Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
|---|---|---|---|---|---|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Shing Tak Lam, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathli... | Mathlib/Data/Nat/Digits.lean | 167 | 172 | theorem ofDigits_eq_foldr {α : Type*} [Semiring α] (b : α) (L : List ℕ) :
ofDigits b L = List.foldr (fun x y => ↑x + b * y) 0 L := by |
induction' L with d L ih
· rfl
· dsimp [ofDigits]
rw [ih]
|
/-
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.Interval.Set.Basic
import Mathlib.Data.Set.Lattice
import Mathlib.Data.SetLike.Basic
#align_import order.interval from "leanprover-community/mathlib... | Mathlib/Order/Interval/Basic.lean | 507 | 507 | theorem mem_pure : b ∈ pure a ↔ b = a := by | rw [← SetLike.mem_coe, coe_pure, mem_singleton_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.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducib... | Mathlib/Topology/Connected/Basic.lean | 678 | 683 | theorem connectedComponentIn_eq {x y : α} {F : Set α} (h : y ∈ connectedComponentIn F x) :
connectedComponentIn F x = connectedComponentIn F y := by |
have hx : x ∈ F := connectedComponentIn_nonempty_iff.mp ⟨y, h⟩
simp_rw [connectedComponentIn_eq_image hx] at h ⊢
obtain ⟨⟨y, hy⟩, h2y, rfl⟩ := h
simp_rw [connectedComponentIn_eq_image hy, connectedComponent_eq h2y]
|
/-
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
-/
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d... | Mathlib/Order/Interval/Set/Disjoint.lean | 122 | 123 | theorem iUnion_Ioo_right [NoMaxOrder α] (a : α) : ⋃ b, Ioo a b = Ioi a := by |
simp only [← Ioi_inter_Iio, ← inter_iUnion, iUnion_Iio, inter_univ]
|
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Data.Real.Sqrt
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedS... | Mathlib/Analysis/RCLike/Basic.lean | 337 | 339 | theorem conj_ofReal (r : ℝ) : conj (r : K) = (r : K) := by |
rw [ext_iff]
simp only [ofReal_im, conj_im, eq_self_iff_true, conj_re, and_self_iff, neg_zero]
|
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov, Hunter Monroe
-/
import Mathlib.Combinatorics.SimpleGraph.Init
import Mathlib.Data.Rel
import Mathlib... | Mathlib/Combinatorics/SimpleGraph/Basic.lean | 963 | 964 | theorem edgeSet_deleteEdges (s : Set (Sym2 V)) : (G.deleteEdges s).edgeSet = G.edgeSet \ s := by |
simp [deleteEdges]
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability... | Mathlib/Probability/Variance.lean | 100 | 103 | theorem _root_.MeasureTheory.Memℒp.ofReal_variance_eq [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
ENNReal.ofReal (variance X μ) = evariance X μ := by |
rw [variance, ENNReal.ofReal_toReal]
exact hX.evariance_lt_top.ne
|
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Algebra.Category.ModuleCat.Free
import Mathlib.Topology.Category.Profinite.CofilteredLimit
import Mathlib.Topology.Category.Profinite.Product
impor... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 378 | 392 | theorem eval_eq (l : Products I) (x : C) :
l.eval C x = if ∀ i, i ∈ l.val → (x.val i = true) then 1 else 0 := by |
change LocallyConstant.evalMonoidHom x (l.eval C) = _
rw [eval, map_list_prod]
split_ifs with h
· simp only [List.map_map]
apply List.prod_eq_one
simp only [List.mem_map, Function.comp_apply]
rintro _ ⟨i, hi, rfl⟩
exact if_pos (h i hi)
· simp only [List.map_map, List.prod_eq_zero_iff, List.me... |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
#align_import data.int.bitwise from "leanprover-community/mathlib"@"0743cc... | Mathlib/Data/Int/Bitwise.lean | 428 | 434 | theorem testBit_bitwise (f : Bool → Bool → Bool) (m n k) :
testBit (bitwise f m n) k = f (testBit m k) (testBit n k) := by |
cases m <;> cases n <;> simp only [testBit, bitwise, natBitwise]
· by_cases h : f false false <;> simp [h]
· by_cases h : f false true <;> simp [h]
· by_cases h : f true false <;> simp [h]
· by_cases h : f true true <;> simp [h]
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.LinearAlgebra.AffineSpace.Ordered
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib... | Mathlib/Topology/UrysohnsLemma.lean | 327 | 344 | theorem exists_continuous_zero_one_of_isClosed [NormalSpace X]
{s t : Set X} (hs : IsClosed s) (ht : IsClosed t)
(hd : Disjoint s t) : ∃ f : C(X, ℝ), EqOn f 0 s ∧ EqOn f 1 t ∧ ∀ x, f x ∈ Icc (0 : ℝ) 1 := by |
-- The actual proof is in the code above. Here we just repack it into the expected format.
let P : Set X → Prop := fun _ ↦ True
set c : Urysohns.CU P :=
{ C := s
U := tᶜ
P_C := trivial
closed_C := hs
open_U := ht.isOpen_compl
subset := disjoint_left.1 hd
hP := by
rintro c u c_clos... |
/-
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.HashMap.Basic
import Batteries.Data.Array.Lemmas
import Batteries.Data.Nat.Lemmas
namespace Batteries.HashMap
namespace Imp
attribute [-simp] ... | .lake/packages/batteries/Batteries/Data/HashMap/WF.lean | 324 | 371 | theorem WF.filterMap {α β γ} {f : α → β → Option γ} [BEq α] [Hashable α]
{m : Imp α β} (H : WF m) : WF (filterMap f m) := by |
let g₁ (l : AssocList α β) := l.toList.filterMap (fun x => (f x.1 x.2).map (x.1, ·))
have H1 (l n acc) : filterMap.go f acc l n =
(((g₁ l).reverse ++ acc.toList).toAssocList, ⟨n.1 + (g₁ l).length⟩) := by
induction l generalizing n acc with simp [filterMap.go, g₁, *]
| cons a b l => match f a b with
... |
/-
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.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.Antichain
import Mathlib.Order.SetNotation
#align_import data.set.intervals.ord_connect... | Mathlib/Order/Interval/Set/OrdConnected.lean | 353 | 355 | theorem ordConnected_iff_uIcc_subset_right (hx : x ∈ s) :
OrdConnected s ↔ ∀ ⦃y⦄, y ∈ s → [[y, x]] ⊆ s := by |
simp_rw [ordConnected_iff_uIcc_subset_left hx, uIcc_comm]
|
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Plus
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
#align_import category_theory.sites.sheafification from "leanprover-com... | Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean | 189 | 211 | theorem toPlus_apply {X : C} {P : Cᵒᵖ ⥤ D} (S : J.Cover X) (x : Meq P S) (I : S.Arrow) :
(J.toPlus P).app _ (x I) = (J.plusObj P).map I.f.op (mk x) := by |
dsimp only [toPlus, plusObj]
delta Cover.toMultiequalizer
dsimp [mk]
erw [← comp_apply]
rw [ι_colimMap_assoc, colimit.ι_pre, comp_apply, comp_apply]
dsimp only [Functor.op]
let e : (J.pullback I.f).obj (unop (op S)) ⟶ ⊤ := homOfLE (OrderTop.le_top _)
rw [← colimit.w _ e.op]
erw [comp_apply]
apply c... |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Tactic.GCongr
#align_import order.filter.archimedean fr... | Mathlib/Order/Filter/Archimedean.lean | 129 | 131 | theorem tendsto_ratCast_atBot_iff [LinearOrderedField R] [Archimedean R] {f : α → ℚ}
{l : Filter α} : Tendsto (fun n => (f n : R)) l atBot ↔ Tendsto f l atBot := by |
rw [← @Rat.comap_cast_atBot R, tendsto_comap_iff]; rfl
|
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Data.ZMod.Quotient
#align_import group_theory.complement from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
/-!
