContinuous f) (h : Injective f) {x y} : f x < f y ↔ x < y := by |
simp only [lt_iff_le_not_le, hf.le_iff h]
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.Factorial.Cast
#align_import data.nat.choose.cast from "leanprover-community/mathlib"@"bb168510ef455e9280a15... | Mathlib/Data/Nat/Choose/Cast.lean | 35 | 38 | theorem cast_choose_eq_ascPochhammer_div (a b : ℕ) :
(a.choose b : K) = (ascPochhammer K b).eval ↑(a - (b - 1)) / b ! := by |
rw [eq_div_iff_mul_eq (cast_ne_zero.2 b.factorial_ne_zero : (b ! : K) ≠ 0), ← cast_mul,
mul_comm, ← descFactorial_eq_factorial_mul_choose, ← cast_descFactorial]
|
/-
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 | 185 | 186 | theorem eqOn_singleton : Set.EqOn f₁ f₂ {a} ↔ f₁ a = f₂ a := by |
simp [Set.EqOn]
|
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean | 59 | 89 | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ := by |
by_cases hf : Integrable f μ
swap; · rw [condexp_undef hf, snorm_zero]; exact zero_le _
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm, snorm_zero]; exact zero_le _
by_cases hsig : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hsig, snorm_zero]; exact zero_le _
calc
snorm (... |
/-
Copyright (c) 2020 Kevin Buzzard, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathl... | Mathlib/CategoryTheory/Sites/Sheaf.lean | 728 | 731 | theorem isSheaf_comp_of_isSheaf (s : A ⥤ B) [PreservesLimitsOfSize.{v₁, max v₁ u₁} s]
(h : IsSheaf J P) : IsSheaf J (P ⋙ s) := by |
rw [isSheaf_iff_isLimit] at h ⊢
apply fun X S hS ↦ (h S hS).map fun t ↦ isLimitOfPreserves s t
|
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.MeasureTheory.Integral.FundThmCalculus
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
import Mathlib.Analysis.SpecialFunction... | Mathlib/Analysis/SpecialFunctions/Integrals.lean | 441 | 444 | theorem integral_id : ∫ x in a..b, x = (b ^ 2 - a ^ 2) / 2 := by |
have := @integral_pow a b 1
norm_num at this
exact this
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Na... | Mathlib/Data/Complex/Exponential.lean | 1,255 | 1,255 | theorem one_lt_exp_iff {x : ℝ} : 1 < exp x ↔ 0 < x := by | rw [← exp_zero, exp_lt_exp]
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Top... | Mathlib/Analysis/Normed/Group/Basic.lean | 720 | 721 | theorem mem_closedBall_iff_norm''' : b ∈ closedBall a r ↔ ‖a / b‖ ≤ r := by |
rw [mem_closedBall', dist_eq_norm_div]
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Batteries.Data.List.Basic
import Batteries.Data.List.Lemmas
/-!
# Counting in lists
T... | .lake/packages/batteries/Batteries/Data/List/Count.lean | 88 | 90 | theorem countP_filter (l : List α) :
countP p (filter q l) = countP (fun a => p a ∧ q a) l := by |
simp only [countP_eq_length_filter, filter_filter]
|
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import comp... | Mathlib/Computability/TMToPartrec.lean | 1,330 | 1,361 | theorem move_ok {p k₁ k₂ q s L₁ o L₂} {S : K' → List Γ'} (h₁ : k₁ ≠ k₂)
(e : splitAtPred p (S k₁) = (L₁, o, L₂)) :
Reaches₁ (TM2.step tr) ⟨some (Λ'.move p k₁ k₂ q), s, S⟩
⟨some q, o, update (update S k₁ L₂) k₂ (L₁.reverseAux (S k₂))⟩ := by |
induction' L₁ with a L₁ IH generalizing S s
· rw [(_ : [].reverseAux _ = _), Function.update_eq_self]
swap
· rw [Function.update_noteq h₁.symm, List.reverseAux_nil]
refine TransGen.head' rfl ?_
simp only [TM2.step, Option.mem_def, TM2.stepAux, Option.elim, ne_eq]
revert e; cases' S k₁ with a Sk... |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Pigeonhole
imp... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 374 | 380 | theorem ceil_lt_mul {x : ℝ} (hx : 50 / 19 ≤ x) : (⌈x⌉₊ : ℝ) < 1.38 * x := by |
refine (ceil_lt_add_one <| hx.trans' <| by norm_num).trans_le ?_
rw [← le_sub_iff_add_le', ← sub_one_mul]
have : (1.38 : ℝ) = 69 / 50 := by norm_num
rwa [this, show (69 / 50 - 1 : ℝ) = (50 / 19)⁻¹ by norm_num1, ←
div_eq_inv_mul, one_le_div]
norm_num1
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
/-!
