openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Classification /DiscreteWeakGromovBridge.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Classification.WeakGromovMainBoundary | |
| import OAI.Probability.CriticalPercolation.Harmonic.WeakGromovHarmonic | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function Topology | |
| open scoped Classical Pointwise | |
| namespace CriticalPercolation.CompactQuotient | |
| lemma virtuallyNilpotent_of_sparse_word_growth {A : Type*} [Group A] | |
| (hg : HasSparsePolynomialWordGrowth A) : Group.IsVirtuallyNilpotent A := by | |
| obtain ⟨S,hSf,h1,hi,hgen,d,hg⟩ := hg | |
| exact SparseGrowth.sparse_polynomial_virtuallyNilpotent d A | |
| ⟨S,hSf,h1,hi,hgen,1,by simpa only [one_mul] using hg⟩ | |
| end CriticalPercolation.CompactQuotient | |
| namespace CriticalPercolation.GraphTopology | |
| lemma no_weak_Gromov_quotient_counterexample {X : Type*} (Γ : SimpleGraph X) : | |
| ¬HasWeakGromovQuotientCounterexample Γ := by | |
| let _ : TopologicalSpace X := ⊥ | |
| let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ | |
| rintro ⟨H,hn,_,_,_,hg,hfail⟩ | |
| let _ := hn | |
| exact hfail (CompactQuotient.virtuallyNilpotent_of_sparse_word_growth hg) | |
| end CriticalPercolation.GraphTopology | |
| end | |
| end | |
| end OAI | |