Datasets:

roseking
/

openAI-math-dataset

import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Expect
import Mathlib.Data.Fin.Tuple.Basic
import Mathlib.Data.Finset.Max
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Sigma
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.List.OfFn
import Mathlib.Basic.Real.Basic
import OAI.Computability.UniqueGames.PCP.AlphabetReductionLemmas
import OAI.Computability.UniqueGames.PCP.InitialGraphLemmas

namespace OAI

section

/-! Full-radius local opinions use a common alphabet indexed by bounded
port addresses. Every opinion is decoded through a fixed address selector;
arbitrary alphabet symbols need not agree on colliding addresses. -/

namespace UniqueGamesTheorem.Foundations.PCP.PoweringLabels

open PoweringWalks
open scoped BigOperators

variable {V D A : Type*}

abbrev PortWords (D : Type*) (t : Nat) := (n : Fin (t + 1)) × (Fin n.val → D)
instance finitePortWords [Finite D] (t : Nat) : Finite (PortWords D t) := by
  let := Fintype.ofFinite D
  exact Finite.of_fintype _
def Ball (G : PortGraph V D) (t : Nat) (v : V) :=
  {u : V // ∃ n, n ≤ t ∧ ∃ p : Fin n → D, wordEnd G n v p = u}
def wordToBall (G : PortGraph V D) (t : Nat) (v : V) (w : PortWords D t) : Ball G t v :=
  ⟨wordEnd G w.1.val v w.2, w.1.val, Nat.le_of_lt_succ w.1.isLt, w.2, rfl⟩
theorem wordToBall_surjective (G : PortGraph V D) (t : Nat) (v : V) :
    Function.Surjective (wordToBall G t v) := by
  intro u
  obtain ⟨n, hn, p, hp⟩ := u.property
  refine ⟨⟨⟨n, Nat.lt_succ_of_le hn⟩, p⟩, ?_⟩
  exact Subtype.ext hp
instance finiteBall [Finite D] (G : PortGraph V D) (t : Nat) (v : V) :
    Finite (Ball G t v) :=
  Finite.of_surjective (wordToBall G t v) (wordToBall_surjective G t v)
theorem card_portWords [Finite D] (t : Nat) :
    Nat.card (PortWords D t) = ∑ n : Fin (t + 1), Nat.card D ^ n.val := by
  simp [PortWords, Nat.card_sigma, Nat.card_fun]
theorem card_ball_le [Finite D] (G : PortGraph V D) (t : Nat) (v : V) :
    Nat.card (Ball G t v) ≤ ∑ n : Fin (t + 1), Nat.card D ^ n.val := by
  calc
    _ ≤ Nat.card (PortWords D t) := Nat.card_le_card_of_surjective
      (wordToBall G t v) (wordToBall_surjective G t v)
    _ = _ := card_portWords t
structure AddressSelector (G : PortGraph V D) (t : Nat) (v : V) where
  address : Ball G t v → PortWords D t
  correct : ∀ u, wordToBall G t v (address u) = u

noncomputable def classicalSelector (G : PortGraph V D) (t : Nat) (v : V) :
    AddressSelector G t v where
  address u := Classical.choose (wordToBall_surjective G t v u)
  correct u := Classical.choose_spec (wordToBall_surjective G t v u)

/-- Executable first-match search, with a proof that a match is present. -/
def scanWitness {A : Type*} (p : A → Prop) [DecidablePred p] :
    (xs : List A) → (∃ a ∈ xs, p a) → {a // p a}
  | [], h => False.elim (by simp at h)
  | a :: xs, h =>
    if ha : p a then ⟨a, ha⟩ else
      scanWitness p xs (by
        obtain ⟨b, hb, hp⟩ := h
        rcases List.mem_cons.mp hb with he | hm
        · exact False.elim (ha (he ▸ hp))
        · exact ⟨b, hm, hp⟩)

