| import Mathlib |
| import OAI.Analysis.HardSphere.Bridges.LocalReference |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
| namespace HardSphere.Comparison |
| open HardSphere.Bridge HardSphere.Wiener HardSphere.GaussianPacking |
| variable {E : Type*} [MeasurableSpace E] |
|
|
| def parameterTargetKernel {B : Set Point} (hB : volume B ≠ ⊤) |
| (A : Set (E×Point)) (hA : MeasurableSet A) : Kernel E Point := by |
| letI : IsFiniteMeasure (volume.restrict B) := ⟨by simpa using hB.lt_top⟩ |
| exact parameterSuccessKernel (Kernel.const E (volume.restrict B)) A hA |
|
|
| lemma parameterTargetKernel_apply {B : Set Point} (hB : volume B ≠ ⊤) |
| (A : Set (E×Point)) (hA : MeasurableSet A) |
| (hAB : ∀e,Prod.mk e ⁻¹' A ⊆ B) (e : E) : |
| parameterTargetKernel hB A hA e=targetUniform (Prod.mk e ⁻¹' A) := by |
| let : IsFiniteMeasure (volume.restrict B) := ⟨by simpa using hB.lt_top⟩ |
| have hs := hA.preimage (@measurable_prodMk_left E Point _ _ e) |
| change ((volume.restrict B) (Prod.mk e ⁻¹' A))⁻¹ • |
| (volume.restrict B).restrict (Prod.mk e ⁻¹' A)=_ |
| rw [Measure.restrict_apply hs,inter_eq_left.mpr (hAB e), |
| Measure.restrict_restrict hs,inter_eq_left.mpr (hAB e)] |
| rfl |
| |
| instance parameterTargetKernel_finite {B : Set Point} (hB : volume B ≠ ⊤) |
| (A : Set (E×Point)) (hA : MeasurableSet A) : |
| IsFiniteKernel (parameterTargetKernel hB A hA) := by |
| let : IsFiniteMeasure (volume.restrict B) := ⟨by simpa using hB.lt_top⟩ |
| exact inferInstanceAs (IsFiniteKernel (parameterSuccessKernel (Kernel.const E (volume.restrict B)) A hA)) |
| |
| lemma parameterTargetKernel_markov {B : Set Point} (hB : volume B ≠ ⊤) |
| (A : Set (E×Point)) (hA : MeasurableSet A) |
| (hAB : ∀e,Prod.mk e ⁻¹' A ⊆ B) |
| (hpos : ∀e,0 < volume (Prod.mk e ⁻¹' A)) : |
| IsMarkovKernel (parameterTargetKernel hB A hA) := by |
| constructor |
| intro e |
| rw [parameterTargetKernel_apply hB A hA hAB e] |
| exact targetUniform_probability (hpos e).ne' |
| (ne_top_of_le_ne_top hB (measure_mono (hAB e))) |
| |
| end HardSphere.Comparison |
| |
| end |
| end |
| |
| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal Topology |
| namespace HardSphere.Comparison |
| variable {E U X Y : Type*} [MeasurableSpace E] [MeasurableSpace U] |
| [MeasurableSpace X] [MeasurableSpace Y] |
| |
| lemma probability_inter_98 (μ : Measure U) [IsProbabilityMeasure μ] |
| {s t : Set U} (_hs : MeasurableSet s) (ht : MeasurableSet t) |
| (hsm : ENNReal.ofReal (99/100:ℝ)≤μ s) (htm : ENNReal.ofReal (99/100:ℝ)≤μ t) : |
| ENNReal.ofReal (49/50:ℝ)≤μ (s∩t) := by |
| have he := congrArg ENNReal.toReal (measure_union_add_inter (μ:=μ) s ht) |
