| import Mathlib |
| import OAI.Analysis.HardSphere.Geometry.TorusNear |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators Classical |
| namespace HardSphere.Lattice |
|
|
| def addedSites {K : ℕ} (H : Finset (DiscreteTorus K)) : Finset (DiscreteTorus K) := |
| H.biUnion fun c ↦ ((torusOffsets 1).erase 0).image (fun d ↦ c+torusProjection K d) |
|
|
| lemma torus_unit_offsets_card : ((torusOffsets 1).erase 0).card=26 := by |
| rw [Finset.card_erase_of_mem (by simp [torusOffsets,Fintype.mem_piFinset]),torusOffsets_card] |
| norm_num |
|
|
| lemma addedSites_card {K : ℕ} (H : Finset (DiscreteTorus K)) : (addedSites H).card≤26*H.card := by |
| calc |
| _ ≤ ∑ c∈H, (((torusOffsets 1).erase 0).image (fun d ↦ c+torusProjection K d)).card := Finset.card_biUnion_le |
| _ ≤ ∑ _c∈H, 26 := by |
| apply Finset.sum_le_sum |
| intro c _ |
| exact Finset.card_image_le.trans torus_unit_offsets_card.le |
| _ = 26*H.card := by simp [Nat.mul_comm] |
|
|
| lemma mem_addedSites {K : ℕ} {H : Finset (DiscreteTorus K)} {c x : DiscreteTorus K} |
| (hc : c∈H) (hne : c≠x) (hcx : torusNear K 1 c x) : x∈addedSites H := by |
| obtain ⟨d,hd,he⟩ := hcx |
| apply Finset.mem_biUnion.mpr |
| refine ⟨c,hc,Finset.mem_image.mpr ⟨d,Finset.mem_erase.mpr ⟨?_,?_⟩,?_⟩⟩ |
| · intro heq |
| apply hne |
| ext i |
| have h := he i |
| simp only [heq,Pi.zero_apply,Int.cast_zero,add_zero] at h |
| exact h.symm |
| · simpa only [torusOffsets,Fintype.mem_piFinset,Finset.mem_Icc,← abs_le] using hd |
| · ext i; exact (he i).symm |
|
|
| end HardSphere.Lattice |
|
|
| open scoped BigOperators Classical |
|
|
| open scoped BigOperators Classical |
| namespace HardSphere.Lattice |
| universe u v |
| variable {V : Type u} [Fintype V] [DecidableEq V] |
|
|
| def encounterSet (e : V → V → Prop) (P Q : Finset V) : Finset (V×V) := |
| (P×ˢQ).filter fun z ↦ e z.1 z.2 |
|
|
| omit [Fintype V] [DecidableEq V] in |
| lemma mem_encounterSet (e : V → V → Prop) (P Q : Finset V) (x y : V) : |
| (x,y)∈encounterSet e P Q ↔ x∈P ∧ y∈Q ∧ e x y := by |
| simp only [encounterSet,Finset.mem_filter,Finset.mem_product] |
| tauto |
|
|
| lemma encounterSet_card_le (e : V → V → Prop) (D : ℕ) |
| (hdeg : ∀ x, (Finset.univ.filter (e x)).card≤D) (P Q : Finset V) : |
| (encounterSet e P Q).card≤P.card*D := by |
| have hsub : encounterSet e P Q ⊆ |
| P.biUnion (fun x ↦ (Finset.univ.filter (e x)).image (fun y ↦ (x,y))) := by |
| rintro ⟨x,y⟩ hxy |
| obtain ⟨hx,_,he⟩ := (mem_encounterSet e P Q x y).mp hxy |
| exact Finset.mem_biUnion.mpr ⟨x,hx,Finset.mem_image.mpr ⟨y,by simp [he],rfl⟩⟩ |
| calc |
| _ ≤ (P.biUnion (fun x ↦ (Finset.univ.filter (e x)).image (fun y ↦ (x,y)))).card := Finset.card_le_card hsub |
| _ ≤ ∑ x∈P, ((Finset.univ.filter (e x)).image (fun y ↦ (x,y))).card := Finset.card_biUnion_le |
| _ ≤ ∑ _x∈P, D := by |
| apply Finset.sum_le_sum |
| intro x _ |
| exact Finset.card_image_le.trans (hdeg x) |
| _ = P.card*D := by simp |
|
|
