openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Classification /QualitativeClassificationBoundary.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Classification.FiniteClassificationObstruction | |
| import OAI.Probability.CriticalPercolation.Nilpotent.FiniteKernelNilpotent | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function Topology | |
| namespace CriticalPercolation.CompactQuotient | |
| variable {A : Type*} [Group A] | |
| lemma virtuallyNilpotent_of_finiteByVirtuallyNilpotent | |
| (h : FiniteByVirtuallyNilpotent A) : Group.IsVirtuallyNilpotent A := by | |
| obtain ⟨F,hn,hF,N,hN,hi⟩ := h | |
| let _ := hn | |
| apply virtuallyNilpotent_of_finite_kernel (QuotientGroup.mk' F) | |
| · simpa only [QuotientGroup.ker_mk'] using hF | |
| · exact ⟨N,hN,⟨hi⟩⟩ | |
| lemma finiteByVirtuallyNilpotent_of_virtuallyNilpotent | |
| (h : Group.IsVirtuallyNilpotent A) : FiniteByVirtuallyNilpotent A := by | |
| obtain ⟨N,hN,hi⟩ := h | |
| let _ := hN | |
| let f : A →* A ⧸ (⊥ : Subgroup A) := QuotientGroup.mk' ⊥ | |
| have hf : Surjective f := QuotientGroup.mk_surjective | |
| have hnil : Group.IsNilpotent (N.map f) := Group.nilpotent_of_surjective | |
| (f.subgroupMap N) (f.subgroupMap_surjective N) | |
| have hindex : (N.map f).index=N.index := by | |
| apply N.index_map_eq hf | |
| change (QuotientGroup.mk' (⊥ : Subgroup A)).ker≤N | |
| rw [QuotientGroup.ker_mk'] | |
| exact bot_le | |
| refine ⟨⊥,inferInstance,by simp,N.map f,hnil,?_⟩ | |
| rw [hindex] | |
| exact hi.index_ne_zero | |
| theorem finiteByVirtuallyNilpotent_iff : | |
| FiniteByVirtuallyNilpotent A ↔ Group.IsVirtuallyNilpotent A := | |
| ⟨virtuallyNilpotent_of_finiteByVirtuallyNilpotent, | |
| finiteByVirtuallyNilpotent_of_virtuallyNilpotent⟩ | |
| end CriticalPercolation.CompactQuotient | |
| namespace CriticalPercolation.GraphTopology | |
| variable {X : Type*} (Γ : SimpleGraph X) | |
| theorem not_virtuallyNilpotent_quotient_of_classification_failure {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 ∧ | |
| ¬Group.IsVirtuallyNilpotent ((Γ ≃g Γ) ⧸ H) := 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,fun hv => hfail | |
| (CompactQuotient.finiteByVirtuallyNilpotent_of_virtuallyNilpotent hv)⟩ | |
| end CriticalPercolation.GraphTopology | |
| end | |
| end | |
| end OAI | |