lean/OAI/Computability/DirectedFeedback/Machines/DummySelectSteps.lean

import Mathlib
import OAI.Computability.DirectedFeedback.Machines.MachineRegularInternalRow

namespace OAI

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namespace DFVSGames.Foundations.Complexity.MachineRegularInternalRow
open Turing MachineComposition
open PCP
open Reduction.MachineSubstitution (pushWord stepAux_pushWord)
variable {K Λ σ : Type} [DecidableEq K]

def dummySelectSteps (j k o m : Nat) : Nat :=
  (2 * (j + 1) + 1) + (2 * (k + 1) + 1) + (k + 1) +
    (2 * (j - k + 1) + 1) + 2 * (o + 1) + 2 * (m + 1) + (j + k + 7)

theorem dummySelectSteps_eq (j k o m : Nat) (h : k ≤ j) :
    dummySelectSteps j k o m = 5 * j + 2 * k + 2 * o + 2 * m + 21 := by
  unfold dummySelectSteps
  omega

theorem selectDummyTrace (graph : List Bool) (j v k o m : Nat) (hk : k ≤ j)
    (ambient : σ) :
    (advance (TM2.step selectProgram))^[dummySelectSteps j k o m]
      (some ⟨some (.inr (.copy .dummyLocal .seed)), selectCleanState ambient none,
        selectMemory graph j v k o m [] [] [] [] [] []⟩) =
      some ⟨none, selectCleanState ambient none,
        selectMemory graph j v k o m [] (encodeWord (m + o + (j - k))) [] [] [] []⟩ := by
  have hjcopy := selectCopyTrace .dummyLocal
    (selectMemory graph j v k o m [] [] [] [] [] []) j rfl rfl ambient none
  simp only [copyDestination, afterCopy, selectMemory_extra, selectExtraMemory,
    update_selectMemory_local, List.append_nil] at hjcopy
  have hkcopy := selectCopyTrace .dummySize
    (selectMemory graph j v k o m [] [] [] [] (encodeWord j) []) k rfl rfl ambient none
  simp only [copyDestination, afterCopy, selectMemory_extra, selectExtraMemory,
    update_selectMemory_size, List.append_nil] at hkcopy
  have scanFrame (left right savedLeft savedRight : List Bool) :
      MachineUnaryLessAt.tapes subtractSlots
        (selectMemory graph j v k o m [] [] [] [] [] [])
        left right savedLeft savedRight =
      selectMemory graph j v k o m [] [] savedLeft savedRight left right := by
    simp only [MachineUnaryLessAt.tapes,
      show subtractSlots 0 = .inr .localWork from rfl,
      show subtractSlots 1 = .inr .sizeWork from rfl,
      show subtractSlots 2 = .inr .savedLeft from rfl,
      show subtractSlots 3 = .inr .savedRight from rfl,
      update_selectMemory_local, update_selectMemory_size,
      update_selectMemory_left, update_selectMemory_right]
  have hscan := MachineUnaryLessAt.scanTrace subtractSlots (.inr .subtract)
    (.inr (.copy .dummyDifference .seed)) selectProgram
    (by rw [selectProgram_control]; rfl)
    (selectMemory graph j v k o m [] [] [] [] [] [])
    j k [] [] [] [] (ambient, MachineCloudSelect.Phase.checking) false none
  simp only [List.append_nil, scanFrame, Nat.min_eq_right hk, Nat.sub_eq_zero_of_le hk,
    show ¬ j < k from Nat.not_lt_of_ge hk, decide_false] at hscan
  have hdiffcopy := selectCopyTrace .dummyDifference
    (selectMemory graph j v k o m [] [] (List.replicate k true) (List.replicate k true)
      (encodeWord (j - k)) (encodeWord 0)) (j - k) rfl rfl ambient none
  simp only [copyDestination, afterCopy, selectMemory_output, update_selectMemory_output, List.append_nil] at hdiffcopy
  have hprefix := selectAddTrace .prefixOffset
    (selectMemory graph j v k o m [] (encodeWord (j - k))
      (List.replicate k true) (List.replicate k true) (encodeWord (j - k)) (encodeWord 0))
    o (j - k) rfl rfl rfl ambient
  simp only [afterAdd, update_selectMemory_output] at hprefix
  have hm := selectAddTrace .darts
    (selectMemory graph j v k o m [] (encodeWord (o + (j - k)))
      (List.replicate k true) (List.replicate k true) (encodeWord (j - k)) (encodeWord 0))
    m (o + (j - k)) rfl rfl rfl ambient
  simp only [afterAdd, update_selectMemory_output, ← Nat.add_assoc] at hm
  have hcleanup := selectCleanupTrace graph j v k o m []
    (encodeWord (m + o + (j - k))) (List.replicate k true) (List.replicate k true)
    (encodeWord (j - k)) (encodeWord 0) ambient
  have hcleanSteps : ([] : List Bool).length + (encodeWord (j - k)).length + (encodeWord 0).length +
      (List.replicate k true).length + (List.replicate k true).length + 5 = j + k + 7 := by
    simp only [List.length_nil, encodeWord_length, List.length_replicate]
    omega
  rw [hcleanSteps] at hcleanup
  exact joinTrace_inline_MachineRegularInternalRow (joinTrace_inline_MachineRegularInternalRow (joinTrace_inline_MachineRegularInternalRow (joinTrace_inline_MachineRegularInternalRow (joinTrace_inline_MachineRegularInternalRow (joinTrace_inline_MachineRegularInternalRow
    hjcopy hkcopy) hscan) hdiffcopy) hprefix) hm) hcleanup

def selectionSteps (t : GraphTables.Table) (padding : Fin t.vertices → Nat)
    (v : Fin t.vertices) (i : Fin (PreprocessingCloudIndex.cloudSize t v + padding v)) : Nat :=
  MachineUnaryLessAt.steps i.val (PreprocessingCloudIndex.cloudSize t v) + 1 +
    if hi : i.val < PreprocessingCloudIndex.cloudSize t v then
      oldSelectSteps t v ⟨i.val, hi⟩
    else dummySelectSteps i.val (PreprocessingCloudIndex.cloudSize t v)
      (PreprocessingPaddingOffsets.offset padding v.val) t.darts

