| import Mathlib |
| import OAI.Analysis.HardSphere.Kernels.ParameterSuccess |
| import OAI.Analysis.HardSphere.Kernels.TrialObserved |
| import OAI.Analysis.HardSphere.Bridges.BridgeReference |
| import OAI.Analysis.HardSphere.Kernels.KernelDensity |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal Topology |
| namespace HardSphere.Bridge |
| open Wiener GaussianPacking Comparison |
|
|
| def targetUniform (A : Set Point) : Measure Point := (volume A)⁻¹ • volume.restrict A |
| lemma targetUniform_probability {A : Set Point} (hA0 : volume A≠0) (hAfinite : volume A≠⊤) : IsProbabilityMeasure (targetUniform A) := by |
| constructor |
| simp only [targetUniform,Measure.smul_apply,Measure.restrict_apply_univ,smul_eq_mul] |
| exact ENNReal.inv_mul_cancel hA0 hAfinite |
|
|
| def midpointLower (t C : ℝ) : ℝ≥0∞ := ENNReal.ofReal (densityLower ((2*t)⁻¹) C) |
| lemma midpointLower_pos {t : ℝ} (ht : 0<t) (C : ℝ) : 0 < midpointLower t C := |
| ENNReal.ofReal_pos.mpr (densityLower_pos (by positivity) C) |
| lemma midpointLower_ne_top (t C : ℝ) : midpointLower t C≠⊤ := ENNReal.ofReal_ne_top |
|
|
| lemma targetUniform_le_midpoint {t C : ℝ} (ht : 0<t) (hC : 0≤C) (y z : Point) |
| {A : Set Point} (hA : MeasurableSet A) (hbound : ∀x∈A,‖x-(fun j ↦ (y j+z j)/2)‖≤C) : |
| targetUniform A≤((volume A)⁻¹*(midpointLower t C)⁻¹) • midpointLaw t y z := by |
| have h := gaussian_dominates_local (v:=Real.toNNReal t) (by simpa [Real.toNNReal_eq_zero] using not_le.mpr ht) |
| (fun j ↦ (y j+z j)/2) hC hA hbound |
| rw [←midpointLaw_gaussian ht.le y z,Real.coe_toNNReal t ht.le] at h |
| change midpointLower t C • volume.restrict A ≤ midpointLaw t y z at h |
| have hd0 := ne_of_gt (midpointLower_pos ht C) |
| intro B |
| have hh := mul_le_mul_right (h B) ((volume A)⁻¹*(midpointLower t C)⁻¹) |
| have he : (volume A)⁻¹*(midpointLower t C)⁻¹*midpointLower t C=(volume A)⁻¹ := by |
| rw [mul_assoc,ENNReal.inv_mul_cancel hd0 (midpointLower_ne_top t C),mul_one] |
| simpa only [targetUniform,Measure.smul_apply,smul_eq_mul,←mul_assoc,he] using hh |
|
|
| def individualReference (t : ℝ) (y z : Point) (S : Set C(Time,Point)) : Measure C(Time,Point) := |
| ((law.map (leg (2*t) y z)) S)⁻¹ • (law.map (leg (2*t) y z)).restrict S |
|
|
| lemma physical_trial_le_reference {t C : ℝ} (ht : 0<t) (hC : 0≤C) (y z : Point) |
| {A : Set Point} (hA : MeasurableSet A) (hAfinite : volume A≠⊤) |
| (hbound : ∀x∈A,‖x-(fun j ↦ (y j+z j)/2)‖≤C) |
| (S : Set C(Time,Point)) (hS : MeasurableSet S) {c : ℝ≥0∞} |
| (hc : ∀ᵐu ∂targetUniform A,c≤trialKernel t y z u S) : |
| ((targetUniform A).compProd (successKernel (trialKernel t y z) S hS)).map Prod.snd ≤ |
| (c⁻¹*((volume A)⁻¹*(midpointLower t C)⁻¹)) • individualReference t y z S := by |
| let : IsFiniteMeasure (volume.restrict A) := ⟨by simp [hAfinite.lt_top]⟩ |
| let : IsFiniteMeasure (targetUniform A) := by |
| by_cases h0 : volume A=0 |
| · have hz : targetUniform A=0 := by simp [targetUniform,Measure.restrict_eq_zero.mpr h0] |
| rw [hz] |
| infer_instance |
| · let : IsProbabilityMeasure (targetUniform A) := targetUniform_probability h0 hAfinite |
| infer_instance |
| have h := successful_marginal_le_reference (targetUniform A) (midpointLaw t y z) |
| (trialKernel t y z) S hS (targetUniform_le_midpoint ht hC y z hA hbound) hc |
| simpa only [full_bridge_midpoint ht.le,individualReference] using h |
|
|
| end HardSphere.Bridge |
|
|
| end |
| end |
|
|
| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal Topology |
| namespace HardSphere.Bridge |
| open Wiener GaussianPacking Comparison |
|
|
| lemma individualReference_probability {t C : ℝ} (ht : 0<t) (hC : 0≤C) (y z : Point) |
| {A : Set Point} (hA : MeasurableSet A) (hA0 : volume A≠0) (hAfinite : volume A≠⊤) |
| (hbound : ∀u∈A,‖u-(fun j ↦ (y j+z j)/2)‖≤C) |
| (S : Set C(Time,Point)) (hS : MeasurableSet S) {c : ℝ≥0∞} (hc0 : 0<c) |
| (hc : ∀ᵐu ∂targetUniform A,c≤trialKernel t y z u S) : |
| IsProbabilityMeasure (individualReference t y z S) := by |
