openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Classification /GraphCompactNormalClassification.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Classification.RelativeClassification | |
| import OAI.Probability.CriticalPercolation.Growth.GraphRelativeGrowth | |
| import OAI.Probability.CriticalPercolation.GraphTopology.CommensurableClosure | |
| import OAI.Probability.CriticalPercolation.Classification.DiscreteWeakGromovBridge | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.GraphQuotientPolynomial | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function Topology | |
| open scoped Classical Pointwise | |
| namespace CriticalPercolation.GraphTopology | |
| variable {X : Type*} (Γ : SimpleGraph X) | |
| theorem compact_open_normal_of_sparse_polynomial | |
| [TopologicalSpace X] [DiscreteTopology X] | |
| (hc : Γ.Connected) (hl : ∀ x, (Γ.neighborSet x).Finite) | |
| (ht : ∀ u v, ∃ g : Γ ≃g Γ, g u=v) (o : X) (d : ℕ) | |
| (hpoly : ∀ R : ℕ, ∃ n≥R, (reachBall Γ o n).ncard≤n^d) : | |
| let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ | |
| ∃ H : Subgroup (Γ ≃g Γ), H.Normal ∧ IsCompact (H:Set (Γ ≃g Γ)) ∧ IsOpen (H:Set (Γ ≃g Γ)) := by | |
| let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ | |
| let K := vertexStabilizer Γ o | |
| let S := displacement Γ o 1 | |
| have h1 : (1:Γ ≃g Γ)∈S := one_mem_displacement Γ o 1 | |
| have hi : S⁻¹=S := by ext g; exact displacement_inv Γ | |
| have hgen : Subgroup.closure S=⊤ := displacement_generates Γ hc ht o | |
| obtain ⟨N,hNn,hNb,hNi⟩ := RelativeGrowth.bounded_normal_commensurable d | |
| (Γ ≃g Γ) K S h1 hi hgen (vertexStabilizer_bounded Γ o) (displacement_relativeDegree Γ hl o d hpoly) | |
| let := hNn | |
| let H := N.topologicalClosure | |
| have hHn : H.Normal := Subgroup.is_normal_topologicalClosure N | |
| have hcH : IsCompact (H:Set (Γ ≃g Γ)) := by | |
| obtain ⟨n,hn⟩ := hNb | |
| have hNs : (N:Set (Γ ≃g Γ))⊆displacement Γ o n := hn.trans (displacement_pow_subset Γ o n) | |
| apply (compact_displacement Γ hc hl o n).of_isClosed_subset N.isClosed_topologicalClosure | |
| exact closure_minimal hNs (isClosed_eval_mem Γ o _) | |
| have hoH : IsOpen (H:Set (Γ ≃g Γ)) := | |
| RelativeGrowth.topologicalClosure_open_of_relIndex K N (open_stabilizer Γ o) hNi | |
| exact ⟨H,hHn,hcH,hoH⟩ | |
| theorem compact_open_virtually_nilpotent_quotient_of_sparse_polynomial | |
| (hc : Γ.Connected) (hl : ∀ x, (Γ.neighborSet x).Finite) | |
| (ht : ∀ u v, ∃ g : Γ ≃g Γ, g u=v) (o : X) (d : ℕ) | |
| (hpoly : ∀ R : ℕ, ∃ n≥R, (reachBall Γ o n).ncard≤n^d) : | |
| HasCompactOpenVirtuallyNilpotentQuotient Γ := by | |
| let _ : TopologicalSpace X := ⊥ | |
| let _ : DiscreteTopology X := ⟨rfl⟩ | |
| let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ | |
| obtain ⟨H,hHn,hcH,hoH⟩ := compact_open_normal_of_sparse_polynomial Γ hc hl ht o d hpoly | |
| let := hHn | |
| have hg := quotient_sparse_polynomial_word_growth Γ hc hl ht o d hpoly H hoH | |
| obtain ⟨N,hNn,hNi⟩ := CompactQuotient.virtuallyNilpotent_of_sparse_word_growth hg | |
| exact ⟨H,hHn,hcH,hoH,N,hNn,hNi.index_ne_zero⟩ | |
| end CriticalPercolation.GraphTopology | |
| end | |
| end | |
| end OAI | |