| lemma encounterSet_card_le_right (e : V → V → Prop) (hsym : ∀ x y, e x y → e y x) (D : ℕ) |
| (hdeg : ∀ x, (Finset.univ.filter (e x)).card≤D) (P Q : Finset V) : |
| (encounterSet e P Q).card≤Q.card*D := by |
| have hsub : encounterSet e P Q ⊆ (encounterSet e Q P).image Prod.swap := by |
| rintro ⟨x,y⟩ hxy |
| obtain ⟨hx,hy,he⟩ := (mem_encounterSet e P Q x y).mp hxy |
| exact Finset.mem_image.mpr ⟨(y,x),(mem_encounterSet e Q P y x).mpr ⟨hy,hx,hsym x y he⟩,rfl⟩ |
| exact (Finset.card_le_card hsub).trans (Finset.card_image_le.trans (encounterSet_card_le e D hdeg Q P)) |
|
|
| theorem encounter_charge_bound {H : Type v} [DecidableEq H] |
| (e : V → V → Prop) (hsym : ∀ x y, e x y → e y x) (D b : ℕ) |
| (hdeg : ∀ x, (Finset.univ.filter (e x)).card≤D) |
| (Γ Γ' P Q : Finset V) (Hs : Finset H) (added : H → Finset V) (size : H → ℕ) |
| (hsize : ∀ h∈Hs, (added h).card≤b*size h) |
| (hcharge : ∀ x∈P, ∀ y∈Q, e x y → |
| (x∈Γ ∧ y∈Γ') ∨ ∃ h∈Hs, x∈added h ∨ y∈added h) : |
| (encounterSet e P Q).card ≤ (encounterSet e Γ Γ').card+(2*b*D)*∑ h∈Hs, size h := by |
| let A := Hs.biUnion (fun h ↦ encounterSet e (added h) Q ∪ encounterSet e P (added h)) |
| have hsubset : encounterSet e P Q ⊆ encounterSet e Γ Γ' ∪ A := by |
| rintro ⟨x,y⟩ hxy |
| obtain ⟨hx,hy,he⟩ := (mem_encounterSet e P Q x y).mp hxy |
| rcases hcharge x hx y hy he with hnom|⟨h,hh,hadd⟩ |
| · exact Finset.mem_union_left A ((mem_encounterSet e Γ Γ' x y).mpr ⟨hnom.1,hnom.2,he⟩) |
| · apply Finset.mem_union_right |
| apply Finset.mem_biUnion.mpr |
| refine ⟨h,hh,?_⟩ |
| rcases hadd with hadd|hadd |
| · exact Finset.mem_union_left _ ((mem_encounterSet e (added h) Q x y).mpr ⟨hadd,hy,he⟩) |
| · exact Finset.mem_union_right _ ((mem_encounterSet e P (added h) x y).mpr ⟨hx,hadd,he⟩) |
| have hA : A.card ≤ (2*b*D)*∑ h∈Hs, size h := by |
| calc |
| _ ≤ ∑ h∈Hs, (encounterSet e (added h) Q ∪ encounterSet e P (added h)).card := Finset.card_biUnion_le |
| _ ≤ ∑ h∈Hs, (2*b*D)*size h := by |
| apply Finset.sum_le_sum |
| intro h hh |
| calc |
| _ ≤ (encounterSet e (added h) Q).card+(encounterSet e P (added h)).card := Finset.card_union_le _ _ |
| _ ≤ (added h).card*D+(added h).card*D := |
| Nat.add_le_add (encounterSet_card_le e D hdeg _ _) (encounterSet_card_le_right e hsym D hdeg _ _) |
| _ ≤ (b*size h)*D+(b*size h)*D := Nat.add_le_add |
| (Nat.mul_le_mul_right D (hsize h hh)) (Nat.mul_le_mul_right D (hsize h hh)) |
| _ = (2*b*D)*size h := by ring |
| _ = (2*b*D)*∑ h∈Hs, size h := by rw [Finset.mul_sum] |
| exact (Finset.card_le_card hsubset).trans ((Finset.card_union_le _ _).trans (Nat.add_le_add_left hA _)) |
| |
| end HardSphere.Lattice |
| |
| namespace HardSphere.Lattice |
| |
| def torusContact {K : ℕ} (R : ℕ) (H Γ : Finset (DiscreteTorus K)) : Prop := |
| ∃ c∈H, ∃ z∈Γ, torusNear K R c z |
| |
| def shellAttachment {K : ℕ} [NeZero K] (R : ℕ) (B Γ : Finset (DiscreteTorus K)) |
| (x : DiscreteTorus K) : Prop := |
| ∃ c∈B, torusNear K 1 c x ∧ |