/-- A concrete complete address list supplies a computational selector. Its
length depends on the degree and radius, not on the ambient vertex count. -/
def listSelector [DecidableEq V] (G : PortGraph V D) (t : Nat) (v : V)
    (addresses : List (PortWords D t)) (complete : ∀ w, w ∈ addresses) :
    AddressSelector G t v where
  address u := (scanWitness (fun w => (wordToBall G t v w).val = u.val) addresses (by
    obtain ⟨w, hw⟩ := wordToBall_surjective G t v u
    exact ⟨w, complete w, congrArg Subtype.val hw⟩)).val
  correct u := Subtype.ext (scanWitness
    (fun w => (wordToBall G t v w).val = u.val) addresses (by
      obtain ⟨w, hw⟩ := wordToBall_surjective G t v u
      exact ⟨w, complete w, congrArg Subtype.val hw⟩)).property

abbrev PaddedLabel (D : Type*) (t : Nat) (A : Type*) := PortWords D t → A

def encode (G : PortGraph V D) (t : Nat) (v : V) (ℓ : Ball G t v → A) :
    PaddedLabel D t A := fun w => ℓ (wordToBall G t v w)

def decode {G : PortGraph V D} {t : Nat} {v : V} (S : AddressSelector G t v)
    (a : PaddedLabel D t A) : Ball G t v → A := fun u => a (S.address u)

theorem decode_encode {G : PortGraph V D} {t : Nat} {v : V}
    (S : AddressSelector G t v) (ℓ : Ball G t v → A) :
    decode S (encode G t v ℓ) = ℓ := by
  funext u
  change ℓ (wordToBall G t v (S.address u)) = ℓ u
  rw [S.correct]

theorem decode_surjective {G : PortGraph V D} {t : Nat} {v : V}
    (S : AddressSelector G t v) : Function.Surjective (decode (A := A) S) :=
  fun ℓ => ⟨encode G t v ℓ, decode_encode S ℓ⟩

theorem encode_injective (G : PortGraph V D) (t : Nat) (v : V) :
    Function.Injective (encode (A := A) G t v) := by
  intro a b h
  funext u
  obtain ⟨w, rfl⟩ := wordToBall_surjective G t v u
  exact congrFun h w

theorem decoded_word_collision {G : PortGraph V D} {t : Nat} {v : V}
    (S : AddressSelector G t v) (a : PaddedLabel D t A) (w z : PortWords D t)
    (h : (wordToBall G t v w).val = (wordToBall G t v z).val) :
    decode S a (wordToBall G t v w) = decode S a (wordToBall G t v z) :=
  congrArg (decode S a) (Subtype.ext h)

theorem card_paddedLabel [Finite D] [Finite A] (t : Nat) :
    Nat.card (PaddedLabel D t A) =
      Nat.card A ^ (∑ n : Fin (t + 1), Nat.card D ^ n.val) := by
  rw [Nat.card_fun, card_portWords]

end UniqueGamesTheorem.Foundations.PCP.PoweringLabels

end

section

/-!
# Executable bounded port-address enumeration

For a port alphabet `Fin d`, the lists below enumerate every word of every
length at most the fixed radius. The resulting first-match selector searches
these addresses and never enumerates the ambient vertex type.
-/

namespace UniqueGamesTheorem.Foundations.PCP.PoweringAddresses

open PoweringWalks PoweringLabels
open scoped BigOperators

variable {V : Type*}

/-- All words of a specified length, ordered first by their head port. -/
def allWords (d : Nat) : (n : Nat) → List (Fin n → Fin d)
  | 0 => [Fin.elim0]
  | n + 1 => (List.finRange d).flatMap fun a =>
      (allWords d n).map fun p => Fin.cases a p

theorem mem_allWords (d : Nat) :
    ∀ n (p : Fin n → Fin d), p ∈ allWords d n := by
  intro n
  induction n with
  | zero =>
    intro p
    simp only [allWords, List.mem_singleton]
    funext i
    exact Fin.elim0 i
  | succ n ih =>
    intro p
    simp only [allWords, List.mem_flatMap, List.mem_map]
    refine ⟨p 0, List.mem_finRange (p 0), (fun j => p j.succ), ih _, ?_⟩
    funext j
    exact Fin.cases rfl (fun _ => rfl) j

