lean/OAI/Computability/BinPacking/PCP/RoundTableGap.lean

import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.List.OfFn
import OAI.Computability.BinPacking.PCP.PCPIteration

namespace OAI

namespace BinPackingGames.Foundations.PCP.AlphabetTable.Enumeration

open BinPackingGames.Foundations.Hastad
open scoped BigOperators

def bitValue (bit : Bool) : Nat := if bit then 1 else 0

def boolEquiv : Bool ≃ Fin 2 := finTwoEquiv.symm

@[simp] theorem boolEquiv_val (bit : Bool) : (boolEquiv bit).val = bitValue bit := by
  cases bit <;> rfl

def productEquiv {A B : Type*} {a b : Nat} (left : A ≃ Fin a) (right : B ≃ Fin b) :
    A × B ≃ Fin (a * b) :=
  (Equiv.prodCongr left right).trans finProdFinEquiv

@[simp] theorem productEquiv_val {A B : Type*} {a b : Nat}
    (left : A ≃ Fin a) (right : B ≃ Fin b) (x : A) (y : B) :
    (productEquiv left right (x, y)).val = (left x).val * b + (right y).val := by
  change (right y).val + b * (left x).val = _
  simp [Nat.mul_comm, Nat.add_comm]

def sumEquiv {A B : Type*} {a b : Nat} (left : A ≃ Fin a) (right : B ≃ Fin b) :
    A ⊕ B ≃ Fin (a + b) :=
  (Equiv.sumCongr left right).trans finSumFinEquiv

@[simp] theorem sumEquiv_inl_val {A B : Type*} {a b : Nat}
    (left : A ≃ Fin a) (right : B ≃ Fin b) (x : A) :
    (sumEquiv left right (.inl x)).val = (left x).val := rfl

@[simp] theorem sumEquiv_inr_val {A B : Type*} {a b : Nat}
    (left : A ≃ Fin a) (right : B ≃ Fin b) (y : B) :
    (sumEquiv left right (.inr y)).val = a + (right y).val := rfl

def tapeCount (q : Nat) : Nat := 2 ^ q
def pairTapeCount (q : Nat) : Nat := 2 ^ (q * q)
def fiveTapeCount (q : Nat) : Nat := pairTapeCount q ^ 5
def localCount (q : Nat) : Nat := 4 * (2 * tapeCount q) * fiveTapeCount q
def eventCount (m q : Nat) : Nat := m * localCount q
def addressCount (n m q : Nat) : Nat := n * tapeCount q + m * pairTapeCount q
def vertexCount (n m q : Nat) : Nat := eventCount m q + addressCount n m q
def dartCount (m q : Nat) : Nat := eventCount m q * 12

def cubeEquiv (q : Nat) : Cube (Fin q) ≃ Fin (tapeCount q) :=
  (Equiv.arrowCongr (Equiv.refl (Fin q)) boolEquiv).trans finFunctionFinEquiv

def cubeRank {q : Nat} (tape : Cube (Fin q)) : Fin (tapeCount q) := cubeEquiv q tape
def cubeUnrank {q : Nat} (index : Fin (tapeCount q)) : Cube (Fin q) :=
  (cubeEquiv q).symm index

@[simp] theorem cubeUnrank_cubeRank {q : Nat} (tape : Cube (Fin q)) :
    cubeUnrank (cubeRank tape) = tape := (cubeEquiv q).symm_apply_apply tape

@[simp] theorem cubeRank_cubeUnrank {q : Nat} (index : Fin (tapeCount q)) :
    cubeRank (cubeUnrank index) = index := (cubeEquiv q).apply_symm_apply index

theorem cubeUnrank_digit {q : Nat} (index : Fin (tapeCount q)) (i : Fin q) :
    bitValue (cubeUnrank index i) = index.val / 2 ^ i.val % 2 := by
  have digit : boolEquiv (cubeUnrank index i) =
      (⟨index.val / 2 ^ i.val % 2, Nat.mod_lt _ (by decide)⟩ : Fin 2) := by
    change boolEquiv (boolEquiv.symm _) = _
    exact boolEquiv.apply_symm_apply _
  simpa only [boolEquiv_val] using congrArg Fin.val digit

theorem cubeEquiv_val (q : Nat) (tape : Cube (Fin q)) :
    (cubeEquiv q tape).val = ∑ i : Fin q, bitValue (tape i) * 2 ^ i.val := by
  change (finFunctionFinEquiv (fun i => boolEquiv (tape i))).val = _
  rw [finFunctionFinEquiv_apply]
  simp only [boolEquiv_val]

def pairIndex (q : Nat) : Fin q × Fin q ≃ Fin (q * q) := finProdFinEquiv

theorem pairIndex_val (q : Nat) (i j : Fin q) :
    (pairIndex q (i, j)).val = i.val * q + j.val := by
  change j.val + q * i.val = _
  simp [Nat.mul_comm, Nat.add_comm]

def pairCubeEquiv (q : Nat) : Cube (Fin q × Fin q) ≃ Fin (pairTapeCount q) :=
  (Equiv.arrowCongr (pairIndex q) (Equiv.refl Bool)).trans (cubeEquiv (q * q))

def pairCubeRank {q : Nat} (tape : Cube (Fin q × Fin q)) : Fin (pairTapeCount q) :=
  pairCubeEquiv q tape

def pairCubeUnrank {q : Nat} (index : Fin (pairTapeCount q)) : Cube (Fin q × Fin q) :=
  (pairCubeEquiv q).symm index

@[simp] theorem pairCubeUnrank_pairCubeRank {q : Nat} (tape : Cube (Fin q × Fin q)) :
    pairCubeUnrank (pairCubeRank tape) = tape := (pairCubeEquiv q).symm_apply_apply tape

@[simp] theorem pairCubeRank_pairCubeUnrank {q : Nat} (index : Fin (pairTapeCount q)) :
    pairCubeRank (pairCubeUnrank index) = index := (pairCubeEquiv q).apply_symm_apply index

theorem pairCubeEquiv_val (q : Nat) (tape : Cube (Fin q × Fin q)) :
    (pairCubeEquiv q tape).val =
      ∑ i : Fin (q * q), bitValue (tape ((pairIndex q).symm i)) * 2 ^ i.val :=
  cubeEquiv_val (q * q) _

def inputCoordinateEquiv (q : Nat) :
    AlphabetReduction.InputCoordinate (Fin q) ≃ Fin (2 * tapeCount q) :=
  productEquiv boolEquiv (cubeEquiv q)

def fivePairTapesEquiv (q : Nat) :
    (Cube (Fin q × Fin q) × Cube (Fin q × Fin q) × Cube (Fin q × Fin q) ×
      Cube (Fin q × Fin q) × Cube (Fin q × Fin q)) ≃ Fin (fiveTapeCount q) :=
  (productEquiv (pairCubeEquiv q)
    (productEquiv (pairCubeEquiv q)
      (productEquiv (pairCubeEquiv q)
        (productEquiv (pairCubeEquiv q) (pairCubeEquiv q))))).trans
    (finCongr (by simp [fiveTapeCount, pow_succ, Nat.mul_assoc]))

def localEventEquiv (q : Nat) : AlphabetGraph.LocalEvent (Fin q) ≃ Fin (localCount q) :=
  (productEquiv (Equiv.refl (Fin 4))
    (productEquiv (inputCoordinateEquiv q) (fivePairTapesEquiv q))).trans
    (finCongr (by simp [localCount, Nat.mul_assoc]))

def eventEquiv (m q : Nat) :
    AlphabetGraph.Event (Fin m) (Fin q) ≃ Fin (eventCount m q) :=
  productEquiv (Equiv.refl (Fin m)) (localEventEquiv q)

def addressEquiv (n m q : Nat) :
    AlphabetGraph.Address (Fin n) (Fin m) (Fin q) ≃ Fin (addressCount n m q) :=
  sumEquiv (productEquiv (Equiv.refl (Fin n)) (cubeEquiv q))
    (productEquiv (Equiv.refl (Fin m)) (pairCubeEquiv q))

def vertexEquiv (n m q : Nat) :
    QueryIncidence.Vertex (AlphabetGraph.Event (Fin m) (Fin q))
      (AlphabetGraph.Address (Fin n) (Fin m) (Fin q)) ≃ Fin (vertexCount n m q) :=
  sumEquiv (eventEquiv m q) (addressEquiv n m q)

def dartEquiv (m q : Nat) :
    QueryIncidence.Dart (AlphabetGraph.Event (Fin m) (Fin q)) 6 ≃ Fin (dartCount m q) :=
  (productEquiv (productEquiv (eventEquiv m q) (Equiv.refl (Fin 6))) boolEquiv).trans
    (finCongr (by simp [dartCount, Nat.mul_assoc]))

def labelEquiv : QueryIncidence.Label 6 ≃ Fin 64 := cubeEquiv 6

theorem labelEquiv_val (label : QueryIncidence.Label 6) :
    (labelEquiv label).val = ∑ i : Fin 6, bitValue (label i) * 2 ^ i.val :=
  cubeEquiv_val 6 label