| rw [ENNReal.toReal_add (measure_ne_top _ _) (measure_ne_top _ _), |
| ENNReal.toReal_add (measure_ne_top _ _) (measure_ne_top _ _)] at he |
| have hu : (μ (s∪t)).toReal≤1 := by |
| exact_mod_cast ENNReal.toReal_mono ENNReal.one_ne_top (prob_le_one : μ (s∪t)≤1) |
| have hsr := ENNReal.toReal_mono (measure_ne_top _ _) hsm |
| have htr := ENNReal.toReal_mono (measure_ne_top _ _) htm |
| apply (ENNReal.toReal_le_toReal ENNReal.ofReal_ne_top (measure_ne_top _ _)).mp |
| norm_num at hsr htr ⊢ |
| linarith |
| |
| def accessibleTarget (κ : Kernel (E×U) X) (S : Set ((E×U)×X)) (c : ℝ≥0∞) : Set (E×U) := |
| {p | c≤κ p (Prod.mk p ⁻¹' S)} |
| lemma accessibleTarget_measurable (κ : Kernel (E×U) X) [IsFiniteKernel κ] |
| (S : Set ((E×U)×X)) (hS : MeasurableSet S) (c : ℝ≥0∞) : |
| MeasurableSet (accessibleTarget κ S c) := |
| measurableSet_le measurable_const (Kernel.measurable_kernel_prodMk_left hS) |
| |
| def commonAccessible (κ : Kernel (E×U) X) (η : Kernel (E×U) Y) |
| (S : Set ((E×U)×X)) (T : Set ((E×U)×Y)) (c d : ℝ≥0∞) (B : Set U) : Set (E×U) := |
| (univ×ˢB)∩(accessibleTarget κ S c∩accessibleTarget η T d) |
| lemma commonAccessible_measurable (κ : Kernel (E×U) X) (η : Kernel (E×U) Y) |
| [IsFiniteKernel κ] [IsFiniteKernel η] |
| (S : Set ((E×U)×X)) (T : Set ((E×U)×Y)) (hS : MeasurableSet S) (hT : MeasurableSet T) |
| (c d : ℝ≥0∞) (B : Set U) (hB : MeasurableSet B) : |
| MeasurableSet (commonAccessible κ η S T c d B) := |
| (MeasurableSet.univ.prod hB).inter |
| ((accessibleTarget_measurable κ S hS c).inter (accessibleTarget_measurable η T hT d)) |
| |
| lemma commonAccessible_volume (μ : Measure U) (κ : Kernel (E×U) X) (η : Kernel (E×U) Y) |
| [IsFiniteKernel κ] [IsFiniteKernel η] |
| (S : Set ((E×U)×X)) (T : Set ((E×U)×Y)) (hS : MeasurableSet S) (hT : MeasurableSet T) |
| (c d : ℝ≥0∞) (B : Set U) (_hB : MeasurableSet B) (hm : μ B=1) (e : E) |
| (hκ : ENNReal.ofReal (99/100:ℝ)≤(μ.restrict B) (Prod.mk e ⁻¹' accessibleTarget κ S c)) |
| (hη : ENNReal.ofReal (99/100:ℝ)≤(μ.restrict B) (Prod.mk e ⁻¹' accessibleTarget η T d)) : |
| ENNReal.ofReal (49/50:ℝ)≤μ (Prod.mk e ⁻¹' commonAccessible κ η S T c d B) := by |
| let _ : IsProbabilityMeasure (μ.restrict B) := ⟨by simp [hm]⟩ |
| have hs := (accessibleTarget_measurable κ S hS c).preimage (@measurable_prodMk_left E U _ _ e) |
| have ht := (accessibleTarget_measurable η T hT d).preimage (@measurable_prodMk_left E U _ _ e) |
| have hh := probability_inter_98 (μ.restrict B) hs ht hκ hη |
| rw [Measure.restrict_apply (hs.inter ht)] at hh |
| convert hh using 1 |
| congr 1 |
| ext u |
| simp [commonAccessible,and_comm] |
| |
| end HardSphere.Comparison |
| |
| end |
| end |
| |
| end |
| |
| end OAI |
| |
