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import OAI.Analysis.HardSphere.Collisions.PhysicalCollision
import OAI.Analysis.HardSphere.Bridges.RouteMarginals
namespace OAI
section
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal NNReal
namespace HardSphere.Groups
open Wiener Bridge Sampling Comparison GaussianPacking PathCutoff CellOrbital
attribute [local instance] Classical.propDecidable
variable {J : Type*} [DecidableEq J]
def physicalCollisionReference {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) (d : PhysicalDatum N K M G L) (j : J) :
Measure (physicalSlotAllowed hL g m r R C S u particle d (some j)) :=
if datumAccessible hL g m t r R C S u (particle j) d then
physicalSlotOn hL g m t r R C S u particle (datumHome hL d (particle j)) d (some j) else 0
omit [DecidableEq J] in
lemma physicalCollisionReference_eq {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) (d : PhysicalDatum N K M G L) (j : J)
(ha : datumAccessible hL g m t r R C S u (particle j) d) (v : Fin 3→Fin K) :
physicalCollisionReference hL g m t r R C S u particle d j=
physicalSlotOn hL g m t r R C S u particle v d (some j) := by
change (if datumAccessible hL g m t r R C S u (particle j) d then _ else _) = _
rw [ite_eq_left ha]
rfl
lemma routeOwner_mem_real {n : ℕ} (labels : Fin n↪J) (b : Bool) (j : J)
(h : routeOwner (routePermutation labels) some b j∈routeSlots labels) :
some j∈routeSlots labels := by
cases b
· exact h
· exact (move_mem_iff _ _ (routePermutation_fixed labels) _).mp h
lemma reverse_routeOwner_mem_real {n : ℕ} (labels : Fin n↪J) (b : Bool) (j : J)
(h : routeOwner (routePermutation labels).symm some b j∈routeSlots labels) :
some j∈routeSlots labels := by
cases b
· exact h
· exact (move_mem_iff _ _ (reverse_fix _ _ (routePermutation_fixed labels)) _).mp h
omit [DecidableEq J] in
theorem physicalCollisionReference_collision {N K M G m : ℕ} {L : Length}
(hL : L.val=(K:ℝ)) (h50 : 50 < L.val) (hm : 0 < m)
(g : Fin G) (t r R C S h : ℝ) (u : Time)
(ht : 0 < t) (hr : 0 < r) (hr1 : r≤1) (hR : 0≤R) (hC : 0≤C) (hh : 0 < h)
(hu : (u:ℝ)=h/(2*t)) (hsmall : r*R≤1)
(hτr : (2*t)/(m:ℝ)≤r^2) (hmscale : (m:ℝ)*r^2≤2*(2*t))
(hwindow : 2*((2*t)/(m:ℝ))≤h) (hlarge : 8*(2*t)≤(m:ℝ)*h) (hdrift : 26/(m:ℝ)≤r)
(particle : J↪Fin N) (d : PhysicalDatum N K M G L)
(hy : ∀j,‖datumStart hL d (particle j)-anchorPoint (datumHome hL d (particle j))‖≤3)
(hz : ∀j,‖datumEnd hL d (particle j)-anchorPoint (datumHome hL d (particle j))‖≤3)
(hsep0 : ∀ i j, i ≠ j → r + 2*S < torusDistance L (projectPoint L (datumStart hL d (particle i)))
(projectPoint L (datumStart hL d (particle j))))
(hsep1 : ∀ i j, i ≠ j → r + 2*S < torusDistance L (projectPoint L (datumEnd hL d (particle i)))
(projectPoint L (datumEnd hL d (particle j)))) :
∀ j k, j ≠ k → ∀ f : physicalSlotAllowed hL g m r R C S u particle d (some k),
physicalCollisionReference hL g m t r R C S u particle d j
{p | TrialCollision r (projectCurve L f.val) p.val}≤datumCollisionBound t r h R C := by
intro j k hjk f
by_cases hj : datumAccessible hL g m t r R C S u (particle j) d
· simp only [physicalCollisionReference,ite_eq_left hj]
exact datum_reference_collision hL h50 hm g t r R C S h u ht hr hr1 hR hC hh hu hsmall
hτr hmscale hwindow hlarge hdrift d (particle j) (particle k) hj (hy j) (hz j)
(hsep0 j k hjk) (hsep1 j k hjk) f
· simp only [physicalCollisionReference,ite_eq_right hj,Measure.coe_zero,Pi.zero_apply,zero_le]
omit [DecidableEq J] in
lemma physical_primary_domination {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) (d : PhysicalDatum N K M G L) (j : J)
(ha : datumAccessible hL g m t r R C S u (particle j) d) (v : Fin 3→Fin K) :
(physicalSlotOn hL g m t r R C S u particle v d (some j)).map (fun f ↦ f)≤
routeDensityBound t • physicalCollisionReference hL g m t r R C S u particle d j := by
rw [Measure.map_id',physicalCollisionReference_eq hL g m t r R C S u particle d j ha v]
intro A
rw [Measure.smul_apply]
exact le_trans (le_of_eq (one_mul _).symm) (mul_le_mul_left (one_le_routeDensityBound t) _)
theorem physical_auxiliary_collision_bounds {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)
(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)) :
let σ := routePermutation labels
let s := routeSlots labels
let A := fun j ↦ physicalSlotAllowed hL g m r R C S u particle d (some j)
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
(∀b j,routeOwner σ some b j∈s→
((μ (routeOwner σ some b j)).compProd (κ (routeOwner σ some b j))).map
(routePath (X:=fun i ↦ physicalSlotAllowed hL g m r R C S u particle d i) σ s (routePermutation_fixed labels) A some (fun _ f ↦ f) b j)≤
routeDensityBound t • physicalCollisionReference hL g m t r R C S u particle d j) ∧
(∀b j,routeOwner σ.symm some b j∈s→
((ν (routeOwner σ.symm some b j)).compProd (η (routeOwner σ.symm some b j))).map
(routePath (X:=fun i ↦ physicalSlotAllowed hL g m r R C S u particle d i) σ.symm s (reverse_fix σ s (routePermutation_fixed labels)) A some (fun _ f ↦ f) b j)≤
routeDensityBound t • physicalCollisionReference hL g m t r R C S u particle d j) := by
dsimp only
obtain ⟨h₀,h₁⟩ := physical_auxiliary_path_bounds hL g m t r R C S ht u particle labels v steps hc d hhome haccess κ η he hr
constructor
· intro b j hj
rw [physicalCollisionReference_eq hL g m t r R C S u particle d j
(route_selected_accessible hL g m t r R C S u particle labels d haccess j
(routeOwner_mem_real labels b j hj)) (v (Fin.last (n+1)))]
exact h₀ b j hj
· intro b j hj
rw [physicalCollisionReference_eq hL g m t r R C S u particle d j
(route_selected_accessible hL g m t r R C S u particle labels d haccess j
(reverse_routeOwner_mem_real labels b j hj)) (v 0)]
exact h₁ b j hj
end HardSphere.Groups
end
end
end OAI
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