lean/OAI/Computability/BinPacking/Information/SourceNonempty.lean

import Mathlib.Algebra.Polynomial.Eval.Defs
import Mathlib.Tactic.Ring
import OAI.Computability.BinPacking.CookLevin.PolynomialMachine
import OAI.Computability.BinPacking.Information.SourceHonest
import OAI.Computability.BinPacking.Information.SourceLoopInit
import OAI.Computability.BinPacking.PCP.FinalBooleanVerifier
import OAI.Computability.BinPacking.PCP.SourceAmplification

namespace OAI

noncomputable section

namespace BinPackingGames.Foundations.Hastad.SourceBounds

open Target
open BinPackingGames.Reduction

theorem formulaBits_length (F : Formula) :
    (Complexity.formulaBits F).length = F.variables + F.clauses.length + 2 +
      (Complexity.encodeWords (F.clauses.flatMap Complexity.clauseWords)).length := by
  simp only [Complexity.formulaBits, Complexity.formulaWords,
    Complexity.encodeWords_append, List.length_append, Complexity.encodeWords,
    Complexity.encodeWord_length, List.length_nil]
  omega

theorem formulaBits_length_ge_variables (F : Formula) :
    F.variables ≤ (Complexity.formulaBits F).length := by
  rw [formulaBits_length]
  omega

theorem formulaBits_length_ge_clauses (F : Formula) :
    F.clauses.length ≤ (Complexity.formulaBits F).length := by
  rw [formulaBits_length]
  omega

theorem formulaBits_length_ge_two (F : Formula) :
    2 ≤ (Complexity.formulaBits F).length := by
  rw [formulaBits_length]
  omega

def bitCountFormula (n m u : ℕ) : ℕ :=
  n ^ u * 2 ^ (2 ^ u) + m ^ u * 2 ^ (8 ^ u) + 1

def bitCoefficient (u : ℕ) : ℕ := 2 ^ (2 ^ u) + 2 ^ (8 ^ u) + 1

def testCoefficient (u D : ℕ) : ℕ :=
  3 ^ u * 2 ^ (2 ^ u) * D ^ (8 ^ u) * 2 ^ (8 ^ u)

def occurrenceCoefficient (u D : ℕ) : ℕ := testCoefficient u D + 1

theorem one_le_input_power (F : Formula) (u : ℕ) :
    1 ≤ (Complexity.formulaBits F).length ^ u := by
  apply Nat.one_le_pow
  have h := formulaBits_length_ge_two F
  omega

theorem bitCountFormula_le_input (F : Formula) (u : ℕ) :
    bitCountFormula F.variables F.clauses.length u ≤
      bitCoefficient u * (Complexity.formulaBits F).length ^ u := by
  have hn := Nat.pow_le_pow_left (formulaBits_length_ge_variables F) u
  have hm := Nat.pow_le_pow_left (formulaBits_length_ge_clauses F) u
  have hpos := one_le_input_power F u
  unfold bitCountFormula bitCoefficient
  calc
    _ ≤ (Complexity.formulaBits F).length ^ u * 2 ^ (2 ^ u) +
        (Complexity.formulaBits F).length ^ u * 2 ^ (8 ^ u) +
        (Complexity.formulaBits F).length ^ u :=
      Nat.add_le_add
        (Nat.add_le_add (Nat.mul_le_mul_right _ hn) (Nat.mul_le_mul_right _ hm)) hpos
    _ = _ := by ring

theorem testCountFormula_le_input (F : Formula) (u D : ℕ) :
    (F.clauses.length * 3) ^ u * 2 ^ (2 ^ u) * D ^ (8 ^ u) * 2 ^ (8 ^ u) ≤
      testCoefficient u D * (Complexity.formulaBits F).length ^ u := by
  have hm := Nat.pow_le_pow_left (formulaBits_length_ge_clauses F) u
  calc
    _ = testCoefficient u D * F.clauses.length ^ u := by
      simp only [testCoefficient, Nat.mul_pow]
      ring
    _ ≤ _ := Nat.mul_le_mul_left _ hm

theorem testCountFormula_add_one_le_input (F : Formula) (u D : ℕ) :
    (F.clauses.length * 3) ^ u * 2 ^ (2 ^ u) * D ^ (8 ^ u) * 2 ^ (8 ^ u) + 1 ≤
      occurrenceCoefficient u D * (Complexity.formulaBits F).length ^ u := by
  calc
    _ ≤ testCoefficient u D * (Complexity.formulaBits F).length ^ u +
        (Complexity.formulaBits F).length ^ u :=
      Nat.add_le_add (testCountFormula_le_input F u D) (one_le_input_power F u)
    _ = _ := by simp only [occurrenceCoefficient, Nat.add_mul, Nat.one_mul]

def sourceSizeBound (V E u L : ℕ) : ℕ :=
  V * L ^ u + E * L ^ u + 2 + E * L ^ u * (3 * (V * L ^ u) + 2)

theorem sourceBits_length_le (input : SourceEncoding.Input) (V E u L : ℕ)
    (hv : input.variables ≤ V * L ^ u)
    (he : input.equations.length ≤ E * L ^ u) :
    (SourceEncoding.inputBits input).length ≤ sourceSizeBound V E u L := by
  apply (SourceEncoding.inputBits_length_le input).trans
  unfold sourceSizeBound
  exact Nat.add_le_add
    (Nat.add_le_add_right (Nat.add_le_add hv he) 2)
    (Nat.mul_le_mul he (Nat.add_le_add_right (Nat.mul_le_mul_left 3 hv) 2))

noncomputable def sourcePolynomial (u D : ℕ) : Polynomial ℕ :=
  let V := Polynomial.C (bitCoefficient u) * Polynomial.X ^ u
  let E := Polynomial.C (occurrenceCoefficient u D) * Polynomial.X ^ u
  V + E + Polynomial.C 2 + E * (Polynomial.C 3 * V + Polynomial.C 2)

theorem sourcePolynomial_eval (u D L : ℕ) :
    (sourcePolynomial u D).eval L =
      sourceSizeBound (bitCoefficient u) (occurrenceCoefficient u D) u L := by
  simp [sourcePolynomial, sourceSizeBound]

open SourceOccurrences SourceContexts

theorem card_testTape (u D : ℕ) :
    Fintype.card (SourceTape.TestTape (I u) (J u) D) =
      2 ^ (2 ^ u) * (D ^ (8 ^ u) * 2 ^ (8 ^ u)) := by
  rw [SourceTape.card_testTape, card_I, card_J]

theorem testTapeEncoding_size (u D : ℕ) :
    (SourceOccurrences.testTapeEncoding u D).size =
      2 ^ (2 ^ u) * (D ^ (8 ^ u) * 2 ^ (8 ^ u)) := rfl

theorem card_sourceIndex (F : Formula) (u D : ℕ) :
    Fintype.card (SourceIndex F u D) =
      (F.clauses.length ^ u * 3 ^ u) *
        (2 ^ (2 ^ u) * (D ^ (8 ^ u) * 2 ^ (8 ^ u))) := by
  rw [(sourceIndexEncoding F u D).card_eq_size]
  rfl

theorem nBits_le_input (F : Formula) (u : ℕ) :
    nBits F u ≤ bitCoefficient u * (Complexity.formulaBits F).length ^ u := by
  have h := bitCountFormula_le_input F u
  simpa only [nBits_eq, bitCountFormula, Nat.add_assoc] using h

theorem rawSourceList_length_eq (F : Formula) (u D : ℕ) :
    (rawSourceList F u D).length = testCoefficient u D * F.clauses.length ^ u := by
  rw [rawSourceList_length]
  unfold testCoefficient
  ac_rfl

theorem rawSourceList_length_le_input (F : Formula) (u D : ℕ) :
    (rawSourceList F u D).length ≤
      testCoefficient u D * (Complexity.formulaBits F).length ^ u := by
  rw [rawSourceList_length_eq]
  exact Nat.mul_le_mul_left _
    (Nat.pow_le_pow_left (formulaBits_length_ge_clauses F) u)

