import Mathlib
import OAI.Computability.VertexCover.PCP.AssignmentTester
import OAI.Computability.VertexCover.PCP.CodeComposition

namespace OAI

                                                                                          

noncomputable section

namespace UniqueGames.Foundations.PCP.AlphabetReduction

open UniqueGames.Foundations.Hastad
open CodeComposition

variable {A : Type*} [Fintype A] [DecidableEq A] [Nonempty A]

abbrev LegalPair (accepts : A → A → Bool) :=
  {p : A × A // accepts p.1 p.2 = true}

abbrev InputCoordinate (A : Type*) := Bool × Cube A

def pairEncoding (accepts : A → A → Bool) (p : LegalPair accepts) :
    InputCoordinate A → Bool :=
  pairWord (codeword p.1.1) (codeword p.1.2)

def edgeTesterReject (accepts : A → A → Bool) (left right : Cube A → Bool)
    (F : Cube (LegalPair accepts) → Bool) : ℝ :=
  AssignmentTester.explicitTesterReject (pairEncoding accepts) (pairWord left right) F

theorem rejected_edge_tester_bound (accepts : A → A → Bool)
    (left right : Cube A → Bool)
    (F : Cube (LegalPair accepts) → Bool)
    (hbad : accepts (nearest left) (nearest right) = false) :
    1 / 2048 ≤ edgeTesterReject accepts left right F := by
  have hfar (p : LegalPair accepts) :
      (1 / 8 : ℝ) ≤ AssignmentTester.distance (pairWord left right) (pairEncoding accepts p) := by
    have h := rejected_pair_far accepts left right hbad p.1.1 p.1.2 p.2
    convert h using 1
    rfl
  have h := AssignmentTester.explicit_proximity_soundness (pairEncoding accepts)
    (pairWord left right) F (δ := 1 / 8) (by norm_num) (by norm_num) hfar
  norm_num at h
  exact h

omit [Nonempty A] in
theorem honest_edge_tester (accepts : A → A → Bool) (a b : A)
    (hab : accepts a b = true) :
    edgeTesterReject accepts (codeword a) (codeword b)
      (fun f => f ⟨(a, b), hab⟩) = 0 := by
  unfold edgeTesterReject
  rw [AssignmentTester.explicitTesterReject_eq]
  exact AssignmentTester.tester_perfect_completeness
    (pairEncoding accepts) ⟨(a, b), hab⟩

end UniqueGames.Foundations.PCP.AlphabetReduction
end

end OAI