| 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 |
|
|
