Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | num_lines int64 1 150 |
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import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Tactic.TFAE
import Mathlib.Topology.Order.Monotone
#align_import set_theory.ordinal.topology from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
noncomputable section
universe u v
open Cardinal Order Topology
namespace Ordina... | Mathlib/SetTheory/Ordinal/Topology.lean | 86 | 124 | theorem mem_closure_tfae (a : Ordinal.{u}) (s : Set Ordinal) :
TFAE [a ∈ closure s,
a ∈ closure (s ∩ Iic a),
(s ∩ Iic a).Nonempty ∧ sSup (s ∩ Iic a) = a,
∃ t, t ⊆ s ∧ t.Nonempty ∧ BddAbove t ∧ sSup t = a,
∃ (o : Ordinal.{u}), o ≠ 0 ∧ ∃ (f : ∀ x < o, Ordinal),
(∀ x hx, f x hx ∈ s) ∧ b... |
tfae_have 1 → 2
· simp only [mem_closure_iff_nhdsWithin_neBot, inter_comm s, nhdsWithin_inter', nhds_left_eq_nhds]
exact id
tfae_have 2 → 3
· intro h
rcases (s ∩ Iic a).eq_empty_or_nonempty with he | hne
· simp [he] at h
· refine ⟨hne, (isLUB_of_mem_closure ?_ h).csSup_eq hne⟩
exact fun x... | 31 |
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.Topology.Instances.AddCircle
#align_import analysis.normed.group.add_circle from "leanprover-community/mathlib"@"084f76e20c88eae536222583331abd9468b08e1c"
noncomputable section
open Set
open Int hiding mem_zmultiples_iff
open AddSubgroup
namespace A... | Mathlib/Analysis/Normed/Group/AddCircle.lean | 86 | 117 | theorem norm_eq {x : ℝ} : ‖(x : AddCircle p)‖ = |x - round (p⁻¹ * x) * p| := by |
suffices ∀ x : ℝ, ‖(x : AddCircle (1 : ℝ))‖ = |x - round x| by
rcases eq_or_ne p 0 with (rfl | hp)
· simp
have hx := norm_coe_mul p x p⁻¹
rw [abs_inv, eq_inv_mul_iff_mul_eq₀ ((not_congr abs_eq_zero).mpr hp)] at hx
rw [← hx, inv_mul_cancel hp, this, ← abs_mul, mul_sub, mul_inv_cancel_left₀ hp, mul... | 31 |
import Mathlib.Topology.Separation
open Topology Filter Set TopologicalSpace
section Basic
variable {α : Type*} [TopologicalSpace α] {C : Set α}
theorem AccPt.nhds_inter {x : α} {U : Set α} (h_acc : AccPt x (𝓟 C)) (hU : U ∈ 𝓝 x) :
AccPt x (𝓟 (U ∩ C)) := by
have : 𝓝[≠] x ≤ 𝓟 U := by
rw [le_princ... | Mathlib/Topology/Perfect.lean | 186 | 218 | theorem exists_countable_union_perfect_of_isClosed [SecondCountableTopology α]
(hclosed : IsClosed C) : ∃ V D : Set α, V.Countable ∧ Perfect D ∧ C = V ∪ D := by |
obtain ⟨b, bct, _, bbasis⟩ := TopologicalSpace.exists_countable_basis α
let v := { U ∈ b | (U ∩ C).Countable }
let V := ⋃ U ∈ v, U
let D := C \ V
have Vct : (V ∩ C).Countable := by
simp only [V, iUnion_inter, mem_sep_iff]
apply Countable.biUnion
· exact Countable.mono inter_subset_left bct
· ... | 31 |
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.RingTheory.Localization.Basic
#align_import ring_theory.localization.ideal from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
namespace IsLocalization
section CommRing
variable {R : Type*} [CommRing R] (M : Submonoid R... | Mathlib/RingTheory/Localization/Ideal.lean | 171 | 204 | theorem surjective_quotientMap_of_maximal_of_localization {I : Ideal S} [I.IsPrime] {J : Ideal R}
{H : J ≤ I.comap (algebraMap R S)} (hI : (I.comap (algebraMap R S)).IsMaximal) :
Function.Surjective (Ideal.quotientMap I (algebraMap R S) H) := by |
intro s
obtain ⟨s, rfl⟩ := Ideal.Quotient.mk_surjective s
obtain ⟨r, ⟨m, hm⟩, rfl⟩ := mk'_surjective M s
by_cases hM : (Ideal.Quotient.mk (I.comap (algebraMap R S))) m = 0
· have : I = ⊤ := by
rw [Ideal.eq_top_iff_one]
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_comap] at hM
convert I.mul... | 31 |
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.IsAdjoinRoot
#align_import number_theory.kummer_dedekind from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
variable (R : Type*) {S : Type*} [CommRing R] [CommRing S] [Algebra R S]
open Ideal Polynomial DoubleQuo... | Mathlib/NumberTheory/KummerDedekind.lean | 152 | 186 | theorem comap_map_eq_map_adjoin_of_coprime_conductor
(hx : (conductor R x).comap (algebraMap R S) ⊔ I = ⊤)
(h_alg : Function.Injective (algebraMap R<x> S)) :
(I.map (algebraMap R S)).comap (algebraMap R<x> S) = I.map (algebraMap R R<x>) := by |
apply le_antisymm
· -- This is adapted from [Neukirch1992]. Let `C = (conductor R x)`. The idea of the proof
-- is that since `I` and `C ∩ R` are coprime, we have
-- `(I * S) ∩ R<x> ⊆ (I + C) * ((I * S) ∩ R<x>) ⊆ I * R<x> + I * C * S ⊆ I * R<x>`.
intro y hy
obtain ⟨z, hz⟩ := y
obtain ⟨p, hp, q,... | 31 |
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Topology.Algebra.Module.Multilinear.Basic
open Bornology Filter Set Function
open scoped Topology
namespace Bornology.IsVonNBounded
variable {ι 𝕜 F : Type*} {E : ι → Type*} [NormedField 𝕜]
[∀ i, AddCommGroup (E i)] [∀ i, Module 𝕜 (E i)] [∀ i, Topol... | Mathlib/Topology/Algebra/Module/Multilinear/Bounded.lean | 44 | 83 | theorem image_multilinear' [Nonempty ι] {s : Set (∀ i, E i)} (hs : IsVonNBounded 𝕜 s)
(f : ContinuousMultilinearMap 𝕜 E F) : IsVonNBounded 𝕜 (f '' s) := fun V hV ↦ by
classical
if h₁ : ∀ c : 𝕜, ‖c‖ ≤ 1 then
exact absorbs_iff_norm.2 ⟨2, fun c hc ↦ by linarith [h₁ c]⟩
else
let _ : NontriviallyNormed... |
have hfV : f ⁻¹' V ∈ 𝓝 0 := (map_continuous f).tendsto' _ _ f.map_zero hV
rwa [nhds_pi, Filter.mem_pi, exists_finite_iff_finset] at hfV
have : ∀ i, ∃ c : 𝕜, c ≠ 0 ∧ ∀ c' : 𝕜, ‖c'‖ ≤ ‖c‖ → ∀ x ∈ s, c' • x i ∈ t i := fun i ↦ by
rw [isVonNBounded_pi_iff] at hs
have := (hs i).tendsto_smallSe... | 31 |
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical
open NNReal ENNReal MeasureTheory
namespace Measur... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 146 | 178 | theorem hasSum_of_disjoint_iUnion [Countable β] {f : β → Set α} (hf₁ : ∀ i, MeasurableSet (f i))
(hf₂ : Pairwise (Disjoint on f)) : HasSum (fun i => v (f i)) (v (⋃ i, f i)) := by |
cases nonempty_encodable β
set g := fun i : ℕ => ⋃ (b : β) (_ : b ∈ Encodable.decode₂ β i), f b with hg
have hg₁ : ∀ i, MeasurableSet (g i) :=
fun _ => MeasurableSet.iUnion fun b => MeasurableSet.iUnion fun _ => hf₁ b
have hg₂ : Pairwise (Disjoint on g) := Encodable.iUnion_decode₂_disjoint_on hf₂
have :=... | 31 |
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.SetTheory.Cardinal.Subfield
import Mathlib.LinearAlgebra.Dimension.RankNullity
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomput... | Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 123 | 166 | theorem linearIndependent_of_top_le_span_of_card_eq_finrank {ι : Type*} [Fintype ι] {b : ι → V}
(spans : ⊤ ≤ span K (Set.range b)) (card_eq : Fintype.card ι = finrank K V) :
LinearIndependent K b :=
linearIndependent_iff'.mpr fun s g dependent i i_mem_s => by
classical
by_contra gx_ne_zero
-- We'l... |
rw [Set.toFinset_card, Fintype.card_ofFinset]
_ ≤ (Set.univ \ {i}).toFinset.card := Finset.card_image_le
_ = (Finset.univ.erase i).card := (congr_arg Finset.card (Finset.ext (by simp [and_comm])))
_ < Finset.univ.card := Finset.card_erase_lt_of_mem (Finset.mem_univ i)
_ = finr... | 32 |
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.SetTheory.Cardinal.Continuum
#align_import measure_theory.card_measurable_space from "leanprover-community/mathlib"@"f2b108e8e97ba393f22bf794989984ddcc1da89b"
universe u
variable {α : Type u}
open Cardi... | Mathlib/MeasureTheory/MeasurableSpace/Card.lean | 117 | 151 | theorem generateMeasurable_eq_rec (s : Set (Set α)) :
{ t | GenerateMeasurable s t } =
⋃ (i : (Quotient.out (aleph 1).ord).α), generateMeasurableRec s i := by |
ext t; refine ⟨fun ht => ?_, fun ht => ?_⟩
· inhabit ω₁
induction' ht with u hu u _ IH f _ IH
· exact mem_iUnion.2 ⟨default, self_subset_generateMeasurableRec s _ hu⟩
· exact mem_iUnion.2 ⟨default, empty_mem_generateMeasurableRec s _⟩
· rcases mem_iUnion.1 IH with ⟨i, hi⟩
obtain ⟨j, hj⟩ := ex... | 32 |
import Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform
import Mathlib.Analysis.Fourier.PoissonSummation
open Real Set MeasureTheory Filter Asymptotics intervalIntegral
open scoped Real Topology FourierTransform RealInnerProductSpace
open Complex hiding exp continuous_exp abs_of_nonneg sq_abs
noncomp... | Mathlib/Analysis/SpecialFunctions/Gaussian/PoissonSummation.lean | 88 | 122 | theorem Complex.tsum_exp_neg_quadratic {a : ℂ} (ha : 0 < a.re) (b : ℂ) :
(∑' n : ℤ, cexp (-π * a * n ^ 2 + 2 * π * b * n)) =
1 / a ^ (1 / 2 : ℂ) * ∑' n : ℤ, cexp (-π / a * (n + I * b) ^ 2) := by |
let f : ℝ → ℂ := fun x ↦ cexp (-π * a * x ^ 2 + 2 * π * b * x)
have hCf : Continuous f := by
refine Complex.continuous_exp.comp (Continuous.add ?_ ?_)
· exact continuous_const.mul (Complex.continuous_ofReal.pow 2)
· exact continuous_const.mul Complex.continuous_ofReal
have hFf : 𝓕 f = fun x : ℝ ↦ 1 ... | 32 |
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.Game.Birthday
import Mathlib.SetTheory.Surreal.Basic
#align_import set_theory.surreal.dyadic from "leanprover-community/mathlib"@"9... | 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 ... | 32 |
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Set.Finite
#align_import combinatorics.hall.finite from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
open Finset
universe u v
namespace HallMarriageTheorem
variable {ι : Type u} {α : Type v} [DecidableEq α] {t : ι → Finset α}
s... | Mathlib/Combinatorics/Hall/Finite.lean | 78 | 121 | theorem hall_hard_inductive_step_A {n : ℕ} (hn : Fintype.card ι = n + 1)
(ht : ∀ s : Finset ι, s.card ≤ (s.biUnion t).card)
(ih :
∀ {ι' : Type u} [Fintype ι'] (t' : ι' → Finset α),
Fintype.card ι' ≤ n →
(∀ s' : Finset ι', s'.card ≤ (s'.biUnion t').card) →
∃ f : ι' → α, Functi... |
haveI : Nonempty ι := Fintype.card_pos_iff.mp (hn.symm ▸ Nat.succ_pos _)
haveI := Classical.decEq ι
-- Choose an arbitrary element `x : ι` and `y : t x`.
let x := Classical.arbitrary ι
have tx_ne : (t x).Nonempty := by
rw [← Finset.card_pos]
calc
0 < 1 := Nat.one_pos
_ ≤ (Finset.biUnion {... | 33 |
import Mathlib.Topology.UniformSpace.CompactConvergence
import Mathlib.Topology.UniformSpace.Equicontinuity
import Mathlib.Topology.UniformSpace.Equiv
open Set Filter Uniformity Topology Function UniformConvergence
variable {ι X Y α β : Type*} [TopologicalSpace X] [UniformSpace α] [UniformSpace β]
variable {F : ι ... | Mathlib/Topology/UniformSpace/Ascoli.lean | 163 | 199 | theorem Equicontinuous.tendsto_uniformFun_iff_pi [CompactSpace X]
(F_eqcont : Equicontinuous F) (ℱ : Filter ι) (f : X → α) :
Tendsto (UniformFun.ofFun ∘ F) ℱ (𝓝 <| UniformFun.ofFun f) ↔
Tendsto F ℱ (𝓝 f) := by |
-- Assume `ℱ` is non trivial.
rcases ℱ.eq_or_neBot with rfl | ℱ_ne
· simp
constructor <;> intro H
-- The forward direction is always true, the interesting part is the converse.
