lean/OAI/Computability/BinPacking/PCP/AlphabetRetraction.lean

import Mathlib.Data.Fin.Tuple.Basic
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Sum
import OAI.Computability.BinPacking.PCP.BinaryFormula
import OAI.Computability.BinPacking.PCP.CloudRounding

namespace OAI

namespace BinPackingGames.Foundations.PCP.QueryIncidence

structure Verifier (E X : Type*) (q : Nat) where
  query : E → Fin q → X
  accepts : E → (Fin q → Bool) → Bool

abbrev Label (q : Nat) := Fin q → Bool
abbrev Vertex (E X : Type*) := E ⊕ X
abbrev Dart (E : Type*) (q : Nat) := (E × Fin q) × Bool

variable {E X : Type*} {q : Nat}

def response (T : Verifier E X q) (assignment : X → Bool) (event : E) : Label q :=
  fun i => assignment (T.query event i)

def zeroSlot (positive : 0 < q) : Fin q := ⟨0, positive⟩

def decodeRight (positive : 0 < q) (label : Label q) : Bool := label (zeroSlot positive)

def encodeRight (bit : Bool) : Label q := fun _ => bit

@[simp] theorem decode_encodeRight (positive : 0 < q) (bit : Bool) :
    decodeRight positive (encodeRight bit) = bit := rfl

def canonicalSlot [DecidableEq X] (T : Verifier E X q) (event : E) (i : Fin q) : Fin q :=
  Fin.find (fun j => T.query event j = T.query event i) ⟨i, rfl⟩

theorem canonicalSlot_query [DecidableEq X] (T : Verifier E X q) (event : E) (i : Fin q) :
    T.query event (canonicalSlot T event i) = T.query event i :=
  Fin.find_spec (p := fun j => T.query event j = T.query event i) ⟨i, rfl⟩

def RepeatedConsistent (T : Verifier E X q) (event : E) (label : Label q) : Prop :=
  ∀ i j, T.query event i = T.query event j → label i = label j

instance repeatedConsistentDecidable [DecidableEq X]
    (T : Verifier E X q) (event : E) (label : Label q) :
    Decidable (RepeatedConsistent T event label) := by
  unfold RepeatedConsistent
  infer_instance

def LeftValid (T : Verifier E X q) (event : E) (label : Label q) : Prop :=
  T.accepts event label = true ∧ RepeatedConsistent T event label

instance leftValidDecidable [DecidableEq X]
    (T : Verifier E X q) (event : E) (label : Label q) :
    Decidable (LeftValid T event label) := by
  unfold LeftValid
  infer_instance

def incidenceAccept [DecidableEq X] (T : Verifier E X q) (positive : 0 < q)
    (event : E) (slot : Fin q) (left right : Label q) : Bool :=
  decide (LeftValid T event left) &&
    decide (left (canonicalSlot T event slot) = decodeRight positive right)

def reverse : Dart E q → Dart E q := fun d => (d.1, !d.2)

theorem reverse_involutive : Function.Involutive (reverse : Dart E q → Dart E q) := by
  rintro ⟨⟨event, slot⟩, orientation⟩
  cases orientation <;> rfl

def reverseEquiv : Dart E q ≃ Dart E q where
  toFun := reverse
  invFun := reverse
  left_inv := reverse_involutive
  right_inv := reverse_involutive

def tail (T : Verifier E X q) : Dart E q → Vertex E X
  | ((event, _), false) => .inl event
  | ((event, slot), true) => .inr (T.query event slot)

def predicate [DecidableEq X] (T : Verifier E X q) (positive : 0 < q) :
    Dart E q → Label q → Label q → Bool
  | ((event, slot), false), a, b => incidenceAccept T positive event slot a b
  | ((event, slot), true), a, b => incidenceAccept T positive event slot b a

def graph [DecidableEq X] (T : Verifier E X q) (positive : 0 < q) :
    ConstraintGraph (Vertex E X) (Dart E q) (Label q) where
  reverse := reverseEquiv
  reverse_involutive := reverse_involutive
  tail := tail T
  accepts := predicate T positive
  reverse_accepts := by
    rintro ⟨⟨event, slot⟩, orientation⟩ a b
    cases orientation <;> rfl

theorem edgeSatisfied_eq_incidence [DecidableEq X] (T : Verifier E X q)
    (positive : 0 < q) (labeling : Vertex E X → Label q)
    (event : E) (slot : Fin q) (orientation : Bool) :
    (graph T positive).edgeSatisfied labeling ((event, slot), orientation) =
      incidenceAccept T positive event slot (labeling (.inl event))
        (labeling (.inr (T.query event slot))) := by
  cases orientation <;> rfl

