lean/OAI/Computability/BinPacking/Computation/MachineCanonicalOutput.lean

import Mathlib.Data.List.OfFn
import OAI.Computability.BinPacking.Computation.MachineFiniteAlphabet

namespace OAI

namespace BinPackingGames.Foundations.Complexity.MachineFiniteSequence

open Turing MachineComposition

variable {Command : Type} (LocalLabel : Command → Type)

def Label : List Command → Type
  | [] => Empty
  | command :: commands => LocalLabel command ⊕ Label commands

instance labelFintype [∀ command, Fintype (LocalLabel command)]
    (commands : List Command) : Fintype (Label LocalLabel commands) := by
  induction commands with
  | nil => exact inferInstanceAs (Fintype Empty)
  | cons command commands ih =>
      letI := ih
      exact inferInstanceAs (Fintype (LocalLabel command ⊕ Label LocalLabel commands))

instance labelDecidableEq [∀ command, DecidableEq (LocalLabel command)]
    (commands : List Command) : DecidableEq (Label LocalLabel commands) := by
  induction commands with
  | nil => exact inferInstanceAs (DecidableEq Empty)
  | cons command commands ih =>
      letI := ih
      exact inferInstanceAs (DecidableEq (LocalLabel command ⊕ Label LocalLabel commands))

variable (main : ∀ command, LocalLabel command)
variable {K Λ σ : Type} {Γ : K → Type}

def entry : (commands : List Command) → (Label LocalLabel commands → Λ) → Option Λ → Option Λ
  | [], _, exit => exit
  | command :: _, labels, _ => some (labels (.inl (main command)))

variable (localInstruction : ∀ command, (LocalLabel command → Λ) → Option Λ →
  LocalLabel command → TM2.Stmt Γ Λ σ)

def instruction : (commands : List Command) → (Label LocalLabel commands → Λ) → Option Λ →
    Label LocalLabel commands → TM2.Stmt Γ Λ σ
  | [], _, _, label => nomatch label
  | command :: commands, labels, exit, .inl label =>
      localInstruction command (fun l => labels (.inl l))
        (entry LocalLabel main commands (fun l => labels (.inr l)) exit) label
  | _ :: commands, labels, exit, .inr label =>
      instruction commands (fun l => labels (.inr l)) exit label

variable {Data : Type} (result : Command → Data → Data) (cost : Command → Data → Nat)

def resultOf : List Command → Data → Data
  | [], data => data
  | command :: commands, data => resultOf commands (result command data)

def steps : List Command → Data → Nat
  | [], _ => 0
  | command :: commands, data => cost command data + steps commands (result command data)

variable [DecidableEq K]

theorem trace (program : Λ → TM2.Stmt Γ Λ σ)
    (invariant : Data → Prop) (state : Data → σ) (tapes : Data → ∀ k, List (Γ k))
    (commands : List Command)
    (localInvariant : ∀ command ∈ commands, ∀ data, invariant data →
      invariant (result command data))
    (localTrace : ∀ command ∈ commands, ∀ (labels : LocalLabel command → Λ) (exit : Option Λ),
      (∀ l, program (labels l) = localInstruction command labels exit l) →
      ∀ data, invariant data →
      (advance (TM2.step program))^[cost command data]
        (some ⟨some (labels (main command)), state data, tapes data⟩) =
      some ⟨exit, state (result command data), tapes (result command data)⟩)
    (labels : Label LocalLabel commands → Λ) (exit : Option Λ)
    (atLabels : ∀ l, program (labels l) =
      instruction LocalLabel main localInstruction commands labels exit l)
    (data : Data) (valid : invariant data) :
    (advance (TM2.step program))^[steps result cost commands data]
      (some ⟨entry LocalLabel main commands labels exit, state data, tapes data⟩) =
    some ⟨exit, state (resultOf result commands data), tapes (resultOf result commands data)⟩ := by
  induction commands generalizing data with
  | nil => rfl
  | cons command commands ih =>
      have firstRun := localTrace command (by simp)
        (fun l => labels (.inl l))
        (entry LocalLabel main commands (fun l => labels (.inr l)) exit)
        (fun l => atLabels (.inl l)) data valid
      have nextValid := localInvariant command (by simp) data valid
      have tailRun := ih
        (fun c hc d hd => localInvariant c (by simp [hc]) d hd)
        (fun c hc => localTrace c (by simp [hc]))
        (fun l => labels (.inr l)) (fun l => atLabels (.inr l))
        (result command data) nextValid
      rw [steps, Nat.add_comm, Function.iterate_add_apply]
      change (advance (TM2.step program))^[steps result cost commands (result command data)]
        ((advance (TM2.step program))^[cost command data]
          (some ⟨some (labels (.inl (main command))), state data, tapes data⟩)) = _
      rw [firstRun]
      exact tailRun