/-- The complete address list, including the unique empty word. -/
def allAddresses (d t : Nat) : List (PortWords (Fin d) t) :=
  (List.finRange (t + 1)).flatMap fun n =>
    (allWords d n.val).map fun p => Sigma.mk n p

theorem mem_allAddresses (d t : Nat) (w : PortWords (Fin d) t) :
    w ∈ allAddresses d t := by
  rcases w with ⟨n, p⟩
  unfold allAddresses
  apply List.mem_flatMap.mpr
  refine ⟨n, List.mem_finRange n, ?_⟩
  exact List.mem_map.mpr ⟨p, mem_allWords d n.val p, rfl⟩

theorem length_allWords (d : Nat) :
    ∀ n, (allWords d n).length = d ^ n := by
  intro n
  induction n with
  | zero => rfl
  | succ n ih =>
    simp only [allWords, List.length_flatMap, List.length_map, ih,
      List.map_const', List.length_finRange, List.sum_replicate_nat]
    exact (Nat.mul_comm d (d ^ n)).trans (Nat.pow_succ d n).symm

/-- The selector's search-list bound depends only on degree and radius. -/
theorem length_allAddresses (d t : Nat) :
    (allAddresses d t).length = ∑ n : Fin (t + 1), d ^ n.val := by
  unfold allAddresses
  simp only [List.length_flatMap, List.length_map, length_allWords]
  exact (Fin.sum_univ_def (fun n : Fin (t + 1) => d ^ n.val)).symm

/-- A fully executable selector from the explicit bounded address list. -/
def finitePortSelector {d : Nat} [DecidableEq V] (G : PortGraph V (Fin d))
    (t : Nat) (v : V) : AddressSelector G t v :=
  listSelector G t v (allAddresses d t) (mem_allAddresses d t)

end UniqueGamesTheorem.Foundations.PCP.PoweringAddresses

end

section

/-! Modal decoding is an actual maximum of finite empirical letter masses.
Its reciprocal-alphabet lower bound is derived from their total mass one. -/

namespace UniqueGamesTheorem.Foundations.PCP.PoweringDecoding

open scoped BigOperators

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

noncomputable def mass (opinion : Ω → A) (a : A) : ℝ := by
  classical
  exact Finset.univ.expect (fun ω => if opinion ω = a then 1 else 0)

omit [Nonempty A] in
theorem mass_sum (opinion : Ω → A) : (∑ a, mass opinion a) = 1 := by
  classical
  calc
    (∑ a, mass opinion a) =
        (Finset.univ : Finset Ω).expect
          (fun ω => ∑ a : A, if opinion ω = a then (1 : ℝ) else 0) :=
      (Finset.expect_sum_comm (Finset.univ : Finset Ω) (Finset.univ : Finset A)
        (fun ω a => if opinion ω = a then (1 : ℝ) else 0)).symm
    _ = (Finset.univ : Finset Ω).expect (fun _ => (1 : ℝ)) := by
      apply Finset.expect_congr rfl
      intro ω _
      simp
    _ = 1 := Finset.expect_const Finset.univ_nonempty 1

omit [Nonempty Ω] in
theorem exists_mode (opinion : Ω → A) :
    ∃ a : A, ∀ b : A, mass opinion b ≤ mass opinion a := by
  obtain ⟨a, _, ha⟩ := Finset.exists_max_image (Finset.univ : Finset A)
    (mass opinion) Finset.univ_nonempty
  exact ⟨a, fun b => ha b (Finset.mem_univ b)⟩

noncomputable def mode (opinion : Ω → A) : A := Classical.choose (exists_mode opinion)

omit [Nonempty Ω] in
theorem mode_max (opinion : Ω → A) (b : A) :
    mass opinion b ≤ mass opinion (mode opinion) :=
  Classical.choose_spec (exists_mode opinion) b