theorem sourceList_length_le_input (F : Formula) (u D : ℕ) :
    (sourceList F u D).length ≤
      occurrenceCoefficient u D * (Complexity.formulaBits F).length ^ u := by
  calc
    _ ≤ (rawSourceList F u D).length + 1 := sourceList_length_le_raw_add_one F u D
    _ ≤ testCoefficient u D * (Complexity.formulaBits F).length ^ u +
        (Complexity.formulaBits F).length ^ u :=
      Nat.add_le_add (rawSourceList_length_le_input F u D) (one_le_input_power F u)
    _ = _ := by simp only [occurrenceCoefficient, Nat.add_mul, Nat.one_mul]

theorem sourceInput_bits_polynomial (F : Formula) (u D : ℕ) (hD : 0 < D) :
    (SourceEncoding.inputBits (sourceInput F u D hD)).length ≤
      (sourcePolynomial u D).eval (Complexity.formulaBits F).length := by
  rw [sourcePolynomial_eval]
  exact sourceBits_length_le (sourceInput F u D hD)
    (bitCoefficient u) (occurrenceCoefficient u D) u (Complexity.formulaBits F).length
    (nBits_le_input F u) (sourceList_length_le_input F u D)

theorem fixed_parameters_source_size (u D : ℕ) (hD : 0 < D) :
    ∃ p : Polynomial ℕ, ∀ F : Formula,
      (SourceEncoding.inputBits (sourceInput F u D hD)).length ≤
        p.eval (Complexity.formulaBits F).length :=
  ⟨sourcePolynomial u D, fun F => sourceInput_bits_polynomial F u D hD⟩

end BinPackingGames.Foundations.Hastad.SourceBounds

namespace BinPackingGames.Foundations.Hastad.SourceSoundness

open scoped BigOperators
open Target SourceContexts SourceOccurrences SourceGame

def leftTable (F : Formula) (u : ℕ) (bits : Fin (nBits F u) → Bool)
    (v : VariableContext F u) : HalfCube (leftAnchor u) → Bool :=
  fun h => bits ((proofEncoding F u).code (.inl (v, h.val)))

def rightOracle (F : Formula) (u : ℕ) (bits : Fin (nBits F u) → Bool)
    (c : ClauseContext F u) : ConditionedOracle (validJ F c) :=
  match h : rightAnchor F c with
  | none => .empty ((rightAnchor_none_iff F c).mp h)
  | some j₀ => .stored j₀
      (fun h => bits ((proofEncoding F u).code
        (.inr (.inl (c, extendRestricted (validJ F c) h.val)))))

@[simp] theorem rightOracle_answer (F : Formula) (u : ℕ)
    (bits : Fin (nBits F u) → Bool) (c : ClauseContext F u) :
    (rightOracle F u bits c).answer = rightResponse F u bits c := by
  unfold rightOracle
  split <;> simp_all [rightResponse, ConditionedOracle.answer]

@[simp] theorem leftTable_answer (F : Formula) (u : ℕ)
    (bits : Fin (nBits F u) → Bool) (v : VariableContext F u) :
    foldedAnswer (leftAnchor u) (leftTable F u bits v) = leftResponse F u bits v := rfl

theorem sourceList_sound (F : Formula) (hne : F.clauses ≠ []) (u D : ℕ)
    (hD : 0 < D) (hDtwo : 2 ≤ D) (δ : ℝ) (hδ : 0 ≤ δ)
    (hvalue : ((baseGame F hne).repetition u).value ≤
      4 * (D : ℝ)⁻¹ * δ ^ 2)
    (bits : Fin (nBits F u) → Bool) :
    ((sourceList F u D).countP (fun e => BinPackingGames.Reduction.CloneGap.satisfied e bits) : ℝ) /
      (sourceList F u D).length ≤ (1 + δ) / 2 := by
  rw [sourceList_nonempty F u D hne, rawSourceList_acceptance F u D hD]
  have hε : (0 : ℝ) < (D : ℝ)⁻¹ := inv_pos.mpr (Nat.cast_pos.mpr hD)
  have hε' : (D : ℝ)⁻¹ ≤ (1 : ℝ) / 2 := by
    have hd : (2 : ℝ) ≤ D := by exact_mod_cast hDtwo
    simpa only [one_div] using
      (inv_le_inv₀ (Nat.cast_pos.mpr hD) (by norm_num : (0 : ℝ) < 2)).mpr hd
  have h := source_acceptance_le_split F hne u ((D : ℝ)⁻¹) δ
    (fun _ => leftAnchor u) (leftTable F u bits) (rightOracle F u bits)
    hε hε' hδ hvalue
  simpa only [leftTable_answer, rightOracle_answer] using h

theorem sourceList_sound_rat (F : Formula) (hne : F.clauses ≠ []) (u D : ℕ)
    (hD : 0 < D) (hDtwo : 2 ≤ D) (δ : ℚ) (hδ : 0 ≤ δ)
    (hvalue : ((baseGame F hne).repetition u).value ≤
      4 * (D : ℝ)⁻¹ * (δ : ℝ) ^ 2)
    (bits : Fin (nBits F u) → Bool) :
    ((sourceList F u D).countP (fun e => BinPackingGames.Reduction.CloneGap.satisfied e bits) : ℚ) /
      (sourceList F u D).length ≤ (1 + δ) / 2 := by
  have h := sourceList_sound F hne u D hD hDtwo (δ : ℝ)
    (by exact_mod_cast hδ) hvalue bits
  apply (Rat.cast_le (K := ℝ)).mp
  push_cast
  exact h

end BinPackingGames.Foundations.Hastad.SourceSoundness

namespace BinPackingGames.Foundations.Hastad.SourceHeaderCounts

open Turing Complexity
open MachineComposition

namespace Eval

open CookLevin.PolynomialMachine

variable {K Λ σ : Type} [DecidableEq K]

abbrev Label (p : Polynomial Nat) := Unit ⊕ (MachineHorner.Label (width p) ⊕ Unit)

def start (p : Polynomial Nat) : Label p := .inl ()
def inner (p : Polynomial Nat) (l : MachineHorner.Label (width p)) : Label p := .inr (.inl l)
def finish (p : Polynomial Nat) : Label p := .inr (.inr ())

def word (p : Polynomial Nat) (slots : MachineHorner.Layout (width p) ↪ K) : K → List Bool :=
  Function.extend slots (fieldWord p) (fun _ => [])

def indices (p : Polynomial Nat) (slots : MachineHorner.Layout (width p) ↪ K) : List K :=
  (fieldIndices p).map slots

omit [DecidableEq K] in
theorem indices_nodup (p : Polynomial Nat) (slots : MachineHorner.Layout (width p) ↪ K) :
    (indices p slots).Nodup := (fieldIndices_nodup p).map slots.injective

omit [DecidableEq K] in
@[simp] theorem word_slot (p : Polynomial Nat) (slots : MachineHorner.Layout (width p) ↪ K)
    (i : MachineHorner.Layout (width p)) : word p slots (slots i) = fieldWord p i :=
  slots.injective.extend_apply _ _ i

omit [DecidableEq K] in
@[simp] theorem control_not_mem (p : Polynomial Nat)
    (slots : MachineHorner.Layout (width p) ↪ K) (i : Fin 6) :
    slots (.inl i) ∉ indices p slots := by
  simp [indices, slots.injective.eq_iff]

omit [DecidableEq K] in
@[simp] theorem digit_mem (p : Polynomial Nat)
    (slots : MachineHorner.Layout (width p) ↪ K) (i : Fin (width p)) :
    slots (.inr i) ∈ indices p slots := by
  simp [indices, slots.injective.eq_iff]

def prepared (p : Polynomial Nat) (slots : MachineHorner.Layout (width p) ↪ K)
    (base : K → List Bool) : K → List Bool :=
  prependFieldsTapes (word p slots) (indices p slots) base