theorem selectionTrace (t : GraphTables.Table) (padding : Fin t.vertices → Nat)
    (v : Fin t.vertices) (i : Fin (PreprocessingCloudIndex.cloudSize t v + padding v))
    (ambient : σ) (register : Option Bool) :
    (advance (TM2.step selectProgram))^[selectionSteps t padding v i]
      (some ⟨some (.inr (.compare .scan)), selectCleanState ambient register,
        selectMemory (GraphTables.tableBits t) i.val v.val
          (PreprocessingCloudIndex.cloudSize t v)
          (PreprocessingPaddingOffsets.offset padding v.val) t.darts [] [] [] [] [] []⟩) =
      some ⟨none, selectCleanState ambient none,
        selectMemory (GraphTables.tableBits t) i.val v.val
          (PreprocessingCloudIndex.cloudSize t v)
          (PreprocessingPaddingOffsets.offset padding v.val) t.darts []
          (encodeWord (PreprocessingInternalRows.selectedIndex t padding v i)) [] [] [] []⟩ := by
  have hc := selectComparisonTrace
    (selectMemory (GraphTables.tableBits t) i.val v.val
      (PreprocessingCloudIndex.cloudSize t v)
      (PreprocessingPaddingOffsets.offset padding v.val) t.darts [] [] [] [] [] [])
    i.val (PreprocessingCloudIndex.cloudSize t v) rfl rfl rfl rfl ambient register
  by_cases hi : i.val < PreprocessingCloudIndex.cloudSize t v
  · simp only [ite_eq_left hi] at hc
    have hb := selectOldTrace t v ⟨i.val, hi⟩
      (PreprocessingPaddingOffsets.offset padding v.val) ambient
    simpa only [selectionSteps, dite_eq_left hi,
      PreprocessingInternalRows.selectedIndex_old t padding v i hi] using joinTrace_inline_MachineRegularInternalRow hc hb
  · simp only [ite_eq_right hi] at hc
    have hb := selectDummyTrace (GraphTables.tableBits t) i.val v.val
      (PreprocessingCloudIndex.cloudSize t v)
      (PreprocessingPaddingOffsets.offset padding v.val) t.darts (Nat.le_of_not_gt hi) ambient
    simpa only [selectionSteps, dite_eq_right hi,
      PreprocessingInternalRows.selectedIndex_dummy t padding v i (Nat.le_of_not_gt hi)]
      using joinTrace_inline_MachineRegularInternalRow hc hb

theorem selectionSteps_le (t : GraphTables.Table) (padding : Fin t.vertices → Nat)
    (v : Fin t.vertices) (i : Fin (PreprocessingCloudIndex.cloudSize t v + padding v)) :
    selectionSteps t padding v i ≤
      10 * (GraphTables.tableBits t).length + 16 * i.val + 13 * v.val +
      5 * PreprocessingCloudIndex.cloudSize t v +
      2 * PreprocessingPaddingOffsets.offset padding v.val + 2 * t.darts + 57 := by
  unfold selectionSteps
  split
  · rename_i hi
    have hb := MachineCloudSelect.cloudSelectSteps_le t v ⟨i.val, hi⟩ []
    have hmin : min i.val (PreprocessingCloudIndex.cloudSize t v) = i.val := Nat.min_eq_left hi.le
    simp only [oldSelectSteps, MachineUnaryLessAt.steps, hmin]
    simp only [MachineCloudSelect.queryWord, MachineCloudRank.queryWord,
      List.length_append, encodeWord_length, List.length_nil] at hb
    omega
  · rename_i hi
    rw [dummySelectSteps_eq _ _ _ _ (Nat.le_of_not_gt hi)]
    have hmin := Nat.min_le_right i.val (PreprocessingCloudIndex.cloudSize t v)
    unfold MachineUnaryLessAt.steps
    omega

def selectorMachine : FinTM2 where
  K := SelectTape
  k₀ := .inl (.inl .original)
  k₁ := selectOutput
  Γ := SelectAlphabet
  Λ := SelectLabel
  main := .inr (.compare .scan)
  σ := SelectState Unit
  initialState := selectCleanState () none
  m := selectProgram

def selectionInTime (t : GraphTables.Table) (padding : Fin t.vertices → Nat)
    (v : Fin t.vertices) (i : Fin (PreprocessingCloudIndex.cloudSize t v + padding v))
    (register : Option Bool) :
    StateTransition.EvalsToInTime selectorMachine.step
      ⟨some (.inr (.compare .scan)), selectCleanState () register,
        selectMemory (GraphTables.tableBits t) i.val v.val
          (PreprocessingCloudIndex.cloudSize t v)
          (PreprocessingPaddingOffsets.offset padding v.val) t.darts [] [] [] [] [] []⟩
      (some ⟨none, selectCleanState () none,
        selectMemory (GraphTables.tableBits t) i.val v.val
          (PreprocessingCloudIndex.cloudSize t v)
          (PreprocessingPaddingOffsets.offset padding v.val) t.darts []
          (encodeWord (PreprocessingInternalRows.selectedIndex t padding v i)) [] [] [] []⟩)
      (10 * (GraphTables.tableBits t).length + 16 * i.val + 13 * v.val +
        5 * PreprocessingCloudIndex.cloudSize t v +
        2 * PreprocessingPaddingOffsets.offset padding v.val + 2 * t.darts + 57) where
  steps := selectionSteps t padding v i
  evals_in_steps := selectionTrace t padding v i () register
  steps_le_m := selectionSteps_le t padding v i

def coreRotorTapes : Fin 8 → CoreTape := ![3, 7, 10, 11, 12, 9, 13, 14]

def coreFinishTapes : Fin 8 → CoreTape := ![1, 15, 14, 9, 24, 25, 26, 8]

def coreSelectTape : SelectTape → CoreTape
  | .inl (.inl .original) => 0
  | .inl (.inl .work) => 17
  | .inl (.inl .target) => 16
  | .inl (.inl .scratch) => 9
  | .inl (.inl .count) => 18
  | .inl (.inl .spare) => 19
  | .inl (.inr ()) => 15
  | .inr .localIndex => 13
  | .inr .owner => 2
  | .inr .cloudSize => 4
  | .inr .prefixOffset => 5
  | .inr .darts => 6
  | .inr .savedLeft => 20
  | .inr .savedRight => 21
  | .inr .localWork => 22
  | .inr .sizeWork => 23

def coreSelectView : CoreTape → Option SelectTape :=
  ![some (.inl (.inl .original)), none, some (.inr .owner), none,
    some (.inr .cloudSize), some (.inr .prefixOffset), some (.inr .darts), none, none,
    some (.inl (.inl .scratch)), none, none, none, some (.inr .localIndex), none,
    some (.inl (.inr ())), some (.inl (.inl .target)), some (.inl (.inl .work)),
    some (.inl (.inl .count)), some (.inl (.inl .spare)),
    some (.inr .savedLeft), some (.inr .savedRight),
    some (.inr .localWork), some (.inr .sizeWork), none, none, none]

