lean/OAI/Computability/BinPacking/Computation/GraphCounterStart.lean

import Mathlib.Tactic.FinCases
import OAI.Computability.BinPacking.Computation.PoweringMasterState

namespace OAI

namespace BinPackingGames.Foundations.Complexity.MachineLogCounter

open Turing

def bitLength (n : Nat) : Nat := if n = 0 then 0 else n.log2 + 1

@[simp] theorem bitLength_zero : bitLength 0 = 0 := rfl

theorem bitLength_positive {n : Nat} (h : n ≠ 0) : bitLength n = n.log2 + 1 := by
  simp [bitLength, h]

theorem bitLength_half {n : Nat} (h : n ≠ 0) : bitLength n = bitLength (n / 2) + 1 := by
  by_cases hone : n = 1
  · subst n; rfl
  have hn : 2 ≤ n := by omega
  have hh : n / 2 ≠ 0 := by omega
  rw [bitLength_positive h, bitLength_positive hh, Nat.log2_def n, ite_eq_left hn]

def increments (n : Nat) (started : Bool) : Nat :=
  if n = 0 then (if started then 0 else 1) else bitLength n

@[simp] theorem increments_true (n : Nat) : increments n true = bitLength n := by
  by_cases hn : n = 0 <;> simp [increments, hn]

theorem increments_half {n : Nat} (h : n ≠ 0) (started : Bool) :
    increments n started = increments (n / 2) true + 1 := by
  rw [increments, ite_eq_right h, increments_true, bitLength_half h]

theorem increments_false (n : Nat) : increments n false = n.log2 + 1 := by
  by_cases hn : n = 0
  · subst n; rfl
  · simp [increments, bitLength, hn]

def coreTime (n : Nat) : Nat :=
  if h : n = 0 then 1 else n + n / 2 + 3 + coreTime (n / 2)
termination_by n
decreasing_by omega

@[simp] theorem coreTime_zero : coreTime 0 = 1 := by rw [coreTime]; rfl

theorem coreTime_positive {n : Nat} (h : n ≠ 0) :
    coreTime n = n + n / 2 + 3 + coreTime (n / 2) := by
  rw [coreTime, dite_eq_right h]

theorem coreTime_bound (n : Nat) : coreTime n ≤ 6 * n + 1 := by
  induction n using Nat.strong_induction_on with
  | h n ih =>
    by_cases hn : n = 0
    · subst n; simp
    · rw [coreTime_positive hn]
      have smaller : n / 2 < n := by omega
      have bound := ih (n / 2) smaller
      omega

abbrev Alphabet (_ : Fin 3) := Bool
abbrev State := (Bool × Bool) × Option Bool

def initialState : State := ((false, false), none)

def rawTapes (input scratch output : List Bool) : Fin 3 → List Bool
  | 0 => input
  | 1 => scratch
  | 2 => output

private theorem update_input (input scratch output replacement : List Bool) :
    Function.update (rawTapes input scratch output) 0 replacement = rawTapes replacement scratch output := by
  funext k
  fin_cases k <;> rfl

private theorem update_scratch (input scratch output replacement : List Bool) :
    Function.update (rawTapes input scratch output) 1 replacement = rawTapes input replacement output := by
  funext k
  fin_cases k <;> rfl

private theorem update_output (input scratch output replacement : List Bool) :
    Function.update (rawTapes input scratch output) 2 replacement = rawTapes input scratch replacement := by
  funext k
  fin_cases k <;> rfl

def stripLoop : TM2.Stmt Alphabet (Fin 5) State :=
  .pop 0 (fun state head => (state.1, head))
    (.branch (fun state => state.2.getD false)
      (.push 1 (fun _ => true) (.goto fun _ => (1 : Fin 5)))
      (.load (fun _ => initialState) (.goto fun _ => 4)))

def guardLoop : TM2.Stmt Alphabet (Fin 5) State :=
  .peek 0 (fun state head => (state.1, head))
    (.branch (fun state => state.2.isSome)
      (.push 2 (fun _ => true)
        (.load (fun _ => ((false, true), none)) (.goto fun _ => 3)))
      (.branch (fun state => state.1.2)
        (.load (fun _ => initialState) .halt)
        (.push 2 (fun _ => true) (.load (fun _ => initialState) .halt))))

def halfLoop : TM2.Stmt Alphabet (Fin 5) State :=
  .pop 0 (fun state head => (state.1, head))
    (.branch (fun state => state.2.isSome)
      (.branch (fun state => state.1.1)
        (.push 1 (fun _ => true)
          (.load (fun state => ((false, state.1.2), state.2)) (.goto fun _ => 3)))
        (.load (fun state => ((true, state.1.2), state.2)) (.goto fun _ => 3)))
      (.load (fun state => ((false, state.1.2), none)) (.goto fun _ => 4)))

def program : Fin 5 → TM2.Stmt Alphabet (Fin 5) State
  | 0 => .push 2 (fun _ => false) (.goto fun _ => 1)
  | 1 => stripLoop
  | 2 => guardLoop
  | 3 => halfLoop
  | 4 => Reduction.MachineTransfer.loopAt 1 0 id false 4 (some 2)

def machine : FinTM2 where
  K := Fin 3
  k₀ := 0
  k₁ := 2
  Γ := Alphabet
  Λ := Fin 5
  main := 0
  σ := State
  initialState := initialState
  m := program

def configuration (label : Option (Fin 5)) (state : State)
    (input scratch output : List Bool) : machine.Cfg :=
  ⟨label, state, rawTapes input scratch output⟩

