lean/OAI/Computability/DirectedFeedback/Machines/MachineBinaryTotalInputMachine.lean

import Mathlib
import OAI.Computability.DirectedFeedback.RankGraph.MachineClone100Finish

namespace OAI

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namespace DFVSGames.BinaryTotalInputMachine

open Turing
open DFVSGames.Foundations.Complexity
open MachineComposition
open DFVSGames.Reduction.MachineTransfer
open DFVSGames.Reduction.MachineSubstitution
open BinaryValidatorMachine (Mode nextMode run)

inductive Label
  | scan
  | restore
  | drain
  | rejectOutput
  deriving DecidableEq

instance : Fintype Label := derive_fintype% Label

abbrev Alphabet (_ : Fin 3) := Bool
abbrev State := Mode × Option Bool

def rejectedWord : List Bool := BinaryEncoding.formulaBits BinaryLanguage.rejectBinary

def instruction : Label → TM2.Stmt Alphabet Label State
  | .scan =>
      .pop 0 (fun state head => (state.1, head))
        (.branch (fun state => state.2.isSome)
          (.push 2 (fun state => state.2.getD false)
            (.load (fun state => (nextMode state.1 (state.2.getD false), none))
              (.goto fun _ => .scan)))
          (.branch (fun state => decide (state.1 = .done))
            (.load (fun _ => (.clause, none)) (.goto fun _ => .restore))
            (.load (fun _ => (.clause, none)) (.goto fun _ => .drain))))
  | .restore => loopAt 2 1 id false .restore none
  | .drain => MachineDrain.drain 2 .drain (some .rejectOutput)
  | .rejectOutput => pushWord 1 rejectedWord.reverse .halt

abbrev machine : FinTM2 where
  K := Fin 3
  k₀ := 0
  k₁ := 1
  Γ := Alphabet
  Λ := Label
  main := .scan
  σ := State
  initialState := (.clause, none)
  m := instruction

def tapes (input output saved : List Bool) : Fin 3 → List Bool
  | 0 => input
  | 1 => output
  | _ => saved

def cfg (label : Option Label) (input output saved : List Bool) (mode : Mode := .clause)
    (register : Option Bool := none) : machine.Cfg :=
  ⟨label, (mode, register), tapes input output saved⟩

@[simp] theorem tapes_input (input output saved : List Bool) :
    tapes input output saved 0 = input := rfl

@[simp] theorem tapes_output (input output saved : List Bool) :
    tapes input output saved 1 = output := rfl

@[simp] theorem tapes_saved (input output saved : List Bool) :
    tapes input output saved 2 = saved := rfl

private theorem update_input_inline_MachineBinaryTotalInputMachine (input output saved replacement : List Bool) :
    Function.update (tapes input output saved) 0 replacement =
      tapes replacement output saved := by
  funext k
  fin_cases k <;> rfl

private theorem update_output_inline_MachineBinaryTotalInputMachine (input output saved replacement : List Bool) :
    Function.update (tapes input output saved) 1 replacement =
      tapes input replacement saved := by
  funext k
  fin_cases k <;> rfl

private theorem update_saved_inline_MachineBinaryTotalInputMachine (input output saved replacement : List Bool) :
    Function.update (tapes input output saved) 2 replacement =
      tapes input output replacement := by
  funext k
  fin_cases k <;> rfl

def afterScan (mode : Mode) : Label := if mode = .done then .restore else .drain

theorem step_scan_empty (output saved : List Bool) (mode : Mode) (register : Option Bool) :
    machine.step (cfg (some .scan) [] output saved mode register) =
      some (cfg (some (afterScan mode)) [] output saved) := by
  change some (TM2.stepAux (instruction .scan) _ _) = _
  by_cases accepted : mode = .done <;>
    simp [instruction, cfg, TM2.stepAux, update_input_inline_MachineBinaryTotalInputMachine, afterScan, accepted]
  all_goals rfl

theorem step_scan_cons (bit : Bool) (input output saved : List Bool)
    (mode : Mode) (register : Option Bool) :
    machine.step (cfg (some .scan) (bit :: input) output saved mode register) =
      some (cfg (some .scan) input output (bit :: saved) (nextMode mode bit)) := by
  change some (TM2.stepAux (instruction .scan) _ _) = _
  simp [instruction, cfg, TM2.stepAux, update_input_inline_MachineBinaryTotalInputMachine, update_saved_inline_MachineBinaryTotalInputMachine]
  rfl

theorem scanTrace (input output saved : List Bool) (mode : Mode) (register : Option Bool) :
    (advance machine.step)^[input.length + 1]
      (some (cfg (some .scan) input output saved mode register)) =
      some (cfg (some (afterScan (run mode input))) [] output (input.reverse ++ saved)) := by
  induction input generalizing saved mode register with
  | nil =>
      simpa only [List.length_nil, Nat.zero_add, Function.iterate_one,
        advance_some, run, List.reverse_nil, List.nil_append] using
          step_scan_empty output saved mode register
  | cons bit input ih =>
      rw [List.length_cons, Function.iterate_succ_apply]
      simp only [advance_some]
      rw [step_scan_cons]
      simpa only [run, List.reverse_cons, List.append_assoc, List.singleton_append] using
        ih (bit :: saved) (nextMode mode bit) none

