| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Criticality.AutDisplacement | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.UniformGeometry | |
| import OAI.Probability.CriticalPercolation.ProductMeasure.BernoulliErgodicity | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set MeasureTheory | |
| open scoped Classical | |
| namespace CriticalPercolation.BondGraph | |
| variable {V E : Type*} (G : BondGraph V E) | |
| theorem invariant_event_zero_one [Infinite V] (hc : G.fullGraph.Connected) | |
| (hl : G.LocallyFinite) (hq : G.QuasiTransitive) (p : unitInterval) | |
| (A : Set (Set E)) (hA : MeasurableSet A) | |
| (hinv : ∀ φ : G.Aut, (fun ω : Set E => φ.edge '' ω) ⁻¹' A = A) : | |
| G.law p A = 0 ∨ G.law p A = 1 := by | |
| have hh := BernoulliProduct.invariant_zero_one p (fun φ : G.Aut => (Aut.symm G φ).edge) | |
| (fun S => by | |
| obtain ⟨φ,hφ⟩ := G.aut_displaces_finite_bonds hc hl hq (S : Set E) S.finite_toSet | |
| exact ⟨φ,hφ⟩) A hA (fun φ => hinv (Aut.symm G φ)) | |
| simpa only [BernoulliProduct.law_const,law] using hh | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end OAI | |