end BinPackingGames.Foundations.Complexity.MachineFiniteSequence

namespace BinPackingGames.Foundations.Complexity.MachineDrainMany

open Turing MachineComposition

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

abbrev Alphabet (_ : K) := Bool
abbrev Tapes (K : Type) := K → List Bool
abbrev LocalLabel (_ : K) := Unit
abbrev Label (chosen : List K) := MachineFiniteSequence.Label (LocalLabel (K := K)) chosen
abbrev Data (K : Type) := Tapes K × Option Bool

def entry (chosen : List K) (labels : Label chosen → Λ) (exit : Option Λ) : Option Λ :=
  MachineFiniteSequence.entry (LocalLabel (K := K)) (fun _ => ()) chosen labels exit

def localInstruction (source : K) (labels : Unit → Λ) (exit : Option Λ) (_ : Unit) :
    TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool) :=
  MachineDrain.drain source (labels ()) exit

def instruction (chosen : List K) (labels : Label chosen → Λ) (exit : Option Λ) :
    Label chosen → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool) :=
  MachineFiniteSequence.instruction (LocalLabel (K := K)) (fun _ => ())
    localInstruction chosen labels exit

def result (source : K) (data : Data K) : Data K :=
  (Function.update data.1 source [], none)

def cost (source : K) (data : Data K) : Nat := (data.1 source).length + 1

def finalTapes : List K → Tapes K → Tapes K
  | [], base => base
  | source :: chosen, base => finalTapes chosen (Function.update base source [])

def finalRegister : List K → Option Bool → Option Bool
  | [], register => register
  | _ :: _, _ => none

omit [DecidableEq K] in
@[simp] theorem finalRegister_none (chosen : List K) : finalRegister chosen none = none := by
  cases chosen <;> rfl

def steps : List K → Tapes K → Nat
  | [], _ => 0
  | source :: chosen, base => (base source).length + 1 +
      steps chosen (Function.update base source [])

@[simp] theorem sequence_result (chosen : List K) (base : Tapes K) (register : Option Bool) :
    MachineFiniteSequence.resultOf result chosen (base, register) =
      (finalTapes chosen base, finalRegister chosen register) := by
  induction chosen generalizing base register with
  | nil => rfl
  | cons source chosen ih =>
      change MachineFiniteSequence.resultOf result chosen (Function.update base source [], none) =
        (finalTapes chosen (Function.update base source []), none)
      rw [ih, finalRegister_none]

@[simp] theorem sequence_steps (chosen : List K) (base : Tapes K) (register : Option Bool) :
    MachineFiniteSequence.steps result cost chosen (base, register) = steps chosen base := by
  induction chosen generalizing base register with
  | nil => rfl
  | cons source chosen ih =>
      simp only [MachineFiniteSequence.steps, result, cost, steps, ih]

theorem finalTapes_apply (chosen : List K) (base : Tapes K) (k : K) :
    finalTapes chosen base k = if k ∈ chosen then [] else base k := by
  induction chosen generalizing base with
  | nil => simp [finalTapes]
  | cons source chosen ih =>
      rw [finalTapes, ih]
      by_cases h : k = source
      · subst k
        simp
      · simp [List.mem_cons, h]

theorem finalTapes_mem (chosen : List K) (base : Tapes K) (k : K) (h : k ∈ chosen) :
    finalTapes chosen base k = [] := by rw [finalTapes_apply, ite_eq_left h]

theorem finalTapes_not_mem (chosen : List K) (base : Tapes K) (k : K) (h : k ∉ chosen) :
    finalTapes chosen base k = base k := by rw [finalTapes_apply, ite_eq_right h]

def lengthSum (chosen : List K) (base : Tapes K) : Nat :=
  (chosen.map (fun k => (base k).length)).sum

omit [DecidableEq K] in
theorem lengthSum_mono (chosen : List K) (first second : Tapes K)
    (h : ∀ k, (first k).length ≤ (second k).length) :
    lengthSum chosen first ≤ lengthSum chosen second := by
  induction chosen with
  | nil => exact Nat.le_refl 0
  | cons source chosen ih =>
      simp only [lengthSum, List.map_cons, List.sum_cons] at ih ⊢
      exact Nat.add_le_add (h source) ih

theorem steps_le (chosen : List K) (base : Tapes K) :
    steps chosen base ≤ lengthSum chosen base + chosen.length := by
  induction chosen generalizing base with
  | nil => simp [steps, lengthSum]
  | cons source chosen ih =>
      have hshort : ∀ k, ((Function.update base source []) k).length ≤ (base k).length := by
        intro k
        by_cases h : k = source
        · subst k
          simp
        · simp [Function.update_of_ne h]
      have hsum := lengthSum_mono chosen (Function.update base source []) base hshort
      have htail := ih (Function.update base source [])
      simp only [steps, lengthSum, List.map_cons, List.sum_cons, List.length_cons] at hsum htail ⊢
      omega

