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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 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 | import Mathlib
import OAI.Analysis.HardSphere.Allocation.PhysicalObservedMass
import OAI.Analysis.HardSphere.Comparison.SumMoment
namespace OAI
section
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal BigOperators
namespace HardSphere.Groups
open Wiener Bridge Sampling Comparison GaussianPacking PathCutoff CellOrbital HardSphere.Lattice
attribute [local instance] Classical.propDecidable
variable {K : ℕ} [NeZero K] {x y : DiscreteTorus K}
def physicalGroupReference {N M G : ℕ} {L : Length} (hL : L.val=(K:ℝ))
(ψ : Jet (N+1) L) (g : Fin G) (m : ℕ) (t r R C S s₀ : ℝ) (u : Time) (b : Bool) :=
∑flags,∑a : GridAllocation (N+1) K (G*M),
physicalObservedReference (x:=x) (y:=y) hL g m t r R C S u (branchParticles g a) b
(physicalBranchLaw hL (endpointWeightedBase ψ (2*t)) g flags a m r R C S s₀ (2*t) u)
theorem physical_group_common_comparison {N M G : ℕ} {L : Length}
(hL : L.val=(K:ℝ)) (hK : 1 ≤ K) (h50 : 50<L.val)
{ψ : Jet (N+1) L} {r : ℝ} (hψ : IsGroundVector (N+1) L r ψ)
(g : Fin G) (m : ℕ) (t R C S s₀ h : ℝ) (u : Time) (hs : PhysicalScaleBounds m t r R C S s₀ h u)
(hxy : x≠y) (E : Set (PhysicalDatum (N+1) K M G L)) (hE : MeasurableSet E)
(ν : Kernel (PhysicalDatum (N+1) K M G L) (SimplePath x y)) [IsMarkovKernel ν]
(hgood : ∀d∈E,ν d {p | simplePathGood (latticeDatumBad hL g m t r R C S u d) p}=1)
(hM : 0<M) (δ B κ Cpath : ℝ) (hδ : 0<δ) (hB : 1 ≤ B) (hCpath : 0 ≤ Cpath)
(hBt : routeDensityBound t ≤ ENNReal.ofReal B)
(hκ : 2*(B-1)/(M:ℝ)-8*Real.log δ ≤ κ)
(hsmall : ENNReal.ofReal δ+(4*(49^3*(M+1)):ℕ)*(routeDensityBound t*datumCollisionBound t r h R C) ≤ 1)
(henc : (∫⁻d in E,∫⁻p,∫⁻p',ENNReal.ofReal (Real.exp (κ*(simpleEncounter 30 p p':ℝ))) ∂ν d ∂ν d
∂physicalDatumLaw g ψ m r R C S s₀ (2*t) u) ≤ ENNReal.ofReal Cpath) :
∃σ : Measure (Torus L×Torus L×Configuration N L), IsFiniteMeasure σ ∧
σ univ=(physicalDatumLaw g ψ m r R C S s₀ (2*t) u) E ∧
∀b,
let P := physicalGroupReference (M:=M) (x:=x) (y:=y) hL ψ g m t r R C S s₀ u b
IsFiniteMeasure P ∧ σ≪P ∧ Integrable (fun z ↦ (σ.rnDeriv P z).toReal^2) P ∧
(∫z,(σ.rnDeriv P z).toReal^2 ∂P) ≤ B*Cpath := by
classical
let I := (Fin (N+1)→Bool)×GridAllocation (N+1) K (G*M)
let μ : I→Measure (PhysicalDatum (N+1) K M G L) := fun i ↦
physicalBranchLaw hL (endpointWeightedBase ψ (2*t)) g i.1 i.2 m r R C S s₀ (2*t) u
have hT : 0 ≤ 2*t := by linarith [hs.time_pos]
let (i : I) : IsFiniteMeasure (μ i) :=
physicalBranchLaw_finite hL hK hψ g i.1 i.2 m R C S s₀ (2*t) hT u
let σ : I→Measure (Torus L×Torus L×Configuration N L) := fun i ↦
physicalCommonObserved hL g m t r R C S u (branchParticles g i.2) hxy ν (μ i) E