# Compl... | Mathlib/GroupTheory/Complement.lean | 216 | 227 | theorem mem_leftTransversals_iff_existsUnique_inv_mul_mem :
S ∈ leftTransversals T ↔ ∀ g : G, ∃! s : S, (s : G)⁻¹ * g ∈ T := by |
rw [leftTransversals, Set.mem_setOf_eq, isComplement_iff_existsUnique]
refine ⟨fun h g => ?_, fun h g => ?_⟩
· obtain ⟨x, h1, h2⟩ := h g
exact
⟨x.1, (congr_arg (· ∈ T) (eq_inv_mul_of_mul_eq h1)).mp x.2.2, fun y hy =>
(Prod.ext_iff.mp (h2 ⟨y, (↑y)⁻¹ * g, hy⟩ (mul_inv_cancel_left ↑y g))).1⟩
· o... |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
/-!
# Noncomputable... | Mathlib/Data/Set/Card.lean | 1,089 | 1,099 | theorem eq_insert_of_ncard_eq_succ {n : ℕ} (h : s.ncard = n + 1) :
∃ a t, a ∉ t ∧ insert a t = s ∧ t.ncard = n := by |
classical
have hsf := finite_of_ncard_pos (n.zero_lt_succ.trans_eq h.symm)
rw [ncard_eq_toFinset_card _ hsf, Finset.card_eq_succ] at h
obtain ⟨a, t, hat, hts, rfl⟩ := h
simp only [Finset.ext_iff, Finset.mem_insert, Finite.mem_toFinset] at hts
refine ⟨a, t, hat, ?_, ?_⟩
· simp only [Finset.mem_coe, ext_if... |
/-
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.Topology.Algebra.Polynomial
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Topology.UnitInterval
import Mathlib.Algebra.Star.Subalge... | Mathlib/Topology/ContinuousFunction/Polynomial.lean | 219 | 231 | theorem polynomialFunctions.eq_adjoin_X (s : Set R) :
polynomialFunctions s = Algebra.adjoin R {toContinuousMapOnAlgHom s X} := by |
refine le_antisymm ?_
(Algebra.adjoin_le fun _ h => ⟨X, trivial, (Set.mem_singleton_iff.1 h).symm⟩)
rintro - ⟨p, -, rfl⟩
rw [AlgHom.coe_toRingHom]
refine p.induction_on (fun r => ?_) (fun f g hf hg => ?_) fun n r hn => ?_
· rw [Polynomial.C_eq_algebraMap, AlgHomClass.commutes]
exact Subalgebra.algebr... |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
#align_import topology.uniform_space.equicontinuity from "leanprover-community/mathlib"@"f2ce6086... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 262 | 264 | theorem equicontinuous_finite [Finite ι] {F : ι → X → α} :
Equicontinuous F ↔ ∀ i, Continuous (F i) := by |
simp only [Equicontinuous, equicontinuousAt_finite, continuous_iff_continuousAt, @forall_swap ι]
|
/-
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.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducib... | Mathlib/Topology/Connected/Basic.lean | 413 | 421 | theorem IsPreconnected.subset_or_subset (hu : IsOpen u) (hv : IsOpen v) (huv : Disjoint u v)
(hsuv : s ⊆ u ∪ v) (hs : IsPreconnected s) : s ⊆ u ∨ s ⊆ v := by |
specialize hs u v hu hv hsuv
obtain hsu | hsu := (s ∩ u).eq_empty_or_nonempty
· exact Or.inr ((Set.disjoint_iff_inter_eq_empty.2 hsu).subset_right_of_subset_union hsuv)
· replace hs := mt (hs hsu)
simp_rw [Set.not_nonempty_iff_eq_empty, ← Set.disjoint_iff_inter_eq_empty,
disjoint_iff_inter_eq_empty.1... |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Analysis.InnerProductSpace.... | Mathlib/Analysis/Fourier/AddCircle.lean | 344 | 353 | theorem fourierCoeffOn_eq_integral {a b : ℝ} (f : ℝ → E) (n : ℤ) (hab : a < b) :
fourierCoeffOn hab f n =
(1 / (b - a)) • ∫ x in a..b, fourier (-n) (x : AddCircle (b - a)) • f x := by |
haveI := Fact.mk (by linarith : 0 < b - a)
rw [fourierCoeffOn, fourierCoeff_eq_intervalIntegral _ _ a, add_sub, add_sub_cancel_left]
congr 1
simp_rw [intervalIntegral.integral_of_le hab.le]
refine setIntegral_congr measurableSet_Ioc fun x hx => ?_
rw [liftIoc_coe_apply]
rwa [add_sub, add_sub_cancel_left]... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic fro... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 585 | 585 | theorem zero_sub (a : Ordinal) : 0 - a = 0 := by | rw [← Ordinal.le_zero]; apply sub_le_self
|
/-
Copyright (c) 2022 Hanting Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hanting Zhang
-/
import Mathlib.Topology.MetricSpace.Antilipschitz
import Mathlib.Topology.MetricSpace.Isometry
import Mathlib.Topology.MetricSpace.Lipschitz
import Mathlib.Data.FunLike... | Mathlib/Topology/MetricSpace/Dilation.lean | 315 | 321 | theorem ratio_id : ratio (Dilation.id α) = 1 := by |
by_cases h : ∀ x y : α, edist x y = 0 ∨ edist x y = ∞
· rw [ratio, if_pos h]
· push_neg at h
rcases h with ⟨x, y, hne⟩
refine (ratio_unique hne.1 hne.2 ?_).symm
simp
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.RowCol
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
#align_import data.matri... | Mathlib/Data/Matrix/Notation.lean | 481 | 493 | theorem mul_fin_three [AddCommMonoid α] [Mul α]
(a₁₁ a₁₂ a₁₃ a₂₁ a₂₂ a₂₃ a₃₁ a₃₂ a₃₃ b₁₁ b₁₂ b₁₃ b₂₁ b₂₂ b₂₃ b₃₁ b₃₂ b₃₃ : α) :
!![a₁₁, a₁₂, a₁₃;
a₂₁, a₂₂, a₂₃;
a₃₁, a₃₂, a₃₃] * !![b₁₁, b₁₂, b₁₃;
b₂₁, b₂₂, b₂₃;
b₃₁, b₃₂, b₃₃] =
!![a₁₁*b₁₁ +... |
ext i j
fin_cases i <;> fin_cases j
<;> simp [Matrix.mul_apply, dotProduct, Fin.sum_univ_succ, ← add_assoc]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Scott Morrison, Jakob von Raumer
-/
import Mathlib.Algebra.Category.ModuleCat.Basic
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.CategoryTheory.Monoid... | Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean | 165 | 176 | theorem leftUnitor_naturality {M N : ModuleCat R} (f : M ⟶ N) :
tensorHom (𝟙 (ModuleCat.of R R)) f ≫ (leftUnitor N).hom = (leftUnitor M).hom ≫ f := by |
-- Porting note (#11041): broken ext
apply TensorProduct.ext
apply LinearMap.ext_ring
apply LinearMap.ext; intro x
-- Porting note (#10934): used to be dsimp
change ((leftUnitor N).hom) ((tensorHom (𝟙 (of R R)) f) ((1 : R) ⊗ₜ[R] x)) =
f (((leftUnitor M).hom) (1 ⊗ₜ[R] x))
erw [TensorProduct.lid_tmul,... |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability... | Mathlib/Probability/Variance.lean | 261 | 275 | theorem meas_ge_le_evariance_div_sq {X : Ω → ℝ} (hX : AEStronglyMeasurable X ℙ) {c : ℝ≥0}
(hc : c ≠ 0) : ℙ {ω | ↑c ≤ |X ω - 𝔼[X]|} ≤ eVar[X] / c ^ 2 := by |
have A : (c : ℝ≥0∞) ≠ 0 := by rwa [Ne, ENNReal.coe_eq_zero]
have B : AEStronglyMeasurable (fun _ : Ω => 𝔼[X]) ℙ := aestronglyMeasurable_const
convert meas_ge_le_mul_pow_snorm ℙ two_ne_zero ENNReal.two_ne_top (hX.sub B) A using 1
· congr
simp only [Pi.sub_apply, ENNReal.coe_le_coe, ← Real.norm_eq_abs, ← co... |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.GroupWithZero.Indicator