theorem restoreTrace (saved output : List Bool) (register : Option Bool) :
    (advance machine.step)^[saved.length + 1]
      (some (cfg (some .restore) [] output saved .clause register)) =
      some (cfg none [] (saved.reverse ++ output) []) := by
  change (nextAt (1 : Fin 3) instruction)^[saved.length + 1]
    (some ⟨some .restore, (Mode.clause, register), tapes [] output saved⟩) =
    some ⟨none, (Mode.clause, none), tapes [] (saved.reverse ++ output) []⟩
  have trace := transferAt_steps (2 : Fin 3) 1 (by decide) id false .restore none
    instruction rfl (tapes [] [] []) saved output Mode.clause register
  simpa only [tapesAt, update_saved_inline_MachineBinaryTotalInputMachine, update_output_inline_MachineBinaryTotalInputMachine,
    List.map_id_fun, id_eq] using trace

theorem drainTrace (saved output : List Bool) (register : Option Bool) :
    (advance machine.step)^[saved.length + 1]
      (some (cfg (some .drain) [] output saved .clause register)) =
      some (cfg (some .rejectOutput) [] output []) := by
  change (advance (TM2.step instruction))^[saved.length + 1]
    (some ⟨some .drain, (Mode.clause, register), tapes [] output saved⟩) =
    some ⟨some .rejectOutput, (Mode.clause, none), tapes [] output []⟩
  have trace := MachineDrain.drainTrace (2 : Fin 3) .drain (some .rejectOutput)
    instruction rfl (tapes [] output []) saved Mode.clause register
  simpa only [update_saved_inline_MachineBinaryTotalInputMachine] using trace

theorem step_rejectOutput (output : List Bool) :
    machine.step (cfg (some .rejectOutput) [] output []) =
      some (cfg none [] (rejectedWord ++ output) []) := by
  change some (TM2.stepAux
    (pushWord (Γ := Alphabet) (Λ := Label) (σ := State)
      (1 : Fin 3) rejectedWord.reverse .halt) _ _) = _
  rw [stepAux_pushWord]
  simp only [cfg, List.reverse_reverse, tapes_output, update_output_inline_MachineBinaryTotalInputMachine, TM2.stepAux]
  rfl

private theorem joinTrace_inline_MachineBinaryTotalInputMachine {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]

theorem acceptedTrace (input : List Bool) (accepted : run .clause input = .done) :
    (advance machine.step)^[2 * input.length + 2]
      (some (cfg (some .scan) input [] [])) = some (cfg none [] input []) := by
  have first := scanTrace input [] [] .clause none
  simp only [afterScan, accepted, ↓reduceIte, List.append_nil] at first
  have second := restoreTrace input.reverse [] none
  simp only [List.length_reverse, List.reverse_reverse, List.append_nil] at second
  have time : 2 * input.length + 2 = (input.length + 1) + (input.length + 1) := by omega
  rw [time]
  exact joinTrace_inline_MachineBinaryTotalInputMachine first second

theorem rejectedTrace (input : List Bool) (rejected : run .clause input ≠ .done) :
    (advance machine.step)^[2 * input.length + 3]
      (some (cfg (some .scan) input [] [])) = some (cfg none [] rejectedWord []) := by
  have first := scanTrace input [] [] .clause none
  simp only [afterScan, rejected, ↓reduceIte, List.append_nil] at first
  have second := drainTrace input.reverse [] none
  simp only [List.length_reverse] at second
  have third : (advance machine.step)^[1]
      (some (cfg (some .rejectOutput) [] [] [])) = some (cfg none [] rejectedWord []) := by
    simpa only [Function.iterate_one, advance_some, List.append_nil] using
      step_rejectOutput []
  have time : 2 * input.length + 3 = (input.length + 1) + (input.length + 1) + 1 := by omega
  rw [time]
  exact joinTrace_inline_MachineBinaryTotalInputMachine (joinTrace_inline_MachineBinaryTotalInputMachine first second) third

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

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

def steps (input : List Bool) : Nat :=
  if run .clause input = .done then 2 * input.length + 2 else 2 * input.length + 3

theorem steps_le (input : List Bool) : steps input ≤ 2 * input.length + 3 := by
  unfold steps
  split <;> omega