let P : Bool→I→Measure (Torus L×Torus L×Configuration N L) := fun b i ↦
physicalObservedReference (x:=x) (y:=y) hL g m t r R C S u (branchParticles g i.2) b (μ i)
have hsum : (∑i,μ i)=physicalDatumLaw g ψ m r R C S s₀ (2*t) u := by
rw [Fintype.sum_prod_type]
exact physicalBranchLaw_sum hL hK ψ g m r R C S s₀ (2*t) u
let F := E.indicator (fun d ↦ ∫⁻p,∫⁻p',ENNReal.ofReal (Real.exp (κ*(simpleEncounter 30 p p':ℝ))) ∂ν d ∂ν d)
let e : I→ℝ≥0∞ := fun i ↦ ∫⁻d,F d ∂μ i
have hesum : ∑i,e i ≤ ENNReal.ofReal Cpath := by
rw [←lintegral_finsetSum_measure,hsum]
simpa only [F,lintegral_indicator hE] using henc
have hei (i : I) : e i≠⊤ := by
apply ne_top_of_le_ne_top ENNReal.ofReal_ne_top
exact (Finset.single_le_sum (fun _ _ ↦ bot_le) (Finset.mem_univ i)).trans hesum
have hbranch (i : I) :
σ i univ=μ i E ∧ ∀b,(σ i≪P b i ∧
Integrable (fun z ↦ ((σ i).rnDeriv (P b i) z).toReal^2) (P b i) ∧
(∫z,((σ i).rnDeriv (P b i) z).toReal^2 ∂P b i) ≤ B*(e i).toReal) := by
apply physical_common_observed_estimates hL g m t r R C S hs.time_pos hs.radius_le u
(branchParticles g i.2) hxy ν (μ i) E hE hM (datumCollisionBound t r h R C)
δ B κ (e i).toReal hδ hB ENNReal.toReal_nonneg hBt hκ hsmall
· exact physical_branch_moment_data hL hK h50 (endpointWeightedBase ψ (2*t)) g i.1 i.2
m t r R C S s₀ h u hs E hE ν hgood
· exact ae_restrict_of_ae (physicalBranchLaw_primary hL hK h50 (endpointWeightedBase ψ (2*t))
m r R C S s₀ t u g i.1 i.2 (siteIndex y))
· rw [ENNReal.ofReal_toReal (hei i)]
exact le_of_eq (lintegral_indicator hE _).symm
let (i : I) : IsFiniteMeasure (σ i) := by dsimp only [σ]; infer_instance
let (b : Bool) (i : I) : IsFiniteMeasure (P b i) := by dsimp only [P]; infer_instance
refine ⟨∑i,σ i,inferInstance,?_,?_⟩
· rw [Measure.finsetSum_apply]
simp_rw [fun i ↦ (hbranch i).1]
rw [←Measure.finsetSum_apply,hsum]
· intro b
have hp : physicalGroupReference (M:=M) (x:=x) (y:=y) hL ψ g m t r R C S s₀ u b=∑i,P b i := by
rw [Fintype.sum_prod_type]; rfl
dsimp only
rw [hp]
refine ⟨inferInstance,?_,?_⟩
· have H : Measure.sum σ≪∑i,P b i := Measure.absolutelyContinuous_sum_left (fun i ↦
((hbranch i).2 b).1.trans (Measure.absolutelyContinuous_of_le
(Finset.single_le_sum (fun _ _ ↦ bot_le) (Finset.mem_univ i))))
simpa only [Measure.sum_fintype] using H
· have H := finite_sum_second_moment (P b) σ (fun i ↦ ((hbranch i).2 b).1)
(fun i ↦ ((hbranch i).2 b).2.1)
refine ⟨H.1,H.2.trans ?_⟩
calc
_ ≤ ∑i,B*(e i).toReal := Finset.sum_le_sum (fun i _ ↦ ((hbranch i).2 b).2.2)
_ = B*(∑i,e i).toReal := by rw [←Finset.mul_sum,ENNReal.toReal_sum (fun i _ ↦ hei i)]
_ ≤ B*Cpath := mul_le_mul_of_nonneg_left
(by simpa only [ENNReal.toReal_ofReal hCpath] using ENNReal.toReal_mono ENNReal.ofReal_ne_top hesum)
(by linarith)
end HardSphere.Groups
end
end
end OAI
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