lean/OAI/Computability/MaxCut/PCP/PortTables.lean

import Mathlib
import OAI.Computability.MaxCut.PCP.Generator

namespace OAI

/-!
# Executable regular constraint tables on the fixed 64-label alphabet

Every pair `(vertex,port)` has one stored reverse index and one complete
4096-bit relation. Flat row `port + ports * vertex` has its tail implicitly
fixed by that pair. Rotation uses the stored vector lookup, followed by the
inverse row-index bijection. Loops and repeated directed occurrences remain
distinct rows.

Serialization uses the existing `GraphTables` codec: header `[vertices,darts]`
with `darts=vertices*ports`, then tail/reverse/4096 bit words per row.
-/

namespace MaxCutGames.Foundations.PCP.PortTables

open PoweringWalks

abbrev Label := GraphTables.Label

def rowIndex (vertices ports : Nat) :
    Fin vertices × Fin ports ≃ Fin (vertices * ports) := finProdFinEquiv

@[simp] theorem rowIndex_val (n d : Nat) (e : Fin n × Fin d) :
    (rowIndex n d e).val = e.2.val + d * e.1.val := rfl

/-- Finite executable input data, with the two graph laws certified on lookup. -/
structure Table (vertices ports : Nat) where
  reverseIndex : Vector (Fin (vertices * ports)) (vertices * ports)
  relations : Vector GraphTables.RelationTable (vertices * ports)
  involutive : Function.Involutive
    (fun i : Fin (vertices * ports) => reverseIndex[i])
  transpose : ∀ (i : Fin (vertices * ports)) (a b : Label),
    GraphTables.relationAt relations[reverseIndex[i]] b a =
      GraphTables.relationAt relations[i] a b

variable {n d : Nat}

/-- One reverse-index lookup and the fixed mixed-radix decoding. -/
def rotation (table : Table n d) (e : Fin n × Fin d) : Fin n × Fin d :=
  (rowIndex n d).symm table.reverseIndex[rowIndex n d e]

/-- Lookup of the full stored binary predicate. -/
def accepts (table : Table n d) (e : Fin n × Fin d) (a b : Label) : Bool :=
  GraphTables.relationAt table.relations[rowIndex n d e] a b

@[simp] theorem rowIndex_rotation (table : Table n d) (e : Fin n × Fin d) :
    rowIndex n d (rotation table e) = table.reverseIndex[rowIndex n d e] := by
  exact (rowIndex n d).apply_symm_apply _

theorem rotation_involutive (table : Table n d) : Function.Involutive (rotation table) := by
  intro e
  simp only [rotation, Equiv.apply_symm_apply]
  exact (congrArg (rowIndex n d).symm (table.involutive (rowIndex n d e))).trans
    ((rowIndex n d).symm_apply_apply e)

theorem accepts_rotation (table : Table n d) (e : Fin n × Fin d) (a b : Label) :
    accepts table (rotation table e) b a = accepts table e a b := by
  simpa only [accepts, rowIndex_rotation] using table.transpose (rowIndex n d e) a b

/-- The actual reversible regular port graph represented by the vectors. -/
def portGraph (table : Table n d) : PortGraph (Fin n) (Fin d) where
  rot :=
    { toFun := rotation table
      invFun := rotation table
      left_inv := rotation_involutive table
      right_inv := rotation_involutive table }
  rot_involutive := rotation_involutive table

@[simp] theorem portGraph_rot (table : Table n d) (e : Fin n × Fin d) :
    (portGraph table).rot e = rotation table e := rfl

/-- Constraint-graph semantics, with each vertex-port pair a distinct dart. -/
def baseGraph (table : Table n d) : ConstraintGraph (Fin n) (Fin n × Fin d) Label where
  reverse := (portGraph table).rot
  reverse_involutive := rotation_involutive table
  tail := Prod.fst
  accepts := accepts table
  reverse_accepts := accepts_rotation table

@[simp] theorem baseGraph_reverse (table : Table n d) (e : Fin n × Fin d) :
    (baseGraph table).reverse e = rotation table e := rfl

@[simp] theorem baseGraph_tail (table : Table n d) (e : Fin n × Fin d) :
    (baseGraph table).tail e = e.1 := rfl

@[simp] theorem baseGraph_head (table : Table n d) (e : Fin n × Fin d) :
    (baseGraph table).head e = (rotation table e).1 := rfl

@[simp] theorem baseGraph_accepts (table : Table n d) (e : Fin n × Fin d) (a b : Label) :
    (baseGraph table).accepts e a b = accepts table e a b := rfl

/-- Materialize a reversible port graph and its predicates into the fixed rows. -/
def ofPortGraph (G : PortGraph (Fin n) (Fin d))
    (predicate : (Fin n × Fin d) → Label → Label → Bool)
    (htranspose : ∀ e a b, predicate (G.rot e) b a = predicate e a b) : Table n d where
  reverseIndex := Vector.ofFn (fun i => rowIndex n d (G.rot ((rowIndex n d).symm i)))
  relations := Vector.ofFn (fun i => GraphTables.relationOf (predicate ((rowIndex n d).symm i)))
  involutive := by
    intro i
    simp only [Fin.getElem_fin, Vector.getElem_ofFn, Fin.eta,
      Equiv.symm_apply_apply]
    exact (congrArg (rowIndex n d) (G.rot_involutive ((rowIndex n d).symm i))).trans
      ((rowIndex n d).apply_symm_apply i)
  transpose := by
    intro i a b
    simp only [Fin.getElem_fin, Vector.getElem_ofFn, Fin.eta,
      Equiv.symm_apply_apply, GraphTables.relationAt_relationOf]
    exact htranspose ((rowIndex n d).symm i) a b

@[simp] theorem rotation_ofPortGraph (G : PortGraph (Fin n) (Fin d))
    (predicate : (Fin n × Fin d) → Label → Label → Bool)
    (htranspose : ∀ e a b, predicate (G.rot e) b a = predicate e a b)
    (e : Fin n × Fin d) : rotation (ofPortGraph G predicate htranspose) e = G.rot e := by
  simp only [rotation, ofPortGraph, Fin.getElem_fin, Vector.getElem_ofFn, Fin.eta,
    Equiv.symm_apply_apply]

@[simp] theorem accepts_ofPortGraph (G : PortGraph (Fin n) (Fin d))
    (predicate : (Fin n × Fin d) → Label → Label → Bool)
    (htranspose : ∀ e a b, predicate (G.rot e) b a = predicate e a b)
    (e : Fin n × Fin d) (a b : Label) :
    accepts (ofPortGraph G predicate htranspose) e a b = predicate e a b := by
  simp only [accepts, ofPortGraph, Fin.getElem_fin, Vector.getElem_ofFn, Fin.eta,
    Equiv.symm_apply_apply, GraphTables.relationAt_relationOf]

def flatRows (table : Table n d) : GraphTables.Rows n (n * d) :=
  Vector.ofFn (fun i =>
    ⟨((rowIndex n d).symm i).1, table.reverseIndex[i], table.relations[i]⟩)