theorem mode_mass_lower (opinion : Ω → A) :
    1 / (Fintype.card A : ℝ) ≤ mass opinion (mode opinion) := by
  have hk : 0 < (Fintype.card A : ℝ) := Nat.cast_pos.mpr Fintype.card_pos
  have hsum : (1 : ℝ) ≤ mass opinion (mode opinion) * (Fintype.card A : ℝ) := by
    calc
      (1 : ℝ) = ∑ a : A, mass opinion a := (mass_sum opinion).symm
      _ ≤ ∑ _a : A, mass opinion (mode opinion) :=
        Finset.sum_le_sum (fun a _ => mode_max opinion a)
      _ = mass opinion (mode opinion) * (Fintype.card A : ℝ) := by simp [mul_comm]
  exact (div_le_iff₀ hk).2 hsum

theorem exists_letter_mass_lower (opinion : Ω → A) :
    ∃ a : A, 1 / (Fintype.card A : ℝ) ≤ mass opinion a :=
  ⟨mode opinion, mode_mass_lower opinion⟩

end UniqueGamesTheorem.Foundations.PCP.PoweringDecoding

end

section

/-!
# Executable integer encodings for the powering construction

All encodings below have explicit inverses. They use the arithmetic finite
function, sigma, and product equivalences rather than choosing enumerations
from cardinality proofs. Word digits use the head-major convention; addresses
are grouped by increasing word length. A dart is numbered first by its vertex,
then its word, with its two orientations adjacent.
-/

namespace UniqueGamesTheorem.Foundations.PCP.PoweringEnumeration

open PoweringLabels
open scoped BigOperators

/-- The number of words with length at most `t`, including the empty word. -/
def addressCount (d t : Nat) : Nat := ∑ i : Fin (t + 1), d ^ i.val

/-- Reverse the digit positions before using the little-endian arithmetic
encoding from mathlib. Both directions are the same executable operation. -/
def reverseWordEquiv (d n : Nat) : (Fin n → Fin d) ≃ (Fin n → Fin d) where
  toFun p i := p i.rev
  invFun p i := p i.rev
  left_inv p := by funext i; simp only [Fin.rev_rev]
  right_inv p := by funext i; simp only [Fin.rev_rev]

/-- An executable bijection from fixed-length port words to their ranks. -/
def wordEquiv (d n : Nat) : (Fin n → Fin d) ≃ Fin (d ^ n) :=
  (reverseWordEquiv d n).trans finFunctionFinEquiv

theorem wordEquiv_val (d n : Nat) (p : Fin n → Fin d) :
    (wordEquiv d n p).val = ∑ i : Fin n, (p i.rev).val * d ^ i.val := rfl

/-- Arithmetic prefix sums give a rank for every bounded-length address. -/
def addressEquiv (d t : Nat) : PortWords (Fin d) t ≃ Fin (addressCount d t) :=
  (Equiv.sigmaCongrRight (fun i : Fin (t + 1) => wordEquiv d i.val)).trans
    finSigmaFinEquiv

theorem addressEquiv_val (d t : Nat) (w : PortWords (Fin d) t) :
    (addressEquiv d t w).val =
      (∑ i : Fin w.1.val, d ^ i.val) + (wordEquiv d w.1.val w.2).val := by
  change (finSigmaFinEquiv
    (⟨w.1, wordEquiv d w.1.val w.2⟩ : (i : Fin (t + 1)) × Fin (d ^ i.val))).val = _
  rw [finSigmaFinEquiv_apply]
  rfl

/-- A padded label is a base-`q` integer whose digits are indexed by the
explicit bounded-address equivalence. -/
def paddedLabelEquiv (d t q : Nat) :
    PaddedLabel (Fin d) t (Fin q) ≃ Fin (q ^ addressCount d t) :=
  ((addressEquiv d t).arrowCongr (Equiv.refl (Fin q))).trans
    (wordEquiv q (addressCount d t))

theorem paddedLabelEquiv_val (d t q : Nat) (a : PaddedLabel (Fin d) t (Fin q)) :
    (paddedLabelEquiv d t q a).val =
      ∑ i : Fin (addressCount d t),
        (a ((addressEquiv d t).symm i.rev)).val * q ^ i.val := rfl

