lean/OAI/Computability/UniqueGames/Machines/MachineBinaryRenameMachine.lean

import Mathlib.Data.List.Basic
import OAI.Computability.UniqueGames.Machines.GraphCounterStartLemmas
import OAI.Computability.UniqueGames.Machines.MachineBinaryHeaderMachine
import OAI.Computability.UniqueGames.Machines.MachineBinaryRenameLoop
import OAI.Computability.UniqueGames.Machines.MachineBinaryRenameWords
import OAI.Computability.UniqueGames.Machines.MachineFiniteAlphabet

namespace OAI

/-!
The finite Boolean-stack driver for first-occurrence renaming. Its permanent
input is copied to a cursor, each literal is translated by the checked search
loop, and the physical output is restored, prefixed with its count headers,
and cleaned up. Every tape uses Bool and every control component is finite.
-/

namespace UniqueGamesTheorem.BinaryRenameMachine

open Turing UniqueGamesTheorem.Foundations
open Complexity Complexity.MachineComposition
open UniqueGamesTheorem.Reduction.MachineTransfer

abbrev Tape := Fin 12
abbrev Alphabet (_ : Tape) := Bool
abbrev State := BinaryRenameLoop.State Unit

inductive Label
  | copyOut | copyBack
  | loop (label : BinaryRenameLoop.Label)
  | restore | clauseStart | clauseLoop | variableStart | variableLoop
  | cleanup | reject
  deriving DecidableEq, Fintype

def loopTapes : Fin 11 → Tape := Fin.castSucc

theorem loopTapes_injective : Function.Injective loopTapes := by
  intro i j h
  apply Fin.ext
  exact congrArg (fun k : Tape => k.val) h

def program : Label → TM2.Stmt Alphabet Label State
  | .copyOut => loopAt 1 6 id false .copyOut (some .copyBack)
  | .copyBack => MachineCopy.forkLoop 6 1 0 false .copyBack (some (.loop .entry))
  | .loop label => BinaryRenameLoop.instruction loopTapes Label.loop
      (some .restore) (some .reject) label
  | .restore => loopAt 8 11 id false .restore (some .clauseStart)
  | .clauseStart => BinaryHeaderMachine.delimiter 11 .clauseLoop
  | .clauseLoop => loopAt 10 11 id false .clauseLoop (some .variableStart)
  | .variableStart => BinaryHeaderMachine.delimiter 11 .variableLoop
  | .variableLoop => loopAt 9 11 id false .variableLoop (some .cleanup)
  | .cleanup => MachineDrain.drain 1 .cleanup none
  | .reject => .halt

abbrev machine : FinTM2 where
  K := Tape
  k₀ := 1
  k₁ := 11
  Γ := Alphabet
  Λ := Label
  main := .copyOut
  σ := State
  initialState := BinaryRenameLoop.clean () 0
  m := program

def tapes (permanent cursor reversed variableCounter clauseCounter output : List Bool) :
    Tape → List Bool := fun k =>
  if k = 0 then cursor
  else if k = 1 then permanent
  else if k = 8 then reversed
  else if k = 9 then variableCounter
  else if k = 10 then clauseCounter
  else if k = 11 then output
  else []

def cfg (label : Option Label)
    (permanent cursor reversed variableCounter clauseCounter output : List Bool) : machine.Cfg :=
  ⟨label, BinaryRenameLoop.clean () 0,
    tapes permanent cursor reversed variableCounter clauseCounter output⟩

theorem initList_eq (input : List Bool) :
    initList machine input = cfg (some .copyOut) input [] [] [] [] [] := by
  unfold initList cfg
  congr 1
  funext k
  fin_cases k <;> simp [tapes, machine]

theorem haltList_eq (output : List Bool) :
    haltList machine output = cfg none [] [] [] [] [] output := by
  unfold haltList cfg
  congr 1
  funext k
  fin_cases k <;> simp [tapes, machine]