def next := MachineComposition.advance machine.step

theorem stripStep_true (input scratch output : List Bool) (state : State) :
    machine.step (configuration (some 1) state (true :: input) scratch output) =
      some (configuration (some 1) (state.1, some true) input (true :: scratch) output) := by
  change some (TM2.stepAux stripLoop state (rawTapes (true :: input) scratch output)) = _
  simp [stripLoop, TM2.stepAux, rawTapes, configuration]
  rw [update_input, update_scratch]
  rfl

theorem stripStep_false (scratch output : List Bool) (state : State) :
    machine.step (configuration (some 1) state [false] scratch output) =
      some (configuration (some 4) initialState [] scratch output) := by
  change some (TM2.stepAux stripLoop state (rawTapes [false] scratch output)) = _
  simp [stripLoop, TM2.stepAux, rawTapes, configuration]
  rw [update_input]
  rfl

theorem stripTrace (n : Nat) (scratch output : List Bool) (state : State) :
    next^[n + 1] (some (configuration (some 1) state (encodeWord n) scratch output)) =
      some (configuration (some 4) initialState [] (List.replicate n true ++ scratch) output) := by
  induction n generalizing scratch state with
  | zero =>
    simpa only [encodeWord, List.replicate_zero, List.nil_append, Nat.zero_add,
      Function.iterate_one, next, MachineComposition.advance_some] using
      stripStep_false scratch output state
  | succ n ih =>
    rw [Function.iterate_succ_apply]
    change next^[n + 1]
      (machine.step (configuration (some 1) state (true :: encodeWord n) scratch output)) = _
    rw [stripStep_true, ih]
    congr 2
    simp only [List.replicate_add, List.replicate_one, List.append_assoc, List.singleton_append]

theorem halfStep_nil (scratch output : List Bool) (parity started : Bool) (register : Option Bool) :
    machine.step (configuration (some 3) ((parity, started), register) [] scratch output) =
      some (configuration (some 4) ((false, started), none) [] scratch output) := by
  change some (TM2.stepAux halfLoop ((parity, started), register) (rawTapes [] scratch output)) = _
  simp [halfLoop, TM2.stepAux, rawTapes, configuration]
  rw [update_input]
  rfl

theorem halfStep_false (input scratch output : List Bool) (started : Bool) (register : Option Bool) :
    machine.step (configuration (some 3) ((false, started), register) (true :: input) scratch output) =
      some (configuration (some 3) ((true, started), some true) input scratch output) := by
  change some (TM2.stepAux halfLoop ((false, started), register)
    (rawTapes (true :: input) scratch output)) = _
  simp [halfLoop, TM2.stepAux, rawTapes, configuration]
  rw [update_input]
  rfl

theorem halfStep_true (input scratch output : List Bool) (started : Bool) (register : Option Bool) :
    machine.step (configuration (some 3) ((true, started), register) (true :: input) scratch output) =
      some (configuration (some 3) ((false, started), some true) input (true :: scratch) output) := by
  change some (TM2.stepAux halfLoop ((true, started), register)
    (rawTapes (true :: input) scratch output)) = _
  simp [halfLoop, TM2.stepAux, rawTapes, configuration]
  rw [update_input, update_scratch]
  rfl

theorem halfTrace (n m : Nat) (output : List Bool) (parity started : Bool)
    (register : Option Bool) :
    next^[n + 1] (some (configuration (some 3) ((parity, started), register)
      (List.replicate n true) (List.replicate m true) output)) =
      some (configuration (some 4) ((false, started), none) []
        (List.replicate (m + (n + if parity then 1 else 0) / 2) true) output) := by
  induction n generalizing m parity register with
  | zero =>
    cases parity <;>
      simpa only [Nat.zero_add, Function.iterate_one, next, MachineComposition.advance_some,
        List.replicate_zero, Bool.false_eq_true, ite_false, ite_true, Nat.zero_div,
        Nat.add_zero, Nat.reduceDiv] using
        halfStep_nil (List.replicate m true) output _ started register
  | succ n ih =>
    rw [Function.iterate_succ_apply]
    change next^[n + 1]
      (machine.step (configuration (some 3) ((parity, started), register)
        (true :: List.replicate n true) (List.replicate m true) output)) = _
    cases parity with
    | false =>
      rw [halfStep_false, ih]
      congr 2
    | true =>
      rw [halfStep_true]
      change next^[n + 1] (some (configuration (some 3) ((false, started), some true)
        (List.replicate n true) (List.replicate (m + 1) true) output)) = _
      rw [ih]
      congr 2
      congr 1
      change m + 1 + n / 2 = m + (n + 1 + 1) / 2
      omega

theorem restoreTrace (n : Nat) (output : List Bool) (started : Bool) :
    next^[n + 1] (some (configuration (some 4) ((false, started), none)
      [] (List.replicate n true) output)) =
      some (configuration (some 2) ((false, started), none) (List.replicate n true) [] output) := by
  have run := Reduction.MachineTransfer.transferAt_fromTapes (1 : Fin 3) 0 (by decide)
    id false (4 : Fin 5) (some 2) program rfl
    (rawTapes [] (List.replicate n true) output) (false, started) none
  simp only [rawTapes, List.length_replicate, List.reverse_replicate, List.map_id,
    List.append_nil] at run
  have tapesEq : Reduction.MachineTransfer.tapesAt (1 : Fin 3) 0
      (rawTapes [] (List.replicate n true) output) [] (List.replicate n true) =
      rawTapes (List.replicate n true) [] output := by
    funext k
    fin_cases k <;> rfl
  rw [tapesEq] at run
  exact run

theorem guardStep_zero (output : List Bool) (started : Bool) :
    machine.step (configuration (some 2) ((false, started), none) [] [] output) =
      some (configuration none initialState [] [] (if started then output else true :: output)) := by
  change some (TM2.stepAux guardLoop ((false, started), none) (rawTapes [] [] output)) = _
  cases started <;> simp [guardLoop, TM2.stepAux, rawTapes, configuration]
  rw [update_output]
  all_goals rfl

