lean/OAI/Computability/DirectedFeedback/Machines/MachinePaddedExpanderFamily.lean

import Mathlib
import OAI.Computability.DirectedFeedback.Machines.MachineExpanderFamilyLoop

namespace OAI

section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section

section

namespace DFVSGames.Foundations.Complexity.MachinePaddedExpanderFamily

open Turing
open MachineCloudPadding
open MachineComposition

variable {ρ : Type} [Fintype ρ]

noncomputable def ceilingInTime (H : SmallTable) (requested : Nat)
    (state : MachineExpanderFamily.State ρ fixedDegree) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step (program H))
      ⟨some main, initialState state register, initialTapes requested⟩
      (some ⟨some familyEntry, (state, (false, none)), handoffTapes requested⟩)
      ((MachineCeilingPower.timePolynomial PCP.ExpanderFamily.growth).eval
        (encodeWord requested).length) := by
  let original := MachineCeilingPower.paddingInTime requested state register
  let renamed := MachineStateEquiv.execution (ceilingStates ρ)
    (MachineCeilingPower.program PCP.ExpanderFamily.growth) original
  let placed := liftExecutionInTime (TM2.step (ceilingSource (ρ := ρ)))
    (TM2.step (program H))
    (Placement.configuration ceilingView ceilingLabel (some familyEntry) (fun _ => []))
    (Placement.step_simulation ceilingTape ceilingView ceilingView_left ceilingView_right
      ceilingLabel (some familyEntry) (fun _ => []) (ceilingSource (ρ := ρ))
      (program H) (program_ceiling H)) renamed
  have initialEq := ceiling_initial_configuration requested state register
  have finalEq := ceiling_handoff_configuration requested state
  unfold ceilingInput at initialEq
  unfold ceilingOutput at finalEq
  simpa only [initialEq, finalEq] using placed

noncomputable def materializeInTime (H : SmallTable) (requested : Nat)
    (state : MachineExpanderFamily.State ρ fixedDegree) :
    StateTransition.EvalsToInTime (TM2.step (program H))
      ⟨some familyEntry, (state, (false, none)), handoffTapes requested⟩
      (some ⟨none, finalState H state, finalTapes H requested⟩)
      (MachineExpanderFamilyBounds.inputCoefficient * (requested + 1)^5) := by
  let original := MachineExpanderFamily.paddedFamilyInTime H requested state []
  let boolean := MachineAlphabetTransport.executionInTime MachineExpanderFamily.alphabet_eq
    (MachineExpanderFamily.program MachineExpanderFamily.baseDegree_positive H
      MachineExpanderFamily.baseDegree_cloud_gt_one) original
  let framed := MachineStateFrame.frameExecution
    (MachineExpanderFamily.boolView MachineExpanderFamily.baseDegree_positive H
      MachineExpanderFamily.baseDegree_cloud_gt_one) (false, (none : Option Bool)) boolean
  let placed := liftExecutionInTime (TM2.step (familySource H)) (TM2.step (program H))
    (Placement.configuration familyView familyLabel none (retainedFrame requested))
    (Placement.step_simulation familyTape familyView familyView_left familyView_right
      familyLabel none (retainedFrame requested) (familySource H) (program H)
      (program_family H)) framed
  simpa only [family_initial_configuration, family_final_configuration] using placed

noncomputable def paddedInTime (H : SmallTable) (requested : Nat)
    (state : MachineExpanderFamily.State ρ fixedDegree) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step (program H))
      ⟨some main, initialState state register, initialTapes requested⟩
      (some ⟨none, finalState H state, finalTapes H requested⟩)
      (MachinePaddedExpanderFamilyBounds.timePolynomial.eval (encodeWord requested).length) := by
  let first := ceilingInTime H requested state register
  let second := materializeInTime H requested state
  let all := StateTransition.EvalsToInTime.trans (TM2.step (program H))
    _ _ _ _ _ first second
  exact
    { toEvalsTo := all.toEvalsTo
      steps_le_m := all.steps_le_m.trans
        (by simpa only [Nat.add_comm] using
          MachinePaddedExpanderFamilyBounds.combinedCost_le requested) }

noncomputable def placedInTime {K Λ : Type} [DecidableEq K]
    (tape : Tape → K) (view : K → Option Tape)
    (left : ∀ t, view (tape t) = some t)
    (right : ∀ j t, view j = some t → tape t = j)
    (labels : Label → Λ) (exit : Option Λ) (extra : K → List Bool)
    (H : SmallTable) (target : Λ → TM2.Stmt (fun _ : K => Bool) Λ (State ρ))
    (code : ∀ l, target (labels l) = Placement.statement tape labels exit (program H l))
    (requested : Nat) (state : MachineExpanderFamily.State ρ fixedDegree)
    (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step target)
      (Placement.configuration view labels exit extra
        ⟨some main, initialState state register, initialTapes requested⟩)
      (some (Placement.configuration view labels exit extra
        ⟨none, finalState H state, finalTapes H requested⟩))
      (MachinePaddedExpanderFamilyBounds.timePolynomial.eval (encodeWord requested).length) :=
  liftExecutionInTime (TM2.step (program H)) (TM2.step target)
    (Placement.configuration view labels exit extra)
    (Placement.step_simulation tape view left right labels exit extra (program H) target code)
    (paddedInTime H requested state register)

end DFVSGames.Foundations.Complexity.MachinePaddedExpanderFamily
end

section

namespace DFVSGames.Foundations.Complexity.MachineRegularFamily

open Turing MachineComposition
open MachineCloudPadding

abbrev CoreTape := Fin 27
abbrev NativeTape := MachinePaddedExpanderFamily.Tape
abbrev Tape := CoreTape ⊕ NativeTape
abbrev Alphabet (_ : Tape) := Bool
abbrev SmallTable := MachinePaddedExpanderFamily.SmallTable
abbrev FamilyState := MachineExpanderFamily.State Unit MachinePaddedExpanderFamily.fixedDegree
abbrev State := (FamilyState × Bool) × Option Bool