theorem copyTrace (input : List Bool) :
    (advance machine.step)^[2 * (input.length + 1)]
      (some (cfg (some .copyOut) input [] [] [] [] [])) =
      some (cfg (some (.loop .entry)) input input [] [] [] []) := by
  have updated : Function.update (tapes input [] [] [] [] []) 0 input =
      tapes input input [] [] [] [] := by
    funext k
    fin_cases k <;> simp [tapes]
  have h := MachineCopy.copyTrace (1 : Tape) 0 6 (by decide) (by decide) (by decide)
    false .copyOut .copyBack (some (.loop .entry)) program rfl rfl
    (tapes input [] [] [] [] []) (by simp [tapes])
    (BinaryRenameLoop.clean () 0).1 none
  have first : tapes input [] [] [] [] [] 1 = input := rfl
  have second : tapes input [] [] [] [] [] 0 = [] := rfl
  rw [first, second, List.append_nil, updated] at h
  exact h

theorem restoreTrace (input body variableCounter clauseCounter : List Bool) :
    (advance machine.step)^[body.length + 1]
      (some (cfg (some .restore) input [] body.reverse variableCounter clauseCounter [])) =
      some (cfg (some .clauseStart) input [] [] variableCounter clauseCounter body) := by
  change (nextAt (11 : Tape) program)^[body.length + 1]
    (some (cfg (some .restore) input [] body.reverse variableCounter clauseCounter [])) = _
  have updated : tapesAt (8 : Tape) 11
      (tapes input [] body.reverse variableCounter clauseCounter []) [] body =
      tapes input [] [] variableCounter clauseCounter body := by
    funext k
    fin_cases k <;> simp [tapesAt, tapes]
  have h := transferAt_fromTapes (Γ := Alphabet) (8 : Tape) 11 (by decide) id false
    .restore (some .clauseStart) program rfl
    (tapes input [] body.reverse variableCounter clauseCounter [])
    (BinaryRenameLoop.clean () 0).1 none
  have source : tapes input [] body.reverse variableCounter clauseCounter [] 8 = body.reverse := rfl
  have output : tapes input [] body.reverse variableCounter clauseCounter [] 11 = [] := rfl
  rw [source, output, List.length_reverse, List.reverse_reverse, List.map_id_fun,
    List.append_nil, id_eq, updated] at h
  exact h

theorem headerTrace (input body : List Bool) (n m : Nat) :
    (advance machine.step)^[n + m + 4]
      (some (cfg (some .clauseStart) input [] []
        (List.replicate n true) (List.replicate m true) body)) =
      some (cfg (some .cleanup) input [] [] [] []
        (encodeWord n ++ encodeWord m ++ body)) := by
  change (advance (TM2.step program))^[n + m + 4]
    (some ⟨some .clauseStart, ((BinaryRenameLoop.clean () 0).1, none),
      tapes input [] [] (List.replicate n true) (List.replicate m true) body⟩) =
    some ⟨some .cleanup, ((BinaryRenameLoop.clean () 0).1, none),
      tapes input [] [] [] [] (encodeWord n ++ encodeWord m ++ body)⟩
  have updated (variableWord clauses output : List Bool) :
      BinaryHeaderMachine.tapes (9 : Tape) 10 11 (tapes input [] [] [] [] [])
        variableWord clauses output = tapes input [] [] variableWord clauses output := by
    funext k
    fin_cases k <;> simp [BinaryHeaderMachine.tapes, tapes]
  have h := BinaryHeaderMachine.headerTrace (9 : Tape) 10 11
    (by decide) (by decide) (by decide)
    .clauseStart .clauseLoop .variableStart .variableLoop (some .cleanup) program
    rfl rfl rfl rfl (tapes input [] [] [] [] [])
    (BinaryRenameLoop.clean () 0).1 n m body none
  simp only [updated] at h
  exact h

theorem cleanupTrace (input output : List Bool) :
    (advance machine.step)^[input.length + 1]
      (some (cfg (some .cleanup) input [] [] [] [] output)) =
      some (cfg none [] [] [] [] [] output) := by
  change (advance (TM2.step program))^[input.length + 1]
    (some ⟨some .cleanup, ((BinaryRenameLoop.clean () 0).1, none),
      tapes input [] [] [] [] output⟩) =
    some ⟨none, ((BinaryRenameLoop.clean () 0).1, none), tapes [] [] [] [] [] output⟩
  have updated (word : List Bool) :
      Function.update (tapes [] [] [] [] [] output) 1 word =
        tapes word [] [] [] [] output := by
    funext k
    fin_cases k <;> simp [tapes]
  have h := MachineDrain.drainTrace (1 : Tape) .cleanup none program rfl
    (tapes [] [] [] [] [] output) input (BinaryRenameLoop.clean () 0).1 none
  simp only [updated] at h
  exact h