# Finite int... | Mathlib/Order/Interval/Finset/Fin.lean | 210 | 211 | theorem map_valEmbedding_Iio : (Iio b).map Fin.valEmbedding = Iio ↑b := by |
simp [Iio_eq_finset_subtype, Finset.fin, Finset.map_map]
|
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Johan Commelin, Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Constructions.Pullbacks
import Mathlib.CategoryTheory.Preadditive.Biproducts
import Mathlib.Categor... | Mathlib/CategoryTheory/Abelian/Basic.lean | 724 | 727 | theorem epi_fst_of_isLimit [Epi g] {s : PullbackCone f g} (hs : IsLimit s) : Epi s.fst := by |
haveI : Epi (NatTrans.app (limit.cone (cospan f g)).π WalkingCospan.left) :=
Abelian.epi_pullback_of_epi_g f g
apply epi_of_epi_fac (IsLimit.conePointUniqueUpToIso_hom_comp (limit.isLimit _) hs _)
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.Finiteness
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
import Mathlib.LinearAlgebra.Dimension.DivisionRing
#align_import... | Mathlib/LinearAlgebra/FiniteDimensional.lean | 510 | 514 | theorem finrank_add_finrank_le_of_disjoint [FiniteDimensional K V]
{s t : Submodule K V} (hdisjoint : Disjoint s t) :
finrank K s + finrank K t ≤ finrank K V := by |
rw [← Submodule.finrank_sup_add_finrank_inf_eq s t, hdisjoint.eq_bot, finrank_bot, add_zero]
exact Submodule.finrank_le _
|
/-
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.FreeNonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.NonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.UniversalEnveloping
import Mathlib.GroupT... | Mathlib/Algebra/Lie/Free.lean | 262 | 265 | theorem hom_ext {F₁ F₂ : FreeLieAlgebra R X →ₗ⁅R⁆ L} (h : ∀ x, F₁ (of R x) = F₂ (of R x)) :
F₁ = F₂ :=
have h' : (lift R).symm F₁ = (lift R).symm F₂ := by | ext; simp [h]
(lift R).symm.injective h'
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Ta... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 1,249 | 1,256 | theorem Chain.imp' {R S : α → α → Prop} (HRS : ∀ ⦃a b⦄, R a b → S a b) {a b : α}
(Hab : ∀ ⦃c⦄, R a c → S b c) {l : List α} (p : Chain R a l) : Chain S b l := by |
induction p generalizing b with
| nil => constructor
| cons r _ ih =>
constructor
· exact Hab r
· exact ih (@HRS _)
|
/-
Copyright (c) 2022 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Junyan Xu
-/
import Mathlib.RingTheory.Localization.LocalizationLocalization
#align_import ring_theory.localization.as_subring from "leanprover-community/mathlib"@"649ca66bf4d... | Mathlib/RingTheory/Localization/AsSubring.lean | 31 | 32 | theorem map_isUnit_of_le (hS : S ≤ A⁰) (s : S) : IsUnit (algebraMap A K s) := by |
apply IsLocalization.map_units K (⟨s.1, hS s.2⟩ : A⁰)
|
/-
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, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathli... | Mathlib/Data/Set/Lattice.lean | 398 | 400 | theorem iInter_setOf (P : ι → α → Prop) : ⋂ i, { x : α | P i x } = { x : α | ∀ i, P i x } := by |
ext
exact mem_iInter
|
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Mea... | Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | 761 | 764 | theorem restrict_prod_eq_prod_univ (s : Set α) :
(μ.restrict s).prod ν = (μ.prod ν).restrict (s ×ˢ univ) := by |