/-- False and true are the adjacent orientation digits zero and one. -/
def orientationEquiv : Bool ≃ Fin 2 := finTwoEquiv.symm

@[simp] theorem orientationEquiv_false : orientationEquiv false = 0 := rfl
@[simp] theorem orientationEquiv_true : orientationEquiv true = 1 := rfl

/-- A vertex's block consists of one adjacent orientation pair per word. -/
def dartBlockEquiv (d n : Nat) :
    ((Fin (n + 1) → Fin d) × Bool) ≃ Fin (2 * d ^ (n + 1)) :=
  ((wordEquiv d (n + 1)).prodCongr orientationEquiv).trans
    (finProdFinEquiv.trans (finCongr (Nat.mul_comm (d ^ (n + 1)) 2)))

theorem dartBlockEquiv_val (d n : Nat) (p : Fin (n + 1) → Fin d) (b : Bool) :
    (dartBlockEquiv d n (p, b)).val =
      2 * (wordEquiv d (n + 1) p).val + (orientationEquiv b).val := by
  change (orientationEquiv b).val + 2 * (wordEquiv d (n + 1) p).val = _
  exact Nat.add_comm _ _

/-- The tuple permutation putting the vertex before its word and orientation. -/
def dartVertexEquiv (vertices d n : Nat) :
    (Bool × (Fin vertices × (Fin (n + 1) → Fin d))) ≃
      (Fin vertices × ((Fin (n + 1) → Fin d) × Bool)) where
  toFun x := (x.2.1, (x.2.2, x.1))
  invFun x := (x.2.2, (x.1, x.2.1))
  left_inv x := by rcases x with ⟨b, v, p⟩; rfl
  right_inv x := by rcases x with ⟨v, p, b⟩; rfl

/-- Executable vertex-major dart numbering, with adjacent orientations. -/
def dartEquiv (vertices d n : Nat) :
    (Bool × (Fin vertices × (Fin (n + 1) → Fin d))) ≃
      Fin (2 * vertices * d ^ (n + 1)) :=
  (dartVertexEquiv vertices d n).trans
    (((Equiv.refl (Fin vertices)).prodCongr (dartBlockEquiv d n)).trans
      (finProdFinEquiv.trans
        (finCongr (show vertices * (2 * d ^ (n + 1)) =
          2 * vertices * d ^ (n + 1) by ac_rfl))))

theorem dartEquiv_val (vertices d n : Nat) (b : Bool) (v : Fin vertices)
    (p : Fin (n + 1) → Fin d) :
    (dartEquiv vertices d n (b, (v, p))).val =
      (2 * d ^ (n + 1)) * v.val +
        2 * (wordEquiv d (n + 1) p).val + (orientationEquiv b).val := by
  change (dartBlockEquiv d n (p, b)).val + (2 * d ^ (n + 1)) * v.val = _
  rw [dartBlockEquiv_val]
  omega

/-- The true orientation immediately follows the false orientation. -/
theorem dartEquiv_true_eq_false_add_one (vertices d n : Nat) (v : Fin vertices)
    (p : Fin (n + 1) → Fin d) :
    (dartEquiv vertices d n (true, (v, p))).val =
      (dartEquiv vertices d n (false, (v, p))).val + 1 := by
  rw [dartEquiv_val, dartEquiv_val, orientationEquiv_true, orientationEquiv_false]
  rfl

def encodeWord (d n : Nat) : (Fin n → Fin d) → Fin (d ^ n) := wordEquiv d n
def decodeWord (d n : Nat) : Fin (d ^ n) → (Fin n → Fin d) := (wordEquiv d n).symm