@[simp] theorem reverseAt_flatRows (table : Table n d) (i : Fin (n * d)) :
    GraphTables.reverseAt (flatRows table) i = table.reverseIndex[i] := by
  simp [GraphTables.reverseAt, flatRows]

@[simp] theorem acceptsAt_flatRows (table : Table n d) (i : Fin (n * d)) (a b : Label) :
    GraphTables.acceptsAt (flatRows table) i a b = GraphTables.relationAt table.relations[i] a b := by
  simp [GraphTables.acceptsAt, flatRows]

theorem flatRows_valid (table : Table n d) : GraphTables.Valid (flatRows table) := by
  constructor
  · intro i
    simpa only [reverseAt_flatRows] using table.involutive i
  · intro i a b
    simpa only [reverseAt_flatRows, acceptsAt_flatRows] using table.transpose i a b

def graphTable (table : Table n d) : GraphTables.Table :=
  ⟨n, n * d, flatRows table, flatRows_valid table⟩

/-- Existing graph codec, with header `[n,n*d]` and 4098 words in every row. -/
def tableWords (table : Table n d) : List Nat := GraphTables.tableWords (graphTable table)

def tableBits (table : Table n d) : List Bool := GraphTables.tableBits (graphTable table)

/-- Variable vertex count, with the port count fixed by the input family. -/
abbrev Input (ports : Nat) := (vertices : Nat) × Table vertices ports

def inputBits {ports : Nat} (input : Input ports) : List Bool := tableBits input.2

theorem tableWords_eq (table : Table n d) :
    tableWords table = [n, n * d] ++ (flatRows table).toList.flatMap GraphTables.rowWords := rfl

theorem tableWords_length (table : Table n d) :
    (tableWords table).length = 2 + 4098 * (n * d) :=
  GraphTables.tableWords_length (graphTable table)

theorem vertices_le_tableBits_length (table : Table n d) : n ≤ (tableBits table).length :=
  GraphTables.vertices_le_tableBits_length (graphTable table)

end MaxCutGames.Foundations.PCP.PortTables

section

/-!
Cloud replacement for arbitrary finite constraint graphs. The new vertices are
the old darts; internal ports enforce equality inside each cloud, and one
external port applies the original constraint. Dummy loop darts allow padding
each cloud independently before replacement.

The supplied cloud port graphs are explicit parameters; no expansion or
soundness claim is assumed.
-/

open scoped BigOperators

namespace MaxCutGames.Foundations.PCP.DegreeReplacement

open PoweringWalks

variable {V E A D : Type*}

/-- The darts departing a specified vertex. -/
abbrev Cloud (G : ConstraintGraph V E A) (v : V) := {e : E // G.tail e = v}

def cloudIndexEquiv (G : ConstraintGraph V E A) :
    (E × D) ≃ (Σ v : V, Cloud G v × D) where
  toFun p := ⟨G.tail p.1, (⟨p.1, rfl⟩, p.2)⟩
  invFun p := (p.2.1.val, p.2.2)
  left_inv _ := rfl
  right_inv := by
    rintro ⟨v, ⟨⟨e, he⟩, d⟩⟩
    cases he
    rfl

def cloudRotation (G : ConstraintGraph V E A)
    (H : ∀ v, PortGraph (Cloud G v) D) :
    (Σ v : V, Cloud G v × D) ≃ (Σ v : V, Cloud G v × D) :=
  Equiv.sigmaCongrRight (fun v => (H v).rot)

theorem cloudRotation_involutive (G : ConstraintGraph V E A)
    (H : ∀ v, PortGraph (Cloud G v) D) :
    Function.Involutive (cloudRotation G H) := by
  rintro ⟨v, p⟩
  change (⟨v, (H v).rot ((H v).rot p)⟩ : Σ v : V, Cloud G v × D) = ⟨v, p⟩
  rw [(H v).rot_involutive p]

/-- Conjugate the disjoint union of cloud rotations into global dart coordinates. -/
def innerRotation (G : ConstraintGraph V E A)
    (H : ∀ v, PortGraph (Cloud G v) D) : (E × D) ≃ (E × D) :=
  ((cloudIndexEquiv G).trans (cloudRotation G H)).trans (cloudIndexEquiv G).symm

theorem innerRotation_involutive (G : ConstraintGraph V E A)
    (H : ∀ v, PortGraph (Cloud G v) D) :
    Function.Involutive (innerRotation G H) := by
  intro p
  apply (cloudIndexEquiv G).injective
  simp only [innerRotation, Equiv.trans_apply, Equiv.apply_symm_apply]
  exact cloudRotation_involutive G H _

theorem innerRotation_tail (G : ConstraintGraph V E A)
    (H : ∀ v, PortGraph (Cloud G v) D) (e : E) (d : D) :
    G.tail (innerRotation G H (e, d)).1 = G.tail e := by
  change G.tail (((H (G.tail e)).rot (⟨e, rfl⟩, d)).1.val) = G.tail e
  exact ((H (G.tail e)).rot (⟨e, rfl⟩, d)).1.property

/-- Internal ports rotate inside a cloud; the extra port reverses an old dart. -/
def replacementStep (G : ConstraintGraph V E A)
    (H : ∀ v, PortGraph (Cloud G v) D) : E × (D ⊕ Unit) → E × (D ⊕ Unit)
  | (e, Sum.inl d) =>
      let p := innerRotation G H (e, d)
      (p.1, Sum.inl p.2)
  | (e, Sum.inr u) => (G.reverse e, Sum.inr u)

theorem replacementStep_involutive (G : ConstraintGraph V E A)
    (H : ∀ v, PortGraph (Cloud G v) D) :
    Function.Involutive (replacementStep G H) := by
  rintro ⟨e, p⟩
  rcases p with d | u
  · change ((innerRotation G H (innerRotation G H (e, d))).1,
        (Sum.inl ((innerRotation G H (innerRotation G H (e, d))).2) : D ⊕ Unit)) =
      (e, Sum.inl d)
    rw [innerRotation_involutive G H (e, d)]
  · change (G.reverse (G.reverse e), (Sum.inr u : D ⊕ Unit)) = (e, Sum.inr u)
    rw [G.reverse_involutive e]

def replacementPortGraph (G : ConstraintGraph V E A)
    (H : ∀ v, PortGraph (Cloud G v) D) : PortGraph E (D ⊕ Unit) where
  rot :=
    { toFun := replacementStep G H
      invFun := replacementStep G H
      left_inv := replacementStep_involutive G H
      right_inv := replacementStep_involutive G H }
  rot_involutive := replacementStep_involutive G H

