| import Mathlib |
| import OAI.Probability.CriticalPercolation.ProductMeasure.Delta |
| import OAI.Probability.CriticalPercolation.GraphGeometry.ForceFalseSubset |
|
|
| namespace OAI |
|
|
| section |
| section |
| open Set MeasureTheory ProbabilityTheory |
| open Set |
| open scoped BigOperators |
| open scoped Classical |
|
|
| namespace CriticalPercolation.BondGraph |
| open scoped Classical |
| open FiniteProduct |
|
|
| variable {V E : Type*} |
|
|
| theorem contact_pivot_iff (G : BondGraph V E) (ω : Set E) (B : Set V) |
| (a u v : V) (f e : E) (hf : G.ends f = s(u,v)) (hef : e ≠ f) |
| (hn : ¬∃ b ∈ B, (G.openGraph ω).Reachable a b) |
| (hau : (G.openGraph ω).Reachable a u) |
| (hv : ∃ b ∈ B, (G.openGraph ω).Reachable v b) |
| (he : ∃ r ∈ G.ends e, (G.openGraph ω).Reachable a r) : |
| openPivot (fun σ => ∃ b ∈ B, (G.openGraph σ).Reachable a b) |
| e (Function.update ω f True) ↔ |
| ω e ∧ ¬(G.openGraph (Function.update ω e False)).Reachable a u := by |
| let F := fun σ : Set E => ∃ b ∈ B, (G.openGraph σ).Reachable a b |
| have hav : ¬(G.openGraph ω).Reachable a v := by |
| intro h |
| obtain ⟨b,hb,hvb⟩ := hv |
| exact hn ⟨b,hb,h.trans hvb⟩ |
| have hopen : F (Function.update ω f True) := |
| (G.target_after_open_iff ω B a u v f hf hn hv hav).2 hau |
| have hmon : G.openGraph (Function.update ω e False) ≤ G.openGraph ω := |
| SimpleGraph.fromEdgeSet_mono (Set.image_mono (force_false_subset ω e)) |
| have hn' : ¬∃ b ∈ B, (G.openGraph (Function.update ω e False)).Reachable a b := by |
| rintro ⟨b,hb,h⟩ |
| exact hn ⟨b,hb,h.mono hmon⟩ |
| have hav' : ¬(G.openGraph (Function.update ω e False)).Reachable a v := |
| fun h => hav (h.mono hmon) |
| have hv' : ∃ b ∈ B, (G.openGraph (Function.update ω e False)).Reachable v b := by |
| obtain ⟨b,hb,hvb⟩ := hv |
| obtain ⟨r,hr,har⟩ := he |
| refine ⟨b,hb,G.reachable_force_false_of_not_reachable ω e r hr ?_ hvb⟩ |
| intro hvr |
| exact hav (har.trans hvr.symm) |
| have hcomm : Function.update (Function.update ω f True) e False = |
| Function.update (Function.update ω e False) f True := by |
| funext g |
| by_cases hge : g = e |
| · subst g; erw [Function.update_self,Function.update_of_ne hef,Function.update_self] |
| · by_cases hgf : g = f |
| · subst g; erw [Function.update_of_ne (Ne.symm hef),Function.update_self,Function.update_self] |
| · erw [Function.update_of_ne hge,Function.update_of_ne hgf, |
| Function.update_of_ne hgf,Function.update_of_ne hge] |
| have hdelete : F (Function.update (Function.update ω f True) e False) ↔ |
| (G.openGraph (Function.update ω e False)).Reachable a u := by |
| rw [hcomm] |
| exact G.target_after_open_iff _ B a u v f hf hn' hv' hav' |
| change Function.update ω f True e ∧ F (Function.update ω f True) ∧ |
| ¬F (Function.update (Function.update ω f True) e False) ↔ _ |
| erw [Function.update_of_ne hef,hdelete] |
| exact and_congr_right (fun _ => and_iff_right hopen) |
|
|
| end CriticalPercolation.BondGraph |
|
|
| end |
| end |
|
|
| end OAI |
|
|
