| import Mathlib |
| import OAI.Analysis.HardSphere.Conditioning.SymmetricTag |
|
|
| namespace OAI |
|
|
| section |
|
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| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal BigOperators |
| namespace HardSphere.Comparison |
| variable {X Y Z : Type*} [MeasurableSpace X] [MeasurableSpace Y] [MeasurableSpace Z] |
|
|
| lemma restrict_kernel_comp (κ : Kernel X Y) (μ : Measure X) {s : Set Y} |
| (hs : MeasurableSet s) : κ.restrict hs ∘ₘ μ=(κ ∘ₘ μ).restrict s := by |
| ext t ht |
| rw [Measure.bind_apply ht (κ.restrict hs).aemeasurable, |
| Measure.restrict_apply ht,Measure.bind_apply (ht.inter hs) κ.aemeasurable] |
| congr 1 |
| funext x |
| exact Kernel.restrict_apply' κ hs x ht |
|
|
| def cellExtraKernel (ν : Measure X) (A B : Set X) (hA : MeasurableSet A) : |
| Kernel (X×Y) ((X×Y)×X) := |
| (Kernel.id ×ₖ Kernel.const _ (ν.restrict B)).restrict |
| (hA.preimage (measurable_fst.comp measurable_fst)) |
|
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| lemma cellExtraKernel_comp (μ : Measure (X×Y)) [SFinite μ] |
| (ν : Measure X) [SFinite ν] (A B : Set X) (hA : MeasurableSet A) |
| (hB : MeasurableSet B) : |
| cellExtraKernel (Y:=Y) ν A B hA ∘ₘ μ= |
| (μ.prod ν).restrict {p | p.1.1∈A ∧ p.2∈B} := by |
| rw [cellExtraKernel,restrict_kernel_comp,←Measure.compProd_eq_comp_prod, |
| Measure.compProd_const] |
| have hp : μ.prod (ν.restrict B)=(μ.prod ν).restrict (univ×ˢB) := by |
| simpa only [Measure.restrict_univ] using Measure.prod_restrict (μ:=μ) (ν:=ν) univ B |
| rw [hp,Measure.restrict_restrict' (MeasurableSet.univ.prod hB)] |
| congr 1 |
| ext p |
| simp |
|
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| lemma cellExtraKernel_density (μ : Measure (X×Y)) [SFinite μ] |
| (ν : Measure X) [SFinite ν] (g : X×Y→ℝ≥0∞) (hg : Measurable g) |
| (A B : Set X) (hA : MeasurableSet A) (hB : MeasurableSet B) : |
| cellExtraKernel (Y:=Y) ν A B hA ∘ₘ μ.withDensity g= |
| ((μ.prod ν).withDensity (fun p ↦ g p.1)).restrict {p | p.1.1∈A ∧ p.2∈B} := by |
| rw [cellExtraKernel_comp _ ν A B hA hB,prod_withDensity_left hg] |
|
|
| end HardSphere.Comparison |
|
|
| namespace HardSphere.Groups |
| open Sampling Wiener PathCutoff Comparison |
| attribute [local instance] HardSphere.Comparison.permSpace HardSphere.Comparison.permSingleton |
|
|
| lemma physical_group_cell_reference {n K M G : ℕ} {L : Length} {a : ℝ} |
| {u : Jet (n+1) L} (hu : IsGroundVector (n+1) L a u) |
| (hL : L.val=(K:ℝ)) (hK : 1≤K) (m : ℕ) (r R C S s₀ T : ℝ) |
| (hT : 0≤T) (q : Time) |
| (β : Measure (Equiv.Perm (Fin n))) [IsProbabilityMeasure β] |
| (A B : Set (Torus L)) (hA : MeasurableSet A) (hB : MeasurableSet B) : |
| let Q := groundPathLaw (n+1) L a T u ⊗ₘ |
| physicalAllocationKernel (N:=n+1) (K:=K) m (M*G) r R C S s₀ q |
| let read := fun i : Fin (n+1) ↦ ((cellExtraKernel volume A B hA) ∘ₖ orderedKernel β i).comap |
| (fun p : ConfigurationPath (n+1) L×((Fin 3→Fin K)→Allocation (M*G) (Fin (n+1))) ↦ p.1 q) |
| ((continuous_eval_const q).measurable.comp measurable_fst) |
| (∑g : Fin G,allocatedTagLaw Q (allocatedPhysicalGroup m r R C S s₀ q) read g)≤ |
| (n+1) • (((((volume : Measure (Torus L)).prod (volume : Measure (Configuration n L))).prod volume).withDensity |
| (fun p : (Torus L×Configuration n L)×Torus L ↦ ENNReal.ofReal (‖u.1 (Fin.cons p.1.1 p.1.2)‖^2))).restrict |
| {p | p.1.1∈A ∧ p.2∈B}) := by |
| have hc : Measurable (fun p : Torus L×Configuration n L ↦ (Fin.cons p.1 p.2 : Configuration (n+1) L)) := by fun_prop |
| have hg : Measurable (fun p : Torus L×Configuration n L ↦ |
| ENNReal.ofReal (‖u.1 (Fin.cons p.1 p.2)‖^2)) := |
| (((Lp.stronglyMeasurable u.1).measurable.comp hc).norm.pow_const 2).ennreal_ofReal |
| have h := physical_group_tag_reference (M:=M) (G:=G) hu hL hK m r R C S s₀ T hT q β |
| (cellExtraKernel volume A B hA) |
| dsimp only at h ⊢ |
| rw [cellExtraKernel_density _ _ _ hg A B hA hB] at h |
| exact h |
|
|
| end HardSphere.Groups |
|
|
| end |
| end |
|
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| end |
|
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| end OAI |
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