import Mathlib.Tactic.FinCases
import Mathlib.Topology.Sets.Closeds
#align_import topology.locally_constant.basic from... | Mathlib/Topology/LocallyConstant/Basic.lean | 398 | 402 | theorem exists_eq_const [PreconnectedSpace X] [Nonempty Y] (f : LocallyConstant X Y) :
∃ y, f = const X y := by |
rcases Classical.em (Nonempty X) with (⟨⟨x⟩⟩ | hX)
· exact ⟨f x, f.eq_const x⟩
· exact ⟨Classical.arbitrary Y, ext fun x => (hX ⟨x⟩).elim⟩
|
/-
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.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Set.Subsingle... | Mathlib/Combinatorics/Enumerative/Composition.lean | 267 | 267 | theorem card_boundaries_eq_succ_length : c.boundaries.card = c.length + 1 := by | simp [boundaries]
|
/-
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 | 1,515 | 1,519 | theorem SimpleFunc.integral_eq_integral (f : α →ₛ E) (hfi : Integrable f μ) :
f.integral μ = ∫ x, f x ∂μ := by |
rw [MeasureTheory.integral_eq f hfi, ← L1.SimpleFunc.toLp_one_eq_toL1,
L1.SimpleFunc.integral_L1_eq_integral, L1.SimpleFunc.integral_eq_integral]
exact SimpleFunc.integral_congr hfi (Lp.simpleFunc.toSimpleFunc_toLp _ _).symm
|
/-
Copyright (c) 2017 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Mario Carneiro
-/
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Sqrt
#align_import data.complex.basic from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed... | Mathlib/Data/Complex/Abs.lean | 34 | 34 | theorem abs_conj (z : ℂ) : (abs conj z) = abs z := by | simp
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, 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 | 507 | 508 | theorem cosh_mul_I : cosh (x * I) = cos x := by |
rw [← mul_right_inj' (two_ne_zero' ℂ), two_cosh, two_cos, neg_mul_eq_neg_mul]
|
/-
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 | 625 | 628 | theorem lineMap_apply_one_sub (p₀ p₁ : P1) (c : k) : lineMap p₀ p₁ (1 - c) = lineMap p₁ p₀ c := by |
rw [lineMap_symm p₀, comp_apply]
congr
simp [lineMap_apply]
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.AlgebraicGeometry.Pullbacks
import Mathlib.CategoryTheory.MorphismProperty.Limits
import Mathlib.Data.List.TFAE... | Mathlib/AlgebraicGeometry/Morphisms/Basic.lean | 434 | 446 | theorem IsLocal.targetAffineLocallyPullbackFstOfRightOfStableUnderBaseChange
{P : AffineTargetMorphismProperty} (hP : P.IsLocal) (hP' : P.StableUnderBaseChange)
{X Y S : Scheme} (f : X ⟶ S) (g : Y ⟶ S) [IsAffine S] (H : P g) :
targetAffineLocally P (pullback.fst : pullback f g ⟶ X) := by |
-- Porting note: rewrite `(hP.affine_openCover_TFAE ...).out 0 1` doesn't work
have h01 := (hP.affine_openCover_TFAE (pullback.fst : pullback f g ⟶ X)).out 0 1
rw [h01]
use X.affineCover, inferInstance
intro i
let e := pullbackSymmetry _ _ ≪≫ pullbackRightPullbackFstIso f g (X.affineCover.map i)
have : e... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp
-/
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
i... | Mathlib/LinearAlgebra/Basis.lean | 897 | 902 | theorem Module.card_fintype [Fintype ι] (b : Basis ι R M) [Fintype R] [Fintype M] :
card M = card R ^ card ι := by |
classical
calc
card M = card (ι → R) := card_congr b.equivFun.toEquiv
_ = card R ^ card ι := card_fun
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib... | Mathlib/Algebra/Polynomial/RingDivision.lean | 775 | 820 | theorem exists_multiset_roots [DecidableEq R] :
∀ {p : R[X]} (_ : p ≠ 0), ∃ s : Multiset R,
(Multiset.card s : WithBot ℕ) ≤ degree p ∧ ∀ a, s.count a = rootMultiplicity a p
| p, hp =>
haveI := Classical.propDecidable (∃ x, IsRoot p x)
if h : ∃ x, IsRoot p x then
let ⟨x, hx⟩ := h
have hpd... |
rw [degree_X_sub_C, degree_eq_natDegree hp]
rw [degree_eq_natDegree hp] at hpd
exact WithBot.coe_le_coe.2 (WithBot.coe_lt_coe.1 hpd)
have hdiv0 : p /ₘ (X - C x) ≠ 0 :=
mt (divByMonic_eq_zero_iff (monic_X_sub_C x)).1 <| not_lt.2 hdeg
⟨x ::ₘ t,
calc
(card (x ... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Ring.Prod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Tactic.FinCases
#align_import data.zmod.basic from "leanprover-community/mathli... | Mathlib/Data/ZMod/Basic.lean | 84 | 88 | theorem val_natCast {n : ℕ} (a : ℕ) : (a : ZMod n).val = a % n := by |
cases n
· rw [Nat.mod_zero]
exact Int.natAbs_ofNat a
· apply Fin.val_natCast
|
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.FieldTheory.Perfect
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.AnnihilatingPolynomial
import Mathlib.Order.CompleteSublattic... | Mathlib/LinearAlgebra/Semisimple.lean | 119 | 132 | theorem isSemisimple_of_squarefree_aeval_eq_zero {p : K[X]}
(hp : Squarefree p) (hpf : aeval f p = 0) : f.IsSemisimple := by |
rw [← RingHom.mem_ker, ← AEval.annihilator_eq_ker_aeval (M := M), mem_annihilator,
← IsTorsionBy, ← isTorsionBySet_singleton_iff, isTorsionBySet_iff_is_torsion_by_span] at hpf
let R := K[X] ⧸ Ideal.span {p}
have : IsReduced R :=
(Ideal.isRadical_iff_quotient_reduced _).mp (isRadical_iff_span_singleton.... |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Data.Option.Defs
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Sigma.Basic
import Mathlib... | Mathlib/Logic/Equiv/Basic.lean | 1,604 | 1,606 | theorem swapCore_self (r a : α) : swapCore a a r = r := by |
unfold swapCore
split_ifs <;> simp [*]
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calc... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 166 | 169 | theorem comp_fderiv {f : G → E} {x : G} :
fderiv 𝕜 (iso ∘ f) x = (iso : E →L[𝕜] F).comp (fderiv 𝕜 f x) := by |
rw [← fderivWithin_univ, ← fderivWithin_univ]
exact iso.comp_fderivWithin uniqueDiffWithinAt_univ
|
/-
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 | 141 | 152 | theorem add_algebraMap {B : Type*} [CommRing B] [Algebra A B] {x : B} (hx : IsIntegral A x)
(a : A) : minpoly A (x + algebraMap A B a) = (minpoly A x).comp (X - C a) := by |
refine (minpoly.unique _ _ ((minpoly.monic hx).comp_X_sub_C _) ?_ fun q qmo hq => ?_).symm
· simp [aeval_comp]
· have : (Polynomial.aeval x) (q.comp (X + C a)) = 0 := by simpa [aeval_comp] using hq
have H := minpoly.min A x (qmo.comp_X_add_C _) this
