Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Computability.DirectedFeedback.Games.AlphabetReduction

namespace OAI

namespace DFVSGames.Foundations.PCP.VerifierToCNF

open Target

abbrev BoolTriple := Bool × Bool × Bool

def tripleValue (t : BoolTriple) : Bool := (t.1 || t.2.1) || t.2.2

def chainView : List Bool → List Bool → List BoolTriple
  | [a, b, c], [] => [(a, b, c)]
  | a :: b :: rest, y :: ys => (a, b, y) :: chainView ((!y) :: rest) ys
  | _, _ => []

def fillBits (bits : List Bool) : List Bool :=
  match bits with
  | a :: b :: c :: d :: rest =>
      (!(a || b)) :: fillBits ((a || b) :: c :: d :: rest)
  | _ => []
termination_by bits.length
decreasing_by simp_wf

theorem chainView_length (bits ys : List Bool)
    (size : bits.length = ys.length + 3) :
    (chainView bits ys).length = ys.length + 1 := by
  induction ys generalizing bits with
  | nil =>
    match bits with
    | [a, b, c] => rfl
    | [] => simp at size
    | [_] => simp at size
    | [_, _] => simp at size
    | _ :: _ :: _ :: _ :: rest =>
      simp only [List.length_cons, List.length_nil] at size
      omega
  | cons y ys ih =>
    match bits with
    | [] => simp at size
    | [_] => simp only [List.length_cons, List.length_nil] at size; omega
    | a :: b :: rest =>
      have smaller : ((!y) :: rest).length = ys.length + 3 := by
        simp only [List.length_cons] at *
        omega
      have h := ih ((!y) :: rest) smaller
      simpa only [chainView, List.length_cons] using congrArg (fun n => n + 1) h

theorem chainView_sound (bits ys : List Bool)
    (size : bits.length = ys.length + 3)
    (accepted : (chainView bits ys).all tripleValue = true) : bits.any id = true := by
  induction ys generalizing bits with
  | nil =>
    match bits with
    | [a, b, c] => simpa [chainView, tripleValue, Bool.or_assoc] using accepted
    | [] => simp at size
    | [_] => simp at size
    | [_, _] => simp at size
    | _ :: _ :: _ :: _ :: rest =>
      simp only [List.length_cons, List.length_nil] at size
      omega
  | cons y ys ih =>
    match bits with
    | [] => simp at size
    | [_] => simp only [List.length_cons, List.length_nil] at size; omega
    | a :: b :: rest =>
      have smaller : ((!y) :: rest).length = ys.length + 3 := by
        simp only [List.length_cons] at *
        omega
      have accepted' : tripleValue (a, b, y) = true ∧
          (chainView ((!y) :: rest) ys).all tripleValue = true := by
        simpa only [chainView, List.all_cons, Bool.and_eq_true] using accepted
      have tailAccepted := ih ((!y) :: rest) smaller accepted'.2
      cases a <;> cases b <;> cases y <;> simp_all [tripleValue]

theorem fillBits_length (bits : List Bool) (size : 3 ≤ bits.length) :
    (fillBits bits).length + 3 = bits.length := by
  match bits with
  | [] => simp at size
  | [_] => simp at size
  | [_, _] => simp at size
  | [a, b, c] => simp [fillBits]
  | a :: b :: c :: d :: rest =>
    have ih := fillBits_length ((a || b) :: c :: d :: rest) (by simp)
    rw [fillBits]
    change (fillBits ((a || b) :: c :: d :: rest)).length + 1 + 3 = rest.length + 4
    simp only [List.length_cons] at ih
    omega
termination_by bits.length
decreasing_by simp_wf

theorem fillBits_correct (bits : List Bool) (size : 3 ≤ bits.length) :
    (chainView bits (fillBits bits)).all tripleValue = true ↔ bits.any id = true := by
  match bits with
  | [] => simp at size
  | [_] => simp at size
  | [_, _] => simp at size
  | [a, b, c] => simp [fillBits, chainView, tripleValue, Bool.or_assoc]
  | a :: b :: c :: d :: rest =>
    have ih := fillBits_correct ((a || b) :: c :: d :: rest) (by simp)
    have tautology : tripleValue (a, b, !(a || b)) = true := by
      cases a <;> cases b <;> rfl
    rw [fillBits]
    simp only [chainView, List.all_cons, Bool.not_not]
    rw [tautology]
    simp only [Bool.true_and]
    simpa [Bool.or_assoc] using ih
termination_by bits.length
decreasing_by simp_wf

theorem zero_false_count_iff_all (bits : List Bool) :
    bits.count false = 0 ↔ bits.all id = true := by
  induction bits with
  | nil => simp
  | cons b bits ih => cases b <;> simp [ih]

theorem chainView_failed (bits ys : List Bool)
    (size : bits.length = ys.length + 3) (failed : bits.any id = false) :
    1 ≤ ((chainView bits ys).map tripleValue).count false := by
  by_contra none
  have zero : ((chainView bits ys).map tripleValue).count false = 0 := by omega
  have all := (zero_false_count_iff_all _).mp zero
  have accepted : (chainView bits ys).all tripleValue = true := by
    simpa only [List.all_map, Function.comp_def, id_eq] using all
  have impossible := chainView_sound bits ys size accepted
  rw [failed] at impossible
  contradiction

abbrev Pattern (q : Nat) := Fin q → Bool

def patterns : (q : Nat) → List (Pattern q)
  | 0 => [Fin.elim0]
  | q + 1 => (patterns q).flatMap (fun p => [Fin.cases false p, Fin.cases true p])

theorem flatMap_pair_length {α β : Type*} (xs : List α) (f g : α → β) :
    (xs.flatMap (fun x => [f x, g x])).length = 2 * xs.length := by
  induction xs with
  | nil => rfl
  | cons x xs ih =>
    simp only [List.flatMap_cons, List.length_append, List.length_cons,
      List.length_nil, ih]
    omega

theorem patterns_length (q : Nat) : (patterns q).length = 2 ^ q := by
  induction q with
  | zero => rfl
  | succ q ih =>
    rw [patterns, flatMap_pair_length, ih, Nat.pow_succ]
    exact Nat.mul_comm _ _