· exact UniformFun.uniformContinuous_toFun.continuous.tendsto _|>.comp H
-- To prove it, assume that `F` tends to `f` *pointwise... | 33 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
import Mathlib.Tactic.IntervalCases
#align_import geometry.euclidean.triangle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped Classica... | Mathlib/Geometry/Euclidean/Triangle.lean | 109 | 143 | theorem cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
Real.cos (angle x (x - y) + angle y (y - x)) = -Real.cos (angle x y) := by |
by_cases hxy : x = y
· rw [hxy, angle_self hy]
simp
· rw [Real.cos_add, cos_angle, cos_angle, cos_angle]
have hxn : ‖x‖ ≠ 0 := fun h => hx (norm_eq_zero.1 h)
have hyn : ‖y‖ ≠ 0 := fun h => hy (norm_eq_zero.1 h)
have hxyn : ‖x - y‖ ≠ 0 := fun h => hxy (eq_of_sub_eq_zero (norm_eq_zero.1 h))
app... | 33 |
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.RingTheory.Ideal.Basic
#align_import algebra.monoid_algebra.ideal from "leanprover-community/mathlib"@"72c366d0475675f1309d3027d3d7d47ee4423951"
variable {k A G : Type*}
| Mathlib/Algebra/MonoidAlgebra/Ideal.lean | 23 | 58 | theorem MonoidAlgebra.mem_ideal_span_of_image [Monoid G] [Semiring k] {s : Set G}
{x : MonoidAlgebra k G} :
x ∈ Ideal.span (MonoidAlgebra.of k G '' s) ↔ ∀ m ∈ x.support, ∃ m' ∈ s, ∃ d, m = d * m' := by |
let RHS : Ideal (MonoidAlgebra k G) :=
{ carrier := { p | ∀ m : G, m ∈ p.support → ∃ m' ∈ s, ∃ d, m = d * m' }
add_mem' := fun {x y} hx hy m hm => by
classical exact (Finset.mem_union.1 <| Finsupp.support_add hm).elim (hx m) (hy m)
zero_mem' := fun m hm => by cases hm
smul_mem' := fun x... | 33 |
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.NumberTheory.Liouville.Residual
import Mathlib.NumberTheory.Liouville.LiouvilleWith
import Mathlib.Analysis.PSeries
#align_import number_theory.liouville.measure from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open sc... | Mathlib/NumberTheory/Liouville/Measure.lean | 34 | 71 | theorem setOf_liouvilleWith_subset_aux :
{ x : ℝ | ∃ p > 2, LiouvilleWith p x } ⊆
⋃ m : ℤ, (· + (m : ℝ)) ⁻¹' ⋃ n > (0 : ℕ),
{ x : ℝ | ∃ᶠ b : ℕ in atTop, ∃ a ∈ Finset.Icc (0 : ℤ) b,
|x - (a : ℤ) / b| < 1 / (b : ℝ) ^ (2 + 1 / n : ℝ) } := by |
rintro x ⟨p, hp, hxp⟩
rcases exists_nat_one_div_lt (sub_pos.2 hp) with ⟨n, hn⟩
rw [lt_sub_iff_add_lt'] at hn
suffices ∀ y : ℝ, LiouvilleWith p y → y ∈ Ico (0 : ℝ) 1 → ∃ᶠ b : ℕ in atTop,
∃ a ∈ Finset.Icc (0 : ℤ) b, |y - a / b| < 1 / (b : ℝ) ^ (2 + 1 / (n + 1 : ℕ) : ℝ) by
simp only [mem_iUnion, mem_pre... | 33 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.LinearAlgebra.AffineSpace.Slope
#align_import analysis.calculus.deriv.slope from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open Topology Filter TopologicalSpace
open Filter Set
secti... | Mathlib/Analysis/Calculus/Deriv/Slope.lean | 99 | 134 | theorem range_derivWithin_subset_closure_span_image
(f : 𝕜 → F) {s t : Set 𝕜} (h : s ⊆ closure (s ∩ t)) :
range (derivWithin f s) ⊆ closure (Submodule.span 𝕜 (f '' t)) := by |
rintro - ⟨x, rfl⟩
rcases eq_or_neBot (𝓝[s \ {x}] x) with H|H
· simp [derivWithin, fderivWithin, H]
exact subset_closure (zero_mem _)
by_cases H' : DifferentiableWithinAt 𝕜 f s x; swap
· rw [derivWithin_zero_of_not_differentiableWithinAt H']
exact subset_closure (zero_mem _)
have I : (𝓝[(s ∩ t) \... | 33 |
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.finsubgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b"
open Set CategoryTheory
universe u v
variable {V : Type u} {W : Type v} {G : Simple... | Mathlib/Combinatorics/SimpleGraph/Finsubgraph.lean | 119 | 153 | theorem nonempty_hom_of_forall_finite_subgraph_hom [Finite W]
(h : ∀ G' : G.Subgraph, G'.verts.Finite → G'.coe →g F) : Nonempty (G →g F) := by |
-- Obtain a `Fintype` instance for `W`.
cases nonempty_fintype W
-- Establish the required interface instances.
haveI : ∀ G' : G.Finsubgraphᵒᵖ, Nonempty ((finsubgraphHomFunctor G F).obj G') := fun G' =>
⟨h G'.unop G'.unop.property⟩
haveI : ∀ G' : G.Finsubgraphᵒᵖ, Fintype ((finsubgraphHomFunctor G F).obj ... | 33 |
import Mathlib.Algebra.Polynomial.Degree.CardPowDegree
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue
import Mathlib.RingTheory.Ideal.LocalRing
#align_import number_theory.class_number.admissible_card_pow_degree from "leanprover-community/mathlib"@"0b... | Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean | 63 | 98 | theorem exists_approx_polynomial_aux [Ring Fq] {d : ℕ} {m : ℕ} (hm : Fintype.card Fq ^ d ≤ m)
(b : Fq[X]) (A : Fin m.succ → Fq[X]) (hA : ∀ i, degree (A i) < degree b) :
∃ i₀ i₁, i₀ ≠ i₁ ∧ degree (A i₁ - A i₀) < ↑(natDegree b - d) := by |
have hb : b ≠ 0 := by
rintro rfl
specialize hA 0
rw [degree_zero] at hA
exact not_lt_of_le bot_le hA
-- Since there are > q^d elements of A, and only q^d choices for the highest `d` coefficients,
-- there must be two elements of A with the same coefficients at
-- `degree b - 1`, ... `degree b -... | 33 |
import Mathlib.Data.Set.Function
import Mathlib.Analysis.BoundedVariation
#align_import analysis.constant_speed from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped NNReal ENNReal
open Set MeasureTheory Classical
variable {α : Type*} [LinearOrder α] {E : Type*} [PseudoEMetr... | Mathlib/Analysis/ConstantSpeed.lean | 102 | 137 | theorem HasConstantSpeedOnWith.union {t : Set ℝ} (hfs : HasConstantSpeedOnWith f s l)
(hft : HasConstantSpeedOnWith f t l) {x : ℝ} (hs : IsGreatest s x) (ht : IsLeast t x) :
HasConstantSpeedOnWith f (s ∪ t) l := by |
rw [hasConstantSpeedOnWith_iff_ordered] at hfs hft ⊢
rintro z (zs | zt) y (ys | yt) zy
· have : (s ∪ t) ∩ Icc z y = s ∩ Icc z y := by
ext w; constructor
· rintro ⟨ws | wt, zw, wy⟩
· exact ⟨ws, zw, wy⟩
· exact ⟨(le_antisymm (wy.trans (hs.2 ys)) (ht.2 wt)).symm ▸ hs.1, zw, wy⟩
· r... | 33 |
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.Data.ZMod.Algebra
#align_import ring_theory.polynomial.cyclotomic.expand from "leanprover-community/mathlib"@"0723536a0522d24fc2f159a096fb3304bef77472"
namespace Polynomial
@[simp]
| Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean | 36 | 72 | theorem cyclotomic_expand_eq_cyclotomic_mul {p n : ℕ} (hp : Nat.Prime p) (hdiv : ¬p ∣ n)
(R : Type*) [CommRing R] :
expand R p (cyclotomic n R) = cyclotomic (n * p) R * cyclotomic n R := by |
rcases Nat.eq_zero_or_pos n with (rfl | hnpos)
· simp
haveI := NeZero.of_pos hnpos
suffices expand ℤ p (cyclotomic n ℤ) = cyclotomic (n * p) ℤ * cyclotomic n ℤ by
rw [← map_cyclotomic_int, ← map_expand, this, Polynomial.map_mul, map_cyclotomic_int,
map_cyclotomic]
refine eq_of_monic_of_dvd_of_natDe... | 34 |
import Mathlib.Data.List.Duplicate
import Mathlib.Data.List.Sort
#align_import data.list.nodup_equiv_fin from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace List
variable {α : Type*}
section Sublist
theorem sublist_of_orderEmbedding_get?_eq {l l' : List α} (f : ℕ ↪o ℕ)
... | Mathlib/Data/List/NodupEquivFin.lean | 168 | 205 | theorem sublist_iff_exists_fin_orderEmbedding_get_eq {l l' : List α} :
l <+ l' ↔
∃ f : Fin l.length ↪o Fin l'.length,
∀ ix : Fin l.length, l.get ix = l'.get (f ix) := by |
rw [sublist_iff_exists_orderEmbedding_get?_eq]
constructor
· rintro ⟨f, hf⟩
have h : ∀ {i : ℕ}, i < l.length → f i < l'.length := by
intro i hi
specialize hf i
rw [get?_eq_get hi, eq_comm, get?_eq_some] at hf
obtain ⟨h, -⟩ := hf
exact h
refine ⟨OrderEmbedding.ofMapLEIff (fun... | 34 |
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Combinatorics.SetFamily.Compression.Down
import Mathlib.Order.UpperLower.Basic
import Mathlib.Data.Fintype.Powerset
#align_import combinatorics.set_family.harris_kleitman from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b205"
open Finset... | Mathlib/Combinatorics/SetFamily/HarrisKleitman.lean | 55 | 91 | theorem IsLowerSet.le_card_inter_finset' (h𝒜 : IsLowerSet (𝒜 : Set (Finset α)))
(hℬ : IsLowerSet (ℬ : Set (Finset α))) (h𝒜s : ∀ t ∈ 𝒜, t ⊆ s) (hℬs : ∀ t ∈ ℬ, t ⊆ s) :
𝒜.card * ℬ.card ≤ 2 ^ s.card * (𝒜 ∩ ℬ).card := by |
induction' s using Finset.induction with a s hs ih generalizing 𝒜 ℬ
· simp_rw [subset_empty, ← subset_singleton_iff', subset_singleton_iff] at h𝒜s hℬs
obtain rfl | rfl := h𝒜s
· simp only [card_empty, zero_mul, empty_inter, mul_zero, le_refl]
obtain rfl | rfl := hℬs
· simp only [card_empty, inter... | 34 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.Module.Defs
import Mathlib.Tactic.Abel
namespace Finset
variable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}
-- The partial sum of `g`, starting from zero
local notation "G " n:80 => ∑ i ∈ range n, g i
... | Mathlib/Algebra/BigOperators/Module.lean | 21 | 57 | theorem sum_Ico_by_parts (hmn : m < n) :
∑ i ∈ Ico m n, f i • g i =
f (n - 1) • G n - f m • G m - ∑ i ∈ Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by |
have h₁ : (∑ i ∈ Ico (m + 1) n, f i • G i) = ∑ i ∈ Ico m (n - 1), f (i + 1) • G (i + 1) := by
rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn), ← sum_Ico_add']
simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,
tsub_eq_zero_iff_le, add_tsub_cancel_right]
have h₂ :
... | 34 |
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.Topology.NoetherianSpace
#align_import algebraic_geometry.prime_spectrum.noetherian from "leanprover-community/mathlib"@"052f6013363326d50cb99c6939814a4b8eb7b301"
universe u v
namespace PrimeSpectrum
open Submodule
variable (R : Type u) [CommR... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Noetherian.lean | 60 | 97 | theorem exists_primeSpectrum_prod_le_and_ne_bot_of_domain (h_fA : ¬IsField A) {I : Ideal A}
(h_nzI : I ≠ ⊥) :
∃ Z : Multiset (PrimeSpectrum A),
Multiset.prod (Z.map asIdeal) ≤ I ∧ Multiset.prod (Z.map asIdeal) ≠ ⊥ := by |
revert h_nzI
-- Porting note: Need to specify `P` explicitly
refine IsNoetherian.induction (P := fun I => I ≠ ⊥ → ∃ Z : Multiset (PrimeSpectrum A),
Multiset.prod (Z.map asIdeal) ≤ I ∧ Multiset.prod (Z.map asIdeal) ≠ ⊥)
(fun (M : Ideal A) hgt => ?_) I
intro h_nzM
have hA_nont : Nontrivial A := IsDom... | 34 |
import Mathlib.RingTheory.DedekindDomain.Dvr
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.pid from "leanprover-community/mathlib"@"6010cf523816335f7bae7f8584cb2edaace73940"
variable {R : Type*} [CommRing R]
open Ideal
open UniqueFactorizationMonoid
open scoped nonZer... | Mathlib/RingTheory/DedekindDomain/PID.lean | 38 | 74 | theorem Ideal.eq_span_singleton_of_mem_of_not_mem_sq_of_not_mem_prime_ne {P : Ideal R}
(hP : P.IsPrime) [IsDedekindDomain R] {x : R} (x_mem : x ∈ P) (hxP2 : x ∉ P ^ 2)
(hxQ : ∀ Q : Ideal R, IsPrime Q → Q ≠ P → x ∉ Q) : P = Ideal.span {x} := by |
letI := Classical.decEq (Ideal R)
have hx0 : x ≠ 0 := by
rintro rfl
exact hxP2 (zero_mem _)
by_cases hP0 : P = ⊥
· subst hP0
-- Porting note: was `simpa using hxP2` but that hypothesis didn't even seem relevant in Lean 3
rwa [eq_comm, span_singleton_eq_bot, ← mem_bot]
have hspan0 : span ({x} ... | 34 |
import Mathlib.GroupTheory.Solvable
import Mathlib.FieldTheory.PolynomialGaloisGroup
import Mathlib.RingTheory.RootsOfUnity.Basic
#align_import field_theory.abel_ruffini from "leanprover-community/mathlib"@"e3f4be1fcb5376c4948d7f095bec45350bfb9d1a"
noncomputable section
open scoped Classical Polynomial Intermedi... | Mathlib/FieldTheory/AbelRuffini.lean | 118 | 153 | theorem gal_X_pow_sub_C_isSolvable_aux (n : ℕ) (a : F)
(h : (X ^ n - 1 : F[X]).Splits (RingHom.id F)) : IsSolvable (X ^ n - C a).Gal := by |
by_cases ha : a = 0
· rw [ha, C_0, sub_zero]
exact gal_X_pow_isSolvable n
have ha' : algebraMap F (X ^ n - C a).SplittingField a ≠ 0 :=
mt ((injective_iff_map_eq_zero _).mp (RingHom.injective _) a) ha
by_cases hn : n = 0
· rw [hn, pow_zero, ← C_1, ← C_sub]
exact gal_C_isSolvable (1 - a)
have hn... | 34 |
import Mathlib.CategoryTheory.Category.Grpd
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Homotopy.Path
import Mathlib.Data.Set.Subsingleton
#align_import algebraic_topology.fundamental_groupoid.basic from "leanprover-community/mathlib"@"3d7987cda72abc473... | Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean | 214 | 253 | theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
(p.trans q).trans r =
(p.trans (q.trans r)).reparam
(fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
(Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAss... |
ext x
simp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left₀, not_le,
Function.comp_apply, Ne, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk,
Path.coe_reparam]
-- TODO: why does split_ifs not reduce the ifs??????