/-- The equality constraints and inherited original constraints on the actual rotation. -/
def replacementGraph [DecidableEq A] (G : ConstraintGraph V E A)
    (H : ∀ v, PortGraph (Cloud G v) D) :
    ConstraintGraph E (E × (D ⊕ Unit)) A where
  reverse := (replacementPortGraph G H).rot
  reverse_involutive := (replacementPortGraph G H).rot_involutive
  tail := Prod.fst
  accepts p a b := match p.2 with
    | Sum.inl _ => decide (a = b)
    | Sum.inr _ => G.accepts p.1 a b
  reverse_accepts := by
    rintro ⟨e, p⟩ a b
    rcases p with d | u
    · change decide (b = a) = decide (a = b)
      simp [eq_comm]
    · exact G.reverse_accepts e a b

def liftLabel (G : ConstraintGraph V E A) (labeling : V → A) : E → A :=
  fun e => labeling (G.tail e)

@[simp] theorem replacement_satisfied_inl [DecidableEq A]
    (G : ConstraintGraph V E A) (H : ∀ v, PortGraph (Cloud G v) D)
    (labeling : V → A) (e : E) (d : D) :
    (replacementGraph G H).edgeSatisfied (liftLabel G labeling) (e, Sum.inl d) =
      true := by
  change decide (labeling (G.tail e) =
    labeling (G.tail (innerRotation G H (e, d)).1)) = true
  rw [innerRotation_tail G H e d]
  simp

@[simp] theorem replacement_satisfied_inr [DecidableEq A]
    (G : ConstraintGraph V E A) (H : ∀ v, PortGraph (Cloud G v) D)
    (labeling : V → A) (e : E) (u : Unit) :
    (replacementGraph G H).edgeSatisfied (liftLabel G labeling) (e, Sum.inr u) =
      G.edgeSatisfied labeling e := rfl

theorem replacement_complete [DecidableEq A]
    (G : ConstraintGraph V E A) (H : ∀ v, PortGraph (Cloud G v) D)
    (labeling : V → A) (h : ∀ e, G.edgeSatisfied labeling e = true) :
    ∀ p, (replacementGraph G H).edgeSatisfied (liftLabel G labeling) p = true := by
  rintro ⟨e, p⟩
  rcases p with d | u
  · exact replacement_satisfied_inl G H labeling e d
  · rw [replacement_satisfied_inr]
    exact h e

theorem replacement_satisfiable [DecidableEq A]
    (G : ConstraintGraph V E A) (H : ∀ v, PortGraph (Cloud G v) D)
    (h : G.Satisfiable) : (replacementGraph G H).Satisfiable := by
  obtain ⟨labeling, hlabeling⟩ := h
  exact ⟨liftLabel G labeling, replacement_complete G H labeling hlabeling⟩

theorem rejectionCount_eq_sum [Fintype E] (G : ConstraintGraph V E A)
    (labeling : V → A) :
    G.rejectionCount labeling =
      ∑ e, if G.edgeSatisfied labeling e = false then 1 else 0 := by
  classical
  simp only [ConstraintGraph.rejectionCount, ConstraintGraph.rejectedDarts,
    Finset.card_eq_sum_ones, Finset.sum_filter]

/-- Lifted labels reject precisely one external dart for each rejected original dart. -/
theorem replacement_rejectionCount [Fintype E] [Fintype D] [DecidableEq A]
    (G : ConstraintGraph V E A) (H : ∀ v, PortGraph (Cloud G v) D)
    (labeling : V → A) :
    (replacementGraph G H).rejectionCount (liftLabel G labeling) =
      G.rejectionCount labeling := by
  classical
  simp only [rejectionCount_eq_sum, Fintype.sum_prod_type, Fintype.sum_sum_type]
  simp
  simp only [Finset.card_filter]
  congr 1

/-- Additional darts are attached to their specified original vertex. -/
abbrev PaddedDart (_G : ConstraintGraph V E A) (dummy : V → Type*) :=
  E ⊕ (Σ v : V, dummy v)

def paddedOwner (G : ConstraintGraph V E A) (dummy : V → Type*) :
    PaddedDart G dummy → V
  | Sum.inl e => G.tail e
  | Sum.inr z => z.1

def paddedReverse (G : ConstraintGraph V E A) (dummy : V → Type*) :
    PaddedDart G dummy ≃ PaddedDart G dummy :=
  Equiv.sumCongr G.reverse (Equiv.refl (Σ v : V, dummy v))

theorem paddedReverse_involutive (G : ConstraintGraph V E A) (dummy : V → Type*) :
    Function.Involutive (paddedReverse G dummy) := by
  intro z
  cases z with
  | inl e =>
    change (Sum.inl (G.reverse (G.reverse e)) : PaddedDart G dummy) = Sum.inl e
    rw [G.reverse_involutive e]
  | inr z => rfl

/-- Dummy darts are self-reversing loops with an always-accepting constraint. -/
def paddedGraph (G : ConstraintGraph V E A) (dummy : V → Type*) :
    ConstraintGraph V (PaddedDart G dummy) A where
  reverse := paddedReverse G dummy
  reverse_involutive := paddedReverse_involutive G dummy
  tail := paddedOwner G dummy
  accepts e a b := match e with
    | Sum.inl e => G.accepts e a b
    | Sum.inr _ => true
  reverse_accepts := by
    intro e a b
    cases e with
    | inl e => exact G.reverse_accepts e a b
    | inr z => rfl

@[simp] theorem padded_satisfied_inl (G : ConstraintGraph V E A)
    (dummy : V → Type*) (labeling : V → A) (e : E) :
    (paddedGraph G dummy).edgeSatisfied labeling (Sum.inl e) =
      G.edgeSatisfied labeling e := rfl

@[simp] theorem padded_satisfied_inr (G : ConstraintGraph V E A)
    (dummy : V → Type*) (labeling : V → A) (z : Σ v : V, dummy v) :
    (paddedGraph G dummy).edgeSatisfied labeling (Sum.inr z) = true := rfl

theorem padded_complete (G : ConstraintGraph V E A) (dummy : V → Type*)
    (labeling : V → A) (h : ∀ e, G.edgeSatisfied labeling e = true) :
    ∀ c, (paddedGraph G dummy).edgeSatisfied labeling c = true := by
  intro c
  cases c with
  | inl e => exact h e
  | inr z => rfl

theorem padded_satisfiable (G : ConstraintGraph V E A) (dummy : V → Type*)
    (h : G.Satisfiable) : (paddedGraph G dummy).Satisfiable := by
  obtain ⟨labeling, hlabeling⟩ := h
  exact ⟨labeling, padded_complete G dummy labeling hlabeling⟩

theorem card_paddedDart [Fintype V] [Fintype E] (G : ConstraintGraph V E A)
    (dummy : V → Type*) [∀ v, Fintype (dummy v)] :
    Fintype.card (PaddedDart G dummy) =
      Fintype.card E + ∑ v, Fintype.card (dummy v) := by
  rw [Fintype.card_sum, Fintype.card_sigma]