def honestLabeling (T : Verifier E X q) (assignment : X → Bool) : Vertex E X → Label q
  | .inl event => response T assignment event
  | .inr address => encodeRight (assignment address)

theorem honest_leftValid (T : Verifier E X q) (assignment : X → Bool) (event : E)
    (accepted : T.accepts event (response T assignment event) = true) :
    LeftValid T event (response T assignment event) := by
  refine ⟨accepted, ?_⟩
  intro i j hij
  exact congrArg assignment hij

theorem honest_incidence [DecidableEq X] (T : Verifier E X q) (positive : 0 < q)
    (assignment : X → Bool) (event : E)
    (accepted : T.accepts event (response T assignment event) = true) (slot : Fin q) :
    incidenceAccept T positive event slot
      (honestLabeling T assignment (.inl event))
      (honestLabeling T assignment (.inr (T.query event slot))) = true := by
  simp only [incidenceAccept, Bool.and_eq_true, honestLabeling,
    decode_encodeRight]
  refine ⟨_root_.decide_eq_true (honest_leftValid T assignment event accepted), ?_⟩
  exact _root_.decide_eq_true (congrArg assignment (canonicalSlot_query T event slot))

theorem perfect_completeness [DecidableEq X] (T : Verifier E X q) (positive : 0 < q)
    (assignment : X → Bool)
    (accepted : ∀ event, T.accepts event (response T assignment event) = true) :
    ∀ dart, (graph T positive).edgeSatisfied (honestLabeling T assignment) dart = true := by
  rintro ⟨⟨event, slot⟩, orientation⟩
  rw [edgeSatisfied_eq_incidence]
  exact honest_incidence T positive assignment event (accepted event) slot

theorem satisfiable_of_verifier_satisfiable [DecidableEq X] (T : Verifier E X q)
    (positive : 0 < q)
    (sat : ∃ assignment : X → Bool,
      ∀ event, T.accepts event (response T assignment event) = true) :
    (graph T positive).Satisfiable := by
  obtain ⟨assignment, h⟩ := sat
  exact ⟨honestLabeling T assignment, perfect_completeness T positive assignment h⟩

def decodedAssignment (positive : 0 < q) (labeling : Vertex E X → Label q) : X → Bool :=
  fun address => decodeRight positive (labeling (.inr address))

theorem incidenceAccept_true [DecidableEq X] (T : Verifier E X q) (positive : 0 < q)
    (event : E) (slot : Fin q) (left right : Label q)
    (accepted : incidenceAccept T positive event slot left right = true) :
    T.accepts event left = true ∧ RepeatedConsistent T event left ∧
      left (canonicalSlot T event slot) = decodeRight positive right := by
  simp only [incidenceAccept, Bool.and_eq_true] at accepted
  have hvalid : LeftValid T event left := _root_.of_decide_eq_true accepted.1
  exact ⟨hvalid.1, hvalid.2, _root_.of_decide_eq_true accepted.2⟩

theorem all_incidences_imply_event [DecidableEq X] (T : Verifier E X q)
    (positive : 0 < q) (labeling : Vertex E X → Label q) (event : E)
    (accepted : ∀ slot, incidenceAccept T positive event slot (labeling (.inl event))
      (labeling (.inr (T.query event slot))) = true) :
    T.accepts event (response T (decodedAssignment positive labeling) event) = true := by
  have first := incidenceAccept_true T positive event (zeroSlot positive) _ _
    (accepted (zeroSlot positive))
  have labels_equal : labeling (.inl event) = response T (decodedAssignment positive labeling) event := by
    funext slot
    have info := incidenceAccept_true T positive event slot _ _ (accepted slot)
    exact (first.2.1 slot (canonicalSlot T event slot)
      (canonicalSlot_query T event slot).symm).trans info.2.2
  rw [← labels_equal]
  exact first.1

theorem rejected_event_forces_slot [DecidableEq X] (T : Verifier E X q)
    (positive : 0 < q) (labeling : Vertex E X → Label q) (event : E)
    (rejected : T.accepts event (response T (decodedAssignment positive labeling) event) = false) :
    ∃ slot, incidenceAccept T positive event slot (labeling (.inl event))
      (labeling (.inr (T.query event slot))) = false := by
  classical
  by_contra none
  have all : ∀ slot, incidenceAccept T positive event slot (labeling (.inl event))
      (labeling (.inr (T.query event slot))) = true := by
    intro slot
    cases h : incidenceAccept T positive event slot (labeling (.inl event))
        (labeling (.inr (T.query event slot))) with
    | false => exact False.elim (none ⟨slot, h⟩)
    | true => rfl
  have h := all_incidences_imply_event T positive labeling event all
  rw [rejected] at h
  cases h

