lean/OAI/Computability/DirectedFeedback/Machines/MachineEmbedding.lean

import Mathlib

namespace OAI


namespace DFVSGames.Foundations.Complexity.MachineEmbedding

open Turing.TM2

variable {K E Λ Λextra σ τ : Type} {Γ : K → Type} {Δ : E → Type}

abbrev Alphabet (Γ : K → Type) (Δ : E → Type) : K ⊕ E → Type
  | .inl k => Γ k
  | .inr e => Δ e

def tapes (source : ∀ k, List (Γ k)) (extra : ∀ e, List (Δ e)) :
    ∀ j, List (Alphabet Γ Δ j)
  | .inl k => source k
  | .inr e => extra e

@[simp] theorem tapes_inl (source : ∀ k, List (Γ k)) (extra : ∀ e, List (Δ e))
    (k : K) : tapes source extra (.inl k) = source k := rfl

@[simp] theorem tapes_inr (source : ∀ k, List (Γ k)) (extra : ∀ e, List (Δ e))
    (e : E) : tapes source extra (.inr e) = extra e := rfl

def label (haltTarget : Option (Λ ⊕ Λextra)) : Option Λ → Option (Λ ⊕ Λextra)
  | none => haltTarget
  | some l => some (.inl l)

def configuration (haltTarget : Option (Λ ⊕ Λextra)) (extraState : τ)
    (extraTapes : ∀ e, List (Δ e)) (c : Cfg Γ Λ σ) :
    Cfg (Alphabet Γ Δ) (Λ ⊕ Λextra) (σ × τ) where
  l := label haltTarget c.l
  var := (c.var, extraState)
  stk := tapes c.stk extraTapes

def statement (haltTarget : Option (Λ ⊕ Λextra)) :
    Stmt Γ Λ σ → Stmt (Alphabet Γ Δ) (Λ ⊕ Λextra) (σ × τ)
  | .push k f next => .push (.inl k) (fun st => f st.1) (statement haltTarget next)
  | .peek k f next => .peek (.inl k) (fun st v => (f st.1 v, st.2))
      (statement haltTarget next)
  | .pop k f next => .pop (.inl k) (fun st v => (f st.1 v, st.2))
      (statement haltTarget next)
  | .load f next => .load (fun st => (f st.1, st.2)) (statement haltTarget next)
  | .branch f yes no => .branch (fun st => f st.1)
      (statement haltTarget yes) (statement haltTarget no)
  | .goto f => .goto (fun st => .inl (f st.1))
  | .halt => match haltTarget with
      | none => .halt
      | some l => .goto (fun _ => l)

variable [DecidableEq K] [DecidableEq E]

theorem tapes_update (source : ∀ k, List (Γ k)) (extra : ∀ e, List (Δ e))
    (k : K) (value : List (Γ k)) :
    tapes (Function.update source k value) extra =
      Function.update (tapes source extra) (.inl k) value := by
  funext j
  cases j with
  | inl j =>
    by_cases h : j = k
    · subst j
      simp [Function.update]
    · simp [Function.update, h]
  | inr e => simp [Function.update]

theorem stepAux_simulation (haltTarget : Option (Λ ⊕ Λextra))
    (extraState : τ) (extraTapes : ∀ e, List (Δ e))
    (q : Stmt Γ Λ σ) (state : σ) (source : ∀ k, List (Γ k)) :
    stepAux (statement haltTarget q) (state, extraState) (tapes source extraTapes) =
      configuration haltTarget extraState extraTapes (stepAux q state source) := by
  induction q generalizing state source with
  | push k f next ih =>
    simp only [statement, stepAux, tapes_inl]
    rw [← tapes_update]
    exact ih state (Function.update source k (f state :: source k))
  | peek k f next ih =>
    simpa only [statement, stepAux, tapes_inl] using
      ih (f state (source k).head?) source
  | pop k f next ih =>
    simp only [statement, stepAux, tapes_inl]
    rw [← tapes_update]
    exact ih (f state (source k).head?) (Function.update source k (source k).tail)
  | load f next ih =>
    simpa only [statement, stepAux] using ih (f state) source
  | branch f yes no ihYes ihNo =>
    cases h : f state with
    | false => simpa only [statement, stepAux, h, Bool.cond_false] using ihNo state source
    | true => simpa only [statement, stepAux, h, Bool.cond_true] using ihYes state source
  | goto f => rfl
  | halt => cases haltTarget <;> rfl