have : ν = ν.restrict Set.univ := Measure.restrict_univ.symm
rw [this, Measure.prod_restrict, ← this]
|
/-
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.Group.ConjFinite
import Mathlib.GroupTheory.Perm.Fin
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.Tactic.IntervalCases
#align_imp... | Mathlib/GroupTheory/SpecificGroups/Alternating.lean | 89 | 91 | theorem finRotate_bit1_mem_alternatingGroup {n : ℕ} :
finRotate (bit1 n) ∈ alternatingGroup (Fin (bit1 n)) := by |
rw [mem_alternatingGroup, bit1, sign_finRotate, pow_bit0', Int.units_mul_self, one_pow]
|
/-
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.Algebra.Group.Submonoid.Pointwise
#align_import group_theory.submonoid.inverses from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782b... | Mathlib/GroupTheory/Submonoid/Inverses.lean | 126 | 127 | theorem fromLeftInv_mul (x : S.leftInv) : (S.fromLeftInv x : M) * x = 1 := by |
rw [mul_comm, mul_fromLeftInv]
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
im... | Mathlib/Data/List/Basic.lean | 846 | 848 | theorem head!_mem_self [Inhabited α] {l : List α} (h : l ≠ nil) : l.head! ∈ l := by |
have h' := mem_cons_self l.head! l.tail
rwa [cons_head!_tail h] at h'
|
/-
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.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dea... | Mathlib/Data/Multiset/Bind.lean | 166 | 167 | theorem bind_congr {f g : α → Multiset β} {m : Multiset α} :
(∀ a ∈ m, f a = g a) → bind m f = bind m g := by | simp (config := { contextual := true }) [bind]
|
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Devon Tuma
-/
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Asymptotics.SpecificAsymptoti... | Mathlib/Analysis/SpecialFunctions/Polynomials.lean | 42 | 54 | theorem isEquivalent_atTop_lead :
(fun x => eval x P) ~[atTop] fun x => P.leadingCoeff * x ^ P.natDegree := by |
by_cases h : P = 0
· simp [h, IsEquivalent.refl]
· simp only [Polynomial.eval_eq_sum_range, sum_range_succ]
exact
IsLittleO.add_isEquivalent
(IsLittleO.sum fun i hi =>
IsLittleO.const_mul_left
((IsLittleO.const_mul_right fun hz => h <| leadingCoeff_eq_zero.mp hz) <|
... |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Fin
import Mathlib... | Mathlib/Data/Fin/Tuple/Basic.lean | 677 | 680 | theorem append_right_cons {n m} {α : Type*} (xs : Fin n → α) (y : α) (ys : Fin m → α) :
Fin.append xs (Fin.cons y ys) =
Fin.append (Fin.snoc xs y) ys ∘ Fin.cast (Nat.succ_add_eq_add_succ ..).symm := by |
rw [append_left_snoc]; 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.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.pi_nat from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5... | Mathlib/Topology/MetricSpace/PiNat.lean | 291 | 300 | theorem dist_triangle_nonarch (x y z : ∀ n, E n) : dist x z ≤ max (dist x y) (dist y z) := by |
rcases eq_or_ne x z with (rfl | hxz)
· simp [PiNat.dist_self x, PiNat.dist_nonneg]
rcases eq_or_ne x y with (rfl | hxy)
· simp
rcases eq_or_ne y z with (rfl | hyz)
· simp
simp only [dist_eq_of_ne, hxz, hxy, hyz, inv_le_inv, one_div, inv_pow, zero_lt_two, Ne,
not_false_iff, le_max_iff, pow_le_pow_iff_... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Separation
/-!