@[simp] theorem prepared_control (p : Polynomial Nat)
    (slots : MachineHorner.Layout (width p) ↪ K) (base : K → List Bool) (i : Fin 6) :
    prepared p slots base (slots (.inl i)) = base (slots (.inl i)) := by
  rw [prepared, prependFieldsTapes_apply _ _ (indices_nodup p slots)]
  simp

@[simp] theorem prepared_digit (p : Polynomial Nat)
    (slots : MachineHorner.Layout (width p) ↪ K) (base : K → List Bool)
    (i : Fin (width p)) (hi : base (slots (.inr i)) = []) :
    prepared p slots base (slots (.inr i)) = encodeWord (digit p i.val) := by
  rw [prepared, prependFieldsTapes_apply _ _ (indices_nodup p slots)]
  simp [fieldWord, hi]

theorem clear_result (p : Polynomial Nat) (slots : MachineHorner.Layout (width p) ↪ K)
    (base : K → List Bool) (result : Nat)
    (hcoeff : ∀ i : Fin (width p), base (slots (.inr i)) = []) :
    clearFieldsTapes (fun k => (word p slots k).length) (indices p slots)
      (MachineHorner.resultTapes slots (prepared p slots base) result) =
      MachineHorner.resultTapes slots base result := by
  funext k
  rw [clearFieldsTapes_apply _ _ (indices_nodup p slots)]
  by_cases h : k ∈ indices p slots
  · obtain ⟨q, hq, rfl⟩ := List.mem_map.mp h
    obtain ⟨i, hi, rfl⟩ := List.mem_map.mp hq
    simp [MachineHorner.resultTapes, slots.injective.eq_iff, prepared_digit,
      fieldWord, hcoeff]
  · by_cases hd : k = slots (.inl 3)
    · subst k
      simp [MachineHorner.resultTapes]
    · simp [h, MachineHorner.resultTapes, hd, prepared,
        prependFieldsTapes_apply _ _ (indices_nodup p slots)]

def statement (p : Polynomial Nat) (slots : MachineHorner.Layout (width p) ↪ K)
    (labels : Label p → Λ) (exit : Option Λ) :
    Label p → TM2.Stmt (fun _ : K => Bool) Λ (MachineHorner.State σ)
  | .inl _ => prependFields (word p slots) (indices p slots)
      (.goto fun _ => labels (inner p .start))
  | .inr (.inl l) => MachineHorner.statement slots (fun l => labels (inner p l))
      (some (labels (finish p))) l
  | .inr (.inr _) => clearFields (fun k => (word p slots k).length) (indices p slots)
      (Reduction.MachineTransfer.exitAt (slots (.inl 3)) exit)

def steps (p : Polynomial Nat) (n : Nat) : Nat :=
  MachineHorner.steps n (digit p) (width p) + 2

theorem trace (p : Polynomial Nat) (slots : MachineHorner.Layout (width p) ↪ K)
    (labels : Label p → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (fun _ : K => Bool) Λ (MachineHorner.State σ))
    (atLabels : ∀ l, program (labels l) = statement p slots labels exit l)
    (base : K → List Bool) (n : Nat)
    (hinput : base (slots (.inl 0)) = encodeWord n)
    (hcoeff : ∀ i : Fin (width p), base (slots (.inr i)) = [])
    (clean : MachineHorner.Clean slots base) (ambient : σ) (register : Option Bool) :
    (advance (TM2.step program))^[steps p n]
      (some ⟨some (labels (start p)), ((ambient, ()), register), base⟩) =
      some ⟨exit, ((ambient, ()), none), MachineHorner.resultTapes slots base (p.eval n)⟩ := by
  have hp : (advance (TM2.step program))^[1]
      (some ⟨some (labels (start p)), ((ambient, ()), register), base⟩) =
      some ⟨some (labels (inner p .start)), ((ambient, ()), register), prepared p slots base⟩ := by
    change some (TM2.stepAux (program (labels (start p))) _ _) = _
    rw [atLabels]
    change some (TM2.stepAux (prependFields _ _ _) _ _) = _
    rw [stepAux_prependFields]
    rfl
  have hh := MachineHorner.hornerTrace slots (fun l => labels (inner p l))
    (some (labels (finish p))) program (fun l => atLabels (inner p l))
    (prepared p slots base) n (digit p) (by simpa using hinput)
    (fun i => prepared_digit p slots base i (hcoeff i))
    (by
      constructor
      · simpa using clean.accA
      · simpa using clean.accB
      · simpa using clean.counter
      · simpa using clean.scratch) ambient register
  rw [hornerValue_eq_eval] at hh
  have hc : (advance (TM2.step program))^[1]
      (some ⟨some (labels (finish p)), ((ambient, ()), none),
        MachineHorner.resultTapes slots (prepared p slots base) (p.eval n)⟩) =
      some ⟨exit, ((ambient, ()), none), MachineHorner.resultTapes slots base (p.eval n)⟩ := by
    change some (TM2.stepAux (program (labels (finish p))) _ _) = _
    rw [atLabels]
    change some (TM2.stepAux (clearFields _ _ _) _ _) = _
    rw [stepAux_clearFields, clear_result p slots base _ hcoeff]
    cases exit <;> rfl
  rw [show steps p n = 1 + (MachineHorner.steps n (digit p) (width p) + 1) by
    unfold steps; omega, Function.iterate_add_apply, Function.iterate_add_apply, hp, hh, hc]

def inTime (p : Polynomial Nat) (slots : MachineHorner.Layout (width p) ↪ K)
    (labels : Label p → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (fun _ : K => Bool) Λ (MachineHorner.State σ))
    (atLabels : ∀ l, program (labels l) = statement p slots labels exit l)
    (base : K → List Bool) (n : Nat)
    (hinput : base (slots (.inl 0)) = encodeWord n)
    (hcoeff : ∀ i : Fin (width p), base (slots (.inr i)) = [])
    (clean : MachineHorner.Clean slots base) (ambient : σ) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some (labels (start p)), ((ambient, ()), register), base⟩
      (some ⟨exit, ((ambient, ()), none), MachineHorner.resultTapes slots base (p.eval n)⟩)
      ((timePolynomial p).eval n) where
  steps := steps p n
  evals_in_steps := trace p slots labels exit program atLabels base n hinput hcoeff clean ambient register
  steps_le_m := by rw [timePolynomial_eval]; unfold steps; omega

end Eval

def leftValue (u n : Nat) : Nat := 2 ^ (2 ^ u) * n ^ u
def rightValue (u m : Nat) : Nat := 2 ^ (8 ^ u) * m ^ u
def dummyValue (u n m : Nat) : Nat := leftValue u n + rightValue u m
def bitValue (u n m : Nat) : Nat := dummyValue u n m + 1
def occurrenceValue (u D m : Nat) : Nat :=
  3 ^ u * (SourceOccurrences.testTapeEncoding u D).size * m ^ u

inductive Kind
  | left | right | occurrences
  deriving DecidableEq

protected abbrev Kind.enumList : List Kind := [.left, .right, .occurrences]

protected theorem Kind.enumList_getElem?_ctorIdx_eq (x : Kind) :
    Kind.enumList[x.ctorIdx]? = some x := by
  cases x <;> rfl

protected theorem Kind.enumList_nodup : Kind.enumList.Nodup := by decide

instance : Fintype Kind where
  elems := ⟨Kind.enumList, Kind.enumList_nodup⟩
  complete x := by cases x <;> decide

def polynomial (u D : Nat) : Kind → Polynomial Nat
  | .left => Polynomial.C (2 ^ (2 ^ u)) * Polynomial.X ^ u
  | .right => Polynomial.C (2 ^ (8 ^ u)) * Polynomial.X ^ u
  | .occurrences => Polynomial.C (3 ^ u * (SourceOccurrences.testTapeEncoding u D).size) *
      Polynomial.X ^ u

