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import OAI.Analysis.HardSphere.Geometry.PathSlots
namespace OAI
section
section
noncomputable section
open scoped Classical
namespace HardSphere.Comparison
open HardSphere.Lattice
variable {I V : Type*} [DecidableEq V]
lemma nearSlots_card_of_fiber (e : V→V→Prop) (c : I→V) (s t : Finset I)
(B : ℕ) (hB : ∀v,(s.filter (fun i ↦ c i=v)).card≤B) :
(nearSlots e c s t).card≤B*(encounterSet e (s.image c) (t.image c)).card := by
have hm : (nearSlots e c s t).image c⊆
(encounterSet e (s.image c) (t.image c)).image Prod.fst := by
intro v hv
obtain ⟨i,hi,rfl⟩ := Finset.mem_image.mp hv
obtain ⟨his,j,hjt,hij⟩ := (mem_nearSlots e c s t i).mp hi
exact Finset.mem_image.mpr ⟨(c i,c j),(mem_encounterSet e _ _ _ _).mpr
⟨Finset.mem_image.mpr ⟨i,his,rfl⟩,Finset.mem_image.mpr ⟨j,hjt,rfl⟩,hij⟩,rfl⟩
have hc : ((nearSlots e c s t).image c).card≤(encounterSet e (s.image c) (t.image c)).card :=
(Finset.card_le_card hm).trans Finset.card_image_le
calc
_=∑v∈(nearSlots e c s t).image c,((nearSlots e c s t).filter (fun i ↦ c i=v)).card :=
Finset.card_eq_sum_card_image c _
_≤∑_v∈(nearSlots e c s t).image c,B := by
apply Finset.sum_le_sum
intro v _
exact (Finset.card_le_card (Finset.filter_subset_filter _ (Finset.filter_subset _ _))).trans (hB v)
_=((nearSlots e c s t).image c).card*B := by simp
_≤B*(encounterSet e (s.image c) (t.image c)).card := by simpa only [Nat.mul_comm] using Nat.mul_le_mul_right B hc
variable {K : ℕ} {x y : DiscreteTorus K} {J : Type*}
lemma departure_anchor_injective (p : SimplePath x y) (anchor : J→DiscreteTorus K)
(pick : Fin p.val.length→J)
(hpick : ∀i,anchor (pick i)=pathVertexEmbedding p i.castSucc) :
Set.InjOn anchor (Set.range pick) := by
rintro a ⟨i,rfl⟩ b ⟨j,rfl⟩ hab
rw [hpick i,hpick j] at hab
have h := (pathVertexEmbedding p).injective hab
have he : i=j := Fin.ext (congrArg (fun k : Fin (p.val.length+1) ↦ k.val) h)
exact congrArg pick he
lemma routeSlots_start_fiber (p : SimplePath x y) (anchor : J→DiscreteTorus K)
(pick : Fin p.val.length→J)
(hpick : ∀i,anchor (pick i)=pathVertexEmbedding p i.castSucc) (v : DiscreteTorus K) :
((routeSlots (departureEmbedding p anchor pick hpick)).filter
(fun i ↦ extendedAnchor anchor x i=v)).card≤2 := by
let s := (routeSlots (departureEmbedding p anchor pick hpick)).filter
(fun i ↦ extendedAnchor anchor x i=v)
have hone : (s.erase none).card≤1 := by
apply Finset.card_le_one.mpr
intro a ha b hb
have hna := Finset.mem_erase.mp ha
have hnb := Finset.mem_erase.mp hb
obtain ⟨a,rfl⟩ := Option.ne_none_iff_exists'.mp hna.1
obtain ⟨b,rfl⟩ := Option.ne_none_iff_exists'.mp hnb.1
have has := Finset.mem_filter.mp hna.2
have hbs := Finset.mem_filter.mp hnb.2
rw [routeSlots_exact] at has hbs
have hai : a∈Set.range pick := by
rcases Finset.mem_insert.mp has.1 with h|h
· contradiction
· obtain ⟨i,_,he⟩ := Finset.mem_image.mp h
exact ⟨i,Option.some.inj he⟩
have hbi : b∈Set.range pick := by
rcases Finset.mem_insert.mp hbs.1 with h|h
· contradiction
· obtain ⟨i,_,he⟩ := Finset.mem_image.mp h
exact ⟨i,Option.some.inj he⟩
exact congrArg some (departure_anchor_injective p anchor pick hpick hai hbi (has.2.trans hbs.2.symm))
have hcard : s.card≤(s.erase none).card+1 := by
have hin : s⊆insert none (s.erase none) := by intro a ha;by_cases h:a=none <;> simp_all
exact (Finset.card_le_card hin).trans (Finset.card_insert_le _ _)
exact hcard.trans (by omega)
end HardSphere.Comparison
end
end
section
noncomputable section
open scoped Classical
namespace HardSphere.Comparison
open HardSphere.Lattice
variable {K : ℕ} {x y : DiscreteTorus K} {J : Type*}
lemma routeSlots_start_sites (p : SimplePath x y) (anchor : J→DiscreteTorus K)
(pick : Fin p.val.length→J)
(hpick : ∀i,anchor (pick i)=pathVertexEmbedding p i.castSucc) :
(routeSlots (departureEmbedding p anchor pick hpick)).image (extendedAnchor anchor x)⊆p.val.support.toFinset := by
intro v hv
obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hv
rw [routeSlots_exact] at ha
rcases Finset.mem_insert.mp ha with rfl|ha
· exact List.mem_toFinset.mpr p.val.start_mem_support
· obtain ⟨i,_,rfl⟩ := Finset.mem_image.mp ha
change anchor (pick i)∈_
rw [hpick i]
exact List.mem_toFinset.mpr (p.val.getVert_mem_support i.val)
lemma second_primary_nearSlots_bound (R : ℕ) (p q : SimplePath x y) (anchor : J→DiscreteTorus K)
(f : Fin p.val.length→J) (g : Fin q.val.length→J)
(hf : ∀i,anchor (f i)=pathVertexEmbedding p i.castSucc)
(hg : ∀i,anchor (g i)=pathVertexEmbedding q i.castSucc) :
(nearSlots (torusNear K R) (extendedAnchor anchor x)
(routeSlots (departureEmbedding p anchor f hf)) (routeSlots (departureEmbedding q anchor g hg))).card≤
2*simpleEncounter R p q := by
have h := nearSlots_card_of_fiber (torusNear K R) (extendedAnchor anchor x)
(routeSlots (departureEmbedding p anchor f hf)) (routeSlots (departureEmbedding q anchor g hg))
2 (routeSlots_start_fiber p anchor f hf)
apply h.trans
apply Nat.mul_le_mul_left
apply Finset.card_le_card
intro v hv
obtain ⟨ha,hb,he⟩ := (mem_encounterSet _ _ _ _ _).mp hv
exact (mem_encounterSet _ _ _ _ _).mpr
⟨routeSlots_start_sites p anchor f hf ha,routeSlots_start_sites q anchor g hg hb,he⟩
end HardSphere.Comparison
end
end
end
end OAI
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