def inputNative : NativeTape := .inr .input
def outputNative : NativeTape := .inl MachineExpanderFamily.tableTape
def powerNative : NativeTape := .inr .power
def sizeNative : NativeTape := .inl (.inr .currentSize)
def levelNative : NativeTape := .inl (.inr .remainingLevel)

def powerTape : Tape := .inr powerNative
def sizeTape : Tape := .inr sizeNative
def levelTape : Tape := .inr levelNative

def familyTape (t : NativeTape) : Tape :=
  if t = inputNative then .inl 4 else if t = outputNative then .inl 7 else .inr t

def familyView : Tape → Option NativeTape
  | .inl i => if i = 4 then some inputNative else if i = 7 then some outputNative else none
  | .inr t => if t = inputNative then none else if t = outputNative then none else some t

theorem familyView_left (t : NativeTape) : familyView (familyTape t) = some t := by
  by_cases hi : t = inputNative
  · subst t; simp [familyTape, familyView]
  · by_cases ho : t = outputNative
    · subst t; simp [familyTape, familyView, hi]
    · simp [familyTape, familyView, hi, ho]

theorem familyView_right (j : Tape) (t : NativeTape)
    (h : familyView j = some t) : familyTape t = j := by
  cases j with
  | inl i =>
    by_cases hi : i = 4
    · subst i
      have ht : inputNative = t := by simpa [familyView] using h
      subst t; simp [familyTape]
    · by_cases ho : i = 7
      · subst i
        have ht : outputNative = t := by simpa [familyView] using h
        subst t; simp [familyTape, inputNative, outputNative]
      · simp [familyView, hi, ho] at h
  | inr j =>
    by_cases hi : j = inputNative
    · simp [familyView, hi] at h
    · by_cases ho : j = outputNative
      · simp [familyView, ho] at h
      · have ht : j = t := by simpa [familyView, hi, ho] using h
        subst t; simp [familyTape, hi, ho]

def stateEquiv : MachinePaddedExpanderFamily.State Unit ≃ State :=
  (Equiv.prodAssoc FamilyState Bool (Option Bool)).symm

def seedState (H : SmallTable) : FamilyState :=
  MachineExpanderFamily.initialState MachineExpanderFamily.baseDegree_positive H ()

def readyState (H : SmallTable) : State := ((seedState H, false), none)

def sourceProgram (H : SmallTable) : MachinePaddedExpanderFamily.Label →
    TM2.Stmt (fun _ : NativeTape => Bool) MachinePaddedExpanderFamily.Label State :=
  MachineStateEquiv.program stateEquiv (MachinePaddedExpanderFamily.program (ρ := Unit) H)

inductive Label
  | start
  | family (label : MachinePaddedExpanderFamily.Label)
  | drainPower | drainSize | drainLevel
  deriving DecidableEq, Fintype

def program (H : SmallTable) : Label → TM2.Stmt Alphabet Label State
  | .start => .load (fun _ => readyState H) (.goto fun _ => .family MachinePaddedExpanderFamily.main)
  | .family l => Placement.statement familyTape Label.family (some .drainPower) (sourceProgram H l)
  | .drainPower => MachineDrain.drain powerTape .drainPower (some .drainSize)
  | .drainSize => MachineDrain.drain sizeTape .drainSize (some .drainLevel)
  | .drainLevel => MachineDrain.drain levelTape .drainLevel none

def frame (core : CoreTape → List Bool) : Tape → List Bool
  | .inl i => core i
  | .inr _ => []

def rotor (H : SmallTable) (k : Nat) : List Bool :=
  MachinePaddedExpanderFamilyBounds.outputWord H k

def resultCore (H : SmallTable) (k : Nat) (core : CoreTape → List Bool) :
    CoreTape → List Bool := Function.update core 7 (rotor H k)

def working (core : CoreTape → List Bool) (power size level : List Bool) : Tape → List Bool
  | .inl i => core i
  | .inr t => if t = powerNative then power
      else if t = sizeNative then size else if t = levelNative then level else []

@[simp] theorem working_power (core : CoreTape → List Bool) (p s l : List Bool) :
    working core p s l powerTape = p := by simp [working, powerTape]

@[simp] theorem working_size (core : CoreTape → List Bool) (p s l : List Bool) :
    working core p s l sizeTape = s := by
  simp [working, sizeTape, powerNative, sizeNative]

@[simp] theorem working_level (core : CoreTape → List Bool) (p s l : List Bool) :
    working core p s l levelTape = l := by
  simp [working, levelTape, powerNative, sizeNative, levelNative]

theorem update_working_power (core : CoreTape → List Bool) (p s l p' : List Bool) :
    Function.update (working core p s l) powerTape p' = working core p' s l := by
  funext t
  cases t with
  | inl i => simp [working, powerTape]
  | inr t => by_cases h : t = powerNative <;> simp [working, powerTape, h]

theorem update_working_size (core : CoreTape → List Bool) (p s l s' : List Bool) :
    Function.update (working core p s l) sizeTape s' = working core p s' l := by
  funext t
  cases t with
  | inl i => simp [working, sizeTape]
  | inr t =>
    by_cases h : t = sizeNative
    · subst t; simp [working, sizeTape, powerNative, sizeNative]
    · simp [working, sizeTape, h]

theorem update_working_level (core : CoreTape → List Bool) (p s l l' : List Bool) :
    Function.update (working core p s l) levelTape l' = working core p s l' := by
  funext t
  cases t with
  | inl i => simp [working, levelTape]
  | inr t =>
    by_cases h : t = levelNative
    · subst t; simp [working, levelTape, powerNative, sizeNative, levelNative]
    · simp [working, levelTape, h]