theorem steps_le_uniform (chosen : List K) (base : Tapes K) (bound : Nat)
    (h : ∀ k, (base k).length ≤ bound) :
    steps chosen base ≤ chosen.length * (bound + 1) := by
  have hsum : lengthSum chosen base ≤ chosen.length * bound := by
    induction chosen with
    | nil => simp [lengthSum]
    | cons source chosen ih =>
        simp only [lengthSum, List.map_cons, List.sum_cons, List.length_cons, Nat.add_mul,
          Nat.one_mul] at ih ⊢
        have hs := h source
        omega
  have hc := steps_le chosen base
  rw [Nat.mul_add, Nat.mul_one]
  omega

theorem trace (chosen : List K) (labels : Label chosen → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool))
    (atLabels : ∀ l, program (labels l) = instruction chosen labels exit l)
    (base : Tapes K) (ambient : σ) (register : Option Bool) :
    (advance (TM2.step program))^[steps chosen base]
      (some ⟨entry chosen labels exit, (ambient, register), base⟩) =
    some ⟨exit, (ambient, finalRegister chosen register), finalTapes chosen base⟩ := by
  have run := MachineFiniteSequence.trace
    (LocalLabel := LocalLabel (K := K)) (main := fun _ => ())
    (localInstruction := localInstruction (σ := σ)) (result := result) (cost := cost)
    program (fun _ : Data K => True) (fun data => (ambient, data.2)) (fun data => data.1)
    chosen (by intros; trivial)
    (by
      intro source _ localLabels localExit atLocal data _
      rcases data with ⟨tapes, reg⟩
      simpa only [cost, result, Function.update_eq_self] using
        MachineDrain.drainTrace source (localLabels ()) localExit program (atLocal ())
          tapes (tapes source) ambient reg)
    labels exit atLabels (base, register) trivial
  simpa only [sequence_steps, sequence_result, entry] using run

def execution (chosen : List K) (labels : Label chosen → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool))
    (atLabels : ∀ l, program (labels l) = instruction chosen labels exit l)
    (base : Tapes K) (ambient : σ) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨entry chosen labels exit, (ambient, register), base⟩
      (some ⟨exit, (ambient, finalRegister chosen register), finalTapes chosen base⟩)
      (lengthSum chosen base + chosen.length) where
  steps := steps chosen base
  evals_in_steps := trace chosen labels exit program atLabels base ambient register
  steps_le_m := steps_le chosen base

def workTapes {N : Nat} (enumeration : Fin N ≃ K) (output : K) : List K :=
  (List.ofFn (fun i => enumeration i)).filter (fun k => decide (k ≠ output))

@[simp] theorem mem_workTapes {N : Nat} (enumeration : Fin N ≃ K) (output k : K) :
    k ∈ workTapes enumeration output ↔ k ≠ output := by
  have hmem : k ∈ List.ofFn (fun i => enumeration i) := by
    apply List.mem_ofFn.mpr
    exact ⟨enumeration.symm k, enumeration.apply_symm_apply k⟩
  simp [workTapes, hmem]

theorem workTapes_length_le {N : Nat} (enumeration : Fin N ≃ K) (output : K) :
    (workTapes enumeration output).length ≤ N := by
  simpa only [workTapes, List.length_ofFn] using
    List.length_filter_le (fun k => decide (k ≠ output)) (List.ofFn (fun i => enumeration i))

def haltTapes (output : K) (word : List Bool) : Tapes K :=
  fun k => if k = output then word else []

theorem finalTapes_workTapes {N : Nat} (enumeration : Fin N ≃ K) (output : K)
    (base : Tapes K) :
    finalTapes (workTapes enumeration output) base = haltTapes output (base output) := by
  funext k
  rw [finalTapes_apply]
  by_cases h : k = output
  · subst k
    simp [haltTapes]
  · simp [haltTapes, h]

theorem cleanupTrace {N : Nat} (enumeration : Fin N ≃ K) (output : K)
    (labels : Label (workTapes enumeration output) → Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool))
    (atLabels : ∀ l, program (labels l) =
      instruction (workTapes enumeration output) labels none l)
    (base : Tapes K) (ambient : σ) :
    (advance (TM2.step program))^[steps (workTapes enumeration output) base]
      (some ⟨entry (workTapes enumeration output) labels none, (ambient, none), base⟩) =
    some ⟨none, (ambient, none), haltTapes output (base output)⟩ := by
  simpa only [finalRegister_none, finalTapes_workTapes] using
    trace (workTapes enumeration output) labels none program atLabels base ambient none

end BinPackingGames.Foundations.Complexity.MachineDrainMany

namespace BinPackingGames.Foundations.Complexity.PoweringMachineFinish

open Turing MachineComposition

variable {K Λ σ : Type} [DecidableEq K] {N : Nat}

abbrev Alphabet (_ : K) := Bool

abbrev Label (enumeration : Fin N ≃ K) (output : K) :=
  MachineUnaryAffineAt.Label ⊕ (MachineUnaryAffineAt.Label ⊕
    MachineDrainMany.Label (MachineDrainMany.workTapes enumeration output))