split_ifs with h₁ h₂ h₃ h₄ h₅
· rfl
· exfalso
... | 35 |
import Mathlib.Geometry.Manifold.SmoothManifoldWithCorners
import Mathlib.Geometry.Manifold.LocalInvariantProperties
#align_import geometry.manifold.cont_mdiff from "leanprover-community/mathlib"@"e5ab837fc252451f3eb9124ae6e7b6f57455e7b9"
open Set Function Filter ChartedSpace SmoothManifoldWithCorners
open scope... | Mathlib/Geometry/Manifold/ContMDiff/Defs.lean | 116 | 154 | theorem contDiffWithinAt_localInvariantProp (n : ℕ∞) :
(contDiffGroupoid ∞ I).LocalInvariantProp (contDiffGroupoid ∞ I')
(ContDiffWithinAtProp I I' n) where
is_local {s x u f} u_open xu := by |
have : I.symm ⁻¹' (s ∩ u) ∩ range I = I.symm ⁻¹' s ∩ range I ∩ I.symm ⁻¹' u := by
simp only [inter_right_comm, preimage_inter]
rw [ContDiffWithinAtProp, ContDiffWithinAtProp, this]
symm
apply contDiffWithinAt_inter
have : u ∈ 𝓝 (I.symm (I x)) := by
rw [ModelWithCorners.left_inv]
... | 35 |
import Mathlib.CategoryTheory.Sites.Coherent.Basic
import Mathlib.CategoryTheory.EffectiveEpi.Comp
import Mathlib.CategoryTheory.EffectiveEpi.Extensive
namespace CategoryTheory
open Limits GrothendieckTopology Sieve
variable (C : Type*) [Category C]
instance [Precoherent C] [HasFiniteCoproducts C] : Preregular C ... | Mathlib/CategoryTheory/Sites/Coherent/Comparison.lean | 57 | 94 | theorem extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by |
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· induction h with
| of Y T hT =>
apply Coverage.saturate.of
simp only [Coverage.sup_covering, Set.mem_union] at hT
exact Or.elim hT
(fun ⟨α, x, X, π, ⟨h, _⟩⟩ ↦ ⟨α, x, X, π, ⟨h, inferInstance⟩⟩)
(fun ⟨Z, f, ⟨h, _⟩⟩ ↦ ⟨Unit, inferInstan... | 35 |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
section B... | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | 158 | 196 | theorem snorm_le_snorm_top_mul_snorm (p : ℝ≥0∞) (f : α → E) {g : α → F}
(hg : AEStronglyMeasurable g μ) (b : E → F → G)
(h : ∀ᵐ x ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊) :
snorm (fun x => b (f x) (g x)) p μ ≤ snorm f ∞ μ * snorm g p μ := by |
by_cases hp_top : p = ∞
· simp_rw [hp_top, snorm_exponent_top]
refine le_trans (essSup_mono_ae <| h.mono fun a ha => ?_) (ENNReal.essSup_mul_le _ _)
simp_rw [Pi.mul_apply, ← ENNReal.coe_mul, ENNReal.coe_le_coe]
exact ha
by_cases hp_zero : p = 0
· simp only [hp_zero, snorm_exponent_zero, mul_zero, l... | 35 |
import Mathlib.Analysis.Complex.AbelLimit
import Mathlib.Analysis.SpecialFunctions.Complex.Arctan
#align_import data.real.pi.leibniz from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
namespace Real
open Filter Finset
open scoped Topology
| Mathlib/Data/Real/Pi/Leibniz.lean | 21 | 57 | theorem tendsto_sum_pi_div_four :
Tendsto (fun k => ∑ i ∈ range k, (-1 : ℝ) ^ i / (2 * i + 1)) atTop (𝓝 (π / 4)) := by |
-- The series is alternating with terms of decreasing magnitude, so it converges to some limit
obtain ⟨l, h⟩ :
∃ l, Tendsto (fun n ↦ ∑ i ∈ range n, (-1 : ℝ) ^ i / (2 * i + 1)) atTop (𝓝 l) := by
apply Antitone.tendsto_alternating_series_of_tendsto_zero
· exact antitone_iff_forall_lt.mpr fun _ _ _ ↦ b... | 35 |
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic
#align_import measure_theory.function.conditional_expectation.indicator from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open TopologicalSpace MeasureTheory.Lp Filter ContinuousLinearMap
open s... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean | 75 | 112 | theorem condexp_indicator (hf_int : Integrable f μ) (hs : MeasurableSet[m] s) :
μ[s.indicator f|m] =ᵐ[μ] s.indicator (μ[f|m]) := by |
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm, Set.indicator_zero']; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm, Set.indicator_zero']; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
-- use `have` to perform what should be the first calc step becau... | 36 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Independence.Basic
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic
#align_import probability.conditional_expectation from "leanprover-community/mathlib"@"2f8347015b12b0864dfaf366ec4909eb70c78740"
open TopologicalSpace Filter
open s... | Mathlib/Probability/ConditionalExpectation.lean | 40 | 77 | theorem condexp_indep_eq (hle₁ : m₁ ≤ m) (hle₂ : m₂ ≤ m) [SigmaFinite (μ.trim hle₂)]
(hf : StronglyMeasurable[m₁] f) (hindp : Indep m₁ m₂ μ) : μ[f|m₂] =ᵐ[μ] fun _ => μ[f] := by |
by_cases hfint : Integrable f μ
swap; · rw [condexp_undef hfint, integral_undef hfint]; rfl
refine (ae_eq_condexp_of_forall_setIntegral_eq hle₂ hfint
(fun s _ hs => integrableOn_const.2 (Or.inr hs)) (fun s hms hs => ?_)
stronglyMeasurable_const.aeStronglyMeasurable').symm
rw [setIntegral_const]
rw ... | 36 |
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.NormedSpace.Completion
#align_import analysis.analytic.uniqueness from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type... | Mathlib/Analysis/Analytic/Uniqueness.lean | 32 | 70 | theorem eqOn_zero_of_preconnected_of_eventuallyEq_zero_aux [CompleteSpace F] {f : E → F} {U : Set E}
(hf : AnalyticOn 𝕜 f U) (hU : IsPreconnected U) {z₀ : E} (h₀ : z₀ ∈ U) (hfz₀ : f =ᶠ[𝓝 z₀] 0) :
EqOn f 0 U := by |
/- Let `u` be the set of points around which `f` vanishes. It is clearly open. We have to show
that its limit points in `U` still belong to it, from which the inclusion `U ⊆ u` will follow
by connectedness. -/
let u := {x | f =ᶠ[𝓝 x] 0}
suffices main : closure u ∩ U ⊆ u by
have Uu : U ⊆ u :=
h... | 36 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import analysis.normed_space.lp_space from "leanprover-community/mathlib"@"de83b43717abe3... | Mathlib/Analysis/NormedSpace/lpSpace.lean | 175 | 211 | theorem of_exponent_ge {p q : ℝ≥0∞} {f : ∀ i, E i} (hfq : Memℓp f q) (hpq : q ≤ p) : Memℓp f p := by |
rcases ENNReal.trichotomy₂ hpq with
(⟨rfl, rfl⟩ | ⟨rfl, rfl⟩ | ⟨rfl, hp⟩ | ⟨rfl, rfl⟩ | ⟨hq, rfl⟩ | ⟨hq, _, hpq'⟩)
· exact hfq
· apply memℓp_infty
obtain ⟨C, hC⟩ := (hfq.finite_dsupport.image fun i => ‖f i‖).bddAbove
use max 0 C
rintro x ⟨i, rfl⟩
by_cases hi : f i = 0
· simp [hi]
· ex... | 36 |
import Mathlib.Geometry.Manifold.ChartedSpace
#align_import geometry.manifold.local_invariant_properties from "leanprover-community/mathlib"@"431589bce478b2229eba14b14a283250428217db"
noncomputable section
open scoped Classical
open Manifold Topology
open Set Filter TopologicalSpace
variable {H M H' M' X : Typ... | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 605 | 643 | theorem isLocalStructomorphWithinAt_localInvariantProp [ClosedUnderRestriction G] :
LocalInvariantProp G G (IsLocalStructomorphWithinAt G) :=
{ is_local := by |
intro s x u f hu hux
constructor
· rintro h hx
rcases h hx.1 with ⟨e, heG, hef, hex⟩
have : s ∩ u ∩ e.source ⊆ s ∩ e.source := by mfld_set_tac
exact ⟨e, heG, hef.mono this, hex⟩
· rintro h hx
rcases h ⟨hx, hux⟩ with ⟨e, heG, hef, hex⟩
refine ⟨e.restr (int... | 36 |
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.Unitization
#align_import analysis.normed_space.star.mul from "leanprover-community/mathlib"@"b2ff9a3d7a15fd5b0f060b135421d6a89a999c2f"
open ContinuousLinearMap
local postfix:max "⋆" => star
variable (𝕜 : Type*) {E : Type*}
varia... | Mathlib/Analysis/NormedSpace/Star/Unitization.lean | 87 | 124 | theorem Unitization.norm_splitMul_snd_sq (x : Unitization 𝕜 E) :
‖(Unitization.splitMul 𝕜 E x).snd‖ ^ 2 ≤ ‖(Unitization.splitMul 𝕜 E (star x * x)).snd‖ := by |
/- The key idea is that we can use `sSup_closed_unit_ball_eq_norm` to make this about
applying this linear map to elements of norm at most one. There is a bit of `sqrt` and `sq`
shuffling that needs to occur, which is primarily just an annoyance. -/
refine (Real.le_sqrt (norm_nonneg _) (norm_nonneg _)).mp ?_
... | 36 |
import Mathlib.MeasureTheory.Integral.Periodic
import Mathlib.Data.ZMod.Quotient
#align_import measure_theory.group.add_circle from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Function Filter MeasureTheory MeasureTheory.Measure Metric
open scoped MeasureTheory Pointwise Top... | Mathlib/MeasureTheory/Group/AddCircle.lean | 54 | 92 | theorem isAddFundamentalDomain_of_ae_ball (I : Set <| AddCircle T) (u x : AddCircle T)
(hu : IsOfFinAddOrder u) (hI : I =ᵐ[volume] ball x (T / (2 * addOrderOf u))) :
IsAddFundamentalDomain (AddSubgroup.zmultiples u) I := by |
set G := AddSubgroup.zmultiples u
set n := addOrderOf u
set B := ball x (T / (2 * n))
have hn : 1 ≤ (n : ℝ) := by norm_cast; linarith [hu.addOrderOf_pos]
refine IsAddFundamentalDomain.mk_of_measure_univ_le ?_ ?_ ?_ ?_
· -- `NullMeasurableSet I volume`
exact measurableSet_ball.nullMeasurableSet.congr hI... | 36 |
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Order.Basic
import Mathlib.Tactic.NoncommRing
#align_import analysis.normed_space.M_structure from "leanprover-community/mathlib"@"d11893b411025250c8e61ff2f12ccbd7ee35ab15"
variable (X : Type*) [NormedAddCommGroup X]
... | Mathlib/Analysis/NormedSpace/MStructure.lean | 105 | 144 | theorem commute [FaithfulSMul M X] {P Q : M} (h₁ : IsLprojection X P) (h₂ : IsLprojection X Q) :
Commute P Q := by |
have PR_eq_RPR : ∀ R : M, IsLprojection X R → P * R = R * P * R := fun R h₃ => by
-- Porting note: Needed to fix function, which changes indent of following lines
refine @eq_of_smul_eq_smul _ X _ _ _ _ fun x => by
rw [← norm_sub_eq_zero_iff]
have e1 : ‖R • x‖ ≥ ‖R • x‖ + 2 • ‖(P * R) • x - (R * P... | 37 |
import Mathlib.Data.Real.Basic
import Mathlib.Combinatorics.Pigeonhole
import Mathlib.Algebra.Order.EuclideanAbsoluteValue