@[simp] theorem working_empty (core : CoreTape → List Bool) :
    working core [] [] [] = frame core := by
  funext t; cases t <;> simp [working, frame]

private theorem initial_other_inline_MachineRegularFamily (k : Nat) (t : NativeTape) (h : t ≠ inputNative) :
    MachinePaddedExpanderFamily.initialTapes k t = [] := by
  cases t with
  | inl t => rfl
  | inr t => cases t <;> simp_all [MachinePaddedExpanderFamily.initialTapes, inputNative]

private theorem final_other_inline_MachineRegularFamily (H : SmallTable) (k : Nat) (t : NativeTape)
    (hi : t ≠ inputNative) (ho : t ≠ outputNative) (hp : t ≠ powerNative)
    (hs : t ≠ sizeNative) (hl : t ≠ levelNative) :
    MachinePaddedExpanderFamily.finalTapes H k t = [] := by
  rcases t with ((row | tableExtra) | familyExtra) | ceiling
  · cases row <;>
      simp_all [inputNative, outputNative, MachineExpanderFamily.tableTape,
        MachinePaddedExpanderFamily.finalTapes, MachineExpanderFamily.toBoolTapes,
        MachineExpanderFamily.toBoolWord, MachineExpanderFamily.familyTapes,
        MachineExpanderFamily.boundaryTapes, MachineExpanderFamily.tableFrame,
        MachineEmbedding.tapes]
  · cases tableExtra <;> rfl
  · cases familyExtra <;>
      simp_all [sizeNative, levelNative, MachinePaddedExpanderFamily.finalTapes,
        MachineExpanderFamily.toBoolTapes, MachineExpanderFamily.toBoolWord,
        MachineExpanderFamily.familyTapes, MachineExpanderFamily.boundaryTapes,
        MachineExpanderFamily.extraFrame, MachineEmbedding.tapes]
  · cases ceiling <;>
      simp_all [inputNative, powerNative, MachinePaddedExpanderFamily.finalTapes,
        MachinePaddedExpanderFamily.retainedCeilingTapes, MachineCeilingPower.memory]

theorem initial_placement (k : Nat) (core : CoreTape → List Bool)
    (hinput : core 4 = encodeWord k) (houtput : core 7 = []) :
    Placement.tapes familyView (MachinePaddedExpanderFamily.initialTapes k) (frame core) =
      frame core := by
  funext t
  cases t with
  | inl i =>
    by_cases hi : i = 4
    · subst i; simp [Placement.tapes, familyView, frame, inputNative, hinput]
    · by_cases ho : i = 7
      · subst i
        simp [Placement.tapes, familyView, frame, outputNative,
          MachinePaddedExpanderFamily.initialTapes, houtput]
      · simp [Placement.tapes, familyView, frame, hi, ho]
  | inr t =>
    by_cases hi : t = inputNative
    · simp [Placement.tapes, familyView, frame, hi]
    · by_cases ho : t = outputNative
      · simp [Placement.tapes, familyView, frame, ho]
      · simp [Placement.tapes, familyView, frame, hi, ho, initial_other_inline_MachineRegularFamily k t hi]

theorem final_placement (H : SmallTable) (k : Nat) (core : CoreTape → List Bool)
    (hinput : core 4 = encodeWord k) :
    Placement.tapes familyView (MachinePaddedExpanderFamily.finalTapes H k) (frame core) =
      working (resultCore H k core)
        (encodeWord (PCP.PreprocessingLevels.paddedSize k))
        (encodeWord (PCP.PreprocessingLevels.paddedSize k)) (encodeWord 0) := by
  funext t
  cases t with
  | inl i =>
    by_cases hi : i = 4
    · subst i
      simp [Placement.tapes, familyView, working, resultCore, inputNative, hinput]
    · by_cases ho : i = 7
      · subst i
        simp [Placement.tapes, familyView, working, resultCore, outputNative,
          rotor, MachinePaddedExpanderFamilyBounds.outputWord]
        change MachinePaddedExpanderFamily.finalTapes H k
          (MachinePaddedExpanderFamily.familyTape MachineExpanderFamily.tableTape) = _
        exact MachinePaddedExpanderFamily.finalTapes_table H k
      · simp [Placement.tapes, familyView, working, resultCore, frame, hi, ho]
  | inr t =>
    by_cases hi : t = inputNative
    · subst t
      simp [Placement.tapes, familyView, working, frame,
        inputNative, powerNative, sizeNative, levelNative]
    · by_cases ho : t = outputNative
      · subst t
        simp [Placement.tapes, familyView, working, frame,
          outputNative, MachineExpanderFamily.tableTape, inputNative,
          powerNative, sizeNative, levelNative]
      · by_cases hp : t = powerNative
        · subst t
          simp [Placement.tapes, familyView, working, powerNative, inputNative, outputNative]
        · by_cases hs : t = sizeNative
          · subst t
            simpa [Placement.tapes, familyView, working, sizeNative,
              powerNative, inputNative, outputNative, MachineExpanderFamily.tableTape,
              MachinePaddedExpanderFamily.familyTape] using
              MachinePaddedExpanderFamily.finalTapes_currentSize H k
          · by_cases hl : t = levelNative
            · subst t
              simpa [Placement.tapes, familyView, working, levelNative,
                sizeNative, powerNative, inputNative, outputNative,
                MachineExpanderFamily.tableTape, MachinePaddedExpanderFamily.ceilingTape] using
                MachinePaddedExpanderFamily.finalTapes_remainingLevel H k
            · simp [Placement.tapes, familyView, working, hi, ho, hp, hs, hl,
                final_other_inline_MachineRegularFamily H k t hi ho hp hs hl]