@[simp] theorem polynomial_left_eval (u D n : Nat) :
    (polynomial u D .left).eval n = leftValue u n := by simp [polynomial, leftValue]
@[simp] theorem polynomial_right_eval (u D m : Nat) :
    (polynomial u D .right).eval m = rightValue u m := by simp [polynomial, rightValue]
@[simp] theorem polynomial_occurrences_eval (u D m : Nat) :
    (polynomial u D .occurrences).eval m = occurrenceValue u D m := by
  simp [polynomial, occurrenceValue]

theorem bitValue_eq_nBits (F : Target.Formula) (u : Nat) :
    bitValue u F.variables F.clauses.length = SourceOccurrences.nBits F u := by
  rw [SourceOccurrences.nBits_eq]
  simp [bitValue, dummyValue, leftValue, rightValue, Nat.mul_comm, Nat.add_assoc]

theorem occurrenceValue_eq_length (F : Target.Formula) (u D : Nat)
    (hm : 0 < F.clauses.length) :
    occurrenceValue u D F.clauses.length = (SourceOccurrences.sourceList F u D).length := by
  have hne : F.clauses ≠ [] := by intro h; simp [h] at hm
  have hfalse : ¬ F.clauses.isEmpty = true := by simpa using hne
  simp only [SourceOccurrences.sourceList, ite_eq_right hfalse]
  rw [SourceOccurrences.rawSourceList_length]
  unfold occurrenceValue
  rw [SourceBounds.testTapeEncoding_size]
  ac_rfl

inductive Control
  | variableCount | clauseCount | leftBlockSize | rightTemporary | dummyIndex
  | bitCount | occurrenceCount | accA | accB | counter | scratch
  deriving DecidableEq

protected abbrev Control.enumList : List Control := [.variableCount, .clauseCount, .leftBlockSize,
  .rightTemporary, .dummyIndex, .bitCount, .occurrenceCount, .accA, .accB, .counter, .scratch]

protected theorem Control.enumList_getElem?_ctorIdx_eq (x : Control) :
    Control.enumList[x.ctorIdx]? = some x := by
  cases x <;> rfl

protected theorem Control.enumList_nodup : Control.enumList.Nodup := by decide

instance : Fintype Control where
  elems := ⟨Control.enumList, Control.enumList_nodup⟩
  complete x := by cases x <;> decide

abbrev Coefficients (u D : Nat) :=
  Fin (CookLevin.PolynomialMachine.width (polynomial u D .left)) ⊕
    (Fin (CookLevin.PolynomialMachine.width (polynomial u D .right)) ⊕
      Fin (CookLevin.PolynomialMachine.width (polynomial u D .occurrences)))

abbrev Layout (u D : Nat) := Control ⊕ Coefficients u D

def coefficientBlock (u D : Nat) (kind : Kind) :
    Fin (CookLevin.PolynomialMachine.width (polynomial u D kind)) ↪ Coefficients u D :=
  match kind with
  | .left => ⟨Sum.inl, Sum.inl_injective⟩
  | .right => ⟨fun i => .inr (.inl i), Sum.inr_injective.comp Sum.inl_injective⟩
  | .occurrences => ⟨fun i => .inr (.inr i), Sum.inr_injective.comp Sum.inr_injective⟩

def inputControl : Kind → Control
  | .left => .variableCount | _ => .clauseCount

def outputControl : Kind → Control
  | .left => .leftBlockSize | .right => .rightTemporary | .occurrences => .occurrenceCount

def controlSlots (kind : Kind) : Fin 6 ↪ Control where
  toFun i := match i.val with
    | 0 => inputControl kind | 1 => .accA | 2 => .accB
    | 3 => outputControl kind | 4 => .counter | _ => .scratch
  inj' := by
    intro i j h
    cases kind <;> fin_cases i <;> fin_cases j <;> cases h <;> rfl

def localSlots (u D : Nat) (kind : Kind) :
    MachineHorner.Layout (CookLevin.PolynomialMachine.width (polynomial u D kind)) ↪ Layout u D where
  toFun
    | .inl i => .inl (controlSlots kind i)
    | .inr i => .inr (coefficientBlock u D kind i)
  inj' := by
    intro a b h
    cases a with
    | inl a =>
      cases b with
      | inl b => exact congrArg Sum.inl ((controlSlots kind).injective (Sum.inl.inj h))
      | inr b => cases h
    | inr a =>
      cases b with
      | inl b => cases h
      | inr b => exact congrArg Sum.inr ((coefficientBlock u D kind).injective (Sum.inr.inj h))

variable {u D : Nat} {K Λ σ : Type}

def control (slots : Layout u D ↪ K) (c : Control) : K := slots (.inl c)

def evalSlots (slots : Layout u D ↪ K) (kind : Kind) :
    MachineHorner.Layout (CookLevin.PolynomialMachine.width (polynomial u D kind)) ↪ K :=
  (localSlots u D kind).trans slots

@[simp] theorem evalSlots_radix (slots : Layout u D ↪ K) (kind : Kind) :
    evalSlots slots kind (.inl 0) = control slots (inputControl kind) := rfl
@[simp] theorem evalSlots_accA (slots : Layout u D ↪ K) (kind : Kind) :
    evalSlots slots kind (.inl 1) = control slots .accA := rfl
@[simp] theorem evalSlots_accB (slots : Layout u D ↪ K) (kind : Kind) :
    evalSlots slots kind (.inl 2) = control slots .accB := rfl
@[simp] theorem evalSlots_destination (slots : Layout u D ↪ K) (kind : Kind) :
    evalSlots slots kind (.inl 3) = control slots (outputControl kind) := rfl
@[simp] theorem evalSlots_counter (slots : Layout u D ↪ K) (kind : Kind) :
    evalSlots slots kind (.inl 4) = control slots .counter := rfl
@[simp] theorem evalSlots_scratch (slots : Layout u D ↪ K) (kind : Kind) :
    evalSlots slots kind (.inl 5) = control slots .scratch := rfl
@[simp] theorem evalSlots_digit (slots : Layout u D ↪ K) (kind : Kind)
    (i : Fin (CookLevin.PolynomialMachine.width (polynomial u D kind))) :
    evalSlots slots kind (.inr i) = slots (.inr (coefficientBlock u D kind i)) := rfl

@[simp] theorem control_eq_iff (slots : Layout u D ↪ K) (a b : Control) :
    control slots a = control slots b ↔ a = b := by
  simp [control, slots.injective.eq_iff]

structure Clean (slots : Layout u D ↪ K) (base : K → List Bool) : Prop where
  left : base (control slots .leftBlockSize) = []
  right : base (control slots .rightTemporary) = []
  dummy : base (control slots .dummyIndex) = []
  bits : base (control slots .bitCount) = []
  occurrences : base (control slots .occurrenceCount) = []
  accA : base (control slots .accA) = []
  accB : base (control slots .accB) = []
  counter : base (control slots .counter) = []
  scratch : base (control slots .scratch) = []
  coefficients : ∀ i : Coefficients u D, base (slots (.inr i)) = []

inductive Label (u D : Nat)
  | evaluate (kind : Kind) (localLabel : Eval.Label (polynomial u D kind))
  | copyLeftOut | copyLeftBack | addRight | clearTemporary
  | copyDummyOut | copyDummyBack | incrementBits
  deriving DecidableEq, Fintype

def evalStart (kind : Kind) : Label u D := .evaluate kind (Eval.start (polynomial u D kind))

def afterEval : Kind → Label u D
  | .left => evalStart .right | .right => evalStart .occurrences | .occurrences => .copyLeftOut

variable [DecidableEq K]