# Order-closed topologies
In this file we introduce 3 typeclass mixins that relate topology ... | Mathlib/Topology/Order/OrderClosed.lean | 193 | 195 | theorem isOpen_Ioi : IsOpen (Ioi a) := by |
rw [← compl_Iic]
exact isClosed_Iic.isOpen_compl
|
/-
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.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.charpoly.linear_map from "leanprover-commu... | Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean | 100 | 111 | theorem Matrix.represents_iff' {A : Matrix ι ι R} {f : Module.End R M} :
A.Represents b f ↔ ∀ j, ∑ i : ι, A i j • b i = f (b j) := by |
constructor
· intro h i
have := LinearMap.congr_fun h (Pi.single i 1)
rwa [PiToModule.fromEnd_apply_single_one, PiToModule.fromMatrix_apply_single_one] at this
· intro h
-- Porting note: was `ext`
refine LinearMap.pi_ext' (fun i => LinearMap.ext_ring ?_)
simp_rw [LinearMap.comp_apply, LinearM... |
/-
Copyright (c) 2020 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Valuation.PrimeMultiplicity
import Mathlib.RingTheory... | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | 402 | 407 | theorem addVal_def (r : R) (u : Rˣ) {ϖ : R} (hϖ : Irreducible ϖ) (n : ℕ) (hr : r = u * ϖ ^ n) :
addVal R r = n := by |
rw [addVal, addValuation_apply, hr, eq_of_associated_left
(associated_of_irreducible R hϖ (Classical.choose_spec (exists_prime R)).irreducible),
eq_of_associated_right (Associated.symm ⟨u, mul_comm _ _⟩),
multiplicity_pow_self_of_prime (irreducible_iff_prime.1 hϖ)]
|
/-
Copyright (c) 2023 Bulhwi Cha. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bulhwi Cha, Mario Carneiro
-/
import Batteries.Data.Char
import Batteries.Data.List.Lemmas
import Batteries.Data.String.Basic
import Batteries.Tactic.Lint.Misc
import Batteries.Tactic.SeqF... | .lake/packages/batteries/Batteries/Data/String/Lemmas.lean | 216 | 226 | theorem utf8PrevAux_of_valid {cs cs' : List Char} {c : Char} {i p : Nat}
(hp : i + (utf8Len cs + csize c) = p) :
utf8PrevAux (cs ++ c :: cs') ⟨i⟩ ⟨p⟩ = ⟨i + utf8Len cs⟩ := by |
match cs with
| [] => simp [utf8PrevAux, ← hp, Pos.addChar_eq]
| c'::cs =>
simp [utf8PrevAux, Pos.addChar_eq, ← hp]; rw [if_neg]
case hnc =>
simp [Pos.ext_iff]; rw [Nat.add_right_comm, Nat.add_left_comm]; apply ne_add_csize_add_self
refine (utf8PrevAux_of_valid (by simp [Nat.add_assoc, Nat.add_... |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, François Dupuis
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Filter.Extr
import Mathlib.Tactic.GCongr
#align_import analysis.convex.function fr... | Mathlib/Analysis/Convex/Function.lean | 638 | 646 | theorem ConvexOn.le_on_segment' (hf : ConvexOn 𝕜 s f) {x y : E} (hx : x ∈ s) (hy : y ∈ s) {a b : 𝕜}
(ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1) : f (a • x + b • y) ≤ max (f x) (f y) :=
calc
f (a • x + b • y) ≤ a • f x + b • f y := hf.2 hx hy ha hb hab
_ ≤ a • max (f x) (f y) + b • max (f x) (f y) := by |
gcongr
· apply le_max_left
· apply le_max_right
_ = max (f x) (f y) := Convex.combo_self hab _
|
/-
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 | 213 | 215 | theorem gcd_union (s₁ s₂ : Multiset α) : (s₁ ∪ s₂).gcd = GCDMonoid.gcd s₁.gcd s₂.gcd := by |
rw [← gcd_dedup, dedup_ext.2, gcd_dedup, gcd_add]
simp
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Finsupp.Encodable
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Span
import Mathlib.Data.Set.Countable
#align_import linear_algebr... | Mathlib/LinearAlgebra/Finsupp.lean | 1,225 | 1,230 | theorem mem_span_range_iff_exists_fun :
x ∈ span R (range v) ↔ ∃ c : α → R, ∑ i, c i • v i = x := by |
-- Porting note: `Finsupp.equivFunOnFinite.surjective.exists` should be come before `simp`.
