import Mathlib

namespace OAI

open Set MeasureTheory ProbabilityTheory

namespace CriticalPercolation

/-- An undirected graph with labelled bonds, including loops and parallel bonds. -/
structure BondGraph (V E : Type*) where
  ends : E → Sym2 V

namespace BondGraph
variable {V E : Type*} (G : BondGraph V E)

/-- The graph of retained bonds. `fromEdgeSet` discards loops. -/
def openGraph (ω : Set E) : SimpleGraph V :=
  SimpleGraph.fromEdgeSet (G.ends '' ω)

/-- The underlying connectivity graph with every bond retained. -/
def fullGraph : SimpleGraph V := G.openGraph Set.univ

/-- Local finiteness counts bonds, not just distinct neighbors. -/
def LocallyFinite : Prop := ∀ v : V, {e : E | v ∈ G.ends e}.Finite

/-- Automorphisms preserve the vertex and bond structure, including multiplicities. -/
structure Aut where
  vertex : V ≃ V
  edge : E ≃ E
  preserves : ∀ e, G.ends (edge e) = Sym2.map vertex (G.ends e)

/-- Finitely many vertex orbits under the full automorphism group. -/
def QuasiTransitive : Prop :=
  ∃ S : Set V, S.Finite ∧ ∀ v : V, ∃ s ∈ S, ∃ φ : G.Aut, φ.vertex s = v

/-- Finite open paths, including the path of length zero, define the open cluster. -/
def cluster (ω : Set E) (v : V) : Set V := {w | (G.openGraph ω).Reachable v w}

/-- The event that at least one open cluster is infinite. -/
def percolates : Set (Set E) := {ω | ∃ v : V, (G.cluster ω v).Infinite}

/-- Independent retention with parameter `p ∈ [0,1]` of each labelled bond. -/
noncomputable def law (_G : BondGraph V E) (p : unitInterval) : Measure (Set E) :=
  ProbabilityTheory.setBernoulli Set.univ p

/-- The infimum of parameters with positive probability of an infinite cluster. -/
noncomputable def criticalProbability : unitInterval :=
  sInf {p : unitInterval | 0 < G.law p G.percolates}

end BondGraph
end CriticalPercolation

end OAI