openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Classification /WeakGromovMainBoundary.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Classification.RawClassificationBoundary | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.GraphQuotientPolynomial | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function Topology | |
| namespace CriticalPercolation | |
| namespace GraphTopology | |
| variable {X : Type*} (Γ : SimpleGraph X) | |
| def HasWeakGromovQuotientCounterexample : 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)) ∧ | |
| Group.FG ((Γ ≃g Γ) ⧸ H) ∧ | |
| CompactQuotient.HasSparsePolynomialWordGrowth ((Γ ≃g Γ) ⧸ H) ∧ | |
| ¬Group.IsVirtuallyNilpotent ((Γ ≃g Γ) ⧸ H) | |
| end GraphTopology | |
| namespace BondGraph | |
| variable {V E : Type*} (G : BondGraph V E) | |
| theorem main_failure_raw_Carolino_or_weak_Gromov [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)) ∧ | |
| ((¬(let _ : TopologicalSpace (MulAction.orbit G.Aut o) := ⊥ | |
| let _ : TopologicalSpace (G.orbitPower o (2*C+1) ≃g G.orbitPower o (2*C+1)) := | |
| GraphTopology.autTopology (G.orbitPower o (2*C+1)) | |
| CompactQuotient.HasUniformCompactLocallyConnectedModels | |
| (G.orbitPower o (2*C+1) ≃g G.orbitPower o (2*C+1)) (5^(d+2)))) ∨ | |
| GraphTopology.HasWeakGromovQuotientCounterexample (G.orbitPower o (2*C+1))) := by | |
| obtain ⟨d,hd,C,hconn,hinf,hloc,ht,hpoly,hpow,hfail⟩ := | |
| G.main_failure_raw_Carolino_or_finite_BGT hc hl hq hpc hper o | |
| refine ⟨d,hd,C,hconn,hinf,hloc,ht,hpoly,hpow,?_⟩ | |
| rcases hfail with hcar | hfin | |
| · exact Or.inl hcar | |
| · right | |
| let u : MulAction.orbit G.Aut o := ⟨o,MulAction.mem_orbit_self o⟩ | |
| exact GraphTopology.weak_Gromov_counterexample_of_finite_classification_failure | |
| _ hconn hloc ht u (d+1) (fun R => by | |
| obtain ⟨n,hn,hp⟩ := hpoly R | |
| exact ⟨n,hn,hp u⟩) hfin | |
| end BondGraph | |
| end CriticalPercolation | |
| end | |
| end | |
| end OAI | |