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67aee1d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 | import Mathlib
import OAI.Analysis.HardSphere.Conditioning.SymmetricTag
namespace OAI
section
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))
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
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
end
end OAI
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