theorem coreSelectView_left (k : SelectTape) : coreSelectView (coreSelectTape k) = some k := by
  cases k with
  | inl k =>
    cases k with
    | inl k => cases k <;> rfl
    | inr k => cases k; rfl
  | inr k => cases k <;> rfl

theorem coreSelectView_right (j : CoreTape) (k : SelectTape)
    (h : coreSelectView j = some k) : coreSelectTape k = j := by
  fin_cases j <;> simp [coreSelectView] at h
  all_goals subst k; rfl

def coreCleanupTape : Fin 10 → CoreTape := ![10, 11, 12, 13, 14, 15, 9, 16, 17, 18]

def coreRotorStateEquiv (σ : Type) (q : Nat) :
    MachineFixedDivMod.State
      ((σ × MachineFixedBlockMap.Buffer 4096) × (Bool × MachineCloudSelect.Phase)) q ≃
        CoreState σ q where
  toFun s := ((s.1.1.1.1, (s.1.1.1.2, (s.1.2, s.1.1.2))), s.2)
  invFun s := ((((s.1.1, s.1.2.1), s.1.2.2.2), s.1.2.2.1), s.2)
  left_inv s := by rcases s with ⟨⟨⟨⟨a,b⟩,⟨c,d⟩⟩,e⟩,f⟩; rfl
  right_inv s := by rcases s with ⟨⟨a,⟨b,⟨c,⟨d,e⟩⟩⟩⟩,f⟩; rfl

def coreSelectStateEquiv (σ : Type) (q : Nat) :
    SelectState (σ × (MachineFixedBlockMap.Buffer 4096 × Fin q)) ≃ CoreState σ q where
  toFun s := ((s.1.1.1.1, (s.1.1.1.2.1, (s.1.1.1.2.2, (s.1.2, s.1.1.2)))), s.2)
  invFun s := ((((s.1.1, (s.1.2.1, s.1.2.2.1)), s.1.2.2.2.2), s.1.2.2.2.1), s.2)
  left_inv s := by rcases s with ⟨⟨⟨⟨a,⟨b,c⟩⟩,d⟩,e⟩,f⟩; rfl
  right_inv s := by rcases s with ⟨⟨a,⟨b,⟨c,⟨d,e⟩⟩⟩⟩,f⟩; rfl

def coreAfterCleanup {q : Nat} (labels : CoreLabel q → Λ) (exit : Option Λ) (i : Fin 10) :
    Option Λ :=
  if h : i.val + 1 < 10 then some (labels (.cleanup ⟨i.val + 1,h⟩)) else exit

def coreInstruction (q : Nat) (positive : 0 < q) (p : Fin q)
    (labels : CoreLabel q → Λ) (exit : Option Λ) :
    CoreLabel q → TM2.Stmt (fun _ : CoreTape => Bool) Λ (CoreState σ q)
  | .rotor stage => MachineStateEquiv.statement (coreRotorStateEquiv σ q)
      (rotorInstruction q positive p coreRotorTapes
        (fun l => labels (.rotor l)) (some (labels (.select (.inr (.compare .scan))))) stage)
  | .select stage => MachineCloudPadding.Placement.statement coreSelectTape
      (fun l => labels (.select l)) (some (labels (.finish .seedReturn)))
      (MachineStateEquiv.statement (coreSelectStateEquiv σ q) (selectProgram stage))
  | .finish stage => finishInstruction q coreFinishTapes
      (fun l => labels (.finish l)) (some (labels (.cleanup 0))) stage
  | .cleanup i => MachineDrain.drain (coreCleanupTape i) (labels (.cleanup i))
      (coreAfterCleanup labels exit i)

def coreInitialState (q : Nat) (positive : 0 < q) (ambient : σ) : CoreState σ q :=
  ((ambient, (MachineFixedBlockMap.emptyBuffer 4096,
    (MachineFixedDivMod.residue q positive 0, (false, MachineCloudSelect.Phase.checking)))), none)

def coreMachine (q : Nat) (positive : 0 < q) (p : Fin q) : FinTM2 where
  K := CoreTape
  k₀ := 0
  k₁ := 8
  Γ _ := Bool
  Λ := CoreLabel q
  main := coreEntry q
  σ := CoreState Unit q
  initialState := coreInitialState q positive ()
  m := coreInstruction q positive p id none

def coreMemory (graph : List Bool) (x v i k o m : Nat) (rotor output : List Bool)
    (query scan flat localIndex returnPort selected : List Bool) : CoreTape → List Bool :=
  ![graph, encodeWord x, encodeWord v, encodeWord i, encodeWord k, encodeWord o,
    encodeWord m, rotor, output, [], query, scan, flat, localIndex, returnPort, selected,
    [], [], [], [], [], [], [], [], [], [], []]

@[simp] theorem coreMemory_update_output (graph : List Bool) (x v i k o m : Nat)
    (rotor output query scan flat localIndex returnPort selected newOutput : List Bool) :
    Function.update (coreMemory graph x v i k o m rotor output query scan flat
      localIndex returnPort selected) (8 : CoreTape) newOutput =
      coreMemory graph x v i k o m rotor newOutput query scan flat localIndex returnPort selected := by
  funext z
  fin_cases z <;> rfl

theorem coreRotorTapes_injective : Function.Injective coreRotorTapes := by
  intro i j h
  fin_cases i <;> fin_cases j <;> simp_all [coreRotorTapes]

theorem coreFinishTapes_injective : Function.Injective coreFinishTapes := by
  intro i j h
  fin_cases i <;> fin_cases j <;> simp_all [coreFinishTapes]

def coreCleaned (base : CoreTape → List Bool) : CoreTape → List Bool :=
  Function.update (Function.update (Function.update (Function.update (Function.update
    (Function.update (Function.update (Function.update (Function.update
      (Function.update base 10 []) 11 []) 12 []) 13 []) 14 []) 15 []) 9 []) 16 []) 17 []) 18 []

def coreCleanupSteps (base : CoreTape → List Bool) : Nat :=
  (base 10).length + (base 11).length + (base 12).length + (base 13).length +
  (base 14).length + (base 15).length + (base 9).length + (base 16).length +
  (base 17).length + (base 18).length + 10