theorem stepAux_preserves_extra_state (haltTarget : Option (Λ ⊕ Λextra))
    (extraState : τ) (extraTapes : ∀ e, List (Δ e))
    (q : Stmt Γ Λ σ) (state : σ) (source : ∀ k, List (Γ k)) :
    (stepAux (statement haltTarget q) (state, extraState)
      (tapes source extraTapes)).var.2 = extraState := by
  rw [stepAux_simulation]
  rfl

theorem stepAux_preserves_extra_tape (haltTarget : Option (Λ ⊕ Λextra))
    (extraState : τ) (extraTapes : ∀ e, List (Δ e))
    (q : Stmt Γ Λ σ) (state : σ) (source : ∀ k, List (Γ k)) (e : E) :
    (stepAux (statement haltTarget q) (state, extraState)
      (tapes source extraTapes)).stk (.inr e) = extraTapes e := by
  rw [stepAux_simulation]
  rfl

def program (haltTarget : Option (Λ ⊕ Λextra)) (source : Λ → Stmt Γ Λ σ)
    (extra : Λextra → Stmt (Alphabet Γ Δ) (Λ ⊕ Λextra) (σ × τ)) :
    Λ ⊕ Λextra → Stmt (Alphabet Γ Δ) (Λ ⊕ Λextra) (σ × τ)
  | .inl l => statement haltTarget (source l)
  | .inr l => extra l

theorem step_running (haltTarget : Option (Λ ⊕ Λextra))
    (extraState : τ) (extraTapes : ∀ e, List (Δ e))
    (source : Λ → Stmt Γ Λ σ)
    (extra : Λextra → Stmt (Alphabet Γ Δ) (Λ ⊕ Λextra) (σ × τ))
    (l : Λ) (state : σ) (sourceTapes : ∀ k, List (Γ k)) :
    step (program haltTarget source extra)
      (configuration haltTarget extraState extraTapes ⟨some l, state, sourceTapes⟩) =
      some (configuration haltTarget extraState extraTapes
        (stepAux (source l) state sourceTapes)) := by
  change some (stepAux (statement haltTarget (source l)) (state, extraState)
    (tapes sourceTapes extraTapes)) = _
  rw [stepAux_simulation]

theorem step_simulation (haltTarget : Option (Λ ⊕ Λextra))
    (extraState : τ) (extraTapes : ∀ e, List (Δ e))
    (source : Λ → Stmt Γ Λ σ)
    (extra : Λextra → Stmt (Alphabet Γ Δ) (Λ ⊕ Λextra) (σ × τ))
    (a b : Cfg Γ Λ σ) (h : step source a = some b) :
    step (program haltTarget source extra)
      (configuration haltTarget extraState extraTapes a) =
      some (configuration haltTarget extraState extraTapes b) := by
  cases a with
  | mk l state sourceTapes =>
    cases l with
    | none => simp [step] at h
    | some l =>
      have hb : stepAux (source l) state sourceTapes = b := Option.some.inj h
      rw [← hb]
      exact step_running haltTarget extraState extraTapes source extra l state sourceTapes

end DFVSGames.Foundations.Complexity.MachineEmbedding


namespace DFVSGames.Foundations.Complexity.MachineComposition

def advance {σ : Type} (step : σ → Option σ) (state : Option σ) : Option σ :=
  state.bind step

@[simp] theorem advance_none {σ : Type} (step : σ → Option σ) :
    advance step none = none := rfl

@[simp] theorem advance_some {σ : Type} (step : σ → Option σ) (state : σ) :
    advance step (some state) = step state := rfl