rw [degree_eq_natDegree qmo.ne_zero,
degree_eq_na... |
/-
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 | 506 | 512 | theorem StrictConvexOn.add_convexOn (hf : StrictConvexOn 𝕜 s f) (hg : ConvexOn 𝕜 s g) :
StrictConvexOn 𝕜 s (f + g) :=
⟨hf.1, fun x hx y hy hxy a b ha hb hab =>
calc
f (a • x + b • y) + g (a • x + b • y) < a • f x + b • f y + (a • g x + b • g y) :=
add_lt_add_of_lt_of_le (hf.2 hx hy hxy ha hb ... | rw [smul_add, smul_add, add_add_add_comm]⟩
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Nor... | Mathlib/Analysis/Convex/Normed.lean | 66 | 67 | theorem convex_closedBall (a : E) (r : ℝ) : Convex ℝ (Metric.closedBall a r) := by |
simpa only [Metric.closedBall, sep_univ] using (convexOn_univ_dist a).convex_le r
|
/-
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 | 392 | 393 | theorem sqrt_div {x : ℝ} (hx : 0 ≤ x) (y : ℝ) : √(x / y) = √x / √y := by |
rw [division_def, sqrt_mul hx, sqrt_inv, division_def]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | Mathlib/Topology/Sheaves/Stalks.lean | 419 | 425 | theorem germ_eq (F : X.Presheaf C) {U V : Opens X} (x : X) (mU : x ∈ U) (mV : x ∈ V)
(s : F.obj (op U)) (t : F.obj (op V)) (h : germ F ⟨x, mU⟩ s = germ F ⟨x, mV⟩ t) :
∃ (W : Opens X) (_m : x ∈ W) (iU : W ⟶ U) (iV : W ⟶ V), F.map iU.op s = F.map iV.op t := by |
obtain ⟨W, iU, iV, e⟩ :=
(Types.FilteredColimit.isColimit_eq_iff.{v, v} _
(isColimitOfPreserves _ (colimit.isColimit ((OpenNhds.inclusion x).op ⋙ F)))).mp h
exact ⟨(unop W).1, (unop W).2, iU.unop, iV.unop, e⟩
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Rémy Degenne
-/
import Mathlib.Probability.Process.Adapted
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import probability.process.stopping from "leanp... | Mathlib/Probability/Process/Stopping.lean | 234 | 239 | theorem isStoppingTime_of_measurableSet_eq [Preorder ι] [Countable ι] {f : Filtration ι m}
{τ : Ω → ι} (hτ : ∀ i, MeasurableSet[f i] {ω | τ ω = i}) : IsStoppingTime f τ := by |
intro i
rw [show {ω | τ ω ≤ i} = ⋃ k ≤ i, {ω | τ ω = k} by ext; simp]
refine MeasurableSet.biUnion (Set.to_countable _) fun k hk => ?_
exact f.mono hk _ (hτ k)
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kevin Buzzard, Scott Morrison, Johan Commelin, Chris Hughes,
Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Hom.De... | Mathlib/Algebra/Group/Hom/Basic.lean | 110 | 113 | theorem comp_mul [Mul M] [CommSemigroup N] [CommSemigroup P] (g : N →ₙ* P) (f₁ f₂ : M →ₙ* N) :
g.comp (f₁ * f₂) = g.comp f₁ * g.comp f₂ := by |
ext
simp only [mul_apply, Function.comp_apply, map_mul, coe_comp]
|
/-
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 | 600 | 611 | theorem image_projection_prod {ι : Type*} {α : ι → Type*} {v : ∀ i : ι, Set (α i)}
(hv : (pi univ v).Nonempty) (i : ι) :
((fun x : ∀ i : ι, α i => x i) '' ⋂ k, (fun x : ∀ j : ι, α j => x k) ⁻¹' v k) = v i := by |
classical
apply Subset.antisymm
· simp [iInter_subset]
· intro y y_in
simp only [mem_image, mem_iInter, mem_preimage]
rcases hv with ⟨z, hz⟩
refine ⟨Function.update z i y, ?_, update_same i y z⟩
rw [@forall_update_iff ι α _ z i y fun i t => t ∈ v i]
exact ⟨y_in, fun j _ => b... |
/-
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.Algebra.Field.Opposite
import Mathlib.Algebra.Group.Subgroup.ZPowers
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Rin... | Mathlib/Algebra/Periodic.lean | 419 | 422 | theorem Antiperiodic.odd_zsmul_antiperiodic [AddGroup α] [InvolutiveNeg β] (h : Antiperiodic f c)
(n : ℤ) : Antiperiodic f ((2 * n + 1) • c) := by |
intro x
rw [add_zsmul, one_zsmul, ← add_assoc, h, h.even_zsmul_periodic]
|
/-
Copyright (c) 2021 Apurva Nakade. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Apurva Nakade
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Localization.Basic
import... | Mathlib/SetTheory/Surreal/Dyadic.lean | 124 | 156 | theorem add_powHalf_succ_self_eq_powHalf (n) : powHalf (n + 1) + powHalf (n + 1) ≈ powHalf n := by |
induction' n using Nat.strong_induction_on with n hn
constructor <;> rw [le_iff_forall_lf] <;> constructor
· rintro (⟨⟨⟩⟩ | ⟨⟨⟩⟩) <;> apply lf_of_lt
· calc
0 + powHalf n.succ ≈ powHalf n.succ := zero_add_equiv _
_ < powHalf n := powHalf_succ_lt_powHalf n
· calc
powHalf n.succ + 0 ... |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
import Mathlib.Algebra.Group.Defs
#align_import data.part from "lean... | Mathlib/Data/Part.lean | 800 | 800 | theorem some_mod_some [Mod α] (a b : α) : some a % some b = some (a % b) := by | simp [mod_def]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Ring.Prod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Tactic.FinCases
#align_import data.zmod.basic from "leanprover-community/mathli... | Mathlib/Data/ZMod/Basic.lean | 684 | 688 | theorem ker_intCastAddHom (n : ℕ) :
(Int.castAddHom (ZMod n)).ker = AddSubgroup.zmultiples (n : ℤ) := by |
ext
rw [Int.mem_zmultiples_iff, AddMonoidHom.mem_ker, Int.coe_castAddHom,
intCast_zmod_eq_zero_iff_dvd]
|
/-
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, Yaël Dillies
-/
import Mathlib.Data.Finset.NAry
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Set.Pointwise.Finite
import Mathlib.Data.Set.Pointwise.SMul
... | Mathlib/Data/Finset/Pointwise.lean | 2,102 | 2,103 | theorem mem_inv_smul_finset_iff : b ∈ a⁻¹ • s ↔ a • b ∈ s := by |
rw [← smul_mem_smul_finset_iff a, smul_inv_smul]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algeb... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 639 | 642 | theorem cos_lt_cos_of_nonneg_of_le_pi {x y : ℝ} (hx₁ : 0 ≤ x) (hy₂ : y ≤ π) (hxy : x < y) :
cos y < cos x := by |
rw [← sin_pi_div_two_sub, ← sin_pi_div_two_sub]
apply sin_lt_sin_of_lt_of_le_pi_div_two <;> linarith
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Moritz Doll
-/
import Mathlib.LinearAlgebra.FinsuppVectorSpace
import Mathlib.LinearAlgebra.Matrix.Basis
import Mathlib.LinearAlgebra.Matrix.Nondegenerate
import... | Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean | 618 | 621 | theorem mem_selfAdjointMatricesSubmodule :
A₁ ∈ selfAdjointMatricesSubmodule J ↔ J.IsSelfAdjoint A₁ := by |
erw [mem_pairSelfAdjointMatricesSubmodule]
rfl
|