rw [Finsupp.equivFunOnFinite.surjective.exists]
simp only [Finsupp.mem_span_range_iff_exists_finsupp, Finsupp.equivFunOnFinite_apply]
exact exists_congr fun c => Eq.congr_left <| Finsupp.sum_fintype _ _ fun i => zero_sm... |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
import Mathlib.Tactic.IntervalCases
#a... | Mathlib/Geometry/Euclidean/Triangle.lean | 207 | 241 | theorem angle_add_angle_sub_add_angle_sub_eq_pi {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
angle x y + angle x (x - y) + angle y (y - x) = π := by |
have hcos := cos_angle_add_angle_sub_add_angle_sub_eq_neg_one hx hy
have hsin := sin_angle_add_angle_sub_add_angle_sub_eq_zero hx hy
rw [Real.sin_eq_zero_iff] at hsin
cases' hsin with n hn
symm at hn
have h0 : 0 ≤ angle x y + angle x (x - y) + angle y (y - x) :=
add_nonneg (add_nonneg (angle_nonneg _ _... |
/-
Copyright (c) 2023 Adrian Wüthrich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adrian Wüthrich
-/
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.LinearAlgebra.Matrix.PosDef
/-!
# Laplacian Matrix
This module defines the Laplacian matrix of a... | Mathlib/Combinatorics/SimpleGraph/LapMatrix.lean | 90 | 96 | theorem posSemidef_lapMatrix [LinearOrderedField R] [StarRing R] [StarOrderedRing R]
[TrivialStar R] : PosSemidef (G.lapMatrix R) := by |
constructor
· rw [IsHermitian, conjTranspose_eq_transpose_of_trivial, isSymm_lapMatrix]
· intro x
rw [star_trivial, ← toLinearMap₂'_apply', lapMatrix_toLinearMap₂']
positivity
|
/-
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.NNReal
#align_import anal... | Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 141 | 146 | theorem tendsto_exp_mul_div_rpow_atTop (s : ℝ) (b : ℝ) (hb : 0 < b) :
Tendsto (fun x : ℝ => exp (b * x) / x ^ s) atTop atTop := by |
refine ((tendsto_rpow_atTop hb).comp (tendsto_exp_div_rpow_atTop (s / b))).congr' ?_
filter_upwards [eventually_ge_atTop (0 : ℝ)] with x hx₀
simp [Real.div_rpow, (exp_pos x).le, rpow_nonneg, ← Real.rpow_mul, ← exp_mul,
mul_comm x, hb.ne', *]
|
/-
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.Algebra.Algebra.Hom
import Mathlib.RingTheory.Ideal.Quotient
#align_import algebra.ring_quot from "leanprover-community/mathlib"@"e5820f6c8fcf1b75bcd7... | Mathlib/Algebra/RingQuot.lean | 75 | 76 | theorem Rel.sub_right {R : Type uR} [Ring R] {r : R → R → Prop} ⦃a b c : R⦄ (h : Rel r b c) :
Rel r (a - b) (a - c) := by | simp only [sub_eq_add_neg, h.neg.add_right]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Module.LinearMap.Basic
#align_import algebra.algebra.basic from "leanprover-community/... | Mathlib/Algebra/Algebra/Defs.lean | 351 | 353 | theorem left_comm (x : A) (r : R) (y : A) :
x * (algebraMap R A r * y) = algebraMap R A r * (x * y) := by |
rw [← mul_assoc, ← commutes, mul_assoc]
|
/-
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.RBMap.Alter
import Batteries.Data.List.Lemmas
/-!