theorem outputBody_eq (formula : BinaryFormula.Formula) :
    BinaryRenameLoop.outputBody (BinaryRenameWords.tokens formula.clauses)
      (BinaryRenameWords.tokens formula.clauses) = BinaryRenameWords.body formula := by
  change ((BinaryRenameWords.literals formula.clauses).map BinaryRenameWords.token).flatMap
    (fun t => BinaryLiteralMachine.outputWord (BinaryRenameWords.tokens formula.clauses)
      t.2 t.1) = (BinaryRenameWords.literals formula.clauses).flatMap
        (BinaryRenameWords.outputLiteral (BinaryFormula.sourceNames formula))
  rw [List.flatMap_map]
  apply List.flatMap_congr
  intro literal _
  simp only [BinaryRenameWords.token, BinaryLiteralMachine.outputWord,
    BinaryLiteralMachine.index, BinaryLiteralMachine.signWord,
    BinaryRenameWords.payload_index, BinaryRenameWords.outputLiteral]
  rfl

theorem loopTrace (formula : BinaryFormula.Formula) :
    (advance machine.step)^[BinaryRenameLoop.steps (BinaryRenameWords.tokens formula.clauses)
        (BinaryRenameWords.tokens formula.clauses)]
      (some (cfg (some (.loop .entry)) (BinaryTokenMachine.tokens formula.clauses)
        (BinaryTokenMachine.tokens formula.clauses) [] [] [] [])) =
      some (cfg (some .restore) (BinaryTokenMachine.tokens formula.clauses) []
        (BinaryRenameWords.body formula).reverse (List.replicate (3 * formula.clauses.length) true)
        (List.replicate formula.clauses.length true) []) := by
  change (advance (TM2.step program))^[BinaryRenameLoop.steps
      (BinaryRenameWords.tokens formula.clauses) (BinaryRenameWords.tokens formula.clauses)]
    (some ⟨some (.loop .entry), BinaryRenameLoop.clean () 0,
      tapes (BinaryTokenMachine.tokens formula.clauses) (BinaryTokenMachine.tokens formula.clauses)
        [] [] [] []⟩) =
    some ⟨some .restore, BinaryRenameLoop.clean () 0,
      tapes (BinaryTokenMachine.tokens formula.clauses) [] (BinaryRenameWords.body formula).reverse
        (List.replicate (3 * formula.clauses.length) true)
        (List.replicate formula.clauses.length true) []⟩
  let input := BinaryTokenMachine.tokens formula.clauses
  have updated (cursor output variableWord clauseWord : List Bool) :
      BinaryRenameLoop.loopTapes loopTapes (tapes input [] [] [] [] [])
        cursor output variableWord clauseWord =
      tapes input cursor output variableWord clauseWord [] := by
    funext k
    fin_cases k <;> simp [BinaryRenameLoop.loopTapes, loopTapes, tapes]
  have scratch (i : Fin 11) (lo : 2 ≤ i.val) (hi : i.val ≤ 7) :
      tapes input [] [] [] [] [] (loopTapes i) = [] := by
    have n0 : loopTapes i ≠ 0 := by intro h; have := congrArg Fin.val h; simp [loopTapes] at this; omega
    have n1 : loopTapes i ≠ 1 := by intro h; have := congrArg Fin.val h; simp [loopTapes] at this; omega
    have n8 : loopTapes i ≠ 8 := by intro h; have := congrArg Fin.val h; simp [loopTapes] at this; omega
    have n9 : loopTapes i ≠ 9 := by intro h; have := congrArg Fin.val h; simp [loopTapes] at this; omega
    have n10 : loopTapes i ≠ 10 := by intro h; have := congrArg Fin.val h; simp [loopTapes] at this; omega
    have n11 : loopTapes i ≠ 11 := by intro h; have := congrArg Fin.val h; simp [loopTapes] at this; omega
    simp [tapes, n0, n1, n8, n9, n10, n11]
  have run := BinaryRenameLoop.triplesTrace loopTapes loopTapes_injective Label.loop
    (some .restore) (some .reject) program (fun _ => rfl)
    (tapes input [] [] [] [] []) ()
    (BinaryRenameWords.tokens formula.clauses) (BinaryRenameWords.tokens formula.clauses)
    formula.clauses.length (BinaryRenameWords.tokens_length formula.clauses) [] [] []
    (BinaryRenameWords.tokens_canonical formula.clauses) (fun _ h => h)
    (by change input = _; rw [BinaryRenameWords.tokens_stream]) scratch
  simp only [updated, BinaryRenameWords.tokens_stream, outputBody_eq, List.append_nil, input] at run
  exact run