@[simp] theorem source_initial_state (H : SmallTable) :
    stateEquiv (MachinePaddedExpanderFamily.initialState (seedState H) none) =
      readyState H := rfl

@[simp] theorem source_final_state (H : SmallTable) :
    stateEquiv (MachinePaddedExpanderFamily.finalState H (seedState H)) =
      readyState H := by
  simp [stateEquiv, MachinePaddedExpanderFamily.finalState, readyState, seedState]

noncomputable def familyInTime (H : SmallTable) (k : Nat) (core : CoreTape → List Bool)
    (hinput : core 4 = encodeWord k) (houtput : core 7 = []) :
    StateTransition.EvalsToInTime (TM2.step (program H))
      ⟨some (.family MachinePaddedExpanderFamily.main), readyState H, frame core⟩
      (some ⟨some .drainPower, readyState H,
        working (resultCore H k core)
          (encodeWord (PCP.PreprocessingLevels.paddedSize k))
          (encodeWord (PCP.PreprocessingLevels.paddedSize k)) (encodeWord 0)⟩)
      (MachinePaddedExpanderFamilyBounds.timePolynomial.eval (encodeWord k).length) := by
  let raw := MachinePaddedExpanderFamily.paddedInTime H k (seedState H) none
  let renamed := MachineStateEquiv.execution stateEquiv
    (MachinePaddedExpanderFamily.program (ρ := Unit) H) raw
  let placed := liftExecutionInTime (TM2.step (sourceProgram H)) (TM2.step (program H))
    (Placement.configuration familyView Label.family (some .drainPower) (frame core))
    (Placement.step_simulation familyTape familyView familyView_left familyView_right
      Label.family (some .drainPower) (frame core) (sourceProgram H) (program H)
      (fun _ => rfl)) renamed
  simpa only [MachineStateEquiv.configuration, Placement.configuration, Placement.label,
    source_initial_state, source_final_state, initial_placement k core hinput houtput,
    final_placement H k core hinput] using placed

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

theorem cleanupTrace (H : SmallTable) (m : Nat) (core : CoreTape → List Bool) :
    (advance (TM2.step (program H)))^[2*m+6]
      (some ⟨some .drainPower, readyState H,
        working core (encodeWord m) (encodeWord m) (encodeWord 0)⟩) =
      some ⟨none, readyState H, frame core⟩ := by
  have first := (MachineDrain.drainInTime powerTape .drainPower (some .drainSize)
    (program H) rfl (working core (encodeWord m) (encodeWord m) (encodeWord 0))
    (seedState H, false) none).evals_in_steps
  have second := (MachineDrain.drainInTime sizeTape .drainSize (some .drainLevel)
    (program H) rfl (working core [] (encodeWord m) (encodeWord 0))
    (seedState H, false) none).evals_in_steps
  have third := (MachineDrain.drainInTime levelTape .drainLevel none
    (program H) rfl (working core [] [] (encodeWord 0))
    (seedState H, false) none).evals_in_steps
  simp only [MachineDrain.drainInTime, working_power, working_size, working_level,
    encodeWord_length, update_working_power, update_working_size,
    update_working_level, working_empty] at first second third
  have all := appendTrace_inline_MachineRegularFamily _ (appendTrace_inline_MachineRegularFamily _ first second) third
  have count : 2*m+6 = (m+1+1)+(m+1+1)+(0+1+1) := by omega
  rw [count]
  exact all

def cleanupInTime (H : SmallTable) (m : Nat) (core : CoreTape → List Bool) :
    StateTransition.EvalsToInTime (TM2.step (program H))
      ⟨some .drainPower, readyState H,
        working core (encodeWord m) (encodeWord m) (encodeWord 0)⟩
      (some ⟨none, readyState H, frame core⟩) (2*m+6) where
  steps := 2*m+6
  evals_in_steps := cleanupTrace H m core
  steps_le_m := le_rfl

def startInTime (H : SmallTable) (core : CoreTape → List Bool) (state : State) :
    StateTransition.EvalsToInTime (TM2.step (program H))
      ⟨some .start, state, frame core⟩
      (some ⟨some (.family MachinePaddedExpanderFamily.main), readyState H, frame core⟩) 1 where
  steps := 1
  evals_in_steps := by
    change some (TM2.stepAux (program H .start) state (frame core)) = _
    rfl
  steps_le_m := le_rfl

noncomputable def timePolynomial : Polynomial Nat :=
  MachinePaddedExpanderFamilyBounds.timePolynomial +
    Polynomial.C (2 * PCP.ExpanderFamily.growth) * Polynomial.X + 7

theorem totalBudget_le (k : Nat) :
    1 + MachinePaddedExpanderFamilyBounds.timePolynomial.eval (encodeWord k).length +
        (2 * PCP.PreprocessingLevels.paddedSize k + 6) ≤
      timePolynomial.eval (encodeWord k).length := by
  have hs := MachineExpanderFamilyBounds.paddedSize_le_succ_input k
  have hm := Nat.mul_le_mul_left 2 hs
  simp only [timePolynomial, Polynomial.eval_add, Polynomial.eval_mul,
    Polynomial.eval_C, Polynomial.eval_X, Polynomial.eval_ofNat, encodeWord_length]
  simp only [Nat.mul_assoc] at hm ⊢
  omega