theorem guardStep_positive (n : Nat) (output : List Bool) (started : Bool) :
    machine.step (configuration (some 2) ((false, started), none)
      (List.replicate (n + 1) true) [] output) =
      some (configuration (some 3) ((false, true), none)
        (List.replicate (n + 1) true) [] (true :: output)) := by
  change some (TM2.stepAux guardLoop ((false, started), none)
    (rawTapes (true :: List.replicate n true) [] output)) = _
  simp [guardLoop, TM2.stepAux, rawTapes, configuration]
  rw [update_output]
  simp only [List.replicate_succ]
  rfl

theorem coreTrace (n : Nat) (output : List Bool) (started : Bool) :
    next^[coreTime n] (some (configuration (some 2) ((false, started), none)
      (List.replicate n true) [] output)) =
      some (configuration none initialState [] []
        (List.replicate (increments n started) true ++ output)) := by
  induction n using Nat.strong_induction_on generalizing output started with
  | h n ih =>
    cases n with
    | zero =>
      cases started with
      | false => simpa [increments, next] using! guardStep_zero output false
      | true => simpa [increments, next] using! guardStep_zero output true
    | succ n =>
      have nonzero : n + 1 ≠ 0 := by omega
      have smaller : (n + 1) / 2 < n + 1 := by omega
      have recursive := ih ((n + 1) / 2) smaller (true :: output) true
      have half := halfTrace (n + 1) 0 (true :: output) false true none
      simp only [Bool.false_eq_true, ↓reduceIte, Nat.add_zero, Nat.zero_add,
        List.replicate_zero] at half
      have restore := restoreTrace ((n + 1) / 2) (true :: output) true
      rw [coreTime_positive nonzero]
      rw [show n + 1 + (n + 1) / 2 + 3 + coreTime ((n + 1) / 2) =
        (coreTime ((n + 1) / 2) + ((n + 1) / 2 + 1) + (n + 1 + 1)) + 1 by omega,
        Function.iterate_succ_apply]
      change next^[coreTime ((n + 1) / 2) + ((n + 1) / 2 + 1) + (n + 1 + 1)]
        (machine.step (configuration (some 2) ((false, started), none)
          (List.replicate (n + 1) true) [] output)) = _
      rw [guardStep_positive, Function.iterate_add_apply, half,
        Function.iterate_add_apply, restore, recursive]
      rw [increments_half nonzero started]
      simp only [List.replicate_add, List.replicate_one, List.append_assoc, List.singleton_append]

theorem initList_eq (input : List Bool) :
    initList machine input = configuration (some 0) initialState input [] [] := by
  unfold initList configuration
  congr 1
  funext k
  change Fin 3 at k
  fin_cases k <;> rfl

theorem haltList_eq (output : List Bool) :
    haltList machine output = configuration none initialState [] [] output := by
  unfold haltList configuration
  congr 1
  funext k
  change Fin 3 at k
  fin_cases k <;> rfl

theorem initialStep (input : List Bool) :
    machine.step (configuration (some 0) initialState input [] []) =
      some (configuration (some 1) initialState input [] [false]) := by
  change some (TM2.stepAux (.push (2 : Fin 3) (fun _ : State => false) (.goto fun _ => (1 : Fin 5)))
    initialState (rawTapes input [] [])) = _
  simp only [TM2.stepAux, rawTapes]
  rw [update_output]
  rfl

def totalTime (n : Nat) : Nat := 2 * n + 3 + coreTime n

theorem totalTime_bound (n : Nat) : totalTime n ≤ 8 * n + 4 := by
  have bound := coreTime_bound n
  unfold totalTime
  omega

theorem machineTrace (n : Nat) :
    next^[totalTime n] (some (initList machine (encodeWord n))) =
      some (haltList machine (encodeWord (n.log2 + 1))) := by
  rw [initList_eq, haltList_eq]
  rw [show totalTime n = (coreTime n + (n + 1) + (n + 1)) + 1 by
    unfold totalTime; omega, Function.iterate_succ_apply]
  change next^[coreTime n + (n + 1) + (n + 1)]
    (machine.step (configuration (some 0) initialState (encodeWord n) [] [])) = _
  rw [initialStep, Function.iterate_add_apply, stripTrace]
  simp only [List.append_nil]
  rw [Function.iterate_add_apply]
  dsimp only [initialState]
  rw [restoreTrace n [false] false, coreTrace, increments_false]
  rfl

def outputsInTime (n : Nat) :
    TM2OutputsInTime machine (encodeWord n) (some (encodeWord (n.log2 + 1))) (8 * n + 4) where
  steps := totalTime n
  evals_in_steps := machineTrace n
  steps_le_m := totalTime_bound n

noncomputable def computableInPolyTime :
    TM2ComputableInPolyTime encodeWord encodeWord (fun n => n.log2 + 1) where
  tm := machine
  inputAlphabet := Equiv.refl Bool
  outputAlphabet := Equiv.refl Bool
  time := 8 * Polynomial.X + 4
  outputsFun n := by
    change TM2OutputsInTime machine ((encodeWord n).map id)
      (some ((encodeWord (n.log2 + 1)).map id))
      ((8 * Polynomial.X + 4 : Polynomial Nat).eval (encodeWord n).length)
    have hi := @List.map_id (machine.Γ machine.k₀) (encodeWord n)
    have ho := @List.map_id (machine.Γ machine.k₁) (encodeWord (n.log2 + 1))
    rw [hi, ho]
    have execution := outputsInTime n
    refine {
      toEvalsTo := execution.toEvalsTo
      steps_le_m := Nat.le_trans execution.steps_le_m ?_
    }
    simp only [Polynomial.eval_add, Polynomial.eval_mul, Polynomial.eval_ofNat, Polynomial.eval_X]
    simp [encodeWord]