private theorem joinTrace {X : Type*} {f : X → X} {a b c : X} {n m : Nat}
    (first : f^[n] a = b) (second : f^[m] b = c) : f^[n + m] a = c := by
  rw [Nat.add_comm, Function.iterate_add_apply, first, second]

def steps (formula : BinaryFormula.Formula) : Nat :=
  2 * ((BinaryTokenMachine.tokens formula.clauses).length + 1) +
    BinaryRenameLoop.steps (BinaryRenameWords.tokens formula.clauses)
      (BinaryRenameWords.tokens formula.clauses) +
    ((BinaryRenameWords.body formula).length + 1) + (4 * formula.clauses.length + 4) +
    ((BinaryTokenMachine.tokens formula.clauses).length + 1)

theorem renameTrace (formula : BinaryFormula.Formula) :
    (advance machine.step)^[steps formula]
      (some (initList machine (BinaryTokenMachine.tokens formula.clauses))) =
      some (haltList machine (formulaBits (BinaryOccurrenceRename.renamed formula))) := by
  have first := copyTrace (BinaryTokenMachine.tokens formula.clauses)
  have second := loopTrace formula
  have third := restoreTrace (BinaryTokenMachine.tokens formula.clauses)
    (BinaryRenameWords.body formula) (List.replicate (3 * formula.clauses.length) true)
    (List.replicate formula.clauses.length true)
  have fourth := headerTrace (BinaryTokenMachine.tokens formula.clauses)
    (BinaryRenameWords.body formula) (3 * formula.clauses.length) formula.clauses.length
  have fifth := cleanupTrace (BinaryTokenMachine.tokens formula.clauses)
    (encodeWord (3 * formula.clauses.length) ++ encodeWord formula.clauses.length ++
      BinaryRenameWords.body formula)
  have full := joinTrace (joinTrace (joinTrace (joinTrace first second) third) fourth) fifth
  have count : 3 * formula.clauses.length + formula.clauses.length + 4 =
      4 * formula.clauses.length + 4 := by omega
  simpa only [steps, initList_eq, haltList_eq, BinaryRenameWords.encoded_renamed, count] using full

theorem clause_count_le_stream (formula : BinaryFormula.Formula) :
    formula.clauses.length ≤ (BinaryTokenMachine.tokens formula.clauses).length := by
  rw [BinaryTokenMachine.tokens_length]
  omega

theorem body_length_le (formula : BinaryFormula.Formula) :
    (BinaryRenameWords.body formula).length ≤
      9 * (BinaryTokenMachine.tokens formula.clauses).length ^ 2 +
        10 * (BinaryTokenMachine.tokens formula.clauses).length + 2 := by
  have initial := BinaryOccurrenceRename.renamed_encoding_bound formula
  rw [BinaryRenameWords.encoded_renamed] at initial
  simp only [List.length_append] at initial
  have count := clause_count_le_stream formula
  have square := Nat.mul_self_le_mul_self count
  nlinarith

noncomputable def timePolynomial : Polynomial Nat :=
  Polynomial.C 12 * Polynomial.X ^ 3 + Polynomial.C 35 * Polynomial.X ^ 2 +
    Polynomial.C 28 * Polynomial.X + Polynomial.C 11

