openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Classification /RelativeClassification.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Growth.CosetIndicability | |
| import OAI.Probability.CriticalPercolation.Growth.CosetIntegerKernel | |
| import OAI.Probability.CriticalPercolation.Growth.CosetKernelDescent | |
| import OAI.Probability.CriticalPercolation.Growth.CosetNormalPromotion | |
| import OAI.Probability.CriticalPercolation.Growth.CosetSubgroupGenerators | |
| import OAI.Probability.CriticalPercolation.Growth.CosetFiniteCore | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function | |
| open scoped Classical Pointwise | |
| namespace CriticalPercolation.RelativeGrowth | |
| variable {A : Type*} [Group A] | |
| lemma bounded_top_of_finiteIndex (K : Subgroup A) (S : Set A) | |
| (h1 : (1:A)∈S) (hi : S⁻¹=S) (hgen : Subgroup.closure S=⊤) | |
| (hK : Bounded S (K:Set A)) [K.FiniteIndex] : Bounded S (⊤:Subgroup A) := by | |
| let := K.fintypeQuotientOfFiniteIndex | |
| apply bounded_of_finite_coset_image S K h1 hi hgen hK | |
| exact Set.toFinite _ | |
| lemma relIndex_map_subgroupOf_of_le (K L : Subgroup A) (hKL : K≤L) | |
| (N : Subgroup L) (hN : N.relIndex (K.subgroupOf L)≠0) : | |
| (N.map L.subtype).relIndex K≠0 := by | |
| have he : (K.subgroupOf L).map L.subtype=K := by | |
| rw [Subgroup.subgroupOf,Subgroup.map_comap_eq,Subgroup.range_subtype,inf_eq_right.mpr hKL] | |
| rw [←he,Subgroup.relIndex_map_map_of_injective N (K.subgroupOf L) L.subtype_injective] | |
| exact hN | |
| universe u | |
| theorem bounded_normal_commensurable (d : ℕ) : | |
| ∀ (A : Type u) [Group A] (K : Subgroup A) (S : Set A), | |
| (1:A)∈S → S⁻¹=S → Subgroup.closure S=⊤ → Bounded S (K:Set A) → HasDegree K S d → | |
| ∃ N : Subgroup A, N.Normal ∧ Bounded S (N:Set A) ∧ N.relIndex K≠0 := by | |
| induction d with | |
| | zero => | |
| intro A _ K S h1 hi hgen hK hg | |
| let := hg.zero_finiteIndex h1 hi hgen | |
| exact ⟨⊤,inferInstance,bounded_top_of_finiteIndex K S h1 hi hgen hK,by simp⟩ | |
| | succ d ih => | |
| intro A _ K S h1 hi hgen hK hg | |
| rcases finite_or_infinite (A ⧸ K) with hfin | hinf | |
| · let : K.FiniteIndex := ⟨by let := hfin; exact Nat.card_pos.ne'⟩ | |
| exact ⟨⊤,inferInstance,bounded_top_of_finiteIndex K S h1 hi hgen hK,by simp⟩ | |
| let := hinf | |
| obtain ⟨H,hHi,f,hf,hKf⟩ := virtually_indicable h1 hi hgen hK hg | |
| let := hHi | |
| let J := K.subgroupOf H | |
| obtain ⟨T,hT1,hTi,hTgen,hTb,hKT⟩ := | |
| finiteIndex_generators_containing_stabilizer K S h1 hi hgen hK H | |
| have hgT : HasDegree J T (d+1) := hg.subgroup_bounded h1 H T hTb | |
| obtain ⟨U,hU1,hUi,hUgen,hUb,hKU⟩ := | |
| hgT.integer_kernel_bounded_generators hT1 hTi hTgen hKT f hf hKf | |
| have hgU : HasDegree (J.subgroupOf f.ker) U d := | |
| hgT.integer_kernel_descent hT1 hTi hTgen f hf hKf U hUb | |
| obtain ⟨N,hNn,hNb,hNi⟩ := ih f.ker (J.subgroupOf f.ker) U hU1 hUi hUgen | |
| ⟨1,by simpa only [pow_one] using hKU⟩ hgU | |
| let := hNn | |
| let P := N.map f.ker.subtype | |
| have hPL : P≤f.ker := by rintro _ ⟨n,hn,rfl⟩; exact n.property | |
| have hPN : f.ker≤Subgroup.normalizer (P:Set H) := normalizer_of_normal_subgroup f.ker N | |
| have hPb : Bounded T (P:Set H) := Bounded.image_trans f.ker.subtype hUb hNb | |
| have hPi : P.relIndex J≠0 := relIndex_map_subgroupOf_of_le J f.ker hKf N hNi | |
| obtain ⟨M,hMn,hMb,hMi⟩ := hgT.promote_integer_normal hT1 hTi hTgen f hf hKf P hPL hPN hPb hPi | |
| let := hMn | |
| let Q := M.map H.subtype | |
| have hQn : H≤Subgroup.normalizer (Q:Set A) := normalizer_of_normal_subgroup H M | |
| have hQb : Bounded S (Q:Set A) := Bounded.image_trans H.subtype hTb hMb | |
| have hQi : Q.relIndex K≠0 := relIndex_map_subgroupOf_nonzero K H M hMi | |
| exact hg.finiteIndex_normal_core h1 hi hgen hK H Q hQn hQb hQi | |
| end CriticalPercolation.RelativeGrowth | |
| end | |
| end | |
| end OAI | |