theorem totalTrace (input : List Bool) :
    (advance machine.step)^[steps input] (some (initList machine input)) =
      some (haltList machine (BinaryEncoding.formulaBits (BinaryLanguage.totalParsed input))) := by
  rw [initList_eq, haltList_eq]
  cases parsed : BinaryEncoding.decodeFormula input with
  | none =>
      have rejected : run .clause input ≠ .done := by
        intro accepted
        obtain ⟨formula, encoded⟩ := (BinaryValidatorMachine.run_clause_iff input).mp accepted
        have roundtrip := BinaryEncoding.decodeFormula_encoded formula
        rw [← encoded, parsed] at roundtrip
        cases roundtrip
      simpa only [steps, rejected, ↓reduceIte, BinaryLanguage.totalParsed, parsed,
        Option.getD_none, rejectedWord] using rejectedTrace input rejected
  | some formula =>
      have encoded := BinaryParsing.decodeFormula_sound input formula parsed
      have accepted := (BinaryValidatorMachine.run_clause_iff input).mpr ⟨formula, encoded⟩
      simp only [steps, accepted, ↓reduceIte, BinaryLanguage.totalParsed, parsed,
        Option.getD_some]
      rw [← encoded]
      exact acceptedTrace input accepted

def outputsInTime (input : List Bool) :
    TM2OutputsInTime machine input
      (some (BinaryEncoding.formulaBits (BinaryLanguage.totalParsed input)))
      (2 * input.length + 3) where
  steps := steps input
  evals_in_steps := totalTrace input
  steps_le_m := steps_le input

noncomputable def computableInPolyTime :
    TM2ComputableInPolyTime (id : List Bool → List Bool) BinaryEncoding.formulaBits
      BinaryLanguage.totalParsed where
  tm := machine
  inputAlphabet := Equiv.refl Bool
  outputAlphabet := Equiv.refl Bool
  time := 2 * Polynomial.X + 3
  outputsFun input := by
    change TM2OutputsInTime machine (input.map id)
      (some ((BinaryEncoding.formulaBits (BinaryLanguage.totalParsed input)).map id))
      ((2 * Polynomial.X + 3 : Polynomial Nat).eval input.length)
    simpa only [List.map_id_fun, id_eq, Polynomial.eval_add, Polynomial.eval_mul,
      Polynomial.eval_X, Polynomial.eval_ofNat] using outputsInTime input

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

theorem computation_finiteAlphabet : MachineFiniteAlphabet.FiniteAlphabet computableInPolyTime.tm :=
  machine_finiteAlphabet

end DFVSGames.BinaryTotalInputMachine

end

section

namespace DFVSGames.BinaryNameCompare

open Turing
open DFVSGames.Foundations.Complexity
open MachineComposition

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

abbrev Alphabet (_ : K) := Bool
abbrev State (A : Type) := (A × Bool × Option Bool) × Option Bool

def clean (ambient : A) : State A := ((ambient, false, none), none)

def scanStateEquiv (A : Type) : BinaryNameMachine.State (A × Bool) ≃ State A where
  toFun s := ((s.1.1, s.1.2, s.2.1), s.2.2)
  invFun s := ((s.1.1, s.1.2.1), s.1.2.2, s.2)
  left_inv := by rintro ⟨⟨a, b⟩, c, d⟩; rfl
  right_inv := by rintro ⟨⟨a, b, c⟩, d⟩; rfl

def compareStateEquiv (A : Type) : MachineCompare.State A ≃ State A where
  toFun s := ((s.1, s.2.1, s.2.2.1), s.2.2.2)
  invFun s := (s.1.1, s.1.2.1, s.1.2.2, s.2)
  left_inv := by rintro ⟨a, b, c, d⟩; rfl
  right_inv := by rintro ⟨⟨a, b, c⟩, d⟩; rfl

inductive Label
  | scan | copyOut | copyBack | compare
  deriving DecidableEq

instance : Fintype Label := derive_fintype% Label

def instruction (tape : Fin 5 → K) (labels : Label → Λ)
    (equalExit differentExit malformedExit : Option Λ) :
    Label → TM2.Stmt (Alphabet (K := K)) Λ (State A)
  | .scan => MachineStateEquiv.statement (scanStateEquiv A)
      (BinaryNameMachine.scan (tape 0) (tape 2) (labels .scan)
        (some (labels .copyOut)) malformedExit)
  | .copyOut => DFVSGames.Reduction.MachineTransfer.loopAt (tape 1) (tape 4) id false
      (labels .copyOut) (some (labels .copyBack))
  | .copyBack => MachineCopy.forkLoop (tape 4) (tape 1) (tape 3) false
      (labels .copyBack) (some (labels .compare))
  | .compare => MachineStateEquiv.statement (compareStateEquiv A)
      (MachineCompare.loop (tape 2) (tape 3) (labels .compare) equalExit differentExit)

def steps (payloadLength keyLength : Nat) : Nat :=
  (payloadLength + 1) + 2 * (keyLength + 1) + (max payloadLength keyLength + 1)

theorem steps_le (payloadLength keyLength : Nat) :
    steps payloadLength keyLength ≤ 2 * payloadLength + 3 * keyLength + 4 := by
  unfold steps
  omega

private theorem joinTrace_inline_MachineBinaryNameCompare {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]