def statement (slots : Layout u D ↪ K) (labels : Label u D → Λ) (exit : Option Λ) :
    Label u D → TM2.Stmt (fun _ : K => Bool) Λ (MachineHorner.State σ)
  | .evaluate kind l => Eval.statement (polynomial u D kind) (evalSlots slots kind)
      (fun l => labels (.evaluate kind l)) (some (labels (afterEval kind))) l
  | .copyLeftOut => Reduction.MachineTransfer.loopAt (control slots .leftBlockSize)
      (control slots .scratch) id false (labels .copyLeftOut) (some (labels .copyLeftBack))
  | .copyLeftBack => MachineCopy.forkLoop (control slots .scratch)
      (control slots .leftBlockSize) (control slots .dummyIndex) false
      (labels .copyLeftBack) (some (labels .addRight))
  | .addRight => MachineUnaryAddAt.loop (control slots .rightTemporary)
      (control slots .dummyIndex) (labels .addRight) (some (labels .clearTemporary))
  | .clearTemporary => .pop (control slots .rightTemporary) (fun state _ => state)
      (.goto fun _ => labels .copyDummyOut)
  | .copyDummyOut => Reduction.MachineTransfer.loopAt (control slots .dummyIndex)
      (control slots .scratch) id false (labels .copyDummyOut) (some (labels .copyDummyBack))
  | .copyDummyBack => MachineCopy.forkLoop (control slots .scratch)
      (control slots .dummyIndex) (control slots .bitCount) false
      (labels .copyDummyBack) (some (labels .incrementBits))
  | .incrementBits => .push (control slots .bitCount) (fun _ => true)
      (Reduction.MachineTransfer.exitAt (control slots .bitCount) exit)

def resultTapes (slots : Layout u D ↪ K) (base : K → List Bool) (n m : Nat) : K → List Bool :=
  Function.update (Function.update (Function.update (Function.update base
    (control slots .leftBlockSize) (encodeWord (leftValue u n)))
    (control slots .occurrenceCount) (encodeWord (occurrenceValue u D m)))
    (control slots .dummyIndex) (encodeWord (dummyValue u n m)))
    (control slots .bitCount) (encodeWord (bitValue u n m))

def steps (u D n m : Nat) : Nat :=
  Eval.steps (polynomial u D .left) n + Eval.steps (polynomial u D .right) m +
    Eval.steps (polynomial u D .occurrences) m +
    2 * leftValue u n + rightValue u m + 2 * dummyValue u n m + 11

structure WorkClean (slots : Layout u D ↪ K) (base : K → List Bool) : Prop where
  accA : base (control slots .accA) = []
  accB : base (control slots .accB) = []
  counter : base (control slots .counter) = []
  scratch : base (control slots .scratch) = []
  coefficients : ∀ i : Coefficients u D, base (slots (.inr i)) = []

omit [DecidableEq K] in
theorem Clean.work {slots : Layout u D ↪ K} {base : K → List Bool} (clean : Clean slots base) :
    WorkClean slots base :=
  ⟨clean.accA, clean.accB, clean.counter, clean.scratch, clean.coefficients⟩

theorem WorkClean.update_output {slots : Layout u D ↪ K} {base : K → List Bool}
    (clean : WorkClean slots base) (kind : Kind) (value : List Bool) :
    WorkClean slots (Function.update base (control slots (outputControl kind)) value) := by
  cases kind <;> constructor
  all_goals try { simpa [outputControl] using clean.accA }
  all_goals try { simpa [outputControl] using clean.accB }
  all_goals try { simpa [outputControl] using clean.counter }
  all_goals try { simpa [outputControl] using clean.scratch }
  all_goals intro i; simpa [control, slots.injective.eq_iff] using clean.coefficients i

theorem evalPhaseTrace (slots : Layout u D ↪ K) (labels : Label u D → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (fun _ : K => Bool) Λ (MachineHorner.State σ))
    (atLabels : ∀ l, program (labels l) = statement slots labels exit l)
    (kind : Kind) (base : K → List Bool) (n : Nat)
    (hinput : base (control slots (inputControl kind)) = encodeWord n)
    (hdest : base (control slots (outputControl kind)) = [])
    (clean : WorkClean slots base) (ambient : σ) (register : Option Bool) :
    (advance (TM2.step program))^[Eval.steps (polynomial u D kind) n]
      (some ⟨some (labels (evalStart kind)), ((ambient, ()), register), base⟩) =
      some ⟨some (labels (afterEval kind)), ((ambient, ()), none),
        Function.update base (control slots (outputControl kind))
          (encodeWord ((polynomial u D kind).eval n))⟩ := by
  have h := Eval.trace (polynomial u D kind) (evalSlots slots kind)
    (fun l => labels (.evaluate kind l)) (some (labels (afterEval kind))) program
    (fun l => atLabels (.evaluate kind l)) base n (by simpa using hinput)
    (fun i => by simpa using clean.coefficients (coefficientBlock u D kind i))
    ⟨by simpa using clean.accA, by simpa using clean.accB,
      by simpa using clean.counter, by simpa using clean.scratch⟩ ambient register
  simpa only [evalStart, MachineHorner.resultTapes, evalSlots_destination,
    hdest, List.append_nil] using h

private theorem trace_trans {α : Type*} (f : α → α) {a b : Nat} {x y z : α}
    (first : f^[a] x = y) (second : f^[b] y = z) : f^[a + b] x = z := by
  rw [Nat.add_comm, Function.iterate_add_apply, first, second]