theorem padded_rejectionCount [Fintype V] [Fintype E]
    (G : ConstraintGraph V E A) (dummy : V → Type*) [∀ v, Fintype (dummy v)]
    (labeling : V → A) :
    (paddedGraph G dummy).rejectionCount labeling = G.rejectionCount labeling := by
  classical
  simp only [rejectionCount_eq_sum, Fintype.sum_sum_type]
  simp
  simp only [Finset.card_filter]
  congr 1

def paddedReplacementPortGraph (G : ConstraintGraph V E A) (dummy : V → Type*)
    (H : ∀ v, PortGraph (Cloud (paddedGraph G dummy) v) D) :
    PortGraph (PaddedDart G dummy) (D ⊕ Unit) :=
  replacementPortGraph (paddedGraph G dummy) H

def paddedReplacementGraph [DecidableEq A]
    (G : ConstraintGraph V E A) (dummy : V → Type*)
    (H : ∀ v, PortGraph (Cloud (paddedGraph G dummy) v) D) :
    ConstraintGraph (PaddedDart G dummy) (PaddedDart G dummy × (D ⊕ Unit)) A :=
  replacementGraph (paddedGraph G dummy) H

theorem paddedReplacement_rejectionCount
    [Fintype V] [Fintype E] [Fintype D] [DecidableEq A]
    (G : ConstraintGraph V E A) (dummy : V → Type*) [∀ v, Fintype (dummy v)]
    (H : ∀ v, PortGraph (Cloud (paddedGraph G dummy) v) D) (labeling : V → A) :
    (paddedReplacementGraph G dummy H).rejectionCount
        (liftLabel (paddedGraph G dummy) labeling) = G.rejectionCount labeling :=
  (replacement_rejectionCount (paddedGraph G dummy) H labeling).trans
    (padded_rejectionCount G dummy labeling)

/-- Omitting all dummy darts recovers exactly the original dart set. -/
def noDummyEquiv (G : ConstraintGraph V E A) :
    PaddedDart G (fun _ : V => Empty) ≃ E where
  toFun z := match z with
    | Sum.inl e => e
    | Sum.inr z => Empty.elim z.2
  invFun := Sum.inl
  left_inv := by
    intro z
    cases z with
    | inl e => rfl
    | inr z => exact Empty.elim z.2
  right_inv _ := rfl

end MaxCutGames.Foundations.PCP.DegreeReplacement

end

/-! A concrete constant-query proximity tester for a finite relation.

The proof oracle is a Boolean function on the Boolean functions on the finite
set of legal assignments. The tests are BLR linearity, self-corrected AND,
self-corrected unit, and comparison with a queried input coordinate.
The oracle's size depends exponentially on the number of legal assignments;
this is a fixed-relation construction, not a polynomial-size tester for an
arbitrary input formula. All tests below are explicit finite expectations.
-/

noncomputable section

namespace MaxCutGames.Foundations.PCP.AssignmentTester

open MaxCutGames.Foundations.Hastad
open scoped BigOperators
open Finset

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

def truth (b : Bool) : ℝ := if b then 1 else 0

def disagree (a b : Bool) : ℝ := if a = b then 0 else 1

theorem truth_nonneg (b : Bool) : 0 ≤ truth b := by cases b <;> norm_num [truth]

theorem disagree_nonneg (a b : Bool) : 0 ≤ disagree a b := by
  cases a <;> cases b <;> norm_num [disagree]

theorem disagree_eq (a b : Bool) :
    disagree a b = (1 - bitSign a * bitSign b) / 2 := by
  cases a <;> cases b <;> norm_num [disagree, bitSign]

theorem disagree_xor_le (a b c d : Bool) :
    disagree (a ^^ b) (c ^^ d) ≤ disagree a c + disagree b d := by
  cases a <;> cases b <;> cases c <;> cases d <;> norm_num [disagree]

def parity (s f : Cube A) : Bool :=
  decide (Odd ((Finset.univ.filter fun a => s a && f a = true).card))

theorem bitSign_parity (s f : Cube A) : bitSign (parity s f) = walsh s f := by
  have hw : walsh s f = (-1 : ℝ) ^
      (Finset.univ.filter fun a => s a && f a = true).card := by
    simp [walsh, bitSign, Finset.prod_ite]
  rw [hw]
  simpa [parity, bitSign] using
    (sign_power_parity true (Finset.univ.filter fun a => s a && f a = true).card).symm

theorem bitSign_injective : Function.Injective bitSign := by
  intro a b h
  cases a <;> cases b <;> simp_all [bitSign] <;> linarith

theorem parity_xor (s f g : Cube A) :
    parity s (cubeXor f g) = (parity s f ^^ parity s g) := by
  apply bitSign_injective
  rw [bitSign_parity, bitSign_xor, bitSign_parity, bitSign_parity, walsh_xor]

omit [Fintype A] [DecidableEq A] in
theorem cubeXor_comm (f g : Cube A) : cubeXor f g = cubeXor g f := by
  funext a
  simp [cubeXor, Bool.xor_comm]

omit [Fintype A] [DecidableEq A] in
@[simp] theorem cubeXor_right_twice (f g : Cube A) :
    cubeXor (cubeXor f g) g = f := by
  funext a
  change ((f a ^^ g a) ^^ g a) = f a
  cases f a <;> cases g a <;> rfl

theorem expect_cubeXor (F : Cube A → ℝ) (g : Cube A) :
    (𝔼 f, F (cubeXor f g)) = 𝔼 f, F f := by
  rw [Fintype.expect_eq_sum_div_card, Fintype.expect_eq_sum_div_card]
  congr 1
  refine Finset.sum_bij (fun f _ => cubeXor f g) ?_ ?_ ?_ ?_
  · intro f _
    exact Finset.mem_univ _
  · intro f _ h _ heq
    have hh := congrArg (fun x => cubeXor x g) heq
    simpa only [cubeXor_right_twice] using hh
  · intro f _
    exact ⟨cubeXor f g, Finset.mem_univ _, cubeXor_right_twice f g⟩
  · intro f _
    rfl

def distance {K : Type*} [Fintype K] (f g : K → Bool) : ℝ :=
  𝔼 k, disagree (f k) (g k)

theorem distance_nonneg {K : Type*} [Fintype K] (f g : K → Bool) :
    0 ≤ distance f g := by
  apply Finset.expect_nonneg
  intro k _
  exact disagree_nonneg _ _

theorem distance_parity (F : Cube A → Bool) (s : Cube A) :
    distance F (parity s) = (1 - coefficient (fun f => bitSign (F f)) s) / 2 := by
  unfold distance
  simp_rw [disagree_eq, bitSign_parity, div_eq_mul_inv,
    ← Finset.expect_mul, Finset.expect_sub_distrib]
  simp only [Fintype.expect_const]
  rfl