theorem compareTrace (tape : Fin 5 → K) (distinct : Function.Injective tape)
    (labels : Label → Λ) (equalExit differentExit malformedExit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State A))
    (atLabels : ∀ l, program (labels l) =
      instruction tape labels equalExit differentExit malformedExit l)
    (base : K → List Bool) (bits suffix key : List Bool)
    (canonical : BinaryNameMachine.canonical bits = true)
    (sourceWord : base (tape 0) = BinaryNameMachine.frame bits ++ suffix)
    (keyWord : base (tape 1) = key)
    (candidateEmpty : base (tape 2) = []) (copyEmpty : base (tape 3) = [])
    (scratchEmpty : base (tape 4) = []) (ambient : A) :
    (advance (TM2.step program))^[steps bits.length key.length]
      (some ⟨some (labels .scan), clean ambient, base⟩) =
      some ⟨if bits.reverse = key then equalExit else differentExit,
        clean ambient, Function.update base (tape 0) suffix⟩ := by
  have hd (i j : Fin 5) (hne : i ≠ j) : tape i ≠ tape j := fun h => hne (distinct h)
  let finalBase := Function.update base (tape 0) suffix
  let afterScan := Function.update finalBase (tape 2) bits.reverse
  let afterCopy := Function.update afterScan (tape 3) key
  have scanRun : (advance (TM2.step program))^[bits.length + 1]
      (some ⟨some (labels .scan), clean ambient, base⟩) =
      some ⟨some (labels .copyOut), clean ambient, afterScan⟩ := by
    let back := MachineStateEquiv.program (scanStateEquiv A).symm program
    have atScan : back (labels .scan) =
        BinaryNameMachine.scan (tape 0) (tape 2) (labels .scan)
          (some (labels .copyOut)) malformedExit := by
      change MachineStateEquiv.statement (scanStateEquiv A).symm
        (program (labels .scan)) = _
      rw [atLabels .scan]
      exact MachineStateEquiv.statement_symm_statement (scanStateEquiv A) _
    have backForward : MachineStateEquiv.program (scanStateEquiv A) back = program := by
      funext l
      change MachineStateEquiv.statement (scanStateEquiv A)
        (MachineStateEquiv.statement (scanStateEquiv A).symm (program l)) = _
      simpa only [Equiv.symm_symm] using
        MachineStateEquiv.statement_symm_statement (scanStateEquiv A).symm (program l)
    have initialTapes : DFVSGames.Reduction.MachineTransfer.tapesAt (tape 0) (tape 2)
        base (BinaryNameMachine.frame bits ++ suffix) [] = base := by
      rw [← sourceWord, ← candidateEmpty]
      exact DFVSGames.Reduction.MachineTransfer.tapesAt_self _ _ _
    have native := BinaryNameMachine.framedTrace (tape 0) (tape 2) (hd 0 2 (by decide))
      (labels .scan) (some (labels .copyOut)) malformedExit back atScan base (ambient, false)
      bits suffix [] none
    simp only [canonical, ↓reduceIte, initialTapes, List.append_nil] at native
    have transported := MachineStateEquiv.trace (scanStateEquiv A) back _ _ _ native
    rw [backForward] at transported
    simpa only [MachineStateEquiv.configuration, scanStateEquiv, Equiv.coe_fn_mk, BinaryNameMachine.clean,
      clean, afterScan, finalBase, DFVSGames.Reduction.MachineTransfer.tapesAt] using transported
  have afterScanKey : afterScan (tape 1) = key := by
    simpa [afterScan, finalBase, hd 1 2 (by decide), hd 1 0 (by decide)] using keyWord
  have afterScanCopy : afterScan (tape 3) = [] := by
    simpa [afterScan, finalBase, hd 3 2 (by decide), hd 3 0 (by decide)] using copyEmpty
  have afterScanScratch : afterScan (tape 4) = [] := by
    simpa [afterScan, finalBase, hd 4 2 (by decide), hd 4 0 (by decide)] using scratchEmpty
  have copyRun : (advance (TM2.step program))^[2 * (key.length + 1)]
      (some ⟨some (labels .copyOut), clean ambient, afterScan⟩) =
      some ⟨some (labels .compare), clean ambient, afterCopy⟩ := by
    have h := MachineCopy.copyTrace (tape 1) (tape 3) (tape 4)
      (hd 1 3 (by decide)) (hd 1 4 (by decide)) (hd 3 4 (by decide)) false
      (labels .copyOut) (labels .copyBack) (some (labels .compare)) program
      (atLabels .copyOut) (atLabels .copyBack) afterScan afterScanScratch
      (ambient, false, none) none
    simpa only [afterScanKey, afterScanCopy, List.append_nil, clean, afterCopy] using h
  have candidateWord : afterCopy (tape 2) = bits.reverse := by
    simp [afterCopy, afterScan, hd 2 3 (by decide)]
  have copiedWord : afterCopy (tape 3) = key := by simp [afterCopy]
  have restored : DFVSGames.Reduction.MachineTransfer.tapesAt (tape 2) (tape 3)
      afterCopy [] [] = finalBase := by
    funext k
    by_cases hc : k = tape 2
    · subst k
      simp [DFVSGames.Reduction.MachineTransfer.tapesAt, afterCopy, afterScan, finalBase,
        hd 2 3 (by decide), hd 2 0 (by decide), candidateEmpty]
    · by_cases hk : k = tape 3
      · subst k
        simp [DFVSGames.Reduction.MachineTransfer.tapesAt, afterCopy, afterScan, finalBase,
          hd 3 0 (by decide), copyEmpty]
      · simp [DFVSGames.Reduction.MachineTransfer.tapesAt, afterCopy, afterScan, hc, hk]
  have comparisonRun : (advance (TM2.step program))^[max bits.length key.length + 1]
      (some ⟨some (labels .compare), clean ambient, afterCopy⟩) =
      some ⟨if bits.reverse = key then equalExit else differentExit,
        clean ambient, finalBase⟩ := by
    let back := MachineStateEquiv.program (compareStateEquiv A).symm program
    have atCompare : back (labels .compare) =
        MachineCompare.loop (tape 2) (tape 3) (labels .compare) equalExit differentExit := by
      change MachineStateEquiv.statement (compareStateEquiv A).symm
        (program (labels .compare)) = _
      rw [atLabels .compare]
      exact MachineStateEquiv.statement_symm_statement (compareStateEquiv A) _
    have backForward : MachineStateEquiv.program (compareStateEquiv A) back = program := by
      funext l
      change MachineStateEquiv.statement (compareStateEquiv A)
        (MachineStateEquiv.statement (compareStateEquiv A).symm (program l)) = _
      simpa only [Equiv.symm_symm] using
        MachineStateEquiv.statement_symm_statement (compareStateEquiv A).symm (program l)
    have native := MachineCompare.compareTrace_fromTapes (tape 2) (tape 3)
      (hd 2 3 (by decide)) (labels .compare) equalExit differentExit back atCompare afterCopy
      ambient none none
    rw [candidateWord, copiedWord, restored, List.length_reverse] at native
    have transported := MachineStateEquiv.trace (compareStateEquiv A) back _ _ _ native
    rw [backForward] at transported
    simpa only [MachineStateEquiv.configuration, compareStateEquiv, Equiv.coe_fn_mk, clean] using transported
  exact joinTrace_inline_MachineBinaryNameCompare (joinTrace_inline_MachineBinaryNameCompare scanRun copyRun) comparisonRun

