| import Mathlib |
| 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 |
|
|