theorem steps_le (formula : BinaryFormula.Formula) :
    steps formula ≤ timePolynomial.eval (BinaryTokenMachine.tokens formula.clauses).length := by
  have loopBound := BinaryRenameLoop.steps_le
    (BinaryRenameWords.tokens formula.clauses) (BinaryRenameWords.tokens formula.clauses)
    (fun _ h => h)
  rw [BinaryRenameWords.tokens_stream, BinaryRenameWords.tokens_length] at loopBound
  have literalCount : 3 * formula.clauses.length ≤
      (BinaryTokenMachine.tokens formula.clauses).length := by
    rw [BinaryTokenMachine.tokens_length]
    omega
  have product := Nat.mul_le_mul_left
    (BinaryRenameLoop.perLiteralBound (BinaryTokenMachine.tokens formula.clauses).length)
    literalCount
  have output := body_length_le formula
  have count := clause_count_le_stream formula
  simp only [timePolynomial, Polynomial.eval_add, Polynomial.eval_mul, Polynomial.eval_pow,
    Polynomial.eval_C, Polynomial.eval_X]
  unfold steps BinaryRenameLoop.perLiteralBound at *
  nlinarith

def outputsInTime (formula : BinaryFormula.Formula) :
    TM2OutputsInTime machine (BinaryTokenMachine.tokens formula.clauses)
      (some (formulaBits (BinaryOccurrenceRename.renamed formula)))
      (timePolynomial.eval (BinaryTokenMachine.tokens formula.clauses).length) where
  steps := steps formula
  evals_in_steps := renameTrace formula
  steps_le_m := steps_le formula

/-- The token-stream phase has a displayed finite machine and actual execution
bound; its polynomial argument is the token stream's physical bit length. -/
noncomputable def tokenComputation :
    TM2ComputableInPolyTime (fun formula : BinaryFormula.Formula =>
      BinaryTokenMachine.tokens formula.clauses) formulaBits BinaryOccurrenceRename.renamed where
  tm := machine
  inputAlphabet := Equiv.refl Bool
  outputAlphabet := Equiv.refl Bool
  time := timePolynomial
  outputsFun formula := by
    change TM2OutputsInTime machine ((BinaryTokenMachine.tokens formula.clauses).map id)
      (some ((formulaBits (BinaryOccurrenceRename.renamed formula)).map id))
      (timePolynomial.eval (BinaryTokenMachine.tokens formula.clauses).length)
    simpa only [List.map_id_fun, id_eq] using outputsInTime formula

/-- Repackage the checked marker-removal machine without changing any machine
or word: the intermediate semantic value is the same formula. -/
noncomputable def preprocessing :
    TM2ComputableInPolyTime BinaryEncoding.formulaBits
      (fun formula : BinaryFormula.Formula => BinaryTokenMachine.tokens formula.clauses)
      (id : BinaryFormula.Formula → BinaryFormula.Formula) where
  tm := BinaryTokenMachine.computableInPolyTime.tm
  inputAlphabet := BinaryTokenMachine.computableInPolyTime.inputAlphabet
  outputAlphabet := BinaryTokenMachine.computableInPolyTime.outputAlphabet
  time := BinaryTokenMachine.computableInPolyTime.time
  outputsFun := BinaryTokenMachine.computableInPolyTime.outputsFun

/-- Actual finite-alphabet polynomial-time sparse binary to dense unary
renaming. Sparse name magnitudes are never expanded in unary. -/
noncomputable def computableInPolyTime :
    TM2ComputableInPolyTime BinaryEncoding.formulaBits formulaBits
      BinaryOccurrenceRename.renamed :=
  MachineSequential.composeBits preprocessing tokenComputation

theorem machine_finiteAlphabet (k : machine.K) : Finite (machine.Γ k) := by
  change Finite Bool
  infer_instance

theorem computation_finiteAlphabet : MachineFiniteAlphabet.FiniteAlphabet computableInPolyTime.tm :=
  MachineFiniteAlphabet.composeBits preprocessing tokenComputation
    BinaryTokenMachine.machine_finiteAlphabet machine_finiteAlphabet

end UniqueGamesTheorem.BinaryRenameMachine

end OAI