theorem coreCleanupTrace (q : Nat) (positive : 0 < q) (p : Fin q)
    (labels : CoreLabel q → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (fun _ : CoreTape => Bool) Λ (CoreState σ q))
    (code : ∀ l, program (labels l) = coreInstruction q positive p labels exit l)
    (base : CoreTape → List Bool)
    (ambient : σ × (MachineFixedBlockMap.Buffer 4096 ×
      (Fin q × (Bool × MachineCloudSelect.Phase)))) :
    (advance (TM2.step program))^[coreCleanupSteps base]
      (some ⟨some (labels (.cleanup 0)), (ambient, none), base⟩) =
      some ⟨exit, (ambient, none), coreCleaned base⟩ := by
  let b1 := Function.update base (10 : CoreTape) []
  have h0 := MachineDrain.drainTrace (10 : CoreTape) (labels (.cleanup 0))
    (some (labels (.cleanup 1))) program (code (.cleanup 0)) base (base 10) ambient none
  rw [Function.update_eq_self] at h0
  let b2 := Function.update b1 (11 : CoreTape) []
  have h1 := MachineDrain.drainTrace (11 : CoreTape) (labels (.cleanup 1))
    (some (labels (.cleanup 2))) program (code (.cleanup 1)) b1 (b1 11) ambient none
  rw [Function.update_eq_self] at h1
  have hw1 : b1 11 = base 11 := by simp [b1]
  rw [hw1] at h1
  let b3 := Function.update b2 (12 : CoreTape) []
  have h2 := MachineDrain.drainTrace (12 : CoreTape) (labels (.cleanup 2))
    (some (labels (.cleanup 3))) program (code (.cleanup 2)) b2 (b2 12) ambient none
  rw [Function.update_eq_self] at h2
  have hw2 : b2 12 = base 12 := by simp [b1, b2]
  rw [hw2] at h2
  let b4 := Function.update b3 (13 : CoreTape) []
  have h3 := MachineDrain.drainTrace (13 : CoreTape) (labels (.cleanup 3))
    (some (labels (.cleanup 4))) program (code (.cleanup 3)) b3 (b3 13) ambient none
  rw [Function.update_eq_self] at h3
  have hw3 : b3 13 = base 13 := by simp [b1, b2, b3]
  rw [hw3] at h3
  let b5 := Function.update b4 (14 : CoreTape) []
  have h4 := MachineDrain.drainTrace (14 : CoreTape) (labels (.cleanup 4))
    (some (labels (.cleanup 5))) program (code (.cleanup 4)) b4 (b4 14) ambient none
  rw [Function.update_eq_self] at h4
  have hw4 : b4 14 = base 14 := by simp [b1, b2, b3, b4]
  rw [hw4] at h4
  let b6 := Function.update b5 (15 : CoreTape) []
  have h5 := MachineDrain.drainTrace (15 : CoreTape) (labels (.cleanup 5))
    (some (labels (.cleanup 6))) program (code (.cleanup 5)) b5 (b5 15) ambient none
  rw [Function.update_eq_self] at h5
  have hw5 : b5 15 = base 15 := by simp [b1, b2, b3, b4, b5]
  rw [hw5] at h5
  let b7 := Function.update b6 (9 : CoreTape) []
  have h6 := MachineDrain.drainTrace (9 : CoreTape) (labels (.cleanup 6))
    (some (labels (.cleanup 7))) program (code (.cleanup 6)) b6 (b6 9) ambient none
  rw [Function.update_eq_self] at h6
  have hw6 : b6 9 = base 9 := by simp [b1, b2, b3, b4, b5, b6]
  rw [hw6] at h6
  let b8 := Function.update b7 (16 : CoreTape) []
  have h7 := MachineDrain.drainTrace (16 : CoreTape) (labels (.cleanup 7))
    (some (labels (.cleanup 8))) program (code (.cleanup 7)) b7 (b7 16) ambient none
  rw [Function.update_eq_self] at h7
  have hw7 : b7 16 = base 16 := by simp [b1, b2, b3, b4, b5, b6, b7]
  rw [hw7] at h7
  let b9 := Function.update b8 (17 : CoreTape) []
  have h8 := MachineDrain.drainTrace (17 : CoreTape) (labels (.cleanup 8))
    (some (labels (.cleanup 9))) program (code (.cleanup 8)) b8 (b8 17) ambient none
  rw [Function.update_eq_self] at h8
  have hw8 : b8 17 = base 17 := by simp [b1, b2, b3, b4, b5, b6, b7, b8]
  rw [hw8] at h8
  let b10 := Function.update b9 (18 : CoreTape) []
  have h9 := MachineDrain.drainTrace (18 : CoreTape) (labels (.cleanup 9))
    exit program (code (.cleanup 9)) b9 (b9 18) ambient none
  rw [Function.update_eq_self] at h9
  have hw9 : b9 18 = base 18 := by simp [b1, b2, b3, b4, b5, b6, b7, b8, b9]
  rw [hw9] at h9
  have total := (joinTrace_inline_MachineRegularInternalRow (joinTrace_inline_MachineRegularInternalRow (joinTrace_inline_MachineRegularInternalRow (joinTrace_inline_MachineRegularInternalRow (joinTrace_inline_MachineRegularInternalRow (joinTrace_inline_MachineRegularInternalRow (joinTrace_inline_MachineRegularInternalRow (joinTrace_inline_MachineRegularInternalRow (joinTrace_inline_MachineRegularInternalRow h0 h1) h2) h3) h4) h5) h6) h7) h8) h9)
  have hn : coreCleanupSteps base =
      (base 10).length + 1 + ((base 11).length + 1) + ((base 12).length + 1) +
      ((base 13).length + 1) + ((base 14).length + 1) + ((base 15).length + 1) +
      ((base 9).length + 1) + ((base 16).length + 1) + ((base 17).length + 1) +
      ((base 18).length + 1) := by unfold coreCleanupSteps; omega
  rw [hn]
  exact total

theorem coreCleaned_memory (graph : List Bool) (x v i k o m : Nat)
    (rotor output query scan flat localIndex returnPort selected : List Bool) :
    coreCleaned (coreMemory graph x v i k o m rotor output query scan flat
      localIndex returnPort selected) =
      coreMemory graph x v i k o m rotor output [] [] [] [] [] [] := by
  funext z
  fin_cases z <;> rfl

