| 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 |
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|
| end OAI |
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|