end BinPackingGames.Foundations.Complexity.MachineLogCounter

namespace BinPackingGames.Foundations.Complexity.GraphCounterModel

open Turing

inductive ExtraTape where
  | input | archive | scratch | output
  deriving DecidableEq

protected abbrev ExtraTape.enumList : List ExtraTape := [.input, .archive, .scratch, .output]

protected theorem ExtraTape.enumList_getElem?_ctorIdx_eq (x : ExtraTape) :
    ExtraTape.enumList[x.ctorIdx]? = some x := by
  cases x <;> rfl

protected theorem ExtraTape.enumList_nodup : ExtraTape.enumList.Nodup := by decide

instance : Fintype ExtraTape where
  elems := ⟨ExtraTape.enumList, ExtraTape.enumList_nodup⟩
  complete x := by cases x <;> decide

inductive ExtraLabel where
  | copyFirst | copySecond | seed | headerFirst | headerSecond
  | clearInput | clockCopy | archiveCopy | finalReverse
  deriving DecidableEq

protected abbrev ExtraLabel.enumList : List ExtraLabel := [.copyFirst, .copySecond, .seed,
  .headerFirst, .headerSecond, .clearInput, .clockCopy, .archiveCopy, .finalReverse]

protected theorem ExtraLabel.enumList_getElem?_ctorIdx_eq (x : ExtraLabel) :
    ExtraLabel.enumList[x.ctorIdx]? = some x := by
  cases x <;> rfl

protected theorem ExtraLabel.enumList_nodup : ExtraLabel.enumList.Nodup := by decide

instance : Fintype ExtraLabel where
  elems := ⟨ExtraLabel.enumList, ExtraLabel.enumList_nodup⟩
  complete x := by cases x <;> decide

abbrev Tape := Fin 3 ⊕ ExtraTape
abbrev Label := Fin 5 ⊕ ExtraLabel
abbrev Alphabet (_ : Tape) := Bool
abbrev State := MachineLogCounter.State × Option Bool

def initialState : State := (MachineLogCounter.initialState, none)

def clockLabel : Option (Fin 5) → Option Label
  | none => some (.inr .clearInput)
  | some l => some (.inl l)

def clockStatement : TM2.Stmt MachineLogCounter.Alphabet (Fin 5) MachineLogCounter.State →
    TM2.Stmt Alphabet Label State
  | .push k f next => .push (.inl k) (fun state => f state.1) (clockStatement next)
  | .peek k f next => .peek (.inl k) (fun state x => (f state.1 x, state.2)) (clockStatement next)
  | .pop k f next => .pop (.inl k) (fun state x => (f state.1 x, state.2)) (clockStatement next)
  | .load f next => .load (fun state => (f state.1, state.2)) (clockStatement next)
  | .branch f yes no => .branch (fun state => f state.1) (clockStatement yes) (clockStatement no)
  | .goto f => .goto (fun state => .inl (f state.1))
  | .halt => .goto (fun _ => .inr .clearInput)

def readHeader (again next : Label) : TM2.Stmt Alphabet Label State :=
  .pop (.inr .input) (fun state head => (state.1, head))
    (.branch (fun state => state.2.getD false)
      (.push (.inl 0) (fun _ => true) (.goto fun _ => again))
      (.load (fun state => (state.1, none)) (.goto fun _ => next)))

def program : Label → TM2.Stmt Alphabet Label State
  | .inl l => clockStatement (MachineLogCounter.program l)
  | .inr .copyFirst => Reduction.MachineTransfer.loopAt (.inr .input) (.inr .scratch)
      id false (.inr .copyFirst) (some (.inr .copySecond))
  | .inr .copySecond => MachineCopy.forkLoop (.inr .scratch) (.inr .input) (.inr .archive)
      false (.inr .copySecond) (some (.inr .seed))
  | .inr .seed => .push (.inl 0) (fun _ => false) (.goto fun _ => .inr .headerFirst)
  | .inr .headerFirst => readHeader (.inr .headerFirst) (.inr .headerSecond)
  | .inr .headerSecond => readHeader (.inr .headerSecond) (.inl 0)
  | .inr .clearInput => MachineDrain.drain (.inr .input) (.inr .clearInput)
      (some (.inr .clockCopy))
  | .inr .clockCopy => Reduction.MachineTransfer.loopAt (.inl 2) (.inr .scratch)
      id false (.inr .clockCopy) (some (.inr .archiveCopy))
  | .inr .archiveCopy => Reduction.MachineTransfer.loopAt (.inr .archive) (.inr .scratch)
      id false (.inr .archiveCopy) (some (.inr .finalReverse))
  | .inr .finalReverse => Reduction.MachineTransfer.loopAt (.inr .scratch) (.inr .output)
      id false (.inr .finalReverse) none

def machine : FinTM2 where
  K := Tape
  k₀ := .inr .input
  k₁ := .inr .output
  Γ := Alphabet
  Λ := Label
  main := .inr .copyFirst
  σ := State
  initialState := initialState
  m := program

def clockTapes (tapes : Fin 3 → List Bool) (extra : ExtraTape → List Bool) : Tape → List Bool
  | .inl k => tapes k
  | .inr k => extra k

def clockConfiguration (extra : ExtraTape → List Bool) (register : Option Bool)
    (c : TM2.Cfg MachineLogCounter.Alphabet (Fin 5) MachineLogCounter.State) :
    TM2.Cfg Alphabet Label State := ⟨clockLabel c.l, (c.var, register), clockTapes c.stk extra⟩