@[simp] theorem advance_iterate_none {σ : Type} (step : σ → Option σ) (n : Nat) :
    (advance step)^[n] none = none := by
  induction n with
  | zero => rfl
  | succ n ih => rw [Function.iterate_succ_apply', ih, advance_none]

theorem liftSuccessfulTrace {σ τ : Type} (source : σ → Option σ)
    (target : τ → Option τ) (embed : σ → τ)
    (simulation : ∀ a b, source a = some b → target (embed a) = some (embed b))
    (n : Nat) (start finish : σ)
    (trace : (advance source)^[n] (some start) = some finish) :
    (advance target)^[n] (some (embed start)) = some (embed finish) := by
  induction n generalizing start with
  | zero =>
      have same := Option.some.inj trace
      cases same
      rfl
  | succ n ih =>
      rw [Function.iterate_succ_apply] at trace ⊢
      change (advance source)^[n] (source start) = some finish at trace
      change (advance target)^[n] (target (embed start)) = some (embed finish)
      cases firstStep : source start with
      | none =>
          rw [firstStep, advance_iterate_none] at trace
          contradiction
      | some intermediate =>
          rw [firstStep] at trace
          rw [simulation start intermediate firstStep]
          exact ih intermediate trace

def liftExecutionInTime {σ τ : Type} (source : σ → Option σ)
    (target : τ → Option τ) (embed : σ → τ)
    (simulation : ∀ a b, source a = some b → target (embed a) = some (embed b))
    {start finish : σ} {budget : Nat}
    (execution : StateTransition.EvalsToInTime source start (some finish) budget) :
    StateTransition.EvalsToInTime target (embed start) (some (embed finish)) budget where
  steps := execution.steps
  evals_in_steps := by
    have sourceTrace := execution.evals_in_steps
    change (advance source)^[execution.steps] (some start) = some finish at sourceTrace
    change (advance target)^[execution.steps] (some (embed start)) = some (embed finish)
    exact liftSuccessfulTrace source target embed simulation execution.steps start finish sourceTrace
  steps_le_m := execution.steps_le_m

@[simp] theorem liftExecutionInTime_steps {σ τ : Type} (source : σ → Option σ)
    (target : τ → Option τ) (embed : σ → τ)
    (simulation : ∀ a b, source a = some b → target (embed a) = some (embed b))
    {start finish : σ} {budget : Nat}
    (execution : StateTransition.EvalsToInTime source start (some finish) budget) :
    (liftExecutionInTime source target embed simulation execution).steps = execution.steps := rfl

def embeddedExecution {K E Λ Λextra σ τ : Type} {Γ : K → Type} {Δ : E → Type}
    [DecidableEq K] [DecidableEq E]
    (haltTarget : Option (Λ ⊕ Λextra)) (extraState : τ)
    (extraTapes : ∀ e, List (Δ e))
    (source : Λ → Turing.TM2.Stmt Γ Λ σ)
    (extra : Λextra → Turing.TM2.Stmt (MachineEmbedding.Alphabet Γ Δ)
      (Λ ⊕ Λextra) (σ × τ))
    {start finish : Turing.TM2.Cfg Γ Λ σ} {budget : Nat}
    (execution : StateTransition.EvalsToInTime (Turing.TM2.step source)
      start (some finish) budget) :
    StateTransition.EvalsToInTime (Turing.TM2.step (MachineEmbedding.program haltTarget source extra))
      (MachineEmbedding.configuration haltTarget extraState extraTapes start)
      (some (MachineEmbedding.configuration haltTarget extraState extraTapes finish)) budget :=
  liftExecutionInTime (Turing.TM2.step source)
    (Turing.TM2.step (MachineEmbedding.program haltTarget source extra))
    (MachineEmbedding.configuration haltTarget extraState extraTapes)
    (MachineEmbedding.step_simulation haltTarget extraState extraTapes source extra) execution

theorem natPolynomial_eval_mono (p : Polynomial Nat) {a b : Nat} (h : a ≤ b) :
    p.eval a ≤ p.eval b := by
  induction p using Polynomial.induction_on' with
  | add p q hp hq =>
      simpa only [Polynomial.eval_add] using Nat.add_le_add hp hq
  | monomial degree coefficient =>
      simp only [Polynomial.eval_monomial]
      exact Nat.mul_le_mul_left coefficient (Nat.pow_le_pow_left h degree)

noncomputable def compositionPolynomial (first second intermediate : Polynomial Nat) : Polynomial Nat :=
  first + Polynomial.C 2 * (intermediate + 1) + second.comp intermediate

