| import Mathlib |
| import OAI.Analysis.HardSphere.Conditioning.SlotMultiplicity |
| import OAI.Analysis.HardSphere.Allocation.PhysicalRouteGeometry |
| import OAI.Analysis.HardSphere.Allocation.RouteCenters |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal |
| namespace HardSphere.Groups |
| open Wiener Bridge Sampling Comparison GaussianPacking PathCutoff CellOrbital HardSphere.Lattice |
| attribute [local instance] Classical.propDecidable |
| variable {K : ℕ} [NeZero K] {x y : DiscreteTorus K} {J : Type*} |
|
|
| def simpleRouteVertices (p : SimplePath x y) (n : ℕ) : Fin (n+2)→Fin 3→Fin K := |
| fun i ↦ siteIndex (p.val.getVert i.val) |
|
|
| lemma simpleRouteVertices_zero (p : SimplePath x y) (n : ℕ) : |
| simpleRouteVertices p n 0=siteIndex x := by simp [simpleRouteVertices] |
| lemma simpleRouteVertices_last (p : SimplePath x y) (n : ℕ) (hn : n+1=p.val.length) : |
| simpleRouteVertices p n (Fin.last (n+1))=siteIndex y := by |
| simp only [simpleRouteVertices,Fin.val_last,hn,SimpleGraph.Walk.getVert_length] |
|
|
| omit [NeZero K] in |
| lemma simpleRouteVertices_adjacent (p : SimplePath x y) (n : ℕ) (hn : n+1=p.val.length) |
| (i : Fin (n+1)) : |
| (torusNearestGraph K).Adj (p.val.getVert i.castSucc.val) (p.val.getVert i.succ.val) := |
| p.val.adj_getVert_succ (by simpa only [Fin.val_castSucc, ←hn] using i.isLt) |
|
|
| lemma simpleRouteVertices_distance {L : Length} (hL : L.val=(K:ℝ)) |
| (p : SimplePath x y) (n : ℕ) (hn : n+1=p.val.length) : |
| ∀i:Fin (n+1),torusDistance L (gridCenter L (simpleRouteVertices p n i.castSucc)) |
| (gridCenter L (simpleRouteVertices p n i.succ))≤1 := |
| fun i ↦ adjacent_center_distance hL (simpleRouteVertices_adjacent p n hn i) |
|
|
| def simpleRouteSteps {L : Length} (hL : L.val=(K:ℝ)) |
| (p : SimplePath x y) (n : ℕ) (hn : n+1=p.val.length) : Fin (n+1)→Fin 7 := |
| fun i ↦ (adjacent_routeCellPeriod hL (simpleRouteVertices_adjacent p n hn i)).choose |
|
|
| lemma simpleRouteSteps_period {L : Length} (hL : L.val=(K:ℝ)) |
| (p : SimplePath x y) (n : ℕ) (hn : n+1=p.val.length) : |
| ∀ i : Fin (n+1), projectPoint L (routeCellPeriod (simpleRouteVertices p n i.succ) |
| (simpleRouteVertices p n i.castSucc) (simpleRouteSteps hL p n hn i))=0 := |
| fun i ↦ (adjacent_routeCellPeriod hL (simpleRouteVertices_adjacent p n hn i)).choose_spec |
|
|
| lemma gridSite_slotCenter {home : J→Fin 3→Fin K} (w : DiscreteTorus K) : |
| (fun i ↦ gridSite (slotCenter home (siteIndex w) i))= |
| extendedAnchor (fun j ↦ gridSite (home j)) w := by |
| funext i |
| cases i with |
| | none => exact indexSite_siteIndex w |
| | some j => rfl |
|
|
| lemma routeSlots_reparam {d e : ℕ} (f : Fin d↪J) (h : e=d) : |
| routeSlots ((finCongr h).toEmbedding.trans f)=routeSlots f := by |
| subst e |
| rfl |
|
|
| lemma simpleRouteSlots {n : ℕ} (p : SimplePath x y) (hn : n+1=p.val.length) |
| (home : J→Fin 3→Fin K) (f : Fin (n+1)↪J) |
| (hf : ∀i,home (f i)=simpleRouteVertices p n i.castSucc) : |
| ∃ pick : Fin p.val.length→J, ∃ H : ∀i,gridSite (home (pick i))=pathVertexEmbedding p i.castSucc, |
| routeSlots f=routeSlots (departureEmbedding p (fun j ↦ gridSite (home j)) pick H) := by |
| let pick : Fin p.val.length→J := fun i ↦ f (Fin.cast hn.symm i) |
| have hp : ∀i,gridSite (home (pick i))=pathVertexEmbedding p i.castSucc := by |
| intro i |
| dsimp only [pick] |
| rw [hf] |
| exact indexSite_siteIndex _ |
| refine ⟨pick,hp,?_⟩ |
| have he : departureEmbedding p (fun j ↦ gridSite (home j)) pick hp= |
| (finCongr hn.symm).toEmbedding.trans f := by ext i; rfl |
| rw [he,routeSlots_reparam] |
| |
| lemma simpleRoute_near_first {n l : ℕ} (p q : SimplePath x y) |
| (hn : n+1=p.val.length) (hl : l+1=q.val.length) |
| (home : J→Fin 3→Fin K) (f : Fin (n+1)↪J) (g : Fin (l+1)↪J) |
| (hf : ∀i,home (f i)=simpleRouteVertices p n i.castSucc) |
| (hg : ∀i,home (g i)=simpleRouteVertices q l i.castSucc) (R : ℕ) : |
| (nearSlots (torusNear K R) (fun i ↦ gridSite (slotCenter home (siteIndex y) i)) |
| (routeSlots f) (routeSlots g)).card≤ simpleEncounter R p q := by |
| obtain ⟨pick,hpick,hp⟩ := simpleRouteSlots p hn home f hf |
| obtain ⟨choose,hchoose,hq⟩ := simpleRouteSlots q hl home g hg |
| rw [gridSite_slotCenter,hp,hq] |
| exact simple_nearSlots_bound R p q _ pick choose hpick hchoose |
|
|
| lemma simpleRoute_near_second {n l : ℕ} (p q : SimplePath x y) |
| (hn : n+1=p.val.length) (hl : l+1=q.val.length) |
| (home : J→Fin 3→Fin K) (f : Fin (n+1)↪J) (g : Fin (l+1)↪J) |
| (hf : ∀i,home (f i)=simpleRouteVertices p n i.castSucc) |
| (hg : ∀i,home (g i)=simpleRouteVertices q l i.castSucc) (R : ℕ) : |
| (nearSlots (torusNear K R) (fun i ↦ gridSite (slotCenter home (siteIndex x) i)) |
| (routeSlots f) (routeSlots g)).card≤2*simpleEncounter R p q := by |
| obtain ⟨pick,hpick,hp⟩ := simpleRouteSlots p hn home f hf |
| obtain ⟨choose,hchoose,hq⟩ := simpleRouteSlots q hl home g hg |
| rw [gridSite_slotCenter,hp,hq] |
| exact second_primary_nearSlots_bound R p q _ pick choose hpick hchoose |
|
|
| end HardSphere.Groups |
| end |
| end |
|
|
| end OAI |
|
|