def copyInstruction (source scratch output : K)
    (labels : MachineUnaryAffineAt.Label → Λ) (exit : Option Λ) :
    MachineUnaryAffineAt.Label → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool)
  | .seed => MachineUnaryAffineAt.seed output 0 (labels .scan)
  | .scan => MachineUnaryAffineAt.scan source scratch output 1 (labels .scan) (labels .restore)
  | .restore => Reduction.MachineTransfer.loopAt scratch source id false (labels .restore) exit

def instruction (enumeration : Fin N ≃ K) (headerV headerD output scratch : K)
    (labels : Label enumeration output → Λ) :
    Label enumeration output → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool)
  | .inl l => copyInstruction headerD scratch output (fun q => labels (.inl q))
      (some (labels (.inr (.inl .seed)))) l
  | .inr (.inl l) => copyInstruction headerV scratch output
      (fun q => labels (.inr (.inl q)))
      (MachineDrainMany.entry (MachineDrainMany.workTapes enumeration output)
        (fun q => labels (.inr (.inr q))) none) l
  | .inr (.inr l) => MachineDrainMany.instruction
      (MachineDrainMany.workTapes enumeration output) (fun q => labels (.inr (.inr q))) none l

def program (enumeration : Fin N ≃ K) (headerV headerD output scratch : K) :
    Label enumeration output →
      TM2.Stmt (Alphabet (K := K)) (Label enumeration output) (σ × Option Bool) :=
  instruction enumeration headerV headerD output scratch id

def afterD (output : K) (base : K → List Bool) (m : Nat) : K → List Bool :=
  Function.update base output (encodeWord m ++ base output)

def afterHeaders (output : K) (base : K → List Bool) (n m : Nat) : K → List Bool :=
  Function.update base output (encodeWords [n, m] ++ base output)

@[simp] theorem afterHeaders_output (output : K) (base : K → List Bool) (n m : Nat) :
    afterHeaders output base n m output = encodeWords [n, m] ++ base output := by
  simp only [afterHeaders, Function.update_self]

theorem afterHeaders_other (output : K) (base : K → List Bool) (n m : Nat)
    (k : K) (hk : k ≠ output) : afterHeaders output base n m k = base k :=
  Function.update_of_ne hk _ _

def copySteps (n m : Nat) : Nat := (2 * (m + 1) + 1) + (2 * (n + 1) + 1)

theorem copySteps_eq (n m : Nat) : copySteps n m = 2 * (n + m) + 6 := by
  unfold copySteps
  omega

def steps (enumeration : Fin N ≃ K) (output : K) (base : K → List Bool) (n m : Nat) : Nat :=
  copySteps n m + MachineDrainMany.steps (MachineDrainMany.workTapes enumeration output)
    (afterHeaders output base n m)

def budget (enumeration : Fin N ≃ K) (output : K) (base : K → List Bool) (n m : Nat) : Nat :=
  copySteps n m + (MachineDrainMany.lengthSum (MachineDrainMany.workTapes enumeration output)
    (afterHeaders output base n m) + (MachineDrainMany.workTapes enumeration output).length)

private theorem trace_trans {α : Type*} (f : α → α) {a b : Nat} {x y z : α}
    (first : f^[a] x = y) (second : f^[b] y = z) : f^[a + b] x = z := by
  rw [Nat.add_comm, Function.iterate_add_apply, first, second]

theorem traceAt (enumeration : Fin N ≃ K) (headerV headerD output scratch : K)
    (hvs : headerV ≠ scratch) (hvo : headerV ≠ output)
    (hds : headerD ≠ scratch) (hdo : headerD ≠ output) (hso : scratch ≠ output)
    (labels : Label enumeration output → Λ)
    (target : Λ → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool))
    (atLabels : ∀ l, target (labels l) =
      instruction enumeration headerV headerD output scratch labels l)
    (base : K → List Bool) (n m : Nat) (suffixV suffixD : List Bool)
    (wordV : base headerV = encodeWord n ++ suffixV)
    (wordD : base headerD = encodeWord m ++ suffixD)
    (scratchEmpty : base scratch = []) (ambient : σ) (register : Option Bool) :
    (advance (TM2.step target))^[steps enumeration output base n m]
      (some ⟨some (labels (.inl .seed)), (ambient, register), base⟩) =
      some ⟨none, (ambient, none),
        MachineDrainMany.haltTapes output (encodeWords [n, m] ++ base output)⟩ := by
  have hd := MachineUnaryAffineAt.seededAffineTrace headerD scratch output hds hdo hso 1 0
    (labels (.inl .seed)) (labels (.inl .scan)) (labels (.inl .restore))
    (some (labels (.inr (.inl .seed)))) target
    (atLabels (.inl .seed)) (atLabels (.inl .scan)) (atLabels (.inl .restore))
    base m suffixD wordD scratchEmpty ambient register
  simp only [Nat.one_mul, Nat.add_zero] at hd
  change (advance (TM2.step target))^[2 * (m + 1) + 1]
    (some ⟨some (labels (.inl .seed)), (ambient, register), base⟩) =
    some ⟨some (labels (.inr (.inl .seed))), (ambient, none), afterD output base m⟩ at hd
  have wordV' : afterD output base m headerV = encodeWord n ++ suffixV := by
    simpa only [afterD, Function.update_of_ne hvo] using wordV
  have scratchEmpty' : afterD output base m scratch = [] := by
    simpa only [afterD, Function.update_of_ne hso] using scratchEmpty
  have hv := MachineUnaryAffineAt.seededAffineTrace headerV scratch output hvs hvo hso 1 0
    (labels (.inr (.inl .seed))) (labels (.inr (.inl .scan)))
    (labels (.inr (.inl .restore)))
    (MachineDrainMany.entry (MachineDrainMany.workTapes enumeration output)
      (fun q => labels (.inr (.inr q))) none)
    target (atLabels (.inr (.inl .seed))) (atLabels (.inr (.inl .scan)))
    (atLabels (.inr (.inl .restore))) (afterD output base m) n suffixV
    wordV' scratchEmpty' ambient none
  simp only [Nat.one_mul, Nat.add_zero] at hv
  have copied : Function.update (afterD output base m) output
      (encodeWord n ++ afterD output base m output) = afterHeaders output base n m := by
    simp only [afterD, afterHeaders, Function.update_self, Function.update_idem,
      encodeWords, List.append_nil, List.append_assoc]
  rw [copied] at hv
  have cleaned := MachineDrainMany.cleanupTrace enumeration output
    (fun q => labels (.inr (.inr q))) target (fun l => atLabels (.inr (.inr l)))
    (afterHeaders output base n m) ambient
  rw [afterHeaders_output] at cleaned
  have total := trace_trans _ (trace_trans _ hd hv) cleaned
  simpa only [steps, copySteps] using total