theorem mem_patterns (q : Nat) (p : Pattern q) : p ∈ patterns q := by
  induction q with
  | zero =>
    have h : p = Fin.elim0 := by funext i; exact Fin.elim0 i
    simp [patterns, h]
  | succ q ih =>
    let tail : Pattern q := fun i => p i.succ
    have htail : tail ∈ patterns q := ih tail
    have prefixEquality : Fin.cases (p 0) tail = p := by
      funext i
      exact Fin.cases rfl (fun _ => rfl) i
    apply List.mem_flatMap.mpr
    refine ⟨tail, htail, ?_⟩
    cases h : p 0 with
    | false =>
      have hp : Fin.cases false tail = p := by simpa only [h] using prefixEquality
      exact List.mem_cons.mpr (Or.inl hp.symm)
    | true =>
      have hp : Fin.cases true tail = p := by simpa only [h] using prefixEquality
      exact List.mem_cons.mpr (Or.inr (List.mem_cons.mpr (Or.inl hp.symm)))

abbrev PatternIndex (q : Nat) := Fin (patterns q).length
def patternAt (q : Nat) (p : PatternIndex q) : Pattern q := (patterns q)[p.val]

theorem patternAt_surjective (q : Nat) (bits : Pattern q) :
    ∃ p : PatternIndex q, patternAt q p = bits := by
  obtain ⟨i, hi, h⟩ := List.getElem_of_mem (mem_patterns q bits)
  exact ⟨⟨i, hi⟩, h⟩

structure FiniteVerifier (q : Nat) where
  «variables» : Nat
  events : Nat
  query : Fin events → Fin q → Fin «variables»
  accepts : Fin events → Pattern q → Bool

def eventValue {q : Nat} (V : FiniteVerifier q) (A : Fin V.«variables» → Bool)
    (e : Fin V.events) : Bool := V.accepts e (fun i => A (V.query e i))

def rejectedEventCount {q : Nat} (V : FiniteVerifier q) (A : Fin V.«variables» → Bool) : Nat :=
  ((List.finRange V.events).map (eventValue V A)).count false

def auxiliaryCount {q : Nat} (V : FiniteVerifier q) : Nat :=
  V.events * (patterns q).length * (q - 3)

def outputVariables {q : Nat} (V : FiniteVerifier q) : Nat := V.«variables» + auxiliaryCount V

def oldIndex {q : Nat} (V : FiniteVerifier q) (v : Fin V.«variables») : Fin (outputVariables V) :=
  Fin.castAdd (auxiliaryCount V) v

def freshIndex {q : Nat} (V : FiniteVerifier q) (e : Fin V.events)
    (p : PatternIndex q) (j : Fin (q - 3)) : Fin (outputVariables V) :=
  Fin.natAdd V.«variables» (finProdFinEquiv (finProdFinEquiv (e, p), j))

theorem freshIndex_value {q : Nat} (V : FiniteVerifier q) (e : Fin V.events)
    (p : PatternIndex q) (j : Fin (q - 3)) :
    (freshIndex V e p j).val =
      V.«variables» + ((e.val * (patterns q).length + p.val) * (q - 3) + j.val) := by
  simp [freshIndex, Fin.natAdd, finProdFinEquiv, Nat.mul_comm, Nat.add_comm]

def liftLiteral {q : Nat} (V : FiniteVerifier q) (l : Literal V.«variables») :
    Literal (outputVariables V) := ⟨oldIndex V l.variableIndex, l.positive⟩

def forbiddenLiteral {q : Nat} (V : FiniteVerifier q) (e : Fin V.events)
    (p : PatternIndex q) (i : Fin q) : Literal V.«variables» :=
  ⟨V.query e i, !(patternAt q p i)⟩

def forbiddenClause {q : Nat} (V : FiniteVerifier q) (e : Fin V.events)
    (p : PatternIndex q) : List (Literal (outputVariables V)) :=
  List.ofFn (fun i => liftLiteral V (forbiddenLiteral V e p i))

def auxiliaryNames {q : Nat} (V : FiniteVerifier q) (e : Fin V.events)
    (p : PatternIndex q) : List (Fin (outputVariables V)) :=
  List.ofFn (freshIndex V e p)

def splitLong {n : Nat} : List (Literal n) → List (Fin n) → List (Clause n)
  | [a, b, c], [] => [#v[a, b, c]]
  | a :: b :: rest, y :: ys =>
      #v[a, b, ⟨y, true⟩] :: splitLong (⟨y, false⟩ :: rest) ys
  | _, _ => []

def tautology {q : Nat} (V : FiniteVerifier q) (hq : 3 ≤ q) (e : Fin V.events) :
    Clause (outputVariables V) :=
  let v := oldIndex V (V.query e ⟨0, by omega⟩)
  #v[⟨v, true⟩, ⟨v, false⟩, ⟨v, true⟩]

def block {q : Nat} (V : FiniteVerifier q) (hq : 3 ≤ q) (e : Fin V.events)
    (p : PatternIndex q) : List (Clause (outputVariables V)) :=
  if V.accepts e (patternAt q p) then List.replicate (q - 2) (tautology V hq e)
  else splitLong (forbiddenClause V e p) (auxiliaryNames V e p)

def eventBlock {q : Nat} (V : FiniteVerifier q) (hq : 3 ≤ q) (e : Fin V.events) :
    List (Clause (outputVariables V)) :=
  (List.finRange (patterns q).length).flatMap (block V hq e)

def convert {q : Nat} (V : FiniteVerifier q) (hq : 3 ≤ q) : Formula where
  «variables» := outputVariables V
  clauses := (List.finRange V.events).flatMap (eventBlock V hq)

def clauseFailures {n : Nat} (clauses : List (Clause n)) (A : Fin n → Bool) : Nat :=
  (clauses.map (fun c => c.eval A)).count false

