lean/OAI/Computability/BinPacking/Reductions/MachineTransfer.lean

import Mathlib.Computability.TuringMachine.Computable

namespace OAI

namespace BinPackingGames.Reduction.MachineTransfer

open Turing

abbrev alphabet (α β : Type) : Bool → Type
  | false => α
  | true => β

variable {α β : Type} [Fintype α] [Fintype β]

def loop (f : α → β) (fallback : β) :
    TM2.Stmt (alphabet α β) Unit (Option β) :=
  .pop false (fun _ head => head.map f)
    (.branch Option.isSome
      (.push true (fun state => state.getD fallback) (.goto fun _ => ()))
      .halt)

def machine (f : α → β) (fallback : β) : FinTM2 where
  K := Bool
  k₀ := false
  k₁ := true
  Γ := alphabet α β
  Λ := Unit
  main := ()
  σ := Option β
  initialState := none
  Γk₀Fin := inferInstanceAs (Fintype α)
  m _ := loop f fallback

def tapeStacks (input : List α) (output : List β) :
    (side : Bool) → List (alphabet α β side)
  | false => input
  | true => output

def running (f : α → β) (fallback : β) (input : List α) (output : List β)
    (state : Option β) : (machine f fallback).Cfg :=
  ⟨some (), state, tapeStacks input output⟩

def halted (f : α → β) (fallback : β) (output : List β) : (machine f fallback).Cfg :=
  ⟨none, none, tapeStacks (α := α) [] output⟩

omit [Fintype α] [Fintype β] in
private theorem update_input (input replacement : List α) (output : List β) :
    Function.update (tapeStacks input output) false replacement = tapeStacks replacement output := by
  funext side
  cases side <;> rfl

omit [Fintype α] [Fintype β] in
private theorem update_output (input : List α) (output replacement : List β) :
    Function.update (tapeStacks input output) true replacement = tapeStacks input replacement := by
  funext side
  cases side <;> rfl

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

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

def next (f : α → β) (fallback : β) (configuration : Option (machine f fallback).Cfg) :
    Option (machine f fallback).Cfg := configuration.bind (machine f fallback).step

theorem transfer_steps (f : α → β) (fallback : β) (input : List α) (output : List β)
    (state : Option β) :
    (next f fallback)^[input.length + 1] (some (running f fallback input output state)) =
      some (halted f fallback (input.reverse.map f ++ 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.map_nil, List.nil_append]
      using step_empty f fallback output state
  | cons head input ih =>
    rw [List.length_cons, Function.iterate_succ_apply]
    change (next f fallback)^[input.length + 1]
      ((machine f fallback).step (running f fallback (head :: input) output state)) = _
    rw [step_cons, ih]
    simp only [List.reverse_cons, List.map_append, List.map_singleton,
      List.append_assoc, List.singleton_append]

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

theorem haltList_eq (f : α → β) (fallback : β) (output : List β) :
    haltList (machine f fallback) output = halted f fallback output := by
  unfold haltList halted
  congr 1
  funext side
  cases side <;> rfl

theorem transfer_init_steps (f : α → β) (fallback : β) (input : List α) :
    (next f fallback)^[input.length + 1] (some (initList (machine f fallback) input)) =
      some (haltList (machine f fallback) (input.reverse.map f)) := by
  erw [initList_eq (α := α) (β := β) f fallback input,
    haltList_eq (α := α) (β := β) f fallback (input.reverse.map f)]
  simpa only [List.append_nil] using transfer_steps f fallback input [] none

def outputsInTime (f : α → β) (fallback : β) (input : List α) :
    TM2OutputsInTime (machine f fallback) input (some (input.reverse.map f))
      (input.length + 1) where
  steps := input.length + 1
  evals_in_steps := transfer_init_steps f fallback input
  steps_le_m := Nat.le_refl _

@[simp] theorem outputsInTime_steps (f : α → β) (fallback : β) (input : List α) :
    (outputsInTime f fallback input).steps = input.length + 1 := rfl

