| import Mathlib |
| import OAI.Probability.CriticalPercolation.Growth.CosetGeometry |
| import OAI.Probability.CriticalPercolation.Growth.CosetBounded |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set Function |
| open scoped Classical Pointwise |
|
|
| namespace CriticalPercolation.RelativeGrowth |
| variable {A : Type*} [Group A] |
|
|
| lemma normalized_generators (K : Subgroup A) (S : Set A) |
| (h1 : (1:A)∈S) (hi : S⁻¹=S) (hgen : Subgroup.closure S=⊤) |
| (hK : Bounded S (K:Set A)) : |
| ∃ P : Presentation A, P.K=K ∧ Bounded S P.S ∧ S⊆P.S := by |
| let T := (K:Set A)*S*(K:Set A) |
| have hs : S⊆T := by |
| intro s hs |
| have hh := Set.mul_mem_mul (Set.mul_mem_mul K.one_mem hs) K.one_mem |
| simpa only [one_mul,mul_one] using hh |
| have hT1 : (1:A)∈T := hs h1 |
| have hTi : T⁻¹=T := by |
| dsimp [T] |
| rw [mul_inv_rev,mul_inv_rev,inv_coe_set,hi,mul_assoc] |
| have hTgen : Subgroup.closure T=⊤ := top_unique (hgen ▸ Subgroup.closure_mono hs) |
| have hleft : (K:Set A)*T⊆T := by |
| rintro _ ⟨k,hk,_,⟨_,⟨l,hl,s,hs,rfl⟩,m,hm,rfl⟩,rfl⟩ |
| have hh := Set.mul_mem_mul (Set.mul_mem_mul (K.mul_mem hk hl) hs) hm |
| simpa only [T,mul_assoc] using hh |
| refine ⟨⟨K,T,hT1,hTi,hTgen,hleft⟩,rfl,?_,hs⟩ |
| exact (hK.mul ⟨1,by simp⟩).mul hK |
|
|
| end CriticalPercolation.RelativeGrowth |
| end |
| end |
|
|
| end OAI |
|
|