theorem trace (slots : Layout u D ↪ K) (labels : Label u D → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (fun _ : K => Bool) Λ (MachineHorner.State σ))
    (atLabels : ∀ l, program (labels l) = statement slots labels exit l)
    (base : K → List Bool) (n m : Nat)
    (hn : base (control slots .variableCount) = encodeWord n)
    (hm : base (control slots .clauseCount) = encodeWord m)
    (clean : Clean slots base) (ambient : σ) (register : Option Bool) :
    (advance (TM2.step program))^[steps u D n m]
      (some ⟨some (labels (evalStart .left)), ((ambient, ()), register), base⟩) =
      some ⟨exit, ((ambient, ()), none), resultTapes slots base n m⟩ := by
  let t1 := Function.update base (control slots .leftBlockSize) (encodeWord (leftValue u n))
  let t2 := Function.update t1 (control slots .rightTemporary) (encodeWord (rightValue u m))
  let t3 := Function.update t2 (control slots .occurrenceCount) (encodeWord (occurrenceValue u D m))
  let t4 := Function.update t3 (control slots .dummyIndex) (encodeWord (leftValue u n))
  let t5 := Function.update (Function.update t4 (control slots .rightTemporary) [false])
    (control slots .dummyIndex) (encodeWord (dummyValue u n m))
  let t6 := Function.update t5 (control slots .rightTemporary) []
  let t7 := Function.update t6 (control slots .bitCount) (encodeWord (dummyValue u n m))
  have h1 := evalPhaseTrace slots labels exit program atLabels .left base n hn clean.left
    clean.work ambient register
  change (advance (TM2.step program))^[Eval.steps (polynomial u D .left) n]
      (some ⟨some (labels (evalStart .left)), ((ambient, ()), register), base⟩) =
      some ⟨some (labels (evalStart .right)), ((ambient, ()), none), _⟩ at h1
  simp only [polynomial_left_eval] at h1
  have cw1 : WorkClean slots t1 := clean.work.update_output .left _
  have h2 := evalPhaseTrace slots labels exit program atLabels .right t1 m
    (by simpa [t1, inputControl] using hm)
    (by simpa [t1, outputControl] using clean.right) cw1 ambient none
  change (advance (TM2.step program))^[Eval.steps (polynomial u D .right) m]
      (some ⟨some (labels (evalStart .right)), ((ambient, ()), none), t1⟩) =
      some ⟨some (labels (evalStart .occurrences)), ((ambient, ()), none), _⟩ at h2
  simp only [polynomial_right_eval] at h2
  have cw2 : WorkClean slots t2 := cw1.update_output .right _
  have h3 := evalPhaseTrace slots labels exit program atLabels .occurrences t2 m
    (by simpa [t2, t1, inputControl] using hm)
    (by simpa [t2, t1, outputControl] using clean.occurrences) cw2 ambient none
  change (advance (TM2.step program))^[Eval.steps (polynomial u D .occurrences) m]
      (some ⟨some (labels (evalStart .occurrences)), ((ambient, ()), none), t2⟩) =
      some ⟨some (labels .copyLeftOut), ((ambient, ()), none), _⟩ at h3
  simp only [polynomial_occurrences_eval] at h3
  have cw3 : WorkClean slots t3 := cw2.update_output .occurrences _
  have copyingLeft := MachineCopy.copyTrace (control slots .leftBlockSize)
    (control slots .dummyIndex) (control slots .scratch)
    (by simp) (by simp) (by simp) false (labels .copyLeftOut) (labels .copyLeftBack)
    (some (labels .addRight)) program (atLabels .copyLeftOut) (atLabels .copyLeftBack)
    t3 cw3.scratch (ambient, ()) none
  have hleft : t3 (control slots .leftBlockSize) = encodeWord (leftValue u n) := by
    simp [t3, t2, t1]
  have hdummy : t3 (control slots .dummyIndex) = [] := by
    simpa [t3, t2, t1] using clean.dummy
  rw [hleft, hdummy, List.append_nil, encodeWord_length] at copyingLeft
  have adding := MachineUnaryAddAt.addFromTapes (control slots .rightTemporary)
    (control slots .dummyIndex) (by simp) (labels .addRight) (some (labels .clearTemporary))
    program (atLabels .addRight) t4 (rightValue u m) (leftValue u n) [] []
    (by simp [t4, t3, t2]) (by simp [t4]) (ambient, ()) none
  have ha : MachineUnaryAddAt.unaryTapes (control slots .rightTemporary)
      (control slots .dummyIndex) t4 0 (rightValue u m + leftValue u n) [] [] = t5 := by
    simp [MachineUnaryAddAt.unaryTapes, Reduction.MachineTransfer.tapesAt, t5,
      encodeWord, dummyValue, Nat.add_comm]
  rw [ha] at adding
  have clearing : (advance (TM2.step program))^[1]
      (some ⟨some (labels .clearTemporary), ((ambient, ()), none), t5⟩) =
      some ⟨some (labels .copyDummyOut), ((ambient, ()), none), t6⟩ := by
    change some (TM2.stepAux (program (labels .clearTemporary)) _ _) = _
    rw [atLabels]
    simp [statement, TM2.stepAux, t6, t5]
  have copyingDummy := MachineCopy.copyTrace (control slots .dummyIndex)
    (control slots .bitCount) (control slots .scratch)
    (by simp) (by simp) (by simp) false (labels .copyDummyOut) (labels .copyDummyBack)
    (some (labels .incrementBits)) program (atLabels .copyDummyOut) (atLabels .copyDummyBack)
    t6 (by simpa [t6, t5, t4] using cw3.scratch) (ambient, ()) none
  have hdi : t6 (control slots .dummyIndex) = encodeWord (dummyValue u n m) := by
    simp [t6, t5]
  have hbi : t6 (control slots .bitCount) = [] := by
    simpa [t6, t5, t4, t3, t2, t1] using clean.bits
  rw [hdi, hbi, List.append_nil, encodeWord_length] at copyingDummy
  have increment : (advance (TM2.step program))^[1]
      (some ⟨some (labels .incrementBits), ((ambient, ()), none), t7⟩) =
      some ⟨exit, ((ambient, ()), none),
        Function.update t7 (control slots .bitCount) (encodeWord (bitValue u n m))⟩ := by
    change some (TM2.stepAux (program (labels .incrementBits)) _ _) = _
    rw [atLabels]
    cases exit <;> simp [statement, TM2.stepAux, Reduction.MachineTransfer.exitAt,
      t7, bitValue, encodeWord, List.replicate_succ]
  have total := trace_trans _ (trace_trans _ (trace_trans _ (trace_trans _
    (trace_trans _ (trace_trans _ (trace_trans _ h1 h2) h3) copyingLeft) adding) clearing)
      copyingDummy) increment
  have htime :
      Eval.steps (polynomial u D .left) n + Eval.steps (polynomial u D .right) m +
        Eval.steps (polynomial u D .occurrences) m + 2 * (leftValue u n + 1 + 1) +
        (rightValue u m + 1) + 1 + 2 * (dummyValue u n m + 1 + 1) + 1 =
      steps u D n m := by unfold steps; omega
  rw [htime] at total
  have frame : Function.update t7 (control slots .bitCount) (encodeWord (bitValue u n m)) =
      resultTapes slots base n m := by
    funext k
    by_cases hr : k = control slots .rightTemporary
    · subst k
      simp [t7, t6, t5, t4, t3, t2, t1, resultTapes, clean.right]
    · simp [t7, t6, t5, t4, t3, t2, t1, resultTapes, Function.update_apply, hr]
      split_ifs <;> rfl
  rw [frame] at total
  exact total

def timePolynomial (u D : Nat) : Polynomial Nat :=
  CookLevin.PolynomialMachine.timePolynomial (polynomial u D .left) +
    CookLevin.PolynomialMachine.timePolynomial (polynomial u D .right) +
    CookLevin.PolynomialMachine.timePolynomial (polynomial u D .occurrences) +
    Polynomial.C 2 * polynomial u D .left + polynomial u D .right +
    Polynomial.C 2 * (polynomial u D .left + polynomial u D .right) + Polynomial.C 11

theorem steps_le_timePolynomial (u D n m L : Nat) (hn : n ≤ L) (hm : m ≤ L) :
    steps u D n m ≤ (timePolynomial u D).eval L := by
  have hEval (p : Polynomial Nat) (a : Nat) (ha : a ≤ L) :
      Eval.steps p a ≤ (CookLevin.PolynomialMachine.timePolynomial p).eval L := by
    apply Nat.le_trans (m := (CookLevin.PolynomialMachine.timePolynomial p).eval a)
    · rw [CookLevin.PolynomialMachine.timePolynomial_eval]
      unfold Eval.steps
      omega
    · exact natPolynomial_eval_mono _ ha
  have h1 := hEval (polynomial u D .left) n hn
  have h2 := hEval (polynomial u D .right) m hm
  have h3 := hEval (polynomial u D .occurrences) m hm
  have hl : leftValue u n ≤ leftValue u L :=
    Nat.mul_le_mul_left _ (Nat.pow_le_pow_left hn u)
  have hr : rightValue u m ≤ rightValue u L :=
    Nat.mul_le_mul_left _ (Nat.pow_le_pow_left hm u)
  simp only [timePolynomial, Polynomial.eval_add, Polynomial.eval_mul, Polynomial.eval_C,
    polynomial_left_eval, polynomial_right_eval]
  unfold steps dummyValue
  omega

def inTime (slots : Layout u D ↪ K) (labels : Label u D → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (fun _ : K => Bool) Λ (MachineHorner.State σ))
    (atLabels : ∀ l, program (labels l) = statement slots labels exit l)
    (base : K → List Bool) (n m : Nat)
    (hn : base (control slots .variableCount) = encodeWord n)
    (hm : base (control slots .clauseCount) = encodeWord m)
    (clean : Clean slots base) (ambient : σ) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some (labels (evalStart .left)), ((ambient, ()), register), base⟩
      (some ⟨exit, ((ambient, ()), none), resultTapes slots base n m⟩)
      (steps u D n m) where
  steps := steps u D n m
  evals_in_steps := trace slots labels exit program atLabels base n m hn hm clean ambient register
  steps_le_m := Nat.le_refl _

