import Mathlib
import OAI.Computability.MaxCut.PCP.PoweringTableSemantics

namespace OAI

/-!
The explicit assignment tester applied to the encoded satisfying pairs of an
actual finite edge predicate.  The resulting soundness loss is independent of
the alphabet size.  Uniform padding of the finite sample space and assembly of
the global incidence graph are separate constructions.
-/

noncomputable section

namespace MaxCutGames.Foundations.PCP.AlphabetReduction

open MaxCutGames.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 MaxCutGames.Foundations.PCP.AlphabetReduction
end

end OAI