openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Classification /FiniteClassificationObstruction.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Classification.ClassificationProducerBoundary | |
| import OAI.Probability.CriticalPercolation.Classification.FiniteBGTBridge | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function Topology | |
| open scoped Pointwise | |
| namespace CriticalPercolation.CompactQuotient | |
| variable {A : Type*} [Group A] | |
| def FiniteBGTPowerOutput (S : Set A) (n : ℕ) : Prop := | |
| ∃ H N : Subgroup A, ∃ hH : H.Normal, let _ := hH | |
| N.Normal ∧ H≤N ∧ (H : Set A)⊆S^(4*n) ∧ N.index≠0 ∧ | |
| Group.IsNilpotent (N ⧸ H.subgroupOf N) | |
| def HasUniformFiniteBGTModels (A : Type*) [Group A] (K : ℝ) : Prop := | |
| ∃ R : ℕ, ∀ S : Set A, S.Finite → (1 : A)∈S → S⁻¹=S → | |
| Subgroup.closure S=⊤ → ∀ n≥R, IsApproximateSubgroup K (S^n) → | |
| FiniteBGTPowerOutput S n | |
| lemma finiteByVirtuallyNilpotent_of_finiteBGTPowerOutput | |
| (S : Set A) (hS : S.Finite) (n : ℕ) (h : FiniteBGTPowerOutput S n) : | |
| FiniteByVirtuallyNilpotent A := by | |
| obtain ⟨H,N,hn,_,hHN,hH,hN,hNil⟩ := h | |
| let _ := hn | |
| let _ := hNil | |
| exact ⟨H,hn,finite_by_nilpotent_of_BGT_output S hS n H N hHN hH hN⟩ | |
| lemma one_le_of_uniformFiniteApproximatePowers {K : ℝ} | |
| (h : UniformFiniteApproximatePowers A K) : 1≤K := by | |
| obtain ⟨S,_,_,_,_,hpow⟩ := h | |
| obtain ⟨n,_,hn⟩ := hpow 0 | |
| exact hn.one_le | |
| theorem finiteByVirtuallyNilpotent_of_uniform_models {K : ℝ} | |
| (hpow : UniformFiniteApproximatePowers A K) | |
| (hbgt : HasUniformFiniteBGTModels A K) : FiniteByVirtuallyNilpotent A := by | |
| obtain ⟨R,hR⟩ := hbgt | |
| obtain ⟨S,hS,h1,hinv,hgen,hscales⟩ := hpow | |
| obtain ⟨n,hn,ha⟩ := hscales R | |
| exact finiteByVirtuallyNilpotent_of_finiteBGTPowerOutput S hS n | |
| (hR S hS h1 hinv hgen n hn ha) | |
| theorem no_uniform_finite_BGT_models {K : ℝ} | |
| (hpow : UniformFiniteApproximatePowers A K) | |
| (hf : ¬FiniteByVirtuallyNilpotent A) : ¬HasUniformFiniteBGTModels A K := | |
| fun h => hf (finiteByVirtuallyNilpotent_of_uniform_models hpow h) | |
| end CriticalPercolation.CompactQuotient | |
| namespace CriticalPercolation.GraphTopology | |
| variable {X : Type*} (Γ : SimpleGraph X) | |
| theorem finite_classification_obstruction {K : ℝ} | |
| (h : HasFiniteApproximateQuotientCounterexample Γ K) : | |
| 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)) ∧ | |
| 1≤K ∧ Group.FG ((Γ ≃g Γ) ⧸ H) ∧ | |
| CompactQuotient.UniformFiniteApproximatePowers ((Γ ≃g Γ) ⧸ H) K ∧ | |
| ¬CompactQuotient.HasUniformFiniteBGTModels ((Γ ≃g Γ) ⧸ H) K := by | |
| let _ : TopologicalSpace X := ⊥ | |
| let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ | |
| obtain ⟨H,hn,hc,ho,hfg,hpow,hfail⟩ := h | |
| let _ := hn | |
| exact ⟨H,hn,hc,ho,CompactQuotient.one_le_of_uniformFiniteApproximatePowers hpow, | |
| hfg,hpow,CompactQuotient.no_uniform_finite_BGT_models hpow hfail⟩ | |
| end CriticalPercolation.GraphTopology | |
| end | |
| end | |
| end OAI | |