| import Mathlib |
| import OAI.Analysis.HardSphere.Bridges.PhysicalSuccess |
| import OAI.Analysis.HardSphere.Bridges.LocalReference |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal Topology BigOperators |
| namespace HardSphere.Bridge |
| open Wiener GaussianPacking PathCutoff Comparison |
|
|
| lemma pairedTrialLaw_other (t : ℝ) (y z y' z' : Point) (A : Set Point) |
| (hA0 : volume A≠0) (hAfinite : volume A≠⊤) |
| (S T : Set C(Time,Point)) (hS : MeasurableSet S) (hT : MeasurableSet T) |
| (hκ : ∀ᵐu ∂targetUniform A,0<trialKernel t y z u S) : |
| (pairedTrialLaw t y z y' z' A S T hS hT).map (fun p ↦ p.2.2)= |
| ((targetUniform A).compProd (successKernel (trialKernel t y' z') T hT)).map Prod.snd := by |
| let _ : IsProbabilityMeasure (targetUniform A) := targetUniform_probability hA0 hAfinite |
| have h := congrArg (fun μ : Measure (Point×C(Time,Point)) ↦ μ.map Prod.snd) |
| (pairedFactor_snd (targetUniform A) (trialKernel t y z) (trialKernel t y' z') S T hS hT hκ) |
| rw [Measure.map_map measurable_snd (by fun_prop)] at h |
| unfold pairedTrialLaw |
| rw [Measure.map_map (by fun_prop) (by fun_prop)] |
| exact h |
|
|
| lemma pairedTrialLaw_marginals {t C C' : ℝ} (ht : 0<t) (hC : 0≤C) (hC' : 0≤C') |
| (y z y' z' : Point) {A : Set Point} (hA : MeasurableSet A) |
| (hA0 : volume A≠0) (hAfinite : volume A≠⊤) |
| (hb : ∀u∈A,‖u-(fun j ↦ (y j+z j)/2)‖≤C) |
| (hb' : ∀u∈A,‖u-(fun j ↦ (y' j+z' j)/2)‖≤C') |
| (S T : Set C(Time,Point)) (hS : MeasurableSet S) (hT : MeasurableSet T) |
| {c d : ℝ≥0∞} (hc0 : 0<c) (hd0 : 0<d) |
| (hc : ∀ᵐu ∂targetUniform A,c≤trialKernel t y z u S) |
| (hd : ∀ᵐu ∂targetUniform A,d≤trialKernel t y' z' u T) : |
| (pairedTrialLaw t y z y' z' A S T hS hT).fst≤ |
| (c⁻¹*((volume A)⁻¹*(midpointLower t C)⁻¹)) • individualReference t y z S ∧ |
| (pairedTrialLaw t y z y' z' A S T hS hT).map (fun p ↦ p.2.2)≤ |
| (d⁻¹*((volume A)⁻¹*(midpointLower t C')⁻¹)) • individualReference t y' z' T := by |
| constructor |
| · rw [pairedTrialLaw_fst t y z y' z' A hA0 hAfinite S T hS hT (hd.mono (fun _ h ↦ hd0.trans_le h))] |
| exact physical_trial_le_reference ht hC y z hA hAfinite hb S hS hc |
| · rw [pairedTrialLaw_other t y z y' z' A hA0 hAfinite S T hS hT (hc.mono (fun _ h ↦ hc0.trans_le h))] |
| exact physical_trial_le_reference ht hC' y' z' hA hAfinite hb' T hT hd |
|
|
| lemma measure_collision_marginals {L : Length} |
| (ρ : Measure (C(Time,Point)×(Point×C(Time,Point)))) |
| (P Q : Measure C(Time,Point)) {G H p q : ℝ≥0∞} |
| (hP : ρ.fst≤G • P) (hQ : ρ.map (fun p ↦ p.2.2)≤H • Q) |
| (r : ℝ) (ω ω' : ConfigurationPath 1 L) |
| (hp : P {f | TrialCollision r ω f}≤p) (hq : Q {f | TrialCollision r ω' f}≤q) : |
| ρ {f | TrialCollision r ω f.1 ∨ TrialCollision r ω' f.2.2}≤G*p+H*q := by |
| have h₁ : MeasurableSet {f : C(Time,Point) | TrialCollision r ω f} := (trialCollision_closed r ω).measurableSet |
| have h₂ : MeasurableSet {f : C(Time,Point) | TrialCollision r ω' f} := (trialCollision_closed r ω').measurableSet |
| calc |
| _ ≤ ρ {f | TrialCollision r ω f.1}+ρ {f | TrialCollision r ω' f.2.2} := measure_union_le _ _ |
| _ ≤ G*p+H*q := by |
| apply add_le_add |
| · have hm := hP {f | TrialCollision r ω f} |
| rw [Measure.fst,Measure.map_apply measurable_fst h₁,Measure.smul_apply,smul_eq_mul] at hm |
| exact hm.trans (mul_le_mul' le_rfl hp) |
| · have hm := hQ {f | TrialCollision r ω' f} |
| rw [Measure.map_apply (by fun_prop) h₂,Measure.smul_apply,smul_eq_mul] at hm |
| exact hm.trans (mul_le_mul' le_rfl hq) |
| |
| end HardSphere.Bridge |
| |
| end |
| end |
| |
| end |
| |
| end OAI |
| |