theorem coreCleanupSteps_memory (graph : List Bool) (x v i k o m : Nat)
    (rotor output scan : List Bool) (j r z : Nat) :
    coreCleanupSteps (coreMemory graph x v i k o m rotor output
      (encodeWord 0) scan (encodeWord 0) (encodeWord j) (encodeWord r) (encodeWord z)) =
      scan.length + j + r + z + 15 := by
  change (encodeWord 0).length + scan.length + (encodeWord 0).length +
    (encodeWord j).length + (encodeWord r).length + (encodeWord z).length +
    ([] : List Bool).length + ([] : List Bool).length + ([] : List Bool).length +
    ([] : List Bool).length + 10 = _
  simp only [encodeWord_length, List.length_nil]
  omega

theorem coreRotorFinalFrame {n q : Nat} (table : ExpanderTables.Table n q) (i : Fin n) (p : Fin q)
    (graph : List Bool) (x v k o m : Nat) (output : List Bool) :
    rotorFinalTapes table i p coreRotorTapes
      (coreMemory graph x v i.val k o m
        (encodeWords (ExpanderTableWords.rotationWords table)) output [] [] [] [] [] []) =
    coreMemory graph x v i.val k o m
      (encodeWords (ExpanderTableWords.rotationWords table)) output (encodeWord 0)
      (encodeWords ((ExpanderTableWords.rotationWords table).drop (q * i.val + p.val + 1)))
      (encodeWord 0) (encodeWord (ExpanderTables.lookup table (i, p)).1.val)
      (encodeWord (ExpanderTables.lookup table (i, p)).2.val) [] := by
  funext z
  fin_cases z <;>
    simp [rotorFinalTapes, coreRotorTapes, rotorLookupTapes, coreMemory,
      MachineFixedDivMod.unaryTapes, MachineFixedDivMod.tapes, MachineCopy.forkTapes,
      MachineAffineLookup.finalTapes, MachinePreservingLookup.finalTapes, MachineLookup.tapes]

theorem coreRotorTrace {n q : Nat} (positive : 0 < q) (table : ExpanderTables.Table n q)
    (i : Fin n) (p : Fin q) (labels : CoreLabel q → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (fun _ : CoreTape => Bool) Λ (CoreState σ q))
    (code : ∀ l, program (labels l) = coreInstruction q positive p labels exit l)
    (graph : List Bool) (x v k o m : Nat) (output : List Bool) (ambient : σ) :
    (advance (TM2.step program))^[rotorSteps table i p]
      (some ⟨some (labels (coreEntry q)), coreInitialState q positive ambient,
        coreMemory graph x v i.val k o m
          (encodeWords (ExpanderTableWords.rotationWords table)) output [] [] [] [] [] []⟩) =
      some ⟨some (labels (.select (.inr (.compare .scan)))), coreInitialState q positive ambient,
        coreMemory graph x v i.val k o m
          (encodeWords (ExpanderTableWords.rotationWords table)) output (encodeWord 0)
          (encodeWords ((ExpanderTableWords.rotationWords table).drop (q * i.val + p.val + 1)))
          (encodeWord 0) (encodeWord (ExpanderTables.lookup table (i, p)).1.val)
          (encodeWord (ExpanderTables.lookup table (i, p)).2.val) []⟩ := by
  let e := coreRotorStateEquiv σ q
  let backwards := MachineStateEquiv.program e.symm program
  have atRotor (l : RotorLabel q) :
      backwards (labels (.rotor l)) = rotorInstruction q positive p coreRotorTapes
        (fun l => labels (.rotor l)) (some (labels (.select (.inr (.compare .scan))))) l := by
    change MachineStateEquiv.statement e.symm (program (labels (.rotor l))) = _
    rw [code]
    exact MachineStateEquiv.statement_symm_statement e _
  have run := rotorTrace positive table i p coreRotorTapes coreRotorTapes_injective
    (fun l => labels (.rotor l)) (some (labels (.select (.inr (.compare .scan)))))
    backwards atRotor
    (coreMemory graph x v i.val k o m
      (encodeWords (ExpanderTableWords.rotationWords table)) output [] [] [] [] [] [])
    rfl rfl rfl rfl rfl rfl
    ((ambient, MachineFixedBlockMap.emptyBuffer 4096), (false, MachineCloudSelect.Phase.checking)) none
  have h := MachineStateEquiv.trace e backwards _ _ _ run
  have backForward : MachineStateEquiv.program e backwards = program := by
    have hp := MachineStateEquiv.program_symm_program e.symm program
    simpa only [Equiv.symm_symm, backwards] using hp
  rw [backForward] at h
  change (advance (TM2.step program))^[rotorSteps table i p]
    (some ⟨some (labels (coreEntry q)), coreInitialState q positive ambient,
      coreMemory graph x v i.val k o m
        (encodeWords (ExpanderTableWords.rotationWords table)) output [] [] [] [] [] []⟩) =
    some ⟨some (labels (.select (.inr (.compare .scan)))), coreInitialState q positive ambient,
      rotorFinalTapes table i p coreRotorTapes
        (coreMemory graph x v i.val k o m
          (encodeWords (ExpanderTableWords.rotationWords table)) output [] [] [] [] [] [])⟩ at h
  rw [coreRotorFinalFrame] at h
  exact h

theorem coreSelectFrame (graph : List Bool) (x v i k o m j : Nat)
    (rotor output query scan flat returnPort selected : List Bool) :
    MachineCloudPadding.Placement.tapes coreSelectView
      (selectMemory graph j v k o m [] selected [] [] [] [])
      (coreMemory graph x v i k o m rotor output query scan flat (encodeWord j) returnPort []) =
      coreMemory graph x v i k o m rotor output query scan flat (encodeWord j) returnPort selected := by
  funext z
  fin_cases z <;> rfl