@[simp] theorem decodeWord_encodeWord (d n : Nat) (p : Fin n → Fin d) :
    decodeWord d n (encodeWord d n p) = p := (wordEquiv d n).symm_apply_apply p

@[simp] theorem encodeWord_decodeWord (d n : Nat) (i : Fin (d ^ n)) :
    encodeWord d n (decodeWord d n i) = i := (wordEquiv d n).apply_symm_apply i

def encodeAddress (d t : Nat) : PortWords (Fin d) t → Fin (addressCount d t) :=
  addressEquiv d t

def decodeAddress (d t : Nat) : Fin (addressCount d t) → PortWords (Fin d) t :=
  (addressEquiv d t).symm

@[simp] theorem decodeAddress_encodeAddress (d t : Nat) (w : PortWords (Fin d) t) :
    decodeAddress d t (encodeAddress d t w) = w := (addressEquiv d t).symm_apply_apply w

@[simp] theorem encodeAddress_decodeAddress (d t : Nat) (i : Fin (addressCount d t)) :
    encodeAddress d t (decodeAddress d t i) = i := (addressEquiv d t).apply_symm_apply i

def encodeLabel (d t q : Nat) :
    PaddedLabel (Fin d) t (Fin q) → Fin (q ^ addressCount d t) := paddedLabelEquiv d t q

def decodeLabel (d t q : Nat) :
    Fin (q ^ addressCount d t) → PaddedLabel (Fin d) t (Fin q) :=
  (paddedLabelEquiv d t q).symm

@[simp] theorem decodeLabel_encodeLabel (d t q : Nat) (a : PaddedLabel (Fin d) t (Fin q)) :
    decodeLabel d t q (encodeLabel d t q a) = a := (paddedLabelEquiv d t q).symm_apply_apply a

@[simp] theorem encodeLabel_decodeLabel (d t q : Nat) (i : Fin (q ^ addressCount d t)) :
    encodeLabel d t q (decodeLabel d t q i) = i := (paddedLabelEquiv d t q).apply_symm_apply i

@[simp] theorem decodeLabel_apply (d t q : Nat) (i : Fin (q ^ addressCount d t))
    (w : PortWords (Fin d) t) :
    decodeLabel d t q i w = decodeWord q (addressCount d t) i (encodeAddress d t w) := rfl

def encodeDart (vertices d n : Nat) :
    (Bool × (Fin vertices × (Fin (n + 1) → Fin d))) →
      Fin (2 * vertices * d ^ (n + 1)) := dartEquiv vertices d n

def decodeDart (vertices d n : Nat) :
    Fin (2 * vertices * d ^ (n + 1)) →
      (Bool × (Fin vertices × (Fin (n + 1) → Fin d))) := (dartEquiv vertices d n).symm

@[simp] theorem decodeDart_encodeDart (vertices d n : Nat)
    (e : Bool × (Fin vertices × (Fin (n + 1) → Fin d))) :
    decodeDart vertices d n (encodeDart vertices d n e) = e :=
  (dartEquiv vertices d n).symm_apply_apply e

@[simp] theorem encodeDart_decodeDart (vertices d n : Nat)
    (i : Fin (2 * vertices * d ^ (n + 1))) :
    encodeDart vertices d n (decodeDart vertices d n i) = i :=
  (dartEquiv vertices d n).apply_symm_apply i

theorem card_words (d n : Nat) : Nat.card (Fin n → Fin d) = d ^ n := by
  simpa only [Nat.card_fin] using Nat.card_congr (wordEquiv d n)

theorem card_addresses (d t : Nat) :
    Nat.card (PortWords (Fin d) t) = addressCount d t := by
  simpa only [Nat.card_fin] using Nat.card_congr (addressEquiv d t)

theorem card_labels (d t q : Nat) :
    Nat.card (PaddedLabel (Fin d) t (Fin q)) = q ^ addressCount d t := by
  simpa only [Nat.card_fin] using Nat.card_congr (paddedLabelEquiv d t q)