#align_import number_theory.class_number.admissible_absolute_value from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
local infixl:50 " ≺ " => EuclideanDomain.r
na... | Mathlib/NumberTheory/ClassNumber/AdmissibleAbsoluteValue.lean | 73 | 112 | theorem exists_approx_aux (n : ℕ) (h : abv.IsAdmissible) :
∀ {ε : ℝ} (_hε : 0 < ε) {b : R} (_hb : b ≠ 0) (A : Fin (h.card ε ^ n).succ → Fin n → R),
∃ i₀ i₁, i₀ ≠ i₁ ∧ ∀ k, (abv (A i₁ k % b - A i₀ k % b) : ℝ) < abv b • ε := by |
haveI := Classical.decEq R
induction' n with n ih
· intro ε _hε b _hb A
refine ⟨0, 1, ?_, ?_⟩
· simp
rintro ⟨i, ⟨⟩⟩
intro ε hε b hb A
let M := h.card ε
-- By the "nicer" pigeonhole principle, we can find a collection `s`
-- of more than `M^n` remainders where the first components lie close to... | 37 |
import Mathlib.Order.Atoms
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.RelIso.Set
import Mathlib.Order.SupClosed
import Mathlib.Order.SupIndep
import Mathlib.Order.Zorn
import Mathlib.Data.Finset.Order
import Mathlib.Order.Interval.Set.OrderIso
import Mathlib.Data.Finite.Set
import Mathlib.Tactic.TFAE
#alig... | Mathlib/Order/CompactlyGenerated/Basic.lean | 110 | 149 | theorem isCompactElement_iff_le_of_directed_sSup_le (k : α) :
IsCompactElement k ↔
∀ s : Set α, s.Nonempty → DirectedOn (· ≤ ·) s → k ≤ sSup s → ∃ x : α, x ∈ s ∧ k ≤ x := by |
classical
constructor
· intro hk s hne hdir hsup
obtain ⟨t, ht⟩ := hk s hsup
-- certainly every element of t is below something in s, since ↑t ⊆ s.
have t_below_s : ∀ x ∈ t, ∃ y ∈ s, x ≤ y := fun x hxt => ⟨x, ht.left hxt, le_rfl⟩
obtain ⟨x, ⟨hxs, hsupx⟩⟩ := Finset.sup_le_of_le_directe... | 37 |
import Mathlib.MeasureTheory.Measure.Doubling
import Mathlib.MeasureTheory.Covering.Vitali
import Mathlib.MeasureTheory.Covering.Differentiation
#align_import measure_theory.covering.density_theorem from "leanprover-community/mathlib"@"5f6e827d81dfbeb6151d7016586ceeb0099b9655"
noncomputable section
open Set Filt... | Mathlib/MeasureTheory/Covering/DensityTheorem.lean | 71 | 109 | theorem closedBall_mem_vitaliFamily_of_dist_le_mul {K : ℝ} {x y : α} {r : ℝ} (h : dist x y ≤ K * r)
(rpos : 0 < r) : closedBall y r ∈ (vitaliFamily μ K).setsAt x := by |
let R := scalingScaleOf μ (max (4 * K + 3) 3)
simp only [vitaliFamily, VitaliFamily.enlarge, Vitali.vitaliFamily, mem_union, mem_setOf_eq,
isClosed_ball, true_and_iff, (nonempty_ball.2 rpos).mono ball_subset_interior_closedBall,
measurableSet_closedBall]
/- The measure is doubling on scales smaller than ... | 37 |
import Mathlib.Data.Fintype.Order
import Mathlib.Data.Set.Finite
import Mathlib.Order.Category.FinPartOrd
import Mathlib.Order.Category.LinOrd
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheory.Limits.Shapes.RegularMono
import Mathlib.Data.Set.Subsingleton
#align_import order.category.No... | Mathlib/Order/Category/NonemptyFinLinOrd.lean | 171 | 209 | theorem epi_iff_surjective {A B : NonemptyFinLinOrd.{u}} (f : A ⟶ B) :
Epi f ↔ Function.Surjective f := by |
constructor
· intro
dsimp only [Function.Surjective]
by_contra! hf'
rcases hf' with ⟨m, hm⟩
let Y := NonemptyFinLinOrd.of (ULift (Fin 2))
let p₁ : B ⟶ Y :=
⟨fun b => if b < m then ULift.up 0 else ULift.up 1, fun x₁ x₂ h => by
simp only
split_ifs with h₁ h₂ h₂
any_g... | 37 |
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.Calculus.Deriv.Inv
#align_import analysis.calculus.lhopital from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
open Filter Set
open scoped Filter Topology Pointwise
variable {a b : ℝ} (hab : a < b) {l : Filter ℝ} {f f... | Mathlib/Analysis/Calculus/LHopital.lean | 51 | 92 | theorem lhopital_zero_right_on_Ioo (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x)
(hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Ioo a b, g' x ≠ 0)
(hfa : Tendsto f (𝓝[>] a) (𝓝 0)) (hga : Tendsto g (𝓝[>] a) (𝓝 0))
(hdiv : Tendsto (fun x => f' x / g' x) (𝓝[>] a) l) :
Tendsto (fun x => f x... |
have sub : ∀ x ∈ Ioo a b, Ioo a x ⊆ Ioo a b := fun x hx =>
Ioo_subset_Ioo (le_refl a) (le_of_lt hx.2)
have hg : ∀ x ∈ Ioo a b, g x ≠ 0 := by
intro x hx h
have : Tendsto g (𝓝[<] x) (𝓝 0) := by
rw [← h, ← nhdsWithin_Ioo_eq_nhdsWithin_Iio hx.1]
exact ((hgg' x hx).continuousAt.continuousWithi... | 37 |
import Mathlib.Combinatorics.Hall.Finite
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Data.Rel
#align_import combinatorics.hall.basic from "leanprover-community/mathlib"@"8195826f5c428fc283510bc67303dd4472d78498"
open Finset CategoryTheory
universe u v
def hallMatchingsOn {ι : Type u} {α : Typ... | Mathlib/Combinatorics/Hall/Basic.lean | 123 | 163 | theorem Finset.all_card_le_biUnion_card_iff_exists_injective {ι : Type u} {α : Type v}
[DecidableEq α] (t : ι → Finset α) :
(∀ s : Finset ι, s.card ≤ (s.biUnion t).card) ↔
∃ f : ι → α, Function.Injective f ∧ ∀ x, f x ∈ t x := by |
constructor
· intro h
-- Set up the functor
haveI : ∀ ι' : (Finset ι)ᵒᵖ, Nonempty ((hallMatchingsFunctor t).obj ι') := fun ι' =>
hallMatchingsOn.nonempty t h ι'.unop
classical
haveI : ∀ ι' : (Finset ι)ᵒᵖ, Finite ((hallMatchingsFunctor t).obj ι') := by
intro ι'
rw [hallMatchi... | 37 |
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.centering from "leanprover-community/mathlib"@"bea6c853b6edbd15e9d0941825abd04d77933ed0"
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory
namespace MeasureTheory
variable {Ω E : Type*} {m0 : ... | Mathlib/Probability/Martingale/Centering.lean | 93 | 131 | theorem martingale_martingalePart (hf : Adapted ℱ f) (hf_int : ∀ n, Integrable (f n) μ)
[SigmaFiniteFiltration μ ℱ] : Martingale (martingalePart f ℱ μ) ℱ μ := by |
refine ⟨adapted_martingalePart hf, fun i j hij => ?_⟩
-- ⊢ μ[martingalePart f ℱ μ j | ℱ i] =ᵐ[μ] martingalePart f ℱ μ i
have h_eq_sum : μ[martingalePart f ℱ μ j|ℱ i] =ᵐ[μ]
f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|ℱ i] - μ[μ[f (k + 1) - f k|ℱ k]|ℱ i]) := by
rw [martingalePart_eq_sum]
refine (c... | 37 |
import Mathlib.Topology.Instances.RealVectorSpace
import Mathlib.Analysis.NormedSpace.AffineIsometry
#align_import analysis.normed_space.mazur_ulam from "leanprover-community/mathlib"@"78261225eb5cedc61c5c74ecb44e5b385d13b733"
variable {E PE F PF : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [MetricSpace PE]
... | Mathlib/Analysis/NormedSpace/MazurUlam.lean | 45 | 83 | theorem midpoint_fixed {x y : PE} :
∀ e : PE ≃ᵢ PE, e x = x → e y = y → e (midpoint ℝ x y) = midpoint ℝ x y := by |
set z := midpoint ℝ x y
-- Consider the set of `e : E ≃ᵢ E` such that `e x = x` and `e y = y`
set s := { e : PE ≃ᵢ PE | e x = x ∧ e y = y }
haveI : Nonempty s := ⟨⟨IsometryEquiv.refl PE, rfl, rfl⟩⟩
-- On the one hand, `e` cannot send the midpoint `z` of `[x, y]` too far
have h_bdd : BddAbove (range fun e :... | 37 |
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.Layercake
#align_import analysis.special_functions.japanese_bracket from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
op... | Mathlib/Analysis/SpecialFunctions/JapaneseBracket.lean | 100 | 139 | theorem finite_integral_one_add_norm {r : ℝ} (hnr : (finrank ℝ E : ℝ) < r) :
(∫⁻ x : E, ENNReal.ofReal ((1 + ‖x‖) ^ (-r)) ∂μ) < ∞ := by |
have hr : 0 < r := lt_of_le_of_lt (finrank ℝ E).cast_nonneg hnr
-- We start by applying the layer cake formula
have h_meas : Measurable fun ω : E => (1 + ‖ω‖) ^ (-r) :=
-- Porting note: was `by measurability`
(measurable_norm.const_add _).pow_const _
have h_pos : ∀ x : E, 0 ≤ (1 + ‖x‖) ^ (-r) := fun x ... | 38 |
import Mathlib.AlgebraicTopology.DoldKan.GammaCompN
import Mathlib.AlgebraicTopology.DoldKan.NReflectsIso
#align_import algebraic_topology.dold_kan.n_comp_gamma from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
noncomputable section
open CategoryTheory CategoryTheory.Category Categor... | Mathlib/AlgebraicTopology/DoldKan/NCompGamma.lean | 83 | 124 | theorem Γ₀_obj_termwise_mapMono_comp_PInfty (X : SimplicialObject C) {Δ Δ' : SimplexCategory}
(i : Δ ⟶ Δ') [Mono i] :
Γ₀.Obj.Termwise.mapMono (AlternatingFaceMapComplex.obj X) i ≫ PInfty.f Δ.len =
PInfty.f Δ'.len ≫ X.map i.op := by |
induction' Δ using SimplexCategory.rec with n
induction' Δ' using SimplexCategory.rec with n'
dsimp
-- We start with the case `i` is an identity
by_cases h : n = n'
· subst h
simp only [SimplexCategory.eq_id_of_mono i, Γ₀.Obj.Termwise.mapMono_id, op_id, X.map_id]
dsimp
simp only [id_comp, comp_... | 38 |
import Mathlib.LinearAlgebra.Charpoly.Basic
import Mathlib.LinearAlgebra.Matrix.Basis
#align_import linear_algebra.charpoly.to_matrix from "leanprover-community/mathlib"@"baab5d3091555838751562e6caad33c844bea15e"
universe u v w
variable {R M M₁ M₂ : Type*} [CommRing R] [Nontrivial R]
variable [AddCommGroup M] [M... | Mathlib/LinearAlgebra/Charpoly/ToMatrix.lean | 48 | 87 | theorem charpoly_toMatrix {ι : Type w} [DecidableEq ι] [Fintype ι] (b : Basis ι R M) :
(toMatrix b b f).charpoly = f.charpoly := by |
let A := toMatrix b b f
let b' := chooseBasis R M
let ι' := ChooseBasisIndex R M
let A' := toMatrix b' b' f
let e := Basis.indexEquiv b b'
let φ := reindexLinearEquiv R R e e
let φ₁ := reindexLinearEquiv R R e (Equiv.refl ι')
let φ₂ := reindexLinearEquiv R R (Equiv.refl ι') (Equiv.refl ι')
let φ₃ := ... | 38 |
import Batteries.Tactic.SeqFocus
import Batteries.Data.List.Lemmas
import Batteries.Data.List.Init.Attach
namespace Std.Range
def numElems (r : Range) : Nat :=
if r.step = 0 then
-- This is a very weird choice, but it is chosen to coincide with the `forIn` impl
if r.stop ≤ r.start then 0 else r.stop
