openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Classification /RawClassificationBoundary.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Classification.ClassificationProducerBoundary | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.SubgroupOrbitKernel | |
| import OAI.Probability.CriticalPercolation.Growth.CosetBallIndex | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function Topology | |
| open scoped Pointwise | |
| namespace CriticalPercolation | |
| namespace CompactQuotient | |
| def HasUniformCompactLocallyConnectedModels (A : Type*) [Group A] | |
| [TopologicalSpace A] (K : ℝ) : Prop := | |
| ∃ B : ℕ, ∀ U : Set A, IsOpen U → IsCompact U → IsApproximateSubgroup K U → | |
| ∃ L : Subgroup A, ∃ H : Subgroup L, ∃ hn : H.Normal, let _ := hn | |
| IsCompact (H : Set L) ∧ | |
| ((Subtype.val : L → A) '' (H : Set L)⊆U^4) ∧ | |
| LocallyCompactSpace (L ⧸ H) ∧ LocallyConnectedSpace (L ⧸ H) ∧ | |
| ∃ F : Finset A, F.card≤B ∧ U⊆(F : Set A)*(L : Set A) | |
| end CompactQuotient | |
| namespace GraphTopology | |
| variable {X : Type*} (Γ : SimpleGraph X) | |
| theorem compact_open_normal_of_raw_models {K : ℝ} | |
| (hpow : HasUniformCompactApproximatePowers Γ K) | |
| (hcar : let _ : TopologicalSpace X := ⊥ | |
| let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ | |
| CompactQuotient.HasUniformCompactLocallyConnectedModels (Γ ≃g Γ) K) : | |
| let _ : TopologicalSpace X := ⊥ | |
| let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ | |
| ∃ J : Subgroup (Γ ≃g Γ), J.Normal ∧ | |
| IsCompact (J : Set (Γ ≃g Γ)) ∧ IsOpen (J : Set (Γ ≃g Γ)) := by | |
| let _ : TopologicalSpace X := ⊥ | |
| let _ : DiscreteTopology X := ⟨rfl⟩ | |
| let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ | |
| obtain ⟨S,hSc,hSo,h1,hinv,hgen,hscales⟩ := hpow | |
| obtain ⟨B,hB⟩ := hcar | |
| obtain ⟨n,hn,hcomp,hopen,happrox⟩ := hscales B | |
| obtain ⟨L,H,hHN,hH,_,hLC,hConn,F,hF,hcover⟩ := hB (S^n) hopen hcomp happrox | |
| let _ := hHN | |
| let _ := hLC | |
| let _ := hConn | |
| let _ : L.FiniteIndex := CosetGrowth.finiteIndex_of_few_cosets S L h1 hinv hgen | |
| n F (hF.trans hn) hcover | |
| exact compact_open_normal_of_finiteIndex_subquotient Γ L H hH | |
| theorem raw_classification_failure_dichotomy {K : ℝ} | |
| (hpow : HasUniformCompactApproximatePowers Γ K) | |
| (hfail : ¬HasCompactOpenVirtuallyNilpotentQuotient Γ) : | |
| (¬(let _ : TopologicalSpace X := ⊥ | |
| let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ | |
| CompactQuotient.HasUniformCompactLocallyConnectedModels (Γ ≃g Γ) K)) ∨ | |
| HasFiniteApproximateQuotientCounterexample Γ K := by | |
| let _ : TopologicalSpace X := ⊥ | |
| let _ : DiscreteTopology X := ⟨rfl⟩ | |
| let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ | |
| by_cases hc : CompactQuotient.HasUniformCompactLocallyConnectedModels (Γ ≃g Γ) K | |
| · right | |
| obtain ⟨J,hn,hJc,hJo⟩ := compact_open_normal_of_raw_models Γ hpow hc | |
| let _ := hn | |
| 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 hJo K S hS h1 hinv hgen | |
| intro R | |
| obtain ⟨n,hn,_,_,ha⟩ := hscales R | |
| exact ⟨n,hn,ha⟩ | |
| refine ⟨J,hn,hJc,hJo,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 hJc hJo F hF N hN) | |
| · exact Or.inl hc | |
| end GraphTopology | |
| namespace BondGraph | |
| variable {V E : Type*} (G : BondGraph V E) | |
| theorem main_failure_raw_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)) ∧ | |
| ((¬(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.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.raw_classification_failure_dichotomy _ hpow hfail⟩ | |
| end BondGraph | |
| end CriticalPercolation | |
| end | |
| end | |
| end OAI | |