def blrReject (F : Cube A → Bool) : ℝ :=
  𝔼 f, 𝔼 g, disagree (F f ^^ F g) (F (cubeXor f g))

theorem blrReject_nonneg (F : Cube A → Bool) : 0 ≤ blrReject F := by
  apply Finset.expect_nonneg
  intro f _
  apply Finset.expect_nonneg
  intro g _
  exact disagree_nonneg _ _

theorem blr_correlation (F : Cube A → Bool) :
    (𝔼 f, 𝔼 g, bitSign (F f) * bitSign (F g) * bitSign (F (cubeXor f g))) =
      ∑ s, coefficient (fun f => bitSign (F f)) s ^ 3 := by
  simp_rw [mul_assoc, ← Finset.mul_expect]
  conv_lhs =>
    enter [2, f, 2, 2, g, 2, 1]
    rw [cubeXor_comm]
  simp_rw [walsh_autocorrelation, Finset.mul_sum]
  rw [Finset.expect_sum_comm]
  apply Finset.sum_congr rfl
  intro s _
  have heq (f : Cube A) : bitSign (F f) *
      (coefficient (fun f => bitSign (F f)) s ^ 2 * walsh s f) =
    coefficient (fun f => bitSign (F f)) s ^ 2 * (bitSign (F f) * walsh s f) := by ring
  simp_rw [heq, ← Finset.mul_expect]
  change coefficient (fun f => bitSign (F f)) s ^ 2 *
    coefficient (fun f => bitSign (F f)) s = _
  ring

theorem blrReject_fourier (F : Cube A → Bool) :
    blrReject F = (1 - ∑ s, coefficient (fun f => bitSign (F f)) s ^ 3) / 2 := by
  unfold blrReject
  simp_rw [disagree_eq, bitSign_xor, div_eq_mul_inv,
    ← Finset.expect_mul, Finset.expect_sub_distrib]
  simp only [Fintype.expect_const]
  rw [blr_correlation]

/-- The actual BLR test yields a nearby parity, with no stability theorem
or Fourier-structure hypothesis assumed. -/
theorem blr_near_parity (F : Cube A → Bool) :
    ∃ s : Cube A, distance F (parity s) ≤ blrReject F := by
  classical
  let c := coefficient (fun f => bitSign (F f))
  have hmass : (∑ s, c s ^ 2) = 1 := sign_parseval F
  have hlarge : ∃ s, (∑ t, c t ^ 3) ≤ c s := by
    by_contra hn
    push Not at hn
    have hpos : ∃ s, 0 < c s ^ 2 := by
      by_contra hh
      push Not at hh
      have hz := Finset.sum_nonpos (s := Finset.univ) (fun s _ => hh s)
      linarith
    obtain ⟨s₀, hs₀⟩ := hpos
    have hle (s : Cube A) (_ : s ∈ Finset.univ) :
        c s ^ 3 ≤ (∑ t, c t ^ 3) * c s ^ 2 := by
      have h := mul_le_mul_of_nonneg_right (hn s).le (sq_nonneg (c s))
      nlinarith
    have hlt : c s₀ ^ 3 < (∑ t, c t ^ 3) * c s₀ ^ 2 := by
      have h := mul_lt_mul_of_pos_right (hn s₀) hs₀
      nlinarith
    have hsum := Finset.sum_lt_sum hle ⟨s₀, Finset.mem_univ _, hlt⟩
    rw [← Finset.mul_sum, hmass, mul_one] at hsum
    exact (lt_irrefl _ hsum)
  obtain ⟨s, hs⟩ := hlarge
  refine ⟨s, ?_⟩
  rw [distance_parity, blrReject_fourier]
  dsimp [c] at hs
  linarith

def selfCorrect (F : Cube A → Bool) (x r : Cube A) : Bool :=
  F r ^^ F (cubeXor r x)

theorem parity_selfCorrect (s x r : Cube A) :
    (parity s r ^^ parity s (cubeXor r x)) = parity s x := by
  rw [parity_xor]
  cases parity s r <;> cases parity s x <;> rfl

/-- The two actual oracle queries recover the parity at every requested x. -/
theorem selfCorrect_error (F : Cube A → Bool) (s x : Cube A) :
    (𝔼 r, disagree (selfCorrect F x r) (parity s x)) ≤ 2 * distance F (parity s) := by
  have hle (r : Cube A) :
      disagree (selfCorrect F x r) (parity s x) ≤
      disagree (F r) (parity s r) +
        disagree (F (cubeXor r x)) (parity s (cubeXor r x)) := by
    rw [← parity_selfCorrect s x r]
    exact disagree_xor_le _ _ _ _
  have h := Finset.expect_le_expect (s := Finset.univ) (fun r _ => hle r)
  rw [Finset.expect_add_distrib,
    expect_cubeXor (fun r => disagree (F r) (parity s r)) x] at h
  change _ ≤ distance F (parity s) + distance F (parity s) at h
  linarith

def singleMask (i : A) : Cube A := fun j => decide (j = i)

def flipAt (i : A) (f : Cube A) : Cube A :=
  fun j => if j = i then !(f j) else f j

omit [Fintype A] in
@[simp] theorem flipAt_twice (i : A) (f : Cube A) : flipAt i (flipAt i f) = f := by
  funext j
  by_cases h : j = i <;> simp [flipAt, h]

omit [Fintype A] in
theorem flipAt_eq_xor (i : A) (f : Cube A) : flipAt i f = cubeXor f (singleMask i) := by
  funext j
  by_cases h : j = i <;> simp [flipAt, cubeXor, singleMask, h]

theorem expect_flipAt (F : Cube A → ℝ) (i : A) :
    (𝔼 f, F (flipAt i f)) = 𝔼 f, F f := by
  simp_rw [flipAt_eq_xor]
  exact expect_cubeXor F _

theorem walsh_singleMask (s : Cube A) (i : A) :
    walsh s (singleMask i) = bitSign (s i) := by
  unfold walsh
  rw [Finset.prod_eq_single i]
  · simp [singleMask]
  · intro j _ hj
    simp [singleMask, hj, bitSign]
  · intro h
    exact False.elim (h (Finset.mem_univ _))

theorem parity_singleMask (s : Cube A) (i : A) : parity s (singleMask i) = s i := by
  apply bitSign_injective
  rw [bitSign_parity, walsh_singleMask]

theorem parity_flipAt (s : Cube A) (i : A) (f : Cube A) (hs : s i = true) :
    parity s (flipAt i f) = !(parity s f) := by
  rw [flipAt_eq_xor, parity_xor, parity_singleMask, hs]
  cases parity s f <;> rfl

def cubeAnd (f g : Cube A) : Cube A := fun a => f a && g a

def andLawFailure (s f g : Cube A) : Bool :=
  parity s (cubeAnd f g) ^^ (parity s f && parity s g)