els... | .lake/packages/batteries/Batteries/Data/Range/Lemmas.lean | 49 | 92 | theorem forIn'_eq_forIn_range' [Monad m] (r : Std.Range)
(init : β) (f : (a : Nat) → a ∈ r → β → m (ForInStep β)) :
forIn' r init f =
forIn
((List.range' r.start r.numElems r.step).pmap Subtype.mk fun _ => mem_range'_elems r)
init (fun ⟨a, h⟩ => f a h) := by |
let ⟨start, stop, step⟩ := r
let L := List.range' start (numElems ⟨start, stop, step⟩) step
let f' : { a // start ≤ a ∧ a < stop } → _ := fun ⟨a, h⟩ => f a h
suffices ∀ H, forIn' [start:stop:step] init f = forIn (L.pmap Subtype.mk H) init f' from this _
intro H; dsimp only [forIn', Range.forIn']
if h : sta... | 38 |
import Mathlib.Analysis.Calculus.LineDeriv.Measurable
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.Analysis.BoundedVariation
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff
import Mathlib.... | Mathlib/Analysis/Calculus/Rademacher.lean | 119 | 160 | theorem integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul'
(hf : LipschitzWith C f) (h'f : HasCompactSupport f) (hg : Continuous g) (v : E) :
Tendsto (fun (t : ℝ) ↦ ∫ x, (t⁻¹ • (f (x + t • v) - f x)) * g x ∂μ) (𝓝[>] 0)
(𝓝 (∫ x, lineDeriv ℝ f x v * g x ∂μ)) := by |
let K := cthickening (‖v‖) (tsupport f)
have K_compact : IsCompact K := IsCompact.cthickening h'f
apply tendsto_integral_filter_of_dominated_convergence
(K.indicator (fun x ↦ (C * ‖v‖) * ‖g x‖))
· filter_upwards with t
apply AEStronglyMeasurable.mul ?_ hg.aestronglyMeasurable
apply aestronglyMeas... | 38 |
import Mathlib.Topology.Constructions
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Order.Filter.ListTraverse
import Mathlib.Tactic.AdaptationNote
#align_import topology.list from "leanprover-community/mathlib"@"48085f140e684306f9e7da907cd5932056d1aded"
open TopologicalSpace Set Filter
open Topology Filt... | Mathlib/Topology/List.lean | 28 | 66 | theorem nhds_list (as : List α) : 𝓝 as = traverse 𝓝 as := by |
refine nhds_mkOfNhds _ _ ?_ ?_
· intro l
induction l with
| nil => exact le_rfl
| cons a l ih =>
suffices List.cons <$> pure a <*> pure l ≤ List.cons <$> 𝓝 a <*> traverse 𝓝 l by
simpa only [functor_norm] using this
exact Filter.seq_mono (Filter.map_mono <| pure_le_nhds a) ih
· i... | 38 |
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.RingTheory.LocalProperties
#align_import algebraic_geometry.morphisms.ring_hom_properties from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were used in this file to improve perfomance #127... | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | 163 | 205 | theorem affineLocally_iff_affineOpens_le
(hP : RingHom.RespectsIso @P) {X Y : Scheme.{u}} (f : X ⟶ Y) :
affineLocally.{u} (@P) f ↔
∀ (U : Y.affineOpens) (V : X.affineOpens) (e : V.1 ≤ (Opens.map f.1.base).obj U.1),
P (Scheme.Hom.appLe f e) := by |
apply forall_congr'
intro U
delta sourceAffineLocally
simp_rw [op_comp, Scheme.Γ.map_comp, Γ_map_morphismRestrict, Category.assoc, Scheme.Γ_map_op,
hP.cancel_left_isIso (Y.presheaf.map (eqToHom _).op)]
constructor
· intro H V e
let U' := (Opens.map f.val.base).obj U.1
have e'' : (Scheme.Hom.ope... | 38 |
import Mathlib.AlgebraicTopology.DoldKan.GammaCompN
import Mathlib.AlgebraicTopology.DoldKan.NReflectsIso
#align_import algebraic_topology.dold_kan.n_comp_gamma from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
noncomputable section
open CategoryTheory CategoryTheory.Category Categor... | Mathlib/AlgebraicTopology/DoldKan/NCompGamma.lean | 38 | 78 | theorem PInfty_comp_map_mono_eq_zero (X : SimplicialObject C) {n : ℕ} {Δ' : SimplexCategory}
(i : Δ' ⟶ [n]) [hi : Mono i] (h₁ : Δ'.len ≠ n) (h₂ : ¬Isδ₀ i) :
PInfty.f n ≫ X.map i.op = 0 := by |
induction' Δ' using SimplexCategory.rec with m
obtain ⟨k, hk⟩ := Nat.exists_eq_add_of_lt (len_lt_of_mono i fun h => by
rw [← h] at h₁
exact h₁ rfl)
simp only [len_mk] at hk
rcases k with _|k
· change n = m + 1 at hk
subst hk
obtain ⟨j, rfl⟩ := eq_δ_of_mono i
rw [Isδ₀.iff] at h₂
... | 38 |
import Mathlib.Algebra.ContinuedFractions.Computation.ApproximationCorollaries
import Mathlib.Algebra.ContinuedFractions.Computation.Translations
import Mathlib.Data.Real.Irrational
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.Basic
#align_import number_theory.diophantine_approximation from "leanpro... | Mathlib/NumberTheory/DiophantineApproximation.lean | 93 | 132 | theorem exists_int_int_abs_mul_sub_le (ξ : ℝ) {n : ℕ} (n_pos : 0 < n) :
∃ j k : ℤ, 0 < k ∧ k ≤ n ∧ |↑k * ξ - j| ≤ 1 / (n + 1) := by |
let f : ℤ → ℤ := fun m => ⌊fract (ξ * m) * (n + 1)⌋
have hn : 0 < (n : ℝ) + 1 := mod_cast Nat.succ_pos _
have hfu := fun m : ℤ => mul_lt_of_lt_one_left hn <| fract_lt_one (ξ * ↑m)
conv in |_| ≤ _ => rw [mul_comm, le_div_iff hn, ← abs_of_pos hn, ← abs_mul]
let D := Icc (0 : ℤ) n
by_cases H : ∃ m ∈ D, f m = ... | 38 |
import Mathlib.Topology.UniformSpace.CompactConvergence
import Mathlib.Topology.UniformSpace.Equicontinuity
import Mathlib.Topology.UniformSpace.Equiv
open Set Filter Uniformity Topology Function UniformConvergence
variable {ι X Y α β : Type*} [TopologicalSpace X] [UniformSpace α] [UniformSpace β]
variable {F : ι ... | Mathlib/Topology/UniformSpace/Ascoli.lean | 85 | 125 | theorem Equicontinuous.comap_uniformFun_eq [CompactSpace X] (F_eqcont : Equicontinuous F) :
(UniformFun.uniformSpace X α).comap F =
(Pi.uniformSpace _).comap F := by |
-- The `≤` inequality is trivial
refine le_antisymm (UniformSpace.comap_mono UniformFun.uniformContinuous_toFun) ?_
-- A bit of rewriting to get a nice intermediate statement.
change comap _ _ ≤ comap _ _
simp_rw [Pi.uniformity, Filter.comap_iInf, comap_comap, Function.comp]
refine ((UniformFun.hasBasis_un... | 38 |
import Mathlib.Data.Nat.Choose.Dvd
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Polynomial.Cyclotomic.Expand
#align_import ring_theory.polynomial.eisenstein.is_integral from "leanprover-community/mathlib"@"5bfbcca0a7ffdd21cf1682e59106d6c942434a32"
universe u ... | Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean | 77 | 117 | theorem cyclotomic_prime_pow_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] (n : ℕ) :
((cyclotomic (p ^ (n + 1)) ℤ).comp (X + 1)).IsEisensteinAt 𝓟 := by |
refine Monic.isEisensteinAt_of_mem_of_not_mem ?_
(Ideal.IsPrime.ne_top <| (Ideal.span_singleton_prime (mod_cast hp.out.ne_zero)).2 <|
Nat.prime_iff_prime_int.1 hp.out) ?_ ?_
· rw [show (X + 1 : ℤ[X]) = X + C 1 by simp]
refine (cyclotomic.monic _ ℤ).comp (monic_X_add_C 1) fun h => ?_
rw [natDe... | 39 |
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.adjoin.fg from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
universe u v w
open Subsemiring Ring Submodule
open Pointwise
na... | Mathlib/RingTheory/Adjoin/FG.lean | 40 | 80 | theorem fg_trans (h1 : (adjoin R s).toSubmodule.FG) (h2 : (adjoin (adjoin R s) t).toSubmodule.FG) :
(adjoin R (s ∪ t)).toSubmodule.FG := by |
rcases fg_def.1 h1 with ⟨p, hp, hp'⟩
rcases fg_def.1 h2 with ⟨q, hq, hq'⟩
refine fg_def.2 ⟨p * q, hp.mul hq, le_antisymm ?_ ?_⟩
· rw [span_le, Set.mul_subset_iff]
intro x hx y hy
change x * y ∈ adjoin R (s ∪ t)
refine Subalgebra.mul_mem _ ?_ ?_
· have : x ∈ Subalgebra.toSubmodule (adjoin R s) :... | 39 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Continuum
#align_import data.real.cardinality from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Nat Set
open Cardinal
no... | Mathlib/Data/Real/Cardinality.lean | 123 | 164 | theorem increasing_cantorFunction (h1 : 0 < c) (h2 : c < 1 / 2) {n : ℕ} {f g : ℕ → Bool}
(hn : ∀ k < n, f k = g k) (fn : f n = false) (gn : g n = true) :
cantorFunction c f < cantorFunction c g := by |
have h3 : c < 1 := by
apply h2.trans
norm_num
induction' n with n ih generalizing f g
· let f_max : ℕ → Bool := fun n => Nat.rec false (fun _ _ => true) n
have hf_max : ∀ n, f n → f_max n := by
intro n hn
cases n
· rw [fn] at hn
contradiction
apply rfl
let g_min : ... | 39 |
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Covering.Besicovitch
import Mathlib.Tactic.AdaptationNote
#align_import measure_theory.covering.besicovitch_vector_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
universe u
open Metric Set Fini... | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | 110 | 150 | theorem card_le_of_separated (s : Finset E) (hs : ∀ c ∈ s, ‖c‖ ≤ 2)
(h : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖) : s.card ≤ 5 ^ finrank ℝ E := by |
/- We consider balls of radius `1/2` around the points in `s`. They are disjoint, and all
contained in the ball of radius `5/2`. A volume argument gives `s.card * (1/2)^dim ≤ (5/2)^dim`,
i.e., `s.card ≤ 5^dim`. -/
borelize E
let μ : Measure E := Measure.addHaar
let δ : ℝ := (1 : ℝ) / 2
let ρ : ℝ := (... | 39 |
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.MeasureTheory.Function.SimpleFuncDense
#align_import measure_theory.function.simple_func_dense_lp from "leanprover-community/mathlib"@"5a2df4cd59cb31e97a516d4603a14bed5c2f9425"
noncomputable section
set_option linter.uppercaseLean3 false
open Set 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... | 39 |
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
import Mathlib.Topology.Algebra.Module.Simple
import Mathlib.Topology.Algebra.Module.Determinant
import Mathlib.RingTheory.Ideal.LocalRing
#align_import topology.algebra.module.finite_dimension from "leanprove... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | 132 | 173 | theorem LinearMap.continuous_of_isClosed_ker (l : E →ₗ[𝕜] 𝕜)
(hl : IsClosed (LinearMap.ker l : Set E)) :
Continuous l := by |
-- `l` is either constant or surjective. If it is constant, the result is trivial.
