| import OAI.Computability.UniqueGames.Games.FinishBoundsLemmas |
| import OAI.Computability.UniqueGames.Repetition.AnalyticCheegerLemmas |
|
|
| namespace OAI |
|
|
| section |
|
|
| /-! |
| The outer error parameters are fixed before the latent gadget or its alphabet. |
| We use the same positive reciprocal integer for the latent noise budget and |
| the rounding tolerance. This permitted specialization makes uniform occurrence |
| rounding an exact fixed-copy operation. The later parity error is chosen only |
| after the tuple length has been fixed. |
| -/ |
|
|
| namespace UniqueGamesTheorem.ParameterSelection |
| noncomputable section |
|
|
| structure OuterParameters (ε δ : ℝ) where |
| epsilon0 : ℚ |
| delta0 : ℚ |
| epsilon0_pos : 0 < epsilon0 |
| delta0_pos : 0 < delta0 |
| epsilon0_le : (epsilon0 : ℝ) ≤ ε |
| delta0_le : (delta0 : ℝ) ≤ δ |
| repetitions : ℕ |
| repetitions_pos : 0 < repetitions |
| soundness_rate : (1 - (1 : ℝ) / 10240000) ^ repetitions ≤ delta0 |
| copies : ℕ |
| copies_large : 200 ≤ copies |
| reciprocal_pos : (0 : ℚ) < 1 / copies |
| reciprocal_small : 1 / (copies : ℚ) ≤ 1 / 200 |
| reciprocal_budget : 1 / (copies : ℚ) ≤ epsilon0 / repetitions |
|
|
| / |
| only the two prescribed real errors. -/ |
| theorem exists_outerParameters (ε δ : ℝ) (hε : 0 < ε) (hδ : 0 < δ) : |
| Nonempty (OuterParameters ε δ) := by |
| obtain ⟨ε₀, hε₀, hεbound⟩ := exists_rat_btwn hε |
| obtain ⟨δ₀, hδ₀, hδbound⟩ := exists_rat_btwn hδ |
| have hεq : (0 : ℚ) < ε₀ := by exact_mod_cast hε₀ |
| have hδq : (0 : ℚ) < δ₀ := by exact_mod_cast hδ₀ |
| obtain ⟨t, ht, hrate⟩ := Repetition.exists_final_repetition_count hδ₀ |
| have htpos : 0 < t := ht |
| obtain ⟨C, hC, hCpos, hCsmall, hCbudget⟩ := |
| Explicit.FinishBounds.exists_reciprocal_tolerance ε₀ hεq t htpos |
| exact ⟨⟨ε₀, δ₀, hεq, hδq, hεbound.le, hδbound.le, |
| t, htpos, hrate, C, hC, hCpos, hCsmall, hCbudget⟩⟩ |
|
|
| namespace OuterParameters |
|
|
| variable {ε δ : ℝ} (P : OuterParameters ε δ) |
|
|
| def pStar : ℚ := 1 / P.copies |
|
|
| theorem pStar_pos : 0 < P.pStar := P.reciprocal_pos |
|
|
| theorem pStar_lt_one : P.pStar < 1 := |
| P.reciprocal_small.trans_lt (by norm_num) |
|
|
| theorem exists_parity_error (k : ℕ) (hk : 0 < k) : |
| ∃ ξ : ℚ, 0 < ξ ∧ ξ < 1 / 100 ∧ |
| ξ < P.epsilon0 / (2 * k * P.repetitions) := by |
| have hdenom : (0 : ℚ) < 2 * k * P.repetitions := by |
| exact mul_pos (mul_pos (by norm_num) (by exact_mod_cast hk)) |
| (by exact_mod_cast P.repetitions_pos) |
| let ξ := min (1 / 100 : ℚ) (P.epsilon0 / (2 * k * P.repetitions)) / 2 |
| have hmin : 0 < min (1 / 100 : ℚ) (P.epsilon0 / (2 * k * P.repetitions)) := |
| lt_min (by norm_num) (div_pos P.epsilon0_pos hdenom) |
| refine ⟨ξ, div_pos hmin (by norm_num), ?_, ?_⟩ |
| · have hle := min_le_left (1 / 100 : ℚ) (P.epsilon0 / (2 * k * P.repetitions)) |
| dsimp [ξ] |
| linarith |
| · have hle := min_le_right (1 / 100 : ℚ) (P.epsilon0 / (2 * k * P.repetitions)) |
| dsimp [ξ] |
| linarith |
|
|
| / |
| alphabet, for the later chosen positive tuple length. -/ |
| theorem matrix_error_budget (k : ℕ) (hk : 0 < k) (ξ : ℚ) |
| (hξ : ξ ≤ P.epsilon0 / (2 * k * P.repetitions)) : |
| (k : ℚ) * ξ + P.pStar / 2 ≤ P.epsilon0 / P.repetitions := by |
| have hkq : (0 : ℚ) < k := by exact_mod_cast hk |
| have htq : (0 : ℚ) < P.repetitions := by exact_mod_cast P.repetitions_pos |
| have hx := (le_div_iff₀ (mul_pos (mul_pos (by norm_num) hkq) htq)).mp hξ |
| have hp := (le_div_iff₀ htq).mp P.reciprocal_budget |
| apply (le_div_iff₀ htq).mpr |
| dsimp [pStar] |
| nlinarith |
|
|
| end OuterParameters |
| end |
| end UniqueGamesTheorem.ParameterSelection |
|
|
| end |
|
|
| end OAI |
|
|
