lean/OAI/Computability/BinPacking/Reductions/MachineReverse.lean

import Mathlib.Computability.TuringMachine.Computable

namespace OAI

namespace BinPackingGames.Reduction.MachineReverse

open Turing

def loop : TM2.Stmt (fun _ : Bool => Bool) Unit (Option Bool) :=
  .pop false (fun _ head => head)
    (.branch Option.isSome
      (.push true (fun state => state.getD false) (.goto fun _ => ()))
      .halt)

def machine : FinTM2 where
  K := Bool
  k₀ := false
  k₁ := true
  Γ _ := Bool
  Λ := Unit
  main := ()
  σ := Option Bool
  initialState := none
  m _ := loop

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

def running (input output : List Bool) (state : Option Bool) : machine.Cfg :=
  ⟨some (), state, tapeStacks input output⟩

def halted (output : List Bool) : machine.Cfg :=
  ⟨none, none, tapeStacks [] output⟩

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]

theorem step_empty (output : List Bool) (state : Option Bool) :
    machine.step (running [] output state) = some (halted output) := by
  change some (TM2.stepAux loop state (tapeStacks [] output)) = some (halted output)
  simp [loop, TM2.stepAux, tapeStacks, halted, Function.update]
  rw [update_input]
  rfl

theorem step_cons (head : Bool) (input output : List Bool) (state : Option Bool) :
    machine.step (running (head :: input) output state) =
      some (running input (head :: output) (some head)) := by
  change some (TM2.stepAux loop state (tapeStacks (head :: input) output)) = _
  simp [loop, TM2.stepAux, tapeStacks, running, Function.update]
  rw [update_input, update_output]
  rfl

def next (configuration : Option machine.Cfg) : Option machine.Cfg :=
  configuration.bind machine.step

theorem reverse_steps (input output : List Bool) (state : Option Bool) :
    next^[input.length + 1] (some (running input output state)) =
      some (halted (input.reverse ++ output)) := by
  induction input generalizing output state with
  | nil =>
    simpa only [List.length_nil, Nat.zero_add, Function.iterate_one, next,
      Option.bind_some, List.reverse_nil, List.nil_append] using step_empty output state
  | cons head input ih =>
    rw [List.length_cons, Function.iterate_succ_apply]
    change next^[input.length + 1]
      (machine.step (running (head :: input) output state)) = _
    rw [step_cons, ih]
    simp only [List.reverse_cons, List.append_assoc, List.singleton_append]

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

theorem haltList_eq (output : List Bool) : haltList machine output = halted output := by
  unfold haltList halted
  congr 1

theorem reverse_init_steps (input : List Bool) :
    next^[input.length + 1] (some (initList machine input)) =
      some (haltList machine input.reverse) := by
  rw [initList_eq, haltList_eq]
  simpa only [List.append_nil] using reverse_steps input [] none

def outputsInTime (input : List Bool) :
    TM2OutputsInTime machine input (some input.reverse) (input.length + 1) where
  steps := input.length + 1
  evals_in_steps := reverse_init_steps input
  steps_le_m := Nat.le_refl _

@[simp] theorem outputsInTime_steps (input : List Bool) :
    (outputsInTime input).steps = input.length + 1 := rfl

noncomputable def computableInPolyTime :
    TM2ComputableInPolyTime (id : List Bool → List Bool) id List.reverse where
  tm := machine
  inputAlphabet := Equiv.refl Bool
  outputAlphabet := Equiv.refl Bool
  time := Polynomial.X + 1
  outputsFun input := by
    change TM2OutputsInTime machine (input.map id) (some (input.reverse.map id))
      ((Polynomial.X + 1 : Polynomial Nat).eval input.length)
    have hi := @List.map_id (machine.Γ machine.k₀) input
    have ho := @List.map_id (machine.Γ machine.k₁) input.reverse
    rw [hi, ho]
    simpa only [Polynomial.eval_add, Polynomial.eval_X, Polynomial.eval_one]
      using outputsInTime input

end BinPackingGames.Reduction.MachineReverse

end OAI