@[simp] theorem inputCoordinateEquiv_val (q : Nat) (side : Bool) (tape : Cube (Fin q)) :
    (inputCoordinateEquiv q (side, tape)).val =
      bitValue side * tapeCount q + (cubeEquiv q tape).val := by
  simp [inputCoordinateEquiv]

theorem fivePairTapesEquiv_val (q : Nat) (f g r₀ r₁ r₂ : Cube (Fin q × Fin q)) :
    (fivePairTapesEquiv q (f, g, r₀, r₁, r₂)).val =
      (pairCubeEquiv q f).val * pairTapeCount q ^ 4 +
        (pairCubeEquiv q g).val * pairTapeCount q ^ 3 +
        (pairCubeEquiv q r₀).val * pairTapeCount q ^ 2 +
        (pairCubeEquiv q r₁).val * pairTapeCount q + (pairCubeEquiv q r₂).val := by
  simp [fivePairTapesEquiv, productEquiv_val, pow_succ, Nat.mul_assoc, Nat.add_assoc]

theorem localEventEquiv_val (q : Nat) (kind : Fin 4)
    (coordinate : AlphabetReduction.InputCoordinate (Fin q))
    (f g r₀ r₁ r₂ : Cube (Fin q × Fin q)) :
    (localEventEquiv q (kind, coordinate, f, g, r₀, r₁, r₂)).val =
      kind.val * (2 * tapeCount q * fiveTapeCount q) +
        (inputCoordinateEquiv q coordinate).val * fiveTapeCount q +
        (fivePairTapesEquiv q (f, g, r₀, r₁, r₂)).val := by
  simp [localEventEquiv, productEquiv_val, Nat.add_assoc]

@[simp] theorem eventEquiv_val (m q : Nat) (edge : Fin m)
    (sample : AlphabetGraph.LocalEvent (Fin q)) :
    (eventEquiv m q (edge, sample)).val = edge.val * localCount q +
      (localEventEquiv q sample).val := by
  change (localEventEquiv q sample).val + localCount q * edge.val = _
  simp [Nat.mul_comm, Nat.add_comm]

@[simp] theorem addressEquiv_inl_val (n m q : Nat) (vertex : Fin n)
    (tape : Cube (Fin q)) :
    (addressEquiv n m q (.inl (vertex, tape))).val =
      vertex.val * tapeCount q + (cubeEquiv q tape).val := by
  change (cubeEquiv q tape).val + tapeCount q * vertex.val = _
  simp [Nat.mul_comm, Nat.add_comm]

@[simp] theorem addressEquiv_inr_val (n m q : Nat) (edge : Fin m)
    (tape : Cube (Fin q × Fin q)) :
    (addressEquiv n m q (.inr (edge, tape))).val =
      n * tapeCount q + (edge.val * pairTapeCount q + (pairCubeEquiv q tape).val) := by
  change n * tapeCount q + ((pairCubeEquiv q tape).val + pairTapeCount q * edge.val) = _
  simp [Nat.mul_comm, Nat.add_comm]

@[simp] theorem vertexEquiv_event_val (n m q : Nat)
    (event : AlphabetGraph.Event (Fin m) (Fin q)) :
    (vertexEquiv n m q (.inl event)).val = (eventEquiv m q event).val := rfl

@[simp] theorem vertexEquiv_vertexAddress_val (n m q : Nat) (vertex : Fin n)
    (tape : Cube (Fin q)) :
    (vertexEquiv n m q (.inr (.inl (vertex, tape)))).val =
      eventCount m q + (vertex.val * tapeCount q + (cubeEquiv q tape).val) := by
  change eventCount m q + (addressEquiv n m q (.inl (vertex, tape))).val = _
  rw [addressEquiv_inl_val]

@[simp] theorem vertexEquiv_edgeAddress_val (n m q : Nat) (edge : Fin m)
    (tape : Cube (Fin q × Fin q)) :
    (vertexEquiv n m q (.inr (.inr (edge, tape)))).val =
      eventCount m q + (n * tapeCount q +
        (edge.val * pairTapeCount q + (pairCubeEquiv q tape).val)) := by
  change eventCount m q + (addressEquiv n m q (.inr (edge, tape))).val = _
  rw [addressEquiv_inr_val]

@[simp] theorem dartEquiv_val (m q : Nat)
    (event : AlphabetGraph.Event (Fin m) (Fin q)) (slot : Fin 6) (orientation : Bool) :
    (dartEquiv m q ((event, slot), orientation)).val =
      (eventEquiv m q event).val * 12 + slot.val * 2 + bitValue orientation := by
  change (boolEquiv orientation).val + 2 * (slot.val + 6 * (eventEquiv m q event).val) = _
  rw [boolEquiv_val]
  omega

theorem dartEquiv_reverse_val (m q : Nat)
    (event : AlphabetGraph.Event (Fin m) (Fin q)) (slot : Fin 6) (orientation : Bool) :
    (dartEquiv m q (QueryIncidence.reverse ((event, slot), orientation))).val =
      (eventEquiv m q event).val * 12 + slot.val * 2 + bitValue (!orientation) :=
  dartEquiv_val m q event slot (!orientation)

def ordered {A : Type*} {count : Nat} (order : A ≃ Fin count) : List A :=
  List.ofFn order.symm

@[simp] theorem ordered_length {A : Type*} {count : Nat} (order : A ≃ Fin count) :
    (ordered order).length = count := by simp [ordered]

theorem mem_ordered {A : Type*} {count : Nat} (order : A ≃ Fin count) (x : A) :
    x ∈ ordered order := by
  simp only [ordered, List.mem_ofFn]
  exact ⟨order x, order.symm_apply_apply x⟩

theorem ordered_nodup {A : Type*} {count : Nat} (order : A ≃ Fin count) :
    (ordered order).Nodup := by
  rw [ordered, List.nodup_ofFn]
  exact order.symm.injective

@[simp] theorem ordered_refl (n : Nat) : ordered (Equiv.refl (Fin n)) = List.finRange n := rfl

@[simp] theorem ordered_bool : ordered boolEquiv = [false, true] := rfl

theorem ordered_trans_finCongr {A : Type*} {a b : Nat}
    (order : A ≃ Fin a) (sameCount : a = b) :
    ordered (order.trans (finCongr sameCount)) = ordered order := by
  cases sameCount
  rfl

theorem ordered_product {A B : Type*} {a b : Nat}
    (left : A ≃ Fin a) (right : B ≃ Fin b) :
    ordered (productEquiv left right) =
      (ordered left).flatMap (fun x => (ordered right).map (fun y => (x, y))) := by
  have atIndex (i : Fin a) (j : Fin b) (h : i.val * b + j.val < a * b) :
      (productEquiv left right).symm ⟨i.val * b + j.val, h⟩ =
        (left.symm i, right.symm j) := by
    have same : (⟨i.val * b + j.val, h⟩ : Fin (a * b)) = finProdFinEquiv (i, j) := by
      apply Fin.ext
      simp [finProdFinEquiv, Nat.mul_comm, Nat.add_comm]
    rw [same]
    simp [productEquiv]
  unfold ordered
  rw [List.ofFn_mul]
  simp only [atIndex, List.flatMap_def, List.map_ofFn, Function.comp_def]

theorem ordered_sum {A B : Type*} {a b : Nat}
    (left : A ≃ Fin a) (right : B ≃ Fin b) :
    ordered (sumEquiv left right) =
      (ordered left).map Sum.inl ++ (ordered right).map Sum.inr := by
  have atLeft (i : Fin a) :
      (sumEquiv left right).symm (i.castLE (Nat.le_add_right a b)) =
        Sum.inl (left.symm i) := by
    change (Equiv.sumCongr left right).symm
      (finSumFinEquiv.symm (Fin.castAdd b i)) = _
    rw [finSumFinEquiv_symm_apply_castAdd]
    rfl
  have atRight (j : Fin b) :
      (sumEquiv left right).symm (j.natAdd a) = Sum.inr (right.symm j) := by
    change (Equiv.sumCongr left right).symm (finSumFinEquiv.symm (j.natAdd a)) = _
    rw [finSumFinEquiv_symm_apply_natAdd]
    rfl
  unfold ordered
  rw [List.ofFn_add]
  simp only [atLeft, atRight, List.map_ofFn, Function.comp_def]

def localEvents (q : Nat) : List (AlphabetGraph.LocalEvent (Fin q)) :=
  ordered (localEventEquiv q)

@[simp] theorem localEvents_length (q : Nat) : (localEvents q).length = localCount q :=
  ordered_length _

theorem ordered_event (m q : Nat) :
    ordered (eventEquiv m q) =
      (List.finRange m).flatMap (fun edge => (localEvents q).map (fun event => (edge, event))) := by
  change ordered (productEquiv (Equiv.refl (Fin m)) (localEventEquiv q)) = _
  rw [ordered_product, ordered_refl]
  rfl

theorem ordered_dart (m q : Nat) :
    ordered (dartEquiv m q) =
      (List.finRange m).flatMap (fun edge =>
        (localEvents q).flatMap (fun event =>
          (List.finRange 6).flatMap (fun slot =>
            [(((edge, event), slot), false), (((edge, event), slot), true)]))) := by
  erw [dartEquiv, ordered_trans_finCongr, ordered_product, ordered_product, ordered_event]
  simp only [ordered_refl, ordered_bool, List.flatMap_assoc, List.flatMap_map,
    List.map_cons, List.map_nil]

@[simp] theorem card_cube (q : Nat) : Fintype.card (Cube (Fin q)) = tapeCount q := by
  simpa using Fintype.card_congr (cubeEquiv q)

@[simp] theorem card_pairCube (q : Nat) :
    Fintype.card (Cube (Fin q × Fin q)) = pairTapeCount q := by
  simpa using Fintype.card_congr (pairCubeEquiv q)

@[simp] theorem card_localEvent (q : Nat) :
    Fintype.card (AlphabetGraph.LocalEvent (Fin q)) = localCount q := by
  simpa using Fintype.card_congr (localEventEquiv q)

@[simp] theorem card_event (m q : Nat) :
    Fintype.card (AlphabetGraph.Event (Fin m) (Fin q)) = eventCount m q := by
  simpa using Fintype.card_congr (eventEquiv m q)

@[simp] theorem card_address (n m q : Nat) :
    Fintype.card (AlphabetGraph.Address (Fin n) (Fin m) (Fin q)) = addressCount n m q := by
  simpa using Fintype.card_congr (addressEquiv n m q)

