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92416a2 | 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 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 | 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
/-- Choose the alphabet-independent exponent and both rational budgets using
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
/-- Both matrix-test completeness losses fit the budget fixed before the
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
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