openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Classification /ConcreteStructureAssembly.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Model | |
| import OAI.Probability.CriticalPercolation.Criticality.MainSparseBoundary | |
| import OAI.Probability.CriticalPercolation.Modular.ModularLevelGeometry | |
| import OAI.Probability.CriticalPercolation.Nilpotent.StructuredVirtuallyNilpotent | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set MeasureTheory | |
| open scoped Classical ENNReal | |
| namespace CriticalPercolation.BondGraph | |
| variable {V E : Type*} (G : BondGraph V E) | |
| theorem critical_null_of_normal_nilpotent_model | |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) | |
| (hg : G.SubexponentialGrowth) | |
| (Λ : Subgroup G.Aut) [Group.FG Λ] | |
| (S : Set V) (hS : S.Finite) (hcover : ∀ v, ∃ s∈S, v∈MulAction.orbit Λ s) | |
| (K : Subgroup Λ) [K.Normal] | |
| (hker : ∀ v : V, (MulAction.orbit K v).Finite) | |
| (hstab : ∀ v : V, ((MulAction.stabilizer Λ v).map (QuotientGroup.mk' K) : | |
| Set (Λ ⧸ K)).Finite) | |
| (N : Subgroup (Λ ⧸ K)) [Group.IsNilpotent N] (hN : N.index≠0) | |
| (hpc : G.criticalProbability<1) : G.law G.criticalProbability G.percolates=0 := by | |
| let _ : MulAction G.Aut E := G.autEdgeAction | |
| apply G.critical_null_of_virtually_nilpotent_quotient hc hl hg | |
| (G.subgroup_preservesEnds G.aut_preservesEnds Λ) S hS hcover | |
| (QuotientGroup.mk' K) (QuotientGroup.mk'_surjective K) _ hstab N hN hpc | |
| intro v | |
| apply (hker v).subset | |
| rintro w ⟨g,rfl⟩ | |
| have hg : g.val∈K := by simpa only [QuotientGroup.ker_mk'] using g.property | |
| exact ⟨⟨g.val,hg⟩,rfl⟩ | |
| theorem main_failure_excludes_normal_nilpotent_model [Infinite V] | |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive) | |
| (hpc : G.criticalProbability<1) | |
| (hper : G.law G.criticalProbability G.percolates≠0) | |
| (Λ : Subgroup G.Aut) [Group.FG Λ] | |
| (S : Set V) (hS : S.Finite) (hcover : ∀ v, ∃ s∈S, v∈MulAction.orbit Λ s) | |
| (K : Subgroup Λ) [K.Normal] | |
| (hker : ∀ v : V, (MulAction.orbit K v).Finite) | |
| (hstab : ∀ v : V, ((MulAction.stabilizer Λ v).map (QuotientGroup.mk' K) : | |
| Set (Λ ⧸ K)).Finite) | |
| (N : Subgroup (Λ ⧸ K)) [Group.IsNilpotent N] (hN : N.index≠0) : False := by | |
| have hg := (G.main_failure_sparse_polynomial hc hl hq hpc hper).1 | |
| exact hper (G.critical_null_of_normal_nilpotent_model hc hl hg Λ S hS hcover K hker hstab N hN hpc) | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end OAI | |