| let _ : IsProbabilityMeasure (targetUniform A) := targetUniform_probability hA0 hAfinite |
| have hp := normalizer_positive_of_uniform_success (targetUniform A) (midpointLaw t y z) |
| (trialKernel t y z) S hS hc0 (ENNReal.mul_ne_top (ENNReal.inv_ne_top.mpr hA0) |
| (ENNReal.inv_ne_top.mpr (midpointLower_pos ht C).ne')) |
| (targetUniform_le_midpoint ht hC y z hA hbound) hc |
| rw [full_bridge_midpoint ht.le y z] at hp |
| let _ : IsProbabilityMeasure (law.map (leg (2*t) y z)) := inferInstance |
| constructor |
| simp only [individualReference,Measure.smul_apply,Measure.restrict_apply_univ,smul_eq_mul] |
| exact ENNReal.inv_mul_cancel hp.ne' (measure_ne_top _ _) |
|
|
| def pairedTrialLaw (t : ℝ) (y z y' z' : Point) (A : Set Point) |
| (S T : Set C(Time,Point)) (hS : MeasurableSet S) (hT : MeasurableSet T) : |
| Measure (C(Time,Point)×(Point×C(Time,Point))) := |
| (pairedFactor (targetUniform A) (trialKernel t y z) (trialKernel t y' z') S T hS hT).map |
| (fun p ↦ (p.2.1,(p.1,p.2.2))) |
|
|
| lemma pairedTrialLaw_probability (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) |
| (hη : ∀ᵐu ∂targetUniform A,0<trialKernel t y' z' u T) : |
| IsProbabilityMeasure (pairedTrialLaw t y z y' z' A S T hS hT) := by |
| let _ : IsProbabilityMeasure (targetUniform A) := targetUniform_probability hA0 hAfinite |
| let _ : IsProbabilityMeasure (pairedFactor (targetUniform A) (trialKernel t y z) (trialKernel t y' z') S T hS hT) := |
| ⟨pairedFactor_mass _ _ _ _ _ _ _ hκ hη⟩ |
| unfold pairedTrialLaw |
| infer_instance |
|
|
| lemma pairedTrialLaw_fst (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) |
| (hη : ∀ᵐu ∂targetUniform A,0<trialKernel t y' z' u T) : |
| (pairedTrialLaw t y z y' z' A S T hS hT).fst= |
| ((targetUniform A).compProd (successKernel (trialKernel t y z) S hS)).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_fst (targetUniform A) (trialKernel t y z) (trialKernel t y' z') S T hS hT hη) |
| rw [Measure.map_map measurable_snd (by fun_prop)] at h |
| unfold pairedTrialLaw Measure.fst |
| rw [Measure.map_map measurable_fst (by fun_prop)] |
| exact h |
|
|
| lemma pairedTrialLaw_midpoint (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) : |
| ∀ᵐp ∂pairedTrialLaw t y z y' z' A S T hS hT, |
| p.1 midpointTime=p.2.1 ∧ p.2.2 midpointTime=p.2.1 := by |
| let _ : IsProbabilityMeasure (targetUniform A) := targetUniform_probability hA0 hAfinite |
| rw [pairedTrialLaw] |
| apply (ae_map_iff (by fun_prop) (by measurability)).mpr |
| exact paired_trial_midpoint (targetUniform A) t y z y' z' S T hS hT |
|
|
| lemma pairedTrial_auxiliary {t C : ℝ} (ht : 0<t) (hC : 0≤C) (y z y' z' : Point) |
| {A : Set Point} (hA : MeasurableSet A) (hA0 : volume A≠0) (hAfinite : volume A≠⊤) |
| (hbound : ∀u∈A,‖u-(fun j ↦ (y j+z j)/2)‖≤C) |
| (S T : Set C(Time,Point)) (hS : MeasurableSet S) (hT : MeasurableSet T) |
| {c : ℝ≥0∞} (hc0 : 0<c) (hc : ∀ᵐu ∂targetUniform A,c≤trialKernel t y z u S) |
| (hη : ∀ᵐu ∂targetUniform A,0<trialKernel t y' z' u T) : |
| ∃ κ : Kernel C(Time,Point) (Point×C(Time,Point)),IsFiniteKernel κ ∧ |
| (individualReference t y z S).compProd κ=pairedTrialLaw t y z y' z' A S T hS hT ∧ |
| (∀f,κ f univ≤c⁻¹*((volume A)⁻¹*(midpointLower t C)⁻¹)) ∧ |
| (∫⁻f,κ f univ ∂individualReference t y z S)=1 := by |
| let _ : IsProbabilityMeasure (individualReference t y z S) := |
| individualReference_probability ht hC y z hA hA0 hAfinite hbound S hS hc0 hc |
| let _ : IsProbabilityMeasure (pairedTrialLaw t y z y' z' A S T hS hT) := |
| pairedTrialLaw_probability t y z y' z' A hA0 hAfinite S T hS hT (hc.mono (fun _ hu ↦ hc0.trans_le hu)) hη |
| apply exists_bounded_auxiliary_kernel |
| · exact ENNReal.mul_ne_top (ENNReal.inv_ne_top.mpr hc0.ne') |
| (ENNReal.mul_ne_top (ENNReal.inv_ne_top.mpr hA0) (ENNReal.inv_ne_top.mpr (midpointLower_pos ht C).ne')) |
| · rw [pairedTrialLaw_fst t y z y' z' A hA0 hAfinite S T hS hT hη] |
| exact physical_trial_le_reference ht hC y z hA hAfinite hbound S hS hc |
|
|
| end HardSphere.Bridge |
|
|
| end |
| end |
|
|
| end |
|
|
| end OAI |
|
|
