| import Mathlib | |
| import OAI.Computability.MaxCut.Model | |
| namespace OAI | |
| /-! | |
| Scalar parameter selection for the v2 repetition rate. | |
| These results concern powers of real numbers only. In particular, they do not | |
| assert a parallel-repetition theorem for games. The exponent depends only on | |
| the gap and the desired error; no alphabet or logical dimension occurs here. | |
| -/ | |
| namespace MaxCutGames.Repetition | |
| /-- The scalar rate appearing in the alphabet-independent repetition estimate. -/ | |
| noncomputable def dsRate (gap : ℝ) : ℝ := 1 - gap ^ 2 / 16 | |
| theorem dsRate_nonneg {gap : ℝ} (hgap₀ : 0 ≤ gap) (hgap₁ : gap ≤ 1) : | |
| 0 ≤ dsRate gap := by | |
| dsimp [dsRate] | |
| nlinarith | |
| theorem dsRate_lt_one {gap : ℝ} (hgap : 0 < gap) : dsRate gap < 1 := by | |
| dsimp [dsRate] | |
| nlinarith [sq_pos_of_pos hgap] | |
| /-- A scalar contraction admits a strictly positive exponent for every positive | |
| target error. This lemma makes no assertion about any game value. -/ | |
| theorem exists_positive_power_lt {rate error : ℝ} | |
| (hrate₀ : 0 ≤ rate) (hrate₁ : rate < 1) (herror : 0 < error) : | |
| ∃ n : ℕ, 1 ≤ n ∧ rate ^ n < error := by | |
| have ht := tendsto_pow_atTop_nhds_zero_of_lt_one hrate₀ hrate₁ | |
| obtain ⟨n, hn⟩ := (ht.eventually (gt_mem_nhds herror)).exists | |
| refine ⟨n + 1, Nat.succ_le_succ (Nat.zero_le n), ?_⟩ | |
| calc | |
| rate ^ (n + 1) = rate ^ n * rate := pow_succ _ _ | |
| _ ≤ rate ^ n := mul_le_of_le_one_right (pow_nonneg hrate₀ n) hrate₁.le | |
| _ < error := hn | |
| /-- Select the repetition count before choosing any alphabet. -/ | |
| theorem exists_dsRate_pow_le {gap error : ℝ} | |
| (hgap₀ : 0 < gap) (hgap₁ : gap < 1) (herror : 0 < error) : | |
| ∃ n : ℕ, 1 ≤ n ∧ dsRate gap ^ n ≤ error := by | |
| obtain ⟨n, hn, hbound⟩ := exists_positive_power_lt | |
| (dsRate_nonneg hgap₀.le hgap₁.le) (dsRate_lt_one hgap₀) herror | |
| exact ⟨n, hn, hbound.le⟩ | |
| theorem dsRate_one_div_eight_hundred : | |
| dsRate (1 / 800) = 1 - (1 : ℝ) / 10240000 := by | |
| norm_num [dsRate] | |
| theorem dsRate_one_div_fifteen : | |
| dsRate (1 / 15) = 1 - (1 : ℝ) / 3600 := by | |
| norm_num [dsRate] | |
| /-- The final soundness exponent depends only on `error`, even when the later | |
| construction chooses its alphabet after this exponent. -/ | |
| theorem exists_final_repetition_count {error : ℝ} (herror : 0 < error) : | |
| ∃ n : ℕ, 1 ≤ n ∧ (1 - (1 : ℝ) / 10240000) ^ n ≤ error := by | |
| simpa only [dsRate_one_div_eight_hundred] using | |
| exists_dsRate_pow_le (gap := (1 / 800 : ℝ)) (by norm_num) (by norm_num) herror | |
| /-- The preliminary constant-gap specialization of the same scalar rate. -/ | |
| theorem exists_preliminary_repetition_count {error : ℝ} (herror : 0 < error) : | |
| ∃ n : ℕ, 1 ≤ n ∧ (1 - (1 : ℝ) / 3600) ^ n ≤ error := by | |
| simpa only [dsRate_one_div_fifteen] using | |
| exists_dsRate_pow_le (gap := (1 / 15 : ℝ)) (by norm_num) (by norm_num) herror | |
| end MaxCutGames.Repetition | |
| end OAI | |