File size: 8,177 Bytes
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 | import Mathlib
import OAI.Analysis.HardSphere.Allocation.PhysicalLifted
import OAI.Analysis.HardSphere.Bridges.RouteAuxiliary
namespace OAI
section
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal NNReal
namespace HardSphere.Groups
open Wiener Bridge Sampling Comparison GaussianPacking PathCutoff CellOrbital
variable {J : Type*} [Fintype J] [DecidableEq J]
omit [Fintype J] [DecidableEq J] in
lemma physicalOn_probability {N K M G : ℕ} {L : Length} (hL : L.val=(K:ℝ))
(g : Fin G) (m : ℕ) (t r R C S : ℝ) (u : Time)
(particle : J→Fin N) (v : Fin 3→Fin K) (d : PhysicalDatum N K M G L)
(hp : ∀j,IsProbabilityMeasure (datumIndividualReference hL g m r R C S t u (particle j) d))
(i : Option J) : IsProbabilityMeasure (physicalSlotOn hL g m t r R C S u particle v d i) := by
have : IsProbabilityMeasure (physicalSlotReference hL g m t r R C S u particle v i d) := by
cases i with
| none => exact inferInstanceAs (IsProbabilityMeasure (cellPathReference v))
| some i => exact hp i
exact physicalSlotOn_probability hL g m t r R C S u particle v d i
omit [Fintype J] in
theorem physical_auxiliary_exists {N K M G n : ℕ} {L : Length} (hL : L.val=(K:ℝ))
(g : Fin G) (m : ℕ) (t r R C S : ℝ) (ht : 0<t) (u : Time)
(particle : J↪Fin N) (labels : Fin (n+1)↪J) (v : Fin (n+2)→(Fin 3→Fin K))
(steps : Fin (n+1)→Fin 7)
(hc : ∀k,projectPoint L (routeCellPeriod (v k.succ) (v k.castSucc) (steps k))=0)
(d : PhysicalDatum N K M G L)
(hhome : ∀k,datumHome hL d (particle (labels k))=v k.castSucc)
(haccess : ∀k,datumAccessible hL g m t r R C S u (particle (labels k)) d)
(hp : ∀j,IsProbabilityMeasure (datumIndividualReference hL g m r R C S t u (particle j) d)) :
let X : Option J → Type := fun i ↦ physicalSlotAllowed hL g m r R C S u particle d i
let σ := routePermutation labels
let μ := physicalSlotOn hL g m t r R C S u particle (v (Fin.last (n+1))) d
let ν := physicalSlotOn hL g m t r R C S u particle (v 0) d
let ρ := physicalSelectedOn hL g m t r R C S u particle labels v steps d
∃ (κ : ∀i,Kernel (X i) (X (σ.symm i))) (η : ∀i,Kernel (X i) (X (σ i))),
(∀i,IsFiniteKernel (κ i)) ∧ (∀i,IsFiniteKernel (η i)) ∧
(∀i:routeSlots labels,(μ i).compProd (κ i)=ρ i) ∧
(∀i,((μ (σ i)).compProd (κ (σ i))).map (reverseFactor (X:=X) σ i)=(ν i).compProd (η i)) ∧
(∀i x,κ i x univ≤routeDensityBound t) ∧ (∀i x,η i x univ≤routeDensityBound t) ∧
(∀i,∫⁻x,κ i x univ ∂μ i=1) ∧ (∀i,∫⁻x,η i x univ ∂ν i=1) := by
dsimp only
let X : Option J → Type := fun i ↦ physicalSlotAllowed hL g m r R C S u particle d i
let σ := routePermutation labels
let s := routeSlots labels
let μ := physicalSlotOn hL g m t r R C S u particle (v (Fin.last (n+1))) d
let ν := physicalSlotOn hL g m t r R C S u particle (v 0) d
let ρ := physicalSelectedOn hL g m t r R C S u particle labels v steps d
have : ∀i,IsProbabilityMeasure (μ i) := physicalOn_probability hL g m t r R C S u particle _ d hp
have : ∀i,IsProbabilityMeasure (ν i) := physicalOn_probability hL g m t r R C S u particle _ d hp
have hρ := physicalSelectedOn_physical hL g m t r R C S ht u particle labels v steps hc d hhome haccess
have : ∀i,IsProbabilityMeasure (ρ i) := fun i ↦ (hρ i).1
let ρ' := extendSelectedPair σ s μ ν ρ
have hh := extendSelectedPair_bounds σ s μ ν ρ (one_le_routeDensityBound t)
(fun i ↦ (hρ i).2.2.2.1) (fun i ↦ (hρ i).2.2.2.2.1)
obtain ⟨κ,η,hκf,hηf,hκe,hηe,hκb,hηb,hκm,hηm⟩ := exists_two_auxiliary σ μ ν ρ'
(routeDensityBound_ne_top ht) (fun i ↦ (hh i).1) (fun i ↦ (hh i).2)
refine ⟨κ,η,hκf,hηf,?_,?_,hκb,hηb,hκm,hηm⟩
· intro i
exact (hκe i).trans (extendSelectedPair_selected σ s μ ν ρ i)
· intro i
rw [hκe,hηe]
theorem physical_two_likelihoods {N K M G n : ℕ} {L : Length} (hL : L.val=(K:ℝ))