theorem card_darts (vertices d n : Nat) :
    Nat.card (Bool × (Fin vertices × (Fin (n + 1) → Fin d))) =
      2 * vertices * d ^ (n + 1) := by
  simpa only [Nat.card_fin] using Nat.card_congr (dartEquiv vertices d n)

theorem length_allAddresses (d t : Nat) :
    (PoweringAddresses.allAddresses d t).length = addressCount d t :=
  PoweringAddresses.length_allAddresses d t

end UniqueGamesTheorem.Foundations.PCP.PoweringEnumeration

end

section

/-!
# Full-radius access to the endpoints of each actual walk edge

The tail of each traversed edge is reachable from the walk's start. Its head
is reachable from the walk's end by reversing the remaining suffix with the
actual arrival ports. These facts supply values in the two neighborhood-ball
types used by the endpoint label decoders.
-/

namespace UniqueGamesTheorem.Foundations.PCP.PoweringReach

open PoweringWalks PoweringLabels

variable {V D : Type*}

def ReachLE (G : PortGraph V D) (t : Nat) (u v : V) : Prop :=
  ∃ l, l ≤ t ∧ ∃ p : Fin l → D, wordEnd G l u p = v

theorem reach_refl (G : PortGraph V D) (t : Nat) (u : V) : ReachLE G t u u :=
  ⟨0, Nat.zero_le t, Fin.elim0, rfl⟩

theorem reach_mono {G : PortGraph V D} {s t : Nat} {u v : V}
    (hst : s ≤ t) (h : ReachLE G s u v) : ReachLE G t u v := by
  obtain ⟨l, hl, p, hp⟩ := h
  exact ⟨l, hl.trans hst, p, hp⟩

theorem reach_prepend (G : PortGraph V D) {t : Nat} (u : V) (d : D) {v : V}
    (h : ReachLE G t (next G u d) v) : ReachLE G (t + 1) u v := by
  obtain ⟨l, hl, p, hp⟩ := h
  refine ⟨l + 1, Nat.succ_le_succ hl,
    (fun i : Fin (l + 1) => Fin.cases d p i), ?_⟩
  simpa only [wordEnd, Fin.cases_zero, Fin.cases_succ] using hp

/-- Reversal records each arrival port, not the original outgoing port. -/
def reversePorts (G : PortGraph V D) : V → List D → List D
  | _, [] => []
  | v, d :: ds => reversePorts G (next G v d) ds ++ [(G.rot (v, d)).2]

theorem reversePorts_length (G : PortGraph V D) (v : V) (ds : List D) :
    (reversePorts G v ds).length = ds.length := by
  induction ds generalizing v with
  | nil => rfl
  | cons d ds ih => simp [reversePorts, ih]

theorem walkEnd_reversePorts (G : PortGraph V D) (v : V) (ds : List D) :
    walkEnd G (walkEnd G v ds) (reversePorts G v ds) = v := by
  induction ds generalizing v with
  | nil => rfl
  | cons d ds ih =>
    change walkEnd G (walkEnd G (next G v d) ds)
      (reversePorts G (next G v d) ds ++ [(G.rot (v, d)).2]) = v
    rw [walkEnd_append, ih]
    change (G.rot (G.rot (v, d))).1 = v
    exact congrArg Prod.fst (G.rot_involutive (v, d))

theorem wordEnd_eq_walkEnd_ofFn (G : PortGraph V D) :
    ∀ n v (p : Fin n → D), wordEnd G n v p = walkEnd G v (List.ofFn p) := by
  intro n
  induction n with
  | zero => intro v p; rfl
  | succ n ih =>
    intro v p
    rw [List.ofFn_succ, walkEnd_cons]
    exact ih (next G v (p 0)) (fun j => p j.succ)

theorem reach_reverse {G : PortGraph V D} {t : Nat} {u v : V}
    (h : ReachLE G t u v) : ReachLE G t v u := by
  obtain ⟨l, hl, p, hp⟩ := h
  refine ⟨(reversePorts G u (List.ofFn p)).length, ?_,
    (reversePorts G u (List.ofFn p)).get, ?_⟩
  · simpa only [reversePorts_length, List.length_ofFn] using hl
  · rw [wordEnd_eq_walkEnd_ofFn, List.ofFn_get]
    have hlist : walkEnd G u (List.ofFn p) = v := by
      rw [← wordEnd_eq_walkEnd_ofFn]
      exact hp
    rw [← hlist, walkEnd_reversePorts]