noncomputable def familyCleanInTime (H : SmallTable) (k : Nat)
    (core : CoreTape → List Bool) (hinput : core 4 = encodeWord k)
    (houtput : core 7 = []) (state : State) :
    StateTransition.EvalsToInTime (TM2.step (program H))
      ⟨some .start, state, frame core⟩
      (some ⟨none, readyState H, frame (resultCore H k core)⟩)
      (timePolynomial.eval (encodeWord k).length) := by
  let first := startInTime H core state
  let second := familyInTime H k core hinput houtput
  let third := cleanupInTime H (PCP.PreprocessingLevels.paddedSize k) (resultCore H k core)
  let pair := StateTransition.EvalsToInTime.trans (TM2.step (program H)) _ _ _ _ _ first second
  let all := StateTransition.EvalsToInTime.trans (TM2.step (program H)) _ _ _ _ _ pair third
  refine { toEvalsTo := all.toEvalsTo, steps_le_m := ?_ }
  exact all.steps_le_m.trans
    (by simpa only [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm] using totalBudget_le k)

@[simp] theorem result_output (H : SmallTable) (k : Nat) (core : CoreTape → List Bool) :
    frame (resultCore H k core) (.inl 7) = rotor H k := by simp [frame, resultCore]

theorem result_preserves_core (H : SmallTable) (k : Nat) (core : CoreTape → List Bool)
    (i : CoreTape) (hi : i ≠ 7) : frame (resultCore H k core) (.inl i) = core i := by
  simp [frame, resultCore, hi]

@[simp] theorem result_private_empty (H : SmallTable) (k : Nat)
    (core : CoreTape → List Bool) (t : NativeTape) :
    frame (resultCore H k core) (.inr t) = [] := rfl

noncomputable def placedInTime {K Λ : Type} [DecidableEq K]
    (tape : Tape → K) (view : K → Option Tape)
    (left : ∀ t, view (tape t) = some t)
    (right : ∀ j t, view j = some t → tape t = j)
    (labels : Label → Λ) (exit : Option Λ) (extra : K → List Bool)
    (H : SmallTable) (target : Λ → TM2.Stmt (fun _ : K => Bool) Λ State)
    (code : ∀ l, target (labels l) = Placement.statement tape labels exit (program H l))
    (k : Nat) (core : CoreTape → List Bool) (hinput : core 4 = encodeWord k)
    (houtput : core 7 = []) (state : State) :
    StateTransition.EvalsToInTime (TM2.step target)
      (Placement.configuration view labels exit extra ⟨some .start, state, frame core⟩)
      (some (Placement.configuration view labels exit extra
        ⟨none, readyState H, frame (resultCore H k core)⟩))
      (timePolynomial.eval (encodeWord k).length) :=
  liftExecutionInTime (TM2.step (program H)) (TM2.step target)
    (Placement.configuration view labels exit extra)
    (Placement.step_simulation tape view left right labels exit extra (program H) target code)
    (familyCleanInTime H k core hinput houtput state)

end DFVSGames.Foundations.Complexity.MachineRegularFamily
end

section

namespace DFVSGames.Foundations.Complexity.MachineUnaryLessAt

open Turing MachineComposition

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

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

def zeroTest (operand : K) (exit : Option Λ) :
    TM2.Stmt (Alphabet (K := K)) Λ (State σ) :=
  .peek operand (fun s head => (s.1, head))
    (.load (fun s => ((s.1.1, !(s.2.getD false)), none))
      (Reduction.MachineTransfer.exitAt operand exit))

theorem zeroTestTrace (operand : K) (testLabel : Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State σ))
    (atTest : program testLabel = zeroTest operand exit)
    (base : K → List Bool) (n : Nat) (suffix : List Bool)
    (word : base operand = encodeWord n ++ suffix)
    (ambient : σ) (flag : Bool) (register : Option Bool) :
    (advance (TM2.step program))^[1]
      (some ⟨some testLabel, ((ambient, flag), register), base⟩) =
      some ⟨exit, ((ambient, decide (n = 0)), none), base⟩ := by
  change some (TM2.stepAux (program testLabel) _ _) = _
  rw [atTest]
  cases n <;> cases exit <;>
    simp [zeroTest, TM2.stepAux, word, encodeWord, List.replicate_succ,
      Reduction.MachineTransfer.exitAt]

def tapes (slots : Fin 4 ↪ K) (base : K → List Bool)
    (left right savedLeft savedRight : List Bool) : K → List Bool :=
  Function.update (Function.update (Function.update (Function.update base
    (slots 0) left) (slots 1) right) (slots 2) savedLeft) (slots 3) savedRight

@[simp] theorem tapes_left (slots : Fin 4 ↪ K) (base : K → List Bool)
    (left right savedLeft savedRight : List Bool) :
    tapes slots base left right savedLeft savedRight (slots 0) = left := by
  simp [tapes, slots.injective.eq_iff]

@[simp] theorem tapes_right (slots : Fin 4 ↪ K) (base : K → List Bool)
    (left right savedLeft savedRight : List Bool) :
    tapes slots base left right savedLeft savedRight (slots 1) = right := by
  simp [tapes, slots.injective.eq_iff]

@[simp] theorem tapes_savedLeft (slots : Fin 4 ↪ K) (base : K → List Bool)
    (left right savedLeft savedRight : List Bool) :
    tapes slots base left right savedLeft savedRight (slots 2) = savedLeft := by
  simp [tapes, slots.injective.eq_iff]

@[simp] theorem tapes_savedRight (slots : Fin 4 ↪ K) (base : K → List Bool)
    (left right savedLeft savedRight : List Bool) :
    tapes slots base left right savedLeft savedRight (slots 3) = savedRight := by
  simp [tapes]

