lean/OAI/Probability/CriticalPercolation/GraphGeometry/CutEdges.lean

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