by_cases H : finrank 𝕜 (LinearMap.range l) = 0
· rw [Submodule.finrank_eq_zero, LinearMap.range_eq_bot] at H
rw [H]
exact continuous_zero
· -- In the case where `l` is surjective, we factor it as `φ : (E ⧸ l.ker) ≃ₗ[𝕜]... | 39 |
import Batteries.Data.List.Lemmas
import Batteries.Data.Array.Basic
import Batteries.Tactic.SeqFocus
import Batteries.Util.ProofWanted
namespace Array
theorem forIn_eq_data_forIn [Monad m]
(as : Array α) (b : β) (f : α → β → m (ForInStep β)) :
forIn as b f = forIn as.data b f := by
let rec loop : ∀ {i h b ... | .lake/packages/batteries/Batteries/Data/Array/Lemmas.lean | 33 | 73 | theorem zipWith_eq_zipWith_data (f : α → β → γ) (as : Array α) (bs : Array β) :
(as.zipWith bs f).data = as.data.zipWith f bs.data := by |
let rec loop : ∀ (i : Nat) cs, i ≤ as.size → i ≤ bs.size →
(zipWithAux f as bs i cs).data = cs.data ++ (as.data.drop i).zipWith f (bs.data.drop i) := by
intro i cs hia hib
unfold zipWithAux
by_cases h : i = as.size ∨ i = bs.size
case pos =>
have : ¬(i < as.size) ∨ ¬(i < bs.size) := by
... | 39 |
import Mathlib.NumberTheory.Zsqrtd.GaussianInt
import Mathlib.NumberTheory.LegendreSymbol.Basic
import Mathlib.Analysis.Normed.Field.Basic
#align_import number_theory.zsqrtd.quadratic_reciprocity from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Zsqrtd Complex
open scoped Comple... | Mathlib/NumberTheory/Zsqrtd/QuadraticReciprocity.lean | 33 | 83 | theorem mod_four_eq_three_of_nat_prime_of_prime (p : ℕ) [hp : Fact p.Prime]
(hpi : Prime (p : ℤ[i])) : p % 4 = 3 :=
hp.1.eq_two_or_odd.elim
(fun hp2 =>
absurd hpi
(mt irreducible_iff_prime.2 fun ⟨_, h⟩ => by
have := h ⟨1, 1⟩ ⟨1, -1⟩ (hp2.symm ▸ rfl)
rw [← norm_eq_one_iff, ← n... |
rw [← Nat.mod_mul_left_mod p 2 2, show 2 * 2 = 4 from rfl] at hp1
have := Nat.mod_lt p (show 0 < 4 by decide)
revert this hp3 hp1
generalize p % 4 = m
intros; interval_cases m <;> simp_all -- Porting note (#11043): was `decide!`
let ⟨k, hk⟩ := (ZMod.exists_sq_eq_neg_one_if... | 39 |
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.Stalks
import Mathlib.CategoryTheory.Limits.Preserves.Filtered
import Mathlib.CategoryTheory.Sites.LocallySurjective
#align_import topology.sheaves.locally_surjective from "leanprover-community/mathlib"@"fb7698eb37544cbb66292b68b40e54d001f8d1a9"... | Mathlib/Topology/Sheaves/LocallySurjective.lean | 78 | 118 | theorem locally_surjective_iff_surjective_on_stalks (T : ℱ ⟶ 𝒢) :
IsLocallySurjective T ↔ ∀ x : X, Function.Surjective ((stalkFunctor C x).map T) := by |
constructor <;> intro hT
· /- human proof:
Let g ∈ Γₛₜ 𝒢 x be a germ. Represent it on an open set U ⊆ X
as ⟨t, U⟩. By local surjectivity, pass to a smaller open set V
on which there exists s ∈ Γ_ ℱ V mapping to t |_ V.
Then the germ of s maps to g -/
-- Let g ∈ Γₛₜ 𝒢 x be a ge... | 39 |
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Combinatorics.Hall.Basic
import Mathlib.Data.Fintype.BigOperators
import Mathlib.SetTheory.Cardinal.Finite
#align_import combinatorics.configuration from "leanprover-community/mathlib"@"d2d8742b0c21426362a9dacebc6005db895ca963"
open Finset
nam... | Mathlib/Combinatorics/Configuration.lean | 125 | 166 | theorem Nondegenerate.exists_injective_of_card_le [Nondegenerate P L] [Fintype P] [Fintype L]
(h : Fintype.card L ≤ Fintype.card P) : ∃ f : L → P, Function.Injective f ∧ ∀ l, f l ∉ l := by |
classical
let t : L → Finset P := fun l => Set.toFinset { p | p ∉ l }
suffices ∀ s : Finset L, s.card ≤ (s.biUnion t).card by
-- Hall's marriage theorem
obtain ⟨f, hf1, hf2⟩ := (Finset.all_card_le_biUnion_card_iff_exists_injective t).mp this
exact ⟨f, hf1, fun l => Set.mem_toFinset.mp (hf2 ... | 40 |
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Order.OmegaCompletePartialOrder
import Mathlib.Topology.Constructions
import Mathlib.MeasureTheory.MeasurableSpace.Basic
open Set
namespace MeasureTheory
variable {ι : Type _} {α : ι → Type _}
section squareCylinders
def squareCylinders (C : ∀ i, Set (Set (α... | Mathlib/MeasureTheory/Constructions/Cylinders.lean | 63 | 105 | theorem isPiSystem_squareCylinders {C : ∀ i, Set (Set (α i))} (hC : ∀ i, IsPiSystem (C i))
(hC_univ : ∀ i, univ ∈ C i) :
IsPiSystem (squareCylinders C) := by |
rintro S₁ ⟨s₁, t₁, h₁, rfl⟩ S₂ ⟨s₂, t₂, h₂, rfl⟩ hst_nonempty
classical
let t₁' := s₁.piecewise t₁ (fun i ↦ univ)
let t₂' := s₂.piecewise t₂ (fun i ↦ univ)
have h1 : ∀ i ∈ (s₁ : Set ι), t₁ i = t₁' i :=
fun i hi ↦ (Finset.piecewise_eq_of_mem _ _ _ hi).symm
have h1' : ∀ i ∉ (s₁ : Set ι), t₁' i = univ :=
... | 40 |
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.MeasureTheory.Integral.Pi
import Mathlib.Analysis.Fourier.FourierTransform
open Real Set MeasureTheory Filter Asymptotics intervalIntegral
open scoped Real Topology FourierTransform Re... | Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | 70 | 112 | theorem verticalIntegral_norm_le (hb : 0 < b.re) (c : ℝ) {T : ℝ} (hT : 0 ≤ T) :
‖verticalIntegral b c T‖ ≤
(2 : ℝ) * |c| * exp (-(b.re * T ^ 2 - (2 : ℝ) * |b.im| * |c| * T - b.re * c ^ 2)) := by |
-- first get uniform bound for integrand
have vert_norm_bound :
∀ {T : ℝ},
0 ≤ T →
∀ {c y : ℝ},
|y| ≤ |c| →
‖cexp (-b * (T + y * I) ^ 2)‖ ≤
exp (-(b.re * T ^ 2 - (2 : ℝ) * |b.im| * |c| * T - b.re * c ^ 2)) := by
intro T hT c y hy
rw [norm_cexp_neg_mul_s... | 40 |
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.Tactic.ByContra
import Mathlib.Topology.Algebra.Polynomial
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Analysis.Complex.Arg
#align_import ring_theory.polynomial.cyclotomic.eval from "leanprover-community/mathlib"@"5bfbcca0a7ffdd21cf16... | Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean | 70 | 111 | theorem cyclotomic_pos {n : ℕ} (hn : 2 < n) {R} [LinearOrderedCommRing R] (x : R) :
0 < eval x (cyclotomic n R) := by |
induction' n using Nat.strong_induction_on with n ih
have hn' : 0 < n := pos_of_gt hn
have hn'' : 1 < n := one_lt_two.trans hn
have := prod_cyclotomic_eq_geom_sum hn' R
apply_fun eval x at this
rw [← cons_self_properDivisors hn'.ne', Finset.erase_cons_of_ne _ hn''.ne', Finset.prod_cons,
eval_mul, eval_... | 40 |
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.Geometry.RingedSpace.SheafedSpace
import Mathlib.Topology.Sheaves.Stalks
import Mathlib.Algebra.Category.Ring.Colimits
import Mathlib.Algebra.Category.Ring.Limits
#align_import algebraic_geometry.ringed_space from "leanprover-community/mathlib"@"5dc... | Mathlib/Geometry/RingedSpace/Basic.lean | 84 | 125 | theorem isUnit_of_isUnit_germ (U : Opens X) (f : X.presheaf.obj (op U))
(h : ∀ x : U, IsUnit (X.presheaf.germ x f)) : IsUnit f := by |
-- We pick a cover of `U` by open sets `V x`, such that `f` is a unit on each `V x`.