def compareInTime (tape : Fin 5 → K) (distinct : Function.Injective tape)
    (labels : Label → Λ) (equalExit differentExit malformedExit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State A))
    (atLabels : ∀ l, program (labels l) =
      instruction tape labels equalExit differentExit malformedExit l)
    (base : K → List Bool) (bits suffix key : List Bool)
    (canonical : BinaryNameMachine.canonical bits = true)
    (sourceWord : base (tape 0) = BinaryNameMachine.frame bits ++ suffix)
    (keyWord : base (tape 1) = key)
    (candidateEmpty : base (tape 2) = []) (copyEmpty : base (tape 3) = [])
    (scratchEmpty : base (tape 4) = []) (ambient : A) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some (labels .scan), clean ambient, base⟩
      (some ⟨if bits.reverse = key then equalExit else differentExit,
        clean ambient, Function.update base (tape 0) suffix⟩)
      (2 * bits.length + 3 * key.length + 4) where
  steps := steps bits.length key.length
  evals_in_steps := compareTrace tape distinct labels equalExit differentExit malformedExit
    program atLabels base bits suffix key canonical sourceWord keyWord
    candidateEmpty copyEmpty scratchEmpty ambient
  steps_le_m := steps_le bits.length key.length

def machine : FinTM2 where
  K := Fin 5
  k₀ := 0
  k₁ := 1
  Γ _ := Bool
  Λ := Label ⊕ Fin 3
  main := .inl .scan
  σ := State Unit
  initialState := clean ()
  m label := match label with
    | .inl l => instruction id Sum.inl (some (.inr 0)) (some (.inr 1)) (some (.inr 2)) l
    | .inr _ => .halt

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

end DFVSGames.BinaryNameCompare

end

section

namespace DFVSGames.BinaryNameSearch

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

abbrev Token := Bool × List Bool

def tokenBits (token : Token) : List Bool :=
  token.1 :: BinaryNameMachine.frame token.2

def stream : List Token → List Bool
  | [] => []
  | token :: tokens => tokenBits token ++ stream tokens

def payloads (tokens : List Token) : List (List Bool) := tokens.map Prod.snd

def afterMatch : List Token → List Bool → List Token
  | [], _ => []
  | token :: tokens, key => if token.2 = key then tokens else afterMatch tokens key

def steps : List Token → List Bool → Nat
  | [], _ => 0
  | token :: tokens, key =>
      1 + BinaryNameCompare.steps token.2.length key.length +
        if token.2 = key then 0 else 1 + steps tokens key

def payloadSize (tokens : List Token) : Nat := (payloads tokens |>.map List.length).sum

