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f4b3ad1 | 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 | import OAI.NumberTheory.Ostmann.Arithmetic.ArithmeticHistorySupportSum
import OAI.NumberTheory.Ostmann.Arithmetic.FrequencyBudgetRates
import OAI.NumberTheory.Ostmann.Arithmetic.LeafFrequencyBudget
namespace OAI
/-! # The asymptotic frequency-support cost under the original Haar law -/
namespace Ostmann
open scoped BigOperators Classical
open Filter
noncomputable def arithmeticHistorySupportSum {Q : ℕ} [NeZero Q]
(S : Finset ℤ) (V n : ℕ) (P : FrequencyTree (S × S) n → (ZMod Q)ˣ)
(data : FrequencyTree (S × S) n → List (ZMod Q)ˣ →
Option (ArithmeticSplitData Q × ArithmeticSplitData Q)) : ℝ :=
∑ t : FrequencyTree (S × S) n,
frequencyLeafWeight (pairedFrequencyLeaf S V) n t *
((Fintype.card (TreeLeafFiber (ZMod Q)ˣ n (P t)) : ℝ)⁻¹ *
∑ x : TreeLeafFiber (ZMod Q)ˣ n (P t),
sequentialSupport (fun past y => arithmeticPairSplitTest (data t past) y) []
(2 ^ n - 1) (treeLeafSplitEquiv n (P t) x))
theorem arithmetic_history_support_rate {Q : ℕ} [NeZero Q]
(n : ℕ) (K C ε : ℝ) (hC : 0 ≤ C) (hε : 0 < ε) :
∀ᶠ m : ℝ in atTop, ∀ N V : ℕ, ∀ Δ : ℝ, ∀ S : Finset ℤ,
(N : ℝ) ≤ Real.exp (C * m) →
(V : ℝ) ≤ Real.exp (Δ + Real.sqrt m) →
(∀ s ∈ S, s ≠ 0 ∧ s.natAbs ≤ N) →
∀ P : FrequencyTree (S × S) n → (ZMod Q)ˣ,
∀ data : FrequencyTree (S × S) n → List (ZMod Q)ˣ →
Option (ArithmeticSplitData Q × ArithmeticSplitData Q),
(∀ t past d e, data t past = some (d, e) →
d.hasFrequencies (treeNodeFrequencies S n t past.length).1 ∧
e.hasFrequencies (treeNodeFrequencies S n t past.length).2) →
Real.exp (-(2 ^ n : ℕ) * Δ + K) * arithmeticHistorySupportSum S V n P data ≤
Real.exp ((2 ^ n : ℕ) * Δ + ε * m) := by
let r : ℕ := 2 ^ n
have hr : (0 : ℝ) < r := by exact_mod_cast (Nat.two_pow_pos n)
let δ : ℝ := ε / (2 * r)
have hδ : 0 < δ := div_pos hε (by positivity)
have hδr : δ * r = ε / 2 := by dsimp [δ]; field_simp
obtain ⟨M, hM⟩ := leaf_frequency_budget_sublinear r K (ε / 2) (by positivity)
filter_upwards [frequency_budget_subexponential C δ hC hδ,
eventually_ge_atTop M, eventually_ge_atTop (0 : ℝ)] with m hrate hm hm0
intro N V Δ S hN hV hS P data hmatch
obtain ⟨D, hD, hdiv, hcost⟩ := hrate
have hdivN (q : ℕ) (hq : q ≠ 0) (hqN : q ≤ N ^ 2) : (q.divisors.card : ℝ) ≤ D := by
apply hdiv q hq
calc
(q : ℝ) ≤ (N : ℝ) ^ 2 := by exact_mod_cast hqN
_ ≤ Real.exp (C * m) ^ 2 := pow_le_pow_left₀ (Nat.cast_nonneg _) hN 2
_ = Real.exp (2 * C * m) := by rw [← Real.exp_nat_mul]; congr 1; ring
have hsum := arithmetic_history_support_sum_le S N V n D hD hS hdivN P data hmatch
change arithmeticHistorySupportSum S V n P data ≤
(8 * D ^ 4 * (1 + Real.log N) ^ 3) ^ (r - 1) * (2 * (V : ℝ)) ^ (2 * r) at hsum
have hC0 : 0 ≤ 8 * D ^ 4 * (1 + Real.log N) ^ 3 := by
have hlog := Real.log_natCast_nonneg N
positivity
have hnode : (8 * D ^ 4 * (1 + Real.log N) ^ 3) ^ (r - 1) ≤
Real.exp ((ε / 2) * m) := by
calc
_ ≤ Real.exp (δ * m) ^ (r - 1) := pow_le_pow_left₀ hC0 (hcost N hN) _
_ = Real.exp ((r - 1 : ℕ) * (δ * m)) := (Real.exp_nat_mul _ _).symm
_ ≤ _ := by
apply Real.exp_le_exp.mpr
have hcount : ((r - 1 : ℕ) : ℝ) ≤ r := by exact_mod_cast Nat.sub_le r 1
have h := mul_le_mul_of_nonneg_right hcount (mul_nonneg hδ.le hm0)
nlinarith [hδr]
have hleaf := hM m hm Δ V hV
change Real.exp (-(r : ℝ) * Δ + K) * arithmeticHistorySupportSum S V n P data ≤ _
calc
_ ≤ Real.exp (-(r : ℝ) * Δ + K) *
((8 * D ^ 4 * (1 + Real.log N) ^ 3) ^ (r - 1) * (2 * (V : ℝ)) ^ (2 * r)) :=
mul_le_mul_of_nonneg_left hsum (Real.exp_pos _).le
_ = (8 * D ^ 4 * (1 + Real.log N) ^ 3) ^ (r - 1) *
((2 * (V : ℝ)) ^ (2 * r) * Real.exp (-(r : ℝ) * Δ + K)) := by ring
_ ≤ Real.exp ((ε / 2) * m) * Real.exp ((r : ℝ) * Δ + (ε / 2) * m) :=
mul_le_mul hnode hleaf (by positivity) (Real.exp_pos _).le
_ = Real.exp ((2 ^ n : ℕ) * Δ + ε * m) := by
rw [← Real.exp_add]
dsimp [r]
congr 1
ring
end Ostmann
end OAI
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