lean/OAI/Computability/UniqueGames/Reduction/MachineTransducer.lean

import Mathlib.Data.Finset.Lattice.Fold
import OAI.Computability.UniqueGames.Machines.GraphCounterStartLemmas
import OAI.Computability.UniqueGames.Reduction.MachineReverse
import OAI.Computability.UniqueGames.Reduction.MachineSubstitution

namespace OAI

namespace UniqueGamesTheorem.Reduction.MachineTransducer

open Turing
open UniqueGamesTheorem.Foundations.Complexity
open MachineSubstitution (pushWord stepAux_pushWord statementPushBound_pushWord)

def tapeStacks (input output : List Bool) : Bool → List Bool :=
  fun side => if side then output else input

private theorem update_input (input output replacement : List Bool) :
    Function.update (tapeStacks input output) false replacement = tapeStacks replacement output := by
  funext side
  cases side <;> simp [tapeStacks]

private theorem update_output (input output replacement : List Bool) :
    Function.update (tapeStacks input output) true replacement = tapeStacks input replacement := by
  funext side
  cases side <;> simp [tapeStacks]

variable {Q : Type} [Fintype Q]

/-- The semantic output, with the control state updated after each input bit. -/
def output (transition : Q → Bool → Q) (emit : Q → Bool → List Bool) :
    Q → List Bool → List Bool
  | _, [] => []
  | state, symbol :: input =>
      emit state symbol ++ output transition emit (transition state symbol) input

def loop (initial : Q) :
    TM2.Stmt (fun _ : Bool => Bool) (Option (Q × Bool)) (Q × Option Bool) :=
  .pop false (fun state head => (state.1, head))
    (.branch (fun state => state.2.isSome)
      (.goto fun state => some (state.1, state.2.getD false))
      (.load (fun _ => (initial, none)) .halt))

def program (initial : Q) (transition : Q → Bool → Q) (emit : Q → Bool → List Bool) :
    Option (Q × Bool) →
      TM2.Stmt (fun _ : Bool => Bool) (Option (Q × Bool)) (Q × Option Bool)
  | none => loop initial
  | some (state, symbol) =>
      .load (fun register => (transition state symbol, register.2))
        (pushWord true (emit state symbol) (.goto fun _ => none))

def machine (initial : Q) (transition : Q → Bool → Q) (emit : Q → Bool → List Bool) :
    FinTM2 where
  K := Bool
  k₀ := false
  k₁ := true
  Γ _ := Bool
  Λ := Option (Q × Bool)
  main := none
  σ := Q × Option Bool
  initialState := (initial, none)
  m := program initial transition emit

/-- A finite maximum depending only on the fixed emission table. -/
noncomputable def maxEmission (emit : Q → Bool → List Bool) : Nat :=
  Finset.univ.sup (fun pair : Q × Bool => (emit pair.1 pair.2).length)

theorem emission_le (emit : Q → Bool → List Bool) (state : Q) (symbol : Bool) :
    (emit state symbol).length ≤ maxEmission emit := by
  exact Finset.le_sup (f := fun pair : Q × Bool => (emit pair.1 pair.2).length)
    (Finset.mem_univ (state, symbol))

theorem statementPushBound_le (initial : Q) (transition : Q → Bool → Q)
    (emit : Q → Bool → List Bool) (label : Option (Q × Bool)) :
    Runtime.statementPushBound (program initial transition emit label) ≤ maxEmission emit := by
  cases label with
  | none => simp [program, loop, Runtime.statementPushBound]
  | some pair =>
    simpa only [program, Runtime.statementPushBound, statementPushBound_pushWord,
      Nat.add_zero] using emission_le emit pair.1 pair.2

theorem programPushBound_le (initial : Q) (transition : Q → Bool → Q)
    (emit : Q → Bool → List Bool) :
    Runtime.programPushBound (machine initial transition emit) ≤ maxEmission emit := by
  have allLabels (labels : List (Option (Q × Bool))) :
      Runtime.maxLabelPushes (program initial transition emit) labels ≤ maxEmission emit := by
    induction labels with
    | nil => exact Nat.zero_le _
    | cons label rest ih =>
      exact max_le (statementPushBound_le initial transition emit label) ih
  exact allLabels (machine initial transition emit).ΛFin.elems.toList