theorem compositionBudget_le (first second intermediate : Polynomial Nat)
    (inputLength intermediateLength : Nat)
    (intermediateBound : intermediateLength ≤ intermediate.eval inputLength) :
    first.eval inputLength + 2 * (intermediateLength + 1) + second.eval intermediateLength ≤
      (compositionPolynomial first second intermediate).eval inputLength := by
  have secondBound := natPolynomial_eval_mono second intermediateBound
  have transferBound := Nat.mul_le_mul_left 2 (Nat.add_le_add_right intermediateBound 1)
  simp only [compositionPolynomial, Polynomial.eval_add, Polynomial.eval_mul,
    Polynomial.eval_C, Polynomial.eval_one, Polynomial.eval_comp]
  omega

def fourPhaseExecution {σ : Type} (step : σ → Option σ)
    (start afterFirst afterTransfer₁ afterTransfer₂ : σ) (finish : Option σ)
    (first second intermediate : Polynomial Nat) (inputLength intermediateLength : Nat)
    (intermediateBound : intermediateLength ≤ intermediate.eval inputLength)
    (runFirst : StateTransition.EvalsToInTime step start (some afterFirst) (first.eval inputLength))
    (transfer₁ : StateTransition.EvalsToInTime step afterFirst (some afterTransfer₁)
      (intermediateLength + 1))
    (transfer₂ : StateTransition.EvalsToInTime step afterTransfer₁ (some afterTransfer₂)
      (intermediateLength + 1))
    (runSecond : StateTransition.EvalsToInTime step afterTransfer₂ finish
      (second.eval intermediateLength)) :
    StateTransition.EvalsToInTime step start finish
      ((compositionPolynomial first second intermediate).eval inputLength) := by
  let firstTwo := StateTransition.EvalsToInTime.trans step _ _ start afterFirst
    (some afterTransfer₁) runFirst transfer₁
  let firstThree := StateTransition.EvalsToInTime.trans step _ _ start afterTransfer₁
    (some afterTransfer₂) firstTwo transfer₂
  let allFour := StateTransition.EvalsToInTime.trans step _ _ start afterTransfer₂
    finish firstThree runSecond
  refine {
    toEvalsTo := allFour.toEvalsTo
    steps_le_m := Nat.le_trans allFour.steps_le_m ?_
  }
  have bounded := compositionBudget_le first second intermediate inputLength intermediateLength
    intermediateBound
  omega

end DFVSGames.Foundations.Complexity.MachineComposition


namespace DFVSGames.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_inline_MachineTransfer (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_inline_MachineTransfer (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_inline_MachineTransfer]
  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_inline_MachineTransfer, update_output_inline_MachineTransfer]
  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
  calc
    _ = (next f fallback)^[input.length + 1]
        (some (running f fallback input [] none)) :=
      congrArg _ (congrArg some (initList_eq f fallback input))
    _ = some (halted f fallback (input.reverse.map f)) := by
      simpa only [List.append_nil] using transfer_steps f fallback input [] none
    _ = _ := congrArg some (haltList_eq f fallback (input.reverse.map f)).symm

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)
    simpa only [machine, List.map_id, 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_inline_MachineTransfer (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_inline_MachineTransfer (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_inline_MachineTransfer]

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_inline_MachineTransfer, update_tapesAt_dst_inline_MachineTransfer]

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 DFVSGames.Reduction.MachineTransfer


namespace DFVSGames.Foundations.Complexity.MachineCopy

open Turing

variable {K Λ σ β : Type} [DecidableEq K]

abbrev Alphabet (β : Type) (_ : K) := β

def forkLoop (source left right : K) (fallback : β) (loopLabel : Λ) (exit : Option Λ) :
    TM2.Stmt (Alphabet (K := K) β) Λ (σ × Option β) :=
  .pop source (fun state head => (state.1, head))
    (.branch (fun state => state.2.isSome)
      (.push left (fun state => state.2.getD fallback)
        (.push right (fun state => state.2.getD fallback) (.goto fun _ => loopLabel)))
      (Reduction.MachineTransfer.exitAt right exit))

def forkTapes (source left right : K) (base : K → List β)
    (input leftOutput rightOutput : List β) : K → List β :=
  Function.update (Function.update (Function.update base source input) left leftOutput)
    right rightOutput

