| import Mathlib |
| import OAI.Analysis.HardSphere.Kernels.ExposedPhysicalPair |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal Topology |
| namespace HardSphere.Bridge |
| open Wiener GaussianPacking Comparison |
| variable {E : Type*} [MeasurableSpace E] |
|
|
| lemma leg_all_continuous (t : ℝ) : |
| Continuous (fun p : (Point×Point)×Noise ↦ leg t p.1.1 p.1.2 p.2) := by |
| apply ContinuousMap.continuous_of_continuous_uncurry |
| change Continuous (fun p : ((Point×Point)×Noise)×Time ↦ fun j ↦ |
| mean p.1.1.1 p.1.1.2 p.2 j+Real.sqrt (2*t)*p.1.2 j p.2) |
| unfold mean |
| fun_prop |
|
|
| def exposedBridgeKernel (t : ℝ) : Kernel (Point×Point) C(Time,Point) := |
| ((Kernel.deterministic id measurable_id).prod (Kernel.const (Point×Point) law)).map |
| (fun p ↦ leg t p.1.1 p.1.2 p.2) |
| instance exposedBridgeKernel_markov (t : ℝ) : IsMarkovKernel (exposedBridgeKernel t) := by |
| unfold exposedBridgeKernel |
| exact Kernel.IsMarkovKernel.map _ (leg_all_continuous t).measurable |
| lemma exposedBridgeKernel_apply (t : ℝ) (y z : Point) : |
| exposedBridgeKernel t (y,z)=law.map (leg t y z) := by |
| rw [exposedBridgeKernel,Kernel.map_apply _ (leg_all_continuous t).measurable, |
| Kernel.prod_apply,Kernel.deterministic_apply,Kernel.const_apply,Measure.dirac_prod, |
| Measure.map_map (leg_all_continuous t).measurable |
| (measurable_const.prodMk measurable_id : Measurable (Prod.mk (id (y,z))))] |
| rfl |
| |
| def parameterBridgeKernel (t : ℝ) (y z : E→Point) (hy : Measurable y) (hz : Measurable z) : |
| Kernel E C(Time,Point) := (exposedBridgeKernel t).comap (fun e ↦ (y e,z e)) (hy.prodMk hz) |
| instance parameterBridgeKernel_markov (t : ℝ) (y z : E→Point) (hy : Measurable y) (hz : Measurable z) : |
| IsMarkovKernel (parameterBridgeKernel t y z hy hz) := inferInstanceAs |
| (IsMarkovKernel ((exposedBridgeKernel t).comap _ _)) |
| lemma parameterBridgeKernel_apply (t : ℝ) (y z : E→Point) (hy : Measurable y) (hz : Measurable z) (e : E) : |
| parameterBridgeKernel t y z hy hz e=law.map (leg t (y e) (z e)) := exposedBridgeKernel_apply _ _ _ |
| |
| def parameterIndividualReference (t : ℝ) (y z : E→Point) (hy : Measurable y) (hz : Measurable z) |
| (S : Set (E×C(Time,Point))) (hS : MeasurableSet S) : Kernel E C(Time,Point) := |
| parameterSuccessKernel (parameterBridgeKernel (2*t) y z hy hz) S hS |
| instance parameterIndividualReference_finite (t : ℝ) (y z : E→Point) (hy : Measurable y) (hz : Measurable z) |
| (S : Set (E×C(Time,Point))) (hS : MeasurableSet S) : |
| IsFiniteKernel (parameterIndividualReference t y z hy hz S hS) := inferInstanceAs |
| (IsFiniteKernel (parameterSuccessKernel _ _ _)) |
| lemma parameterIndividualReference_apply (t : ℝ) (y z : E→Point) (hy : Measurable y) (hz : Measurable z) |
| (S : Set (E×C(Time,Point))) (hS : MeasurableSet S) (e : E) : |
| parameterIndividualReference t y z hy hz S hS e=individualReference t (y e) (z e) (Prod.mk e ⁻¹' S) := by |
| rw [parameterIndividualReference,parameterSuccessKernel_apply,parameterBridgeKernel_apply] |
| rfl |
| |
| end HardSphere.Bridge |
| |
| end |
| end |
| |
| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal Topology |
| namespace HardSphere.Bridge |
| open Wiener GaussianPacking Comparison |
| variable {E : Type*} [MeasurableSpace E] |
