| import OAI.NumberTheory.Ostmann.Arithmetic.VariableModulusSupport |
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| namespace OAI |
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| /-! # The numerical rate after the actual arithmetic frequency sum -/ |
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| namespace Ostmann |
| open Filter |
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| / |
| modulus. This is the numerical estimate used by the same-assignment norm. -/ |
| theorem arithmetic_frequency_budget_rate (n : ℕ) (K C ε : ℝ) (hC : 0 ≤ C) (hε : 0 < ε) : |
| ∀ᶠ m : ℝ in atTop, ∃ D : ℝ, 0 ≤ D ∧ |
| (∀ q : ℕ, q ≠ 0 → (q : ℝ) ≤ Real.exp (2 * C * m) → (q.divisors.card : ℝ) ≤ D) ∧ |
| ∀ N V : ℕ, ∀ Δ : ℝ, (N : ℝ) ≤ Real.exp (C * m) → |
| (V : ℝ) ≤ Real.exp (Δ + Real.sqrt (4 * m)) → |
| Real.exp (-(2 ^ n : ℕ) * Δ + K) * |
| ((8 * D ^ 4 * (1 + Real.log N) ^ 3) ^ (2 ^ n - 1) * |
| (2 * (V : ℝ)) ^ (2 * 2 ^ n)) ≤ |
| 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 / 8) (by positivity) |
| filter_upwards [frequency_budget_subexponential C δ hC hδ, |
| eventually_ge_atTop (M / 4), eventually_ge_atTop (0 : ℝ)] with m hrate hm hm0 |
| obtain ⟨D, hD, hdiv, hcost⟩ := hrate |
| refine ⟨D, hD, hdiv, ?_⟩ |
| intro N V Δ hN hV |
| have hC0 : 0 ≤ 8 * D ^ 4 * (1 + Real.log N) ^ 3 := by |
| have := 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 : (2 * (V : ℝ)) ^ (2 * r) * Real.exp (-(r : ℝ) * Δ + K) ≤ |
| Real.exp ((r : ℝ) * Δ + (ε / 2) * m) := by |
| convert hM (4 * m) (by linarith) Δ V hV using 1 |
| congr 1 |
| ring |
| calc |
| _ = (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 |
| _ = _ := by rw [← Real.exp_add]; dsimp [r]; congr 1; ring |
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| / |
| loss is exactly `2 * 6^(r*m)`, independently of the depth. -/ |
| theorem arithmetic_norm_page_spectator_cost (n m : ℕ) : |
| (2 : ℝ) ^ (2 ^ n * m) * 2 * 3 ^ (2 ^ n * m) = |
| 2 * Real.exp (Real.log 6 * (2 ^ n : ℕ) * m) := by |
| rw [mul_right_comm _ 2, ← mul_pow, show (2 : ℝ) * 3 = 6 by norm_num, mul_comm _ 2] |
| congr 1 |
| rw [show Real.log 6 * (2 ^ n : ℕ) * m = ((2 ^ n * m : ℕ) : ℝ) * Real.log 6 by |
| push_cast |
| ring, Real.exp_nat_mul, Real.exp_log (by norm_num : (0 : ℝ) < 6)] |
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| / |
| `log 6` of the bulk cost. -/ |
| theorem arithmetic_norm_budget_rate (n : ℕ) (A C ε : ℝ) (hA : 0 ≤ A) (hC : 0 ≤ C) (hε : 0 < ε) : |
| ∀ᶠ m : ℝ in atTop, ∃ D : ℝ, 0 ≤ D ∧ |
| (∀ q : ℕ, q ≠ 0 → (q : ℝ) ≤ Real.exp (2 * C * m) → (q.divisors.card : ℝ) ≤ D) ∧ |
| ∀ N V : ℕ, ∀ Δ : ℝ, (N : ℝ) ≤ Real.exp (C * m) → |
| (V : ℝ) ≤ Real.exp (Δ + Real.sqrt (4 * m)) → |
| (A * Real.exp (-(2 ^ n : ℕ) * Δ)) * |
| ((2 * Real.exp (Real.log 6 * (2 ^ n : ℕ) * m)) * |
| ((8 * D ^ 4 * (1 + Real.log N) ^ 3) ^ (2 ^ n - 1) * |
| (2 * (V : ℝ)) ^ (2 * 2 ^ n))) ≤ |
| Real.exp ((2 ^ n : ℕ) * Δ + Real.log 6 * (2 ^ n : ℕ) * m + ε * m) := by |
| let K := Real.log (2 * A + 1) |
| have hK : 2 * A ≤ Real.exp K := by |
| dsimp only [K] |
| rw [Real.exp_log (by positivity : 0 < 2 * A + 1)] |
| linarith |
| filter_upwards [arithmetic_frequency_budget_rate n K C ε hC hε] with m hm |
| obtain ⟨D, hD, hdiv, hcost⟩ := hm |
| refine ⟨D, hD, hdiv, ?_⟩ |
| intro N V Δ hN hV |
| have hb : 0 ≤ (8 * D ^ 4 * (1 + Real.log N) ^ 3) ^ (2 ^ n - 1) * |
| (2 * (V : ℝ)) ^ (2 * 2 ^ n) := by |
| have := Real.log_natCast_nonneg N |
| positivity |
| calc |
| _ = Real.exp (Real.log 6 * (2 ^ n : ℕ) * m) * |
| ((2 * A) * Real.exp (-(2 ^ n : ℕ) * Δ) * |
| ((8 * D ^ 4 * (1 + Real.log N) ^ 3) ^ (2 ^ n - 1) * |
| (2 * (V : ℝ)) ^ (2 * 2 ^ n))) := by ring |
| _ ≤ Real.exp (Real.log 6 * (2 ^ n : ℕ) * m) * |
| (Real.exp (-(2 ^ n : ℕ) * Δ + K) * |
| ((8 * D ^ 4 * (1 + Real.log N) ^ 3) ^ (2 ^ n - 1) * |
| (2 * (V : ℝ)) ^ (2 * 2 ^ n))) := by |
| apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le |
| rw [Real.exp_add] |
| exact mul_le_mul_of_nonneg_right |
| ((mul_le_mul_of_nonneg_right hK (Real.exp_pos _).le).trans_eq (mul_comm _ _)) hb |
| _ ≤ Real.exp (Real.log 6 * (2 ^ n : ℕ) * m) * |
| Real.exp ((2 ^ n : ℕ) * Δ + ε * m) := |
| mul_le_mul_of_nonneg_left (hcost N V Δ hN hV) (Real.exp_pos _).le |
| _ = _ := by rw [← Real.exp_add]; congr 1; ring |
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| end Ostmann |
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| end OAI |
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