@[simp] theorem forkTapes_source (source left right : K)
    (sourceLeft : source ≠ left) (sourceRight : source ≠ right)
    (base : K → List β) (input leftOutput rightOutput : List β) :
    forkTapes source left right base input leftOutput rightOutput source = input := by
  simp [forkTapes, sourceLeft, sourceRight]

@[simp] theorem forkTapes_left (source left right : K) (leftRight : left ≠ right)
    (base : K → List β) (input leftOutput rightOutput : List β) :
    forkTapes source left right base input leftOutput rightOutput left = leftOutput := by
  simp [forkTapes, leftRight]

@[simp] theorem forkTapes_right (source left right : K)
    (base : K → List β) (input leftOutput rightOutput : List β) :
    forkTapes source left right base input leftOutput rightOutput right = rightOutput := by
  simp [forkTapes]

@[simp] theorem forkTapes_self (source left right : K) (base : K → List β) :
    forkTapes source left right base (base source) (base left) (base right) = base := by
  simp [forkTapes]

private theorem update_fork_source_inline_MachineCopy (source left right : K)
    (sourceLeft : source ≠ left) (sourceRight : source ≠ right) (leftRight : left ≠ right)
    (base : K → List β) (input leftOutput rightOutput replacement : List β) :
    Function.update (forkTapes source left right base input leftOutput rightOutput)
      source replacement = forkTapes source left right base replacement leftOutput rightOutput := by
  funext k
  by_cases hs : k = source
  · subst k; simp [forkTapes, sourceLeft, sourceRight]
  · by_cases hl : k = left
    · subst k; simp [forkTapes, Ne.symm sourceLeft, leftRight]
    · by_cases hr : k = right
      · subst k; simp [forkTapes, Ne.symm sourceRight]
      · simp [forkTapes, hs, hl, hr]

private theorem update_fork_left_inline_MachineCopy (source left right : K) (leftRight : left ≠ right)
    (base : K → List β) (input leftOutput rightOutput replacement : List β) :
    Function.update (forkTapes source left right base input leftOutput rightOutput)
      left replacement = forkTapes source left right base input replacement rightOutput := by
  funext k
  by_cases hl : k = left
  · subst k; simp [forkTapes, leftRight]
  · by_cases hr : k = right
    · subst k; simp [forkTapes, Ne.symm leftRight]
    · simp [forkTapes, hl, hr]

private theorem update_fork_right_inline_MachineCopy (source left right : K)
    (base : K → List β) (input leftOutput rightOutput replacement : List β) :
    Function.update (forkTapes source left right base input leftOutput rightOutput)
      right replacement = forkTapes source left right base input leftOutput replacement := by
  simp [forkTapes]

theorem forkStep_empty (source left right : K)
    (sourceLeft : source ≠ left) (sourceRight : source ≠ right) (leftRight : left ≠ right)
    (fallback : β) (loopLabel : Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K) β) Λ (σ × Option β))
    (atLoop : program loopLabel = forkLoop source left right fallback loopLabel exit)
    (base : K → List β) (leftOutput rightOutput : List β) (ambient : σ) (register : Option β) :
    TM2.step program
      ⟨some loopLabel, (ambient, register), forkTapes source left right base [] leftOutput rightOutput⟩ =
        some ⟨exit, (ambient, none), forkTapes source left right base [] leftOutput rightOutput⟩ := by
  change some (TM2.stepAux (program loopLabel) (ambient, register)
    (forkTapes source left right base [] leftOutput rightOutput)) = _
  rw [atLoop]
  cases exit <;>
    simp [forkLoop, Reduction.MachineTransfer.exitAt, TM2.stepAux, sourceLeft, sourceRight,
      leftRight, update_fork_source_inline_MachineCopy]