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

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

def exitAt (dst : K) (exit : Option Λ) : TM2.Stmt Γ Λ (σ × Option (Γ dst)) :=
  match exit with
  | none => .halt
  | some label => .goto fun _ => label

def loopAt (src dst : K) (f : Γ src → Γ dst) (fallback : Γ dst)
    (loopLabel : Λ) (exit : Option Λ) : TM2.Stmt Γ Λ (σ × Option (Γ dst)) :=
  .pop src (fun state head => (state.1, head.map f))
    (.branch (fun state => state.2.isSome)
      (.push dst (fun state => state.2.getD fallback) (.goto fun _ => loopLabel))
      (exitAt dst exit))

def tapesAt (src dst : K) (base : (k : K) → List (Γ k))
    (input : List (Γ src)) (output : List (Γ dst)) : (k : K) → List (Γ k) :=
  Function.update (Function.update base src input) dst output

@[simp] theorem tapesAt_src (src dst : K) (distinct : src ≠ dst)
    (base : (k : K) → List (Γ k)) (input : List (Γ src)) (output : List (Γ dst)) :
    tapesAt src dst base input output src = input := by
  simp [tapesAt, distinct]

@[simp] theorem tapesAt_dst (src dst : K)
    (base : (k : K) → List (Γ k)) (input : List (Γ src)) (output : List (Γ dst)) :
    tapesAt src dst base input output dst = output := by
  simp [tapesAt]

theorem tapesAt_other (src dst k : K) (notSrc : k ≠ src) (notDst : k ≠ dst)
    (base : (k : K) → List (Γ k)) (input : List (Γ src)) (output : List (Γ dst)) :
    tapesAt src dst base input output k = base k := by
  simp [tapesAt, notSrc, notDst]

@[simp] theorem tapesAt_self (src dst : K) (base : (k : K) → List (Γ k)) :
    tapesAt src dst base (base src) (base dst) = base := by
  simp [tapesAt]

private theorem update_tapesAt_src (src dst : K) (distinct : src ≠ dst)
    (base : (k : K) → List (Γ k)) (input replacement : List (Γ src))
    (output : List (Γ dst)) :
    Function.update (tapesAt src dst base input output) src replacement =
      tapesAt src dst base replacement output := by
  funext k
  by_cases hs : k = src
  · subst k
    simp [tapesAt, distinct]
  · by_cases hd : k = dst
    · subst k
      simp [tapesAt, Ne.symm distinct]
    · simp [tapesAt, hs, hd]

private theorem update_tapesAt_dst (src dst : K)
    (base : (k : K) → List (Γ k)) (input : List (Γ src))
    (output replacement : List (Γ dst)) :
    Function.update (tapesAt src dst base input output) dst replacement =
      tapesAt src dst base input replacement := by
  funext k
  by_cases hd : k = dst
  · subst k
    simp [tapesAt]
  · simp [tapesAt, hd]

def nextAt (dst : K) (program : Λ → TM2.Stmt Γ Λ (σ × Option (Γ dst)))
    (configuration : Option (TM2.Cfg Γ Λ (σ × Option (Γ dst)))) :
    Option (TM2.Cfg Γ Λ (σ × Option (Γ dst))) :=
  configuration.bind (TM2.step program)

theorem stepAt_empty (src dst : K) (distinct : src ≠ dst)
    (f : Γ src → Γ dst) (fallback : Γ dst) (loopLabel : Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt Γ Λ (σ × Option (Γ dst)))
    (atLoop : program loopLabel = loopAt src dst f fallback loopLabel exit)
    (base : (k : K) → List (Γ k)) (output : List (Γ dst))
    (ambient : σ) (register : Option (Γ dst)) :
    TM2.step program ⟨some loopLabel, (ambient, register), tapesAt src dst base [] output⟩ =
      some ⟨exit, (ambient, none), tapesAt src dst base [] output⟩ := by
  change some (TM2.stepAux (program loopLabel) (ambient, register)
    (tapesAt src dst base [] output)) = _
  rw [atLoop]
  cases exit <;>
    simp [loopAt, exitAt, TM2.stepAux, tapesAt_src, distinct, update_tapesAt_src]