theorem coreSelectionTrace (q : Nat) (positive : 0 < q) (p : Fin q)
    (labels : CoreLabel q → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (fun _ : CoreTape => Bool) Λ (CoreState σ q))
    (code : ∀ l, program (labels l) = coreInstruction q positive p labels exit l)
    (t : GraphTables.Table) (padding : Fin t.vertices → Nat) (v : Fin t.vertices)
    (j : Fin (PreprocessingCloudIndex.cloudSize t v + padding v))
    (x i : Nat) (rotor output query scan flat returnPort : List Bool) (ambient : σ) :
    (advance (TM2.step program))^[selectionSteps t padding v j]
      (some ⟨some (labels (.select (.inr (.compare .scan)))), coreInitialState q positive ambient,
        coreMemory (GraphTables.tableBits t) x v.val i (PreprocessingCloudIndex.cloudSize t v)
          (PreprocessingPaddingOffsets.offset padding v.val) t.darts rotor output query scan flat
          (encodeWord j.val) returnPort []⟩) =
      some ⟨some (labels (.finish .seedReturn)), coreInitialState q positive ambient,
        coreMemory (GraphTables.tableBits t) x v.val i (PreprocessingCloudIndex.cloudSize t v)
          (PreprocessingPaddingOffsets.offset padding v.val) t.darts rotor output query scan flat
          (encodeWord j.val) returnPort
          (encodeWord (PreprocessingInternalRows.selectedIndex t padding v j))⟩ := by
  let e := coreSelectStateEquiv σ q
  let state := (ambient, (MachineFixedBlockMap.emptyBuffer 4096,
    MachineFixedDivMod.residue q positive 0))
  have sourceRun := selectionTrace t padding v j state none
  have moved := MachineStateEquiv.trace e selectProgram _ _ _ sourceRun
  let base := coreMemory (GraphTables.tableBits t) x v.val i
    (PreprocessingCloudIndex.cloudSize t v)
    (PreprocessingPaddingOffsets.offset padding v.val) t.darts rotor output query scan flat
    (encodeWord j.val) returnPort []
  have placed := MachineCloudPadding.Placement.trace coreSelectTape coreSelectView
    coreSelectView_left coreSelectView_right (fun l => labels (.select l))
    (some (labels (.finish .seedReturn))) base
    (MachineStateEquiv.program e selectProgram) program (fun l => code (.select l))
    _ _ _ moved
  simp only [MachineCloudPadding.Placement.configuration, MachineCloudPadding.Placement.label,
    MachineStateEquiv.configuration, base, coreSelectFrame] at placed
  convert placed using 1 <;> rfl

theorem coreFinishTrace (q : Nat) (positive : 0 < q) (p : Fin q)
    (labels : CoreLabel q → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (fun _ : CoreTape => Bool) Λ (CoreState σ q))
    (code : ∀ l, program (labels l) = coreInstruction q positive p labels exit l)
    (graph : List Bool) (x v i k o m j r z : Nat)
    (rotor output query scan flat : List Bool) (ambient : σ) :
    (advance (TM2.step program))^[finishSteps q x z r output.length]
      (some ⟨some (labels (.finish .seedReturn)), coreInitialState q positive ambient,
        coreMemory graph x v i k o m rotor output query scan flat
          (encodeWord j) (encodeWord r) (encodeWord z)⟩) =
      some ⟨some (labels (.cleanup 0)), coreInitialState q positive ambient,
        coreMemory graph x v i k o m rotor (output ++ emittedBits q x z r) query scan flat
          (encodeWord j) (encodeWord r) (encodeWord z)⟩ := by
  have h := finishTrace q coreFinishTapes coreFinishTapes_injective
    (fun l => labels (.finish l)) (some (labels (.cleanup 0))) program
    (fun l => code (.finish l))
    (coreMemory graph x v i k o m rotor output query scan flat
      (encodeWord j) (encodeWord r) (encodeWord z))
    x z r rfl rfl rfl rfl rfl rfl rfl
    (ambient, (MachineFixedBlockMap.emptyBuffer 4096,
      (MachineFixedDivMod.residue q positive 0, (false, MachineCloudSelect.Phase.checking)))) none
  change (advance (TM2.step program))^[finishSteps q x z r output.length]
    (some ⟨some (labels (.finish .seedReturn)), coreInitialState q positive ambient,
      coreMemory graph x v i k o m rotor output query scan flat
        (encodeWord j) (encodeWord r) (encodeWord z)⟩) =
    some ⟨some (labels (.cleanup 0)), coreInitialState q positive ambient,
      Function.update (coreMemory graph x v i k o m rotor output query scan flat
        (encodeWord j) (encodeWord r) (encodeWord z)) (8 : CoreTape)
        (output ++ emittedBits q x z r)⟩ at h
  rw [coreMemory_update_output] at h
  exact h

def coreInputTapes (t : GraphTables.Table) (padding : Fin t.vertices → Nat) {q : Nat}
    (tables : ∀ v, ExpanderTables.Table (PreprocessingCloudIndex.cloudSize t v + padding v) q)
    (v : Fin t.vertices) (x : PreprocessingCloudIndex.PaddedCloud t padding v)
    (output : List Bool) : CoreTape → List Bool :=
  coreMemory (GraphTables.tableBits t)
    (PreprocessingRegularTables.vertexOrder t padding x.val).val v.val
    (PreprocessingCloudIndex.paddedCloudRank t padding v x).val
    (PreprocessingCloudIndex.cloudSize t v) (PreprocessingPaddingOffsets.offset padding v.val) t.darts
    (encodeWords (ExpanderTableWords.rotationWords (tables v))) output [] [] [] [] [] []

def coreSteps (t : GraphTables.Table) (padding : Fin t.vertices → Nat) {q : Nat}
    (tables : ∀ v, ExpanderTables.Table (PreprocessingCloudIndex.cloudSize t v + padding v) q)
    (v : Fin t.vertices) (x : PreprocessingCloudIndex.PaddedCloud t padding v)
    (p : Fin q) (outputLength : Nat) : Nat :=
  let rank := PreprocessingCloudIndex.paddedCloudRank t padding v x
  let step := ExpanderTables.lookup (tables v) (rank, p)
  let selected := PreprocessingInternalRows.selectedIndex t padding v step.1
  let suffixBits := encodeWords ((ExpanderTableWords.rotationWords (tables v)).drop
    (q * rank.val + p.val + 1))
  rotorSteps (tables v) rank p + selectionSteps t padding v step.1 +
    finishSteps q (PreprocessingRegularTables.vertexOrder t padding x.val).val
      selected step.2.val outputLength +
    (suffixBits.length + step.1.val + step.2.val + selected + 15)