theorem forkStep_cons (source left right : K)
    (sourceLeft : source ≠ left) (sourceRight : source ≠ right) (leftRight : left ≠ right)
    (fallback : β) (loopLabel : Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K) β) Λ (σ × Option β))
    (atLoop : program loopLabel = forkLoop source left right fallback loopLabel exit)
    (base : K → List β) (head : β) (input leftOutput rightOutput : List β)
    (ambient : σ) (register : Option β) :
    TM2.step program
      ⟨some loopLabel, (ambient, register),
        forkTapes source left right base (head :: input) leftOutput rightOutput⟩ =
      some ⟨some loopLabel, (ambient, some head),
        forkTapes source left right base input (head :: leftOutput) (head :: rightOutput)⟩ := by
  change some (TM2.stepAux (program loopLabel) (ambient, register)
    (forkTapes source left right base (head :: input) leftOutput rightOutput)) = _
  rw [atLoop]
  simp [forkLoop, TM2.stepAux, sourceLeft, sourceRight, leftRight,
    update_fork_source_inline_MachineCopy, update_fork_left_inline_MachineCopy, update_fork_right_inline_MachineCopy]

theorem forkTrace (source left right : K)
    (sourceLeft : source ≠ left) (sourceRight : source ≠ right) (leftRight : left ≠ right)
    (fallback : β) (loopLabel : Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K) β) Λ (σ × Option β))
    (atLoop : program loopLabel = forkLoop source left right fallback loopLabel exit)
    (base : K → List β) (input leftOutput rightOutput : List β)
    (ambient : σ) (register : Option β) :
    (MachineComposition.advance (TM2.step program))^[input.length + 1]
      (some ⟨some loopLabel, (ambient, register),
        forkTapes source left right base input leftOutput rightOutput⟩) =
      some ⟨exit, (ambient, none), forkTapes source left right base []
        (input.reverse ++ leftOutput) (input.reverse ++ rightOutput)⟩ := by
  induction input generalizing leftOutput rightOutput register with
  | nil =>
    simpa only [List.length_nil, Nat.zero_add, Function.iterate_one,
      MachineComposition.advance_some, List.reverse_nil, List.nil_append] using
        forkStep_empty source left right sourceLeft sourceRight leftRight fallback loopLabel exit
          program atLoop base leftOutput rightOutput ambient register
  | cons head input ih =>
    rw [List.length_cons, Function.iterate_succ_apply]
    change (MachineComposition.advance (TM2.step program))^[input.length + 1]
      (TM2.step program ⟨some loopLabel, (ambient, register),
        forkTapes source left right base (head :: input) leftOutput rightOutput⟩) = _
    rw [forkStep_cons source left right sourceLeft sourceRight leftRight fallback loopLabel exit
      program atLoop base head input leftOutput rightOutput ambient register, ih]
    simp only [List.reverse_cons, List.append_assoc, List.singleton_append]

def forkInTime (source left right : K)
    (sourceLeft : source ≠ left) (sourceRight : source ≠ right) (leftRight : left ≠ right)
    (fallback : β) (loopLabel : Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K) β) Λ (σ × Option β))
    (atLoop : program loopLabel = forkLoop source left right fallback loopLabel exit)
    (base : K → List β) (ambient : σ) (register : Option β) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some loopLabel, (ambient, register), base⟩
      (some ⟨exit, (ambient, none), forkTapes source left right base []
        ((base source).reverse ++ base left) ((base source).reverse ++ base right)⟩)
      ((base source).length + 1) where
  steps := (base source).length + 1
  evals_in_steps := by
    change (MachineComposition.advance (TM2.step program))^[(base source).length + 1]
      (some ⟨some loopLabel, (ambient, register), base⟩) = _
    simpa only [forkTapes_self] using
      forkTrace source left right sourceLeft sourceRight leftRight fallback loopLabel exit
        program atLoop base (base source) (base left) (base right) ambient register
  steps_le_m := Nat.le_refl _

def drainedTapes (source scratch : K) (base : K → List β) : K → List β :=
  Reduction.MachineTransfer.tapesAt source scratch base [] (base source).reverse