theorem splitLong_values {n : Nat} (ls : List (Literal n)) (ys : List (Fin n))
    (A : Fin n → Bool) :
    (splitLong ls ys).map (fun c => c.eval A) =
      (chainView (ls.map (fun l => l.eval A)) (ys.map A)).map tripleValue := by
  induction ys generalizing ls with
  | nil =>
    match ls with
    | [] => rfl
    | [_] => rfl
    | [_, _] => rfl
    | [a, b, c] => rfl
    | _ :: _ :: _ :: _ :: _ => rfl
  | cons y ys ih =>
    match ls with
    | [] => rfl
    | [_] => rfl
    | a :: b :: rest =>
      have h := ih (⟨y, false⟩ :: rest)
      simp only [splitLong, List.map_cons, chainView]
      apply congrArg₂ List.cons
      · rfl
      · simpa only [List.map_cons, Literal.eval, Bool.false_eq_true, ↓reduceIte] using h

theorem splitLong_length {n : Nat} (ls : List (Literal n)) (ys : List (Fin n))
    (size : ls.length = ys.length + 3) : (splitLong ls ys).length = ys.length + 1 := by
  have values := congrArg List.length (splitLong_values ls ys (fun _ => false))
  simp only [List.length_map] at values
  rw [values]
  have h := chainView_length (ls.map (fun l => l.eval (fun _ => false)))
    (ys.map (fun _ => false)) (by simpa only [List.length_map] using size)
  simpa only [List.length_map] using h

def restrictAssignment {q : Nat} (V : FiniteVerifier q)
    (B : Fin (outputVariables V) → Bool) : Fin V.«variables» → Bool :=
  fun v => B (oldIndex V v)

def mismatchBits {q : Nat} (V : FiniteVerifier q) (A : Fin V.«variables» → Bool)
    (e : Fin V.events) (p : PatternIndex q) : List Bool :=
  List.ofFn (fun i => (forbiddenLiteral V e p i).eval A)

@[simp] theorem mismatchBits_length {q : Nat} (V : FiniteVerifier q)
    (A : Fin V.«variables» → Bool) (e : Fin V.events) (p : PatternIndex q) :
    (mismatchBits V A e p).length = q := by simp [mismatchBits]

theorem forbiddenLiteral_false {q : Nat} (V : FiniteVerifier q) (A : Fin V.«variables» → Bool)
    (e : Fin V.events) (p : PatternIndex q) (i : Fin q) :
    (forbiddenLiteral V e p i).eval A = false ↔ A (V.query e i) = patternAt q p i := by
  cases hp : patternAt q p i <;> cases ha : A (V.query e i) <;>
    simp [forbiddenLiteral, Literal.eval, hp, ha]

theorem forbiddenClause_values {q : Nat} (V : FiniteVerifier q)
    (B : Fin (outputVariables V) → Bool) (e : Fin V.events) (p : PatternIndex q) :
    (forbiddenClause V e p).map (fun l => l.eval B) =
      mismatchBits V (restrictAssignment V B) e p := by
  simp [forbiddenClause, mismatchBits, List.map_ofFn, Function.comp_def, liftLiteral, Literal.eval,
    restrictAssignment]

theorem mismatchBits_false {q : Nat} (V : FiniteVerifier q) (A : Fin V.«variables» → Bool)
    (e : Fin V.events) (p : PatternIndex q)
    (actual : ∀ i, A (V.query e i) = patternAt q p i) :
    (mismatchBits V A e p).any id = false := by
  apply List.any_eq_false.mpr
  intro b hb
  obtain ⟨i, rfl⟩ := List.mem_ofFn.mp hb
  have h := (forbiddenLiteral_false V A e p i).mpr (actual i)
  simp [h]

theorem mismatchBits_true {q : Nat} (V : FiniteVerifier q) (A : Fin V.«variables» → Bool)
    (e : Fin V.events) (p : PatternIndex q) (honest : eventValue V A e = true)
    (rejected : V.accepts e (patternAt q p) = false) :
    (mismatchBits V A e p).any id = true := by
  cases h : (mismatchBits V A e p).any id with
  | true => rfl
  | false =>
    have none := List.any_eq_false.mp h
    have equal : (fun i => A (V.query e i)) = patternAt q p := by
      funext i
      apply (forbiddenLiteral_false V A e p i).mp
      have hi := none ((forbiddenLiteral V e p i).eval A)
        (List.mem_ofFn.mpr ⟨i, rfl⟩)
      cases hv : (forbiddenLiteral V e p i).eval A <;> simp_all
    have impossible : V.accepts e (patternAt q p) = true := by
      simpa only [eventValue, equal] using honest
    rw [rejected] at impossible
    contradiction

theorem mismatchFill_length {q : Nat} (V : FiniteVerifier q) (hq : 3 ≤ q)
    (A : Fin V.«variables» → Bool) (e : Fin V.events) (p : PatternIndex q) :
    (fillBits (mismatchBits V A e p)).length = q - 3 := by
  have h := fillBits_length (mismatchBits V A e p) (by simpa using hq)
  rw [mismatchBits_length] at h
  omega

def mismatchFill {q : Nat} (V : FiniteVerifier q) (hq : 3 ≤ q)
    (A : Fin V.«variables» → Bool) (e : Fin V.events) (p : PatternIndex q)
    (j : Fin (q - 3)) : Bool :=
  (fillBits (mismatchBits V A e p))[j.val]'(by
    rw [mismatchFill_length V hq A e p]
    exact j.isLt)

def extendAssignment {q : Nat} (V : FiniteVerifier q) (hq : 3 ≤ q)
    (A : Fin V.«variables» → Bool) : Fin (outputVariables V) → Bool :=
  Fin.addCases A (fun k : Fin (auxiliaryCount V) =>
    let pair : Fin (V.events * (patterns q).length) × Fin (q - 3) := finProdFinEquiv.symm k
    let ep : Fin V.events × PatternIndex q := finProdFinEquiv.symm pair.1
    mismatchFill V hq A ep.1 ep.2 pair.2)

@[simp] theorem extend_old {q : Nat} (V : FiniteVerifier q) (hq : 3 ≤ q)
    (A : Fin V.«variables» → Bool) (v : Fin V.«variables») :
    extendAssignment V hq A (oldIndex V v) = A v := by
  simp [extendAssignment, oldIndex]