# Additional lemmas for Red-black trees
-/
namespace Batteries
namespace RBNode
open RBColor... | .lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | 658 | 660 | theorem insert_toList {p : Path α} :
(p.insert t v).toList = p.withList (t.setRoot v).toList := by |
simp [insert]; split <;> simp [setRoot]
|
/-
Copyright (c) 2021 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.Tactic.ByContra
import Mathlib.Topology.Algebra.Polynomial
import Mathlib.NumberTheory.Padics.Pad... | Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean | 305 | 314 | theorem sub_one_pow_totient_lt_natAbs_cyclotomic_eval {n : ℕ} {q : ℕ} (hn' : 1 < n) (hq : q ≠ 1) :
(q - 1) ^ totient n < ((cyclotomic n ℤ).eval ↑q).natAbs := by |
rcases hq.lt_or_lt.imp_left Nat.lt_one_iff.mp with (rfl | hq')
· rw [zero_tsub, zero_pow (Nat.totient_pos.2 (pos_of_gt hn')).ne', pos_iff_ne_zero,
Int.natAbs_ne_zero, Nat.cast_zero, ← coeff_zero_eq_eval_zero, cyclotomic_coeff_zero _ hn']
exact one_ne_zero
rw [← @Nat.cast_lt ℝ, Nat.cast_pow, Nat.cast_su... |
/-
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 | 573 | 575 | theorem continuousAt_prod_of_discrete_left [DiscreteTopology α] {f : α × β → γ} {x : α × β} :
ContinuousAt f x ↔ ContinuousAt (f ⟨x.1, ·⟩) x.2 := by |
simp_rw [← continuousWithinAt_univ]; exact continuousWithinAt_prod_of_discrete_left
|
/-
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 | 513 | 513 | theorem whiskerLeft_iff {X Y : C} (f g : X ⟶ Y) : 𝟙_ C ◁ f = 𝟙_ C ◁ g ↔ f = g := by | simp
|
/-
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.tagged from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97... | Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean | 393 | 399 | theorem IsHenstock.disjUnion (h₁ : IsHenstock π₁) (h₂ : IsHenstock π₂)
(h : Disjoint π₁.iUnion π₂.iUnion) : IsHenstock (π₁.disjUnion π₂ h) := by |
refine fun J hJ => (Finset.mem_union.1 hJ).elim (fun hJ => ?_) fun hJ => ?_
· rw [disjUnion_tag_of_mem_left _ hJ]
exact h₁ _ hJ
· rw [disjUnion_tag_of_mem_right _ hJ]
exact h₂ _ hJ
|
/-
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 | 465 | 466 | theorem mem_iInf {ι} (p : ι → LieSubmodule R L M) {x} : (x ∈ ⨅ i, p i) ↔ ∀ i, x ∈ p i := by |
rw [← SetLike.mem_coe, iInf_coe, Set.mem_iInter]; rfl
|
/-
Copyright (c) 2020 Jannis Limperg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jannis Limperg
-/
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Range
#align_import data.list.indexes from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b830696... | Mathlib/Data/List/Indexes.lean | 182 | 188 | theorem mapIdx_eq_enum_map (l : List α) (f : ℕ → α → β) :
l.mapIdx f = l.enum.map (Function.uncurry f) := by |
rw [List.new_def_eq_old_def]
induction' l with hd tl hl generalizing f
· rfl
· rw [List.oldMapIdx, List.oldMapIdxCore, List.oldMapIdxCore_eq, hl]
simp [map, enum_eq_zip_range, map_uncurry_zip_eq_zipWith]
|
/-
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, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathli... | Mathlib/Data/Set/Lattice.lean | 1,999 | 2,002 | theorem iUnion_univ_pi {ι : α → Type*} (t : (a : α) → ι a → Set (π a)) :
⋃ x : (a : α) → ι a, pi univ (fun a => t a (x a)) = pi univ fun a => ⋃ j : ι a, t a j := by |
ext
simp [Classical.skolem]
|
/-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
/-!