@[simp] theorem update_tapes (slots : Fin 4 ↪ K) (base : K → List Bool)
    (left right savedLeft savedRight replacement : List Bool) (i : Fin 4) :
    Function.update (tapes slots base left right savedLeft savedRight) (slots i) replacement =
      tapes slots base (if i = 0 then replacement else left)
        (if i = 1 then replacement else right) (if i = 2 then replacement else savedLeft)
        (if i = 3 then replacement else savedRight) := by
  fin_cases i <;> funext k
  all_goals
    by_cases h₀ : k = slots 0
    · subst k; simp [tapes, slots.injective.eq_iff]
    · by_cases h₁ : k = slots 1
      · subst k; simp [tapes, slots.injective.eq_iff]
      · by_cases h₂ : k = slots 2
        · subst k; simp [tapes, slots.injective.eq_iff]
        · by_cases h₃ : k = slots 3
          · subst k; simp [tapes, slots.injective.eq_iff]
          · simp [tapes, h₀, h₁, h₂, h₃]

@[simp] theorem tapes_self (slots : Fin 4 ↪ K) (base : K → List Bool) :
    tapes slots base (base (slots 0)) (base (slots 1)) (base (slots 2)) (base (slots 3)) = base := by
  simp [tapes]

def stop (result : Bool) (restoreLabel : Λ) :
    TM2.Stmt (Alphabet (K := K)) Λ (State σ) :=
  .load (fun s => ((s.1.1, result), none)) (.goto fun _ => restoreLabel)

def scan (slots : Fin 4 ↪ K) (scanLabel restoreLabel : Λ) :
    TM2.Stmt (Alphabet (K := K)) Λ (State σ) :=
  .peek (slots 0) (fun s head => (s.1, head))
    (.branch (fun s => s.2.getD false)
      (.peek (slots 1) (fun s head => (s.1, head))
        (.branch (fun s => s.2.getD false)
          (.pop (slots 0) (fun s _ => s)
            (.pop (slots 1) (fun s _ => s)
              (.push (slots 2) (fun _ => true)
                (.push (slots 3) (fun _ => true) (.goto fun _ => scanLabel)))))
          (stop false restoreLabel)))
      (.peek (slots 1) (fun s head => (s.1, head))
        (.branch (fun s => s.2.getD false)
          (stop true restoreLabel) (stop false restoreLabel))))

theorem scan_step_zero_left (slots : Fin 4 ↪ K) (scanLabel restoreLabel : Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State σ))
    (atScan : program scanLabel = scan slots scanLabel restoreLabel)
    (base : K → List Bool) (b : Nat) (leftSuffix rightSuffix savedLeft savedRight : List Bool)
    (ambient : σ) (flag : Bool) (register : Option Bool) :
    TM2.step program ⟨some scanLabel, ((ambient, flag), register),
      tapes slots base (encodeWord 0 ++ leftSuffix) (encodeWord b ++ rightSuffix) savedLeft savedRight⟩ =
      some ⟨some restoreLabel, ((ambient, decide (0 < b)), none),
        tapes slots base (encodeWord 0 ++ leftSuffix) (encodeWord b ++ rightSuffix) savedLeft savedRight⟩ := by
  change some (TM2.stepAux (program scanLabel) _ _) = _
  rw [atScan]
  cases b <;> simp [scan, stop, TM2.stepAux, encodeWord, List.replicate_succ]

theorem scan_step_zero_right (slots : Fin 4 ↪ K) (scanLabel restoreLabel : Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State σ))
    (atScan : program scanLabel = scan slots scanLabel restoreLabel)
    (base : K → List Bool) (a : Nat) (leftSuffix rightSuffix savedLeft savedRight : List Bool)
    (ambient : σ) (flag : Bool) (register : Option Bool) :
    TM2.step program ⟨some scanLabel, ((ambient, flag), register),
      tapes slots base (encodeWord (a + 1) ++ leftSuffix) (encodeWord 0 ++ rightSuffix) savedLeft savedRight⟩ =
      some ⟨some restoreLabel, ((ambient, false), none),
        tapes slots base (encodeWord (a + 1) ++ leftSuffix) (encodeWord 0 ++ rightSuffix) savedLeft savedRight⟩ := by
  change some (TM2.stepAux (program scanLabel) _ _) = _
  rw [atScan]
  simp [scan, stop, TM2.stepAux, encodeWord, List.replicate_succ]

theorem scan_step_succ (slots : Fin 4 ↪ K) (scanLabel restoreLabel : Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State σ))
    (atScan : program scanLabel = scan slots scanLabel restoreLabel)
    (base : K → List Bool) (a b : Nat) (leftSuffix rightSuffix savedLeft savedRight : List Bool)
    (ambient : σ) (flag : Bool) (register : Option Bool) :
    TM2.step program ⟨some scanLabel, ((ambient, flag), register),
      tapes slots base (encodeWord (a + 1) ++ leftSuffix) (encodeWord (b + 1) ++ rightSuffix)
        savedLeft savedRight⟩ =
      some ⟨some scanLabel, ((ambient, flag), some true),
        tapes slots base (encodeWord a ++ leftSuffix) (encodeWord b ++ rightSuffix)
          (true :: savedLeft) (true :: savedRight)⟩ := by
  change some (TM2.stepAux (program scanLabel) _ _) = _
  rw [atScan]
  simp [scan, TM2.stepAux, encodeWord, List.replicate_succ]