@[simp] theorem extend_fresh {q : Nat} (V : FiniteVerifier q) (hq : 3 ≤ q)
    (A : Fin V.«variables» → Bool) (e : Fin V.events) (p : PatternIndex q)
    (j : Fin (q - 3)) :
    extendAssignment V hq A (freshIndex V e p j) = mismatchFill V hq A e p j := by
  unfold extendAssignment freshIndex auxiliaryCount
  rw [Fin.addCases_right]
  simp only [Equiv.symm_apply_apply]

@[simp] theorem restrict_extend {q : Nat} (V : FiniteVerifier q) (hq : 3 ≤ q)
    (A : Fin V.«variables» → Bool) : restrictAssignment V (extendAssignment V hq A) = A := by
  funext v
  exact extend_old V hq A v

theorem ofFn_getElem_of_length {α : Type*} (xs : List α) (n : Nat) (h : xs.length = n) :
    List.ofFn (fun i : Fin n => xs[i.val]'(by rw [h]; exact i.isLt)) = xs := by
  subst n
  exact List.ofFn_getElem

theorem extended_auxiliary_values {q : Nat} (V : FiniteVerifier q) (hq : 3 ≤ q)
    (A : Fin V.«variables» → Bool) (e : Fin V.events) (p : PatternIndex q) :
    (auxiliaryNames V e p).map (extendAssignment V hq A) = fillBits (mismatchBits V A e p) := by
  simp only [auxiliaryNames, List.map_ofFn, Function.comp_def, extend_fresh, mismatchFill]
  exact ofFn_getElem_of_length _ _ (mismatchFill_length V hq A e p)

theorem tautology_satisfied {q : Nat} (V : FiniteVerifier q) (hq : 3 ≤ q)
    (e : Fin V.events) (B : Fin (outputVariables V) → Bool) :
    (tautology V hq e).eval B = true := by
  simp [tautology, Clause.eval, Literal.eval]

theorem block_length {q : Nat} (V : FiniteVerifier q) (hq : 3 ≤ q)
    (e : Fin V.events) (p : PatternIndex q) : (block V hq e p).length = q - 2 := by
  by_cases accepted : V.accepts e (patternAt q p) = true
  · simp [block, accepted]
  · simp only [block, accepted, Bool.false_eq_true, ↓reduceIte]
    have size : (forbiddenClause V e p).length = (auxiliaryNames V e p).length + 3 := by
      simp only [forbiddenClause, auxiliaryNames, List.length_ofFn]
      omega
    have h := splitLong_length (forbiddenClause V e p) (auxiliaryNames V e p) size
    have auxLength : (auxiliaryNames V e p).length = q - 3 := by simp [auxiliaryNames]
    rw [auxLength] at h
    omega

theorem block_honest {q : Nat} (V : FiniteVerifier q) (hq : 3 ≤ q)
    (A : Fin V.«variables» → Bool) (e : Fin V.events) (p : PatternIndex q)
    (honest : eventValue V A e = true) :
    (block V hq e p).all (fun c => c.eval (extendAssignment V hq A)) = true := by
  cases accepted : V.accepts e (patternAt q p) with
  | true => simp [block, accepted, tautology_satisfied]
  | false =>
    have good := (fillBits_correct (mismatchBits V A e p) (by simpa using hq)).mpr
      (mismatchBits_true V A e p honest accepted)
    have values := splitLong_values (forbiddenClause V e p) (auxiliaryNames V e p)
      (extendAssignment V hq A)
    rw [forbiddenClause_values, restrict_extend, extended_auxiliary_values] at values
    have all := congrArg (fun bs : List Bool => bs.all id) values
    simp only [List.all_map, Function.comp_def, id_eq] at all
    simpa only [block, accepted, Bool.false_eq_true, ↓reduceIte] using all.trans good

theorem length_flatMap_constant {α β : Type*} (xs : List α) (f : α → List β) (k : Nat)
    (lengths : ∀ x ∈ xs, (f x).length = k) : (xs.flatMap f).length = xs.length * k := by
  induction xs with
  | nil => simp
  | cons x xs ih =>
    have hx := lengths x (by simp)
    have ht := ih (fun y hy => lengths y (by simp [hy]))
    simp only [List.flatMap_cons, List.length_append, List.length_cons, hx, ht,
      Nat.add_mul, Nat.one_mul]
    omega

theorem eventBlock_length {q : Nat} (V : FiniteVerifier q) (hq : 3 ≤ q)
    (e : Fin V.events) : (eventBlock V hq e).length = 2 ^ q * (q - 2) := by
  rw [eventBlock, length_flatMap_constant _ _ (q - 2) (fun p _ => block_length V hq e p)]
  simp only [List.length_finRange, patterns_length]

theorem convert_clause_count {q : Nat} (V : FiniteVerifier q) (hq : 3 ≤ q) :
    (convert V hq).clauses.length = V.events * (2 ^ q * (q - 2)) := by
  change ((List.finRange V.events).flatMap (eventBlock V hq)).length = _
  rw [length_flatMap_constant _ _ _ (fun e _ => eventBlock_length V hq e)]
  rw [List.length_finRange]

theorem convert_nonempty {q : Nat} (V : FiniteVerifier q) (hq : 3 ≤ q)
    (eventsPositive : 0 < V.events) : (convert V hq).clauses ≠ [] := by
  have p : 0 < (convert V hq).clauses.length := by
    rw [convert_clause_count]
    exact Nat.mul_pos eventsPositive (Nat.mul_pos (Nat.pow_pos (by decide)) (by omega))
  intro empty
  simp only [empty, List.length_nil] at p
  omega

theorem convert_completeness {q : Nat} (V : FiniteVerifier q) (hq : 3 ≤ q)
    (A : Fin V.«variables» → Bool) (honest : ∀ e, eventValue V A e = true) :
    (convert V hq).Satisfiable := by
  refine ⟨extendAssignment V hq A, fun c hc => ?_⟩
  obtain ⟨e, _, hc⟩ := List.mem_flatMap.mp hc
  obtain ⟨p, _, hc⟩ := List.mem_flatMap.mp hc
  exact List.all_eq_true.mp (block_honest V hq A e p (honest e)) c hc

theorem clauseFailures_append {n : Nat} (xs ys : List (Clause n)) (A : Fin n → Bool) :
    clauseFailures (xs ++ ys) A = clauseFailures xs A + clauseFailures ys A := by
  simp [clauseFailures, List.map_append, List.count_append]

