openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Classification /FiniteBGTBridge.lean
| import Mathlib | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function | |
| open scoped Pointwise | |
| namespace CriticalPercolation.CompactQuotient | |
| variable {A : Type*} [Group A] | |
| lemma nilpotent_image_of_nilpotent_subquotient (H N : Subgroup A) [H.Normal] | |
| [Group.IsNilpotent (N ⧸ H.subgroupOf N)] : | |
| Group.IsNilpotent (N.map (QuotientGroup.mk' H)) := by | |
| let f : N →* N.map (QuotientGroup.mk' H) := (QuotientGroup.mk' H).subgroupMap N | |
| have hker : H.subgroupOf N≤f.ker := by | |
| intro n hn | |
| apply Subtype.ext | |
| exact (QuotientGroup.eq_one_iff _).mpr hn | |
| let ψ := QuotientGroup.lift (H.subgroupOf N) f hker | |
| have hs : Surjective ψ := by | |
| intro y | |
| obtain ⟨n,rfl⟩ := (QuotientGroup.mk' H).subgroupMap_surjective N y | |
| exact ⟨QuotientGroup.mk' (H.subgroupOf N) n,rfl⟩ | |
| exact Group.nilpotent_of_surjective ψ hs | |
| lemma index_image_of_subquotient (H N : Subgroup A) [H.Normal] (hHN : H≤N) : | |
| (N.map (QuotientGroup.mk' H)).index=N.index := by | |
| apply N.index_map_eq QuotientGroup.mk_surjective | |
| rw [QuotientGroup.ker_mk'] | |
| exact hHN | |
| theorem finite_by_nilpotent_of_BGT_output (S : Set A) (hS : S.Finite) (n : ℕ) | |
| (H N : Subgroup A) [H.Normal] (hHN : H≤N) (hH : (H : Set A)⊆S^(4*n)) | |
| [Group.IsNilpotent (N ⧸ H.subgroupOf N)] (hN : N.index≠0) : | |
| (H : Set A).Finite ∧ | |
| ∃ M : Subgroup (A ⧸ H), Group.IsNilpotent M ∧ M.index≠0 := by | |
| have hpow : ∀ k : ℕ, (S^k).Finite := by | |
| intro k | |
| induction k with | |
| | zero => simpa only [pow_zero] using (Set.finite_one : (1 : Set A).Finite) | |
| | succ k ih => simpa only [pow_succ] using ih.mul hS | |
| refine ⟨(hpow (4*n)).subset hH,N.map (QuotientGroup.mk' H), | |
| nilpotent_image_of_nilpotent_subquotient H N,?_⟩ | |
| rw [index_image_of_subquotient H N hHN] | |
| exact hN | |
| end CriticalPercolation.CompactQuotient | |
| end | |
| end | |
| end OAI | |