| [MeasurableSpace.CountableOrCountablyGenerated E C(Time,Point)] |
| |
| lemma exposed_physical_auxiliary |
| (Q : Kernel E Point) [IsFiniteKernel Q] {t C : ℝ} (ht : 0<t) (hC : 0≤C) |
| (y z y' z' : E→Point) (hy : Measurable y) (hz : Measurable z) |
| (hy' : Measurable y') (hz' : Measurable z') |
| (S T : Set (E×C(Time,Point))) (hS : MeasurableSet S) (hT : MeasurableSet T) |
| (A : E→Set Point) (hA : ∀e,MeasurableSet (A e)) |
| (hAf : ∀e,volume (A e)≠⊤) (hQ : ∀e,Q e=targetUniform (A e)) |
| {a c : ℝ≥0∞} (ha : 0<a) (hc : 0<c) (hvol : ∀e,a≤volume (A e)) |
| (hb : ∀e u,u∈A e → ‖u-(fun j ↦ (y e j+z e j)/2)‖≤C) |
| (hsuccess : ∀e,∀ᵐu ∂targetUniform (A e),c≤trialKernel t (y e) (z e) u (Prod.mk e ⁻¹' S)) |
| (hother : ∀e,∀ᵐu ∂targetUniform (A e),0<trialKernel t (y' e) (z' e) u (Prod.mk e ⁻¹' T)) : |
| ∃ κ : Kernel (E×C(Time,Point)) (Point×C(Time,Point)), IsFiniteKernel κ ∧ |
| (parameterIndividualReference t y z hy hz S hS).compProd κ= |
| exposedPairedTrialKernel Q t y z y' z' hy hz hy' hz' S T hS hT ∧ |
| (∀e f,κ (e,f) univ≤c⁻¹*(a⁻¹*(midpointLower t C)⁻¹)) ∧ |
| (∀e,∫⁻f,κ (e,f) univ ∂parameterIndividualReference t y z hy hz S hS e=1) := by |
| have hA0 (e : E) : volume (A e)≠0 := (ha.trans_le (hvol e)).ne' |
| let P := parameterIndividualReference t y z hy hz S hS |
| let ρ := exposedPairedTrialKernel Q t y z y' z' hy hz hy' hz' S T hS hT |
| let _ : IsMarkovKernel P := by |
| constructor |
| intro e |
| change IsProbabilityMeasure (parameterIndividualReference t y z hy hz S hS e) |
| rw [parameterIndividualReference_apply] |
| exact individualReference_probability ht hC (y e) (z e) (hA e) (hA0 e) (hAf e) (hb e) |
| (Prod.mk e ⁻¹' S) (hS.preimage measurable_prodMk_left) hc (hsuccess e) |
| let _ : IsMarkovKernel ρ := by |
| constructor |
| intro e |
| change IsProbabilityMeasure (exposedPairedTrialKernel Q t y z y' z' hy hz hy' hz' S T hS hT e) |
| rw [exposedPairedTrialKernel_physical Q t y z y' z' hy hz hy' hz' S T hS hT A hQ] |
| exact pairedTrialLaw_probability t (y e) (z e) (y' e) (z' e) (A e) (hA0 e) (hAf e) |
| (Prod.mk e ⁻¹' S) (Prod.mk e ⁻¹' T) |
| (hS.preimage measurable_prodMk_left) (hT.preimage measurable_prodMk_left) |
| ((hsuccess e).mono (fun _ hu ↦ hc.trans_le hu)) (hother e) |
| apply exists_parameter_auxiliary P ρ |
| · exact ENNReal.mul_ne_top (ENNReal.inv_ne_top.mpr hc.ne') |
| (ENNReal.mul_ne_top (ENNReal.inv_ne_top.mpr ha.ne') |
| (ENNReal.inv_ne_top.mpr (midpointLower_pos ht C).ne')) |
| · intro e |
| change (ρ e).fst≤_ • P e |
| dsimp [ρ,P] |
| rw [exposedPairedTrialKernel_physical Q t y z y' z' hy hz hy' hz' S T hS hT A hQ, |
| parameterIndividualReference_apply, |
| pairedTrialLaw_fst t (y e) (z e) (y' e) (z' e) (A e) (hA0 e) (hAf e) |
| (Prod.mk e ⁻¹' S) (Prod.mk e ⁻¹' T) |
| (hS.preimage measurable_prodMk_left) (hT.preimage measurable_prodMk_left) (hother e)] |
| apply (physical_trial_le_reference ht hC (y e) (z e) (hA e) (hAf e) (hb e) |
| (Prod.mk e ⁻¹' S) (hS.preimage measurable_prodMk_left) (hsuccess e)).trans |
| intro B |
| simp only [Measure.smul_apply,smul_eq_mul] |
| exact mul_le_mul' (mul_le_mul' le_rfl |
| (mul_le_mul' (ENNReal.inv_le_inv.mpr (hvol e)) le_rfl)) le_rfl |
| |
| end HardSphere.Bridge |
| |
| end |
| end |
| |
| end |
| |
| end OAI |
| |