theorem clauseFailures_le_flatMap {α : Type*} {n : Nat} (xs : List α)
    (f : α → List (Clause n)) (A : Fin n → Bool) (x : α) (hx : x ∈ xs) :
    clauseFailures (f x) A ≤ clauseFailures (xs.flatMap f) A := by
  induction xs with
  | nil => simp at hx
  | cons y ys ih =>
    simp only [List.mem_cons] at hx
    rcases hx with rfl | hx
    · simp only [List.flatMap_cons, clauseFailures_append]
      omega
    · have h := ih hx
      simp only [List.flatMap_cons, clauseFailures_append]
      omega

theorem block_failed {q : Nat} (V : FiniteVerifier q) (hq : 3 ≤ q)
    (B : Fin (outputVariables V) → Bool) (e : Fin V.events) (p : PatternIndex q)
    (rejected : V.accepts e (patternAt q p) = false)
    (actual : ∀ i, restrictAssignment V B (V.query e i) = patternAt q p i) :
    1 ≤ clauseFailures (block V hq e p) B := by
  have bitsFalse := mismatchBits_false V (restrictAssignment V B) e p actual
  have size : ((forbiddenClause V e p).map (fun l => l.eval B)).length =
      ((auxiliaryNames V e p).map B).length + 3 := by
    simp only [List.length_map, forbiddenClause, auxiliaryNames, List.length_ofFn]
    omega
  have failed := chainView_failed ((forbiddenClause V e p).map (fun l => l.eval B))
    ((auxiliaryNames V e p).map B) size (by
      rw [forbiddenClause_values]
      exact bitsFalse)
  simpa only [block, rejected, Bool.false_eq_true, ↓reduceIte, clauseFailures,
    splitLong_values] using failed

theorem eventBlock_failed {q : Nat} (V : FiniteVerifier q) (hq : 3 ≤ q)
    (B : Fin (outputVariables V) → Bool) (e : Fin V.events)
    (rejected : eventValue V (restrictAssignment V B) e = false) :
    1 ≤ clauseFailures (eventBlock V hq e) B := by
  obtain ⟨p, hp⟩ := patternAt_surjective q (fun i => restrictAssignment V B (V.query e i))
  have rejectedPattern : V.accepts e (patternAt q p) = false := by
    rw [hp]
    exact rejected
  have failed := block_failed V hq B e p rejectedPattern (fun i => (congrFun hp i).symm)
  exact failed.trans (clauseFailures_le_flatMap (List.finRange (patterns q).length)
    (block V hq e) B p (by simp))

theorem count_false_map_sum {α : Type*} (xs : List α) (f : α → Bool) :
    (xs.map f).count false = (xs.map (fun x => if f x then 0 else 1)).sum := by
  induction xs with
  | nil => simp
  | cons x xs ih => cases h : f x <;> simp [h, ih, Nat.add_comm]

theorem clauseFailures_flatMap {α : Type*} {n : Nat} (xs : List α)
    (f : α → List (Clause n)) (A : Fin n → Bool) :
    clauseFailures (xs.flatMap f) A = (xs.map (fun x => clauseFailures (f x) A)).sum := by
  induction xs with
  | nil => rfl
  | cons x xs ih =>
    simp only [List.flatMap_cons, clauseFailures_append, List.map_cons, List.sum_cons, ih]

theorem nat_sum_map_le {α : Type*} (xs : List α) (f g : α → Nat)
    (bound : ∀ x ∈ xs, f x ≤ g x) : (xs.map f).sum ≤ (xs.map g).sum := by
  induction xs with
  | nil => exact Nat.le_refl 0
  | cons x xs ih =>
    simp only [List.map_cons, List.sum_cons]
    exact Nat.add_le_add (bound x (by simp)) (ih (fun y hy => bound y (by simp [hy])))

theorem convert_count_bridge {q : Nat} (V : FiniteVerifier q) (hq : 3 ≤ q)
    (B : Fin (outputVariables V) → Bool) :
    rejectedEventCount V (restrictAssignment V B) ≤ NameCompaction.failedCount (convert V hq) B := by
  change ((List.finRange V.events).map (eventValue V (restrictAssignment V B))).count false ≤
    clauseFailures ((List.finRange V.events).flatMap (eventBlock V hq)) B
  rw [count_false_map_sum, clauseFailures_flatMap]
  apply nat_sum_map_le
  intro e _
  cases rejected : eventValue V (restrictAssignment V B) e with
  | false => simpa only [rejected, Bool.false_eq_true, ↓reduceIte] using
      eventBlock_failed V hq B e rejected
  | true => simp only [↓reduceIte, Nat.zero_le]

theorem convert_gap {q : Nat} (V : FiniteVerifier q) (hq : 3 ≤ q) (a b : Nat)
    (gap : ∀ A, a * V.events ≤ b * rejectedEventCount V A)
    (B : Fin (outputVariables V) → Bool) :
    a * (convert V hq).clauses.length ≤
      (b * (2 ^ q * (q - 2))) * NameCompaction.failedCount (convert V hq) B := by
  have eventGap := gap (restrictAssignment V B)
  have countBound := Nat.mul_le_mul_left b (convert_count_bridge V hq B)
  rw [convert_clause_count]
  calc
    a * (V.events * (2 ^ q * (q - 2))) = (a * V.events) * (2 ^ q * (q - 2)) := by
      rw [Nat.mul_assoc]
    _ ≤ (b * NameCompaction.failedCount (convert V hq) B) * (2 ^ q * (q - 2)) :=
      Nat.mul_le_mul_right _ (eventGap.trans countBound)
    _ = (b * (2 ^ q * (q - 2))) * NameCompaction.failedCount (convert V hq) B := by ac_rfl

theorem twelve_query_factor : 2 ^ 12 * (12 - 2) = 40960 := by decide

theorem twelve_query_clause_count (V : FiniteVerifier 12) :
    (convert V (by decide)).clauses.length = V.events * 40960 := by
  simpa only [twelve_query_factor] using convert_clause_count V (by decide)

def padLiteral {n : Nat} (l : Literal n) : Literal (n + 1) :=
  ⟨Fin.castAdd 1 l.variableIndex, l.positive⟩