@[simp] theorem card_vertex (n m q : Nat) :
    Fintype.card (QueryIncidence.Vertex (AlphabetGraph.Event (Fin m) (Fin q))
      (AlphabetGraph.Address (Fin n) (Fin m) (Fin q))) = vertexCount n m q := by
  simpa using Fintype.card_congr (vertexEquiv n m q)

@[simp] theorem card_dart (m q : Nat) :
    Fintype.card (QueryIncidence.Dart (AlphabetGraph.Event (Fin m) (Fin q)) 6) =
      dartCount m q := by
  simpa using Fintype.card_congr (dartEquiv m q)

@[simp] theorem localCount_formula (q : Nat) :
    localCount q = 4 * (2 * 2 ^ q) * (2 ^ (q * q)) ^ 5 := rfl

end BinPackingGames.Foundations.PCP.AlphabetTable.Enumeration

namespace BinPackingGames.Foundations.PCP.AlphabetTable.Relations

open Hastad
open Queries

variable {q : Nat}

def sameAddress (selfLoop : Bool) : RawQuery q → RawQuery q → Bool
  | .inl (side, f), .inl (side', g) => ((side == side') || selfLoop) && decide (f = g)
  | .inr f, .inr g => decide (f = g)
  | _, _ => false

@[simp] theorem sameAddress_refl (selfLoop : Bool) (x : RawQuery q) :
    sameAddress selfLoop x x = true := by
  rcases x with ⟨side, f⟩ | f <;> simp [sameAddress]

theorem globalize_eq_iff {V E : Type*} [DecidableEq V]
    (tail head : V) (edge : E) (selfLoop : Bool)
    (hloop : selfLoop = decide (tail = head)) (x y : RawQuery q) :
    globalize tail head edge x = globalize tail head edge y ↔
      sameAddress selfLoop x y = true := by
  rcases x with ⟨side, f⟩ | f <;> rcases y with ⟨side', g⟩ | g
  · cases side <;> cases side'
    · simp [globalize, sameAddress]
    · simp [globalize, sameAddress, hloop]
    · simp [globalize, sameAddress, hloop,
        show head = tail ↔ tail = head from eq_comm]
    · simp [globalize, sameAddress]
  · simp [globalize, sameAddress]
  · simp [globalize, sameAddress]
  · simp [globalize, sameAddress]

def eqProfile (selfLoop : Bool) (queries : Fin 6 → RawQuery q)
    (i j : Fin 6) : Bool := sameAddress selfLoop (queries i) (queries j)

@[simp] theorem eqProfile_refl (selfLoop : Bool) (queries : Fin 6 → RawQuery q)
    (i : Fin 6) : eqProfile selfLoop queries i i = true := sameAddress_refl _ _

def canonicalSlot (selfLoop : Bool) (queries : Fin 6 → RawQuery q) (i : Fin 6) : Fin 6 :=
  Fin.find (fun j => eqProfile selfLoop queries j i = true) ⟨i, eqProfile_refl _ _ _⟩

theorem canonicalSlot_profile (selfLoop : Bool) (queries : Fin 6 → RawQuery q)
    (i : Fin 6) : eqProfile selfLoop queries (canonicalSlot selfLoop queries i) i = true := by
  unfold canonicalSlot
  exact Fin.find_spec (p := fun j => eqProfile selfLoop queries j i = true)
    ⟨i, eqProfile_refl _ _ _⟩

theorem canonicalSlot_le (selfLoop : Bool) (queries : Fin 6 → RawQuery q)
    (i j : Fin 6) (h : eqProfile selfLoop queries j i = true) :
    canonicalSlot selfLoop queries i ≤ j :=
  Fin.find_le_of_pos ⟨i, eqProfile_refl _ _ _⟩ h

def RepeatedConsistent (selfLoop : Bool) (queries : Fin 6 → RawQuery q)
    (label : QueryIncidence.Label 6) : Prop :=
  ∀ i j, eqProfile selfLoop queries i j = true → label i = label j

instance repeatedConsistentDecidable (selfLoop : Bool) (queries : Fin 6 → RawQuery q)
    (label : QueryIncidence.Label 6) : Decidable (RepeatedConsistent selfLoop queries label) := by
  unfold RepeatedConsistent
  infer_instance

def incidenceAccept (selfLoop : Bool) (queries : Fin 6 → RawQuery q)
    (accept : QueryIncidence.Label 6 → Bool) (slot : Fin 6)
    (left right : QueryIncidence.Label 6) : Bool :=
  decide (accept left = true ∧ RepeatedConsistent selfLoop queries left) &&
    decide (left (canonicalSlot selfLoop queries slot) = right 0)

def predicate (selfLoop : Bool) (queries : Fin 6 → RawQuery q)
    (accept : QueryIncidence.Label 6 → Bool) (slot : Fin 6) (orientation : Bool)
    (a b : QueryIncidence.Label 6) : Bool :=
  if orientation then incidenceAccept selfLoop queries accept slot b a
  else incidenceAccept selfLoop queries accept slot a b

def relation (P : Fin q → Fin q → Bool) (selfLoop : Bool)
    (event : AlphabetGraph.LocalEvent (Fin q)) (slot : Fin 6) (orientation : Bool) :
    GraphTables.RelationTable :=
  GraphTables.relationOf fun a b =>
    predicate selfLoop (query P event) (AssignmentTester.eventAccepts event.1)
      slot orientation (Enumeration.labelEquiv.symm a) (Enumeration.labelEquiv.symm b)

@[simp] theorem relationAt_relation (P : Fin q → Fin q → Bool) (selfLoop : Bool)
    (event : AlphabetGraph.LocalEvent (Fin q)) (slot : Fin 6) (orientation : Bool)
    (a b : GraphTables.Label) :
    GraphTables.relationAt (relation P selfLoop event slot orientation) a b =
      predicate selfLoop (query P event) (AssignmentTester.eventAccepts event.1)
        slot orientation (Enumeration.labelEquiv.symm a) (Enumeration.labelEquiv.symm b) := by
  exact GraphTables.relationAt_relationOf _ _ _

theorem relation_reverse (P : Fin q → Fin q → Bool) (selfLoop : Bool)
    (event : AlphabetGraph.LocalEvent (Fin q)) (slot : Fin 6) (orientation : Bool)
    (a b : GraphTables.Label) :
    GraphTables.relationAt (relation P selfLoop event slot (!orientation)) b a =
      GraphTables.relationAt (relation P selfLoop event slot orientation) a b := by
  simp only [relationAt_relation]
  cases orientation <;> rfl

def oldPredicate (old : Vector Bool (q * q)) (a b : Fin q) : Bool :=
  old[(finProdFinEquiv (a, b) : Fin (q * q))]

def relationFromTable (old : Vector Bool (q * q)) (selfLoop : Bool)
    (event : AlphabetGraph.LocalEvent (Fin q)) (slot : Fin 6) (orientation : Bool) :
    GraphTables.RelationTable := relation (oldPredicate old) selfLoop event slot orientation

@[simp] theorem oldPredicate_relationAt (old : GraphTables.RelationTable)
    (a b : GraphTables.Label) : oldPredicate old a b = GraphTables.relationAt old a b := rfl

theorem relationFromTable_eq (old : GraphTables.RelationTable) (selfLoop : Bool)
    (event : AlphabetGraph.LocalEvent (Fin 64)) (slot : Fin 6) (orientation : Bool) :
    relationFromTable old selfLoop event slot orientation =
      relation (GraphTables.relationAt old) selfLoop event slot orientation := rfl

variable {V E : Type*} [DecidableEq V] [DecidableEq E]

omit [DecidableEq E] in
theorem eqProfile_query_iff (G : ConstraintGraph V E (Fin q)) (e : E)
    (event : AlphabetGraph.LocalEvent (Fin q)) (i j : Fin 6) :
    eqProfile (decide (G.tail e = G.head e)) (query (G.accepts e) event) i j = true ↔
      (AlphabetGraph.verifier G).query (e, event) i =
        (AlphabetGraph.verifier G).query (e, event) j := by
  rw [← globalize_query G e event i, ← globalize_query G e event j]
  exact (globalize_eq_iff (G.tail e) (G.head e) e _ rfl _ _).symm

theorem canonicalSlot_eq (G : ConstraintGraph V E (Fin q)) (e : E)
    (event : AlphabetGraph.LocalEvent (Fin q)) (slot : Fin 6) :
    canonicalSlot (decide (G.tail e = G.head e)) (query (G.accepts e) event) slot =
      QueryIncidence.canonicalSlot (AlphabetGraph.verifier G) (e, event) slot := by
  unfold canonicalSlot QueryIncidence.canonicalSlot
  apply Fin.find_congr'
  exact eqProfile_query_iff G e event _ slot

omit [DecidableEq E] in
theorem repeatedConsistent_iff (G : ConstraintGraph V E (Fin q)) (e : E)
    (event : AlphabetGraph.LocalEvent (Fin q)) (label : QueryIncidence.Label 6) :
    RepeatedConsistent (decide (G.tail e = G.head e)) (query (G.accepts e) event) label ↔
      QueryIncidence.RepeatedConsistent (AlphabetGraph.verifier G) (e, event) label := by
  constructor
  · intro h i j hij
    exact h i j ((eqProfile_query_iff G e event i j).mpr hij)
  · intro h i j hij
    exact h i j ((eqProfile_query_iff G e event i j).mp hij)

omit [DecidableEq V] [DecidableEq E] in
theorem verifier_accepts (G : ConstraintGraph V E (Fin q)) (e : E)
    (event : AlphabetGraph.LocalEvent (Fin q)) (label : QueryIncidence.Label 6) :
    (AlphabetGraph.verifier G).accepts (e, event) label =
      AssignmentTester.eventAccepts event.1 label := by
  rcases event with ⟨kind, k, f, g, r₀, r₁, r₂⟩
  rfl