def running (initial : Q) (transition : Q → Bool → Q) (emit : Q → Bool → List Bool)
    (state : Q) (input accumulator : List Bool) (register : Option Bool) :
    (machine initial transition emit).Cfg :=
  ⟨some none, (state, register), tapeStacks input accumulator⟩

def emitting (initial : Q) (transition : Q → Bool → Q) (emit : Q → Bool → List Bool)
    (oldState : Q) (symbol : Bool) (input accumulator : List Bool)
    (state : Q × Option Bool) : (machine initial transition emit).Cfg :=
  ⟨some (some (oldState, symbol)), state, tapeStacks input accumulator⟩

def halted (initial : Q) (transition : Q → Bool → Q) (emit : Q → Bool → List Bool)
    (accumulator : List Bool) : (machine initial transition emit).Cfg :=
  ⟨none, (initial, none), tapeStacks [] accumulator⟩

/-- The final empty pop also restores the initial finite control state. -/
theorem step_empty (initial : Q) (transition : Q → Bool → Q)
    (emit : Q → Bool → List Bool) (state : Q) (accumulator : List Bool)
    (register : Option Bool) :
    (machine initial transition emit).step
        (running initial transition emit state [] accumulator register) =
      some (halted initial transition emit accumulator) := by
  change some (TM2.stepAux (loop initial) (state, register) (tapeStacks [] accumulator)) = _
  simp [loop, TM2.stepAux, tapeStacks, halted]
  rw [update_input]
  rfl

theorem step_cons (initial : Q) (transition : Q → Bool → Q)
    (emit : Q → Bool → List Bool) (state : Q) (symbol : Bool)
    (input accumulator : List Bool) (register : Option Bool) :
    (machine initial transition emit).step
        (running initial transition emit state (symbol :: input) accumulator register) =
      some (emitting initial transition emit state symbol input accumulator (state, some symbol)) := by
  change some (TM2.stepAux (loop initial) (state, register)
    (tapeStacks (symbol :: input) accumulator)) = _
  simp [loop, TM2.stepAux, tapeStacks, emitting]
  rw [update_input]
  rfl

/-- Emission reads the old state from its label and installs its successor state. -/
theorem step_emit (initial : Q) (transition : Q → Bool → Q)
    (emit : Q → Bool → List Bool) (oldState : Q) (symbol : Bool)
    (input accumulator : List Bool) (state : Q × Option Bool) :
    (machine initial transition emit).step
        (emitting initial transition emit oldState symbol input accumulator state) =
      some (running initial transition emit (transition oldState symbol) input
        ((emit oldState symbol).reverse ++ accumulator) state.2) := by
  change some (TM2.stepAux (pushWord true (emit oldState symbol) (.goto fun _ => none))
    (transition oldState symbol, state.2) (tapeStacks input accumulator)) = _
  rw [stepAux_pushWord]
  simp only [TM2.stepAux]
  change some (⟨some none, (transition oldState symbol, state.2),
    Function.update (tapeStacks input accumulator) true
      ((emit oldState symbol).reverse ++ accumulator)⟩ :
    (machine initial transition emit).Cfg) = _
  exact congrArg (fun tapes : Bool → List Bool =>
    some (⟨some none, (transition oldState symbol, state.2), tapes⟩ :
      (machine initial transition emit).Cfg))
    (update_output input accumulator ((emit oldState symbol).reverse ++ accumulator))

def next (initial : Q) (transition : Q → Bool → Q) (emit : Q → Bool → List Bool)
    (configuration : Option (machine initial transition emit).Cfg) :
    Option (machine initial transition emit).Cfg :=
  configuration.bind (machine initial transition emit).step

theorem two_steps_cons (initial : Q) (transition : Q → Bool → Q)
    (emit : Q → Bool → List Bool) (state : Q) (symbol : Bool)
    (input accumulator : List Bool) (register : Option Bool) :
    (next initial transition emit)^[2]
        (some (running initial transition emit state (symbol :: input) accumulator register)) =
      some (running initial transition emit (transition state symbol) input
        ((emit state symbol).reverse ++ accumulator) (some symbol)) := by
  change (machine initial transition emit).step
    (running initial transition emit state (symbol :: input) accumulator register) >>=
      (machine initial transition emit).step = _
  rw [step_cons]
  exact step_emit initial transition emit state symbol input accumulator (state, some symbol)

