openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /GraphTopology /CommensurableClosure.lean
| import Mathlib | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function Topology | |
| open scoped Classical | |
| namespace CriticalPercolation.RelativeGrowth | |
| variable {A : Type*} [Group A] [TopologicalSpace A] [IsTopologicalGroup A] | |
| lemma isOpen_of_isClosed_relIndex (K H : Subgroup A) (hK : IsOpen (K:Set A)) | |
| (hH : IsClosed (H:Set A)) (hi : H.relIndex K≠0) : IsOpen (H:Set A) := by | |
| let J := H.subgroupOf K | |
| let : J.FiniteIndex := ⟨hi⟩ | |
| have hJ : IsOpen (J:Set K) := J.isOpen_of_isClosed_of_finiteIndex | |
| (hH.preimage continuous_subtype_val) | |
| have hI : IsOpen ((J.map K.subtype):Set A) := hK.isOpenMap_subtype_val _ hJ | |
| exact Subgroup.isOpen_mono (by rintro _ ⟨k,hk,rfl⟩; exact hk) hI | |
| lemma topologicalClosure_open_of_relIndex (K N : Subgroup A) (hK : IsOpen (K:Set A)) | |
| (hi : N.relIndex K≠0) : IsOpen (N.topologicalClosure:Set A) := by | |
| apply isOpen_of_isClosed_relIndex K N.topologicalClosure hK N.isClosed_topologicalClosure | |
| exact fun h => hi (Subgroup.relIndex_eq_zero_of_le_left N.le_topologicalClosure h) | |
| end CriticalPercolation.RelativeGrowth | |
| end | |
| end | |
| end OAI | |