@[simp] theorem stream_nil : stream [] = [] := rfl

@[simp] theorem stream_cons (token : Token) (tokens : List Token) :
    stream (token :: tokens) = tokenBits token ++ stream tokens := rfl

@[simp] theorem payloads_nil : payloads [] = [] := rfl

@[simp] theorem payloads_cons (token : Token) (tokens : List Token) :
    payloads (token :: tokens) = token.2 :: payloads tokens := rfl

@[simp] theorem payloadSize_nil : payloadSize [] = 0 := rfl

@[simp] theorem payloadSize_cons (token : Token) (tokens : List Token) :
    payloadSize (token :: tokens) = token.2.length + payloadSize tokens := rfl

@[simp] theorem stream_length (tokens : List Token) :
    (stream tokens).length = 2 * payloadSize tokens + 2 * tokens.length := by
  induction tokens with
  | nil => rfl
  | cons token tokens ih =>
      simp only [stream_cons, List.length_append, tokenBits, List.length_cons,
        BinaryNameMachine.frame_length, ih, payloadSize_cons]
      omega

theorem steps_le (tokens : List Token) (key : List Bool) :
    steps tokens key ≤ 2 * payloadSize tokens + (3 * key.length + 6) * tokens.length := by
  induction tokens with
  | nil => simp [steps]
  | cons token tokens ih =>
      have h := BinaryNameCompare.steps_le token.2.length key.length
      simp only [steps, payloadSize_cons, List.length_cons]
      split <;> nlinarith

theorem steps_le_stream (tokens : List Token) (key : List Bool) :
    steps tokens key ≤ (3 * key.length + 7) * (stream tokens).length := by
  have h := steps_le tokens key
  rw [stream_length]
  nlinarith

section Program

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

abbrev Alphabet (_ : K) := Bool
abbrev State (A : Type) := BinaryNameCompare.State A

abbrev clean (ambient : A) : State A := BinaryNameCompare.clean ambient

inductive Label
  | sign
  | comparison (label : BinaryNameCompare.Label)
  | increment
  deriving DecidableEq, Fintype

def compareTapes (tape : Fin 6 → K) : Fin 5 → K := fun i => tape i.castSucc

omit [DecidableEq K] in
theorem compareTapes_injective (tape : Fin 6 → K) (distinct : Function.Injective tape) :
    Function.Injective (compareTapes tape) := by
  intro i j h
  apply Fin.ext
  exact congrArg (fun i : Fin 6 => i.val) (distinct h)

def exitAt (exit : Option Λ) : TM2.Stmt (Alphabet (K := K)) Λ (State A) :=
  match exit with
  | none => .halt
  | some label => .goto fun _ => label

def finish (exit : Option Λ) : TM2.Stmt (Alphabet (K := K)) Λ (State A) :=
  .load (fun state => clean state.1.1) (exitAt exit)

def instruction (tape : Fin 6 → K) (labels : Label → Λ)
    (foundExit missingExit malformedExit : Option Λ) :
    Label → TM2.Stmt (Alphabet (K := K)) Λ (State A)
  | .sign =>
      .pop (tape 0) (fun state head => (state.1, head))
        (.branch (fun state => state.2.isSome)
          (finish (some (labels (.comparison .scan)))) (finish missingExit))
  | .comparison l => BinaryNameCompare.instruction (compareTapes tape)
      (fun l => labels (.comparison l)) foundExit (some (labels .increment)) malformedExit l
  | .increment => .push (tape 5) (fun _ => true) (.goto fun _ => labels .sign)

@[simp] theorem stepAux_finish (exit : Option Λ) (state : State A) (base : K → List Bool) :
    TM2.stepAux (finish exit) state base = ⟨exit, clean state.1.1, base⟩ := by
  cases exit <;> rfl

private theorem update_source_inline_MachineBinaryNameSearch (source counter : K) (distinct : source ≠ counter)
    (base : K → List Bool) (input count replacement : List Bool) :
    Function.update (tapesAt source counter base input count) source replacement =
      tapesAt source counter base replacement count := by
  funext k
  by_cases hs : k = source
  · subst k; simp [tapesAt, distinct]
  · by_cases hc : k = counter
    · subst k; simp [tapesAt, Ne.symm distinct]
    · simp [tapesAt, hs, hc]

private theorem update_counter_inline_MachineBinaryNameSearch (source counter : K)
    (base : K → List Bool) (input count replacement : List Bool) :
    Function.update (tapesAt source counter base input count) counter replacement =
      tapesAt source counter base input replacement := by
  simp [tapesAt]

private theorem joinTrace_inline_MachineBinaryNameSearch {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]

variable (tape : Fin 6 → K) (distinct : Function.Injective tape)
variable (labels : Label → Λ) (foundExit missingExit malformedExit : Option Λ)
variable (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State A))
variable (atLabels : ∀ l, program (labels l) =
  instruction tape labels foundExit missingExit malformedExit l)
variable (base : K → List Bool) (ambient : A)

