| import Mathlib.Analysis.SpecialFunctions.Exp |
| import Mathlib.Tactic |
|
|
| namespace OAI |
|
|
| /-! |
| # The fixed repetition schedule |
|
|
| The numerical amplification estimate from the final section of |
| *A Low-Space Algorithm for Worst-Case Subset Sum*. The per-round failure bound |
| must be proved for the actual trials before this estimate applies. |
| -/ |
|
|
| namespace LowSpaceSubsetSum |
|
|
| / |
| The bound applies to any nonnegative per-round failure probability `q` below |
| the stated threshold. -/ |
| theorem amplification_bound {u : ℕ} (hu : 2 ≤ u) {q : ℝ} |
| (hq0 : 0 ≤ q) (hq : q ≤ 1 - 1 / (u : ℝ) ^ 110) : |
| q ^ (u ^ 120) < (1 / 3 : ℝ) := by |
| have hu0 : (0 : ℝ) < u := by exact_mod_cast (by omega : 0 < u) |
| have hu1 : (1 : ℝ) < u := by exact_mod_cast hu |
| have hp : (u : ℝ) ^ 120 = (u : ℝ) ^ 110 * (u : ℝ) ^ 10 := by |
| rw [← pow_add] |
| have hratio : (u : ℝ) ^ 10 / (u ^ 120 : ℕ) = 1 / (u : ℝ) ^ 110 := by |
| rw [Nat.cast_pow, hp] |
| field_simp |
| have ht : (u : ℝ) ^ 10 ≤ (u ^ 120 : ℕ) := by |
| rw [Nat.cast_pow, hp] |
| have hlarge : (1 : ℝ) ≤ (u : ℝ) ^ 110 := one_le_pow₀ hu1.le |
| nlinarith [pow_pos hu0 10] |
| have hexp := Real.one_sub_div_pow_le_exp_neg (n := u ^ 120) ht |
| rw [hratio] at hexp |
| have hqpow := pow_le_pow_left₀ hq0 hq (u ^ 120) |
| have ht2 : (2 : ℝ) ≤ (u : ℝ) ^ 10 := by |
| have hule : (u : ℝ) ≤ (u : ℝ) ^ 10 := le_self_pow₀ (by linarith) (by decide) |
| have hu2 : (2 : ℝ) ≤ u := by exact_mod_cast hu |
| exact hu2.trans hule |
| have hstrict : 3 < Real.exp ((u : ℝ) ^ 10) := by |
| have h := Real.add_one_lt_exp (by positivity : (u : ℝ) ^ 10 ≠ 0) |
| linarith |
| have hsmall : Real.exp (-((u : ℝ) ^ 10)) < (1 / 3 : ℝ) := by |
| rw [Real.exp_neg] |
| apply (inv_lt_comm₀ (Real.exp_pos _) (by norm_num : (0 : ℝ) < 1 / 3)).mpr |
| norm_num |
| exact hstrict |
| exact (hqpow.trans hexp).trans_lt hsmall |
|
|
| end LowSpaceSubsetSum |
|
|
| end OAI |
|
|
