| import OAI.Probability.GammaPassage.BallTravel |
| import OAI.Probability.GammaPassage.CrossingLocalized |
|
|
| namespace OAI |
|
|
| namespace GammaFPP.CrossingBridge |
| open PlanarFPP GammaFPP.RealDirectionBridge |
| open GammaFPP.IntervalTravelBridge Set MeasureTheory Filter |
| open scoped Topology |
|
|
| def allPairsRegionEvent (V : Finset Vertex) (I J : Set Vertex) |
| (bound : Vertex → Vertex → ℝ) : Set (Edge → ℝ) := |
| {w | ∀ p ∈ I, ∀ q ∈ J, w ∈ boundedLocalizedEvent V |
| (fun a ds => a=p ∧ a+displacement ds=q) (fun _ _ => bound p q)} |
|
|
| theorem mem_allPairsRegionEvent (V : Finset Vertex) (I J : Set Vertex) |
| (bound : Vertex → Vertex → ℝ) (w : Edge → ℝ) : |
| w ∈ allPairsRegionEvent V I J bound ↔ |
| ∀ p ∈ I, ∀ q ∈ J, ∃ ds, p+displacement ds=q ∧ |
| PathIn p ds (V : Set Vertex) ∧ pathCost w p ds ≤ bound p q := by |
| simp [allPairsRegionEvent,boundedLocalizedEvent] |
|
|
| theorem measurableSet_allPairsRegionEvent (V : Finset Vertex) (I J : Set Vertex) |
| (bound : Vertex → Vertex → ℝ) : MeasurableSet (allPairsRegionEvent V I J bound) := by |
| simp only [allPairsRegionEvent,ofPred_forall] |
| exact MeasurableSet.iInter (fun p => MeasurableSet.iInter (fun _ => |
| MeasurableSet.iInter (fun q => MeasurableSet.iInter (fun _ => |
| measurableSet_boundedLocalizedEvent V _ _)))) |
|
|
| theorem allPairsRegionEvent_depends (V : Finset Vertex) (I J : Set Vertex) |
| (bound : Vertex → Vertex → ℝ) : |
| DependsOnEdges (allPairsRegionEvent V I J bound) (internalEdges V) := by |
| intro w w' hh |
| simp only [allPairsRegionEvent,mem_ofPred_eq] |
| apply forall_congr' |
| intro p |
| apply forall_congr' |
| intro hp |
| apply forall_congr' |
| intro q |
| apply forall_congr' |
| intro hq |
| exact boundedLocalizedEvent_depends V _ _ w w' hh |
|
|
| variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {τ : Edge → Ω → ℝ} |
| variable {shape rate : ℝ} |
|
|
| /-- Uniform connection estimates for finite-region endpoint tests from |
| diameter, support-cost, and tube-containment bounds. -/ |
| theorem gamma_allPairsRegionEvent (h : GammaEnvironment shape rate P τ) |
| (hshape : 0 < shape) (hrate : 0 < rate) |
| (D η δ : ℝ) (hD : 0 < D) (hη : 0 < η) (hδ : 0 < δ) : |
| ∀ ε > 0, ∀ᶠ s : ℝ in atTop, ∀ (V : Finset Vertex) (I J : Set Vertex) |
| (bound : Vertex → Vertex → ℝ), |
| (∀ p ∈ I ∪ J, ∀ q ∈ I ∪ J, ‖latticeEmbed (q-p)‖ ≤ D*s) → |
| (∀ p ∈ I, ∀ q ∈ J, gammaMeanNorm h hshape hrate (latticeEmbed (q-p))+η*s ≤ bound p q) → |
| (∀ p ∈ I, ∀ q ∈ J, SegmentTube p q (δ*s) ⊆ (V : Set Vertex)) → |
| P.real {ω | (fun e => τ e ω) ∉ allPairsRegionEvent V I J bound} < ε := by |
| let _ := h.probability |
| intro ε hε |
| filter_upwards [gamma_set_travel h hshape hrate D η δ hD hη hδ ε hε] with s hs |
| intro V I J bound hdiam hcost hregion |
| refine (measureReal_mono (show {ω | (fun e => τ e ω) ∉ allPairsRegionEvent V I J bound} ⊆ |
| {ω | ¬∀ p ∈ I, ∀ q ∈ J, LocalTravel (gammaMeanNorm h hshape hrate) η δ s p q (fun e => τ e ω)} from ?_)).trans_lt |
| (hs I J hdiam) |
| intro ω hbad htravel |
| apply hbad |
| rw [mem_allPairsRegionEvent] |
| intro p hp q hq |
| obtain ⟨ds,hd,hc,hin⟩ := htravel p hp q hq |
| refine ⟨ds,hd,?_,hc.trans (hcost p hp q hq)⟩ |
| intro i hi |
| exact hregion p hp q hq (hin i hi) |
|
|
| end GammaFPP.CrossingBridge |
|
|
| end OAI |
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