theorem trace (enumeration : Fin N ≃ K) (headerV headerD output scratch : K)
    (hvs : headerV ≠ scratch) (hvo : headerV ≠ output)
    (hds : headerD ≠ scratch) (hdo : headerD ≠ output) (hso : scratch ≠ output)
    (base : K → List Bool) (n m : Nat) (suffixV suffixD : List Bool)
    (wordV : base headerV = encodeWord n ++ suffixV)
    (wordD : base headerD = encodeWord m ++ suffixD)
    (scratchEmpty : base scratch = []) (ambient : σ) (register : Option Bool) :
    (advance (TM2.step (program enumeration headerV headerD output scratch)))^[
      steps enumeration output base n m]
      (some ⟨some (.inl .seed), (ambient, register), base⟩) =
      some ⟨none, (ambient, none),
        MachineDrainMany.haltTapes output (encodeWords [n, m] ++ base output)⟩ :=
  traceAt enumeration headerV headerD output scratch hvs hvo hds hdo hso id
    (program enumeration headerV headerD output scratch) (fun _ => rfl)
    base n m suffixV suffixD wordV wordD scratchEmpty ambient register

theorem steps_le_budget (enumeration : Fin N ≃ K) (output : K)
    (base : K → List Bool) (n m : Nat) :
    steps enumeration output base n m ≤ budget enumeration output base n m :=
  Nat.add_le_add_left (MachineDrainMany.steps_le
    (MachineDrainMany.workTapes enumeration output) (afterHeaders output base n m)) _

theorem afterHeaders_length_le (output : K) (base : K → List Bool) (n m bound : Nat)
    (bounded : ∀ k, (base k).length ≤ bound) (k : K) :
    (afterHeaders output base n m k).length ≤ bound + n + m + 2 := by
  by_cases hk : k = output
  · subst k
    rw [afterHeaders_output]
    have hb := bounded output
    simp only [List.length_append, encodeWords, encodeWord_length, List.length_nil]
    omega
  · rw [afterHeaders_other output base n m k hk]
    have hb := bounded k
    omega

theorem steps_le_uniform (enumeration : Fin N ≃ K) (output : K)
    (base : K → List Bool) (n m bound : Nat)
    (bounded : ∀ k, (base k).length ≤ bound) :
    steps enumeration output base n m ≤ 2 * (n + m) + 6 + N * (bound + n + m + 3) := by
  have hc := MachineDrainMany.steps_le_uniform (MachineDrainMany.workTapes enumeration output)
    (afterHeaders output base n m) (bound + n + m + 2)
    (afterHeaders_length_le output base n m bound bounded)
  have hl := MachineDrainMany.workTapes_length_le enumeration output
  have hm := Nat.mul_le_mul_right (bound + n + m + 3) hl
  rw [show bound + n + m + 2 + 1 = bound + n + m + 3 by omega] at hc
  unfold steps
  rw [copySteps_eq]
  exact Nat.add_le_add_left (hc.trans hm) _