theorem coreTrace (q : Nat) (positive : 0 < q) (p : Fin q)
    (labels : CoreLabel q → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (fun _ : CoreTape => Bool) Λ (CoreState σ q))
    (code : ∀ l, program (labels l) = coreInstruction q positive p labels exit l)
    (t : GraphTables.Table) (padding : Fin t.vertices → Nat)
    (tables : ∀ v, ExpanderTables.Table (PreprocessingCloudIndex.cloudSize t v + padding v) q)
    (v : Fin t.vertices) (x : PreprocessingCloudIndex.PaddedCloud t padding v)
    (output : List Bool) (ambient : σ) :
    (advance (TM2.step program))^[coreSteps t padding tables v x p output.length]
      (some ⟨some (labels (coreEntry q)), coreInitialState q positive ambient,
        coreInputTapes t padding tables v x output⟩) =
      some ⟨exit, coreInitialState q positive ambient,
        coreInputTapes t padding tables v x
          (output ++ encodeWords (GraphTables.rowWords
            (PreprocessingInternalRows.row t padding tables v x p)))⟩ := by
  let rank := PreprocessingCloudIndex.paddedCloudRank t padding v x
  let step := ExpanderTables.lookup (tables v) (rank, p)
  let z := PreprocessingInternalRows.selectedIndex t padding v step.1
  let X := (PreprocessingRegularTables.vertexOrder t padding x.val).val
  let o := PreprocessingPaddingOffsets.offset padding v.val
  let bits := encodeWords (ExpanderTableWords.rotationWords (tables v))
  let tailBits := encodeWords ((ExpanderTableWords.rotationWords (tables v)).drop
    (q * rank.val + p.val + 1))
  let graph := GraphTables.tableBits t
  let k := PreprocessingCloudIndex.cloudSize t v
  have hr := coreRotorTrace positive (tables v) rank p labels exit program code
    graph X v.val k o t.darts output ambient
  have hs := coreSelectionTrace q positive p labels exit program code t padding v step.1
    X rank.val bits output (encodeWord 0) tailBits (encodeWord 0) (encodeWord step.2.val) ambient
  have hf := coreFinishTrace q positive p labels exit program code
    graph X v.val rank.val k o t.darts step.1.val step.2.val z bits output
    (encodeWord 0) tailBits (encodeWord 0) ambient
  have hc := coreCleanupTrace q positive p labels exit program code
    (coreMemory graph X v.val rank.val k o t.darts bits
      (output ++ emittedBits q X z step.2.val)
      (encodeWord 0) tailBits (encodeWord 0) (encodeWord step.1.val)
      (encodeWord step.2.val) (encodeWord z))
    (ambient, (MachineFixedBlockMap.emptyBuffer 4096,
      (MachineFixedDivMod.residue q positive 0, (false, MachineCloudSelect.Phase.checking))))
  rw [coreCleanupSteps_memory, coreCleaned_memory] at hc
  have h := joinTrace_inline_MachineRegularInternalRow (joinTrace_inline_MachineRegularInternalRow (joinTrace_inline_MachineRegularInternalRow hr hs) hf) hc
  have hbits : emittedBits q X z step.2.val =
      encodeWords (GraphTables.rowWords (PreprocessingInternalRows.row t padding tables v x p)) :=
    emittedBits_eq_row t padding tables v x p
  rw [hbits] at h
  exact h

theorem selectedIndex_le (t : GraphTables.Table) (padding : Fin t.vertices → Nat)
    (v : Fin t.vertices) (i : Fin (PreprocessingCloudIndex.cloudSize t v + padding v)) :
    PreprocessingInternalRows.selectedIndex t padding v i ≤
      t.darts + PreprocessingPaddingOffsets.offset padding v.val + i.val := by
  unfold PreprocessingInternalRows.selectedIndex
  split
  · rename_i hi
    have h := (PreprocessingCloudIndex.cloudSelect t v ⟨i.val, hi⟩).val.isLt
    omega
  · omega

theorem encodedDrop_length_le (values : List Nat) (n : Nat) :
    (encodeWords (values.drop n)).length ≤ (encodeWords values).length := by
  have h : (encodeWords (values.take n)).length + (encodeWords (values.drop n)).length =
      (encodeWords values).length := by
    rw [← List.length_append, ← encodeWords_append, List.take_append_drop]
  omega

theorem rotorSize_le_encodedLength {n q : Nat} (positive : 0 < q)
    (table : ExpanderTables.Table n q) :
    n ≤ (encodeWords (ExpanderTableWords.rotationWords table)).length := by
  have h : n * q ≤ (encodeWords (ExpanderTableWords.rotationWords table)).length := by
    rw [encodeWords_length, ExpanderTableWords.rotationWords_length]
    omega
  exact (Nat.le_mul_of_pos_right n positive).trans h

def coreTimeBound (q graphLength rotorLength x v i k o m outputLength : Nat) : Nat :=
  (5 * q + 32) * rotorLength + 10 * graphLength + 2 * i + 13 * v + 5 * k +
    (5 * q + 10) * (m + o) + 4 * x + 2 * outputLength + 8 * q + 41066

theorem coreSteps_le (q : Nat) (positive : 0 < q) (t : GraphTables.Table)
    (padding : Fin t.vertices → Nat)
    (tables : ∀ v, ExpanderTables.Table (PreprocessingCloudIndex.cloudSize t v + padding v) q)
    (v : Fin t.vertices) (x : PreprocessingCloudIndex.PaddedCloud t padding v)
    (p : Fin q) (outputLength : Nat) :
    coreSteps t padding tables v x p outputLength ≤
      coreTimeBound q (GraphTables.tableBits t).length
        (encodeWords (ExpanderTableWords.rotationWords (tables v))).length
        (PreprocessingRegularTables.vertexOrder t padding x.val).val v.val
        (PreprocessingCloudIndex.paddedCloudRank t padding v x).val
        (PreprocessingCloudIndex.cloudSize t v)
        (PreprocessingPaddingOffsets.offset padding v.val) t.darts outputLength := by
  let rank := PreprocessingCloudIndex.paddedCloudRank t padding v x
  let step := ExpanderTables.lookup (tables v) (rank, p)
  let z := PreprocessingInternalRows.selectedIndex t padding v step.1
  let X := (PreprocessingRegularTables.vertexOrder t padding x.val).val
  let o := PreprocessingPaddingOffsets.offset padding v.val
  let L := (GraphTables.tableBits t).length
  let R := (encodeWords (ExpanderTableWords.rotationWords (tables v))).length
  let k := PreprocessingCloudIndex.cloudSize t v
  let tailBits := encodeWords ((ExpanderTableWords.rotationWords (tables v)).drop
    (q * rank.val + p.val + 1))
  have hN : k + padding v ≤ R := rotorSize_le_encodedLength positive (tables v)
  have hj : step.1.val ≤ R := step.1.isLt.le.trans hN
  have hr : step.2.val ≤ q := step.2.isLt.le
  have hz : z ≤ t.darts + o + R := by
    have h := selectedIndex_le t padding v step.1
    change z ≤ t.darts + o + step.1.val at h
    omega
  have hsize : (k + padding v) * q ≤ R := by
    dsimp only [R, k]
    rw [encodeWords_length, ExpanderTableWords.rotationWords_length]
    omega
  have hrotor : rotorSteps (tables v) rank p ≤ 2 * rank.val + 6 * R + 8 := by
    have h := rotorSteps_le (tables v) rank p
    change rotorSteps (tables v) rank p ≤ 2 * rank.val + 5 * R + (k + padding v) * q + 8 at h
    omega
  have hselect : selectionSteps t padding v step.1 ≤
      10 * L + 16 * R + 13 * v.val + 5 * k + 2 * o + 2 * t.darts + 57 := by
    have h := selectionSteps_le t padding v step.1
    change selectionSteps t padding v step.1 ≤
      10 * L + 16 * step.1.val + 13 * v.val + 5 * k + 2 * o + 2 * t.darts + 57 at h
    omega
  have hfinish : finishSteps q X z step.2.val outputLength ≤
      (5 * q + 7) * (t.darts + o + R) + 7 * q + 4 * X + 2 * outputLength + 40986 := by
    have h := finishSteps_le q X z step.2.val outputLength
    have hm := Nat.mul_le_mul_left (5 * q + 7) hz
    omega
  have htail : tailBits.length ≤ R := encodedDrop_length_le
    (ExpanderTableWords.rotationWords (tables v)) (q * rank.val + p.val + 1)
  have hcleanup : tailBits.length + step.1.val + step.2.val + z + 15 ≤
      R + R + q + (t.darts + o + R) + 15 := by omega
  calc
    coreSteps t padding tables v x p outputLength
        ≤ (2 * rank.val + 6 * R + 8) +
          (10 * L + 16 * R + 13 * v.val + 5 * k + 2 * o + 2 * t.darts + 57) +
          ((5 * q + 7) * (t.darts + o + R) + 7 * q + 4 * X + 2 * outputLength + 40986) +
          (R + R + q + (t.darts + o + R) + 15) :=
      Nat.add_le_add (Nat.add_le_add (Nat.add_le_add hrotor hselect) hfinish) hcleanup
    _ = _ := by
      change _ = coreTimeBound q L R X v.val rank.val k o t.darts outputLength
      unfold coreTimeBound
      ring