theorem clockTapes_update (tapes : Fin 3 → List Bool) (extra : ExtraTape → List Bool)
    (k : Fin 3) (word : List Bool) :
    clockTapes (Function.update tapes k word) extra =
      Function.update (clockTapes tapes extra) (.inl k) word := by
  funext tape
  cases tape <;> simp [clockTapes, Function.comp_def]

theorem clockStatement_simulation (extra : ExtraTape → List Bool) (register : Option Bool)
    (statement : TM2.Stmt MachineLogCounter.Alphabet (Fin 5) MachineLogCounter.State)
    (state : MachineLogCounter.State) (tapes : Fin 3 → List Bool) :
    TM2.stepAux (clockStatement statement) (state, register) (clockTapes tapes extra) =
      clockConfiguration extra register (TM2.stepAux statement state tapes) := by
  induction statement generalizing state tapes with
  | push k f next ih =>
      simp only [clockStatement, TM2.stepAux, clockTapes]
      rw [← clockTapes_update]
      exact ih state (Function.update tapes k (f state :: tapes k))
  | peek k f next ih =>
      simpa only [clockStatement, TM2.stepAux, clockTapes] using ih (f state (tapes k).head?) tapes
  | pop k f next ih =>
      simp only [clockStatement, TM2.stepAux, clockTapes]
      rw [← clockTapes_update]
      exact ih (f state (tapes k).head?) (Function.update tapes k (tapes k).tail)
  | load f next ih =>
      simpa only [clockStatement, TM2.stepAux] using ih (f state) tapes
  | branch f yes no ihYes ihNo =>
      cases h : f state with
      | false => simpa only [clockStatement, TM2.stepAux, h, Bool.cond_false] using ihNo state tapes
      | true => simpa only [clockStatement, TM2.stepAux, h, Bool.cond_true] using ihYes state tapes
  | goto f => rfl
  | halt => rfl

theorem clockStep (extra : ExtraTape → List Bool) (register : Option Bool)
    (a b : MachineLogCounter.machine.Cfg)
    (h : MachineLogCounter.machine.step a = some b) :
    TM2.step program (clockConfiguration extra register a) =
      some (clockConfiguration extra register b) := by
  cases a with
  | mk label state tapes =>
      cases label with
      | none => cases h
      | some label =>
          have hb := Option.some.inj h
          subst b
          exact congrArg some (clockStatement_simulation extra register
            (MachineLogCounter.program label) state tapes)

def clockInTime (extra : ExtraTape → List Bool) (register : Option Bool) (n : Nat) :
    StateTransition.EvalsToInTime (TM2.step program)
      (clockConfiguration extra register (initList MachineLogCounter.machine (encodeWord n)))
      (some (clockConfiguration extra register
        (haltList MachineLogCounter.machine (encodeWord (n.log2 + 1))))) (8 * n + 4) :=
  MachineComposition.liftExecutionInTime _ _ (clockConfiguration extra register)
    (clockStep extra register) (MachineLogCounter.outputsInTime n)

end BinPackingGames.Foundations.Complexity.GraphCounterModel

namespace BinPackingGames.Foundations.Complexity.PCPIterationMachine

open Turing
open MachineCountedLoop

def rounds (n : Nat) : Nat := n.log2 + 1

theorem rounds_le (n : Nat) : rounds n ≤ n + 1 :=
  Nat.add_le_add_right (Nat.log2_le_self n) 1

theorem two_pow_rounds_le (n : Nat) : 2 ^ rounds n ≤ 2 * (n + 1) := by
  by_cases hn : n = 0
  · subst n; decide
  · have h := Nat.log2_self_le hn
    simp only [rounds, pow_succ]
    omega

theorem growth_pow_rounds_le (growth n : Nat) :
    growth ^ rounds n ≤ (2 * (n + 1)) ^ growth := by
  calc
    growth ^ rounds n ≤ (2 ^ growth) ^ rounds n :=
      Nat.pow_le_pow_left (Nat.le_of_lt (Nat.lt_two_pow_self (n := growth))) _
    _ = (2 ^ rounds n) ^ growth := by simp only [← pow_mul, Nat.mul_comm]
    _ ≤ (2 * (n + 1)) ^ growth := Nat.pow_le_pow_left (two_pow_rounds_le n) growth

def sizeEnvelope (growth n : Nat) : Nat := (n + 1) * (2 * (n + 1)) ^ growth

theorem semanticSize_geometric (sizes : Nat → Nat) (growth n : Nat)
    (initialBound : sizes 0 ≤ n + 1)
    (growthBound : ∀ i, i < rounds n → sizes (i + 1) ≤ growth * sizes i)
    (i : Nat) (hi : i ≤ rounds n) : sizes i ≤ (n + 1) * growth ^ i := by
  induction i with
  | zero => simpa only [pow_zero, Nat.mul_one] using initialBound
  | succ i ih =>
    have ilt : i < rounds n := by omega
    calc
      sizes (i + 1) ≤ growth * sizes i := growthBound i ilt
      _ ≤ growth * ((n + 1) * growth ^ i) :=
        Nat.mul_le_mul_left growth (ih (by omega))
      _ = (n + 1) * growth ^ (i + 1) := by rw [pow_succ]; ac_rfl

theorem semanticSize_bound (sizes : Nat → Nat) (growth n : Nat)
    (growthPositive : 0 < growth)
    (initialBound : sizes 0 ≤ n + 1)
    (growthBound : ∀ i, i < rounds n → sizes (i + 1) ≤ growth * sizes i)
    (i : Nat) (hi : i ≤ rounds n) : sizes i ≤ sizeEnvelope growth n := by
  calc
    sizes i ≤ (n + 1) * growth ^ i :=
      semanticSize_geometric sizes growth n initialBound growthBound i hi
    _ ≤ (n + 1) * growth ^ rounds n :=
      Nat.mul_le_mul_left _ (Nat.pow_le_pow_right growthPositive hi)
    _ ≤ sizeEnvelope growth n :=
      Nat.mul_le_mul_left _ (growth_pow_rounds_le growth n)

noncomputable def sizePolynomial (growth : Nat) : Polynomial Nat :=
  (Polynomial.X + 1) * (2 * (Polynomial.X + 1)) ^ growth