def execution (enumeration : Fin N ≃ K) (headerV headerD output scratch : K)
    (hvs : headerV ≠ scratch) (hvo : headerV ≠ output)
    (hds : headerD ≠ scratch) (hdo : headerD ≠ output) (hso : scratch ≠ output)
    (base : K → List Bool) (n m : Nat) (suffixV suffixD : List Bool)
    (wordV : base headerV = encodeWord n ++ suffixV)
    (wordD : base headerD = encodeWord m ++ suffixD)
    (scratchEmpty : base scratch = []) (ambient : σ) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step (program enumeration headerV headerD output scratch))
      ⟨some (.inl .seed), (ambient, register), base⟩
      (some ⟨none, (ambient, none),
        MachineDrainMany.haltTapes output (encodeWords [n, m] ++ base output)⟩)
      (budget enumeration output base n m) where
  steps := steps enumeration output base n m
  evals_in_steps := trace enumeration headerV headerD output scratch hvs hvo hds hdo hso
    base n m suffixV suffixD wordV wordD scratchEmpty ambient register
  steps_le_m := steps_le_budget enumeration output base n m

end BinPackingGames.Foundations.Complexity.PoweringMachineFinish

namespace BinPackingGames.Foundations.Complexity.MachineCanonicalOutput

open Turing MachineComposition

structure Program (K Λ σ : Type) where
  input : K
  output : K
  main : Λ
  initial : σ
  code : Λ → TM2.Stmt (fun _ : K => Bool) Λ σ

section Structural

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

def extendedLabel (labels : Λ → Λ') (haltTarget : Option Λ') : Option Λ → Option Λ'
  | none => haltTarget
  | some label => some (labels label)

def extendedCfg (labels : Λ → Λ') (haltTarget : Option Λ') (register : Option Bool)
    (cfg : TM2.Cfg (fun _ : K => Bool) Λ σ) :
    TM2.Cfg (fun _ : K => Bool) Λ' (σ × Option Bool) :=
  ⟨extendedLabel labels haltTarget cfg.l, (cfg.var, register), cfg.stk⟩

def extendedStmt (labels : Λ → Λ') (haltTarget : Option Λ') :
    TM2.Stmt (fun _ : K => Bool) Λ σ →
      TM2.Stmt (fun _ : K => Bool) Λ' (σ × Option Bool)
  | .push k f next => .push k (fun state => f state.1)
      (extendedStmt labels haltTarget next)
  | .peek k f next => .peek k (fun state symbol => (f state.1 symbol, state.2))
      (extendedStmt labels haltTarget next)
  | .pop k f next => .pop k (fun state symbol => (f state.1 symbol, state.2))
      (extendedStmt labels haltTarget next)
  | .load f next => .load (fun state => (f state.1, state.2))
      (extendedStmt labels haltTarget next)
  | .branch test yes no => .branch (fun state => test state.1)
      (extendedStmt labels haltTarget yes) (extendedStmt labels haltTarget no)
  | .goto label => .goto (fun state => labels (label state.1))
  | .halt => match haltTarget with
    | none => .halt
    | some label => .goto (fun _ => label)

theorem stepAux_extended (labels : Λ → Λ') (haltTarget : Option Λ')
    (stmt : TM2.Stmt (fun _ : K => Bool) Λ σ) (state : σ)
    (register : Option Bool) (tapes : K → List Bool) :
    TM2.stepAux (extendedStmt labels haltTarget stmt) (state, register) tapes =
      extendedCfg labels haltTarget register (TM2.stepAux stmt state tapes) := by
  induction stmt generalizing state tapes with
  | push k f next ih => exact ih state (Function.update tapes k (f state :: tapes k))
  | peek k f next ih => exact ih (f state (tapes k).head?) tapes
  | pop k f next ih =>
      exact ih (f state (tapes k).head?) (Function.update tapes k (tapes k).tail)
  | load f next ih => exact ih (f state) tapes
  | branch test yes no ihYes ihNo =>
      cases h : test state with
      | false =>
          simpa only [extendedStmt, TM2.stepAux, h, Bool.cond_false] using ihNo state tapes
      | true =>
          simpa only [extendedStmt, TM2.stepAux, h, Bool.cond_true] using ihYes state tapes
  | goto label => rfl
  | halt => cases haltTarget <;> rfl

end Structural

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

def sourceMachine (P : Program K Λ σ) : FinTM2 where
  K := K
  k₀ := P.input
  k₁ := P.output
  Γ _ := Bool
  Λ := Λ
  main := P.main
  σ := σ
  initialState := P.initial
  m := P.code

abbrev Label (Λ : Type) (chosen : List K) := Λ ⊕ (MachineDrainMany.Label chosen ⊕ Unit)

def cleanupLabel (chosen : List K) : MachineDrainMany.Label chosen → Label Λ chosen :=
  fun label => .inr (.inl label)

def resetLabel (chosen : List K) : Label Λ chosen := .inr (.inr ())

def cleanupEntry (chosen : List K) : Option (Label Λ chosen) :=
  MachineDrainMany.entry chosen (cleanupLabel chosen) (some (resetLabel chosen))

def completedProgram (P : Program K Λ σ) (chosen : List K) :
    Label Λ chosen → TM2.Stmt (fun _ : K => Bool) (Label Λ chosen) (σ × Option Bool)
  | .inl label => extendedStmt Sum.inl (cleanupEntry chosen) (P.code label)
  | .inr (.inl label) => MachineDrainMany.instruction chosen (cleanupLabel chosen)
      (some (resetLabel chosen)) label
  | .inr (.inr _) => .load (fun _ => (P.initial, none)) .halt