include distinct atLabels

theorem signStep (sign : Bool) (input counter : List Bool) :
    TM2.step program
      ⟨some (labels .sign), clean ambient,
        tapesAt (tape 0) (tape 5) base (sign :: input) counter⟩ =
      some ⟨some (labels (.comparison .scan)), clean ambient,
        tapesAt (tape 0) (tape 5) base input counter⟩ := by
  have hd : tape 0 ≠ tape 5 := fun h => (by decide : (0 : Fin 6) ≠ 5) (distinct h)
  change some (TM2.stepAux (program (labels .sign)) _ _) = _
  rw [atLabels .sign]
  simp [instruction, TM2.stepAux, clean, BinaryNameCompare.clean, hd, update_source_inline_MachineBinaryNameSearch]

theorem emptyStep (counter : List Bool) :
    TM2.step program
      ⟨some (labels .sign), clean ambient,
        tapesAt (tape 0) (tape 5) base [] counter⟩ =
      some ⟨missingExit, clean ambient, tapesAt (tape 0) (tape 5) base [] counter⟩ := by
  have hd : tape 0 ≠ tape 5 := fun h => (by decide : (0 : Fin 6) ≠ 5) (distinct h)
  change some (TM2.stepAux (program (labels .sign)) _ _) = _
  rw [atLabels .sign]
  simp [instruction, TM2.stepAux, clean, BinaryNameCompare.clean, hd, update_source_inline_MachineBinaryNameSearch]

omit distinct in
theorem incrementStep (input counter : List Bool) :
    TM2.step program
      ⟨some (labels .increment), clean ambient,
        tapesAt (tape 0) (tape 5) base input counter⟩ =
      some ⟨some (labels .sign), clean ambient,
        tapesAt (tape 0) (tape 5) base input (true :: counter)⟩ := by
  change some (TM2.stepAux (program (labels .increment)) _ _) = _
  rw [atLabels .increment]
  simp [instruction, TM2.stepAux, update_counter_inline_MachineBinaryNameSearch]

theorem tokenTrace (token : Token) (key suffix counter : List Bool)
    (canonical : BinaryNameMachine.canonical token.2 = true)
    (keyWord : base (tape 1) = key.reverse)
    (candidateEmpty : base (tape 2) = []) (copyEmpty : base (tape 3) = [])
    (scratchEmpty : base (tape 4) = []) :
    (advance (TM2.step program))^[1 + BinaryNameCompare.steps token.2.length key.length]
      (some ⟨some (labels .sign), clean ambient,
        tapesAt (tape 0) (tape 5) base (tokenBits token ++ suffix) counter⟩) =
      some ⟨if token.2 = key then foundExit else some (labels .increment), clean ambient,
        tapesAt (tape 0) (tape 5) base suffix counter⟩ := by
  have hd (i j : Fin 6) (hne : i ≠ j) : tape i ≠ tape j := fun h => hne (distinct h)
  let initial := tapesAt (tape 0) (tape 5) base
    (BinaryNameMachine.frame token.2 ++ suffix) counter
  have first : (advance (TM2.step program))^[1]
      (some ⟨some (labels .sign), clean ambient,
        tapesAt (tape 0) (tape 5) base (tokenBits token ++ suffix) counter⟩) =
      some ⟨some (labels (.comparison .scan)), clean ambient, initial⟩ := by
    simpa only [Function.iterate_one, advance_some, tokenBits, List.cons_append] using
      signStep tape distinct labels foundExit missingExit malformedExit program atLabels
        base ambient token.1 (BinaryNameMachine.frame token.2 ++ suffix) counter
  have second := BinaryNameCompare.compareTrace (compareTapes tape)
    (compareTapes_injective tape distinct) (fun l => labels (.comparison l))
    foundExit (some (labels .increment)) malformedExit program
    (fun l => atLabels (.comparison l)) initial token.2 suffix key.reverse canonical
    (by simp [initial, compareTapes, hd 0 5 (by decide)])
    (by simpa [initial, compareTapes, tapesAt,
      hd 1 0 (by decide), hd 1 5 (by decide)] using keyWord)
    (by simpa [initial, compareTapes, tapesAt, hd 2 0 (by decide), hd 2 5 (by decide)]
      using candidateEmpty)
    (by simpa [initial, compareTapes, tapesAt,
      hd 3 0 (by decide), hd 3 5 (by decide)] using copyEmpty)
    (by simpa [initial, compareTapes, tapesAt,
      hd 4 0 (by decide), hd 4 5 (by decide)] using scratchEmpty)
    ambient
  have heq : token.2.reverse = key.reverse ↔ token.2 = key := by
    constructor
    · intro h
      have := congrArg List.reverse h
      simpa only [List.reverse_reverse] using this
    · intro h; rw [h]
  simp only [List.length_reverse, heq] at second
  rw [show Function.update initial (compareTapes tape 0) suffix =
      tapesAt (tape 0) (tape 5) base suffix counter by
    exact update_source_inline_MachineBinaryNameSearch _ _ (hd 0 5 (by decide)) base _ counter suffix] at second
  exact joinTrace_inline_MachineBinaryNameSearch first second