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 1,669 | 1,674 | theorem edges_map : (p.map f).edges = p.edges.map (Sym2.map f) := by |
induction p with
| nil => rfl
| cons _ _ ih =>
simp only [Walk.map_cons, edges_cons, List.map_cons, Sym2.map_pair_eq, List.cons.injEq,
true_and, ih]
|
/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Biproducts.lean | 250 | 254 | theorem biproductComparison'_comp_biproductComparison :
biproductComparison' F f ≫ biproductComparison F f = 𝟙 (⨁ F.obj ∘ f) := by |
classical
ext
simp [biproduct.ι_π, ← Functor.map_comp, eqToHom_map]
|
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Nat.Defs
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.Co... | Mathlib/Data/Fin/Basic.lean | 1,238 | 1,240 | theorem rev_pred {i : Fin (n + 1)} (h : i ≠ 0) (h' := rev_ne_iff.mpr ((rev_last _).symm ▸ h)) :
rev (pred i h) = castPred (rev i) h' := by |
rw [← castSucc_inj, castSucc_castPred, ← rev_succ, succ_pred]
|
/-
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 | 641 | 642 | theorem Disjoint.sup_sdiff_cancel_left (h : Disjoint a b) : (a ⊔ b) \ a = b := by |
rw [sup_sdiff, sdiff_self, bot_sup_eq, h.sdiff_eq_right]
|
/-
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.Function.ConditionalExpectation.CondexpL2
#align_import measure_theory.function.conditional_expectation.condexp_L1 from "leanprover-communit... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 246 | 254 | theorem condexpIndL1_disjoint_union (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ∞)
(hμt : μ t ≠ ∞) (hst : s ∩ t = ∅) (x : G) :
condexpIndL1 hm μ (s ∪ t) x = condexpIndL1 hm μ s x + condexpIndL1 hm μ t x := by |
have hμst : μ (s ∪ t) ≠ ∞ :=
((measure_union_le s t).trans_lt (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne
rw [condexpIndL1_of_measurableSet_of_measure_ne_top hs hμs x,
condexpIndL1_of_measurableSet_of_measure_ne_top ht hμt x,
condexpIndL1_of_measurableSet_of_measure_ne_top (hs.union... |
/-
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.Topology.Order.Basic
/-!
# Set neighborhoods of intervals
In this file we prove basic theorems about `𝓝ˢ s`,
where `s` is one of the intervals
`Se... | Mathlib/Topology/Order/NhdsSet.lean | 41 | 42 | theorem nhdsSet_Ico (h : a < b) : 𝓝ˢ (Ico a b) = 𝓝 a ⊔ 𝓟 (Ioo a b) := by |
rw [← Ioo_insert_left h, nhdsSet_insert, nhdsSet_Ioo]
|
/-
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.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calcul... | Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 368 | 372 | theorem Filter.EventuallyEq.lineDerivWithin_eq (hs : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) :
lineDerivWithin 𝕜 f₁ s x v = lineDerivWithin 𝕜 f s x v := by |
apply derivWithin_eq ?_ (by simpa using hx)
have A : Continuous (fun (t : 𝕜) ↦ x + t • v) := by continuity
exact A.continuousWithinAt.preimage_mem_nhdsWithin'' hs (by simp)
|
/-
Copyright (c) 2022 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Heather Macbeth
-/
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.MeasureTheory.Function.SimpleFuncDense
#align_import measure_theory.func... | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 625 | 628 | theorem zero_toSimpleFunc : toSimpleFunc (0 : Lp.simpleFunc E p μ) =ᵐ[μ] 0 := by |
filter_upwards [toSimpleFunc_eq_toFun (0 : Lp.simpleFunc E p μ),
Lp.coeFn_zero E 1 μ] with _ h₁ _
rwa [h₁]
|
/-
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.Data.Num.Lemmas
import Mathlib.Data.Nat.Prime
import Mathlib.Tactic.Ring
#align_import data.num.prime from "leanprover-community/mathlib"@"58581d0fe52... | Mathlib/Data/Num/Prime.lean | 65 | 83 | theorem minFac_to_nat (n : PosNum) : (minFac n : ℕ) = Nat.minFac n := by |
cases' n with n
· rfl
· rw [minFac, Nat.minFac_eq, if_neg]
swap
· simp
rw [minFacAux_to_nat]
· rfl
simp only [cast_one, cast_bit1]
unfold _root_.bit1 _root_.bit0
rw [Nat.sqrt_lt]
calc
(n : ℕ) + (n : ℕ) + 1 ≤ (n : ℕ) + (n : ℕ) + (n : ℕ) := by simp
_ = (n : ℕ) * (1 + 1 +... |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Comp
#align_import analysis.calculus.deriv.inv from "leanpro... | Mathlib/Analysis/Calculus/Deriv/Inv.lean | 118 | 121 | theorem fderivWithin_inv (x_ne_zero : x ≠ 0) (hxs : UniqueDiffWithinAt 𝕜 s x) :
fderivWithin 𝕜 (fun x => x⁻¹) s x = smulRight (1 : 𝕜 →L[𝕜] 𝕜) (-(x ^ 2)⁻¹) := by |
rw [DifferentiableAt.fderivWithin (differentiableAt_inv.2 x_ne_zero) hxs]
exact fderiv_inv
|
/-
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.Compactness.SigmaCompact
import Mathlib.Topology.Connected.TotallyDisconnected
import Mathlib.Topology.Inseparable
#align_imp... | Mathlib/Topology/Separation.lean | 1,036 | 1,038 | theorem isClosed_setOf_specializes : IsClosed { p : X × X | p.1 ⤳ p.2 } := by |
simp only [← isOpen_compl_iff, compl_setOf, ← disjoint_nhds_nhds_iff_not_specializes,
isOpen_setOf_disjoint_nhds_nhds]
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Covering.Bes... | Mathlib/MeasureTheory/Function/Jacobian.lean | 109 | 243 | theorem exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt [SecondCountableTopology F]
(f : E → F) (s : Set E) (f' : E → E →L[ℝ] F) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
(r : (E →L[ℝ] F) → ℝ≥0) (rpos : ∀ A, r A ≠ 0) :
∃ (t : ℕ → Set E) (A : ℕ → E →L[ℝ] F),
(∀ n, IsClosed (t n)) ∧
... |
/- Choose countably many linear maps `f' z`. For every such map, if `f` has a derivative at `x`
close enough to `f' z`, then `f y - f x` is well approximated by `f' z (y - x)` for `y` close
enough to `x`, say on a ball of radius `r` (or even `u n` for some `n`, where `u` is a fixed
sequence tending to `0... |
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Alex Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Gr... | Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 236 | 240 | theorem sum_restrict_of_ac (h : IsFundamentalDomain G s μ) (hν : ν ≪ μ) :
(sum fun g : G => ν.restrict (g • s)) = ν := by |
rw [← restrict_iUnion_ae (h.aedisjoint.mono fun i j h => hν h) fun g =>
(h.nullMeasurableSet_smul g).mono_ac hν,
restrict_congr_set (hν h.iUnion_smul_ae_eq), restrict_univ]
|
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.Algebra.CharP.Reduced
/-!
