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f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | 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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