theorem incidenceAccept_eq (G : ConstraintGraph V E (Fin q)) (e : E)
    (event : AlphabetGraph.LocalEvent (Fin q)) (slot : Fin 6)
    (left right : QueryIncidence.Label 6) :
    incidenceAccept (decide (G.tail e = G.head e)) (query (G.accepts e) event)
      (AssignmentTester.eventAccepts event.1) slot left right =
      QueryIncidence.incidenceAccept (AlphabetGraph.verifier G) (by decide) (e, event) slot
        left right := by
  simp only [incidenceAccept, QueryIncidence.incidenceAccept, QueryIncidence.LeftValid,
    verifier_accepts, repeatedConsistent_iff, canonicalSlot_eq,
    QueryIncidence.decodeRight, QueryIncidence.zeroSlot]
  rfl

theorem relationAt_relation_graph (G : ConstraintGraph V E (Fin q)) (e : E)
    (event : AlphabetGraph.LocalEvent (Fin q)) (slot : Fin 6) (orientation : Bool)
    (a b : GraphTables.Label) :
    GraphTables.relationAt
        (relation (G.accepts e) (decide (G.tail e = G.head e)) event slot orientation) a b =
      (AlphabetGraph.graph G).accepts (((e, event), slot), orientation)
        (Enumeration.labelEquiv.symm a) (Enumeration.labelEquiv.symm b) := by
  rw [relationAt_relation]
  cases orientation <;>
    exact incidenceAccept_eq G e event slot _ _

end BinPackingGames.Foundations.PCP.AlphabetTable.Relations

namespace BinPackingGames.Foundations.PCP.AlphabetTable.Addresses

open BinPackingGames.Foundations.Hastad
open Enumeration

abbrev Terms := List (Fin 5 × Nat)

structure Field (bound : Nat) where
  terms : Terms
  offset : Fin bound

def values (n m edge tail head : Nat) : Fin 5 → Nat := ![n, m, edge, tail, head]

def eval {bound : Nat} (field : Field bound) (inputs : Fin 5 → Nat) : Nat :=
  Emitter.affineValue field.terms inputs field.offset.val

def fieldBound (q : Nat) : Nat :=
  12 * localCount q + tapeCount q + pairTapeCount q + 1

theorem fieldBound_pos (q : Nat) : 0 < fieldBound q := by
  unfold fieldBound
  omega

theorem tapeCount_lt_bound (q : Nat) : tapeCount q < fieldBound q := by
  unfold fieldBound
  omega

theorem pairTapeCount_lt_bound (q : Nat) : pairTapeCount q < fieldBound q := by
  unfold fieldBound
  omega

theorem localCount_lt_bound (q : Nat) : localCount q < fieldBound q := by
  unfold fieldBound
  omega

theorem local_add_pair_lt_bound (q : Nat) : localCount q + pairTapeCount q < fieldBound q := by
  unfold fieldBound
  omega

theorem twelve_local_lt_bound (q : Nat) : 12 * localCount q < fieldBound q := by
  unfold fieldBound
  omega

def zeroOffset (q : Nat) : Fin (fieldBound q) := ⟨0, fieldBound_pos q⟩

def eventOffset (q : Nat) (event : AlphabetGraph.LocalEvent (Fin q)) : Fin (fieldBound q) :=
  ⟨(localEventEquiv q event).val,
    (localEventEquiv q event).isLt.trans (localCount_lt_bound q)⟩

def rawQueryOffset (q : Nat) : Queries.RawQuery q → Fin (fieldBound q)
  | .inl (_, tape) =>
    ⟨(cubeEquiv q tape).val, (cubeEquiv q tape).isLt.trans (tapeCount_lt_bound q)⟩
  | .inr tape =>
    ⟨(pairCubeEquiv q tape).val, (pairCubeEquiv q tape).isLt.trans (pairTapeCount_lt_bound q)⟩

def reverseOffset (q : Nat) (event : AlphabetGraph.LocalEvent (Fin q))
    (slot : Fin 6) (orientation : Bool) : Fin (fieldBound q) :=
  ⟨(localEventEquiv q event).val * 12 + slot.val * 2 + bitValue (!orientation), by
    have he := (localEventEquiv q event).isLt
    have hs := slot.isLt
    have hb : bitValue (!orientation) < 2 := by
      simpa only [boolEquiv_val] using (boolEquiv (!orientation)).isLt
    have hbound := twelve_local_lt_bound q
    omega⟩

def headerVertices (q : Nat) : Field (fieldBound q) :=
  ⟨[(0, tapeCount q), (1, localCount q + pairTapeCount q)], zeroOffset q⟩

def headerDarts (q : Nat) : Field (fieldBound q) :=
  ⟨[(1, 12 * localCount q)], zeroOffset q⟩

def eventVertexField (q : Nat) (event : AlphabetGraph.LocalEvent (Fin q)) :
    Field (fieldBound q) :=
  ⟨[(2, localCount q)], eventOffset q event⟩

def vertexQueryTerms (q : Nat) (side : Bool) : Terms :=
  [(1, localCount q), (if side then 4 else 3, tapeCount q)]

def edgeQueryTerms (q : Nat) : Terms :=
  [(1, localCount q), (0, tapeCount q), (2, pairTapeCount q)]

def rawQueryTerms (q : Nat) : Queries.RawQuery q → Terms
  | .inl (side, _) => vertexQueryTerms q side
  | .inr _ => edgeQueryTerms q

def rawQueryField (q : Nat) (query : Queries.RawQuery q) : Field (fieldBound q) :=
  ⟨rawQueryTerms q query, rawQueryOffset q query⟩

def reverseDartField (q : Nat) (event : AlphabetGraph.LocalEvent (Fin q))
    (slot : Fin 6) (orientation : Bool) : Field (fieldBound q) :=
  ⟨[(2, 12 * localCount q)], reverseOffset q event slot orientation⟩

theorem headerVertices_eval (q n m edge tail head : Nat) :
    eval (headerVertices q) (values n m edge tail head) = vertexCount n m q := by
  simp [eval, headerVertices, values, zeroOffset, Emitter.affineValue,
    vertexCount, eventCount, addressCount, Nat.mul_add,
    Nat.mul_comm, Nat.add_comm, Nat.add_assoc]

theorem headerDarts_eval (q n m edge tail head : Nat) :
    eval (headerDarts q) (values n m edge tail head) = dartCount m q := by
  simp [eval, headerDarts, values, zeroOffset, Emitter.affineValue,
    dartCount, eventCount, Nat.mul_comm, Nat.mul_assoc]

theorem eventVertexField_eval (n m q : Nat) (edge : Fin m)
    (event : AlphabetGraph.LocalEvent (Fin q)) (tail head : Nat) :
    eval (eventVertexField q event) (values n m edge.val tail head) =
      (vertexEquiv n m q (.inl (edge, event))).val := by
  rw [vertexEquiv_event_val, eventEquiv_val]
  simp [eval, eventVertexField, eventOffset, values, Emitter.affineValue, Nat.mul_comm]

theorem rawQueryField_eval (n m q : Nat) (edge : Fin m) (tail head : Fin n)
    (query : Queries.RawQuery q) :
    eval (rawQueryField q query) (values n m edge.val tail.val head.val) =
      (vertexEquiv n m q (.inr (Queries.globalize tail head edge query))).val := by
  cases query with
  | inl query =>
    rcases query with ⟨side, tape⟩
    cases side <;>
      simp [eval, rawQueryField, rawQueryTerms, vertexQueryTerms, rawQueryOffset,
        values, Emitter.affineValue, Queries.globalize, eventCount,
        Nat.mul_comm, Nat.add_assoc]
  | inr tape =>
    simp [eval, rawQueryField, rawQueryTerms, edgeQueryTerms, rawQueryOffset,
      values, Emitter.affineValue, Queries.globalize, eventCount,
      Nat.mul_comm, Nat.add_assoc]

theorem reverseDartField_eval (n m q : Nat) (edge : Fin m)
    (event : AlphabetGraph.LocalEvent (Fin q)) (slot : Fin 6) (orientation : Bool)
    (tail head : Nat) :
    eval (reverseDartField q event slot orientation) (values n m edge.val tail head) =
      (dartEquiv m q (QueryIncidence.reverse (((edge, event), slot), orientation))).val := by
  rw [dartEquiv_reverse_val, eventEquiv_val]
  simp [eval, reverseDartField, reverseOffset, values, Emitter.affineValue,
    Nat.mul_add, Nat.mul_comm, Nat.mul_assoc,
    Nat.add_assoc]

def queryEventTerms (q : Nat) (event : AlphabetGraph.LocalEvent (Fin q))
    (slot : Fin 6) : Terms :=
  rawQueryTerms q (Queries.query (fun _ _ => false) event slot)

theorem rawQueryTerms_query (q : Nat) (P : Fin q → Fin q → Bool)
    (event : AlphabetGraph.LocalEvent (Fin q)) (slot : Fin 6) :
    rawQueryTerms q (Queries.query P event slot) = queryEventTerms q event slot := by
  rcases event with ⟨kind, ⟨side, tape⟩, f, g, r₀, r₁, r₂⟩
  by_cases hk₀ : kind = 0 <;> by_cases hk₁ : kind = 1 <;> by_cases hk₂ : kind = 2 <;>
    simp only [queryEventTerms, Queries.query, hk₀, hk₁, hk₂, ite_true, ite_false]
  all_goals split_ifs <;> rfl

def queryEventOffset (q : Nat) (event : AlphabetGraph.LocalEvent (Fin q))
    (slot : Fin 6) (P : Fin q → Fin q → Bool) : Fin (fieldBound q) :=
  rawQueryOffset q (Queries.query P event slot)