def padClause {n : Nat} (c : Clause n) : Clause (n + 1) :=
  #v[padLiteral (c)[0], padLiteral (c)[1], padLiteral (c)[2]]

def padTautology (n : Nat) : Clause (n + 1) :=
  let v := Fin.natAdd n (0 : Fin 1)
  #v[⟨v, true⟩, ⟨v, false⟩, ⟨v, true⟩]

def padInput (F : Formula) : Formula where
  «variables» := F.«variables» + 1
  clauses := F.clauses.map padClause ++ [padTautology F.«variables»]

theorem padClause_eval {n : Nat} (c : Clause n) (B : Fin (n + 1) → Bool) :
    (padClause c).eval B = c.eval (fun v => B (Fin.castAdd 1 v)) := rfl

theorem padTautology_satisfied (n : Nat) (B : Fin (n + 1) → Bool) :
    (padTautology n).eval B = true := by
  simp [padTautology, Clause.eval, Literal.eval]

theorem padInput_satisfiable_iff (F : Formula) : (padInput F).Satisfiable ↔ F.Satisfiable := by
  change (∃ B : Fin (F.«variables» + 1) → Bool,
    ∀ c ∈ F.clauses.map padClause ++ [padTautology F.«variables»], c.eval B = true) ↔ _
  constructor
  · rintro ⟨B, hB⟩
    refine ⟨fun v => B (Fin.castAdd 1 v), fun c hc => ?_⟩
    have hmem : padClause c ∈ (padInput F).clauses := by
      apply List.mem_append.mpr
      exact Or.inl (List.mem_map.mpr ⟨c, hc, rfl⟩)
    exact (padClause_eval c B).symm.trans (hB _ hmem)
  · rintro ⟨A, hA⟩
    let B : Fin (F.«variables» + 1) → Bool := Fin.addCases A (fun _ => false)
    refine ⟨B, fun c hc => ?_⟩
    rcases List.mem_append.mp hc with original | padding
    · obtain ⟨d, hd, rfl⟩ := List.mem_map.mp original
      have h := hA d hd
      simpa only [padClause_eval, B, Fin.addCases_left] using h
    · have hc : c = padTautology F.«variables» := by simpa using padding
      subst c
      exact padTautology_satisfied F.«variables» B

@[simp] theorem padInput_clause_count (F : Formula) :
    (padInput F).clauses.length = F.clauses.length + 1 := by simp [padInput]

theorem padInput_nonempty (F : Formula) : (padInput F).clauses ≠ [] := by
  intro empty
  have size := padInput_clause_count F
  rw [empty, List.length_nil] at size
  omega

def prepareInput (F : Formula) : Formula := NameCompaction.compact (padInput F)

theorem prepareInput_satisfiable_iff (F : Formula) :
    (prepareInput F).Satisfiable ↔ F.Satisfiable :=
  (NameCompaction.compact_satisfiable_iff (padInput F)).trans (padInput_satisfiable_iff F)

theorem prepareInput_clause_count (F : Formula) :
    (prepareInput F).clauses.length = F.clauses.length + 1 := by
  simp only [prepareInput, NameCompaction.compact_clause_count, padInput_clause_count]

theorem prepareInput_nonempty (F : Formula) : (prepareInput F).clauses ≠ [] := by
  intro empty
  have h := prepareInput_clause_count F
  rw [empty, List.length_nil] at h
  omega

theorem prepareInput_variable_bound (F : Formula) :
    (prepareInput F).«variables» ≤ 3 * (F.clauses.length + 1) := by
  simpa only [prepareInput, padInput_clause_count] using NameCompaction.compact_active_bound (padInput F)

end DFVSGames.Foundations.PCP.VerifierToCNF

namespace DFVSGames.Foundations.PCP.FinalBooleanVerifier

open Target

abbrev Label := Fin 6 → Bool

variable {V E : Type*} {n m : Nat}

def bitIndex (vertices : V ≃ Fin n) (v : V) (j : Fin 6) : Fin (n * 6) :=
  finProdFinEquiv (vertices v, j)

def labelsOfBits (vertices : V ≃ Fin n) (assignment : Fin (n * 6) → Bool) : V → Label :=
  fun v j => assignment (bitIndex vertices v j)

def bitsOfLabels (vertices : V ≃ Fin n) (labeling : V → Label) : Fin (n * 6) → Bool :=
  fun k =>
    let pair : Fin n × Fin 6 := finProdFinEquiv.symm k
    labeling (vertices.symm pair.1) pair.2

@[simp] theorem labelsOfBits_bitsOfLabels (vertices : V ≃ Fin n) (labeling : V → Label) :
    labelsOfBits vertices (bitsOfLabels vertices labeling) = labeling := by
  funext v j
  simp only [labelsOfBits, bitsOfLabels, bitIndex, Equiv.symm_apply_apply]

@[simp] theorem bitsOfLabels_labelsOfBits (vertices : V ≃ Fin n)
    (assignment : Fin (n * 6) → Bool) : bitsOfLabels vertices (labelsOfBits vertices assignment) = assignment := by
  funext k
  simp only [bitsOfLabels, labelsOfBits, bitIndex, Equiv.apply_symm_apply, Prod.eta]

abbrev verifier (G : ConstraintGraph V E Label) (vertices : V ≃ Fin n) (edges : E ≃ Fin m) :
    VerifierToCNF.FiniteVerifier 12 where
  «variables» := n * 6
  events := m
  query := fun e i => Fin.addCases (motive := fun _ : Fin (6 + 6) => Fin (n * 6))
    (fun j : Fin 6 => bitIndex vertices (G.tail (edges.symm e)) j)
    (fun j : Fin 6 => bitIndex vertices (G.head (edges.symm e)) j) i
  accepts := fun e bits => G.accepts (edges.symm e)
    (fun j => bits (Fin.castAdd 6 j)) (fun j => bits (Fin.natAdd 6 j))

theorem query_tail (G : ConstraintGraph V E Label) (vertices : V ≃ Fin n)
    (edges : E ≃ Fin m) (e : Fin m) (j : Fin 6) :
    (verifier G vertices edges).query e (Fin.castAdd 6 j) =
      bitIndex vertices (G.tail (edges.symm e)) j := by
  simp only [verifier, Fin.addCases_left]