/-- Exact execution from any finite control state, accumulator, and symbol register. -/
theorem transduce_steps (initial : Q) (transition : Q → Bool → Q)
    (emit : Q → Bool → List Bool) (state : Q) (input accumulator : List Bool)
    (register : Option Bool) :
    (next initial transition emit)^[2 * input.length + 1]
        (some (running initial transition emit state input accumulator register)) =
      some (halted initial transition emit ((output transition emit state input).reverse ++ accumulator)) := by
  induction input generalizing state accumulator register with
  | nil =>
    simpa only [List.length_nil, Nat.mul_zero, Nat.zero_add, Function.iterate_one, next,
      Option.bind_some, output, List.reverse_nil, List.nil_append]
      using step_empty initial transition emit state accumulator register
  | cons symbol input ih =>
    rw [List.length_cons]
    rw [show 2 * (input.length + 1) + 1 = (2 * input.length + 1) + 2 by omega]
    rw [Function.iterate_add_apply, two_steps_cons, ih]
    simp only [output, List.reverse_append, List.append_assoc]

theorem initList_eq (initial : Q) (transition : Q → Bool → Q)
    (emit : Q → Bool → List Bool) (input : List Bool) :
    initList (machine initial transition emit) input =
      running initial transition emit initial input [] none := by
  unfold initList running
  congr 1
  funext side
  cases side <;> rfl

theorem haltList_eq (initial : Q) (transition : Q → Bool → Q)
    (emit : Q → Bool → List Bool) (accumulator : List Bool) :
    haltList (machine initial transition emit) accumulator =
      halted initial transition emit accumulator := by
  unfold haltList halted
  congr 1

theorem transduce_init_steps (initial : Q) (transition : Q → Bool → Q)
    (emit : Q → Bool → List Bool) (input : List Bool) :
    (next initial transition emit)^[2 * input.length + 1]
        (some (initList (machine initial transition emit) input)) =
      some (haltList (machine initial transition emit) (output transition emit initial input).reverse) := by
  rw [initList_eq, haltList_eq]
  simpa only [List.append_nil] using transduce_steps initial transition emit initial input [] none

def outputsInTime (initial : Q) (transition : Q → Bool → Q)
    (emit : Q → Bool → List Bool) (input : List Bool) :
    TM2OutputsInTime (machine initial transition emit) input
      (some (output transition emit initial input).reverse) (2 * input.length + 1) where
  steps := 2 * input.length + 1
  evals_in_steps := transduce_init_steps initial transition emit input
  steps_le_m := Nat.le_refl _

@[simp] theorem outputsInTime_steps (initial : Q) (transition : Q → Bool → Q)
    (emit : Q → Bool → List Bool) (input : List Bool) :
    (outputsInTime initial transition emit input).steps = 2 * input.length + 1 := rfl

/-- Reversed transducer output in polynomial `2 * X + 1` time. -/
noncomputable def reversedComputableInPolyTime (initial : Q) (transition : Q → Bool → Q)
    (emit : Q → Bool → List Bool) :
    TM2ComputableInPolyTime (id : List Bool → List Bool) id
      (fun input => (output transition emit initial input).reverse) where
  tm := machine initial transition emit
  inputAlphabet := Equiv.refl Bool
  outputAlphabet := Equiv.refl Bool
  time := 2 * Polynomial.X + 1
  outputsFun input := by
    change TM2OutputsInTime (machine initial transition emit) (input.map id)
      (some ((output transition emit initial input).reverse.map id))
      ((2 * Polynomial.X + 1 : Polynomial Nat).eval input.length)
    have hi := @List.map_id ((machine initial transition emit).Γ
      (machine initial transition emit).k₀) input
    have ho := @List.map_id ((machine initial transition emit).Γ
      (machine initial transition emit).k₁) (output transition emit initial input).reverse
    rw [hi, ho]
    simpa only [Polynomial.eval_add, Polynomial.eval_mul, Polynomial.eval_X,
      Polynomial.eval_ofNat, Polynomial.eval_one] using outputsInTime initial transition emit input

/-- Forward transducer output, using the checked actual reversal composition. -/
noncomputable def computableInPolyTime (initial : Q) (transition : Q → Bool → Q)
    (emit : Q → Bool → List Bool) :
    TM2ComputableInPolyTime (id : List Bool → List Bool) id
      (output transition emit initial) := by
  simpa only [List.reverse_reverse] using
    MachineSequential.composeBits (reversedComputableInPolyTime initial transition emit)
      MachineReverse.computableInPolyTime

end UniqueGamesTheorem.Reduction.MachineTransducer

end OAI