def fieldForQueryEvent (q : Nat) (event : AlphabetGraph.LocalEvent (Fin q))
    (slot : Fin 6) (P : Fin q → Fin q → Bool) : Field (fieldBound q) :=
  ⟨queryEventTerms q event slot, queryEventOffset q event slot P⟩

theorem fieldForQueryEvent_eq_rawQueryField (q : Nat)
    (event : AlphabetGraph.LocalEvent (Fin q)) (slot : Fin 6)
    (P : Fin q → Fin q → Bool) :
    fieldForQueryEvent q event slot P = rawQueryField q (Queries.query P event slot) := by
  unfold fieldForQueryEvent queryEventOffset rawQueryField
  rw [rawQueryTerms_query]

theorem fieldForQueryEvent_eval (n m q : Nat) (edge : Fin m) (tail head : Fin n)
    (event : AlphabetGraph.LocalEvent (Fin q)) (slot : Fin 6)
    (P : Fin q → Fin q → Bool) :
    eval (fieldForQueryEvent q event slot P) (values n m edge.val tail.val head.val) =
      (vertexEquiv n m q
        (.inr (Queries.globalize tail head edge (Queries.query P event slot)))).val := by
  rw [fieldForQueryEvent_eq_rawQueryField]
  exact rawQueryField_eval n m q edge tail head _

theorem fieldForQueryEvent_eval_verifier (n m q : Nat)
    (G : ConstraintGraph (Fin n) (Fin m) (Fin q)) (edge : Fin m)
    (event : AlphabetGraph.LocalEvent (Fin q)) (slot : Fin 6) :
    eval (fieldForQueryEvent q event slot (G.accepts edge))
      (values n m edge.val (G.tail edge).val (G.head edge).val) =
      (vertexEquiv n m q (.inr ((AlphabetGraph.verifier G).query (edge, event) slot))).val := by
  rw [← Queries.globalize_query G edge event slot]
  exact fieldForQueryEvent_eval n m q edge (G.tail edge) (G.head edge) event slot _

def dartTailTerms (q : Nat) (event : AlphabetGraph.LocalEvent (Fin q))
    (slot : Fin 6) (orientation : Bool) : Terms :=
  if orientation then queryEventTerms q event slot else [(2, localCount q)]

def dartTailOffset (q : Nat) (event : AlphabetGraph.LocalEvent (Fin q))
    (slot : Fin 6) (orientation : Bool) (P : Fin q → Fin q → Bool) : Fin (fieldBound q) :=
  if orientation then queryEventOffset q event slot P else eventOffset q event

def fieldForDartTail (q : Nat) (event : AlphabetGraph.LocalEvent (Fin q))
    (slot : Fin 6) (orientation : Bool) (P : Fin q → Fin q → Bool) : Field (fieldBound q) :=
  ⟨dartTailTerms q event slot orientation, dartTailOffset q event slot orientation P⟩

theorem fieldForDartTail_eval (n m q : Nat) (edge : Fin m) (tail head : Fin n)
    (event : AlphabetGraph.LocalEvent (Fin q)) (slot : Fin 6) (orientation : Bool)
    (P : Fin q → Fin q → Bool) :
    eval (fieldForDartTail q event slot orientation P)
      (values n m edge.val tail.val head.val) =
      (vertexEquiv n m q (if orientation then
        .inr (Queries.globalize tail head edge (Queries.query P event slot))
        else .inl (edge, event))).val := by
  cases orientation with
  | false => exact eventVertexField_eval n m q edge event tail.val head.val
  | true => exact fieldForQueryEvent_eval n m q edge tail head event slot P

def CoefficientsBounded (q : Nat) (terms : Terms) : Prop :=
  ∀ term ∈ terms, term.2 < fieldBound q

theorem headerVertices_coefficients (q : Nat) : CoefficientsBounded q (headerVertices q).terms := by
  simp only [CoefficientsBounded, headerVertices, List.mem_cons, List.not_mem_nil, or_false]
  intro term h
  rcases h with rfl | rfl
  · exact tapeCount_lt_bound q
  · exact local_add_pair_lt_bound q

theorem headerDarts_coefficients (q : Nat) : CoefficientsBounded q (headerDarts q).terms := by
  intro term h
  have heq : term = (1, 12 * localCount q) := by simpa [headerDarts] using h
  subst term
  exact twelve_local_lt_bound q

theorem eventVertexField_coefficients (q : Nat) (event : AlphabetGraph.LocalEvent (Fin q)) :
    CoefficientsBounded q (eventVertexField q event).terms := by
  intro term h
  have heq : term = (2, localCount q) := by simpa [eventVertexField] using h
  subst term
  exact localCount_lt_bound q

theorem rawQueryTerms_coefficients (q : Nat) (query : Queries.RawQuery q) :
    CoefficientsBounded q (rawQueryTerms q query) := by
  cases query with
  | inl query =>
    rcases query with ⟨side, tape⟩
    intro term h
    simp only [rawQueryTerms, vertexQueryTerms, List.mem_cons, List.not_mem_nil, or_false] at h
    rcases h with rfl | rfl
    · exact localCount_lt_bound q
    · exact tapeCount_lt_bound q
  | inr tape =>
    intro term h
    simp only [rawQueryTerms, edgeQueryTerms, List.mem_cons, List.not_mem_nil, or_false] at h
    rcases h with rfl | rfl | rfl
    · exact localCount_lt_bound q
    · exact tapeCount_lt_bound q
    · exact pairTapeCount_lt_bound q

theorem queryEventTerms_coefficients (q : Nat) (event : AlphabetGraph.LocalEvent (Fin q))
    (slot : Fin 6) : CoefficientsBounded q (queryEventTerms q event slot) :=
  rawQueryTerms_coefficients q _

theorem reverseDartField_coefficients (q : Nat) (event : AlphabetGraph.LocalEvent (Fin q))
    (slot : Fin 6) (orientation : Bool) :
    CoefficientsBounded q (reverseDartField q event slot orientation).terms := by
  intro term h
  have heq : term = (2, 12 * localCount q) := by simpa [reverseDartField] using h
  subst term
  exact twelve_local_lt_bound q

theorem rawQueryTerms_length_le (q : Nat) (query : Queries.RawQuery q) :
    (rawQueryTerms q query).length ≤ 3 := by
  cases query with
  | inl query => rcases query with ⟨side, tape⟩; simp [rawQueryTerms, vertexQueryTerms]
  | inr tape => simp [rawQueryTerms, edgeQueryTerms]

theorem queryEventTerms_length_le (q : Nat) (event : AlphabetGraph.LocalEvent (Fin q))
    (slot : Fin 6) : (queryEventTerms q event slot).length ≤ 3 :=
  rawQueryTerms_length_le q _

theorem dartTailTerms_coefficients (q : Nat) (event : AlphabetGraph.LocalEvent (Fin q))
    (slot : Fin 6) (orientation : Bool) :
    CoefficientsBounded q (dartTailTerms q event slot orientation) := by
  cases orientation with
  | false => exact eventVertexField_coefficients q event
  | true => exact queryEventTerms_coefficients q event slot

theorem dartTailTerms_length_le (q : Nat) (event : AlphabetGraph.LocalEvent (Fin q))
    (slot : Fin 6) (orientation : Bool) :
    (dartTailTerms q event slot orientation).length ≤ 3 := by
  cases orientation with
  | false => simp [dartTailTerms]
  | true => exact queryEventTerms_length_le q event slot

theorem offset_lt {bound : Nat} (field : Field bound) : field.offset.val < bound :=
  field.offset.isLt

end BinPackingGames.Foundations.PCP.AlphabetTable.Addresses

namespace BinPackingGames.Foundations.PCP.AlphabetTable.Table

variable {q : Nat}

def rowFromDart (input : GenericGraphTables.Table q)
    (dart : QueryIncidence.Dart (AlphabetGraph.Event (Fin input.darts) (Fin q)) 6) :
    GraphTables.DartRow (Enumeration.vertexCount input.vertices input.darts q)
      (Enumeration.dartCount input.darts q) :=
  let edge := dart.1.1.1
  let event := dart.1.1.2
  let slot := dart.1.2
  let old := input.rows[edge]
  let head := input.rows[old.reverseIndex].tail
  { tail := Enumeration.vertexEquiv input.vertices input.darts q
      (if dart.2 then
        .inr (Queries.globalize old.tail head edge
          (Queries.query (GenericGraphTables.relationAt old.relation) event slot))
      else .inl (edge, event))
    reverseIndex := Enumeration.dartEquiv input.darts q (dart.1, !dart.2)
    relation := Relations.relationFromTable old.relation (decide (old.tail = head))
      event slot dart.2 }

def rows (input : GenericGraphTables.Table q) :
    GraphTables.Rows (Enumeration.vertexCount input.vertices input.darts q)
      (Enumeration.dartCount input.darts q) :=
  Vector.ofFn (fun index =>
    rowFromDart input ((Enumeration.dartEquiv input.darts q).symm index))