choose V iVU m h_unit using fun x : U => X.isUnit_res_of_isUnit_germ U f x (h x)
have hcover : U ≤ iSup V := by
intro x hxU
-- Porting note: in Lean3 `rw` is sufficient
erw [Opens.mem_iSup]
exact ⟨⟨x, hxU⟩, m ⟨x,... | 40 |
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.MeasureTheory.Integral.FundThmCalculus
#align_import analysis.special_functions.non_integrable from "leanprover-community/mathlib"@"55ec6e9af7d3e0043f57e394cb06a72f6275273e"
open scoped MeasureTheory Topology Interval NNReal ENNReal
open MeasureTh... | Mathlib/Analysis/SpecialFunctions/NonIntegrable.lean | 52 | 96 | theorem not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter_aux
[CompleteSpace E] {f : ℝ → E} {g : ℝ → F}
{k : Set ℝ} (l : Filter ℝ) [NeBot l] [TendstoIxxClass Icc l l]
(hl : k ∈ l) (hd : ∀ᶠ x in l, DifferentiableAt ℝ f x) (hf : Tendsto (fun x => ‖f x‖) l atTop)
(hfg : deriv f =O[l] g) : ¬Inte... |
intro hgi
obtain ⟨C, hC₀, s, hsl, hsub, hfd, hg⟩ :
∃ (C : ℝ) (_ : 0 ≤ C), ∃ s ∈ l, (∀ x ∈ s, ∀ y ∈ s, [[x, y]] ⊆ k) ∧
(∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], DifferentiableAt ℝ f z) ∧
∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], ‖deriv f z‖ ≤ C * ‖g z‖ := by
rcases hfg.exists_nonneg with ⟨C, C₀, hC⟩
have... | 40 |
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.MeasureTheory.Function.AEMeasurableOrder
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Integral.Average
import Mathlib.MeasureTheory.Decomposition.Lebesgue
#align_import measure... | Mathlib/MeasureTheory/Covering/Differentiation.lean | 160 | 201 | theorem ae_eventually_measure_zero_of_singular (hρ : ρ ⟂ₘ μ) :
∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 0) := by |
have A : ∀ ε > (0 : ℝ≥0), ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, ρ a < ε * μ a := by
intro ε εpos
set s := {x | ¬∀ᶠ a in v.filterAt x, ρ a < ε * μ a} with hs
change μ s = 0
obtain ⟨o, _, ρo, μo⟩ : ∃ o : Set α, MeasurableSet o ∧ ρ o = 0 ∧ μ oᶜ = 0 := hρ
apply le_antisymm _ bot_le
calc
μ s ≤ μ (s... | 40 |
import Mathlib.Geometry.Manifold.VectorBundle.Tangent
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Classical Topology Manifold
open Set ChartedSpace
section DerivativesDefinitions
variable {𝕜 : Type*} ... | Mathlib/Geometry/Manifold/MFDeriv/Defs.lean | 134 | 177 | theorem differentiable_within_at_localInvariantProp :
(contDiffGroupoid ⊤ I).LocalInvariantProp (contDiffGroupoid ⊤ I')
(DifferentiableWithinAtProp I I') :=
{ is_local := by |
intro s x u f u_open xu
have : I.symm ⁻¹' (s ∩ u) ∩ Set.range I = I.symm ⁻¹' s ∩ Set.range I ∩ I.symm ⁻¹' u := by
simp only [Set.inter_right_comm, Set.preimage_inter]
rw [DifferentiableWithinAtProp, DifferentiableWithinAtProp, this]
symm
apply differentiableWithinAt_inter
ha... | 40 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Interval.Set.IsoIoo
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.UrysohnsBounded
#align_import topology.tietze_extension from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable {X Y : T... | Mathlib/Topology/TietzeExtension.lean | 220 | 262 | theorem exists_extension_norm_eq_of_closedEmbedding' (f : X →ᵇ ℝ) (e : C(X, Y))
(he : ClosedEmbedding e) : ∃ g : Y →ᵇ ℝ, ‖g‖ = ‖f‖ ∧ g.compContinuous e = f := by |
/- For the proof, we iterate `tietze_extension_step`. Each time we apply it to the difference
between the previous approximation and `f`. -/
choose F hF_norm hF_dist using fun f : X →ᵇ ℝ => tietze_extension_step f e he
set g : ℕ → Y →ᵇ ℝ := fun n => (fun g => g + F (f - g.compContinuous e))^[n] 0
have g0 :... | 41 |
import Mathlib.Probability.Kernel.Disintegration.Integral
open MeasureTheory Set Filter MeasurableSpace
open scoped ENNReal MeasureTheory Topology ProbabilityTheory
namespace ProbabilityTheory
variable {α β Ω : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
[MeasurableSpace Ω] [StandardBorelSpace Ω] ... | Mathlib/Probability/Kernel/Disintegration/Unique.lean | 81 | 124 | theorem eq_condKernel_of_measure_eq_compProd (κ : kernel α Ω) [IsFiniteKernel κ]
(hκ : ρ = ρ.fst ⊗ₘ κ) :
∀ᵐ x ∂ρ.fst, κ x = ρ.condKernel x := by |
-- The idea is to transport the question to `ℝ` from `Ω` using `embeddingReal`
-- and then construct a measure on `α × ℝ`
let f := embeddingReal Ω
have hf := measurableEmbedding_embeddingReal Ω
set ρ' : Measure (α × ℝ) := ρ.map (Prod.map id f) with hρ'def
have hρ' : ρ'.fst = ρ.fst := by
ext s hs
rw... | 41 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
#align_import algebra.char_p.local_ring from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
| Mathlib/Algebra/CharP/LocalRing.lean | 25 | 67 | theorem charP_zero_or_prime_power (R : Type*) [CommRing R] [LocalRing R] (q : ℕ)
[char_R_q : CharP R q] : q = 0 ∨ IsPrimePow q := by |
-- Assume `q := char(R)` is not zero.
apply or_iff_not_imp_left.2
intro q_pos
let K := LocalRing.ResidueField R
haveI RM_char := ringChar.charP K
let r := ringChar K
let n := q.factorization r
-- `r := char(R/m)` is either prime or zero:
cases' CharP.char_is_prime_or_zero K r with r_prime r_zero
· ... | 41 |
import Mathlib.Algebra.Polynomial.Degree.CardPowDegree
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue
import Mathlib.RingTheory.Ideal.LocalRing
#align_import number_theory.class_number.admissible_card_pow_degree from "leanprover-community/mathlib"@"0b... | Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean | 106 | 149 | theorem exists_approx_polynomial {b : Fq[X]} (hb : b ≠ 0) {ε : ℝ} (hε : 0 < ε)
(A : Fin (Fintype.card Fq ^ ⌈-log ε / log (Fintype.card Fq)⌉₊).succ → Fq[X]) :
∃ i₀ i₁, i₀ ≠ i₁ ∧ (cardPowDegree (A i₁ % b - A i₀ % b) : ℝ) < cardPowDegree b • ε := by |
have hbε : 0 < cardPowDegree b • ε := by
rw [Algebra.smul_def, eq_intCast]
exact mul_pos (Int.cast_pos.mpr (AbsoluteValue.pos _ hb)) hε
have one_lt_q : 1 < Fintype.card Fq := Fintype.one_lt_card
have one_lt_q' : (1 : ℝ) < Fintype.card Fq := by assumption_mod_cast
have q_pos : 0 < Fintype.card Fq := by ... | 41 |
import Mathlib.Algebra.Homology.Additive
import Mathlib.AlgebraicTopology.MooreComplex
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Idempotents.FunctorCategories
#align_import algebraic_topology.alternating_face_map_complex from "leanprover-c... | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | 70 | 112 | theorem d_squared (n : ℕ) : objD X (n + 1) ≫ objD X n = 0 := by |
-- we start by expanding d ≫ d as a double sum
dsimp
simp only [comp_sum, sum_comp, ← Finset.sum_product']
-- then, we decompose the index set P into a subset S and its complement Sᶜ
let P := Fin (n + 2) × Fin (n + 3)
let S := Finset.univ.filter fun ij : P => (ij.2 : ℕ) ≤ (ij.1 : ℕ)
erw [← Finset.sum_add... | 42 |
import Mathlib.RingTheory.Adjoin.FG
#align_import ring_theory.adjoin.tower from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Pointwise
universe u v w u₁
variable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁)
section
open scoped Classical
theorem Algebra.fg_trans' {R S... | Mathlib/RingTheory/Adjoin/Tower.lean | 92 | 135 | theorem exists_subalgebra_of_fg (hAC : (⊤ : Subalgebra A C).FG) (hBC : (⊤ : Submodule B C).FG) :
∃ B₀ : Subalgebra A B, B₀.FG ∧ (⊤ : Submodule B₀ C).FG := by |
cases' hAC with x hx
cases' hBC with y hy
have := hy
simp_rw [eq_top_iff', mem_span_finset] at this
choose f hf using this
let s : Finset B := Finset.image₂ f (x ∪ y * y) y
have hxy :
∀ xi ∈ x, xi ∈ span (Algebra.adjoin A (↑s : Set B)) (↑(insert 1 y : Finset C) : Set C) :=
fun xi hxi =>
hf xi... | 42 |
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Order.Filter.Curry
#align_import analysis.calculus.uniform_limits_deriv from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Filter
open scoped uniformity Filter Topology
section L... | Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean | 176 | 220 | theorem uniformCauchySeqOn_ball_of_fderiv {r : ℝ} (hf' : UniformCauchySeqOn f' l (Metric.ball x r))
(hf : ∀ n : ι, ∀ y : E, y ∈ Metric.ball x r → HasFDerivAt (f n) (f' n y) y)
(hfg : Cauchy (map (fun n => f n x) l)) : UniformCauchySeqOn f l (Metric.ball x r) := by |
letI : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 _
have : NeBot l := (cauchy_map_iff.1 hfg).1
rcases le_or_lt r 0 with (hr | hr)
· simp only [Metric.ball_eq_empty.2 hr, UniformCauchySeqOn, Set.mem_empty_iff_false,
IsEmpty.forall_iff, eventually_const, imp_true_iff]
rw [SeminormedAddGroup.unif... | 42 |
import Mathlib.Topology.Separation
import Mathlib.Algebra.Group.Defs
#align_import topology.algebra.semigroup from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
@[to_additive
"Any nonempty compact Hausdorff additive semigroup where right-addition is continuous
contains an ... | Mathlib/Topology/Algebra/Semigroup.lean | 27 | 72 | theorem exists_idempotent_of_compact_t2_of_continuous_mul_left {M} [Nonempty M] [Semigroup M]
[TopologicalSpace M] [CompactSpace M] [T2Space M]
(continuous_mul_left : ∀ r : M, Continuous (· * r)) : ∃ m : M, m * m = m := by |
/- We apply Zorn's lemma to the poset of nonempty closed subsemigroups of `M`.
It will turn out that any minimal element is `{m}` for an idempotent `m : M`. -/
let S : Set (Set M) :=
{ N | IsClosed N ∧ N.Nonempty ∧ ∀ (m) (_ : m ∈ N) (m') (_ : m' ∈ N), m * m' ∈ N }
rsuffices ⟨N, ⟨N_closed, ⟨m, hm⟩, N_mul... | 43 |
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheory.Ordinal.Principal
import Mathlib.Tactic.Linarith
#align_import set_theory.cardinal.ordinal from "leanprover-community/mathlib"@"7c2ce0c2da15516b4e65d0c9e254bb6dc93abd1f"
noncomputa... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 500 | 543 | theorem mul_eq_self {c : Cardinal} (h : ℵ₀ ≤ c) : c * c = c := by |
refine le_antisymm ?_ (by simpa only [mul_one] using mul_le_mul_left' (one_le_aleph0.trans h) c)
-- the only nontrivial part is `c * c ≤ c`. We prove it inductively.
refine Acc.recOn (Cardinal.lt_wf.apply c) (fun c _ => Quotient.inductionOn c fun α IH ol => ?_) h
-- consider the minimal well-order `r` on `α` (... | 43 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Interval.Set.IsoIoo
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.UrysohnsBounded
#align_import topology.tietze_extension from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable {X Y : T... | Mathlib/Topology/TietzeExtension.lean | 169 | 213 | theorem tietze_extension_step (f : X →ᵇ ℝ) (e : C(X, Y)) (he : ClosedEmbedding e) :
∃ g : Y →ᵇ ℝ, ‖g‖ ≤ ‖f‖ / 3 ∧ dist (g.compContinuous e) f ≤ 2 / 3 * ‖f‖ := by |
have h3 : (0 : ℝ) < 3 := by norm_num1
have h23 : 0 < (2 / 3 : ℝ) := by norm_num1
-- In the trivial case `f = 0`, we take `g = 0`
rcases eq_or_ne f 0 with (rfl | hf)
· use 0
simp
replace hf : 0 < ‖f‖ := norm_pos_iff.2 hf
/- Otherwise, the closed sets `e '' (f ⁻¹' (Iic (-‖f‖ / 3)))` and `e '' (f ⁻¹' (I... | 43 |
import Mathlib.Data.Real.Pi.Bounds
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
-- TODO. Rewrite some of the FLT results on the disciminant using the definitions and results of
-- this file
namespace NumberField
open FiniteDimensional NumberField NumberField.InfinitePlace Matrix
open sco... | Mathlib/NumberTheory/NumberField/Discriminant.lean | 105 | 151 | theorem exists_ne_zero_mem_ideal_of_norm_le_mul_sqrt_discr (I : (FractionalIdeal (𝓞 K)⁰ K)ˣ) :
∃ a ∈ (I : FractionalIdeal (𝓞 K)⁰ K), a ≠ 0 ∧
|Algebra.norm ℚ (a:K)| ≤ FractionalIdeal.absNorm I.1 * (4 / π) ^ NrComplexPlaces K *
(finrank ℚ K).factorial / (finrank ℚ K) ^ (finrank ℚ K) * Real.sqrt |discr... |
-- The smallest possible value for `exists_ne_zero_mem_ideal_of_norm_le`
let B := (minkowskiBound K I * (convexBodySumFactor K)⁻¹).toReal ^ (1 / (finrank ℚ K : ℝ))
have h_le : (minkowskiBound K I) ≤ volume (convexBodySum K B) := by
refine le_of_eq ?_
rw [convexBodySum_volume, ← ENNReal.ofReal_pow (by pos... | 43 |
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Limits.Shapes.KernelPair
#align_import category_theory.adhesive from "leanprover-community/mathlib"@"afff1f24a6b68d0077c9d63782a1d093e337758c"
namespace CategoryTheory
open Limits
universe v' u' v u
variable {J : Type v'} [Category.{u'} J] {... | Mathlib/CategoryTheory/Adhesive.lean | 66 | 110 | theorem IsPushout.isVanKampen_iff (H : IsPushout f g h i) :
H.IsVanKampen ↔ IsVanKampenColimit (PushoutCocone.mk h i H.w) := by |
constructor
· intro H F' c' α fα eα hα
refine Iff.trans ?_
((H (F'.map WalkingSpan.Hom.fst) (F'.map WalkingSpan.Hom.snd) (c'.ι.app _) (c'.ι.app _)
(α.app _) (α.app _) (α.app _) fα (by convert hα WalkingSpan.Hom.fst)
(by convert hα WalkingSpan.Hom.snd) ?_ ?_ ?_).trans ?_)
· have ... | 43 |
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.Topology.Instances.Matrix
import Mathlib.Topology.Algebra.Module.FiniteDimension
#align_import number_theory.modular from "leanprover-community/mat... | Mathlib/NumberTheory/Modular.lean | 117 | 161 | theorem tendsto_normSq_coprime_pair :
Filter.Tendsto (fun p : Fin 2 → ℤ => normSq ((p 0 : ℂ) * z + p 1)) cofinite atTop := by |
-- using this instance rather than the automatic `Function.module` makes unification issues in
-- `LinearEquiv.closedEmbedding_of_injective` less bad later in the proof.
