| import Mathlib |
| import OAI.Analysis.HardSphere.Geometry.PathSlots |
|
|
| namespace OAI |
|
|
| section |
|
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| 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 |
|
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| 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⟩ |
|
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| end HardSphere.Comparison |
|
|
| end |
| end |
|
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| end |
|
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| end OAI |
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