(g : Fin G) (m : ℕ) (t r R C S : ℝ) (u : Time)
(particle : J↪Fin N) (labels : Fin (n+1)↪J) (v : Fin (n+2)→(Fin 3→Fin K))
(steps : Fin (n+1)→Fin 7) (d : PhysicalDatum N K M G L)
(hp : ∀j,IsProbabilityMeasure (datumIndividualReference hL g m r R C S t u (particle j) d))
(κ : ∀i,Kernel (physicalSlotAllowed hL g m r R C S u particle d i)
(physicalSlotAllowed hL g m r R C S u particle d ((routePermutation labels).symm i)))
[∀i,IsFiniteKernel (κ i)]
(η : ∀i,Kernel (physicalSlotAllowed hL g m r R C S u particle d i)
(physicalSlotAllowed hL g m r R C S u particle d (routePermutation labels i)))
[∀i,IsFiniteKernel (η i)]
(he : ∀i:routeSlots labels,
(physicalSlotOn hL g m t r R C S u particle (v (Fin.last (n+1))) d i).compProd (κ i)=
physicalSelectedOn hL g m t r R C S u particle labels v steps d i)
(hr : ∀i:routeSlots labels,
((physicalSlotOn hL g m t r R C S u particle (v (Fin.last (n+1))) d (routePermutation labels i)).compProd
(κ (routePermutation labels i))).map (reverseFactor (X:=fun i ↦ physicalSlotAllowed hL g m r R C S u particle d i)
(routePermutation labels) i)=
(physicalSlotOn hL g m t r R C S u particle (v 0) d i).compProd (η i))
(z : ∀i,physicalSlotAllowed hL g m r R C S u particle d ((routePermutation labels).symm i))
(w : ∀i,physicalSlotAllowed hL g m r R C S u particle d (routePermutation labels i)) :
let σ := routePermutation labels
let s := routeSlots labels
let anchor := fun j ↦ projectPoint L (anchorPoint (datumHome hL d (particle j)))
let allowed := fun j ↦ physicalSlotAllowed hL g m r R C S u particle d (some j)
let R₀ := physicalOnChecks hL g m r R C S u particle d
let C₀ := routeChecks σ s (routePermutation_fixed labels) r anchor allowed some (fun _ f ↦ f) (fun _ ↦ measurable_id)
let C₁ := routeChecks σ.symm s (reverse_fix σ s (routePermutation_fixed labels)) r anchor allowed some (fun _ f ↦ f) (fun _ ↦ measurable_id)
let μ := physicalSlotOn hL g m t r R C S u particle (v (Fin.last (n+1))) d
let ν := physicalSlotOn hL g m t r R C S u particle (v 0) d
let T := physicalSharedOn hL g m t r R C S u particle labels v steps d
T.map (ambientFirst s σ)=
(physicalOnPrimary hL g m t r R C S u particle (v (Fin.last (n+1))) d).withDensity
(fun x ↦ conditionalLikelihood R₀ C₀ μ κ s (Finset.univ\s) x z) ∧
T.map (ambientSecond s σ (routePermutation_fixed labels))=
(physicalOnPrimary hL g m t r R C S u particle (v 0) d).withDensity
(fun x ↦ conditionalLikelihood R₀ C₁ ν η s (Finset.univ\s) x w) := by
dsimp only
let X : Option J → Type := fun i ↦ physicalSlotAllowed hL g m r R C S u particle d i
let σ := routePermutation labels
let s := routeSlots labels
let μ := physicalSlotOn hL g m t r R C S u particle (v (Fin.last (n+1))) d
let ν := physicalSlotOn hL g m t r R C S u particle (v 0) d
have : ∀i,IsProbabilityMeasure (μ i) := physicalOn_probability hL g m t r R C S u particle _ d hp
have : ∀i,IsProbabilityMeasure (ν i) := physicalOn_probability hL g m t r R C S u particle _ d hp
have heq : ∀i,i≠none→μ i=ν i := by
intro i hi
cases i with
| none => exact (hi rfl).elim
| some i => rfl
have hs : none∈s := by
exact Finset.mem_map.mpr ⟨Fin.last (n+1),Finset.mem_univ _,routeEmbedding_last labels⟩
have h := ambient_two_primary_laws (X:=X) σ s (routePermutation_fixed labels) r
(fun j ↦ projectPoint L (anchorPoint (datumHome hL d (particle j))))
(fun j ↦ physicalSlotAllowed hL g m r R C S u particle d (some j)) some (fun _ f ↦ f) (fun _ ↦ measurable_id)
none (fun _ ↦ Option.some_ne_none _) hs μ ν heq κ η hr z w ()
have hρ : (fun i:s ↦ Kernel.const Unit ((μ i).compProd (κ i)))=
(fun i:s ↦ Kernel.const Unit (physicalSelectedOn hL g m t r R C S u particle labels v steps d i)) := by
funext i
rw [he i]
dsimp only at h
simp only [hρ] at h
exact h
end HardSphere.Groups
end
end
end OAI
|