| import Mathlib |
| import OAI.Probability.CriticalPercolation.Clusters.ComponentSelf |
| import OAI.Probability.CriticalPercolation.GraphGeometry.Link2 |
|
|
| namespace OAI |
|
|
| section |
| section CurrentModule3 |
| noncomputable section |
| open Set |
| open scoped Classical BigOperators |
|
|
| namespace CriticalPercolation.Sharpness |
| open ClusterDeletion FiniteProduct |
|
|
| variable {V E : Type*} (G : BondGraph V E) |
|
|
| def cutInside (S : Set V) (e : E) : V := |
| if (G.ends e).out.1 ∈ S then (G.ends e).out.1 else (G.ends e).out.2 |
|
|
| def cutOutside (S : Set V) (e : E) : V := |
| if (G.ends e).out.1 ∈ S then (G.ends e).out.2 else (G.ends e).out.1 |
|
|
| def IsCut (S : Set V) (e : E) : Prop := cutInside G S e ∈ S ∧ cutOutside G S e ∉ S |
|
|
| lemma cut_ends (S : Set V) (e : E) : |
| G.ends e = s(cutInside G S e, cutOutside G S e) := by |
| unfold cutInside cutOutside |
| split_ifs |
| · exact (G.ends e).out_eq.symm |
| · exact (G.ends e).out_eq.symm.trans (Sym2.eq_swap) |
|
|
| lemma cut_orientation {S : Set V} {e : E} {x y : V} |
| (he : G.ends e = s(x,y)) (hx : x ∈ S) (hy : y ∉ S) : |
| cutInside G S e = x ∧ cutOutside G S e = y := by |
| have h : s((G.ends e).out.1,(G.ends e).out.2) = s(x,y) := (G.ends e).out_eq.trans he |
| rcases Sym2.eq_iff.mp h with ⟨h1,h2⟩ | ⟨h1,h2⟩ |
| · change (G.ends e).out.1 = x at h1 |
| change (G.ends e).out.2 = y at h2 |
| simp only [cutInside,cutOutside,h1,h2,ite_eq_left hx,and_self] |
| · change (G.ends e).out.1 = y at h1 |
| change (G.ends e).out.2 = x at h2 |
| simp only [cutInside,cutOutside,h1,h2,ite_eq_right hy,and_self] |
|
|
| lemma boundaryReach_iff_cut (S : Set V) (H : SimpleGraph V) (o : V) (e : E) : |
| boundaryReach G Sᶜ H o e ↔ IsCut G S e ∧ H.Reachable o (cutInside G S e) := by |
| constructor |
| · rintro ⟨x,y,he,hx,hy,hr⟩ |
| change y ∉ S at hy |
| have hh := cut_orientation G he (not_not.mp hx) hy |
| exact ⟨by simpa only [IsCut,hh.1,hh.2] using And.intro (not_not.mp hx) hy, |
| by simpa only [hh.1] using hr⟩ |
| · rintro ⟨hc,hr⟩ |
| exact ⟨_,_,cut_ends G S e,not_not.mpr hc.1,hc.2,hr⟩ |
|
|
| def internalIncident (S C : Set V) : Set E := |
| {e | e ∈ G.incidentEdges C ∧ ∀ v ∈ G.ends e, v ∈ S} |
|
|
| def internalCluster (S : Set V) (o : V) (ω : Set E) : Set V := |
| component (delete (G.openGraph ω) Sᶜ) o |
|
|
| lemma internalCluster_self (S : Set V) (o : V) (ω : Set E) : |
| o ∈ internalCluster G S o ω := .refl _ |
|
|
| lemma internalCluster_subset (S : Set V) {o : V} (ho : o ∈ S) (ω : Set E) : |
| internalCluster G S o ω ⊆ S := by |
| intro v hv |
| exact reachable_mem_of_closed (H := delete (G.openGraph ω) Sᶜ) |
| (U := S) (fun {_ _} _ h => not_not.mp h.2.2) ho hv |
|
|
| lemma internal_adj_eq {S C : Set V} {ω ω' : Set E} |
| (hω : ∀ e ∈ internalIncident G S C, e ∈ ω ↔ e ∈ ω') {x y : V} (hx : x ∈ C) : |
| (delete (G.openGraph ω) Sᶜ).Adj x y ↔ (delete (G.openGraph ω') Sᶜ).Adj x y := by |
| have hh (e) (he : G.ends e = s(x,y)) (hxS : x ∈ S) (hyS : y ∈ S) : |
| e ∈ internalIncident G S C := by |
| refine ⟨⟨x,hx,by rw [he]; simp⟩,?_⟩ |
| intro v hv |
| rw [he,Sym2.mem_iff] at hv |
| rcases hv with rfl | rfl <;> assumption |
| constructor |
| · rintro ⟨⟨⟨e,he,hend⟩,hne⟩,hxs,hys⟩ |