@[simp] theorem rows_at (input : GenericGraphTables.Table q)
    (index : Fin (Enumeration.dartCount input.darts q)) :
    (rows input)[index] =
      rowFromDart input ((Enumeration.dartEquiv input.darts q).symm index) := by
  simp [rows]

theorem rows_reverseIndex (input : GenericGraphTables.Table q)
    (index : Fin (Enumeration.dartCount input.darts q)) :
    (rows input)[index].reverseIndex =
      Enumeration.dartEquiv input.darts q
        (((Enumeration.dartEquiv input.darts q).symm index).1,
          !((Enumeration.dartEquiv input.darts q).symm index).2) := by
  simp [rows, rowFromDart]

private theorem dartRow_ext {n m : Nat} {left right : GraphTables.DartRow n m}
    (ht : left.tail = right.tail) (hr : left.reverseIndex = right.reverseIndex)
    (hp : left.relation = right.relation) : left = right := by
  cases left
  cases right
  cases ht
  cases hr
  cases hp
  rfl

private theorem relation_ext {left right : GraphTables.RelationTable}
    (h : ∀ a b, GraphTables.relationAt left a b = GraphTables.relationAt right a b) :
    left = right := by
  apply Vector.ext
  intro i hi
  let labels := GraphTables.relationIndex.symm (⟨i, hi⟩ : Fin 4096)
  have index : GraphTables.relationIndex labels = (⟨i, hi⟩ : Fin 4096) :=
    GraphTables.relationIndex.apply_symm_apply _
  have point := h labels.1 labels.2
  simpa only [GraphTables.relationAt, index, Fin.getElem_fin] using point

theorem rowFromDart_eq_graphRow (input : GenericGraphTables.Table q)
    (dart : QueryIncidence.Dart (AlphabetGraph.Event (Fin input.darts) (Fin q)) 6) :
    rowFromDart input dart =
      { tail := Enumeration.vertexEquiv input.vertices input.darts q
          ((AlphabetGraph.graph (GenericGraphTables.semantics input)).tail dart)
        reverseIndex := Enumeration.dartEquiv input.darts q
          ((AlphabetGraph.graph (GenericGraphTables.semantics input)).reverse dart)
        relation := GraphTables.relationOf (fun a b =>
          (AlphabetGraph.graph (GenericGraphTables.semantics input)).accepts dart
            (Enumeration.labelEquiv.symm a) (Enumeration.labelEquiv.symm b)) } := by
  rcases dart with ⟨⟨⟨edge, event⟩, slot⟩, orientation⟩
  apply dartRow_ext
  · cases orientation with
    | false => rfl
    | true =>
      have query := Queries.globalize_query (GenericGraphTables.semantics input)
        edge event slot
      have predicate : (GenericGraphTables.semantics input).accepts edge =
          GenericGraphTables.relationAt input.rows[edge].relation := rfl
      rw [predicate] at query
      simpa only [rowFromDart, AlphabetGraph.graph, QueryIncidence.graph,
        QueryIncidence.tail, GenericGraphTables.semantics_tail,
        ConstraintGraph.head, GenericGraphTables.semantics_reverse,
        ite_true] using
        congrArg (fun address => Enumeration.vertexEquiv input.vertices input.darts q
          (.inr address)) query
  · rfl
  · apply relation_ext
    intro a b
    rw [GraphTables.relationAt_relationOf]
    have relation := Relations.relationAt_relation_graph (GenericGraphTables.semantics input)
      edge event slot orientation a b
    have predicate : (GenericGraphTables.semantics input).accepts edge =
        Relations.oldPredicate input.rows[edge].relation := rfl
    rw [predicate] at relation
    simpa only [rowFromDart, Relations.relationFromTable,
      GenericGraphTables.semantics_tail, ConstraintGraph.head,
      GenericGraphTables.semantics_reverse] using! relation

theorem rows_eq_graphRows (input : GenericGraphTables.Table q) :
    rows input = GraphTables.graphRows
      (GraphTables.enumeratedGraph (AlphabetGraph.graph (GenericGraphTables.semantics input))
        (Enumeration.vertexEquiv input.vertices input.darts q)
        (Enumeration.dartEquiv input.darts q) Enumeration.labelEquiv) := by
  apply Vector.ext
  intro i hi
  simp only [rows, GraphTables.graphRows, Vector.getElem_ofFn]
  exact rowFromDart_eq_graphRow input
    ((Enumeration.dartEquiv input.darts q).symm ⟨i, hi⟩)

theorem rows_valid (input : GenericGraphTables.Table q) : GraphTables.Valid (rows input) := by
  rw [rows_eq_graphRows]
  exact GraphTables.graphRows_valid _

def build (input : GenericGraphTables.Table q) : GraphTables.Table where
  vertices := Enumeration.vertexCount input.vertices input.darts q
  darts := Enumeration.dartCount input.darts q
  rows := rows input
  valid := rows_valid input

@[simp] theorem build_vertices (input : GenericGraphTables.Table q) :
    (build input).vertices = Enumeration.vertexCount input.vertices input.darts q := rfl

@[simp] theorem build_darts (input : GenericGraphTables.Table q) :
    (build input).darts = Enumeration.dartCount input.darts q := rfl

private theorem table_rows_congr {n m : Nat} {left right : GraphTables.Rows n m}
    (h : left = right) (hl : GraphTables.Valid left) (hr : GraphTables.Valid right) :
    (⟨n, m, left, hl⟩ : GraphTables.Table) = ⟨n, m, right, hr⟩ := by
  subst right
  rfl

private theorem semantics_rows_congr {n m : Nat} {left right : GraphTables.Rows n m}
    (h : left = right) (hl : GraphTables.Valid left) (hr : GraphTables.Valid right) :
    GraphTables.semantics (⟨n, m, left, hl⟩ : GraphTables.Table) =
      GraphTables.semantics (⟨n, m, right, hr⟩ : GraphTables.Table) := by
  subst right
  rfl

theorem build_eq_ofEnumeratedGraph (input : GenericGraphTables.Table q) :
    build input =
      GraphTables.ofEnumeratedGraph (AlphabetGraph.graph (GenericGraphTables.semantics input))
        (Enumeration.vertexEquiv input.vertices input.darts q)
        (Enumeration.dartEquiv input.darts q) Enumeration.labelEquiv := by
  exact table_rows_congr (rows_eq_graphRows input) (rows_valid input)
    (GraphTables.graphRows_valid _)

theorem semantics_build (input : GenericGraphTables.Table q) :
    GraphTables.semantics (build input) =
      GraphTables.enumeratedGraph (AlphabetGraph.graph (GenericGraphTables.semantics input))
        (Enumeration.vertexEquiv input.vertices input.darts q)
        (Enumeration.dartEquiv input.darts q) Enumeration.labelEquiv := by
  exact (semantics_rows_congr (rows_eq_graphRows input) (rows_valid input)
    (GraphTables.graphRows_valid _)).trans (GraphTables.semantics_ofGraph _)

theorem decode_build (input : GenericGraphTables.Table q) :
    GraphTables.decodeTableBits (GraphTables.tableBits (build input)) =
      some (GraphTables.ofEnumeratedGraph
        (AlphabetGraph.graph (GenericGraphTables.semantics input))
        (Enumeration.vertexEquiv input.vertices input.darts q)
        (Enumeration.dartEquiv input.darts q) Enumeration.labelEquiv) := by
  rw [GraphTables.decodeTableBits_encoded, build_eq_ofEnumeratedGraph]

theorem edgeSatisfied_build (input : GenericGraphTables.Table q)
    (labeling : QueryIncidence.Vertex (AlphabetGraph.Event (Fin input.darts) (Fin q))
      (AlphabetGraph.Address (Fin input.vertices) (Fin input.darts) (Fin q)) →
        QueryIncidence.Label 6)
    (dart : QueryIncidence.Dart (AlphabetGraph.Event (Fin input.darts) (Fin q)) 6) :
    (GraphTables.semantics (build input)).edgeSatisfied
      (fun v => Enumeration.labelEquiv
        (labeling ((Enumeration.vertexEquiv input.vertices input.darts q).symm v)))
      (Enumeration.dartEquiv input.darts q dart) =
        (AlphabetGraph.graph (GenericGraphTables.semantics input)).edgeSatisfied labeling dart := by
  rw [semantics_build]
  exact GraphTables.enumeratedGraph_edgeSatisfied _ _ _ _ labeling dart

theorem rejectionCount_build (input : GenericGraphTables.Table q)
    (labeling : QueryIncidence.Vertex (AlphabetGraph.Event (Fin input.darts) (Fin q))
      (AlphabetGraph.Address (Fin input.vertices) (Fin input.darts) (Fin q)) →
        QueryIncidence.Label 6) :
    (GraphTables.semantics (build input)).rejectionCount
      (fun v => Enumeration.labelEquiv
        (labeling ((Enumeration.vertexEquiv input.vertices input.darts q).symm v))) =
        (AlphabetGraph.graph (GenericGraphTables.semantics input)).rejectionCount labeling := by
  rw [semantics_build]
  exact GraphTables.enumeratedGraph_rejectionCount _ _ _ _ labeling

theorem perfect_completeness [Nonempty (Fin q)] (input : GenericGraphTables.Table q)
    (satisfied : (GenericGraphTables.semantics input).Satisfiable) :
    (GraphTables.semantics (build input)).Satisfiable := by
  obtain ⟨labeling, accepted⟩ :=
    AlphabetGraph.perfect_completeness (GenericGraphTables.semantics input) satisfied
  refine ⟨fun v => Enumeration.labelEquiv
    (labeling ((Enumeration.vertexEquiv input.vertices input.darts q).symm v)), ?_⟩
  intro index
  have h := edgeSatisfied_build input labeling
    ((Enumeration.dartEquiv input.darts q).symm index)
  have index_eq := (Enumeration.dartEquiv input.darts q).apply_symm_apply index
  rw [index_eq] at h
  exact h.trans (accepted _)

