openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Growth /CosetNormalPromotion.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Growth.CosetFiniteIndex | |
| import OAI.Probability.CriticalPercolation.Growth.CosetTransfer | |
| import OAI.Probability.CriticalPercolation.Criticality.BoundedNormalJoins | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function | |
| open scoped Classical Pointwise | |
| namespace CriticalPercolation.RelativeGrowth | |
| variable {A : Type*} [Group A] | |
| lemma conjugate_eq_of_integer_value (f : A →* Multiplicative ℤ) | |
| (a : A) (ha : f a=Multiplicative.ofAdd 1) (N : Subgroup A) | |
| (hN : f.ker≤(Subgroup.normalizer (N:Set A))) (g : A) : | |
| N.map (MulAut.conj g).toMonoidHom=N.map (MulAut.conj (a^Multiplicative.toAdd (f g))).toMonoidHom := by | |
| let z := Multiplicative.toAdd (f g) | |
| have hl : (a^z)⁻¹*g∈f.ker := by | |
| change f ((a^z)⁻¹*g)=1 | |
| rw [map_mul,map_inv,map_zpow,ha] | |
| apply Multiplicative.toAdd.injective | |
| simp [z] | |
| have he : N.map (MulAut.conj ((a^z)⁻¹*g)).toMonoidHom=N := | |
| Subgroup.mem_normalizer_iff_map_conj_eq.mp (hN hl) | |
| have hg : g=a^z*((a^z)⁻¹*g) := by simp | |
| conv_lhs => rw [hg] | |
| rw [←conjugate_map_map,he] | |
| lemma finite_bounded_conjugate_family (K : Subgroup A) (S : Set A) | |
| (hf : ∀ n, (ball K S n).Finite) (f : A →* Multiplicative ℤ) (hK : K≤f.ker) | |
| (a : A) (ha : f a=Multiplicative.ofAdd 1) (N : Subgroup A) | |
| (hN : f.ker≤(Subgroup.normalizer (N:Set A))) (n : ℕ) : | |
| ((fun g => N.map (MulAut.conj g).toMonoidHom) '' S^n).Finite := by | |
| have hKi : (f '' (K:Set A)).Finite := by | |
| apply (Set.finite_singleton (1:Multiplicative ℤ)).subset | |
| rintro _ ⟨k,hk,rfl⟩ | |
| exact MonoidHom.mem_ker.mp (hK hk) | |
| have hfi := finite_image_of_finite_coset_image K f hKi (S^n) (hf n) | |
| apply (hfi.image (fun z => N.map (MulAut.conj (a^Multiplicative.toAdd z)).toMonoidHom)).subset | |
| rintro _ ⟨g,hg,rfl⟩ | |
| exact ⟨f g,⟨g,hg,rfl⟩,(conjugate_eq_of_integer_value f a ha N hN g).symm⟩ | |
| lemma HasDegree.promote_integer_normal {K : Subgroup A} {S : Set A} {d : ℕ} | |
| (h1 : (1:A)∈S) (hi : S⁻¹=S) (hgen : Subgroup.closure S=⊤) (hg : HasDegree K S d) | |
| (f : A →* Multiplicative ℤ) (hf : Surjective f) (hK : K≤f.ker) | |
| (N : Subgroup A) (hNL : N≤f.ker) (hLN : f.ker≤(Subgroup.normalizer (N:Set A))) | |
| (hNb : Bounded S (N:Set A)) (hNi : N.relIndex K≠0) : | |
| ∃ M : Subgroup A, M.Normal ∧ Bounded S (M:Set A) ∧ M.relIndex K≠0 := by | |
| let J := K⊓N | |
| have hJi : J.relIndex K≠0 := by simpa only [J,Subgroup.inf_relIndex_left] using hNi | |
| let : Finite (K ⧸ J.subgroupOf K) := Nat.finite_of_card_ne_zero hJi | |
| have hgJ := hg.smaller_stabilizer (show J≤K from inf_le_left) | |
| obtain ⟨L,hL⟩ := hNb | |
| obtain ⟨n,hn⟩ := exists_conjugateStage_stationary S (N:Set A) h1 hi J | |
| (by simpa only [Subgroup.closure_eq] using (show J≤N from inf_le_right)) d hgJ L hL | |
| let M := Subgroup.normalClosure (N:Set A) | |
| have hM : M=conjugateStage S (N:Set A) n := normalClosure_eq_conjugateStage_of_stationary S (N:Set A) h1 hi hgen n hn | |
| obtain ⟨a,ha⟩ := hf (Multiplicative.ofAdd 1) | |
| refine ⟨M,inferInstance,?_,?_⟩ | |
| · rw [hM,conjugateStage_eq_join] | |
| apply finite_normalized_join_bounded S h1 f.ker _ | |
| (finite_bounded_conjugate_family K S hg.1 f hK a ha N hLN n) | |
| · rintro Q ⟨g,hg,rfl⟩ | |
| have hh := Subgroup.map_mono hNL (f:=(MulAut.conj g).toMonoidHom) | |
| have he : f.ker.map (MulAut.conj g).toMonoidHom=f.ker := Subgroup.Normal.map_conj_eq f.ker g | |
| rwa [he] at hh | |
| · rintro Q ⟨g,hg,rfl⟩ | |
| exact normalized_conjugate f.ker N hLN g | |
| · rintro Q ⟨g,hg,rfl⟩ | |
| exact (show Bounded S (N:Set A) from ⟨L,hL⟩).conj h1 hi hgen g | |
| · exact fun he => hNi (Subgroup.relIndex_eq_zero_of_le_left (fun x hx => Subgroup.subset_normalClosure hx) he) | |
| end CriticalPercolation.RelativeGrowth | |
| end | |
| end | |
| end OAI | |