| import Mathlib | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.VertexBall | |
| namespace OAI | |
| section | |
| section | |
| open Set MeasureTheory ProbabilityTheory | |
| open Set | |
| open scoped BigOperators | |
| open scoped Classical | |
| namespace CriticalPercolation.BondGraph | |
| open scoped Classical | |
| variable {V E : Type*} | |
| def cutEdges (G : BondGraph V E) (O : Set V) : Set E := | |
| {e | ∃ x y, G.ends e = s(x,y) ∧ x ∉ O ∧ y ∈ O} | |
| lemma not_reachable_of_cut_closed (G : BondGraph V E) (O : Set V) (ω : Set E) | |
| (hc : ∀ e ∈ G.cutEdges O, ¬ω e) {a b : V} (ha : a ∉ O) (hb : b ∈ O) : | |
| ¬(G.openGraph ω).Reachable a b := by | |
| intro hab | |
| have hclosed : ∀ ⦃v z⦄, v ∈ Oᶜ → (G.openGraph ω).Adj v z → z ∈ Oᶜ := by | |
| intro v z hv hvz hz | |
| obtain ⟨⟨e,he,heq⟩,_⟩ := (SimpleGraph.fromEdgeSet_adj _).mp hvz | |
| exact hc e ⟨v,z,heq,hv,hz⟩ he | |
| exact (ClusterDeletion.reachable_mem_of_closed hclosed ha hab) hb | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end OAI | |