| import Mathlib |
| import OAI.Analysis.HardSphere.Allocation.PhysicalPrimary |
|
|
| 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] |
| |
| instance physicalSlotAllowed_borel {N K M G : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (g : Fin G) (m : ℕ) (r R C S : ℝ) (u : Time) |
| (particle : J→Fin N) (d : PhysicalDatum N K M G L) (i : Option J) : |
| StandardBorelSpace (physicalSlotAllowed hL g m r R C S u particle d i) := |
| (physicalSlotAllowed_measurable hL g m r R C S u particle d i).standardBorel |
| |
| lemma routeDensityBound_ne_top {t : ℝ} (ht : 0<t) : routeDensityBound t≠⊤ := by |
| have ha : (ENNReal.ofReal (9/10:ℝ))≠0 := by positivity |
| have hb : (ENNReal.ofReal (49/50:ℝ))≠0 := by positivity |
| have hc := (midpointLower_pos ht 13).ne' |
| simp only [routeDensityBound,ne_eq,max_eq_top] |
| exact not_or.mpr ⟨by norm_num,ENNReal.mul_ne_top (ENNReal.inv_ne_top.mpr ha) |
| (ENNReal.mul_ne_top (ENNReal.inv_ne_top.mpr hb) (ENNReal.inv_ne_top.mpr hc))⟩ |
| |
| lemma one_le_routeDensityBound (t : ℝ) : 1≤routeDensityBound t := |
| le_trans (by norm_num) (le_max_left _ _) |
| |
| def physicalSelectedOn {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) (i : routeSlots labels) : |
| Measure (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)) := |
| supportedPair (physicalSelectedFactor hL g m t r R C S u particle labels v steps i d) |
| (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)) |
| |
| omit [Fintype J] [DecidableEq J] in |
| theorem physicalSelectedOn_physical {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) |
| (i : routeSlots labels) : |
| let ρ := physicalSelectedOn hL g m t r R C S u particle labels v steps d i |
| let P := physicalSlotOn hL g m t r R C S u particle (v (Fin.last (n+1))) d i |
| let Q := physicalSlotOn hL g m t r R C S u particle (v 0) d ((routePermutation labels).symm i) |
| IsProbabilityMeasure ρ ∧ IsProbabilityMeasure P ∧ IsProbabilityMeasure Q ∧ |
| ρ.fst≤routeDensityBound t • P ∧ ρ.snd≤routeDensityBound t • Q ∧ |
| ρ.map (Prod.map Subtype.val Subtype.val)= |
| physicalSelectedFactor hL g m t r R C S u particle labels v steps i d := by |
| dsimp only |
| obtain ⟨hp,hP,hQ,hfst,hsnd,-⟩ := physicalSelectedFactor_physical |
| hL g m t r R C S ht u particle labels v steps hc d hhome haccess i |
| let := hp |
| let := hP |
| let := hQ |
| rw [←physicalSlotOn_map hL g m t r R C S u particle (v (Fin.last (n+1))) d i] at hfst |
| rw [←physicalSlotOn_map hL g m t r R C S u particle (v 0) d ((routePermutation labels).symm i)] at hsnd |
| have hS := physicalSlotAllowed_measurable hL g m r R C S u particle d i |
| have hT := physicalSlotAllowed_measurable hL g m r R C S u particle d ((routePermutation labels).symm i) |
| have hsupp := supported_of_marginals _ hS hT _ _ hfst hsnd |
| exact ⟨supportedPair_probability _ hS hT hsupp, |
| physicalSlotOn_probability hL g m t r R C S u particle _ d i, |
| physicalSlotOn_probability hL g m t r R C S u particle _ d _, |
| supportedPair_fst_le _ hS hT hsupp _ hfst, |
| supportedPair_snd_le _ hS hT hsupp _ hsnd, |
| supportedPair_map _ hS hT hsupp⟩ |
| |
| end HardSphere.Groups |
| end |
| end |
| |
| end OAI |
| |
