openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /GraphTopology /TopologicalClassificationBoundary.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Model | |
| import OAI.Probability.CriticalPercolation.Classification.FaithfulOrbitalModel | |
| import OAI.Probability.CriticalPercolation.Criticality.MainSparseBoundary | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.FullOrbitalStructureTransfer | |
| import OAI.Probability.CriticalPercolation.GraphTopology.GraphGeneratingNeighborhood | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function MeasureTheory Topology | |
| open scoped Classical ENNReal Pointwise | |
| namespace CriticalPercolation | |
| namespace GraphTopology | |
| variable {X : Type*} (Γ : SimpleGraph X) | |
| def HasCompactOpenVirtuallyNilpotentQuotient : Prop := | |
| let _ : TopologicalSpace X := ⊥ | |
| let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ | |
| ∃ H : Subgroup (Γ ≃g Γ), ∃ hn : H.Normal, | |
| let _ := hn | |
| IsCompact (H : Set (Γ ≃g Γ)) ∧ IsOpen (H : Set (Γ ≃g Γ)) ∧ | |
| ∃ N : Subgroup ((Γ ≃g Γ) ⧸ H), Group.IsNilpotent N ∧ N.index≠0 | |
| def HasUniformCompactApproximatePowers (K : ℝ) : Prop := | |
| let _ : TopologicalSpace X := ⊥ | |
| let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ | |
| ∃ S : Set (Γ ≃g Γ), IsCompact S ∧ IsOpen S ∧ (1 : Γ ≃g Γ)∈S ∧ | |
| S⁻¹=S ∧ Subgroup.closure S=⊤ ∧ | |
| ∀ R : ℕ, ∃ n≥R, IsCompact (S^n) ∧ IsOpen (S^n) ∧ | |
| IsApproximateSubgroup K (S^n) | |
| lemma uniform_compact_approximate_powers_of_sparse_polynomial | |
| (hc : Γ.Connected) (hl : ∀ x, (Γ.neighborSet x).Finite) | |
| (ht : ∀ u v, ∃ g : Γ ≃g Γ, g u=v) (o : X) (d : ℕ) | |
| (hpoly : ∀ R : ℕ, ∃ n≥R, (reachBall Γ o n).ncard≤n^d) : | |
| HasUniformCompactApproximatePowers Γ (5^(d+1)) := by | |
| let _ : TopologicalSpace X := ⊥ | |
| let _ : DiscreteTopology X := ⟨rfl⟩ | |
| let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ | |
| exact ⟨displacement Γ o 1, | |
| compact_open_generating_approximate_powers Γ hc hl ht o d hpoly⟩ | |
| end GraphTopology | |
| namespace BondGraph | |
| variable {V E : Type*} (G : BondGraph V E) | |
| theorem main_failure_topological_classification_counterexample [Infinite V] | |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive) | |
| (hpc : G.criticalProbability<1) | |
| (hper : G.law G.criticalProbability G.percolates≠0) (o : V) : | |
| ∃ d : ℕ, 2≤d ∧ ∃ C : ℕ, | |
| (G.orbitPower o (2*C+1)).Connected ∧ | |
| (MulAction.orbit G.Aut o).Infinite ∧ | |
| (∀ u, ((G.orbitPower o (2*C+1)).neighborSet u).Finite) ∧ | |
| (∀ u v, ∃ g : G.orbitPower o (2*C+1) ≃g G.orbitPower o (2*C+1), g u=v) ∧ | |
| (∀ R : ℕ, ∃ n≥R, ∀ u : MulAction.orbit G.Aut o, | |
| (GraphTopology.reachBall (G.orbitPower o (2*C+1)) u n).ncard≤n^(d+1)) ∧ | |
| GraphTopology.HasUniformCompactApproximatePowers | |
| (G.orbitPower o (2*C+1)) (5^(d+2)) ∧ | |
| ¬GraphTopology.HasCompactOpenVirtuallyNilpotentQuotient | |
| (G.orbitPower o (2*C+1)) := by | |
| obtain ⟨hg,_,d,hd,hpoly⟩ := G.main_failure_sparse_polynomial hc hl hq hpc hper | |
| obtain ⟨C,Λ,hfg,hconn,hinf,hloc,htrans,hker,hreps,hscales⟩ := | |
| G.sparse_polynomial_faithful_orbital_model hc hl hq o d hpoly | |
| let _ : Group.FG Λ := hfg | |
| have ht : ∀ u v, ∃ g : G.orbitPower o (2*C+1) ≃g G.orbitPower o (2*C+1), g u=v := by | |
| intro u v | |
| obtain ⟨g,hg⟩ := htrans u v | |
| exact ⟨g.val,hg⟩ | |
| have hpow : GraphTopology.HasUniformCompactApproximatePowers | |
| (G.orbitPower o (2*C+1)) (5^(d+2)) := by | |
| apply GraphTopology.uniform_compact_approximate_powers_of_sparse_polynomial | |
| _ hconn hloc ht (⟨o,MulAction.mem_orbit_self o⟩ : MulAction.orbit G.Aut o) (d+1) | |
| intro R | |
| obtain ⟨n,hn,hh⟩ := hscales R | |
| exact ⟨n,hn,hh _⟩ | |
| refine ⟨d,hd,C,hconn,hinf,hloc,ht,hscales,hpow,?_⟩ | |
| rintro ⟨H,hn,hcomp,hopen,N,hnil,hN⟩ | |
| let _ := hn | |
| let _ := hnil | |
| obtain ⟨S,hS,hcover⟩ := hreps | |
| exact hper (G.critical_null_of_compact_open_orbital_quotient hc hl hg o (2*C+1) | |
| hconn Λ S hS hcover H N hN ⟨hcomp,hopen⟩ hpc) | |
| end BondGraph | |
| end CriticalPercolation | |
| end | |
| end | |
| end OAI | |