theorem andLawFailure_flipAt_second (s : Cube A) (i : A) (f g : Cube A)
    (hi : s i = true) :
    andLawFailure s f (flipAt i g) =
      (andLawFailure s f g ^^ (parity s f ^^ f i)) := by
  have hAnd : cubeAnd f (flipAt i g) =
      (if f i then flipAt i (cubeAnd f g) else cubeAnd f g) := by
    funext a
    by_cases ha : a = i
    · subst a
      cases hfi : f i <;> simp [cubeAnd, flipAt, hfi]
    · cases hfi : f i <;> simp [cubeAnd, flipAt, ha]
  have hg := parity_flipAt s i g hi
  have hfg := parity_flipAt s i (cubeAnd f g) hi
  cases hfi : f i <;> cases hf : parity s f <;>
    cases hg₀ : parity s g <;> cases ha : parity s (cubeAnd f g) <;>
    simp [andLawFailure, hAnd, hg, hfg, hfi, hf, hg₀, ha]

private theorem four_truth_lower_inline_AssignmentTester (e₀ e₁ d : Bool) :
    (1 : ℝ) ≤ truth e₀ + truth (e₀ ^^ d) + truth e₁ + truth (e₁ ^^ (!d)) := by
  cases e₀ <;> cases e₁ <;> cases d <;> norm_num [truth]

/-- Every parity involving two or more distinct coordinates violates Boolean
AND with probability at least one quarter under two independent queries. -/
theorem parity_and_rejection (s : Cube A) (i j : A)
    (hi : s i = true) (hj : s j = true) (hij : i ≠ j) :
    (1 : ℝ) / 4 ≤ 𝔼 f, 𝔼 g, truth (andLawFailure s f g) := by
  have hpoint (f g : Cube A) : (1 : ℝ) ≤
      truth (andLawFailure s f g) + truth (andLawFailure s f (flipAt i g)) +
      truth (andLawFailure s (flipAt j f) g) +
      truth (andLawFailure s (flipAt j f) (flipAt i g)) := by
    rw [andLawFailure_flipAt_second s i f g hi,
      andLawFailure_flipAt_second s i (flipAt j f) g hi]
    have hD : (parity s (flipAt j f) ^^ (flipAt j f) i) =
        !(parity s f ^^ f i) := by
      rw [parity_flipAt s j f hj]
      simp only [flipAt, ite_eq_right hij]
      cases hf : parity s f <;> cases hfi : f i <;> simp []
    rw [hD]
    exact four_truth_lower_inline_AssignmentTester _ _ _
  have havg := Finset.expect_le_expect (s := Finset.univ) (fun f _ =>
    Finset.expect_le_expect (s := Finset.univ) (fun g _ => hpoint f g))
  simp only [Finset.expect_add_distrib, Fintype.expect_const] at havg
  have hinner (f : Cube A) :
      (𝔼 g, truth (andLawFailure s f (flipAt i g))) =
        𝔼 g, truth (andLawFailure s f g) :=
    expect_flipAt (fun g => truth (andLawFailure s f g)) i
  have houter : (𝔼 f, 𝔼 g, truth (andLawFailure s (flipAt j f) g)) =
      𝔼 f, 𝔼 g, truth (andLawFailure s f g) :=
    expect_flipAt (fun f => 𝔼 g, truth (andLawFailure s f g)) j
  simp only [hinner, houter] at havg
  linarith

theorem truth_xor (a b : Bool) : truth (a ^^ b) = disagree a b := by
  cases a <;> cases b <;> norm_num [truth, disagree]

theorem disagree_triangle (a b c : Bool) :
    disagree a c ≤ disagree a b + disagree b c := by
  cases a <;> cases b <;> cases c <;> norm_num [disagree]

theorem and_disagree_transfer (p₀ p₁ p₂ q₀ q₁ q₂ : Bool) :
    disagree p₀ (p₁ && p₂) ≤ disagree q₀ (q₁ && q₂) +
      disagree q₀ p₀ + disagree q₁ p₁ + disagree q₂ p₂ := by
  cases p₀ <;> cases p₁ <;> cases p₂ <;>
    cases q₀ <;> cases q₁ <;> cases q₂ <;> norm_num [disagree]

/-- Six oracle queries, using three independent self-correction masks. -/
def andReject (F : Cube A → Bool) : ℝ :=
  𝔼 f, 𝔼 g, 𝔼 r₀, 𝔼 r₁, 𝔼 r₂,
    disagree (selfCorrect F (cubeAnd f g) r₀)
      (selfCorrect F f r₁ && selfCorrect F g r₂)

def unitReject (F : Cube A → Bool) : ℝ :=
  𝔼 r, disagree (selfCorrect F (fun _ => true) r) true

variable {K : Type*} [Fintype K] [Nonempty K]

/-- One original input query and two proof-oracle queries. -/
def inputReject (label : A → K → Bool) (σ : K → Bool) (F : Cube A → Bool) : ℝ :=
  𝔼 k, 𝔼 r, disagree (σ k) (selfCorrect F (fun a => label a k) r)

def testerReject (label : A → K → Bool) (σ : K → Bool) (F : Cube A → Bool) : ℝ :=
  (blrReject F + andReject F + unitReject F + inputReject label σ F) / 4

theorem andReject_nonneg (F : Cube A → Bool) : 0 ≤ andReject F := by
  apply Finset.expect_nonneg
  intro f _
  apply Finset.expect_nonneg
  intro g _
  apply Finset.expect_nonneg
  intro r₀ _
  apply Finset.expect_nonneg
  intro r₁ _
  apply Finset.expect_nonneg
  intro r₂ _
  exact disagree_nonneg _ _

theorem unitReject_nonneg (F : Cube A → Bool) : 0 ≤ unitReject F := by
  apply Finset.expect_nonneg
  intro r _
  exact disagree_nonneg _ _

omit [Nonempty K] in
theorem inputReject_nonneg (label : A → K → Bool) (σ : K → Bool) (F : Cube A → Bool) :
    0 ≤ inputReject label σ F := by
  apply Finset.expect_nonneg
  intro k _
  apply Finset.expect_nonneg
  intro r _
  exact disagree_nonneg _ _