def rejectedEvents [Fintype E] (T : Verifier E X q) (assignment : X → Bool) : Finset E :=
  Finset.univ.filter (fun event => T.accepts event (response T assignment event) = false)

def verifierRejectionCount [Fintype E] (T : Verifier E X q) (assignment : X → Bool) : Nat :=
  (rejectedEvents T assignment).card

theorem rejection_count_bound [Fintype E] [DecidableEq X]
    (T : Verifier E X q) (positive : 0 < q) (labeling : Vertex E X → Label q) :
    2 * verifierRejectionCount T (decodedAssignment positive labeling) ≤
      (graph T positive).rejectionCount labeling := by
  classical
  let failed := rejectedEvents T (decodedAssignment positive labeling)
  have witnesses : ∀ event, event ∈ failed →
      ∃ slot, incidenceAccept T positive event slot (labeling (.inl event))
        (labeling (.inr (T.query event slot))) = false := by
    intro event he
    apply rejected_event_forces_slot T positive labeling event
    simpa [failed, rejectedEvents] using he
  let chosen := fun event he => Classical.choose (witnesses event he)
  let inject : (failed.product (Finset.univ : Finset Bool)) →
      (graph T positive).rejectedDarts labeling := fun p =>
    ⟨((p.val.1, chosen p.val.1 (Finset.mem_product.mp p.property).1), p.val.2), by
      apply (ConstraintGraph.mem_rejectedDarts _ _ _).mpr
      rw [edgeSatisfied_eq_incidence]
      exact Classical.choose_spec (witnesses p.val.1 (Finset.mem_product.mp p.property).1)⟩
  have injective : Function.Injective inject := by
    intro a b h
    apply Subtype.ext
    apply Prod.ext
    · exact congrArg (fun d => d.val.1.1) h
    · exact congrArg (fun d => d.val.2) h
  have bound := Finset.card_le_card_of_injective injective
  simpa [failed, verifierRejectionCount, ConstraintGraph.rejectionCount,
    Finset.card_product, Nat.mul_comm] using bound

@[simp] theorem card_label (q : Nat) : Fintype.card (Label q) = 2 ^ q := by
  simp [Label]

@[simp] theorem card_vertex [Fintype E] [Fintype X] :
    Fintype.card (Vertex E X) = Fintype.card E + Fintype.card X := by
  simp [Vertex, Fintype.card_sum]

@[simp] theorem card_dart [Fintype E] :
    Fintype.card (Dart E q) = 2 * q * Fintype.card E := by
  simp [Dart, Fintype.card_prod, Nat.mul_comm, Nat.mul_left_comm, Nat.mul_assoc]

theorem gap_transfer [Fintype E] [DecidableEq X] (T : Verifier E X q)
    (positive : 0 < q) (a b : Nat)
    (sourceGap : ∀ assignment : X → Bool,
      a * Fintype.card E ≤ b * verifierRejectionCount T assignment)
    (labeling : Vertex E X → Label q) :
    a * Fintype.card (Dart E q) ≤ (b * q) * (graph T positive).rejectionCount labeling := by
  have source := Nat.mul_le_mul_left (2 * q) (sourceGap (decodedAssignment positive labeling))
  have incidences := Nat.mul_le_mul_left (b * q) (rejection_count_bound T positive labeling)
  rw [card_dart]
  calc
    a * (2 * q * Fintype.card E) = (2 * q) * (a * Fintype.card E) := by ac_rfl
    _ ≤ (2 * q) * (b * verifierRejectionCount T (decodedAssignment positive labeling)) := source
    _ = (b * q) * (2 * verifierRejectionCount T (decodedAssignment positive labeling)) := by ac_rfl
    _ ≤ (b * q) * (graph T positive).rejectionCount labeling := incidences

theorem six_query_alphabet : Fintype.card (Label 6) = 64 := by
  simp

end BinPackingGames.Foundations.PCP.QueryIncidence

namespace BinPackingGames.Foundations.PCP.InitialGraph

open Target

abbrev Label := Bool × Bool × Bool
abbrev Vertex (F : Formula) := (Fin F.variables ⊕ Fin F.clauses.length) ⊕ Unit
abbrev Dart (F : Formula) := (RandomEvent F × Bool) ⊕ Unit

instance slotFintype : Fintype Slot where
  elems := {Slot.first, Slot.second, Slot.third}
  complete := by intro s; cases s <;> simp