def inPolynomialTime (slots : Layout u D ↪ K) (labels : Label u D → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (fun _ : K => Bool) Λ (MachineHorner.State σ))
    (atLabels : ∀ l, program (labels l) = statement slots labels exit l)
    (base : K → List Bool) (n m L : Nat)
    (hn : base (control slots .variableCount) = encodeWord n)
    (hm : base (control slots .clauseCount) = encodeWord m)
    (clean : Clean slots base) (hnL : n ≤ L) (hmL : m ≤ L)
    (ambient : σ) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some (labels (evalStart .left)), ((ambient, ()), register), base⟩
      (some ⟨exit, ((ambient, ()), none), resultTapes slots base n m⟩)
      ((timePolynomial u D).eval L) where
  steps := steps u D n m
  evals_in_steps := trace slots labels exit program atLabels base n m hn hm clean ambient register
  steps_le_m := steps_le_timePolynomial u D n m L hnL hmL

def formulaInTime (slots : Layout u D ↪ K) (labels : Label u D → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (fun _ : K => Bool) Λ (MachineHorner.State σ))
    (atLabels : ∀ l, program (labels l) = statement slots labels exit l)
    (base : K → List Bool) (F : Target.Formula)
    (hn : base (control slots .variableCount) = encodeWord F.variables)
    (hm : base (control slots .clauseCount) = encodeWord F.clauses.length)
    (clean : Clean slots base) (ambient : σ) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some (labels (evalStart .left)), ((ambient, ()), register), base⟩
      (some ⟨exit, ((ambient, ()), none), resultTapes slots base F.variables F.clauses.length⟩)
      ((timePolynomial u D).eval (formulaBits F).length) :=
  inPolynomialTime slots labels exit program atLabels base F.variables F.clauses.length
    (formulaBits F).length hn hm clean (SourceBounds.formulaBits_length_ge_variables F)
    (SourceBounds.formulaBits_length_ge_clauses F) ambient register

theorem result_frame (slots : Layout u D ↪ K) (base : K → List Bool) (n m : Nat) (k : K)
    (hl : k ≠ control slots .leftBlockSize) (ho : k ≠ control slots .occurrenceCount)
    (hd : k ≠ control slots .dummyIndex) (hb : k ≠ control slots .bitCount) :
    resultTapes slots base n m k = base k := by
  simp [resultTapes, hl, ho, hd, hb]

@[simp] theorem result_left (slots : Layout u D ↪ K) (base : K → List Bool) (n m : Nat) :
    resultTapes slots base n m (control slots .leftBlockSize) = encodeWord (leftValue u n) := by
  simp [resultTapes]
@[simp] theorem result_occurrences (slots : Layout u D ↪ K) (base : K → List Bool) (n m : Nat) :
    resultTapes slots base n m (control slots .occurrenceCount) = encodeWord (occurrenceValue u D m) := by
  simp [resultTapes]
@[simp] theorem result_dummy (slots : Layout u D ↪ K) (base : K → List Bool) (n m : Nat) :
    resultTapes slots base n m (control slots .dummyIndex) = encodeWord (dummyValue u n m) := by
  simp [resultTapes]
@[simp] theorem result_bits (slots : Layout u D ↪ K) (base : K → List Bool) (n m : Nat) :
    resultTapes slots base n m (control slots .bitCount) = encodeWord (bitValue u n m) := by
  simp [resultTapes]
@[simp] theorem result_variableCount (slots : Layout u D ↪ K) (base : K → List Bool) (n m : Nat) :
    resultTapes slots base n m (control slots .variableCount) = base (control slots .variableCount) := by
  simp [resultTapes]
@[simp] theorem result_clauseCount (slots : Layout u D ↪ K) (base : K → List Bool) (n m : Nat) :
    resultTapes slots base n m (control slots .clauseCount) = base (control slots .clauseCount) := by
  simp [resultTapes]

theorem result_source_headers (slots : Layout u D ↪ K) (base : K → List Bool)
    (F : Target.Formula) (hm : 0 < F.clauses.length) :
    resultTapes slots base F.variables F.clauses.length (control slots .bitCount) =
      encodeWord (SourceOccurrences.nBits F u) ∧
    resultTapes slots base F.variables F.clauses.length (control slots .occurrenceCount) =
      encodeWord (SourceOccurrences.sourceList F u D).length := by
  simp only [result_bits, result_occurrences, bitValue_eq_nBits,
    occurrenceValue_eq_length F u D hm, and_self]

def program (slots : Layout u D ↪ K) : Label u D →
    TM2.Stmt (fun _ : K => Bool) (Label u D) (MachineHorner.State σ) :=
  statement slots id none

def machine (u D : Nat) : FinTM2 where
  K := Layout u D
  k₀ := .inl .variableCount
  k₁ := .inl .bitCount
  Γ _ := Bool
  Λ := Label u D
  main := evalStart .left
  σ := MachineHorner.State Unit
  initialState := (((), ()), none)
  m := program (Function.Embedding.refl _)

def afterInitializationInTime {Extra : Type} [DecidableEq Extra]
    (slots : Layout u D ↪ SourceLoopInit.Tape u Extra)
    (hnslot : control slots .variableCount = SourceLoopInit.variableHeader)
    (hmslot : control slots .clauseCount = SourceLoopInit.clauseHeader)
    (labels : Label u D → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (fun _ : SourceLoopInit.Tape u Extra => Bool) Λ
      (MachineHorner.State σ))
    (atLabels : ∀ l, program (labels l) = statement slots labels exit l)
    (base : SourceLoopInit.Tape u Extra → List Bool) (F : Target.Formula)
    (hn : base SourceLoopInit.variableHeader = [])
    (hm : base SourceLoopInit.clauseHeader = [])
    (clean : Clean slots (SourceLoopInit.outputTapes F base))
    (ambient : σ) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some (labels (evalStart .left)), ((ambient, ()), register),
        SourceLoopInit.outputTapes F base⟩
      (some ⟨exit, ((ambient, ()), none),
        resultTapes slots (SourceLoopInit.outputTapes F base) F.variables F.clauses.length⟩)
      ((timePolynomial u D).eval (formulaBits F).length) :=
  formulaInTime slots labels exit program atLabels (SourceLoopInit.outputTapes F base) F
    (by rw [hnslot, SourceLoopInit.output_variableHeader, hn, List.append_nil])
    (by rw [hmslot, SourceLoopInit.output_clauseHeader, hm, List.append_nil])
    clean ambient register

end BinPackingGames.Foundations.Hastad.SourceHeaderCounts

namespace BinPackingGames.Foundations.Hastad.SourceGap

open Target SourceContexts SourceOccurrences SourceGame
open BinPackingGames.Integration.SourceParameters
open BinPackingGames.Reduction

def ClauseGap (F : Formula) (η : ℚ) : Prop :=
  F.clauses ≠ [] ∧ ∀ assignment : Fin F.variables → Bool,
    (η : ℝ) * F.clauses.length ≤
      (PCP.failureCount F assignment (PCP.allIndices F) : ℝ)

theorem third_pos {η : ℚ} (hη : 0 < η) : 0 < η / 3 := by positivity

theorem third_le_one {η : ℚ} (hη : η ≤ 1) : η / 3 ≤ 1 := by linarith

def repetitionCount (η ξ : ℚ) (hη : 0 < η) (hη1 : η ≤ 1) (hξ : 0 < ξ)
    (D : ℕ) (hD : 0 < D) : ℕ :=
  repetitionLength (η / 3) ξ (third_pos hη) (third_le_one hη1) hξ D hD

theorem repetitionCount_pos (η ξ : ℚ) (hη : 0 < η) (hη1 : η ≤ 1)
    (hξ : 0 < ξ) (D : ℕ) (hD : 0 < D) :
    0 < repetitionCount η ξ hη hη1 hξ D hD :=
  (repetitionLength_spec (η / 3) ξ (third_pos hη) (third_le_one hη1) hξ D hD).1