def completedMachine (P : Program K Λ σ) (chosen : List K) : FinTM2 where
  K := K
  k₀ := P.input
  k₁ := P.output
  Γ _ := Bool
  Λ := Label Λ chosen
  main := .inl P.main
  σ := σ × Option Bool
  initialState := (P.initial, none)
  m := completedProgram P chosen

theorem finiteAlphabet (P : Program K Λ σ) (chosen : List K) :
    MachineFiniteAlphabet.FiniteAlphabet (completedMachine P chosen) := by
  intro k
  exact inferInstanceAs (Finite Bool)

def embeddedCfg (P : Program K Λ σ) (chosen : List K)
    (cfg : (sourceMachine P).Cfg) : (completedMachine P chosen).Cfg :=
  extendedCfg Sum.inl (cleanupEntry chosen) none cfg

theorem step_simulation (P : Program K Λ σ) (chosen : List K)
    (a b : (sourceMachine P).Cfg) (step : (sourceMachine P).step a = some b) :
    (completedMachine P chosen).step (embeddedCfg P chosen a) =
      some (embeddedCfg P chosen b) := by
  cases a with
  | mk label state tapes =>
    cases label with
    | none => simp [FinTM2.step, TM2.step] at step
    | some label =>
      have hb : TM2.stepAux (P.code label) state tapes = b := Option.some.inj step
      rw [← hb]
      change some (TM2.stepAux
        (extendedStmt Sum.inl (cleanupEntry chosen) (P.code label)) (state, none) tapes) = _
      erw [stepAux_extended]
      rfl

@[simp] theorem embedded_init (P : Program K Λ σ) (chosen : List K) (input : List Bool) :
    embeddedCfg P chosen (initList (sourceMachine P) input) =
      initList (completedMachine P chosen) input := rfl

omit [Fintype K] [Fintype Λ] [Fintype σ] in
theorem finalTapes_eq (P : Program K Λ σ) (chosen : List K)
    (complete : ∀ k, k ∈ chosen ↔ k ≠ P.output) (base : K → List Bool) :
    MachineDrainMany.finalTapes chosen base =
      MachineDrainMany.haltTapes P.output (base P.output) := by
  funext k
  rw [MachineDrainMany.finalTapes_apply]
  by_cases h : k = P.output
  · subst k
    simp [MachineDrainMany.haltTapes, complete]
  · simp [MachineDrainMany.haltTapes, complete, h]

theorem haltList_eq (P : Program K Λ σ) (chosen : List K) (output : List Bool) :
    haltList (completedMachine P chosen) output =
      ⟨none, (P.initial, none), MachineDrainMany.haltTapes P.output output⟩ := by
  congr 1

def cleanupExecution (P : Program K Λ σ) (chosen : List K)
    (complete : ∀ k, k ∈ chosen ↔ k ≠ P.output) (state : σ)
    (register : Option Bool) (base : K → List Bool) :
    StateTransition.EvalsToInTime (completedMachine P chosen).step
      ⟨cleanupEntry chosen, (state, register), base⟩
      (some (haltList (completedMachine P chosen) (base P.output)))
      (MachineDrainMany.steps chosen base + 1) := by
  let after : (completedMachine P chosen).Cfg :=
    ⟨some (resetLabel chosen), (state, MachineDrainMany.finalRegister chosen register),
      MachineDrainMany.finalTapes chosen base⟩
  have drains : StateTransition.EvalsToInTime (completedMachine P chosen).step
      ⟨cleanupEntry chosen, (state, register), base⟩ (some after)
      (MachineDrainMany.steps chosen base) := {
    steps := MachineDrainMany.steps chosen base
    evals_in_steps := MachineDrainMany.trace chosen (cleanupLabel chosen)
      (some (resetLabel chosen)) (completedProgram P chosen) (fun _ => rfl) base state register
    steps_le_m := le_rfl }
  have reset : StateTransition.EvalsToInTime (completedMachine P chosen).step
      after (some (haltList (completedMachine P chosen) (base P.output))) 1 := {
    steps := 1
    evals_in_steps := by
      change some (⟨none, (P.initial, none), MachineDrainMany.finalTapes chosen base⟩ :
          (completedMachine P chosen).Cfg) =
        some (haltList (completedMachine P chosen) (base P.output))
      erw [finalTapes_eq P chosen complete, haltList_eq]
      rfl
    steps_le_m := le_rfl }
  simpa only [Nat.add_comm] using
    StateTransition.EvalsToInTime.trans _ (MachineDrainMany.steps chosen base) 1
      _ after _ drains reset