# Lindelöf sets and Lindelöf spaces
## Mai... | Mathlib/Topology/Compactness/Lindelof.lean | 302 | 318 | theorem Set.Countable.isLindelof_biUnion {s : Set ι} {f : ι → Set X} (hs : s.Countable)
(hf : ∀ i ∈ s, IsLindelof (f i)) : IsLindelof (⋃ i ∈ s, f i) := by |
apply isLindelof_of_countable_subcover
intro i U hU hUcover
have hiU : ∀ i ∈ s, f i ⊆ ⋃ i, U i :=
fun _ is ↦ _root_.subset_trans (subset_biUnion_of_mem is) hUcover
have iSets := fun i is ↦ (hf i is).elim_countable_subcover U hU (hiU i is)
choose! r hr using iSets
use ⋃ i ∈ s, r i
constructor
· refi... |
/-
Copyright (c) 2018 Violeta Hernández Palacios, Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios, Mario Carneiro
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Exponential
#align_import set_th... | Mathlib/SetTheory/Ordinal/FixedPoint.lean | 128 | 134 | theorem apply_le_nfpFamily [hι : Nonempty ι] {f : ι → Ordinal → Ordinal} (H : ∀ i, IsNormal (f i))
{a b} : (∀ i, f i b ≤ nfpFamily.{u, v} f a) ↔ b ≤ nfpFamily.{u, v} f a := by |
refine ⟨fun h => ?_, fun h i => ?_⟩
· cases' hι with i
exact ((H i).self_le b).trans (h i)
rw [← nfpFamily_fp (H i)]
exact (H i).monotone h
|
/-
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.Data.Set.Pointwise.Interval
import Mathlib.LinearAlgebra.AffineSpace.Basic
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.Pi
import ... | Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean | 845 | 848 | theorem pi_ext_nonempty' [Nonempty ι] (h : ∀ i, f.comp (LinearMap.single i).toAffineMap =
g.comp (LinearMap.single i).toAffineMap) : f = g := by |
refine pi_ext_nonempty fun i x => ?_
convert AffineMap.congr_fun (h i) x
|
/-
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 | 150 | 151 | theorem mem_cylinder_comm (x y : ∀ n, E n) (n : ℕ) : y ∈ cylinder x n ↔ x ∈ cylinder y n := by |
simp [mem_cylinder_iff_eq, eq_comm]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.SmoothManifoldWithCorners
import Mathlib.Geometry.Manifold.LocalInvariantProperties
#align_import geometry.m... | Mathlib/Geometry/Manifold/ContMDiff/Defs.lean | 309 | 310 | theorem contMDiffOn_univ : ContMDiffOn I I' n f univ ↔ ContMDiff I I' n f := by |
simp only [ContMDiffOn, ContMDiff, contMDiffWithinAt_univ, forall_prop_of_true, mem_univ]
|
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
import Mathlib.Data.SetLike.Basic
#align_import data.sym.sym2 from "leanpro... | Mathlib/Data/Sym/Sym2.lean | 265 | 267 | theorem map_id : map (@id α) = id := by |
ext ⟨⟨x, y⟩⟩
rfl
|
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Equiv.Fin
#align_import data.fi... | Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean | 131 | 139 | theorem antidiagonalTuple_one (n : ℕ) : antidiagonalTuple 1 n = [![n]] := by |
simp_rw [antidiagonalTuple, antidiagonal, List.range_succ, List.map_append, List.map_singleton,
tsub_self, List.append_bind, List.bind_singleton, List.map_bind]
conv_rhs => rw [← List.nil_append [![n]]]
congr 1
simp_rw [List.bind_eq_nil, List.mem_range, List.map_eq_nil]
intro x hx
obtain ⟨m, rfl⟩ := Na... |
/-
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.Data.Bundle
import Mathlib.Data.Set.Image
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.Order.Basic
#align_import topology.f... | Mathlib/Topology/FiberBundle/Trivialization.lean | 765 | 768 | theorem frontier_preimage (e : Trivialization F proj) (s : Set B) :
e.source ∩ frontier (proj ⁻¹' s) = proj ⁻¹' (e.baseSet ∩ frontier s) := by |
rw [← (e.isImage_preimage_prod s).frontier.preimage_eq, frontier_prod_univ_eq,