theorem repeated_game_bound (F : Formula) (η ξ : ℚ)
    (hη : 0 < η) (hη1 : η ≤ 1) (hξ : 0 < ξ) (D : ℕ) (hD : 0 < D)
    (hgap : ClauseGap F η) :
    ((baseGame F hgap.1).repetition
      (repetitionCount η ξ hη hη1 hξ D hD)).value ≤
        4 * (D : ℝ)⁻¹ * (ξ : ℝ)^2 := by
  have hb : (baseGame F hgap.1).value ≤ 1 - ((η / 3 : ℚ) : ℝ) := by
    simpa only [Rat.cast_div, Rat.cast_ofNat] using
      base_value_le_of_clause_gap F hgap.1 (η : ℝ) hgap.2
  have hc : Fintype.card Bool * Fintype.card PCP.ClauseAnswer = 16 := by
    simp
  exact (game_repetition_rate (baseGame F hgap.1) (third_pos hη)
    (third_le_one hη1) hc hb _).trans
      (repetitionLength_real_bound (η / 3) ξ (third_pos hη)
        (third_le_one hη1) hξ D hD)

def source (η ξ : ℚ) (hη : 0 < η) (hη1 : η ≤ 1) (hξ : 0 < ξ)
    (D : ℕ) (hD : 0 < D) (F : Formula) : SourceEncoding.Input :=
  sourceInput F (repetitionCount η ξ hη hη1 hξ D hD) D hD

theorem source_complete (η ξ : ℚ) (hη : 0 < η) (hη1 : η ≤ 1) (hξ : 0 < ξ)
    (D : ℕ) (hD : 0 < D) (hnoise : (D : ℚ)⁻¹ ≤ ξ)
    (F : Formula) (hF : F.Satisfiable) :
    ∃ bits : Fin (source η ξ hη hη1 hξ D hD F).variables → Bool,
      1 - ξ ≤ (((source η ξ hη hη1 hξ D hD F).equations.countP
        (fun e => CloneGap.satisfied e bits) : ℚ) /
          (source η ξ hη hη1 hξ D hD F).equations.length) := by
  obtain ⟨assignment, hs⟩ := hF
  refine ⟨SourceHonest.honestBits F _ assignment, ?_⟩
  have h := SourceHonest.sourceList_honest_acceptance_rat F
    (repetitionCount η ξ hη hη1 hξ D hD) D hD assignment hs
  exact (sub_le_sub_left hnoise 1).trans h

theorem source_sound (η ξ : ℚ) (hη : 0 < η) (hη1 : η ≤ 1) (hξ : 0 < ξ)
    (D : ℕ) (hD : 0 < D) (hDtwo : 2 ≤ D)
    (F : Formula) (hgap : ClauseGap F η)
    (bits : Fin (source η ξ hη hη1 hξ D hD F).variables → Bool) :
    ((source η ξ hη hη1 hξ D hD F).equations.countP
      (fun e => CloneGap.satisfied e bits) : ℚ) /
        (source η ξ hη hη1 hξ D hD F).equations.length ≤ (1 + ξ) / 2 :=
  SourceSoundness.sourceList_sound_rat F hgap.1
    (repetitionCount η ξ hη hη1 hξ D hD) D hD hDtwo ξ hξ.le
      (repeated_game_bound F η ξ hη hη1 hξ D hD hgap) bits

theorem source_size (η ξ : ℚ) (hη : 0 < η) (hη1 : η ≤ 1) (hξ : 0 < ξ)
    (D : ℕ) (hD : 0 < D) :
    ∃ p : Polynomial ℕ, ∀ F : Formula,
      (SourceEncoding.inputBits (source η ξ hη hη1 hξ D hD F)).length ≤
        p.eval (Complexity.formulaBits F).length :=
  SourceBounds.fixed_parameters_source_size (repetitionCount η ξ hη hη1 hξ D hD) D hD

end BinPackingGames.Foundations.Hastad.SourceGap

namespace BinPackingGames.Foundations.Hastad.SourceNonempty

open Target PCP

theorem real_clause_gap_of_count (F : Formula) (a b : ℕ) (hb : 0 < b)
    (hgap : ∀ assignment : Fin F.variables → Bool,
      a * F.clauses.length ≤ b * NameCompaction.failedCount F assignment)
    (assignment : Fin F.variables → Bool) :
    (a : ℝ) / (b : ℝ) * F.clauses.length ≤
      (failureCount F assignment (allIndices F) : ℝ) := by
  rw [NameCompaction.verifier_failureCount]
  have hreal : (a : ℝ) * F.clauses.length ≤
      (b : ℝ) * (NameCompaction.failedCount F assignment : ℝ) := by
    exact_mod_cast hgap assignment
  rw [div_mul_eq_mul_div]
  apply (div_le_iff₀ (Nat.cast_pos.mpr hb)).mpr
  simpa only [mul_comm] using hreal

theorem clauseGap_of_count (F : Formula) (a b : ℕ) (hne : F.clauses ≠ [])
    (hb : 0 < b)
    (hgap : ∀ assignment : Fin F.variables → Bool,
      a * F.clauses.length ≤ b * NameCompaction.failedCount F assignment) :
    SourceGap.ClauseGap F ((a : ℚ) / (b : ℚ)) := by
  refine ⟨hne, fun assignment => ?_⟩
  simpa only [Rat.cast_div, Rat.cast_natCast] using
    real_clause_gap_of_count F a b hb hgap assignment

variable {V E : Type*} {n m : ℕ}

theorem cnf_nonempty_of_nonempty_darts [Fintype E] [Nonempty E]
    (G : ConstraintGraph V E FinalBooleanVerifier.Label)
    (vertices : V ≃ Fin n) (edges : E ≃ Fin m) :
    (FinalBooleanVerifier.cnf G vertices edges).clauses ≠ [] := by
  have hcard : Fintype.card E = m := by
    simpa only [Fintype.card_fin] using Fintype.card_congr edges
  have hm : 0 < m := hcard ▸ Fintype.card_pos
  exact FinalBooleanVerifier.cnf_nonempty G vertices edges hm

theorem cnf_clauseGap [Fintype E] [Nonempty E]
    (G : ConstraintGraph V E FinalBooleanVerifier.Label)
    (vertices : V ≃ Fin n) (edges : E ≃ Fin m) (a b : ℕ) (hb : 0 < b)
    (hgap : ∀ labeling : V → FinalBooleanVerifier.Label,
      a * Fintype.card E ≤ b * G.rejectionCount labeling) :
    SourceGap.ClauseGap (FinalBooleanVerifier.cnf G vertices edges)
      ((a : ℚ) / ((b : ℚ) * 40960)) := by
  have h := clauseGap_of_count (FinalBooleanVerifier.cnf G vertices edges) a (b * 40960)
    (cnf_nonempty_of_nonempty_darts G vertices edges)
    (Nat.mul_pos hb (by decide)) (FinalBooleanVerifier.cnf_gap G vertices edges a b hgap)
  simpa only [Nat.cast_mul, Nat.cast_ofNat] using h

theorem cnf_clauseGap_unit [Fintype E] [Nonempty E]
    (G : ConstraintGraph V E FinalBooleanVerifier.Label)
    (vertices : V ≃ Fin n) (edges : E ≃ Fin m) (walkLength : ℕ)
    (hwalk : 0 < walkLength)
    (hgap : ∀ labeling : V → FinalBooleanVerifier.Label,
      Fintype.card E ≤ walkLength * G.rejectionCount labeling) :
    SourceGap.ClauseGap (FinalBooleanVerifier.cnf G vertices edges)
      (1 / (40960 * (walkLength : ℚ))) := by
  have h := cnf_clauseGap G vertices edges 1 walkLength hwalk
    (by simpa only [one_mul] using hgap)
  simpa only [Nat.cast_one, mul_comm] using h

end BinPackingGames.Foundations.Hastad.SourceNonempty

end

end OAI