| import Mathlib |
| import OAI.Probability.CriticalPercolation.Model |
| import OAI.Probability.CriticalPercolation.ProductMeasure.Delta |
| import OAI.Probability.CriticalPercolation.Exploration.ReachableIncidenceImp |
| import OAI.Probability.CriticalPercolation.GraphGeometry.ReachableReplaceTail |
| import OAI.Probability.CriticalPercolation.Gluing.ContactPivot |
|
|
| 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*} |
|
|
| lemma contact_exit_pivot (G : BondGraph V E) (ω : Set E) (B : Set V) |
| (a u v : V) (f : E) (hf : G.ends f = s(u,v)) (hclosed : ¬ω 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) : |
| openPivot (fun σ => ∃ b ∈ B, (G.openGraph σ).Reachable a b) |
| f (Function.update ω f True) := by |
| apply (openPivot_force_true _ _ _).mpr |
| constructor |
| · apply (G.target_after_open_iff ω B a u v f hf hn hv ?_).mpr hau |
| intro hav |
| obtain ⟨b,hb,hvb⟩ := hv |
| exact hn ⟨b,hb,hav.trans hvb⟩ |
| · rw [G.openGraph_force_false_of_closed ω f hclosed] |
| exact hn |
|
|
| theorem contact_pivot_incidence_congr (G : BondGraph V E) (ω η : Set E) (B : Set V) |
| (a u v : V) (f e : E) (hf : G.ends f = s(u,v)) (hclosed : ¬ω 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) |
| (hv' : ∃ b ∈ B, (G.openGraph η).Reachable v b) |
| (h : ∀ g, (∃ r ∈ G.ends g, (G.openGraph ω).Reachable a r) → (ω g ↔ η g)) |
| (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) ↔ |
| openPivot (fun σ => ∃ b ∈ B, (G.openGraph σ).Reachable a b) e (Function.update η f True) := by |
| have hr := G.reachable_incidence_congr ω η a h |
| have hn' : ¬∃ b ∈ B, (G.openGraph η).Reachable a b := by |
| rintro ⟨b,hb,hab⟩; exact hn ⟨b,hb,(hr b).mpr hab⟩ |
| have hau' := (hr u).mp hau |
| by_cases hef : e = f |
| · subst e |
| have hclosed' : ¬η f := fun he => hclosed ((h f ⟨u,by simp [hf],hau⟩).mpr he) |
| exact iff_of_true (G.contact_exit_pivot ω B a u v f hf hclosed hn hau hv) |
| (G.contact_exit_pivot η B a u v f hf hclosed' hn' hau' hv') |
| · have he' : ∃ r ∈ G.ends e, (G.openGraph η).Reachable a r := by |
| obtain ⟨r,her,har⟩ := he; exact ⟨r,her,(hr r).mp har⟩ |
| rw [G.contact_pivot_iff ω B a u v f e hf hef hn hau hv he, |
| G.contact_pivot_iff η B a u v f e hf hef hn' hau' hv' he',h e he, |
| G.reachable_delete_incidence_congr ω η a h e u] |
| |
| end CriticalPercolation.BondGraph |
| |
| end |
| end |
| |
| end OAI |
| |