# Perfect fields and rings
In this f... | Mathlib/FieldTheory/Perfect.lean | 336 | 337 | theorem roots_expand_map_frobenius : (expand R p f).roots.map (frobenius R p) = p • f.roots := by |
simp [roots_expand, Multiset.map_nsmul]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.OuterMeasure.Induced
import Mathlib.MeasureTheory.OuterMeasure.AE
import Mathlib.Order.Filter.CountableInter
#align_impo... | Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean | 165 | 166 | theorem measure_eq_iInf (s : Set α) : μ s = ⨅ (t) (_ : s ⊆ t) (_ : MeasurableSet t), μ t := by |
rw [measure_eq_trim, OuterMeasure.trim_eq_iInf, μ.coe_toOuterMeasure]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.Lattice
#align_import category_theory.subobject.limits from "leanprover-community/mathlib"@"956af7c76589f444f2e... | Mathlib/CategoryTheory/Subobject/Limits.lean | 350 | 353 | theorem factorThruImageSubobject_comp_self_assoc {W W' : C} (k : W ⟶ W') (k' : W' ⟶ X) (h) :
(imageSubobject f).factorThru (k ≫ k' ≫ f) h = k ≫ k' ≫ factorThruImageSubobject f := by |
ext
simp
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fin.Tuple.Basic
import Mathlib.Data.List.Join
#align_import data.list.of_fn from "leanprover-community/mathlib"@"bf27744463e9620ca4e4ebe951fe8353... | Mathlib/Data/List/OfFn.lean | 50 | 54 | theorem get_ofFn_go {n} (f : Fin n → α) (i j h) (k) (hk) :
get (ofFn.go f i j h) ⟨k, hk⟩ = f ⟨j + k, by simp at hk; omega⟩ := by |
let i+1 := i
cases k <;> simp [ofFn.go, get_ofFn_go (i := i)]
congr 2; omega
|
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
#align_import probability.kernel.cond_distrib from "leanprover-community/math... | Mathlib/Probability/Kernel/CondDistrib.lean | 88 | 93 | theorem _root_.MeasureTheory.AEStronglyMeasurable.ae_integrable_condDistrib_map_iff
(hY : AEMeasurable Y μ) (hf : AEStronglyMeasurable f (μ.map fun a => (X a, Y a))) :
(∀ᵐ a ∂μ.map X, Integrable (fun ω => f (a, ω)) (condDistrib Y X μ a)) ∧
Integrable (fun a => ∫ ω, ‖f (a, ω)‖ ∂condDistrib Y X μ a) (μ.map ... |
rw [condDistrib, ← hf.ae_integrable_condKernel_iff, Measure.fst_map_prod_mk₀ hY]
|
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.... | Mathlib/Algebra/GeomSum.lean | 347 | 349 | theorem geom_sum_Ico_mul_neg [Ring α] (x : α) {m n : ℕ} (hmn : m ≤ n) :
(∑ i ∈ Finset.Ico m n, x ^ i) * (1 - x) = x ^ m - x ^ n := by |
rw [sum_Ico_eq_sub _ hmn, sub_mul, geom_sum_mul_neg, geom_sum_mul_neg, sub_sub_sub_cancel_left]
|
/-
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 | 445 | 457 | theorem LinearOrder.strictConvexOn_of_lt (hs : Convex 𝕜 s)
(hf : ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 →
f (a • x + b • y) < a • f x + b • f y) :
StrictConvexOn 𝕜 s f := by |
refine ⟨hs, fun x hx y hy hxy a b ha hb hab => ?_⟩
-- Porting note: without clearing the stray variables, `wlog` gives a bad term.
-- See https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/wlog.20.2316495
clear! α F ι
wlog h : x < y
· rw [add_comm (a • x), add_comm (a • f x)]
rw [add... |
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Limits.Products
#align_import topology.category.Top.limits.pullbacks from "leanp... | Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | 447 | 452 | theorem coinduced_of_isColimit {F : J ⥤ TopCat.{max v u}} (c : Cocone F) (hc : IsColimit c) :
c.pt.str = ⨆ j, (F.obj j).str.coinduced (c.ι.app j) := by |
let homeo := homeoOfIso (hc.coconePointUniqueUpToIso (colimitCoconeIsColimit F))
ext
refine homeo.symm.isOpen_preimage.symm.trans (Iff.trans ?_ isOpen_iSup_iff.symm)
exact isOpen_iSup_iff
|
/-
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 | 717 | 720 | theorem HasBasis.liminf_eq_sSup_univ_of_empty {f : ι → α} {v : Filter ι}
{p : ι' → Prop} {s : ι' → Set ι} (hv : v.HasBasis p s) (i : ι') (hi : p i) (h'i : s i = ∅) :
liminf f v = sSup univ := by |
simp [hv.eq_bot_iff.2 ⟨i, hi, h'i⟩, liminf_eq]
|
/-
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 | 172 | 174 | theorem isSuccLimitRecOn_limit (hs : ∀ a, ¬IsMax a → C (succ a)) (hl : ∀ a, IsSuccLimit a → C a)
(hb : IsSuccLimit b) : @isSuccLimitRecOn α _ _ C b hs hl = hl b hb := by |
classical exact dif_pos hb
|
/-
Copyright (c) 2024 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.SpecificLimits.Normed
/-!
