| import OAI.Analysis.CircleDomains.Selection.PuncturedHitProduct |
| import OAI.Analysis.CircleDomains.Selection.CountableMeetingBound |
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|
| namespace OAI |
|
|
| /-! The quantitative dust meeting estimate for the actual two code laws. -/ |
|
|
| open Set Function MeasureTheory |
| open scoped ENNReal Classical |
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|
| noncomputable section |
| namespace CircleDomainRigidity |
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|
| variable {X Y : Type*} [MetricSpace X] [MetricSpace Y] |
| [MeasurableSpace X] [MeasurableSpace C(PathTime, X)] [BorelSpace C(PathTime, X)] |
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| omit [BorelSpace C(PathTime, X)] in |
| theorem code_dust_meeting_bound |
| (d e : PathCodeData X Y) (P : ProbabilityMeasure (BasePathCode d)) |
| (Q : ProbabilityMeasure (BasePathCode e)) (U D : Set X) (hD : D.Countable) |
| (hcontacts : ∀ᵐ z ∂((P : Measure (BasePathCode d)).prod (Q : Measure (BasePathCode e))), |
| range z.1.val.1 ∩ range z.2.val.1 ∩ d.dust ⊆ D) |
| (p q : X) (hp : p ∈ D ∩ (U ∩ d.dust)) (hq : q ∈ D ∩ (U ∩ d.dust)) (hpq : p ≠ q) |
| {δ : ℝ≥0∞} |
| (hatail : countingSquareNorm (fun x : ↥(D ∩ (U ∩ d.dust)) => |
| if x.val ≠ p then codeHitProbability d P x.val else 0) ≤ δ) |
| (hbtail : countingSquareNorm (fun x : ↥(D ∩ (U ∩ d.dust)) => |
| if x.val ≠ q then codeHitProbability e Q x.val else 0) ≤ δ) : |
| ((P : Measure (BasePathCode d)).prod (Q : Measure (BasePathCode e))) |
| {z | (range z.1.val.1 ∩ range z.2.val.1 ∩ (U ∩ d.dust)).Nonempty} ≤ |
| 2 * δ + δ ^ 2 := by |
| have hcount : (D ∩ (U ∩ d.dust)).Countable := hD.mono inter_subset_left |
| let : Countable ↥(D ∩ (U ∩ d.dust)) := hcount.to_subtype |
| apply (code_pair_meeting_bound d e P Q (U ∩ d.dust) D hD ?_).trans |
| · let p' : ↥(D ∩ (U ∩ d.dust)) := ⟨p, hp⟩ |
| let q' : ↥(D ∩ (U ∩ d.dust)) := ⟨q, hq⟩ |
| have ha : codeHitProbability d P p ≤ 1 := prob_le_one |
| have hb : codeHitProbability e Q q ≤ 1 := prob_le_one |
| have hpa : countingSquareNorm (puncturedHit |
| (fun x : ↥(D ∩ (U ∩ d.dust)) => codeHitProbability d P x.val) univ p') ≤ δ := by |
| convert hatail using 1 |
| congr 1 |
| funext x |
| simp [puncturedHit, p', Subtype.ext_iff] |
| have hqb : countingSquareNorm (puncturedHit |
| (fun x : ↥(D ∩ (U ∩ d.dust)) => codeHitProbability e Q x.val) univ q') ≤ δ := by |
| convert hbtail using 1 |
| congr 1 |
| funext x |
| simp [puncturedHit, q', Subtype.ext_iff] |
| have h := punctured_hit_product_sum_le |
| (fun x : ↥(D ∩ (U ∩ d.dust)) => codeHitProbability d P x.val) |
| (fun x : ↥(D ∩ (U ∩ d.dust)) => codeHitProbability e Q x.val) |
| univ p' q' (mem_univ _) (mem_univ _) |
| (fun h => hpq (congrArg Subtype.val h)) ha hb |
| hpa hqb |
| simpa only [mem_univ, ite_true] using h |
| · filter_upwards [hcontacts] with z hz |
| rintro x ⟨hx, hxU, hxB⟩ |
| exact hz ⟨hx, hxB⟩ |
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| end CircleDomainRigidity |
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| end |
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| end OAI |
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