@[simp] theorem card_slot : Fintype.card Slot = 3 := by decide
@[simp] theorem card_label : Fintype.card Label = 8 := by
  simp [Label, Fintype.card_prod]

def variableVertex (F : Formula) (v : Fin F.variables) : Vertex F := .inl (.inl v)
def clauseVertex (F : Formula) (i : Fin F.clauses.length) : Vertex F := .inl (.inr i)
def dummyVertex (F : Formula) : Vertex F := .inr ()
def dummyDart (F : Formula) : Dart F := .inr ()

instance vertexNonempty (F : Formula) : Nonempty (Vertex F) := ⟨dummyVertex F⟩
instance dartNonempty (F : Formula) : Nonempty (Dart F) := ⟨dummyDart F⟩

def reverse (F : Formula) : Dart F → Dart F
  | .inl (event, orientation) => .inl (event, !orientation)
  | .inr _ => .inr ()

theorem reverse_involutive (F : Formula) : Function.Involutive (reverse F) := by
  intro d
  cases d with
  | inl pair => rcases pair with ⟨e, b⟩; cases b <;> rfl
  | inr u => cases u; rfl

def reverseEquiv (F : Formula) : Dart F ≃ Dart F where
  toFun := reverse F
  invFun := reverse F
  left_inv := reverse_involutive F
  right_inv := reverse_involutive F

def tail (F : Formula) : Dart F → Vertex F
  | .inl (event, false) => clauseVertex F event.1
  | .inl (event, true) => variableVertex F (nameAt (clauseAt F event.1) event.2)
  | .inr _ => dummyVertex F

def toClauseAnswer (label : Label) : ClauseAnswer := ⟨label.1, label.2.1, label.2.2⟩
def fromClauseAnswer (answer : ClauseAnswer) : Label := (answer.first, answer.second, answer.third)
def variableLabel (value : Bool) : Label := (value, false, false)

@[simp] theorem to_fromClauseAnswer (answer : ClauseAnswer) :
    toClauseAnswer (fromClauseAnswer answer) = answer := by
  cases answer; rfl

def variableValid (label : Label) : Bool :=
  decide (label.2.1 = false ∧ label.2.2 = false)

def incidenceAccept (F : Formula) (event : RandomEvent F)
    (clauseLabel variableAnswer : Label) : Bool :=
  variableValid variableAnswer &&
    (localSatisfies (clauseAt F event.1) (toClauseAnswer clauseLabel) &&
      decide (answerAt (toClauseAnswer clauseLabel) event.2 = variableAnswer.1))

def predicate (F : Formula) : Dart F → Label → Label → Bool
  | .inl (event, false), a, b => incidenceAccept F event a b
  | .inl (event, true), a, b => incidenceAccept F event b a
  | .inr _, _, _ => true

def raw (F : Formula) : ConstraintGraph (Vertex F) (Dart F) Label where
  reverse := reverseEquiv F
  reverse_involutive := reverse_involutive F
  tail := tail F
  accepts := predicate F
  reverse_accepts := by
    intro d a b
    cases d with
    | inl pair => rcases pair with ⟨event, orientation⟩; cases orientation <;> rfl
    | inr u => rfl

theorem raw_forward (F : Formula) (labeling : Vertex F → Label) (event : RandomEvent F) :
    (raw F).edgeSatisfied labeling (.inl (event, false)) =
      incidenceAccept F event (labeling (clauseVertex F event.1))
        (labeling (variableVertex F (nameAt (clauseAt F event.1) event.2))) := rfl

theorem incidenceAccept_true (F : Formula) (event : RandomEvent F) (a b : Label)
    (h : incidenceAccept F event a b = true) :
    localSatisfies (clauseAt F event.1) (toClauseAnswer a) = true ∧
      answerAt (toClauseAnswer a) event.2 = b.1 := by
  simp only [incidenceAccept, Bool.and_eq_true, decide_eq_true_eq] at h
  exact h.2

def honestLabeling (F : Formula) (A : Fin F.variables → Bool) : Vertex F → Label
  | .inl (.inl v) => variableLabel (A v)
  | .inl (.inr i) => fromClauseAnswer (honestAnswer (clauseAt F i) A)
  | .inr _ => variableLabel false