@[simp] theorem sizePolynomial_eval (growth n : Nat) :
    (sizePolynomial growth).eval n = sizeEnvelope growth n := by
  simp [sizePolynomial, sizeEnvelope]

noncomputable def timePolynomial (growth : Nat)
    (encodingSize bodyTime : Polynomial Nat) : Polynomial Nat :=
  (Polynomial.X + 1) *
    ((bodyTime.comp encodingSize).comp (sizePolynomial growth) + 1) + 2

@[simp] theorem timePolynomial_eval (growth n : Nat)
    (encodingSize bodyTime : Polynomial Nat) :
    (timePolynomial growth encodingSize bodyTime).eval n =
      (n + 1) * (bodyTime.eval (encodingSize.eval (sizeEnvelope growth n)) + 1) + 2 := by
  simp [timePolynomial]

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

abbrev Alphabet (_ : K) := Bool
abbrev State (σ : Type) := σ × Option Bool
abbrev Labels (Λ : Type) := Bool ⊕ Λ

def program (counter : K) (entry : Λ)
    (body : Λ → TM2.Stmt (Alphabet (K := K)) (Labels Λ) (State σ)) :
    Labels Λ → TM2.Stmt (Alphabet (K := K)) (Labels Λ) (State σ)
  | .inl false => MachineUnaryCounter.guard counter (.inr entry) (.inl true)
  | .inl true => .pop counter (fun state _ => (state.1, none)) .halt
  | .inr label => body label

omit [DecidableEq K] in
@[simp] theorem program_guard (counter : K) (entry : Λ)
    (body : Λ → TM2.Stmt (Alphabet (K := K)) (Labels Λ) (State σ)) :
    program counter entry body (.inl false) =
      MachineUnaryCounter.guard counter (.inr entry) (.inl true) := rfl

def machine [Fintype K] [Fintype Λ] [Fintype σ]
    (counter output : K) (entry : Λ) (initial : σ)
    (body : Λ → TM2.Stmt (Alphabet (K := K)) (Labels Λ) (State σ)) : FinTM2 where
  K := K
  k₀ := counter
  k₁ := output
  Γ := Alphabet
  Λ := Labels Λ
  main := .inl false
  σ := State σ
  initialState := (initial, none)
  m := program counter entry body

def haltedConfiguration (counter : K) (suffix : List Bool)
    (ambient : Nat → σ) (base : Nat → K → List Bool) :
    TM2.Cfg (Alphabet (K := K)) (Labels Λ) (State σ) :=
  ⟨none, (ambient 0, none), Function.update (base 0) counter suffix⟩

theorem haltStep (counter : K) (entry : Λ)
    (body : Λ → TM2.Stmt (Alphabet (K := K)) (Labels Λ) (State σ))
    (suffix : List Bool) (ambient : Nat → σ) (base : Nat → K → List Bool) :
    TM2.step (program counter entry body)
      (exitConfiguration counter (.inl true) suffix ambient base) =
      some (haltedConfiguration counter suffix ambient base) := by
  change some (TM2.stepAux (.pop counter (fun state _ => (state.1, none)) .halt)
    (ambient 0, none) (MachineUnaryCounter.counterTapes counter (base 0) 0 suffix)) = _
  simp [TM2.stepAux, MachineUnaryCounter.counterTapes, encodeWord, haltedConfiguration]

theorem iterationTrace (counter : K) (entry : Λ)
    (body : Λ → TM2.Stmt (Alphabet (K := K)) (Labels Λ) (State σ))
    (suffix : List Bool) (ambient : Nat → σ) (register : Nat → Option Bool)
    (base : Nat → K → List Bool) (cost : Nat → Nat) (n : Nat)
    (bodyTraces : BodyTraces counter (.inl false) (.inr entry)
      (program counter entry body) suffix ambient register base cost (rounds n)) :
    (MachineComposition.advance (TM2.step (program counter entry body)))^[
        totalSteps cost (rounds n) + 1]
      (some (guardConfiguration counter (.inl false) suffix ambient register base (rounds n))) =
      some (haltedConfiguration counter suffix ambient base) := by
  rw [Function.iterate_succ_apply']
  rw [loopTrace counter (.inl false) (.inr entry) (.inl true)
    (program counter entry body) (program_guard counter entry body) suffix
    ambient register base cost (rounds n) bodyTraces]
  exact haltStep counter entry body suffix ambient base

theorem bodyCosts_bound (growth n : Nat) (growthPositive : 0 < growth)
    (sizes encodedSizes cost : Nat → Nat) (encodingSize bodyTime : Polynomial Nat)
    (initialBound : sizes 0 ≤ n + 1)
    (growthBound : ∀ i, i < rounds n → sizes (i + 1) ≤ growth * sizes i)
    (encodingBound : ∀ r, r < rounds n →
      encodedSizes r ≤ encodingSize.eval (sizes (rounds n - (r + 1))))
    (bodyCost : ∀ r, r < rounds n → cost r ≤ bodyTime.eval (encodedSizes r)) :
    ∀ r, r < rounds n →
      cost r ≤ bodyTime.eval (encodingSize.eval (sizeEnvelope growth n)) := by
  intro r hr
  have hs := semanticSize_bound sizes growth n growthPositive initialBound growthBound
    (rounds n - (r + 1)) (Nat.sub_le _ _)
  exact (bodyCost r hr).trans
    (MachineComposition.natPolynomial_eval_mono bodyTime
      ((encodingBound r hr).trans
        (MachineComposition.natPolynomial_eval_mono encodingSize hs)))

