File size: 4,182 Bytes
92416a2 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 | import Mathlib
namespace OAI
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
namespace VertexCover.AffineSelector
open Filter MeasureTheory
open scoped Topology
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
variable {ι : Type*} [Fintype ι]
def Regular (a : ι → E →L[ℝ] ℝ) (b : ι → ℝ) (x : E) : Prop :=
∀ i j, a i x + b i = a j x + b j → a i = a j
theorem locally_affine (a : ι → E →L[ℝ] ℝ) (b : ι → ℝ)
{f : E → ℝ} (hf : Continuous f)
(hselect : ∀ y, ∃ i, f y = a i y + b i) {x : E}
(hx : Regular a b x) :
∃ i, f =ᶠ[𝓝 x] (fun y => a i y + b i) := by
obtain ⟨i, hi⟩ := hselect x
refine ⟨i, ?_⟩
have hj : ∀ j, ∀ᶠ y in 𝓝 x,
f y = a j y + b j → a j y + b j = a i y + b i := by
intro j
by_cases h : f x = a j x + b j
· have ha : a i = a j := hx i j (hi.symm.trans h)
have hb : b i = b j := by rw [ha] at hi; linarith
exact Filter.Eventually.of_forall (fun y _ => by rw [ha, hb])
· have hn : f x - (a j x + b j) ≠ 0 := sub_ne_zero.mpr h
have he := (hf.continuousAt.sub ((a j).continuous.continuousAt.add_const (b j))).eventually_ne hn
filter_upwards [he] with y hy
intro heq
exact (hy (sub_eq_zero.mpr heq)).elim
filter_upwards [Filter.eventually_all.mpr hj] with y hy
obtain ⟨j, hj⟩ := hselect y
exact hj.trans (hy j hj)
theorem hasFDerivAt (a : ι → E →L[ℝ] ℝ) (b : ι → ℝ)
{f : E → ℝ} (hf : Continuous f)
(hselect : ∀ y, ∃ i, f y = a i y + b i) {x : E}
(hx : Regular a b x) : ∃ i, HasFDerivAt f (a i) x := by
obtain ⟨i, hi⟩ := locally_affine a b hf hselect hx
exact ⟨i, ((a i).hasFDerivAt.add_const (b i)).congr_of_eventuallyEq hi⟩
variable [MeasurableSpace E] [BorelSpace E]
theorem regular_measurableSet (a : ι → E →L[ℝ] ℝ) (b : ι → ℝ) :
MeasurableSet {x | Regular a b x} := by
classical
simp only [Regular, Set.ofPred_forall]
apply MeasurableSet.iInter; intro i
apply MeasurableSet.iInter; intro j
by_cases h : a i = a j
· simp [h]
· simp only [h, imp_false]
exact (measurableSet_eq_fun ((a i).continuous.measurable.add_const (b i))
((a j).continuous.measurable.add_const (b j))).compl
theorem affine_tie_null [FiniteDimensional ℝ E] (μ : Measure E) [μ.IsAddHaarMeasure]
(a a' : E →L[ℝ] ℝ) (b b' : ℝ) (ha : a ≠ a') :
μ {x | a x + b = a' x + b'} = 0 := by
let S : AffineSubspace ℝ E := AffineSubspace.comap
(a-a').toLinearMap.toAffineMap (affineSpan ℝ ({b'-b} : Set ℝ))
have hmem : ∀ x, x ∈ S ↔ a x + b = a' x + b' := by
intro x
simp only [S, AffineSubspace.mem_comap, AffineSubspace.mem_affineSpan_singleton]
change a x - a' x = b'-b ↔ _
constructor <;> intro h <;> linarith
have hproper : S ≠ ⊤ := by
intro htop
have hall : ∀ x, a x + b = a' x + b' := fun x =>
(hmem x).mp (by rw [htop]; exact AffineSubspace.mem_top ℝ E x)
have hz : b = b' := by simpa using hall 0
apply ha
ext x
linarith [hall x]
have heq : (S : Set E) = {x | a x + b = a' x + b'} := Set.ext hmem
rw [← heq]
exact Measure.addHaar_affineSubspace μ S hproper
theorem ae_regular [FiniteDimensional ℝ E] (μ : Measure E) [μ.IsAddHaarMeasure]
(a : ι → E →L[ℝ] ℝ) (b : ι → ℝ) : ∀ᵐ x ∂μ, Regular a b x := by
classical
apply Filter.eventually_all.mpr
intro i
apply Filter.eventually_all.mpr
intro j
by_cases hij : a i = a j
· exact Filter.Eventually.of_forall (fun _ _ => hij)
· have hne : ∀ᵐ x ∂μ, a i x + b i ≠ a j x + b j := by
rw [ae_iff]
simpa only [not_not] using affine_tie_null μ (a i) (a j) (b i) (b j) hij
filter_upwards [hne] with x hx
exact fun h => (hx h).elim
end VertexCover.AffineSelector
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end OAI
|