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