| exact ⟨⟨⟨e,(hω e (hh e hend (not_not.mp hxs) (not_not.mp hys))).mp he,hend⟩,hne⟩,hxs,hys⟩ |
| · rintro ⟨⟨⟨e,he,hend⟩,hne⟩,hxs,hys⟩ |
| exact ⟨⟨⟨e,(hω e (hh e hend (not_not.mp hxs) (not_not.mp hys))).mpr he,hend⟩,hne⟩,hxs,hys⟩ |
| |
| lemma internalCluster_eq_of_incident {S C : Set V} {o : V} {ω ω' : Set E} |
| (hC : internalCluster G S o ω = C) |
| (hω : ∀ e ∈ internalIncident G S C, e ∈ ω ↔ e ∈ ω') : |
| internalCluster G S o ω' = C := by |
| change component (delete (G.openGraph ω') Sᶜ) o = C |
| rw [← roots_singleton] |
| exact roots_eq_of_adj_eq_incident |
| (by simpa only [roots_singleton, internalCluster] using hC) |
| (fun {_ _} hx => internal_adj_eq G hω hx) |
|
|
| lemma dependsOn_internalCluster (S C : Set V) (o : V) : |
| DependsOn (fun ω : Set E => internalCluster G S o ω = C) (internalIncident G S C) := by |
| intro ω ω' hω |
| apply propext |
| exact ⟨fun h => internalCluster_eq_of_incident G h (fun e he => (hω e he).to_iff), |
| fun h => internalCluster_eq_of_incident G h (fun e he => (hω e he).symm.to_iff)⟩ |
|
|
| lemma last_exit {H : SimpleGraph V} {C : Set V} {o v : V} |
| (ho : o ∈ C) (hv : v ∉ C) (hr : H.Reachable o v) : |
| ∃ x ∈ C, ∃ y ∉ C, H.Adj x y ∧ (delete H C).Reachable y v := by |
| have hh : ∀ z, H.Reachable v z → |
| (z ∉ C ∧ (delete H C).Reachable v z) ∨ |
| ∃ x ∈ C, ∃ y ∉ C, H.Adj x y ∧ (delete H C).Reachable y v := by |
| intro z hz |
| rw [SimpleGraph.reachable_iff_reflTransGen] at hz |
| induction hz with |
| | refl => exact Or.inl ⟨hv,.refl _⟩ |
| | @tail a b _ hab ih => |
| rcases ih with ⟨ha,hva⟩ | ih |
| · by_cases hb : b ∈ C |
| · exact Or.inr ⟨b,hb,a,ha,hab.symm,hva.symm⟩ |
| · exact Or.inl ⟨hb,hva.trans (SimpleGraph.Adj.reachable ⟨hab,ha,hb⟩)⟩ |
| · exact Or.inr ih |
| exact (hh o hr.symm).resolve_left (fun h => h.1 ho) |
|
|
| lemma internalCluster_exit (S : Set V) (o : V) (ho : o ∈ S) (ω : Set E) {v : V} |
| (hv : v ∉ S) (hr : (G.openGraph ω).Reachable o v) : |
| ∃ e : E, IsCut G S e ∧ cutInside G S e ∈ internalCluster G S o ω ∧ e ∈ ω ∧ |
| (delete (G.openGraph ω) (internalCluster G S o ω)).Reachable (cutOutside G S e) v := by |
| obtain ⟨x,hx,y,hy,hxy,hyv⟩ := last_exit (internalCluster_self G S o ω) |
| (fun hc => hv (internalCluster_subset G S ho ω hc)) hr |
| have hxS : x ∈ S := internalCluster_subset G S ho ω hx |
| have hyS : y ∉ S := by |
| intro hyS |
| exact hy (hx.trans (SimpleGraph.Adj.reachable ⟨hxy,not_not.mpr hxS,not_not.mpr hyS⟩)) |
| obtain ⟨⟨e,he,hend⟩,_hne⟩ := hxy |
| obtain ⟨hi,ho'⟩ := cut_orientation G hend hxS hyS |
| exact ⟨e,by simpa only [IsCut,hi,ho'] using And.intro hxS hyS, |
| by rwa [hi],he,by rwa [ho']⟩ |
|
|
| lemma cut_not_internalIncident {S C : Set V} {e : E} (hc : IsCut G S e) : |
| e ∉ internalIncident G S C := by |
| intro h |
| apply hc.2 |
| apply h.2 |
| rw [cut_ends G S e] |
| simp |
|
|
| lemma delete_update_incident {C : Set V} {e : E} (he : e ∈ G.incidentEdges C) |
| (ω : Set E) (b : Prop) : |
| delete (G.openGraph (Function.update ω e b)) C = delete (G.openGraph ω) C := by |
| apply G.delete_openGraph_dependsOn C |
| intro j hj |
| have hje : j ≠ e := by rintro rfl; exact hj he |
| exact Function.update_of_ne hje _ _ |
| |
| end CriticalPercolation.Sharpness |
| end |
| end CurrentModule3 |
| |
| end |
| |
| end OAI |
| |
