File size: 2,926 Bytes
ea76aac | 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 | 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
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