def outputsInTime (P : Program K Λ σ) (chosen : List K)
    (complete : ∀ k, k ∈ chosen ↔ k ≠ P.output)
    (input output : List Bool) (state : σ) (base : K → List Bool) (budget : Nat)
    (raw : StateTransition.EvalsToInTime (sourceMachine P).step
      (initList (sourceMachine P) input) (some ⟨none, state, base⟩) budget)
    (correctOutput : base P.output = output) :
    TM2OutputsInTime (completedMachine P chosen) input (some output)
      (budget + chosen.length *
        (input.length + budget * Runtime.programPushBound (sourceMachine P) + 1) + 1) := by
  let lifted := liftExecutionInTime (sourceMachine P).step
    (completedMachine P chosen).step (embeddedCfg P chosen) (step_simulation P chosen) raw
  have joined := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ lifted
    (cleanupExecution P chosen complete state none base)
  have stackBound : ∀ k, (base k).length ≤
      input.length + budget * Runtime.programPushBound (sourceMachine P) := by
    intro k
    have h := Runtime.executionSizeBound (sourceMachine P).step
      (fun cfg => (cfg.stk k).length) (Runtime.programPushBound (sourceMachine P))
      (Runtime.stepStackLength (sourceMachine P) k) raw
    exact h.trans (Nat.add_le_add_right (Runtime.initialStackLength (sourceMachine P) input k) _)
  have cleanupBound := MachineDrainMany.steps_le_uniform chosen base _ stackBound
  rw [embedded_init, correctOutput] at joined
  exact { toEvalsTo := joined.toEvalsTo
          steps_le_m := joined.steps_le_m.trans (by omega) }

noncomputable def completedTime (P : Program K Λ σ) (chosen : List K)
    (rawTime : Polynomial Nat) : Polynomial Nat :=
  rawTime + Polynomial.C chosen.length *
    (Polynomial.X + rawTime * Polynomial.C (Runtime.programPushBound (sourceMachine P)) + 1) + 1

theorem completedTime_eval (P : Program K Λ σ) (chosen : List K)
    (rawTime : Polynomial Nat) (n : Nat) :
    (completedTime P chosen rawTime).eval n =
      rawTime.eval n + chosen.length *
        (n + rawTime.eval n * Runtime.programPushBound (sourceMachine P) + 1) + 1 := by
  simp [completedTime]

structure TerminalRun (P : Program K Λ σ) (input output : List Bool) (budget : Nat) where
  state : σ
  tapes : K → List Bool
  execution : StateTransition.EvalsToInTime (sourceMachine P).step
    (initList (sourceMachine P) input) (some ⟨none, state, tapes⟩) budget
  output_eq : tapes P.output = output

noncomputable def computableInPolyTime {α β : Type}
    (P : Program K Λ σ) (chosen : List K)
    (complete : ∀ k, k ∈ chosen ↔ k ≠ P.output)
    (encodeIn : α → List Bool) (encodeOut : β → List Bool) (f : α → β)
    (rawTime : Polynomial Nat)
    (run : ∀ a, TerminalRun P (encodeIn a) (encodeOut (f a))
      (rawTime.eval (encodeIn a).length)) :
    TM2ComputableInPolyTime encodeIn encodeOut f where
  tm := completedMachine P chosen
  inputAlphabet := Equiv.refl Bool
  outputAlphabet := Equiv.refl Bool
  time := completedTime P chosen rawTime
  outputsFun a := by
    change TM2OutputsInTime (completedMachine P chosen) ((encodeIn a).map id)
      (some ((encodeOut (f a)).map id)) ((completedTime P chosen rawTime).eval _)
    erw [List.map_id, List.map_id, completedTime_eval]
    exact outputsInTime P chosen complete _ _ (run a).state (run a).tapes _
      (run a).execution (run a).output_eq

theorem computableInPolyTime_finite_alphabet {α β : Type}
    (P : Program K Λ σ) (chosen : List K)
    (complete : ∀ k, k ∈ chosen ↔ k ≠ P.output)
    (encodeIn : α → List Bool) (encodeOut : β → List Bool) (f : α → β)
    (rawTime : Polynomial Nat)
    (run : ∀ a, TerminalRun P (encodeIn a) (encodeOut (f a))
      (rawTime.eval (encodeIn a).length)) :
    MachineFiniteAlphabet.FiniteAlphabet
      (computableInPolyTime P chosen complete encodeIn encodeOut f rawTime run).tm :=
  finiteAlphabet P chosen

noncomputable def computableInPolyTimeOfEnumeration {α β : Type} {N : Nat}
    (P : Program K Λ σ) (enumeration : Fin N ≃ K)
    (encodeIn : α → List Bool) (encodeOut : β → List Bool) (f : α → β)
    (rawTime : Polynomial Nat)
    (run : ∀ a, TerminalRun P (encodeIn a) (encodeOut (f a))
      (rawTime.eval (encodeIn a).length)) :
    TM2ComputableInPolyTime encodeIn encodeOut f :=
  computableInPolyTime P (MachineDrainMany.workTapes enumeration P.output)
    (fun k => MachineDrainMany.mem_workTapes enumeration P.output k)
    encodeIn encodeOut f rawTime run

end BinPackingGames.Foundations.Complexity.MachineCanonicalOutput

end OAI