theorem query_head (G : ConstraintGraph V E Label) (vertices : V ≃ Fin n)
    (edges : E ≃ Fin m) (e : Fin m) (j : Fin 6) :
    (verifier G vertices edges).query e (Fin.natAdd 6 j) =
      bitIndex vertices (G.head (edges.symm e)) j := by
  simp only [verifier, Fin.addCases_right]

theorem eventValue_eq (G : ConstraintGraph V E Label) (vertices : V ≃ Fin n)
    (edges : E ≃ Fin m) (assignment : Fin (n * 6) → Bool) (e : Fin m) :
    VerifierToCNF.eventValue (verifier G vertices edges) assignment e =
      G.edgeSatisfied (labelsOfBits vertices assignment) (edges.symm e) := by
  simp only [VerifierToCNF.eventValue, Fin.addCases_left, Fin.addCases_right,
    ConstraintGraph.edgeSatisfied]
  rfl

theorem count_false_map_filter {α : Type*} (xs : List α) (f : α → Bool) :
    (xs.map f).count false = (xs.filter (fun x => decide (f x = false))).length := by
  induction xs with
  | nil => rfl
  | cons x xs ih => cases h : f x <;> simp [h, ih, Nat.add_comm]

theorem count_false_finRange (f : Fin m → Bool) :
    ((List.finRange m).map f).count false =
      (Finset.univ.filter (fun e : Fin m => f e = false)).card := by
  classical
  calc
    _ = ((List.finRange m).filter (fun e => decide (f e = false))).length :=
      count_false_map_filter _ _
    _ = (((List.finRange m).filter (fun e => decide (f e = false))).toFinset).card :=
      (List.toFinset_card_of_nodup ((List.nodup_finRange m).filter _)).symm
    _ = _ := by
      congr 1
      ext e
      simp

theorem rejection_count_eq [Fintype E] (G : ConstraintGraph V E Label)
    (vertices : V ≃ Fin n) (edges : E ≃ Fin m) (assignment : Fin (n * 6) → Bool) :
    VerifierToCNF.rejectedEventCount (verifier G vertices edges) assignment =
      G.rejectionCount (labelsOfBits vertices assignment) := by
  classical
  change ((List.finRange m).map
    (VerifierToCNF.eventValue (verifier G vertices edges) assignment)).count false = _
  rw [count_false_finRange]
  change (Finset.univ.filter (fun e : Fin m =>
    VerifierToCNF.eventValue (verifier G vertices edges) assignment e = false)).card =
      (G.rejectedDarts (labelsOfBits vertices assignment)).card
  apply Finset.card_equiv edges.symm
  intro e
  simp only [Finset.mem_filter, Finset.mem_univ, true_and,
    ConstraintGraph.rejectedDarts, eventValue_eq]

theorem verifier_completeness (G : ConstraintGraph V E Label) (vertices : V ≃ Fin n)
    (edges : E ≃ Fin m) (labeling : V → Label)
    (satisfied : ∀ e, G.edgeSatisfied labeling e = true) :
    ∀ e, VerifierToCNF.eventValue (verifier G vertices edges)
      (bitsOfLabels vertices labeling) e = true := by
  intro e
  rw [eventValue_eq, labelsOfBits_bitsOfLabels]
  exact satisfied (edges.symm e)

theorem verifier_reflects (G : ConstraintGraph V E Label) (vertices : V ≃ Fin n)
    (edges : E ≃ Fin m) (assignment : Fin (n * 6) → Bool)
    (accepted : ∀ e, VerifierToCNF.eventValue (verifier G vertices edges) assignment e = true) :
    ∀ e, G.edgeSatisfied (labelsOfBits vertices assignment) e = true := by
  intro e
  have h := accepted (edges e)
  simpa only [eventValue_eq, Equiv.symm_apply_apply] using h

def cnf (G : ConstraintGraph V E Label) (vertices : V ≃ Fin n) (edges : E ≃ Fin m) : Formula :=
  VerifierToCNF.convert (verifier G vertices edges) (by decide)

abbrev OutputAssignment (G : ConstraintGraph V E Label) (vertices : V ≃ Fin n)
    (edges : E ≃ Fin m) :=
  Fin (VerifierToCNF.outputVariables (verifier G vertices edges)) → Bool

def extractLabeling (G : ConstraintGraph V E Label) (vertices : V ≃ Fin n)
    (edges : E ≃ Fin m) (assignment : OutputAssignment G vertices edges) : V → Label :=
  labelsOfBits vertices (VerifierToCNF.restrictAssignment (verifier G vertices edges) assignment)

theorem cnf_completeness (G : ConstraintGraph V E Label) (vertices : V ≃ Fin n)
    (edges : E ≃ Fin m) (sat : G.Satisfiable) : (cnf G vertices edges).Satisfiable := by
  obtain ⟨labeling, satisfied⟩ := sat
  exact VerifierToCNF.convert_completeness (verifier G vertices edges) (by decide)
    (bitsOfLabels vertices labeling) (verifier_completeness G vertices edges labeling satisfied)

theorem cnf_count_bridge [Fintype E] (G : ConstraintGraph V E Label)
    (vertices : V ≃ Fin n) (edges : E ≃ Fin m) (assignment : OutputAssignment G vertices edges) :
    G.rejectionCount (extractLabeling G vertices edges assignment) ≤
      NameCompaction.failedCount (cnf G vertices edges) assignment := by
  have h := VerifierToCNF.convert_count_bridge (verifier G vertices edges) (by decide) assignment
  rw [rejection_count_eq] at h
  exact h

theorem failedCount_zero_of_satisfied (F : Formula) (assignment : Fin F.«variables» → Bool)
    (satisfied : ∀ c ∈ F.clauses, c.eval assignment = true) :
    NameCompaction.failedCount F assignment = 0 := by
  apply (VerifierToCNF.zero_false_count_iff_all _).mpr
  have h := List.all_eq_true.mpr satisfied
  simpa only [NameCompaction.evaluationList, List.all_map, Function.comp_def, id_eq] using h