theorem incidence_honest (F : Formula) (A : Fin F.variables → Bool)
    (sat : ∀ c ∈ F.clauses, c.eval A = true) (event : RandomEvent F) :
    incidenceAccept F event (honestLabeling F A (clauseVertex F event.1))
      (honestLabeling F A (variableVertex F (nameAt (clauseAt F event.1) event.2))) = true := by
  have hc : (clauseAt F event.1).eval A = true := sat _ (List.getElem_mem _)
  simp [incidenceAccept, honestLabeling, clauseVertex, variableVertex, variableLabel,
    variableValid, honest_satisfies, honest_answerAt, hc]

theorem raw_completeness (F : Formula) (sat : F.Satisfiable) : (raw F).Satisfiable := by
  rcases sat with ⟨A, hA⟩
  refine ⟨honestLabeling F A, fun d => ?_⟩
  cases d with
  | inl pair =>
    rcases pair with ⟨event, orientation⟩
    cases orientation <;> exact incidence_honest F A hA event
  | inr u => rfl

theorem clauseAnswer_eq_of_slots (a b : ClauseAnswer)
    (h : ∀ s, answerAt a s = answerAt b s) : a = b := by
  rcases a with ⟨a₀, a₁, a₂⟩
  rcases b with ⟨b₀, b₁, b₂⟩
  have h₀ : a₀ = b₀ := h .first
  have h₁ : a₁ = b₁ := h .second
  have h₂ : a₂ = b₂ := h .third
  cases h₀; cases h₁; cases h₂; rfl

theorem raw_reflects (F : Formula) (sat : (raw F).Satisfiable) : F.Satisfiable := by
  rcases sat with ⟨labeling, hlabeling⟩
  let A : Fin F.variables → Bool := fun v => (labeling (variableVertex F v)).1
  have slot_correct (i : Fin F.clauses.length) (s : Slot) :
      localSatisfies (clauseAt F i) (toClauseAnswer (labeling (clauseVertex F i))) = true ∧
        answerAt (toClauseAnswer (labeling (clauseVertex F i))) s =
          A (nameAt (clauseAt F i) s) := by
    exact incidenceAccept_true F (i, s) _ _ (hlabeling (.inl ((i, s), false)))
  have clause_correct (i : Fin F.clauses.length) : (clauseAt F i).eval A = true := by
    have equal : toClauseAnswer (labeling (clauseVertex F i)) = honestAnswer (clauseAt F i) A :=
      clauseAnswer_eq_of_slots _ _ (fun s => by
        simpa only [honest_answerAt] using (slot_correct i s).2)
    have satisfied := (slot_correct i .first).1
    rw [equal, honest_satisfies] at satisfied
    exact satisfied
  refine ⟨A, fun c hc => ?_⟩
  obtain ⟨i, hi, heq⟩ := List.getElem_of_mem hc
  simpa only [clauseAt, heq] using clause_correct ⟨i, hi⟩

theorem raw_satisfiable_iff (F : Formula) : (raw F).Satisfiable ↔ F.Satisfiable :=
  ⟨raw_reflects F, raw_completeness F⟩

@[simp] theorem card_vertex (F : Formula) :
    Fintype.card (Vertex F) = F.variables + F.clauses.length + 1 := by
  simp [Vertex, Fintype.card_sum]

@[simp] theorem card_dart (F : Formula) :
    Fintype.card (Dart F) = 6 * F.clauses.length + 1 := by
  simp [Dart, RandomEvent, Fintype.card_sum, Fintype.card_prod]
  omega

abbrev CompactVertex (F : Formula) := Vertex (NameCompaction.compact F)
abbrev CompactDart (F : Formula) := Dart (NameCompaction.compact F)

def build (F : Formula) : ConstraintGraph (CompactVertex F) (CompactDart F) Label :=
  raw (NameCompaction.compact F)

theorem build_satisfiable_iff (F : Formula) : (build F).Satisfiable ↔ F.Satisfiable := by
  exact (raw_satisfiable_iff (NameCompaction.compact F)).trans
    (NameCompaction.compact_satisfiable_iff F)

theorem build_vertex_bound (F : Formula) :
    Fintype.card (CompactVertex F) ≤ 4 * F.clauses.length + 1 := by
  have h := NameCompaction.compact_active_bound F
  simp only [CompactVertex, card_vertex, NameCompaction.compact_clause_count]
  omega