| ∃ z∈Γ, z∈badComponent (torusRangeGraph K (R+2)) B c |
| |
| lemma near_badComponent_eq {K R : ℕ} [NeZero K] (B : Finset (DiscreteTorus K)) |
| {c d : DiscreteTorus K} (hc : c∈B) (hd : d∈B) (hcd : torusNear K R c d) : |
| badComponent (torusRangeGraph K R) B c=badComponent (torusRangeGraph K R) B d := by |
| by_cases he : c=d |
| · rw [he] |
| · apply badComponent_eq_of_reachable |
| exact (show (badGraph (torusRangeGraph K R) B).Adj c d from ⟨hc,hd,he,hcd⟩).reachable |
| |
| lemma shell_encounter_charge {K R : ℕ} [NeZero K] |
| (B Γ Γ' : Finset (DiscreteTorus K)) {x y : DiscreteTorus K} |
| (hxgood : x∉B) (hxattach : shellAttachment R B Γ x) |
| (hy : y∈Γ' ∨ shellAttachment R B Γ' y) (hxy : torusNear K R x y) : |
| ∃ H∈badComponents (torusRangeGraph K (R+2)) B, |
| torusContact (R+2) H Γ ∧ torusContact (R+2) H Γ' ∧ x∈addedSites H := by |
| obtain ⟨c,hc,hcx,z,hz,hzc⟩ := hxattach |
| let H := badComponent (torusRangeGraph K (R+2)) B c |
| have hcH : c∈H := mem_own_badComponent _ _ hc |
| refine ⟨H,Finset.mem_image.mpr ⟨c,hc,rfl⟩,?_,?_,?_⟩ |
| · exact ⟨z,hzc,z,hz,torusNear_refl _ _ _⟩ |
| · rcases hy with hy|hy |
| · exact ⟨c,hcH,y,hy,torusNear_mono (by omega) (torusNear_add hcx hxy)⟩ |
| · obtain ⟨d,hd,hdy,w,hw,hwd⟩ := hy |
| have hcd : torusNear K (R+2) c d := by |
| convert torusNear_add (torusNear_add hcx hxy) (torusNear_symm hdy) using 1; omega |
| have he := near_badComponent_eq B hc hd hcd |
| have hwH : w∈H := by dsimp [H]; rw [he]; exact hwd |
| exact ⟨w,hwH,w,hw,torusNear_refl _ _ _⟩ |
| · exact mem_addedSites hcH (fun he ↦ hxgood (he ▸ hc)) hcx |
| |
| theorem torus_detour_charge {K : ℕ} [NeZero K] (R : ℕ) |
| (B Γ Γ' P Q : Finset (DiscreteTorus K)) |
| (hP : ∀ x∈P, x∉B ∧ (x∈Γ ∨ shellAttachment R B Γ x)) |
| (hQ : ∀ y∈Q, y∉B ∧ (y∈Γ' ∨ shellAttachment R B Γ' y)) : |
| (encounterSet (torusNear K R) P Q).card ≤ |
| (encounterSet (torusNear K R) Γ Γ').card+ |
| (52*(2*R+1)^3)*∑ H∈(badComponents (torusRangeGraph K (R+2)) B).filter |
| (fun H ↦ torusContact (R+2) H Γ ∧ torusContact (R+2) H Γ'), H.card := by |
| apply encounter_charge_bound (torusNear K R) (fun _ _ h ↦ torusNear_symm h) |
| ((2*R+1)^3) 26 (torusNear_ball_card R) Γ Γ' P Q |
| ((badComponents (torusRangeGraph K (R+2)) B).filter |
| (fun H ↦ torusContact (R+2) H Γ ∧ torusContact (R+2) H Γ')) addedSites Finset.card |
| · intro H _ |
| exact addedSites_card H |
| · intro x hx y hy hxy |
| have hx' := hP x hx |
| have hy' := hQ y hy |
| rcases hx'.2 with hxnom|hxattach |
| · rcases hy'.2 with hynom|hyattach |
| · exact Or.inl ⟨hxnom,hynom⟩ |
| · obtain ⟨H,hH,hHΓ',hHΓ,hyH⟩ := shell_encounter_charge B Γ' Γ hy'.1 hyattach |
| (Or.inl hxnom) (torusNear_symm hxy) |
| exact Or.inr ⟨H,Finset.mem_filter.mpr ⟨hH,hHΓ,hHΓ'⟩,Or.inr hyH⟩ |
| · obtain ⟨H,hH,hHΓ,hHΓ',hxH⟩ := shell_encounter_charge B Γ Γ' hx'.1 hxattach hy'.2 hxy |
| exact Or.inr ⟨H,Finset.mem_filter.mpr ⟨hH,hHΓ,hHΓ'⟩,Or.inl hxH⟩ |
| |
| end HardSphere.Lattice |
| |
| end |
| |
| end OAI |
| |