def iterationInTime (counter : K) (entry : Λ)
    (body : Λ → TM2.Stmt (Alphabet (K := K)) (Labels Λ) (State σ))
    (suffix : List Bool) (ambient : Nat → σ) (register : Nat → Option Bool)
    (base : Nat → K → List Bool) (cost : Nat → Nat) (growth n : Nat)
    (growthPositive : 0 < growth)
    (bodyTraces : BodyTraces counter (.inl false) (.inr entry)
      (program counter entry body) suffix ambient register base cost (rounds n))
    (sizes encodedSizes : Nat → Nat) (encodingSize bodyTime : Polynomial Nat)
    (initialBound : sizes 0 ≤ n + 1)
    (growthBound : ∀ i, i < rounds n → sizes (i + 1) ≤ growth * sizes i)
    (encodingBound : ∀ r, r < rounds n →
      encodedSizes r ≤ encodingSize.eval (sizes (rounds n - (r + 1))))
    (bodyCost : ∀ r, r < rounds n → cost r ≤ bodyTime.eval (encodedSizes r)) :
    StateTransition.EvalsToInTime (TM2.step (program counter entry body))
      (guardConfiguration counter (.inl false) suffix ambient register base (rounds n))
      (some (haltedConfiguration counter suffix ambient base))
      ((timePolynomial growth encodingSize bodyTime).eval n) where
  steps := totalSteps cost (rounds n) + 1
  evals_in_steps := iterationTrace counter entry body suffix ambient register base cost n bodyTraces
  steps_le_m := by
    rw [timePolynomial_eval]
    have h := totalSteps_le cost (rounds n)
      (bodyTime.eval (encodingSize.eval (sizeEnvelope growth n)))
      (bodyCosts_bound growth n growthPositive sizes encodedSizes cost encodingSize bodyTime
        initialBound growthBound encodingBound bodyCost)
    have hm := Nat.mul_le_mul_right
      (bodyTime.eval (encodingSize.eval (sizeEnvelope growth n)) + 1) (rounds_le n)
    omega

end BinPackingGames.Foundations.Complexity.PCPIterationMachine

namespace BinPackingGames.Foundations.Complexity.GraphCounterModel

open Turing

def startExtra (input archive : List Bool) : ExtraTape → List Bool
  | .input => input
  | .archive => archive
  | _ => []

def startMemory (input archive sum : List Bool) : Tape → List Bool
  | .inl k => if k = 0 then sum else []
  | .inr k => startExtra input archive k

theorem startMemory_input_update (input archive sum replacement : List Bool) :
    Function.update (startMemory input archive sum) (.inr .input) replacement =
      startMemory replacement archive sum := by
  funext tape
  cases tape with
  | inl k => simp [startMemory]
  | inr k => cases k <;> simp [startMemory, startExtra]

theorem startMemory_sum_update (input archive sum replacement : List Bool) :
    Function.update (startMemory input archive sum) (.inl 0) replacement =
      startMemory input archive replacement := by
  funext tape
  cases tape with
  | inl k => by_cases h : k = 0 <;> simp [startMemory, h]
  | inr k => simp [startMemory]

theorem headerStep_zero (again next : Label)
    (atHeader : program again = readHeader again next)
    (suffix archive : List Bool) (sum : Nat)
    (state : MachineLogCounter.State) (register : Option Bool) :
    TM2.step program
      ⟨some again, (state, register), startMemory (encodeWord 0 ++ suffix) archive (encodeWord sum)⟩ =
      some ⟨some next, (state, none), startMemory suffix archive (encodeWord sum)⟩ := by
  change some (TM2.stepAux (program again) _ _) = _
  rw [atHeader]
  simp only [readHeader, TM2.stepAux, startMemory, startExtra, encodeWord,
    List.replicate_zero, List.nil_append, List.singleton_append, List.head?_cons,
    List.tail_cons, Option.getD_some, Bool.cond_false]
  rw [startMemory_input_update]

theorem headerStep_succ (again next : Label)
    (atHeader : program again = readHeader again next)
    (n sum : Nat) (suffix archive : List Bool)
    (state : MachineLogCounter.State) (register : Option Bool) :
    TM2.step program
      ⟨some again, (state, register),
        startMemory (encodeWord (n + 1) ++ suffix) archive (encodeWord sum)⟩ =
      some ⟨some again, (state, some true),
        startMemory (encodeWord n ++ suffix) archive (encodeWord (sum + 1))⟩ := by
  change some (TM2.stepAux (program again) _ _) = _
  rw [atHeader]
  simp only [readHeader, TM2.stepAux, startMemory, startExtra, encodeWord,
    List.replicate_succ, List.cons_append, List.head?_cons, List.tail_cons,
    Option.getD_some, Bool.cond_true]
  rw [startMemory_input_update, startMemory_sum_update]
  rfl

theorem headerTrace (again next : Label)
    (atHeader : program again = readHeader again next)
    (n sum : Nat) (suffix archive : List Bool)
    (state : MachineLogCounter.State) (register : Option Bool) :
    (MachineComposition.advance (TM2.step program))^[n + 1]
      (some ⟨some again, (state, register),
        startMemory (encodeWord n ++ suffix) archive (encodeWord sum)⟩) =
      some ⟨some next, (state, none), startMemory suffix archive (encodeWord (n + sum))⟩ := by
  induction n generalizing sum register with
  | zero =>
      simpa only [Nat.zero_add, Function.iterate_one, MachineComposition.advance_some] using
        headerStep_zero again next atHeader suffix archive sum state register
  | succ n ih =>
      rw [Function.iterate_succ_apply, MachineComposition.advance_some]
      rw [headerStep_succ again next atHeader n sum suffix archive state register]
      simpa only [Nat.add_assoc, Nat.add_comm 1 sum] using ih (sum + 1) (some true)