/-- The actual prefix places each traversed edge's tail in the start ball. -/
theorem tail_reach (G : PortGraph V D) :
    ∀ (n : Nat) (w : Walk V D (n + 1)) (k : Fin (n + 1)),
      ReachLE G (n + 1) w.1 (edgeAt G n w k).1 := by
  intro n
  induction n with
  | zero =>
    intro w k
    exact reach_refl G 1 w.1
  | succ n ih =>
    intro w k
    refine Fin.cases ?_ (fun j => ?_) k
    · exact reach_refl G (n + 2) w.1
    · exact reach_prepend G w.1 (w.2 0) (ih (advanceTail G w) j)

/-- The suffix proceeds from the traversed edge's head to the walk endpoint. -/
theorem head_reach_endpoint (G : PortGraph V D) :
    ∀ (n : Nat) (w : Walk V D (n + 1)) (k : Fin (n + 1)),
      ReachLE G (n + 1) (G.rot (edgeAt G n w k)).1 (endpoint G w) := by
  intro n
  induction n with
  | zero =>
    intro w k
    exact reach_refl G 1 (G.rot (w.1, w.2 0)).1
  | succ n ih =>
    intro w k
    refine Fin.cases ?_ (fun j => ?_) k
    · exact ⟨n + 1, Nat.le_succ (n + 1), (fun j => w.2 j.succ), rfl⟩
    · exact reach_mono (Nat.le_succ (n + 1)) (ih (advanceTail G w) j)

def tailFromStart (G : PortGraph V D) (n : Nat) (w : Walk V D (n + 1))
    (k : Fin (n + 1)) : Ball G (n + 1) w.1 :=
  ⟨(edgeAt G n w k).1, tail_reach G n w k⟩

def headFromEnd (G : PortGraph V D) (n : Nat) (w : Walk V D (n + 1))
    (k : Fin (n + 1)) : Ball G (n + 1) (endpoint G w) :=
  ⟨(G.rot (edgeAt G n w k)).1, reach_reverse (head_reach_endpoint G n w k)⟩

@[simp] theorem tailFromStart_val (G : PortGraph V D) (n : Nat)
    (w : Walk V D (n + 1)) (k : Fin (n + 1)) :
    (tailFromStart G n w k).val = (edgeAt G n w k).1 := rfl

@[simp] theorem headFromEnd_val (G : PortGraph V D) (n : Nat)
    (w : Walk V D (n + 1)) (k : Fin (n + 1)) :
    (headFromEnd G n w k).val = (G.rot (edgeAt G n w k)).1 := rfl

end UniqueGamesTheorem.Foundations.PCP.PoweringReach

end

section

/-!
An actual incidence constraint graph for a finite Boolean verifier with q queries.
Left labels are complete local answer tuples. Their validity includes consistency
at repeated proof addresses. A dart checks the tuple at the least position naming
its proof address. Both sides use the same 2^q-element alphabet; arbitrary right
labels are explicitly decoded by their first coordinate, and honest right labels
are constant tuples. Every incidence is represented by two opposite darts.

The count theorem proves a loss of exactly q in a supplied verifier rejection
fraction. It supplies a concrete construction, not a PCP-hardness premise. The
generic count statements permit an empty event type and introduce no dummy darts.
-/

namespace UniqueGamesTheorem.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

/-- Every rejected event contributes two distinct rejected darts, one in each
orientation. A classical witness is used only in this counting proof. -/
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 UniqueGamesTheorem.Foundations.PCP.QueryIncidence

end

end OAI