(e.isImage_preimage_prod _).preimage_eq, e.source_eq, preimage_inter]
|
/-
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, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathli... | Mathlib/Data/Set/Lattice.lean | 2,177 | 2,181 | theorem sigmaToiUnion_surjective : Surjective (sigmaToiUnion t)
| ⟨b, hb⟩ =>
have : ∃ a, b ∈ t a := by | simpa using hb
let ⟨a, hb⟩ := this
⟨⟨a, b, hb⟩, rfl⟩
|
/-
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 | 397 | 403 | theorem pow_norm_pow_one_div_tendsto_nhds_spectralRadius (a : A) :
Tendsto (fun n : ℕ => ENNReal.ofReal (‖a ^ n‖ ^ (1 / n : ℝ))) atTop
(𝓝 (spectralRadius ℂ a)) := by |
convert pow_nnnorm_pow_one_div_tendsto_nhds_spectralRadius a using 1
ext1
rw [← ofReal_rpow_of_nonneg (norm_nonneg _) _, ← coe_nnnorm, coe_nnreal_eq]
exact one_div_nonneg.mpr (mod_cast zero_le _)
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 726 | 734 | theorem translationNumber_mul_of_commute {f g : CircleDeg1Lift} (h : Commute f g) :
τ (f * g) = τ f + τ g := by |
refine tendsto_nhds_unique ?_
(f.tendsto_translationNumber_aux.add g.tendsto_translationNumber_aux)
simp only [transnumAuxSeq, ← add_div]
refine (f * g).tendsto_translationNumber_of_dist_bounded_aux
(fun n ↦ (f ^ n) 0 + (g ^ n) 0) 1 fun n ↦ ?_
rw [h.mul_pow, dist_comm]
exact le_of_lt ((f ^ n).dist_ma... |
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Devon Tuma
-/
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.JacobsonIdeal
#align_import ring_theory.jacobson from "leanp... | Mathlib/RingTheory/Jacobson.lean | 685 | 690 | theorem quotient_mk_comp_C_isIntegral_of_jacobson {R : Type*} [CommRing R] [IsJacobson R]
(P : Ideal (MvPolynomial (Fin n) R)) [hP : P.IsMaximal] :
RingHom.IsIntegral (RingHom.comp (Quotient.mk P) (MvPolynomial.C)) := by |
change RingHom.IsIntegral (algebraMap R (MvPolynomial (Fin n) R ⧸ P))
apply quotient_mk_comp_C_isIntegral_of_jacobson'
infer_instance
|
/-
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.Group.Basic
import Mathlib.Algebra.GroupWithZero.NeZero
import Mathlib.Logic.Unique
#align_import algebra.group_with_zero.basic from "leanprov... | Mathlib/Algebra/GroupWithZero/Basic.lean | 262 | 262 | theorem right_eq_mul₀ (hb : b ≠ 0) : b = a * b ↔ a = 1 := by | rw [eq_comm, mul_eq_right₀ hb]
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mat... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 1,256 | 1,258 | theorem isBigOWith_zero_right_iff : (IsBigOWith c l f'' fun _x => (0 : F')) ↔ f'' =ᶠ[l] 0 := by |
simp only [IsBigOWith_def, exists_prop, true_and_iff, norm_zero, mul_zero,
norm_le_zero_iff, EventuallyEq, Pi.zero_apply]
|
/-
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 | 173 | 178 | theorem IsBoundedUnder.bddAbove_range_of_cofinite [Preorder β] [IsDirected β (· ≤ ·)] {f : α → β}
(hf : IsBoundedUnder (· ≤ ·) cofinite f) : BddAbove (range f) := by |
rcases hf with ⟨b, hb⟩
haveI : Nonempty β := ⟨b⟩
rw [← image_univ, ← union_compl_self { x | f x ≤ b }, image_union, bddAbove_union]
exact ⟨⟨b, forall_mem_image.2 fun x => id⟩, (hb.image f).bddAbove⟩
|
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Sqrt
#al... | Mathlib/Analysis/Seminorm.lean | 867 | 872 | theorem balanced_ball_zero (r : ℝ) : Balanced 𝕜 (ball p 0 r) := by |
rintro a ha x ⟨y, hy, hx⟩
rw [mem_ball_zero, ← hx, map_smul_eq_mul]
calc
_ ≤ p y := mul_le_of_le_one_left (apply_nonneg p _) ha
_ < r := by rwa [mem_ball_zero] at hy
|
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