def headerInTime (again next : Label)
    (atHeader : program again = readHeader again next)
    (n sum : Nat) (suffix archive : List Bool)
    (state : MachineLogCounter.State) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some again, (state, register), startMemory (encodeWord n ++ suffix) archive (encodeWord sum)⟩
      (some ⟨some next, (state, none), startMemory suffix archive (encodeWord (n + sum))⟩)
      (n + 1) where
  steps := n + 1
  evals_in_steps := headerTrace again next atHeader n sum suffix archive state register
  steps_le_m := le_rfl

theorem initialMemory (word : List Bool) :
    initList machine word =
      ⟨some (.inr .copyFirst), initialState, startMemory word [] []⟩ := by
  have ht : (initList machine word).stk = startMemory word [] [] := by
    funext tape
    cases tape with
    | inl k => simp [initList, machine, startMemory]
    | inr k => (cases k <;> simp [initList, machine, startMemory, startExtra]); rfl
  exact congrArg (TM2.Cfg.mk _ _) ht

theorem clockInitialMemory (word rest : List Bool) (sum : Nat) :
    clockConfiguration (startExtra rest word) none
      (initList MachineLogCounter.machine (encodeWord sum)) =
      ⟨some (.inl 0), initialState, startMemory rest word (encodeWord sum)⟩ := by
  have ht : clockTapes (initList MachineLogCounter.machine (encodeWord sum)).stk
      (startExtra rest word) = startMemory rest word (encodeWord sum) := by
    funext tape
    cases tape with
    | inl k => (fin_cases k <;> simp [clockTapes, initList, MachineLogCounter.machine, startMemory]); rfl
    | inr k => rfl
  exact congrArg (TM2.Cfg.mk _ _) ht

def startInTime (n m : Nat) (rest : List Bool) :
    StateTransition.EvalsToInTime machine.step
      (initList machine (encodeWords [n, m] ++ rest))
      (some (clockConfiguration (startExtra rest (encodeWords [n, m] ++ rest)) none
        (initList MachineLogCounter.machine (encodeWord (n + m)))))
      (2 * ((encodeWords [n, m] ++ rest).length + 1) + 1 + (n + 1) + (m + 1)) := by
  let word := encodeWords [n, m] ++ rest
  let b₀ := startMemory word [] []
  let b₁ := startMemory word word []
  let b₂ := startMemory word word (encodeWord 0)
  have hc : Function.update b₀ (.inr .archive) (b₀ (.inr .input) ++ b₀ (.inr .archive)) = b₁ := by
    funext tape
    cases tape with
    | inl k => simp [b₀, b₁, startMemory]
    | inr k => cases k <;> simp [b₀, b₁, startMemory, startExtra]
  let copy := MachineCopy.copyInTime (.inr ExtraTape.input) (.inr ExtraTape.archive)
    (.inr ExtraTape.scratch) (by decide) (by decide) (by decide) false
    (.inr ExtraLabel.copyFirst) (.inr ExtraLabel.copySecond) (some (.inr ExtraLabel.seed))
    program rfl rfl b₀ rfl MachineLogCounter.initialState none
  have copy' : StateTransition.EvalsToInTime (TM2.step program)
      ⟨some (.inr .copyFirst), initialState, b₀⟩
      (some ⟨some (.inr .seed), initialState, b₁⟩) (2 * (word.length + 1)) := by
    have h := copy
    rw [hc] at h
    simpa only [initialState, show b₀ (.inr .input) = word from rfl] using h
  have seed : StateTransition.EvalsToInTime (TM2.step program)
      ⟨some (.inr .seed), initialState, b₁⟩
      (some ⟨some (.inr .headerFirst), initialState, b₂⟩) 1 := by
    refine ⟨⟨1, ?_⟩, le_rfl⟩
    change some (TM2.stepAux (program (.inr .seed)) initialState b₁) = _
    simp only [program, TM2.stepAux, b₁, startMemory]
    rw [startMemory_sum_update]
    rfl
  have first := headerInTime (.inr .headerFirst) (.inr .headerSecond) rfl
    n 0 (encodeWord m ++ rest) word MachineLogCounter.initialState none
  have first' : StateTransition.EvalsToInTime (TM2.step program)
      ⟨some (.inr .headerFirst), initialState, b₂⟩
      (some ⟨some (.inr .headerSecond), initialState,
        startMemory (encodeWord m ++ rest) word (encodeWord n)⟩) (n + 1) := by
    simpa only [b₂, initialState, word, encodeWords, List.append_nil, List.append_assoc,
      Nat.add_zero] using first
  have second := headerInTime (.inr .headerSecond) (.inl 0) rfl
    m n rest word MachineLogCounter.initialState none
  have second' : StateTransition.EvalsToInTime (TM2.step program)
      ⟨some (.inr .headerSecond), initialState,
        startMemory (encodeWord m ++ rest) word (encodeWord n)⟩
      (some ⟨some (.inl 0), initialState, startMemory rest word (encodeWord (n + m))⟩)
      (m + 1) := by simpa only [initialState, Nat.add_comm m n] using second
  let p₀ := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ copy' seed
  let p₁ := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ p₀ first'
  let p := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ p₁ second'
  rw [initialMemory, clockInitialMemory]
  exact {
    toEvalsTo := p.toEvalsTo
    steps_le_m := by
      have h := p.steps_le_m
      change p.steps ≤ 2 * (word.length + 1) + 1 + (n + 1) + (m + 1)
      omega
  }

end BinPackingGames.Foundations.Complexity.GraphCounterModel

end OAI