theorem andReject_transfer (F : Cube A → Bool) (s : Cube A) :
    (𝔼 f, 𝔼 g, truth (andLawFailure s f g)) ≤
      andReject F + 6 * distance F (parity s) := by
  have hlocal (f g : Cube A) : truth (andLawFailure s f g) ≤
      (𝔼 r₀, 𝔼 r₁, 𝔼 r₂,
        disagree (selfCorrect F (cubeAnd f g) r₀)
          (selfCorrect F f r₁ && selfCorrect F g r₂)) +
      6 * distance F (parity s) := by
    have hp (r₀ r₁ r₂ : Cube A) := and_disagree_transfer
      (parity s (cubeAnd f g)) (parity s f) (parity s g)
      (selfCorrect F (cubeAnd f g) r₀) (selfCorrect F f r₁) (selfCorrect F g r₂)
    have havg := Finset.expect_le_expect (s := Finset.univ) (fun r₀ _ =>
      Finset.expect_le_expect (s := Finset.univ) (fun r₁ _ =>
        Finset.expect_le_expect (s := Finset.univ) (fun r₂ _ => hp r₀ r₁ r₂)))
    simp only [Finset.expect_add_distrib, Fintype.expect_const] at havg
    have h₀ := selfCorrect_error F s (cubeAnd f g)
    have h₁ := selfCorrect_error F s f
    have h₂ := selfCorrect_error F s g
    rw [andLawFailure, truth_xor]
    linarith
  have h := Finset.expect_le_expect (s := Finset.univ) (fun f _ =>
    Finset.expect_le_expect (s := Finset.univ) (fun g _ => hlocal f g))
  simpa only [Finset.expect_add_distrib, Fintype.expect_const, andReject] using h

theorem parity_zero (f : Cube A) : parity (fun _ => false) f = false := by
  apply bitSign_injective
  rw [bitSign_parity]
  simp [walsh, bitSign]

theorem unitReject_zero_parity (F : Cube A → Bool) :
    1 - 2 * distance F (parity (fun _ => false)) ≤ unitReject F := by
  have herr := selfCorrect_error F (fun _ => false) (fun _ => true)
  simp only [parity_zero] at herr
  have heq : unitReject F +
      (𝔼 r, disagree (selfCorrect F (fun _ => true) r) false) = 1 := by
    unfold unitReject
    rw [← Finset.expect_add_distrib]
    have hp (b : Bool) : disagree b true + disagree b false = 1 := by
      cases b <;> norm_num [disagree]
    simp only [hp, Fintype.expect_const]
  linarith

theorem parity_singleMask_eval (a : A) (f : Cube A) :
    parity (singleMask a) f = f a := by
  apply bitSign_injective
  rw [bitSign_parity, walsh_symm, walsh_singleMask]

theorem inputReject_transfer (label : A → K → Bool) (σ : K → Bool)
    (F : Cube A → Bool) (a : A) :
    distance σ (label a) ≤ inputReject label σ F +
      2 * distance F (parity (singleMask a)) := by
  have hlocal (k : K) : disagree (σ k) (label a k) ≤
      (𝔼 r, disagree (σ k) (selfCorrect F (fun a => label a k) r)) +
        2 * distance F (parity (singleMask a)) := by
    have hp (r : Cube A) := disagree_triangle (σ k)
      (selfCorrect F (fun a => label a k) r) (label a k)
    have havg := Finset.expect_le_expect (s := Finset.univ) (fun r _ => hp r)
    simp only [Fintype.expect_const, Finset.expect_add_distrib] at havg
    have herr := selfCorrect_error F (singleMask a) (fun a => label a k)
    simp only [parity_singleMask_eval] at herr
    linarith
  have h := Finset.expect_le_expect (s := Finset.univ) (fun k _ => hlocal k)
  simpa only [Finset.expect_add_distrib, Fintype.expect_const, inputReject, distance] using h

/-- Proximity soundness of the actual four finite tests. The only promise
is the property being tested: the queried input is δ-far from every legal label. -/
theorem tester_proximity_soundness (label : A → K → Bool) (σ : K → Bool)
    (F : Cube A → Bool) {δ : ℝ} (hδ : 0 < δ) (hδ' : δ ≤ 1 / 4)
    (hfar : ∀ a, δ ≤ distance σ (label a)) :
    δ / 256 ≤ testerReject label σ F := by
  have hblr := blrReject_nonneg F
  have hand := andReject_nonneg F
  have hunit := unitReject_nonneg F
  have hin := inputReject_nonneg label σ F
  by_cases hb : δ / 64 ≤ blrReject F
  · unfold testerReject
    linarith
  obtain ⟨s, hs⟩ := blr_near_parity F
  by_cases hz : s = fun _ => false
  · subst s
    have hu := unitReject_zero_parity F
    unfold testerReject
    linarith
  have hsome : ∃ i, s i = true := by
    by_contra hh
    push Not at hh
    apply hz
    funext i
    have h := hh i
    cases hi : s i <;> simp_all
  obtain ⟨i, hi⟩ := hsome
  by_cases htwo : ∃ j, i ≠ j ∧ s j = true
  · obtain ⟨j, hij, hj⟩ := htwo
    have ha := parity_and_rejection s i j hi hj hij
    have ht := andReject_transfer F s
    unfold testerReject
    linarith
  · have heq : s = singleMask i := by
      funext j
      by_cases hji : j = i
      · subst j
        simp [singleMask, hi]
      · have hj : s j = false := by
          cases hv : s j
          · rfl
          · exact False.elim (htwo ⟨j, Ne.symm hji, hv⟩)
        simp [singleMask, hji, hj]
    subst s
    have ht := inputReject_transfer label σ F i
    have hf := hfar i
    unfold testerReject
    linarith

omit [Fintype A] [DecidableEq A] in
theorem selfCorrect_dictator (a : A) (x r : Cube A) :
    selfCorrect (fun f => f a) x r = x a := by
  change (r a ^^ (r a ^^ x a)) = x a
  cases r a <;> cases x a <;> rfl

omit [Nonempty K] in
/-- Every legal assignment has an honest proof accepted on every test. -/
theorem tester_perfect_completeness (label : A → K → Bool) (a : A) :
    testerReject label (label a) (fun f => f a) = 0 := by
  have hb : blrReject (fun f : Cube A => f a) = 0 := by
    simp [blrReject, cubeXor, disagree]
  have ha : andReject (fun f : Cube A => f a) = 0 := by
    simp [andReject, selfCorrect_dictator, cubeAnd, disagree]
  have hu : unitReject (fun f : Cube A => f a) = 0 := by
    simp [unitReject, selfCorrect_dictator, disagree]
  have hi : inputReject label (label a) (fun f => f a) = 0 := by
    simp [inputReject, selfCorrect_dictator, disagree]
  simp [testerReject, hb, ha, hu, hi]