theorem gap_transfer [Nonempty (Fin q)] (input : GenericGraphTables.Table q) (a b : Nat)
    (source : ∀ labeling : Fin input.vertices → Fin q,
      a * input.darts ≤ b * (GenericGraphTables.semantics input).rejectionCount labeling)
    (labeling : Fin (build input).vertices → GraphTables.Label) :
    a * (build input).darts ≤
      (b * 12288) * (GraphTables.semantics (build input)).rejectionCount labeling := by
  classical
  let lifted := fun vertex => Enumeration.labelEquiv.symm
    (labeling (Enumeration.vertexEquiv input.vertices input.darts q vertex))
  have h := AlphabetGraph.gap_transfer (GenericGraphTables.semantics input) a b
    (by simpa only [Fintype.card_fin] using source) lifted
  rw [Enumeration.card_dart] at h
  have counts := rejectionCount_build input lifted
  have restored : (fun v : Fin (build input).vertices => Enumeration.labelEquiv
      (lifted ((Enumeration.vertexEquiv input.vertices input.darts q).symm v))) = labeling := by
    funext v
    dsimp only [lifted]
    rw [Enumeration.labelEquiv.apply_symm_apply]
    exact congrArg labeling
      ((Enumeration.vertexEquiv input.vertices input.darts q).apply_symm_apply v)
  rw [restored] at counts
  rw [← counts] at h
  exact h

theorem rowList_eq_ordered (input : GenericGraphTables.Table q) :
    GraphTables.rowList (build input) =
      (Enumeration.ordered (Enumeration.dartEquiv input.darts q)).map
        (rowFromDart input) := by
  change (Vector.ofFn (fun index => rowFromDart input
      ((Enumeration.dartEquiv input.darts q).symm index))).toList =
    (List.ofFn (Enumeration.dartEquiv input.darts q).symm).map (rowFromDart input)
  simp only [Vector.toList_ofFn, List.map_ofFn, Function.comp_def]

theorem tableWords_eq_ordered (input : GenericGraphTables.Table q) :
    GraphTables.tableWords (build input) =
      [Enumeration.vertexCount input.vertices input.darts q,
        Enumeration.dartCount input.darts q] ++
        (Enumeration.ordered (Enumeration.dartEquiv input.darts q)).flatMap
          (fun dart => GraphTables.rowWords (rowFromDart input dart)) := by
  simp only [GraphTables.tableWords, rowList_eq_ordered, build_vertices, build_darts,
    List.flatMap_map]

@[simp] theorem rowList_length (input : GenericGraphTables.Table q) :
    (GraphTables.rowList (build input)).length = Enumeration.dartCount input.darts q := by
  exact GraphTables.rowList_length (build input)

theorem tableWords_length (input : GenericGraphTables.Table q) :
    (GraphTables.tableWords (build input)).length =
      2 + 4098 * Enumeration.dartCount input.darts q := by
  exact GraphTables.tableWords_length (build input)

theorem tableBits_length_le (input : GenericGraphTables.Table q) :
    (GraphTables.tableBits (build input)).length ≤
      Enumeration.vertexCount input.vertices input.darts q +
        Enumeration.dartCount input.darts q + 2 + Enumeration.dartCount input.darts q *
          (Enumeration.vertexCount input.vertices input.darts q +
            Enumeration.dartCount input.darts q + 8192) := by
  exact GraphTables.tableBits_length_le (build input)

end BinPackingGames.Foundations.PCP.AlphabetTable.Table

namespace BinPackingGames.Foundations.PCP.AlphabetTableBounds

open AlphabetTable

variable {q : Nat}

theorem gap_transfer_real (hq : 0 < q) (input : GenericGraphTables.Table q)
    (eps : ℝ) (eps_nonnegative : 0 ≤ eps)
    (source : ∀ labeling : Fin input.vertices → Fin q,
      eps * (input.darts : ℝ) ≤
        ((GenericGraphTables.semantics input).rejectionCount labeling : ℝ))
    (labeling : Fin (Table.build input).vertices → GraphTables.Label) :
    (eps / 12288) * ((Table.build input).darts : ℝ) ≤
      ((GraphTables.semantics (Table.build input)).rejectionCount labeling : ℝ) := by
  classical
  let : Nonempty (Fin q) := ⟨⟨0, hq⟩⟩
  let lifted := fun vertex => Enumeration.labelEquiv.symm
    (labeling (Enumeration.vertexEquiv input.vertices input.darts q vertex))
  have h := AlphabetGraphBounds.gap_transfer_real
    (GenericGraphTables.semantics input) eps eps_nonnegative
    (by simpa only [Fintype.card_fin] using source) lifted
  rw [Enumeration.card_dart] at h
  have counts := Table.rejectionCount_build input lifted
  have restored : (fun v : Fin (Table.build input).vertices => Enumeration.labelEquiv
      (lifted ((Enumeration.vertexEquiv input.vertices input.darts q).symm v))) = labeling := by
    funext v
    dsimp only [lifted]
    rw [Enumeration.labelEquiv.apply_symm_apply]
    exact congrArg labeling
      ((Enumeration.vertexEquiv input.vertices input.darts q).apply_symm_apply v)
  rw [restored] at counts
  rw [← counts] at h
  exact h

theorem build_vertices (input : GenericGraphTables.Table q) :
    (Table.build input).vertices = input.vertices * 2 ^ q +
      input.darts * AlphabetGraphBounds.vertexFactor q := by
  rw [Table.build_vertices]
  simp only [Enumeration.vertexCount, Enumeration.eventCount, Enumeration.addressCount,
    Enumeration.localCount, Enumeration.tapeCount, Enumeration.pairTapeCount,
    Enumeration.fiveTapeCount, AlphabetGraphBounds.vertexFactor,
    AlphabetGraphBounds.localEventFactor]
  ring

theorem build_darts (input : GenericGraphTables.Table q) :
    (Table.build input).darts = input.darts * AlphabetGraphBounds.dartFactor q := by
  rw [Table.build_darts]
  simp only [Enumeration.dartCount, Enumeration.eventCount, Enumeration.localCount,
    Enumeration.tapeCount, Enumeration.pairTapeCount, Enumeration.fiveTapeCount,
    AlphabetGraphBounds.dartFactor, AlphabetGraphBounds.localEventFactor]
  ring

theorem build_total (input : GenericGraphTables.Table q) :
    (Table.build input).vertices + (Table.build input).darts =
      input.vertices * 2 ^ q + input.darts *
        (AlphabetGraphBounds.vertexFactor q + AlphabetGraphBounds.dartFactor q) := by
  rw [build_vertices, build_darts]
  ring

theorem build_total_le (input : GenericGraphTables.Table q) :
    (Table.build input).vertices + (Table.build input).darts ≤
      AlphabetGraphBounds.sizeFactor q * (input.vertices + input.darts) := by
  rw [build_total]
  have hv : 2 ^ q ≤ AlphabetGraphBounds.sizeFactor q := by
    unfold AlphabetGraphBounds.sizeFactor
    omega
  have he : AlphabetGraphBounds.vertexFactor q + AlphabetGraphBounds.dartFactor q ≤
      AlphabetGraphBounds.sizeFactor q := by
    unfold AlphabetGraphBounds.sizeFactor
    exact Nat.add_le_add_right (Nat.le_add_left _ _) _
  calc
    _ ≤ input.vertices * AlphabetGraphBounds.sizeFactor q +
        input.darts * AlphabetGraphBounds.sizeFactor q :=
      Nat.add_le_add (Nat.mul_le_mul_left _ hv) (Nat.mul_le_mul_left _ he)
    _ = _ := by ring

theorem build_vertices_le (input : GenericGraphTables.Table q) :
    (Table.build input).vertices ≤
      AlphabetGraphBounds.sizeFactor q * (input.vertices + input.darts) :=
  (Nat.le_add_right _ _).trans (build_total_le input)

theorem build_darts_le (input : GenericGraphTables.Table q) :
    (Table.build input).darts ≤
      AlphabetGraphBounds.sizeFactor q * (input.vertices + input.darts) :=
  (Nat.le_add_left _ _).trans (build_total_le input)

theorem sizeFactor_positive : 0 < AlphabetGraphBounds.sizeFactor q :=
  AlphabetGraphBounds.sizeFactor_positive q

end BinPackingGames.Foundations.PCP.AlphabetTableBounds

namespace BinPackingGames.Foundations.PCP.RoundTables

abbrev BaseTable := PreprocessingTables.BaseTable

opaque fixedWalkParameter : {n : Nat // n = 2 * FinalConstants.endpointLength} :=
  ⟨2 * FinalConstants.endpointLength, rfl⟩

def walkParameter : Nat := fixedWalkParameter.val

theorem walkParameter_eq : walkParameter = 2 * FinalConstants.endpointLength :=
  fixedWalkParameter.property

def alphabet : Nat := PoweringTables.labelCount PreprocessingTables.degree walkParameter

theorem alphabet_positive : 0 < alphabet := by
  unfold alphabet PoweringTables.labelCount
  exact Nat.pow_pos (by decide)

def powered (H : BaseTable) (table : GraphTables.Table) : GenericGraphTables.Table alphabet :=
  PoweringTables.table (PreprocessingTables.preprocess H table) walkParameter

def build (H : BaseTable) (table : GraphTables.Table) : GraphTables.Table :=
  AlphabetTable.Table.build (powered H table)

def sizeFactor : Nat := AlphabetGraphBounds.sizeFactor alphabet *
  (ExpanderFamily.growth ^ 2 * (1 + 2 * PreprocessingTables.degree ^ (walkParameter + 1)))

theorem degree_positive : 0 < PreprocessingTables.degree := by
  unfold PreprocessingTables.degree
  omega

theorem dartFactor_positive (q : Nat) : 0 < AlphabetGraphBounds.dartFactor q := by
  unfold AlphabetGraphBounds.dartFactor AlphabetGraphBounds.localEventFactor
  positivity

theorem sizeFactor_positive : 0 < sizeFactor := by
  unfold sizeFactor
  apply Nat.mul_pos (AlphabetGraphBounds.sizeFactor_positive alphabet)
  apply Nat.mul_pos
  · exact Nat.pow_pos (Nat.zero_lt_one.trans ExpanderFamily.growth_gt_one)
  · omega

