| import Mathlib |
| import OAI.Probability.CriticalPercolation.Criticality.NormalClosureGrowth |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set Function Filter Topology |
| open scoped Classical Pointwise |
|
|
| namespace CriticalPercolation.RelativeGrowth |
| variable {A : Type*} [Group A] |
|
|
| def ball (K : Subgroup A) (S : Set A) (n : ℕ) : Set (A ⧸ K) := |
| (QuotientGroup.mk : A → A ⧸ K) '' S^n |
|
|
| def HasDegree (K : Subgroup A) (S : Set A) (d : ℕ) : Prop := |
| (∀ n, (ball K S n).Finite) ∧ ∃ C : ℕ, ∀ R : ℕ, ∃ n≥R, (ball K S n).ncard≤C*n^d |
|
|
| lemma ball_mono (K : Subgroup A) (S : Set A) (h1 : (1:A)∈S) : |
| Monotone (ball K S) := fun _ _ h => Set.image_mono (Set.pow_subset_pow_right h1 h) |
|
|
| lemma not_quadratic_exponential_lower (K : Subgroup A) (S : Set A) (h1 : (1:A)∈S) |
| (d : ℕ) (hg : HasDegree K S d) (L : ℕ) (hL : 0<L) |
| (hlo : ∀ n : ℕ, 2^n ≤ (ball K S (L*n^2)).ncard) : False := by |
| obtain ⟨hfin,C,hC⟩ := hg |
| obtain ⟨N,hN⟩ := SparseGrowth.eventually_exponential_dominates (C*(4*L)^d) (2*d) |
| obtain ⟨r,hr,hbound⟩ := hC (L*(max N 1)^2) |
| let n := Nat.sqrt (r/L) |
| have hn : max N 1 ≤ n := Nat.le_sqrt'.mpr ((Nat.le_div_iff_mul_le hL).mpr (by |
| simpa only [Nat.mul_comm] using hr)) |
| have hn1 : 1 ≤ n := (le_max_right N 1).trans hn |
| have hnr : L*n^2 ≤ r := by |
| calc |
| _ ≤ L*(r/L) := Nat.mul_le_mul_left L (Nat.sqrt_le' (r/L)) |
| _ ≤ r := by simpa only [Nat.mul_comm] using Nat.div_mul_le_self r L |
| have hrn : r ≤ (4*L)*n^2 := by |
| have ht := Nat.lt_mul_div_succ r hL |
| have hs : r/L+1 ≤ (n+1)^2 := Nat.succ_le_of_lt (Nat.lt_succ_sqrt' (r/L)) |
| have hs' : (n+1)^2 ≤ 4*n^2 := by nlinarith |
| have hj := (Nat.mul_le_mul_left L hs).trans (Nat.mul_le_mul_left L hs') |
| nlinarith |
| have hu : 2^n ≤ (C*(4*L)^d)*n^(2*d) := by |
| calc |
| _ ≤ (ball K S (L*n^2)).ncard := hlo n |
| _ ≤ (ball K S r).ncard := Set.ncard_le_ncard (ball_mono K S h1 hnr) (hfin r) |
| _ ≤ C*r^d := hbound |
| _ ≤ C*((4*L)*n^2)^d := Nat.mul_le_mul_left C (Nat.pow_le_pow_left hrn d) |
| _ = (C*(4*L)^d)*n^(2*d) := by rw [mul_pow,pow_mul]; ring |
| exact (not_le_of_gt (hN n ((le_max_left N 1).trans hn))) hu |
|
|
| lemma binaryProducts_coset_inj (K : Subgroup A) (H : ℕ → Subgroup A) (hH : Monotone H) |
| (hK : K≤H 0) (g : ℕ → A) (hg : ∀ n, g n∈H (n+1)) (hout : ∀ n, g n∉H n) : |
| ∀ n, Set.InjOn (QuotientGroup.mk : A → A ⧸ K) (SparseGrowth.binaryProducts g n : Set A) := by |
| intro n |
| induction n with |
| | zero => simp [SparseGrowth.binaryProducts,Set.InjOn] |
| | succ n ih => |
| intro x hx y hy he |
| have hKn : K≤H n := hK.trans (hH (Nat.zero_le n)) |
| have hmem := SparseGrowth.binaryProducts_mem H hH g hg n |