/-- The actual six oracle addresses for each test. Unused slots repeat an
address already queried, so every event has the same fixed six-slot format. -/
def eventQueries (label : A → K → Bool) (kind : Fin 4) (k : K)
    (f g r₀ r₁ r₂ : Cube A) (slot : Fin 6) : K ⊕ Cube A :=
  if kind = 0 then
    if slot = 0 then .inr f else if slot = 1 then .inr g else .inr (cubeXor f g)
  else if kind = 1 then
    if slot = 0 then .inr r₀ else
    if slot = 1 then .inr (cubeXor r₀ (cubeAnd f g)) else
    if slot = 2 then .inr r₁ else
    if slot = 3 then .inr (cubeXor r₁ f) else
    if slot = 4 then .inr r₂ else .inr (cubeXor r₂ g)
  else if kind = 2 then
    if slot = 0 then .inr r₀ else .inr (cubeXor r₀ (fun _ => true))
  else
    if slot = 0 then .inl k else
    if slot = 1 then .inr r₀ else .inr (cubeXor r₀ (fun a => label a k))

def eventAccepts (kind : Fin 4) (b : Fin 6 → Bool) : Bool :=
  if kind = 0 then decide ((b 0 ^^ b 1) = b 2)
  else if kind = 1 then decide ((b 0 ^^ b 1) = ((b 2 ^^ b 3) && (b 4 ^^ b 5)))
  else if kind = 2 then (b 0 ^^ b 1)
  else decide (b 0 = (b 1 ^^ b 2))

def oracleAnswer (σ : K → Bool) (F : Cube A → Bool) : K ⊕ Cube A → Bool :=
  Sum.elim σ F

def eventReject (label : A → K → Bool) (σ : K → Bool) (F : Cube A → Bool)
    (kind : Fin 4) (k : K) (f g r₀ r₁ r₂ : Cube A) : ℝ :=
  truth (!(eventAccepts kind (fun slot =>
    oracleAnswer σ F (eventQueries label kind k f g r₀ r₁ r₂ slot))))

theorem truth_not_decide_eq (a b : Bool) : truth (!(decide (a = b))) = disagree a b := by
  cases a <;> cases b <;> norm_num [truth, disagree]

omit [Fintype A] [DecidableEq A] [Fintype K] [Nonempty K] in
theorem eventReject_blr (label : A → K → Bool) (σ : K → Bool) (F : Cube A → Bool)
    (k : K) (f g r₀ r₁ r₂ : Cube A) :
    eventReject label σ F 0 k f g r₀ r₁ r₂ =
      disagree (F f ^^ F g) (F (cubeXor f g)) := by
  simp [eventReject, eventAccepts, eventQueries, oracleAnswer, truth_not_decide_eq]

omit [Fintype A] [DecidableEq A] [Fintype K] [Nonempty K] in
theorem eventReject_and (label : A → K → Bool) (σ : K → Bool) (F : Cube A → Bool)
    (k : K) (f g r₀ r₁ r₂ : Cube A) :
    eventReject label σ F 1 k f g r₀ r₁ r₂ =
      disagree (selfCorrect F (cubeAnd f g) r₀)
        (selfCorrect F f r₁ && selfCorrect F g r₂) := by
  simp [eventReject, eventAccepts, eventQueries, oracleAnswer,
    truth_not_decide_eq, selfCorrect]

omit [Fintype A] [DecidableEq A] [Fintype K] [Nonempty K] in
theorem eventReject_unit (label : A → K → Bool) (σ : K → Bool) (F : Cube A → Bool)
    (k : K) (f g r₀ r₁ r₂ : Cube A) :
    eventReject label σ F 2 k f g r₀ r₁ r₂ =
      disagree (selfCorrect F (fun _ => true) r₀) true := by
  simp only [eventReject, eventAccepts, eventQueries, oracleAnswer]
  norm_num
  change truth (!(selfCorrect F (fun _ => true) r₀)) =
    disagree (selfCorrect F (fun _ => true) r₀) true
  cases selfCorrect F (fun _ => true) r₀ <;> norm_num [truth, disagree]

omit [Fintype A] [DecidableEq A] [Fintype K] [Nonempty K] in
theorem eventReject_input (label : A → K → Bool) (σ : K → Bool) (F : Cube A → Bool)
    (k : K) (f g r₀ r₁ r₂ : Cube A) :
    eventReject label σ F 3 k f g r₀ r₁ r₂ =
      disagree (σ k) (selfCorrect F (fun a => label a k) r₀) := by
  simp [eventReject, eventAccepts, eventQueries, oracleAnswer,
    truth_not_decide_eq, selfCorrect]

omit [Fintype A] [DecidableEq A] [Fintype K] [Nonempty K] in
/-- The honest proof satisfies each explicit event separately, including all
padding slots, so it supplies actual satisfying labels in the incidence graph. -/
theorem event_perfect_completeness (label : A → K → Bool) (a : A)
    (kind : Fin 4) (k : K) (f g r₀ r₁ r₂ : Cube A) :
    eventAccepts kind (fun slot => oracleAnswer (label a) (fun x => x a)
      (eventQueries label kind k f g r₀ r₁ r₂ slot)) = true := by
  have hkind : kind = 0 ∨ kind = 1 ∨ kind = 2 ∨ kind = 3 := by omega
  have hreject : eventReject label (label a) (fun x => x a) kind k f g r₀ r₁ r₂ = 0 := by
    rcases hkind with h | h | h | h <;> subst kind
    · simp [eventReject_blr, cubeXor, disagree]
    · simp [eventReject_and, selfCorrect_dictator, cubeAnd, disagree]
    · simp [eventReject_unit, selfCorrect_dictator, disagree]
    · simp [eventReject_input, selfCorrect_dictator, disagree]
  unfold eventReject at hreject
  cases h : eventAccepts kind (fun slot => oracleAnswer (label a) (fun x => x a)
    (eventQueries label kind k f g r₀ r₁ r₂ slot)) <;> simp_all [truth]

/-- One finite uniform sample space for the executable six-address test. -/
def explicitTesterReject (label : A → K → Bool) (σ : K → Bool) (F : Cube A → Bool) : ℝ :=
  𝔼 kind : Fin 4, 𝔼 k, 𝔼 f, 𝔼 g, 𝔼 r₀, 𝔼 r₁, 𝔼 r₂,
    eventReject label σ F kind k f g r₀ r₁ r₂

theorem explicitTesterReject_eq (label : A → K → Bool) (σ : K → Bool)
    (F : Cube A → Bool) : explicitTesterReject label σ F = testerReject label σ F := by
  unfold explicitTesterReject
  rw [Fintype.expect_eq_sum_div_card]
  simp [Fin.sum_univ_succ, eventReject_blr, eventReject_and, eventReject_unit,
    eventReject_input, testerReject, blrReject, andReject, unitReject, inputReject, add_assoc]

theorem explicit_proximity_soundness (label : A → K → Bool) (σ : K → Bool)
    (F : Cube A → Bool) {δ : ℝ} (hδ : 0 < δ) (hδ' : δ ≤ 1 / 4)
    (hfar : ∀ a, δ ≤ distance σ (label a)) :
    δ / 256 ≤ explicitTesterReject label σ F := by
  rw [explicitTesterReject_eq]
  exact tester_proximity_soundness label σ F hδ hδ' hfar

end MaxCutGames.Foundations.PCP.AssignmentTester
end

end OAI