theorem powered_vertices (H : BaseTable) (table : GraphTables.Table) :
    (powered H table).vertices = PreprocessingTables.vertices table := rfl

theorem powered_darts (H : BaseTable) (table : GraphTables.Table) :
    (powered H table).darts = 2 * PreprocessingTables.vertices table *
      PreprocessingTables.degree ^ (walkParameter + 1) := rfl

theorem build_darts_positive (H : BaseTable) (table : GraphTables.Table) :
    0 < (build H table).darts := by
  change 0 < (AlphabetTable.Table.build (powered H table)).darts
  rw [AlphabetTableBounds.build_darts, powered_darts]
  exact Nat.mul_pos (Nat.mul_pos
    (Nat.mul_pos (by decide) (PreprocessingTables.vertices_positive table))
    (Nat.pow_pos degree_positive)) (dartFactor_positive alphabet)

theorem build_size_le (H : BaseTable) (table : GraphTables.Table)
    (ht : 0 < table.darts) :
    (build H table).vertices + (build H table).darts ≤
      sizeFactor * (table.vertices + table.darts) := by
  have hv := PreprocessingTables.vertices_le_of_positive table ht
  have he : (powered H table).vertices + (powered H table).darts =
      PreprocessingTables.vertices table *
        (1 + 2 * PreprocessingTables.degree ^ (walkParameter + 1)) := by
    rw [powered_vertices, powered_darts]
    ring
  have hp := Nat.mul_le_mul_right
    (1 + 2 * PreprocessingTables.degree ^ (walkParameter + 1)) hv
  calc
    _ ≤ AlphabetGraphBounds.sizeFactor alphabet *
        ((powered H table).vertices + (powered H table).darts) :=
      AlphabetTableBounds.build_total_le (powered H table)
    _ = AlphabetGraphBounds.sizeFactor alphabet *
        (PreprocessingTables.vertices table *
          (1 + 2 * PreprocessingTables.degree ^ (walkParameter + 1))) := by rw [he]
    _ ≤ AlphabetGraphBounds.sizeFactor alphabet *
        ((ExpanderFamily.growth ^ 2 * table.darts) *
          (1 + 2 * PreprocessingTables.degree ^ (walkParameter + 1))) :=
      Nat.mul_le_mul_left _ hp
    _ = sizeFactor * table.darts := by unfold sizeFactor; ring
    _ ≤ sizeFactor * (table.vertices + table.darts) :=
      Nat.mul_le_mul_left _ (Nat.le_add_left _ _)

theorem build_completeness (H : BaseTable) (table : GraphTables.Table)
    (sat : (GraphTables.semantics table).Satisfiable) :
    (GraphTables.semantics (build H table)).Satisfiable := by
  let : Nonempty (Fin alphabet) := ⟨⟨0, alphabet_positive⟩⟩
  exact AlphabetTable.Table.perfect_completeness (powered H table)
    (PoweringTableSemantics.preserves_satisfiability
      (PreprocessingTables.preprocess H table) walkParameter
      (PreprocessingGuarantees.completeness H table sat))

abbrev Input := {table : GraphTables.Table // 0 < table.darts}

instance (input : Input) : Nonempty (Fin input.val.darts) := ⟨⟨0, input.property⟩⟩

def step (H : BaseTable) (input : Input) : Input :=
  ⟨build H input.val, build_darts_positive H input.val⟩

def initial (F : Target.Formula) : Input :=
  ⟨RawInitialTables.table F, RawInitialTables.table_darts_positive F⟩

def size (input : Input) : Nat := input.val.vertices + input.val.darts

def Satisfiable (input : Input) : Prop := (GraphTables.semantics input.val).Satisfiable

noncomputable def gap (input : Input) : ℝ := (GraphTables.semantics input.val).gap

theorem size_positive (input : Input) : 0 < size input :=
  input.property.trans_le (Nat.le_add_left _ _)

theorem gap_nonnegative (input : Input) : 0 ≤ gap input :=
  (GraphTables.semantics input.val).gap_nonnegative

theorem gap_eq_zero_iff (input : Input) : gap input = 0 ↔ Satisfiable input :=
  (GraphTables.semantics input.val).gap_eq_zero_iff

theorem one_le_size_mul_gap (input : Input) (unsat : ¬ Satisfiable input) :
    1 ≤ (size input : ℝ) * gap input := by
  have h := (GraphTables.semantics input.val).inverse_card_le_gap_of_unsatisfiable unsat
  have hp : (0 : ℝ) < input.val.darts := Nat.cast_pos.mpr input.property
  simp only [Fintype.card_fin] at h
  have hd := (div_le_iff₀ hp).mp h
  have hsize : (input.val.darts : ℝ) ≤ (size input : ℝ) := by
    exact_mod_cast (Nat.le_add_left input.val.darts input.val.vertices)
  exact (show 1 ≤ (input.val.darts : ℝ) * gap input by
    simpa only [gap, mul_comm] using hd).trans
    (mul_le_mul_of_nonneg_right hsize (gap_nonnegative input))

theorem initial_satisfiable_iff (F : Target.Formula) : Satisfiable (initial F) ↔ F.Satisfiable :=
  RawInitialTables.table_satisfiable_iff F

theorem step_completeness (H : BaseTable) (input : Input) :
    Satisfiable input → Satisfiable (step H input) := build_completeness H input.val

theorem step_size (H : BaseTable) (input : Input) :
    size (step H input) ≤ sizeFactor * size input := build_size_le H input.val input.property

end BinPackingGames.Foundations.PCP.RoundTables

namespace BinPackingGames.Foundations.PCP.RoundTableGap

open RoundTables
open PoweringWalks SpectralReturn

variable (H : BaseTable)
  (certificate : SpectralCertificate (ExpanderTables.graph H) (1 / 100 : ℝ))

include certificate

theorem powered_count_gap (input : Input)
    (labeling : Fin (powered H input.val).vertices → Fin alphabet) :
    RoundGap.poweredLower (gap input) * ((powered H input.val).darts : ℝ) ≤
      ((GenericGraphTables.semantics (powered H input.val)).rejectionCount labeling : ℝ) := by
  let old := PreprocessingOverlayTables.overlay
    (PreprocessingTables.padded H input.val) (PreprocessingTables.overlayFamily H input.val)
  have hd : 0 < (PreprocessingRegularTables.internalDegree + 1) +
      PreprocessingRegularTables.internalDegree := by omega
  let : Nonempty (Fin ((PreprocessingRegularTables.internalDegree + 1) +
      PreprocessingRegularTables.internalDegree)) := ⟨⟨0, hd⟩⟩
  have spectral : SpectralCertificate (lazyGraph (PortTables.portGraph old)) (31 / 32 : ℝ) :=
    LazySpectral.lazy_certificate_31_32 (PortTables.portGraph old)
      (PreprocessingGuarantees.overlay_certificate H certificate input.val)
  have lower := PreprocessingGuarantees.gap_transfer_real H certificate input.val input.property
    (gap input) (gap_nonnegative input)
    (by simpa only [Fintype.card_fin] using
      ((GraphTables.semantics input.val).le_gap_iff (gap input)).mp le_rfl)
  have hs : (0 : ℝ) < Preprocessing.sizeFactor := Nat.cast_pos.mpr Preprocessing.sizeFactor_positive
  have hn : walkParameter = 2 * PoweringSoundness.center 64 FinalConstants.windowHalf :=
    walkParameter_eq
  have bound := PoweringPortReindex.lazy_table_uniform_count_gap_at old
    (PreprocessingTables.vertices_positive input.val) hd (31 / 32) spectral
    FinalConstants.windowHalf FinalConstants.windowHalf_positive walkParameter hn
    (gap input / Preprocessing.sizeFactor) (div_nonneg (gap_nonnegative input) hs.le)
    lower labeling
  have hw : walkParameter + 1 = FinalConstants.walkLength := by
    rw [walkParameter_eq]
    rfl
  simpa only [RoundGap.poweredLower, PoweringFinalConstants.card_alphabet,
    hw, FinalConstants.cap, powered, PreprocessingTables.preprocess, old] using! bound

theorem build_count_gap (input : Input)
    (labeling : Fin (build H input.val).vertices → GraphTables.Label) :
    min (2 * gap input) FinalConstants.cap * ((build H input.val).darts : ℝ) ≤
      ((GraphTables.semantics (build H input.val)).rejectionCount labeling : ℝ) := by
  have h := AlphabetTableBounds.gap_transfer_real alphabet_positive (powered H input.val)
    (RoundGap.poweredLower (gap input))
    (RoundGap.poweredLower_nonnegative (gap input) (gap_nonnegative input))
    (powered_count_gap H certificate input) labeling
  have scalar : min (2 * gap input) FinalConstants.cap ≤
      RoundGap.poweredLower (gap input) / 12288 := by
    simpa only [RoundGap.poweredLower, FinalConstants.cap, FinalConstants.walkLength,
      PoweringFinalConstants.center_eq] using
      PoweringFinalConstants.composed_scaled_gap (gap input) (gap_nonnegative input)
  exact (mul_le_mul_of_nonneg_right scalar (Nat.cast_nonneg _)).trans h

theorem step_gap (input : Input) :
    min (2 * gap input) FinalConstants.cap ≤ gap (step H input) := by
  apply ((GraphTables.semantics (step H input).val).le_gap_iff _).mpr
  change ∀ labeling : Fin (build H input.val).vertices → GraphTables.Label,
    min (2 * gap input) FinalConstants.cap *
        (Fintype.card (Fin (build H input.val).darts) : ℝ) ≤
      ((GraphTables.semantics (build H input.val)).rejectionCount labeling : ℝ)
  simpa only [Fintype.card_fin] using build_count_gap H certificate input

end BinPackingGames.Foundations.PCP.RoundTableGap

end OAI