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9eeb9c7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 | 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
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