letI : Module ℝ (Fin 2 → ℝ) := NormedSpace.toModule
let π₀ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 0
let π₁ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := Linear... | 43 |
import Mathlib.MeasureTheory.Integral.FundThmCalculus
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
#align_import analysis.special_functions.integrals from "leanprover-community/mathlib"@"011... | Mathlib/Analysis/SpecialFunctions/Integrals.lean | 120 | 164 | theorem intervalIntegrable_cpow {r : ℂ} (h : 0 ≤ r.re ∨ (0 : ℝ) ∉ [[a, b]]) :
IntervalIntegrable (fun x : ℝ => (x : ℂ) ^ r) μ a b := by |
by_cases h2 : (0 : ℝ) ∉ [[a, b]]
· -- Easy case #1: 0 ∉ [a, b] -- use continuity.
refine (ContinuousAt.continuousOn fun x hx => ?_).intervalIntegrable
exact Complex.continuousAt_ofReal_cpow_const _ _ (Or.inr <| ne_of_mem_of_not_mem hx h2)
rw [eq_false h2, or_false_iff] at h
rcases lt_or_eq_of_le h with... | 43 |
import Mathlib.FieldTheory.Finite.Basic
#align_import field_theory.chevalley_warning from "leanprover-community/mathlib"@"e001509c11c4d0f549d91d89da95b4a0b43c714f"
universe u v
section FiniteField
open MvPolynomial
open Function hiding eval
open Finset FiniteField
variable {K σ ι : Type*} [Fintype K] [Field ... | Mathlib/FieldTheory/ChevalleyWarning.lean | 53 | 97 | theorem MvPolynomial.sum_eval_eq_zero (f : MvPolynomial σ K)
(h : f.totalDegree < (q - 1) * Fintype.card σ) : ∑ x, eval x f = 0 := by |
haveI : DecidableEq K := Classical.decEq K
calc
∑ x, eval x f = ∑ x : σ → K, ∑ d ∈ f.support, f.coeff d * ∏ i, x i ^ d i := by
simp only [eval_eq']
_ = ∑ d ∈ f.support, ∑ x : σ → K, f.coeff d * ∏ i, x i ^ d i := sum_comm
_ = 0 := sum_eq_zero ?_
intro d hd
obtain ⟨i, hi⟩ : ∃ i, d i < q - 1 := ... | 43 |
import Mathlib.NumberTheory.Cyclotomic.Discriminant
import Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral
import Mathlib.RingTheory.Ideal.Norm
#align_import number_theory.cyclotomic.rat from "leanprover-community/mathlib"@"b353176c24d96c23f0ce1cc63efc3f55019702d9"
universe u
open Algebra IsCyclotomicExtensio... | Mathlib/NumberTheory/Cyclotomic/Rat.lean | 74 | 119 | theorem isIntegralClosure_adjoin_singleton_of_prime_pow [hcycl : IsCyclotomicExtension {p ^ k} ℚ K]
(hζ : IsPrimitiveRoot ζ ↑(p ^ k)) : IsIntegralClosure (adjoin ℤ ({ζ} : Set K)) ℤ K := by |
refine ⟨Subtype.val_injective, @fun x => ⟨fun h => ⟨⟨x, ?_⟩, rfl⟩, ?_⟩⟩
swap
· rintro ⟨y, rfl⟩
exact
IsIntegral.algebraMap
((le_integralClosure_iff_isIntegral.1
(adjoin_le_integralClosure (hζ.isIntegral (p ^ k).pos))).isIntegral _)
let B := hζ.subOnePowerBasis ℚ
have hint : IsInte... | 44 |
import Mathlib.ModelTheory.Quotients
import Mathlib.Order.Filter.Germ
import Mathlib.Order.Filter.Ultrafilter
#align_import model_theory.ultraproducts from "leanprover-community/mathlib"@"f1ae620609496a37534c2ab3640b641d5be8b6f0"
universe u v
variable {α : Type*} (M : α → Type*) (u : Ultrafilter α)
open FirstOr... | Mathlib/ModelTheory/Ultraproducts.lean | 96 | 144 | theorem boundedFormula_realize_cast {β : Type*} {n : ℕ} (φ : L.BoundedFormula β n)
(x : β → ∀ a, M a) (v : Fin n → ∀ a, M a) :
(φ.Realize (fun i : β => (x i : (u : Filter α).Product M))
(fun i => (v i : (u : Filter α).Product M))) ↔
∀ᶠ a : α in u, φ.Realize (fun i : β => x i a) fun i => v i a := b... |
letI := (u : Filter α).productSetoid M
induction' φ with _ _ _ _ _ _ _ _ m _ _ ih ih' k φ ih
· simp only [BoundedFormula.Realize, eventually_const]
· have h2 : ∀ a : α, (Sum.elim (fun i : β => x i a) fun i => v i a) = fun i => Sum.elim x v i a :=
fun a => funext fun i => Sum.casesOn i (fun i => rfl) fun ... | 44 |
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Analysis.SpecificLimits.Basic
#align_import analysis.box_integral.box.subbox_induction from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Finset Function Filter Metric Classical Topology Filter ENNReal
noncomputable... | Mathlib/Analysis/BoxIntegral/Box/SubboxInduction.lean | 122 | 170 | theorem subbox_induction_on' {p : Box ι → Prop} (I : Box ι)
(H_ind : ∀ J ≤ I, (∀ s, p (splitCenterBox J s)) → p J)
(H_nhds : ∀ z ∈ Box.Icc I, ∃ U ∈ 𝓝[Box.Icc I] z, ∀ J ≤ I, ∀ (m : ℕ), z ∈ Box.Icc J →
Box.Icc J ⊆ U → (∀ i, J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J) :
p I := by |
by_contra hpI
-- First we use `H_ind` to construct a decreasing sequence of boxes such that `∀ m, ¬p (J m)`.
replace H_ind := fun J hJ ↦ not_imp_not.2 (H_ind J hJ)
simp only [exists_imp, not_forall] at H_ind
choose! s hs using H_ind
set J : ℕ → Box ι := fun m ↦ (fun J ↦ splitCenterBox J (s J))^[m] I
have... | 44 |
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 221 | 266 | theorem convexBodyLT'_volume :
volume (convexBodyLT' K f w₀) = convexBodyLT'Factor K * ∏ w, (f w) ^ (mult w) := by |
have vol_box : ∀ B : ℝ≥0, volume {x : ℂ | |x.re| < 1 ∧ |x.im| < B^2} = 4*B^2 := by
intro B
rw [← (Complex.volume_preserving_equiv_real_prod.symm).measure_preimage]
· simp_rw [Set.preimage_setOf_eq, Complex.measurableEquivRealProd_symm_apply]
rw [show {a : ℝ × ℝ | |a.1| < 1 ∧ |a.2| < B ^ 2} =
... | 44 |
import Mathlib.Analysis.Convex.Combination
import Mathlib.LinearAlgebra.AffineSpace.Independent
import Mathlib.Tactic.FieldSimp
#align_import analysis.convex.caratheodory from "leanprover-community/mathlib"@"e6fab1dc073396d45da082c644642c4f8bff2264"
open Set Finset
universe u
variable {𝕜 : Type*} {E : Type u} ... | Mathlib/Analysis/Convex/Caratheodory.lean | 52 | 98 | theorem mem_convexHull_erase [DecidableEq E] {t : Finset E} (h : ¬AffineIndependent 𝕜 ((↑) : t → E))
{x : E} (m : x ∈ convexHull 𝕜 (↑t : Set E)) :
∃ y : (↑t : Set E), x ∈ convexHull 𝕜 (↑(t.erase y) : Set E) := by |
simp only [Finset.convexHull_eq, mem_setOf_eq] at m ⊢
obtain ⟨f, fpos, fsum, rfl⟩ := m
obtain ⟨g, gcombo, gsum, gpos⟩ := exists_nontrivial_relation_sum_zero_of_not_affine_ind h
replace gpos := exists_pos_of_sum_zero_of_exists_nonzero g gsum gpos
clear h
let s := @Finset.filter _ (fun z => 0 < g z) (fun _ =... | 44 |
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.SetTheory.Cardinal.Subfield
import Mathlib.LinearAlgebra.Dimension.RankNullity
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomput... | Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 239 | 283 | theorem max_aleph0_card_le_rank_fun_nat : max ℵ₀ #K ≤ Module.rank K (ℕ → K) := by |
have aleph0_le : ℵ₀ ≤ Module.rank K (ℕ → K) := (rank_finsupp_self K ℕ).symm.trans_le
(Finsupp.lcoeFun.rank_le_of_injective <| by exact DFunLike.coe_injective)
refine max_le aleph0_le ?_
obtain card_K | card_K := le_or_lt #K ℵ₀
· exact card_K.trans aleph0_le
by_contra!
obtain ⟨⟨ιK, bK⟩⟩ := Module.Free.e... | 44 |
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable s... | Mathlib/MeasureTheory/Constructions/Prod/Integral.lean | 77 | 122 | theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : StronglyMeasurable fun x => ∫ y, f x y ∂ν := by |
by_cases hE : CompleteSpace E; swap; · simp [integral, hE, stronglyMeasurable_const]
borelize E
haveI : SeparableSpace (range (uncurry f) ∪ {0} : Set E) :=
hf.separableSpace_range_union_singleton
let s : ℕ → SimpleFunc (α × β) E :=
SimpleFunc.approxOn _ hf.measurable (range (uncurry f) ∪ {0}) 0 (by sim... | 44 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.Group.Indicator
import Mathlib.Order.LiminfLimsup
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Data.Set.La... | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | 339 | 385 | theorem Antitone.map_limsSup_of_continuousAt {F : Filter R} [NeBot F] {f : R → S}
(f_decr : Antitone f) (f_cont : ContinuousAt f F.limsSup)
(bdd_above : F.IsBounded (· ≤ ·) := by | isBoundedDefault)
(bdd_below : F.IsBounded (· ≥ ·) := by isBoundedDefault) :
f F.limsSup = F.liminf f := by
have cobdd : F.IsCobounded (· ≤ ·) := bdd_below.isCobounded_flip
apply le_antisymm
· rw [limsSup, f_decr.map_sInf_of_continuousAt' f_cont bdd_above cobdd]
apply le_of_forall_lt
intro c hc
... | 45 |
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Join
#align_import analysis.convex.stone_separation from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
open Set
variable {𝕜 E ι : Type*} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s t : Set E}
| Mathlib/Analysis/Convex/StoneSeparation.lean | 30 | 77 | theorem not_disjoint_segment_convexHull_triple {p q u v x y z : E} (hz : z ∈ segment 𝕜 x y)
(hu : u ∈ segment 𝕜 x p) (hv : v ∈ segment 𝕜 y q) :
¬Disjoint (segment 𝕜 u v) (convexHull 𝕜 {p, q, z}) := by |
rw [not_disjoint_iff]
obtain ⟨az, bz, haz, hbz, habz, rfl⟩ := hz
obtain rfl | haz' := haz.eq_or_lt
· rw [zero_add] at habz
rw [zero_smul, zero_add, habz, one_smul]
refine ⟨v, by apply right_mem_segment, segment_subset_convexHull ?_ ?_ hv⟩ <;> simp
obtain ⟨av, bv, hav, hbv, habv, rfl⟩ := hv
obtain r... | 45 |
import Mathlib.Analysis.Convex.Body
import Mathlib.Analysis.Convex.Measure
import Mathlib.MeasureTheory.Group.FundamentalDomain
#align_import measure_theory.group.geometry_of_numbers from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
namespace MeasureTheory
open ENNReal FiniteDimensio... | Mathlib/MeasureTheory/Group/GeometryOfNumbers.lean | 92 | 142 | theorem exists_ne_zero_mem_lattice_of_measure_mul_two_pow_le_measure [NormedAddCommGroup E]
[NormedSpace ℝ E] [BorelSpace E] [FiniteDimensional ℝ E] [Nontrivial E] [IsAddHaarMeasure μ]
{L : AddSubgroup E} [Countable L] [DiscreteTopology L] (fund : IsAddFundamentalDomain L F μ)
(h_symm : ∀ x ∈ s, -x ∈ s) (h_... |
have h_mes : μ s ≠ 0 := by
intro hμ
suffices μ F = 0 from fund.measure_ne_zero (NeZero.ne μ) this
rw [hμ, le_zero_iff, mul_eq_zero] at h
exact h.resolve_right <| pow_ne_zero _ two_ne_zero
have h_nemp : s.Nonempty := nonempty_of_measure_ne_zero h_mes
let u : ℕ → ℝ≥0 := (exists_seq_strictAnti_tends... | 45 |
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