openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Classification /ClassificationProducerBoundary.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Criticality.CompactDiscreteQuotient | |
| import OAI.Probability.CriticalPercolation.GraphTopology.TopologicalClassificationBoundary | |
| import OAI.Probability.CriticalPercolation.GraphTopology.CompactOrbitKernel | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function Topology | |
| open scoped Classical Pointwise | |
| namespace CriticalPercolation | |
| namespace CompactQuotient | |
| def FiniteByVirtuallyNilpotent (A : Type*) [Group A] : Prop := | |
| ∃ F : Subgroup A, ∃ hn : F.Normal, let _ := hn | |
| (F : Set A).Finite ∧ | |
| ∃ N : Subgroup (A ⧸ F), Group.IsNilpotent N ∧ N.index≠0 | |
| def UniformFiniteApproximatePowers (A : Type*) [Group A] (K : ℝ) : Prop := | |
| ∃ T : Set A, T.Finite ∧ (1 : A)∈T ∧ T⁻¹=T ∧ Subgroup.closure T=⊤ ∧ | |
| ∀ R : ℕ, ∃ n≥R, IsApproximateSubgroup K (T^n) | |
| lemma fg_of_uniformFiniteApproximatePowers {A : Type*} [Group A] {K : ℝ} | |
| (h : UniformFiniteApproximatePowers A K) : Group.FG A := by | |
| obtain ⟨T,hT,_,_,hgen,_⟩ := h | |
| exact Group.fg_iff.mpr ⟨T,hgen,hT⟩ | |
| end CompactQuotient | |
| namespace GraphTopology | |
| variable {X : Type*} (Γ : SimpleGraph X) | |
| def HasCompactNormalOpenLocallyConnectedSubquotient : Prop := | |
| let _ : TopologicalSpace X := ⊥ | |
| let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ | |
| ∃ H : Subgroup (Γ ≃g Γ), ∃ hn : H.Normal, let _ := hn | |
| ∃ L : Subgroup (Γ ≃g Γ), L.Normal ∧ H≤L ∧ | |
| IsCompact (H : Set (Γ ≃g Γ)) ∧ IsOpen (L : Set (Γ ≃g Γ)) ∧ L.index≠0 ∧ | |
| LocallyConnectedSpace (L ⧸ H.subgroupOf L) | |
| def HasFiniteApproximateQuotientCounterexample (K : ℝ) : 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.UniformFiniteApproximatePowers ((Γ ≃g Γ) ⧸ H) K ∧ | |
| ¬CompactQuotient.FiniteByVirtuallyNilpotent ((Γ ≃g Γ) ⧸ H) | |
| theorem classification_failure_dichotomy {K : ℝ} | |
| (hpow : HasUniformCompactApproximatePowers Γ K) | |
| (hfail : ¬HasCompactOpenVirtuallyNilpotentQuotient Γ) : | |
| ¬HasCompactNormalOpenLocallyConnectedSubquotient Γ ∨ | |
| HasFiniteApproximateQuotientCounterexample Γ K := by | |
| by_cases hcar : HasCompactNormalOpenLocallyConnectedSubquotient Γ | |
| · right | |
| let _ : TopologicalSpace X := ⊥ | |
| let _ : DiscreteTopology X := ⟨rfl⟩ | |
| let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ | |
| obtain ⟨H,hn,L,_,_,hH,hL,_,hLC⟩ := hcar | |
| let _ := hn | |
| let _ := hLC | |
| let J := orbitKernel Γ H | |
| have hJ := compact_open_orbitKernel_of_open_locallyConnected_subquotient Γ H L hH hL | |
| obtain ⟨S,hS,_,h1,hinv,hgen,hscales⟩ := hpow | |
| have hfin : CompactQuotient.UniformFiniteApproximatePowers ((Γ ≃g Γ) ⧸ J) K := by | |
| apply CompactQuotient.finite_generating_approximate_powers J hJ.2 K S hS h1 hinv hgen | |
| intro R | |
| obtain ⟨n,hn,_,_,ha⟩ := hscales R | |
| exact ⟨n,hn,ha⟩ | |
| refine ⟨J,inferInstance,hJ.1,hJ.2, | |
| CompactQuotient.fg_of_uniformFiniteApproximatePowers hfin,hfin,?_⟩ | |
| rintro ⟨F,hFN,hF,N,hNil,hN⟩ | |
| let _ := hFN | |
| let _ := hNil | |
| exact hfail (CompactQuotient.lift_finite_by_virtually_nilpotent J hJ.1 hJ.2 F hF N hN) | |
| · exact Or.inl hcar | |
| end GraphTopology | |
| namespace BondGraph | |
| variable {V E : Type*} (G : BondGraph V E) | |
| theorem main_failure_Carolino_or_finite_BGT [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.HasCompactNormalOpenLocallyConnectedSubquotient | |
| (G.orbitPower o (2*C+1)) ∨ | |
| GraphTopology.HasFiniteApproximateQuotientCounterexample | |
| (G.orbitPower o (2*C+1)) (5^(d+2))) := by | |
| obtain ⟨d,hd,C,hconn,hinf,hloc,ht,hpoly,hpow,hfail⟩ := | |
| G.main_failure_topological_classification_counterexample hc hl hq hpc hper o | |
| exact ⟨d,hd,C,hconn,hinf,hloc,ht,hpoly,hpow, | |
| GraphTopology.classification_failure_dichotomy _ hpow hfail⟩ | |
| end BondGraph | |
| end CriticalPercolation | |
| end | |
| end | |
| end OAI | |