theorem restoredTapes (source destination scratch : K)
    (sourceDestination : source ≠ destination) (sourceScratch : source ≠ scratch)
    (destinationScratch : destination ≠ scratch) (base : K → List β)
    (scratchEmpty : base scratch = []) :
    forkTapes scratch source destination (drainedTapes source scratch base) []
      (base source) (base source ++ base destination) =
        Function.update base destination (base source ++ base destination) := by
  funext k
  by_cases hs : k = source
  · subst k
    simp [forkTapes, sourceDestination]
  · by_cases hd : k = destination
    · subst k; simp [forkTapes]
    · by_cases ht : k = scratch
      · subst k
        simp [forkTapes, Ne.symm sourceScratch, Ne.symm destinationScratch, scratchEmpty]
      · simp [forkTapes, drainedTapes, Reduction.MachineTransfer.tapesAt, hs, hd, ht]

theorem copyTrace (source destination scratch : K)
    (sourceDestination : source ≠ destination) (sourceScratch : source ≠ scratch)
    (destinationScratch : destination ≠ scratch)
    (fallback : β) (firstLabel secondLabel : Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K) β) Λ (σ × Option β))
    (atFirst : program firstLabel =
      Reduction.MachineTransfer.loopAt source scratch id fallback firstLabel (some secondLabel))
    (atSecond : program secondLabel = forkLoop scratch source destination fallback secondLabel exit)
    (base : K → List β) (scratchEmpty : base scratch = [])
    (ambient : σ) (register : Option β) :
    (MachineComposition.advance (TM2.step program))^[2 * ((base source).length + 1)]
      (some ⟨some firstLabel, (ambient, register), base⟩) =
      some ⟨exit, (ambient, none),
        Function.update base destination (base source ++ base destination)⟩ := by
  have first := Reduction.MachineTransfer.transferAt_fromTapes source scratch sourceScratch
    id fallback firstLabel (some secondLabel) program atFirst base ambient register
  change (MachineComposition.advance (TM2.step program))^[(base source).length + 1]
    (some ⟨some firstLabel, (ambient, register), base⟩) = _ at first
  simp only [List.map_id, scratchEmpty, List.append_nil] at first
  have second := forkTrace scratch source destination (Ne.symm sourceScratch)
    (Ne.symm destinationScratch) sourceDestination fallback secondLabel exit program atSecond
    (drainedTapes source scratch base)
    (drainedTapes source scratch base scratch) (drainedTapes source scratch base source)
    (drainedTapes source scratch base destination) ambient none
  simp only [forkTapes_self] at second
  have scratchWord : drainedTapes source scratch base scratch = (base source).reverse := by
    simp [drainedTapes]
  have sourceWord : drainedTapes source scratch base source = [] := by
    simp [drainedTapes, sourceScratch]
  have destinationWord : drainedTapes source scratch base destination = base destination := by
    simp [drainedTapes, Reduction.MachineTransfer.tapesAt, Ne.symm sourceDestination,
      destinationScratch]
  rw [scratchWord, sourceWord, destinationWord] at second
  simp only [List.reverse_reverse, List.append_nil, List.length_reverse] at second
  rw [restoredTapes source destination scratch sourceDestination sourceScratch destinationScratch
    base scratchEmpty] at second
  rw [show 2 * ((base source).length + 1) =
    ((base source).length + 1) + ((base source).length + 1) by omega,
    Function.iterate_add_apply, first]
  exact second

def copyInTime (source destination scratch : K)
    (sourceDestination : source ≠ destination) (sourceScratch : source ≠ scratch)
    (destinationScratch : destination ≠ scratch)
    (fallback : β) (firstLabel secondLabel : Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K) β) Λ (σ × Option β))
    (atFirst : program firstLabel =
      Reduction.MachineTransfer.loopAt source scratch id fallback firstLabel (some secondLabel))
    (atSecond : program secondLabel = forkLoop scratch source destination fallback secondLabel exit)
    (base : K → List β) (scratchEmpty : base scratch = [])
    (ambient : σ) (register : Option β) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some firstLabel, (ambient, register), base⟩
      (some ⟨exit, (ambient, none),
        Function.update base destination (base source ++ base destination)⟩)
      (2 * ((base source).length + 1)) where
  steps := 2 * ((base source).length + 1)
  evals_in_steps := copyTrace source destination scratch sourceDestination sourceScratch
    destinationScratch fallback firstLabel secondLabel exit program atFirst atSecond base
    scratchEmpty ambient register
  steps_le_m := Nat.le_refl _

end DFVSGames.Foundations.Complexity.MachineCopy

end OAI