theorem build_dart_count (F : Formula) :
    Fintype.card (CompactDart F) = 6 * F.clauses.length + 1 := by
  simp only [CompactDart, card_dart, NameCompaction.compact_clause_count]

theorem build_initial_gap (F : Formula) (unsat : ¬ F.Satisfiable)
    (labeling : CompactVertex F → Label) : 1 ≤ (build F).rejectionCount labeling := by
  apply ConstraintGraph.rejectionCount_positive
  intro h
  exact unsat ((build_satisfiable_iff F).mp h)

theorem build_inverse_size_gap (F : Formula) (unsat : ¬ F.Satisfiable)
    (labeling : CompactVertex F → Label) :
    Fintype.card (CompactDart F) ≤
      (6 * F.clauses.length + 1) * (build F).rejectionCount labeling := by
  rw [build_dart_count]
  simpa only [Nat.mul_one] using
    Nat.mul_le_mul_left (6 * F.clauses.length + 1) (build_initial_gap F unsat labeling)

end BinPackingGames.Foundations.PCP.InitialGraph

namespace BinPackingGames.Foundations.PCP.AlphabetRetraction

variable {V E A B : Type*}

def pullback (G : ConstraintGraph V E A) (decode : B → A) : ConstraintGraph V E B where
  reverse := G.reverse
  reverse_involutive := G.reverse_involutive
  tail := G.tail
  accepts := fun e b c => G.accepts e (decode b) (decode c)
  reverse_accepts := fun e b c => G.reverse_accepts e (decode b) (decode c)

@[simp] theorem edgeSatisfied_pullback (G : ConstraintGraph V E A)
    (decode : B → A) (labeling : V → B) (e : E) :
    (pullback G decode).edgeSatisfied labeling e =
      G.edgeSatisfied (decode ∘ labeling) e := rfl

@[simp] theorem rejectionCount_pullback [Fintype E] (G : ConstraintGraph V E A)
    (decode : B → A) (labeling : V → B) :
    (pullback G decode).rejectionCount labeling =
      G.rejectionCount (decode ∘ labeling) := rfl

theorem satisfiable_pullback_iff (G : ConstraintGraph V E A)
    (decode : B → A) (encode : A → B) (sectionLaw : ∀ a, decode (encode a) = a) :
    (pullback G decode).Satisfiable ↔ G.Satisfiable := by
  constructor
  · rintro ⟨labeling, h⟩
    exact ⟨decode ∘ labeling, h⟩
  · rintro ⟨labeling, h⟩
    refine ⟨encode ∘ labeling, fun e => ?_⟩
    change G.accepts e (decode (encode (labeling (G.tail e))))
      (decode (encode (labeling (G.head e)))) = true
    simpa only [ConstraintGraph.edgeSatisfied, sectionLaw] using h e

abbrev Label64 := QueryIncidence.Label 6

def decode64 (label : Label64) : InitialGraph.Label := (label 0, label 1, label 2)

def encode64 (label : InitialGraph.Label) : Label64 :=
  fun i => if i = 0 then label.1 else if i = 1 then label.2.1 else label.2.2

@[simp] theorem decode_encode64 (label : InitialGraph.Label) :
    decode64 (encode64 label) = label := by
  rcases label with ⟨a, b, c⟩
  rfl

def initial64 (F : BinPackingGames.Foundations.Target.Formula) :
    ConstraintGraph (InitialGraph.CompactVertex F) (InitialGraph.CompactDart F) Label64 :=
  pullback (InitialGraph.build F) decode64

theorem initial64_satisfiable_iff (F : BinPackingGames.Foundations.Target.Formula) :
    (initial64 F).Satisfiable ↔ F.Satisfiable :=
  (satisfiable_pullback_iff (InitialGraph.build F) decode64 encode64 decode_encode64).trans
    (InitialGraph.build_satisfiable_iff F)

theorem initial64_alphabet : Fintype.card Label64 = 64 :=
  QueryIncidence.six_query_alphabet

theorem initial64_inverse_size_gap (F : BinPackingGames.Foundations.Target.Formula)
    (unsat : ¬ F.Satisfiable) (labeling : InitialGraph.CompactVertex F → Label64) :
    Fintype.card (InitialGraph.CompactDart F) ≤
      (6 * F.clauses.length + 1) * (initial64 F).rejectionCount labeling :=
  InitialGraph.build_inverse_size_gap F unsat (decode64 ∘ labeling)

end BinPackingGames.Foundations.PCP.AlphabetRetraction

end OAI