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e320e82 | 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 | import OAI.Combinatorics.HalvingLines.Calculus
namespace OAI
namespace PlanarHalving
namespace RankLimit
open Filter MeasureTheory Set Metric
open scoped Topology
abbrev RatPoint := ℚ × ℚ
def ratPoint (q : RatPoint) : Point := ((q.1 : ℝ), (q.2 : ℝ))
noncomputable def lowerGrid (g : RatPoint → unitInterval) (x : Point) : unitInterval :=
⨆ q : RatPoint, if (q.1 : ℝ) < x.1 ∧ (q.2 : ℝ) < x.2 then g q else ⊥
lemma lowerGrid_monotone (g : RatPoint → unitInterval) : Monotone (lowerGrid g) := by
intro x y hxy
apply iSup_le
intro q
by_cases hq : (q.1 : ℝ) < x.1 ∧ (q.2 : ℝ) < x.2
· have hqy : (q.1 : ℝ) < y.1 ∧ (q.2 : ℝ) < y.2 :=
⟨hq.1.trans_le hxy.1, hq.2.trans_le hxy.2⟩
simpa only [lowerGrid, ite_eq_left hq, ite_eq_left hqy] using
(le_iSup (fun r : RatPoint =>
if (r.1 : ℝ) < y.1 ∧ (r.2 : ℝ) < y.2 then g r else ⊥) q)
· simp only [ite_eq_right hq, bot_le]
lemma lowerGrid_measurable (g : RatPoint → unitInterval) : Measurable (lowerGrid g) := by
apply Measurable.iSup
intro q
exact Measurable.ite
((isOpen_lt continuous_const continuous_fst).inter
(isOpen_lt continuous_const continuous_snd)).measurableSet
measurable_const measurable_const
lemma grid_le_lowerGrid {g : RatPoint → unitInterval} {q : RatPoint} {x : Point}
(hq : (q.1 : ℝ) < x.1 ∧ (q.2 : ℝ) < x.2) : g q ≤ lowerGrid g x := by
simpa only [lowerGrid, ite_eq_left hq] using
(le_iSup (fun r : RatPoint =>
if (r.1 : ℝ) < x.1 ∧ (r.2 : ℝ) < x.2 then g r else ⊥) q)
theorem moving_tendsto_of_grid {f : ℕ → Point → unitInterval}
(hf : ∀ n, Monotone (f n)) {g : RatPoint → unitInterval}
(hg : ∀ q, Tendsto (fun n => f n (ratPoint q)) atTop (𝓝 (g q)))
{x : Point} (hc : ContinuousAt (lowerGrid g) x) {y : ℕ → Point}
(hy : Tendsto y atTop (𝓝 x)) :
Tendsto (fun n => f n (y n)) atTop (𝓝 (lowerGrid g x)) := by
rw [tendsto_order]
constructor
· intro a ha
obtain ⟨q, hq⟩ := lt_iSup_iff.mp ha
by_cases hqx : (q.1 : ℝ) < x.1 ∧ (q.2 : ℝ) < x.2
· simp only [ite_eq_left hqx] at hq
filter_upwards [(tendsto_order.mp (hg q)).1 a hq,
(tendsto_order.mp ((continuous_fst.tendsto x).comp hy)).1 _ hqx.1,
(tendsto_order.mp ((continuous_snd.tendsto x).comp hy)).1 _ hqx.2] with n hn hn₁ hn₂
exact hn.trans_le (hf n ⟨hn₁.le, hn₂.le⟩)
· simp only [ite_eq_right hqx, not_lt_bot] at hq
· intro a ha
have hn := hc.preimage_mem_nhds (isOpen_Iio.mem_nhds ha)
obtain ⟨δ, hδ, hδU⟩ := Metric.mem_nhds_iff.mp hn
let b : Point := x + (δ / 2, δ / 2)
have hb : b ∈ ball x δ := by
rw [Metric.mem_ball, dist_eq_norm]
simp only [b, add_sub_cancel_left, Prod.norm_def, Real.norm_eq_abs,
abs_of_pos (half_pos hδ), max_self]
linarith only [hδ]
have hba : lowerGrid g b < a := hδU hb
obtain ⟨r₁, hxr₁, hr₁b⟩ := exists_rat_btwn
(show x.1 < b.1 by dsimp [b]; linarith only [hδ])
obtain ⟨r₂, hxr₂, hr₂b⟩ := exists_rat_btwn
(show x.2 < b.2 by dsimp [b]; linarith only [hδ])
let r : RatPoint := (r₁, r₂)
have hra : g r < a := (grid_le_lowerGrid ⟨hr₁b, hr₂b⟩).trans_lt hba