theorem stepAt_cons (src dst : K) (distinct : src ≠ dst)
    (f : Γ src → Γ dst) (fallback : Γ dst) (loopLabel : Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt Γ Λ (σ × Option (Γ dst)))
    (atLoop : program loopLabel = loopAt src dst f fallback loopLabel exit)
    (base : (k : K) → List (Γ k)) (head : Γ src) (input : List (Γ src))
    (output : List (Γ dst)) (ambient : σ) (register : Option (Γ dst)) :
    TM2.step program
      ⟨some loopLabel, (ambient, register), tapesAt src dst base (head :: input) output⟩ =
      some ⟨some loopLabel, (ambient, some (f head)),
        tapesAt src dst base input (f head :: output)⟩ := by
  change some (TM2.stepAux (program loopLabel) (ambient, register)
    (tapesAt src dst base (head :: input) output)) = _
  rw [atLoop]
  simp [loopAt, TM2.stepAux, tapesAt_src, tapesAt_dst, distinct,
    update_tapesAt_src, update_tapesAt_dst]

theorem transferAt_steps (src dst : K) (distinct : src ≠ dst)
    (f : Γ src → Γ dst) (fallback : Γ dst) (loopLabel : Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt Γ Λ (σ × Option (Γ dst)))
    (atLoop : program loopLabel = loopAt src dst f fallback loopLabel exit)
    (base : (k : K) → List (Γ k)) (input : List (Γ src)) (output : List (Γ dst))
    (ambient : σ) (register : Option (Γ dst)) :
    (nextAt dst program)^[input.length + 1]
      (some ⟨some loopLabel, (ambient, register), tapesAt src dst base input output⟩) =
      some ⟨exit, (ambient, none), tapesAt src dst base [] (input.reverse.map f ++ output)⟩ := by
  induction input generalizing output register with
  | nil =>
    simpa only [List.length_nil, Nat.zero_add, Function.iterate_one, nextAt,
      Option.bind_some, List.reverse_nil, List.map_nil, List.nil_append]
      using stepAt_empty src dst distinct f fallback loopLabel exit program atLoop base output
        ambient register
  | cons head input ih =>
    rw [List.length_cons, Function.iterate_succ_apply]
    change (nextAt dst program)^[input.length + 1]
      (TM2.step program
        ⟨some loopLabel, (ambient, register), tapesAt src dst base (head :: input) output⟩) = _
    rw [stepAt_cons src dst distinct f fallback loopLabel exit program atLoop, ih]
    simp only [List.reverse_cons, List.map_append, List.map_singleton,
      List.append_assoc, List.singleton_append]

theorem transferAt_fromTapes (src dst : K) (distinct : src ≠ dst)
    (f : Γ src → Γ dst) (fallback : Γ dst) (loopLabel : Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt Γ Λ (σ × Option (Γ dst)))
    (atLoop : program loopLabel = loopAt src dst f fallback loopLabel exit)
    (base : (k : K) → List (Γ k)) (ambient : σ) (register : Option (Γ dst)) :
    (nextAt dst program)^[(base src).length + 1]
      (some ⟨some loopLabel, (ambient, register), base⟩) =
      some ⟨exit, (ambient, none),
        tapesAt src dst base [] ((base src).reverse.map f ++ base dst)⟩ := by
  simpa only [tapesAt_self] using
    transferAt_steps src dst distinct f fallback loopLabel exit program atLoop base
      (base src) (base dst) ambient register

def transferAtInTime (src dst : K) (distinct : src ≠ dst)
    (f : Γ src → Γ dst) (fallback : Γ dst) (loopLabel : Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt Γ Λ (σ × Option (Γ dst)))
    (atLoop : program loopLabel = loopAt src dst f fallback loopLabel exit)
    (base : (k : K) → List (Γ k)) (ambient : σ) (register : Option (Γ dst)) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some loopLabel, (ambient, register), base⟩
      (some ⟨exit, (ambient, none),
        tapesAt src dst base [] ((base src).reverse.map f ++ base dst)⟩)
      ((base src).length + 1) where
  steps := (base src).length + 1
  evals_in_steps := transferAt_fromTapes src dst distinct f fallback loopLabel exit
    program atLoop base ambient register
  steps_le_m := Nat.le_refl _

end BinPackingGames.Reduction.MachineTransfer

end OAI