theorem scanTrace (slots : Fin 4 ↪ K) (scanLabel restoreLabel : Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State σ))
    (atScan : program scanLabel = scan slots scanLabel restoreLabel)
    (base : K → List Bool) (a b : Nat) (leftSuffix rightSuffix savedLeft savedRight : List Bool)
    (ambient : σ) (flag : Bool) (register : Option Bool) :
    (advance (TM2.step program))^[min a b + 1]
      (some ⟨some scanLabel, ((ambient, flag), register),
        tapes slots base (encodeWord a ++ leftSuffix) (encodeWord b ++ rightSuffix) savedLeft savedRight⟩) =
      some ⟨some restoreLabel, ((ambient, decide (a < b)), none),
        tapes slots base (encodeWord (a - b) ++ leftSuffix) (encodeWord (b - a) ++ rightSuffix)
          (List.replicate (min a b) true ++ savedLeft) (List.replicate (min a b) true ++ savedRight)⟩ := by
  induction a generalizing b savedLeft savedRight register with
  | zero =>
    simpa using scan_step_zero_left slots scanLabel restoreLabel program atScan base b
      leftSuffix rightSuffix savedLeft savedRight ambient flag register
  | succ a ih =>
    cases b with
    | zero =>
      simpa using scan_step_zero_right slots scanLabel restoreLabel program atScan base a
        leftSuffix rightSuffix savedLeft savedRight ambient flag register
    | succ b =>
      have hm : min (a + 1) (b + 1) = min a b + 1 := by omega
      rw [hm, Function.iterate_succ_apply]
      simp only [advance_some]
      rw [scan_step_succ slots scanLabel restoreLabel program atScan, ih]
      have hsave (xs : List Bool) :
          List.replicate (min a b) true ++ true :: xs =
            List.replicate (min a b + 1) true ++ xs := by
        simp [List.replicate_add, List.append_assoc]
      simp only [hsave, Nat.add_sub_add_right, Nat.add_lt_add_iff_right]

@[simp] theorem transferLeft_tapes (slots : Fin 4 ↪ K) (base : K → List Bool)
    (left right savedLeft savedRight replacementSaved replacementLeft : List Bool) :
    Reduction.MachineTransfer.tapesAt (slots 2) (slots 0)
      (tapes slots base left right savedLeft savedRight) replacementSaved replacementLeft =
      tapes slots base replacementLeft right replacementSaved savedRight := by
  simp [Reduction.MachineTransfer.tapesAt]

@[simp] theorem transferRight_tapes (slots : Fin 4 ↪ K) (base : K → List Bool)
    (left right savedLeft savedRight replacementSaved replacementRight : List Bool) :
    Reduction.MachineTransfer.tapesAt (slots 3) (slots 1)
      (tapes slots base left right savedLeft savedRight) replacementSaved replacementRight =
      tapes slots base left replacementRight savedLeft replacementSaved := by
  simp [Reduction.MachineTransfer.tapesAt]

inductive Label where
  | scan
  | restoreLeft
  | restoreRight
  deriving DecidableEq

instance : Fintype Label where
  elems := { .scan, .restoreLeft, .restoreRight }
  complete l := by cases l <;> simp

def statement (slots : Fin 4 ↪ K) (labels : Label → Λ) (exit : Option Λ) :
    Label → TM2.Stmt (Alphabet (K := K)) Λ (State σ)
  | .scan => scan slots (labels .scan) (labels .restoreLeft)
  | .restoreLeft => Reduction.MachineTransfer.loopAt (slots 2) (slots 0) id false
      (labels .restoreLeft) (some (labels .restoreRight))
  | .restoreRight => Reduction.MachineTransfer.loopAt (slots 3) (slots 1) id false
      (labels .restoreRight) exit

def steps (a b : Nat) : Nat := 3 * (min a b + 1)

theorem comparisonTrace (slots : Fin 4 ↪ K) (labels : Label → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State σ))
    (atLabels : ∀ l, program (labels l) = statement slots labels exit l)
    (base : K → List Bool) (a b : Nat) (leftSuffix rightSuffix : List Bool)
    (ambient : σ) (flag : Bool) (register : Option Bool) :
    (advance (TM2.step program))^[steps a b]
      (some ⟨some (labels .scan), ((ambient, flag), register),
        tapes slots base (encodeWord a ++ leftSuffix) (encodeWord b ++ rightSuffix) [] []⟩) =
      some ⟨exit, ((ambient, decide (a < b)), none),
        tapes slots base (encodeWord a ++ leftSuffix) (encodeWord b ++ rightSuffix) [] []⟩ := by
  have hscan := scanTrace slots (labels .scan) (labels .restoreLeft) program (atLabels .scan)
    base a b leftSuffix rightSuffix [] [] ambient flag register
  simp only [List.append_nil] at hscan
  have hwordLeft : List.replicate (min a b) true ++ (encodeWord (a - b) ++ leftSuffix) =
      encodeWord a ++ leftSuffix := by
    rw [MachineUnaryAffineAt.prepend_replicate_word,
      show min a b + (a - b) = a by omega]
  have hwordRight : List.replicate (min a b) true ++ (encodeWord (b - a) ++ rightSuffix) =
      encodeWord b ++ rightSuffix := by
    rw [MachineUnaryAffineAt.prepend_replicate_word,
      show min a b + (b - a) = b by omega]
  let middle := tapes slots base (encodeWord (a - b) ++ leftSuffix)
    (encodeWord (b - a) ++ rightSuffix) (List.replicate (min a b) true) (List.replicate (min a b) true)
  have hleft := Reduction.MachineTransfer.transferAt_fromTapes
    (Γ := fun _ : K => Bool) (σ := σ × Bool) (slots 2) (slots 0)
    (slots.injective.ne (by decide)) id false (labels .restoreLeft) (some (labels .restoreRight))
    program (by simpa only [statement] using atLabels .restoreLeft)
    middle (ambient, decide (a < b)) none
  change (advance (TM2.step program))^[(middle (slots 2)).length + 1]
    (some ⟨some (labels .restoreLeft), ((ambient, decide (a < b)), none), middle⟩) = _ at hleft
  simp only [middle, tapes_savedLeft, tapes_left, List.length_replicate, List.reverse_replicate,
    List.map_id, hwordLeft, transferLeft_tapes] at hleft
  let afterLeft := tapes slots base (encodeWord a ++ leftSuffix)
    (encodeWord (b - a) ++ rightSuffix) [] (List.replicate (min a b) true)
  have hright := Reduction.MachineTransfer.transferAt_fromTapes
    (Γ := fun _ : K => Bool) (σ := σ × Bool) (slots 3) (slots 1)
    (slots.injective.ne (by decide)) id false (labels .restoreRight) exit
    program (by simpa only [statement] using atLabels .restoreRight)
    afterLeft (ambient, decide (a < b)) none
  change (advance (TM2.step program))^[(afterLeft (slots 3)).length + 1]
    (some ⟨some (labels .restoreRight), ((ambient, decide (a < b)), none), afterLeft⟩) = _ at hright
  simp only [afterLeft, tapes_savedRight, tapes_right, List.length_replicate, List.reverse_replicate,
    List.map_id, hwordRight, transferRight_tapes] at hright
  have hfirst : (advance (TM2.step program))^[(min a b + 1) + (min a b + 1)]
      (some ⟨some (labels .scan), ((ambient, flag), register),
        tapes slots base (encodeWord a ++ leftSuffix) (encodeWord b ++ rightSuffix) [] []⟩) =
      some ⟨some (labels .restoreRight), ((ambient, decide (a < b)), none), afterLeft⟩ := by
    rw [Function.iterate_add_apply, hscan]
    exact hleft
  rw [show steps a b = (min a b + 1) + ((min a b + 1) + (min a b + 1)) by
    unfold steps; omega, Function.iterate_add_apply, hfirst]
  exact hright