# Abel's limit theorem
If a real or complex power series for a function has radius o... | Mathlib/Analysis/Complex/AbelLimit.lean | 47 | 54 | theorem stolzSet_empty {M : ℝ} (hM : M ≤ 1) : stolzSet M = ∅ := by |
ext z
rw [stolzSet, Set.mem_setOf, Set.mem_empty_iff_false, iff_false, not_and, not_lt, ← sub_pos]
intro zn
calc
_ ≤ 1 * (1 - ‖z‖) := mul_le_mul_of_nonneg_right hM zn.le
_ = ‖(1 : ℂ)‖ - ‖z‖ := by rw [one_mul, norm_one]
_ ≤ _ := norm_sub_norm_le _ _
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Sophie Morel, Yury Kudryashov
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Logic.Embedding.Basic
import Mathlib.Data.Fintype.Car... | Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean | 205 | 235 | theorem norm_image_sub_le_of_bound {C : ℝ} (hC : 0 ≤ C) (H : ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖)
(m₁ m₂ : ∀ i, E i) :
‖f m₁ - f m₂‖ ≤ C * Fintype.card ι * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) * ‖m₁ - m₂‖ := by |
classical
have A :
∀ i : ι,
∏ j, (if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) ≤
‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) := by
intro i
calc
∏ j, (if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) ≤
∏ j : ι, Function.update (fun _ => max ‖m₁‖ ‖m₂‖) i ‖m₁ -... |
/-
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 | 56 | 67 | theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by |
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
rw [eq_top_iff]
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (me... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Set
import Mathlib.Data.Nat.Set
import Mathlib.Data.Set.Prod
import Mathlib.Data.ULift
import Mathlib.Order.Bounds.Basic
import Mathlib.Order... | Mathlib/Order/CompleteLattice.lean | 1,752 | 1,755 | theorem iSup_apply {α : Type*} {β : α → Type*} {ι : Sort*} [∀ i, SupSet (β i)] {f : ι → ∀ a, β a}
{a : α} : (⨆ i, f i) a = ⨆ i, f i a := by |
rw [iSup, sSup_apply, iSup, iSup, ← image_eq_range (fun f : ∀ i, β i => f a) (range f), ←
range_comp]; rfl
|
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Algebra.Order.Floor
import Mathlib.Topology.Algebra.Order.Group
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.floor fro... | Mathlib/Topology/Algebra/Order/Floor.lean | 108 | 110 | theorem tendsto_ceil_right_pure_add_one (n : ℤ) :
Tendsto (ceil : α → ℤ) (𝓝[>] n) (pure (n + 1)) := by |
simpa only [floor_intCast] using tendsto_ceil_right_pure_floor_add_one (n : α)
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Alex Kontorovich
-/
import Mathlib.Order.Filter.Bases
#align_import order.filter.pi from "leanprover-community/mathlib"@"ce64cd319bb6b3e82f31c2d38e79080d377be4... | Mathlib/Order/Filter/Pi.lean | 80 | 88 | theorem mem_pi {s : Set (∀ i, α i)} :
s ∈ pi f ↔ ∃ I : Set ι, I.Finite ∧ ∃ t : ∀ i, Set (α i), (∀ i, t i ∈ f i) ∧ I.pi t ⊆ s := by |
constructor
· simp only [pi, mem_iInf', mem_comap, pi_def]
rintro ⟨I, If, V, hVf, -, rfl, -⟩
choose t htf htV using hVf
exact ⟨I, If, t, htf, iInter₂_mono fun i _ => htV i⟩
· rintro ⟨I, If, t, htf, hts⟩
exact mem_of_superset (pi_mem_pi If fun i _ => htf i) hts
|
/-
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 | 236 | 251 | theorem card_sphere_le_rothNumberNat (n d k : ℕ) :
(sphere n d k).card ≤ rothNumberNat ((2 * d - 1) ^ n) := by |
cases n
· dsimp; refine (card_le_univ _).trans_eq ?_; rfl
cases d
· simp
apply threeAPFree_image_sphere.le_rothNumberNat _ _ (card_image_of_injOn _)
· intro; assumption
· simp only [subset_iff, mem_image, and_imp, forall_exists_index, mem_range,
forall_apply_eq_imp_iff₂, sphere, mem_filter]
rin... |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Wrenna Robson
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebr... | Mathlib/LinearAlgebra/Lagrange.lean | 543 | 544 | theorem eval_nodal {x : R} : (nodal s v).eval x = ∏ i ∈ s, (x - v i) := by |
simp_rw [nodal, eval_prod, eval_sub, eval_X, eval_C]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.Analysis.NormedSpace.IndicatorFunction
#align_import measure_theory.integral.integrable_on... | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 774 | 776 | theorem integrableOn_Icc_iff_integrableOn_Ioo :
IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ioo a b) μ := by |
rw [integrableOn_Icc_iff_integrableOn_Ioc, integrableOn_Ioc_iff_integrableOn_Ioo]
|
/-
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.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integral
#align_import ring_theory.integrally_closed from "leanprover-community/mathlib"@"... | Mathlib/RingTheory/IntegrallyClosed.lean | 236 | 252 | theorem pow_dvd_pow_iff {n : ℕ} (hn : n ≠ 0) {a b : R} : a ^ n ∣ b ^ n ↔ a ∣ b := by |
refine ⟨fun ⟨x, hx⟩ ↦ ?_, fun h ↦ pow_dvd_pow_of_dvd h n⟩
by_cases ha : a = 0
· simpa [ha, hn] using hx
let K := FractionRing R
replace ha : algebraMap R K a ≠ 0 := fun h ↦
ha <| (injective_iff_map_eq_zero _).1 (IsFractionRing.injective R K) _ h
let y := (algebraMap R K b) / (algebraMap R K a)
have h... |
/-
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.FiniteMeasure
import Mathlib.MeasureTheory.Integral.Average
#align_import measure_theory.measure.probability_measure from "leanprove... | Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean | 417 | 422 | theorem _root_.ProbabilityMeasure.toFiniteMeasure_normalize_eq_self {m0 : MeasurableSpace Ω}
(μ : ProbabilityMeasure Ω) : μ.toFiniteMeasure.normalize = μ := by |
apply ProbabilityMeasure.eq_of_forall_apply_eq
intro s _s_mble
rw [μ.toFiniteMeasure.normalize_eq_of_nonzero μ.toFiniteMeasure_nonzero s]
simp only [ProbabilityMeasure.mass_toFiniteMeasure, inv_one, one_mul, μ.coeFn_toFiniteMeasure]
|
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Data.Multiset.Sort
import Mathlib.Data.PNat.Basic
import Mathlib.Data.PN... | Mathlib/NumberTheory/ADEInequality.lean | 198 | 213 | theorem lt_four {q r : ℕ+} (hqr : q ≤ r) (H : 1 < sumInv {2, q, r}) : q < 4 := by |
have h4 : (0 : ℚ) < 4 := by norm_num
contrapose! H
rw [sumInv_pqr]
have h4r := H.trans hqr
have hq: (q : ℚ)⁻¹ ≤ 4⁻¹ := by
rw [inv_le_inv _ h4]
· assumption_mod_cast
· norm_num
have hr: (r : ℚ)⁻¹ ≤ 4⁻¹ := by
rw [inv_le_inv _ h4]
· assumption_mod_cast
· norm_num
calc
(2⁻¹ + (q :... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen,
Scott Morrison, Chris Hughes, Anne Baanen, Junyan Xu
-/
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimen... | Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 342 | 344 | theorem rank_lt_rank_dual {K V : Type u} [Field K] [AddCommGroup V] [Module K V]
(h : ℵ₀ ≤ Module.rank K V) : Module.rank K V < Module.rank K (V →ₗ[K] K) := by |
convert lift_rank_lt_rank_dual h; rw [lift_id]
|
/-
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.LinearAlgebra.ExteriorAlgebra.Basic
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib... | Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean | 176 | 177 | theorem contractRight_ι (x : M) : ι Q x⌊d = algebraMap R _ (d x) := by |
rw [contractRight_eq, reverse_ι, contractLeft_ι, reverse.commutes]
|
/-
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.Order.Filter.CountableInter
/-!