theorem searchTrace (tokens : List Token) (key suffix counter : List Bool)
    (canonical : ∀ token ∈ tokens, BinaryNameMachine.canonical token.2 = true)
    (present : key ∈ payloads tokens)
    (keyWord : base (tape 1) = key.reverse)
    (candidateEmpty : base (tape 2) = []) (copyEmpty : base (tape 3) = [])
    (scratchEmpty : base (tape 4) = []) :
    (advance (TM2.step program))^[steps tokens key]
      (some ⟨some (labels .sign), clean ambient,
        tapesAt (tape 0) (tape 5) base (stream tokens ++ suffix) counter⟩) =
      some ⟨foundExit, clean ambient,
        tapesAt (tape 0) (tape 5) base (stream (afterMatch tokens key) ++ suffix)
          (List.replicate ((payloads tokens).idxOf key) true ++ counter)⟩ := by
  induction tokens generalizing counter with
  | nil => simp at present
  | cons token tokens ih =>
      have headCanonical := canonical token (by simp)
      have tailCanonical : ∀ t ∈ tokens, BinaryNameMachine.canonical t.2 = true :=
        fun t ht => canonical t (by simp [ht])
      have first := tokenTrace tape distinct labels foundExit missingExit malformedExit
        program atLabels base ambient token key (stream tokens ++ suffix) counter
        headCanonical keyWord candidateEmpty copyEmpty scratchEmpty
      by_cases same : token.2 = key
      · simpa only [steps, same, ite_true, Nat.add_zero, stream_cons,
          List.append_assoc, afterMatch, payloads_cons, List.idxOf_cons_self,
          List.replicate_zero, List.nil_append] using first
      · have remaining : key ∈ payloads tokens := by
          rcases List.mem_cons.mp present with equal | remaining
          · exact False.elim (same equal.symm)
          · exact remaining
        have increment : (advance (TM2.step program))^[1]
            (some ⟨some (labels .increment), clean ambient,
              tapesAt (tape 0) (tape 5) base (stream tokens ++ suffix) counter⟩) =
            some ⟨some (labels .sign), clean ambient,
              tapesAt (tape 0) (tape 5) base (stream tokens ++ suffix) (true :: counter)⟩ := by
          simpa only [Function.iterate_one, advance_some] using
            incrementStep tape labels foundExit missingExit malformedExit program
              atLabels base ambient (stream tokens ++ suffix) counter
        have rest := ih (true :: counter) tailCanonical remaining
        simp only [same, ite_false] at first
        have full := joinTrace_inline_MachineBinaryNameSearch first (joinTrace_inline_MachineBinaryNameSearch increment rest)
        have counterEq : List.replicate ((payloads tokens).idxOf key) true ++
            (true :: counter) =
            List.replicate ((payloads tokens).idxOf key + 1) true ++ counter := by
          rw [List.replicate_succ']
          simp only [List.append_assoc, List.singleton_append]
        have indexEq : (token.2 :: payloads tokens).idxOf key =
            (payloads tokens).idxOf key + 1 := by
          exact List.idxOf_cons_ne _ same
        simpa only [steps, same, ite_false, stream_cons, List.append_assoc, afterMatch,
          payloads_cons, indexEq, counterEq] using full

def searchInTime (tokens : List Token) (key suffix counter : List Bool)
    (canonical : ∀ token ∈ tokens, BinaryNameMachine.canonical token.2 = true)
    (present : key ∈ payloads tokens)
    (keyWord : base (tape 1) = key.reverse)
    (candidateEmpty : base (tape 2) = []) (copyEmpty : base (tape 3) = [])
    (scratchEmpty : base (tape 4) = []) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some (labels .sign), clean ambient,
        tapesAt (tape 0) (tape 5) base (stream tokens ++ suffix) counter⟩
      (some ⟨foundExit, clean ambient,
        tapesAt (tape 0) (tape 5) base (stream (afterMatch tokens key) ++ suffix)
          (List.replicate ((payloads tokens).idxOf key) true ++ counter)⟩)
      ((3 * key.length + 7) * (stream tokens).length) where
  steps := steps tokens key
  evals_in_steps := searchTrace tape distinct labels foundExit missingExit malformedExit
    program atLabels base ambient tokens key suffix counter canonical present keyWord
      candidateEmpty copyEmpty scratchEmpty
  steps_le_m := steps_le_stream tokens key

end Program

def machine : FinTM2 where
  K := Fin 6
  k₀ := 0
  k₁ := 5
  Γ _ := Bool
  Λ := Label ⊕ Fin 3
  main := .inl .sign
  σ := State Unit
  initialState := clean ()
  m label := match label with
    | .inl l => instruction id Sum.inl (some (.inr 0)) (some (.inr 1)) (some (.inr 2)) l
    | .inr _ => .halt

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

end DFVSGames.BinaryNameSearch

end

end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end

end OAI