openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Criticality /BoundedNormalJoins.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Growth.CosetBounded | |
| import OAI.Probability.CriticalPercolation.Growth.CosetNormalClosure | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function | |
| open scoped Classical Pointwise | |
| namespace CriticalPercolation.RelativeGrowth | |
| variable {A : Type*} [Group A] | |
| lemma conjugate_map_map (N : Subgroup A) (a b : A) : | |
| (N.map (MulAut.conj b).toMonoidHom).map (MulAut.conj a).toMonoidHom=N.map (MulAut.conj (a*b)).toMonoidHom := by | |
| rw [Subgroup.map_map] | |
| congr 1 | |
| ext x | |
| simp [MulAut.conj_apply,mul_assoc] | |
| lemma Bounded.conj {S : Set A} (h1 : (1:A)∈S) (hi : S⁻¹=S) | |
| (hg : Subgroup.closure S=⊤) {N : Subgroup A} (hN : Bounded S (N:Set A)) (g : A) : | |
| Bounded S (N.map (MulAut.conj g).toMonoidHom : Set A) := by | |
| have hh := ((bounded_singleton S h1 hi hg g).mul hN).mul (bounded_singleton S h1 hi hg g⁻¹) | |
| apply hh.mono | |
| rintro y ⟨x,hx,rfl⟩ | |
| exact Set.mul_mem_mul (Set.mul_mem_mul (Set.mem_singleton g) hx) (Set.mem_singleton g⁻¹) | |
| lemma normalized_conjugate (L N : Subgroup A) [L.Normal] (hN : L≤(Subgroup.normalizer (N:Set A))) (g : A) : | |
| L≤(Subgroup.normalizer ((N.map (MulAut.conj g).toMonoidHom):Set A)) := by | |
| have hh := (Subgroup.map_mono hN (f:=(MulAut.conj g).toMonoidHom)).trans (N.le_normalizer_map (MulAut.conj g).toMonoidHom) | |
| have he : L.map (MulAut.conj g).toMonoidHom=L := Subgroup.Normal.map_conj_eq L g | |
| rwa [he] at hh | |
| lemma finite_normalized_join_bounded (S : Set A) (h1 : (1:A)∈S) | |
| (L : Subgroup A) (F : Set (Subgroup A)) (hf : F.Finite) | |
| (hle : ∀ N∈F, N≤L) (hnorm : ∀ N∈F, L≤(Subgroup.normalizer (N:Set A))) | |
| (hb : ∀ N∈F, Bounded S (N:Set A)) : | |
| Bounded S ((⨆ N∈F, N : Subgroup A) : Set A) := by | |
| induction F, hf using Set.Finite.induction_on with | |
| | empty => exact ⟨1,by simpa using h1⟩ | |
| | @insert N F hN hF ih => | |
| have hbF := ih (fun M hM => hle M (Set.mem_insert_of_mem _ hM)) | |
| (fun M hM => hnorm M (Set.mem_insert_of_mem _ hM)) | |
| (fun M hM => hb M (Set.mem_insert_of_mem _ hM)) | |
| have hleF : (⨆ M∈F, M : Subgroup A)≤L := iSup_le (fun M => iSup_le (fun hM => hle M (Set.mem_insert_of_mem _ hM))) | |
| have hn : (⨆ M∈F, M : Subgroup A)≤(Subgroup.normalizer (N:Set A)) := hleF.trans (hnorm N (Set.mem_insert N F)) | |
| rw [iSup_insert,Subgroup.coe_mul_of_right_le_normalizer_left N _ hn] | |
| exact (hb N (Set.mem_insert N F)).mul hbF | |
| lemma conjugateStage_eq_join (S : Set A) (N : Subgroup A) (n : ℕ) : | |
| conjugateStage S (N:Set A) n=⨆ M∈((fun g => N.map (MulAut.conj g).toMonoidHom) '' S^n), M := by | |
| apply le_antisymm | |
| · apply (Subgroup.closure_le _).mpr | |
| rintro x ⟨g,hg,y,hy,rfl⟩ | |
| have hh : N.map (MulAut.conj g).toMonoidHom ≤ ⨆ M∈((fun g => N.map (MulAut.conj g).toMonoidHom) '' S^n), M := | |
| le_iSup_of_le _ (le_iSup_of_le ⟨g,hg,rfl⟩ le_rfl) | |
| exact hh (Subgroup.mem_map.mpr ⟨y,hy,rfl⟩) | |
| · apply iSup_le | |
| intro M | |
| apply iSup_le | |
| rintro ⟨g,hg,rfl⟩ x ⟨y,hy,rfl⟩ | |
| exact Subgroup.subset_closure ⟨g,hg,y,hy,rfl⟩ | |
| end CriticalPercolation.RelativeGrowth | |
| end | |
| end | |
| end OAI | |