| have hbad (x y : A) (hx : x∈SparseGrowth.binaryProducts g n) |
| (hy : y∈SparseGrowth.binaryProducts g n) |
| (he : (QuotientGroup.mk x : A ⧸ K)=QuotientGroup.mk (g n*y)) : False := by |
| have hxy : x⁻¹*(g n*y)∈H n := hKn (QuotientGroup.eq.mp he) |
| have hz := (H n).mul_mem ((H n).mul_mem (hmem x hx) hxy) ((H n).inv_mem (hmem y hy)) |
| apply hout n |
| simpa only [mul_inv_cancel_left,mul_assoc,mul_inv_cancel,mul_one] using hz |
| rcases Finset.mem_union.mp hx with hx|hx <;> rcases Finset.mem_union.mp hy with hy|hy |
| · exact ih hx hy he |
| · obtain ⟨z,hz,rfl⟩ := Finset.mem_image.mp hy |
| exact False.elim (hbad x z hx hz he) |
| · obtain ⟨z,hz,rfl⟩ := Finset.mem_image.mp hx |
| exact False.elim (hbad y z hy hz he.symm) |
| · obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hx |
| obtain ⟨b,hb,rfl⟩ := Finset.mem_image.mp hy |
| apply congrArg (fun t => g n*t) |
| apply ih ha hb |
| apply QuotientGroup.eq.mpr |
| have hh := QuotientGroup.eq.mp he |
| simpa only [mul_inv_rev,mul_assoc,inv_mul_cancel_left] using hh |
|
|
| lemma binaryProducts_subset_pow (S : Set A) (h1 : (1:A)∈S) |
| (g : ℕ → A) (L : ℕ) (hg : ∀ n, g n∈S^(L*(n+1))) : |
| ∀ n, (SparseGrowth.binaryProducts g n : Set A) ⊆ S^(L*n^2) := by |
| intro n |
| induction n with |
| | zero => simp [SparseGrowth.binaryProducts] |
| | succ n ih => |
| intro x hx |
| rcases Finset.mem_union.mp hx with hx | hx |
| · exact Set.pow_subset_pow_right h1 (Nat.mul_le_mul_left L (Nat.pow_le_pow_left (Nat.le_succ n) 2)) (ih hx) |
| · obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx |
| have hlen : L*(n+1)+L*n^2 ≤ L*(n+1)^2 := by |
| calc |
| _ = L*(n+1+n^2) := by ring |
| _ ≤ L*(n+1)^2 := Nat.mul_le_mul_left L (by nlinarith) |
| apply Set.pow_subset_pow_right h1 hlen |
| rw [pow_add] |
| exact Set.mul_mem_mul (hg n) (ih hy) |
|
|
| lemma no_linear_strict_chain (K : Subgroup A) (S : Set A) (h1 : (1:A)∈S) |
| (d : ℕ) (hg : HasDegree K S d) (H : ℕ → Subgroup A) (hH : Monotone H) (hK : K≤H 0) |
| (g : ℕ → A) (hgin : ∀ n, g n∈H (n+1)) (hgout : ∀ n, g n∉H n) |
| (L : ℕ) (hL : 0<L) (hword : ∀ n, g n∈S^(L*(n+1))) : False := by |
| apply not_quadratic_exponential_lower K S h1 d hg L hL |
| intro n |
| have hin := binaryProducts_coset_inj K H hH hK g hgin hgout n |
| rw [←SparseGrowth.binaryProducts_card H hH g hgin hgout n] |
| have heq := hin.ncard_image |
| rw [Set.ncard_coe_finset] at heq |
| rw [←heq] |
| exact Set.ncard_le_ncard (Set.image_mono (binaryProducts_subset_pow S h1 g L hword n)) (hg.1 _) |
|
|
| end CriticalPercolation.RelativeGrowth |
| end |
| end |
|
|
| end OAI |
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|
