| import Mathlib |
| import OAI.Probability.CriticalPercolation.Growth.CosetTransfer |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set Function |
| open scoped Classical Pointwise |
|
|
| namespace CriticalPercolation.RelativeGrowth |
| variable {A B : Type*} [Group A] [Group B] |
|
|
| lemma exists_finite_coset_cover (K : Subgroup A) (T : Set A) |
| (hf : ((QuotientGroup.mk : A → A ⧸ K) '' T).Finite) : |
| ∃ F : Set A, F.Finite ∧ F.ncard=((QuotientGroup.mk : A → A ⧸ K) '' T).ncard ∧ T⊆F*(K:Set A) := by |
| let Q := (QuotientGroup.mk : A → A ⧸ K) '' T |
| have hh : ∀ q∈Q, ∃ a∈T, (QuotientGroup.mk a : A ⧸ K)=q := fun q hq => hq |
| choose a ha he using hh |
| let F := Set.range (fun q : Q => a q.val q.property) |
| let : Finite Q := hf.to_subtype |
| have hi : Injective (fun q : Q => a q.val q.property) := by |
| intro q r h |
| apply Subtype.ext |
| rw [←he q.val q.property,←he r.val r.property] |
| exact congrArg (QuotientGroup.mk : A → A ⧸ K) h |
| refine ⟨F,Set.finite_range _,?_,?_⟩ |
| · rw [Set.ncard_range_of_injective hi] |
| rfl |
| · intro t ht |
| let q : Q := ⟨QuotientGroup.mk t,⟨t,ht,rfl⟩⟩ |
| refine ⟨a q.val q.property,⟨q,rfl⟩,(a q.val q.property)⁻¹*t,?_,by simp⟩ |
| exact QuotientGroup.eq.mp (he q.val q.property) |
|
|
| lemma image_ncard_bound (K : Subgroup A) (f : A →* B) (hK : (f '' (K:Set A)).Finite) |
| (T : Set A) (hT : ((QuotientGroup.mk : A → A ⧸ K) '' T).Finite) : |
| (f '' T).ncard≤((QuotientGroup.mk : A → A ⧸ K) '' T).ncard*(f '' (K:Set A)).ncard := by |
| obtain ⟨F,hF,hFc,hcov⟩ := exists_finite_coset_cover K T hT |
| have hs : f '' T⊆(f '' F)*(f '' (K:Set A)) := by |
| rintro _ ⟨t,ht,rfl⟩ |
| obtain ⟨a,ha,k,hk,rfl⟩ := hcov ht |
| rw [map_mul] |
| exact Set.mul_mem_mul ⟨a,ha,rfl⟩ ⟨k,hk,rfl⟩ |
| calc |
| _ ≤ ((f '' F)*(f '' (K:Set A))).ncard := Set.ncard_le_ncard hs ((hF.image f).mul hK) |
| _ ≤ (f '' F).ncard*(f '' (K:Set A)).ncard := by |
| have hh := @Finset.card_mul_le B _ _ (hF.image f).toFinset hK.toFinset |
| simpa only [←Set.ncard_coe_finset,Finset.coe_mul,Set.Finite.coe_toFinset] using hh |
| _ ≤ F.ncard*(f '' (K:Set A)).ncard := Nat.mul_le_mul_right _ (Set.ncard_image_le hF) |
| _ = _ := by rw [hFc] |
|
|
| lemma HasDegree.image_polynomial {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 →* B) (hf : Surjective f) (hK : (f '' (K:Set A)).Finite) : |
| SparseGrowth.HasPolynomialDegree B d := by |
| have hSf : (f '' S).Finite := finite_image_of_finite_coset_image K f hK S (by simpa only [ball,pow_one] using hg.1 1) |
| refine ⟨f '' S,hSf,⟨1,h1,map_one f⟩,?_,?_,?_,?_⟩ |
| · rw [←Set.image_inv,hi] |
| · rw [←MonoidHom.map_closure,hgen] |
| exact Subgroup.map_top_of_surjective f hf |
| · exact hg.2.choose*(f '' (K:Set A)).ncard |
| · intro R |
| obtain ⟨n,hn,hb⟩ := hg.2.choose_spec R |
| refine ⟨n,hn,?_⟩ |
| rw [←Set.image_pow] |
| calc |
| _ ≤ (ball K S n).ncard*(f '' (K:Set A)).ncard := image_ncard_bound K f hK (S^n) (hg.1 n) |
| _ ≤ (hg.2.choose*n^d)*(f '' (K:Set A)).ncard := Nat.mul_le_mul_right _ hb |
| _ = _ := by ring |
|
|
| end CriticalPercolation.RelativeGrowth |
| end |
| end |
|
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| end OAI |
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