| import Mathlib |
| import OAI.Computability.VertexCover.PCP.FinalConstants |
| import OAI.Computability.VertexCover.PCP.PoweringSoundness |
|
|
| namespace OAI |
|
|
| |
|
|
| namespace UniqueGames.Foundations.PCP.PoweringFinalConstants |
|
|
| abbrev Alphabet := Fin 6 → Bool |
|
|
| @[simp] theorem card_alphabet : Fintype.card Alphabet = 64 := by |
| norm_num [Alphabet, Fintype.card_fun] |
|
|
| theorem card_alphabet_eq_final : Fintype.card Alphabet = FinalConstants.alphabet := by |
| exact card_alphabet |
|
|
| theorem center_eq : |
| PoweringSoundness.center (Fintype.card Alphabet) FinalConstants.windowHalf = |
| FinalConstants.endpointLength := by |
| rw [card_alphabet] |
| rfl |
|
|
| theorem walkLength_eq : |
| 2 * PoweringSoundness.center (Fintype.card Alphabet) FinalConstants.windowHalf + 1 = |
| FinalConstants.walkLength := by |
| rw [center_eq] |
| rfl |
|
|
| theorem spectral_denominator_eq : |
| (1 + 2 / (1 - (31 / 32 : ℝ))) + 1 = 66 := by |
| norm_num |
|
|
| theorem gain_eq : |
| PoweringSoundness.gain 64 FinalConstants.windowHalf (31 / 32) / 12288 = |
| FinalConstants.gain := by |
| change |
| (1 / (2 * (FinalConstants.alphabet : ℝ))) ^ 4 * |
| (FinalConstants.windowSize : ℝ) / |
| ((1 + 2 / (1 - (31 / 32 : ℝ))) + 1) / |
| (FinalConstants.compositionLoss : ℝ) = |
| (FinalConstants.windowSize : ℝ) / (FinalConstants.denominator : ℝ) |
| rw [spectral_denominator_eq] |
| have hden : (FinalConstants.denominator : ℝ) = |
| 16 * (FinalConstants.alphabet : ℝ) ^ 4 * 66 * |
| (FinalConstants.compositionLoss : ℝ) := by |
| simp only [FinalConstants.denominator, Nat.cast_mul, Nat.cast_pow, Nat.cast_ofNat] |
| rw [hden] |
| simp only [div_eq_mul_inv, mul_inv_rev] |
| ring |
|
|
| theorem gain_card_eq : |
| PoweringSoundness.gain (Fintype.card Alphabet) FinalConstants.windowHalf (31 / 32) / |
| 12288 = FinalConstants.gain := by |
| rw [card_alphabet] |
| exact gain_eq |
|
|
| theorem scaled_bound_eq (ε : ℝ) : |
| (PoweringSoundness.gain (Fintype.card Alphabet) FinalConstants.windowHalf (31 / 32) * |
| min ε |
| (1 / ((2 * PoweringSoundness.center (Fintype.card Alphabet) |
| FinalConstants.windowHalf + 1 : Nat) : ℝ))) / 12288 = |
| FinalConstants.gain * min ε FinalConstants.cap := by |
| rw [walkLength_eq] |
| rw [mul_div_right_comm, gain_card_eq] |
| rfl |
|
|
| theorem composed_scaled_gap (ε : ℝ) (hε : 0 ≤ ε) : |
| min (2 * ε) FinalConstants.cap ≤ |
| (PoweringSoundness.gain (Fintype.card Alphabet) FinalConstants.windowHalf (31 / 32) * |
| min (ε / (Preprocessing.sizeFactor : ℝ)) |
| (1 / ((2 * PoweringSoundness.center (Fintype.card Alphabet) |
| FinalConstants.windowHalf + 1 : Nat) : ℝ))) / 12288 := by |
| rw [scaled_bound_eq] |
| exact FinalConstants.composed_gap ε hε |
|
|
| end UniqueGames.Foundations.PCP.PoweringFinalConstants |
|
|
| end OAI |
|
|