filter_upwards [(tendsto_order.mp (hg r)).2 a hra,
(tendsto_order.mp ((continuous_fst.tendsto x).comp hy)).2 _ hxr₁,
(tendsto_order.mp ((continuous_snd.tendsto x).comp hy)).2 _ hxr₂] with n hn hn₁ hn₂
exact (hf n ⟨hn₁.le, hn₂.le⟩).trans_lt hn
theorem monotone_grid_subsequence {ι : Type*} [Countable ι]
(f : ℕ → ι → Point → unitInterval) (hf : ∀ n i, Monotone (f n i)) :
∃ (φ : ℕ → ℕ) (F : ι → Point → unitInterval), StrictMono φ ∧
(∀ i, Monotone (F i) ∧ Measurable (F i)) ∧
(∀ i x, ContinuousAt (F i) x → ∀ y : ℕ → Point,
Tendsto y atTop (𝓝 x) →
Tendsto (fun n => f (φ n) i (y n)) atTop (𝓝 (F i x))) := by
obtain ⟨g, φ, hφ, hg⟩ := CompactSpace.tendsto_subseq
(fun n i q => f n i (ratPoint q))
refine ⟨φ, fun i => lowerGrid (g i), hφ,
fun i => ⟨lowerGrid_monotone _, lowerGrid_measurable _⟩, ?_⟩
intro i x hx y hy
apply moving_tendsto_of_grid (fun n => hf (φ n) i) _ hx hy
intro q
exact ((tendsto_pi_nhds.mp hg) i).apply_nhds q
end RankLimit
namespace Transversality
open MeasureTheory Set
open scoped MeasureTheory ENNReal
theorem critical_projection_null {Z : Type*} [MeasurableSpace Z]
(σ : Measure Z) (coordinate : Z → Point) (hcoordinate : Measurable coordinate)
(hdom : σ.map coordinate ≪ volume) (p : Z → Point) (F : Point → Point)
(S : Set Z) (hidentity : ∀ z ∈ S, coordinate z = F (p z))
(hdiff : ∀ z ∈ S, DifferentiableAt ℝ F (p z))
(hsing : ∀ z ∈ S, (fderiv ℝ F (p z)).det = 0) : σ S = 0 := by
obtain ⟨N, hsub, hN, hnull⟩ := exists_measurable_superset_of_null (critical_image_null F)
apply measure_mono_null (t := coordinate ⁻¹' N)
· intro z hz
change coordinate z ∈ N
rw [hidentity z hz]
exact hsub ⟨p z, ⟨hdiff z hz, hsing z hz⟩, rfl⟩
· rw [← Measure.map_apply hcoordinate hN]
exact hdom hnull
theorem independent_gradients_ae {Z ι : Type*} [MeasurableSpace Z] [Countable ι]
(ν : Measure Z) (h : Z → ℝ≥0∞) (hh : Measurable h)
(σ : ι → Measure Z) (hν : ν.withDensity h ≪ Measure.sum σ)
(coordinate : ι → Z → Point) (hcoordinate : ∀ i, Measurable (coordinate i))
(hdom : ∀ i, (σ i).map (coordinate i) ≪ volume)
(p : Z → Point) (F : ι → Point → Point) (S : Set Z)
(hidentity : ∀ i z, z ∈ S → coordinate i z = F i (p z))
(hdiff : ∀ i z, z ∈ S → DifferentiableAt ℝ (F i) (p z)) :
∀ᵐ z ∂ν, z ∈ S → h z ≠ 0 → ∃ i, (fderiv ℝ (F i) (p z)).det ≠ 0 := by
have hbad : (ν.withDensity h) {z | z ∈ S ∧ ∀ i, (fderiv ℝ (F i) (p z)).det = 0} = 0 := by
apply hν
rw [Measure.sum_apply_eq_zero]
intro i
apply critical_projection_null (σ i) (coordinate i) (hcoordinate i) (hdom i) p
(F i) _
· intro z hz
exact hidentity i z hz.1
· intro z hz
exact hdiff i z hz.1
· intro z hz
exact hz.2 i
have hgood : ∀ᵐ z ∂ν.withDensity h,
z ∈ S → ∃ i, (fderiv ℝ (F i) (p z)).det ≠ 0 := by
apply ae_iff.mpr
simpa only [Classical.not_imp, not_exists, not_not] using hbad
have hg := (ae_withDensity_iff hh).mp hgood
filter_upwards [hg] with z hz hzS hzh
exact hz hzh hzS
end Transversality
end PlanarHalving
end OAI
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