theorem lessThanTrace (slots : Fin 4 ↪ K) (labels : Label → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State σ))
    (atLabels : ∀ l, program (labels l) = statement slots labels exit l)
    (base : K → List Bool) (a b : Nat) (leftSuffix rightSuffix : List Bool)
    (leftWord : base (slots 0) = encodeWord a ++ leftSuffix)
    (rightWord : base (slots 1) = encodeWord b ++ rightSuffix)
    (leftEmpty : base (slots 2) = []) (rightEmpty : base (slots 3) = [])
    (ambient : σ) (flag : Bool) (register : Option Bool) :
    (advance (TM2.step program))^[steps a b]
      (some ⟨some (labels .scan), ((ambient, flag), register), base⟩) =
      some ⟨exit, ((ambient, decide (a < b)), none), base⟩ := by
  have h := comparisonTrace slots labels exit program atLabels base a b leftSuffix rightSuffix
    ambient flag register
  have hframe := tapes_self slots base
  rw [leftWord, rightWord, leftEmpty, rightEmpty] at hframe
  rw [hframe] at h
  exact h

noncomputable def timePolynomial : Polynomial Nat := Polynomial.C 3 * Polynomial.X + Polynomial.C 3

theorem steps_le_time (a b : Nat) : steps a b ≤ timePolynomial.eval (a + b) := by
  simp only [steps, timePolynomial, Polynomial.eval_add, Polynomial.eval_mul,
    Polynomial.eval_C, Polynomial.eval_X]
  have := Nat.min_le_left a b
  omega

def lessThanInPolynomialTime (slots : Fin 4 ↪ K) (labels : Label → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State σ))
    (atLabels : ∀ l, program (labels l) = statement slots labels exit l)
    (base : K → List Bool) (a b : Nat) (leftSuffix rightSuffix : List Bool)
    (leftWord : base (slots 0) = encodeWord a ++ leftSuffix)
    (rightWord : base (slots 1) = encodeWord b ++ rightSuffix)
    (leftEmpty : base (slots 2) = []) (rightEmpty : base (slots 3) = [])
    (ambient : σ) (flag : Bool) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some (labels .scan), ((ambient, flag), register), base⟩
      (some ⟨exit, ((ambient, decide (a < b)), none), base⟩) (timePolynomial.eval (a + b)) where
  steps := steps a b
  evals_in_steps := lessThanTrace slots labels exit program atLabels base a b leftSuffix rightSuffix
    leftWord rightWord leftEmpty rightEmpty ambient flag register
  steps_le_m := steps_le_time a b

abbrev machine : Turing.FinTM2 where
  K := Fin 4
  k₀ := 0
  k₁ := 1
  Γ _ := Bool
  Λ := Label
  main := .scan
  σ := State Unit
  initialState := (((), false), none)
  m := statement (Function.Embedding.refl _) id none

def machineInPolynomialTime (base : Fin 4 → List Bool) (a b : Nat)
    (leftSuffix rightSuffix : List Bool)
    (leftWord : base 0 = encodeWord a ++ leftSuffix)
    (rightWord : base 1 = encodeWord b ++ rightSuffix)
    (leftEmpty : base 2 = []) (rightEmpty : base 3 = []) :
    StateTransition.EvalsToInTime machine.step
      ⟨some .scan, (((), false), none), base⟩
      (some ⟨none, (((), decide (a < b)), none), base⟩) (timePolynomial.eval (a + b)) :=
  lessThanInPolynomialTime (Function.Embedding.refl _) id none machine.m (fun _ => rfl)
    base a b leftSuffix rightSuffix leftWord rightWord leftEmpty rightEmpty () false none

end DFVSGames.Foundations.Complexity.MachineUnaryLessAt
end

end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end

end OAI