theorem all_satisfied_of_rejectionCount_zero [Fintype E] (G : ConstraintGraph V E Label)
    (labeling : V → Label) (zero : G.rejectionCount labeling = 0) :
    ∀ e, G.edgeSatisfied labeling e = true := by
  intro e
  cases h : G.edgeSatisfied labeling e with
  | true => rfl
  | false =>
    have member : e ∈ G.rejectedDarts labeling := (G.mem_rejectedDarts labeling e).mpr h
    have positive : 0 < G.rejectionCount labeling := Finset.card_pos.mpr ⟨e, member⟩
    omega

theorem cnf_reflects [Fintype E] (G : ConstraintGraph V E Label)
    (vertices : V ≃ Fin n) (edges : E ≃ Fin m) (sat : (cnf G vertices edges).Satisfiable) :
    G.Satisfiable := by
  obtain ⟨assignment, satisfied⟩ := sat
  have zero := failedCount_zero_of_satisfied (cnf G vertices edges) assignment satisfied
  have bound := cnf_count_bridge G vertices edges assignment
  have graphZero : G.rejectionCount (extractLabeling G vertices edges assignment) = 0 := by omega
  exact ⟨extractLabeling G vertices edges assignment,
    all_satisfied_of_rejectionCount_zero G _ graphZero⟩

theorem cnf_satisfiable_iff [Fintype E] (G : ConstraintGraph V E Label)
    (vertices : V ≃ Fin n) (edges : E ≃ Fin m) :
    (cnf G vertices edges).Satisfiable ↔ G.Satisfiable :=
  ⟨cnf_reflects G vertices edges, cnf_completeness G vertices edges⟩

theorem cnf_clause_count (G : ConstraintGraph V E Label) (vertices : V ≃ Fin n)
    (edges : E ≃ Fin m) : (cnf G vertices edges).clauses.length = m * 40960 :=
  VerifierToCNF.twelve_query_clause_count (verifier G vertices edges)

theorem twelve_query_auxiliary_factor : 2 ^ 12 * (12 - 3) = 36864 := by decide

theorem cnf_variable_count (G : ConstraintGraph V E Label) (vertices : V ≃ Fin n)
    (edges : E ≃ Fin m) : (cnf G vertices edges).«variables» = n * 6 + m * 36864 := by
  change n * 6 + m * (VerifierToCNF.patterns 12).length * (12 - 3) = _
  rw [VerifierToCNF.patterns_length, Nat.mul_assoc, twelve_query_auxiliary_factor]

theorem cnf_nonempty (G : ConstraintGraph V E Label) (vertices : V ≃ Fin n)
    (edges : E ≃ Fin m) (nonempty : 0 < m) : (cnf G vertices edges).clauses ≠ [] :=
  VerifierToCNF.convert_nonempty (verifier G vertices edges) (by decide) nonempty

theorem cnf_gap [Fintype E] (G : ConstraintGraph V E Label)
    (vertices : V ≃ Fin n) (edges : E ≃ Fin m) (a b : Nat)
    (source : ∀ labeling : V → Label, a * Fintype.card E ≤ b * G.rejectionCount labeling)
    (assignment : OutputAssignment G vertices edges) :
    a * (cnf G vertices edges).clauses.length ≤
      (b * 40960) * NameCompaction.failedCount (cnf G vertices edges) assignment := by
  have cardEdges : Fintype.card E = m := by simpa using Fintype.card_congr edges
  have testGap (bits : Fin (n * 6) → Bool) :
      a * m ≤ b * VerifierToCNF.rejectedEventCount (verifier G vertices edges) bits := by
    rw [rejection_count_eq]
    simpa only [cardEdges] using source (labelsOfBits vertices bits)
  simpa only [cnf, VerifierToCNF.twelve_query_factor] using
    VerifierToCNF.convert_gap (verifier G vertices edges) (by decide) a b testGap assignment

abbrev ReducedVertex (V E A : Type*) :=
  QueryIncidence.Vertex (AlphabetGraph.Event E A) (AlphabetGraph.Address V E A)

abbrev ReducedDart (E A : Type*) := QueryIncidence.Dart (AlphabetGraph.Event E A) 6

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

noncomputable def alphabetCNF (G : ConstraintGraph V E A)
    (vertices : ReducedVertex V E A ≃ Fin n) (edges : ReducedDart E A ≃ Fin m) : Formula :=
  cnf (AlphabetGraph.graph G) vertices edges

omit [DecidableEq A] [Nonempty A] in
theorem alphabetCNF_completeness (G : ConstraintGraph V E A)
    (vertices : ReducedVertex V E A ≃ Fin n) (edges : ReducedDart E A ≃ Fin m)
    (sat : G.Satisfiable) : (alphabetCNF G vertices edges).Satisfiable :=
  cnf_completeness (AlphabetGraph.graph G) vertices edges (AlphabetGraph.perfect_completeness G sat)

omit [DecidableEq A] [Nonempty A] in
theorem alphabetCNF_clause_count (G : ConstraintGraph V E A)
    (vertices : ReducedVertex V E A ≃ Fin n) (edges : ReducedDart E A ≃ Fin m) :
    (alphabetCNF G vertices edges).clauses.length = m * 40960 :=
  cnf_clause_count (AlphabetGraph.graph G) vertices edges

theorem alphabetCNF_gap [Fintype E] (G : ConstraintGraph V E A)
    (vertices : ReducedVertex V E A ≃ Fin n) (edges : ReducedDart E A ≃ Fin m)
    (a b : Nat)
    (source : ∀ labeling : V → A, a * Fintype.card E ≤ b * G.rejectionCount labeling)
    (assignment : OutputAssignment (AlphabetGraph.graph G) vertices edges) :
    a * (alphabetCNF G vertices edges).clauses.length ≤
      (b * 503316480) * NameCompaction.failedCount (alphabetCNF G vertices edges) assignment := by
  have h := cnf_gap (AlphabetGraph.graph G) vertices edges a (b * 12288)
    (AlphabetGraph.gap_transfer G a b source) assignment
  have factor : (b * 12288) * 40960 = b * 503316480 := by omega
  simpa only [alphabetCNF, factor] using h

end DFVSGames.Foundations.PCP.FinalBooleanVerifier

end OAI