# Filters with countable intersections and countable separating families
In this file we prove some facts about a f... | Mathlib/Order/Filter/CountableSeparatingOn.lean | 128 | 139 | theorem HasCountableSeparatingOn.subtype_iff {α : Type*} {p : Set α → Prop} {t : Set α} :
HasCountableSeparatingOn t (fun u ↦ ∃ v, p v ∧ (↑) ⁻¹' v = u) univ ↔
HasCountableSeparatingOn α p t := by |
constructor <;> intro h
· exact h.of_subtype $ fun s ↦ id
rcases h with ⟨S, Sct, Sp, hS⟩
use {Subtype.val ⁻¹' s | s ∈ S}, Sct.image _, ?_, ?_
· rintro u ⟨t, tS, rfl⟩
exact ⟨t, Sp _ tS, rfl⟩
rintro x - y - hxy
exact Subtype.val_injective $ hS _ (Subtype.coe_prop _) _ (Subtype.coe_prop _)
fun s hs ... |
/-
Copyright (c) 2022 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Heather Macbeth
-/
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.MeasureTheory.Function.SimpleFuncDense
#align_import measure_theory.func... | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 93 | 135 | theorem tendsto_approxOn_Lp_snorm [OpensMeasurableSpace E] {f : β → E} (hf : Measurable f)
{s : Set E} {y₀ : E} (h₀ : y₀ ∈ s) [SeparableSpace s] (hp_ne_top : p ≠ ∞) {μ : Measure β}
(hμ : ∀ᵐ x ∂μ, f x ∈ closure s) (hi : snorm (fun x => f x - y₀) p μ < ∞) :
Tendsto (fun n => snorm (⇑(approxOn f hf s y₀ h₀ n) ... |
by_cases hp_zero : p = 0
· simpa only [hp_zero, snorm_exponent_zero] using tendsto_const_nhds
have hp : 0 < p.toReal := toReal_pos hp_zero hp_ne_top
suffices
Tendsto (fun n => ∫⁻ x, (‖approxOn f hf s y₀ h₀ n x - f x‖₊ : ℝ≥0∞) ^ p.toReal ∂μ) atTop
(𝓝 0) by
simp only [snorm_eq_lintegral_rpow_n... |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.ChartedSpace
#align_import geometry.manifold.local_invariant_properties from "leanprover-community/mathlib"@... | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 255 | 270 | theorem liftPropWithinAt_indep_chart_source_aux (g : M → H') (he : e ∈ G.maximalAtlas M)
(xe : x ∈ e.source) (he' : e' ∈ G.maximalAtlas M) (xe' : x ∈ e'.source) :
P (g ∘ e.symm) (e.symm ⁻¹' s) (e x) ↔ P (g ∘ e'.symm) (e'.symm ⁻¹' s) (e' x) := by |
rw [← hG.right_invariance (compatible_of_mem_maximalAtlas he he')]
swap; · simp only [xe, xe', mfld_simps]
simp_rw [PartialHomeomorph.trans_apply, e.left_inv xe]
rw [hG.congr_iff]
· refine hG.congr_set ?_
refine (eventually_of_mem ?_ fun y (hy : y ∈ e'.symm ⁻¹' e.source) ↦ ?_).set_eq
· refine (e'.sym... |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.ChartedSpace
#align_import geometry.manifold.local_invariant_properties from "leanprover-community/mathlib"@... | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 475 | 478 | theorem liftPropOn_of_liftProp (mono : ∀ ⦃s x t⦄ ⦃f : H → H'⦄, t ⊆ s → P f s x → P f t x)
(h : LiftProp P g) : LiftPropOn P g s := by |
rw [← liftPropOn_univ] at h
exact liftPropOn_mono mono h (subset_univ _)
|
/-
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 | 225 | 230 | theorem directionOfNonempty_eq_direction {s : AffineSubspace k P} (h : (s : Set P).Nonempty) :
directionOfNonempty h = s.direction := by |
refine le_antisymm ?_ (Submodule.span_le.2 Set.Subset.rfl)
rw [← SetLike.coe_subset_coe, directionOfNonempty, direction, Submodule.coe_set_mk,
AddSubmonoid.coe_set_mk]
exact vsub_set_subset_vectorSpan k _
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Top... | Mathlib/Analysis/Normed/Group/Basic.lean | 2,096 | 2,107 | theorem cauchySeq_prod_of_eventually_eq {u v : ℕ → E} {N : ℕ} (huv : ∀ n ≥ N, u n = v n)
(hv : CauchySeq fun n => ∏ k ∈ range (n + 1), v k) :
CauchySeq fun n => ∏ k ∈ range (n + 1), u k := by |
let d : ℕ → E := fun n => ∏ k ∈ range (n + 1), u k / v k
rw [show (fun n => ∏ k ∈ range (n + 1), u k) = d * fun n => ∏ k ∈ range (n + 1), v k
by ext n; simp [d]]
suffices ∀ n ≥ N, d n = d N from (tendsto_atTop_of_eventually_const this).cauchySeq.mul hv
intro n hn
dsimp [d]
rw [eventually_constant_pro... |
/-
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.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors ... | Mathlib/NumberTheory/Divisors.lean | 220 | 221 | theorem one_mem_properDivisors_iff_one_lt : 1 ∈ n.properDivisors ↔ 1 < n := by |
rw [mem_properDivisors, and_iff_right (one_dvd _)]
|
/-
Copyright (c) 2020 Hanting Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hanting Zhang, Johan Commelin
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.MvPolynomial.CommRing
#align_import r... | Mathlib/RingTheory/MvPolynomial/Symmetric.lean | 318 | 319 | theorem psum_one : psum σ R 1 = ∑ i, X i := by |
simp only [psum, _root_.pow_one]
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.AlgebraicGeometry.Pullbacks
import Mathlib.CategoryTheory.MorphismProperty.Limits
import Mathlib.Data.List.TFAE... | Mathlib/AlgebraicGeometry/Morphisms/Basic.lean | 109 | 116 | theorem AffineTargetMorphismProperty.respectsIso_mk {P : AffineTargetMorphismProperty}
(h₁ : ∀ {X Y Z} (e : X ≅ Y) (f : Y ⟶ Z) [IsAffine Z], P f → P (e.hom ≫ f))
(h₂ : ∀ {X Y Z} (e : Y ≅ Z) (f : X ⟶ Y) [h : IsAffine Y],
P f → @P _ _ (f ≫ e.hom) (isAffineOfIso e.inv)) :
P.toProperty.RespectsIso := by |
constructor
· rintro X Y Z e f ⟨a, h⟩; exact ⟨a, h₁ e f h⟩
· rintro X Y Z e f ⟨a, h⟩; exact ⟨isAffineOfIso e.inv, h₂ e f h⟩
|
/-
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 | 214 | 239 | theorem add {f g : ∀ i, E i} (hf : Memℓp f p) (hg : Memℓp g p) : Memℓp (f + g) p := by |
rcases p.trichotomy with (rfl | rfl | hp)
· apply memℓp_zero
refine (hf.finite_dsupport.union hg.finite_dsupport).subset fun i => ?_
simp only [Pi.add_apply, Ne, Set.mem_union, Set.mem_setOf_eq]
contrapose!
rintro ⟨hf', hg'⟩
simp [hf', hg']
· apply memℓp_infty
obtain ⟨A, hA⟩ := hf.bddAbov... |
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