def coreMachineInTime (q : Nat) (positive : 0 < q) (t : GraphTables.Table)
    (padding : Fin t.vertices → Nat)
    (tables : ∀ v, ExpanderTables.Table (PreprocessingCloudIndex.cloudSize t v + padding v) q)
    (v : Fin t.vertices) (x : PreprocessingCloudIndex.PaddedCloud t padding v)
    (p : Fin q) (output : List Bool) :
    StateTransition.EvalsToInTime (coreMachine q positive p).step
      ⟨some (coreEntry q), coreInitialState q positive (), coreInputTapes t padding tables v x output⟩
      (some ⟨none, coreInitialState q positive (), coreInputTapes t padding tables v x
        (output ++ encodeWords (GraphTables.rowWords (PreprocessingInternalRows.row t padding tables v x p)))⟩)
      (coreTimeBound q (GraphTables.tableBits t).length
        (encodeWords (ExpanderTableWords.rotationWords (tables v))).length
        (PreprocessingRegularTables.vertexOrder t padding x.val).val v.val
        (PreprocessingCloudIndex.paddedCloudRank t padding v x).val
        (PreprocessingCloudIndex.cloudSize t v)
        (PreprocessingPaddingOffsets.offset padding v.val) t.darts output.length) where
  steps := coreSteps t padding tables v x p output.length
  evals_in_steps := coreTrace q positive p id none (coreMachine q positive p).m
    (fun _ => rfl) t padding tables v x output ()
  steps_le_m := coreSteps_le q positive t padding tables v x p output.length

end DFVSGames.Foundations.Complexity.MachineRegularInternalRow
end

section

namespace DFVSGames.Foundations.Complexity.MachineRegularOriginalClean

open Turing MachineComposition PCP PCP.GraphTables PCP.PreprocessingRegularTables

abbrev Tape := MachineRegularOriginalRow.Tape
abbrev Alphabet := MachineRegularOriginalRow.Alphabet
abbrev State := MachineRegularOriginalRow.State

inductive Label
  | row (label : MachineRegularOriginalRow.Label)
  | query | scan | rawReverse | computedReverse | relation
  deriving DecidableEq, Fintype

def instruction {σ Λ : Type} (q : Nat) (labels : Label → Λ) (exit : Option Λ) :
    Label → TM2.Stmt Alphabet Λ (State σ)
  | .row l => MachineRegularOriginalRow.instruction q (fun l => labels (.row l))
      (some (labels .query)) l
  | .query => MachineDrain.drain 2 (labels .query) (some (labels .scan))
  | .scan => MachineDrain.drain 3 (labels .scan) (some (labels .rawReverse))
  | .rawReverse => MachineDrain.drain 4 (labels .rawReverse) (some (labels .computedReverse))
  | .computedReverse => MachineDrain.drain 6 (labels .computedReverse) (some (labels .relation))
  | .relation => MachineDrain.drain 7 (labels .relation) exit

def cleanupTapes (base : Tape → List Bool) : Tape → List Bool :=
  Function.update (Function.update (Function.update (Function.update
    (Function.update base 2 []) 3 []) 4 []) 6 []) 7 []

def cleanupSteps (base : Tape → List Bool) : Nat :=
  ((base 2).length + 1) + ((base 3).length + 1) + ((base 4).length + 1) +
    ((base 6).length + 1) + ((base 7).length + 1)

private theorem join_trace_inline_MachineRegularOriginalClean {A : Type*} {f : A → A} {m n : Nat} {a b c : A}
    (first : f^[m] a = b) (second : f^[n] b = c) : f^[m + n] a = c := by
  rw [Nat.add_comm m n, Function.iterate_add_apply, first, second]

theorem cleanupTraceAt {σ Λ : Type} (q : Nat) (labels : Label → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt Alphabet Λ (State σ))
    (code : ∀ l, program (labels l) = instruction q labels exit l)
    (base : Tape → List Bool) (state : State σ) :
    (advance (TM2.step program))^[cleanupSteps base]
      (some ⟨some (labels .query), state, base⟩) =
      some ⟨exit, (state.1, none), cleanupTapes base⟩ := by
  let t1 := Function.update base (2 : Tape) []
  let t2 := Function.update t1 (3 : Tape) []
  let t3 := Function.update t2 (4 : Tape) []
  let t4 := Function.update t3 (6 : Tape) []
  have h0 := MachineDrain.drainTrace (2 : Tape) (labels .query) (some (labels .scan))
    program (code .query) base (base 2) state.1 state.2
  have h1 := MachineDrain.drainTrace (3 : Tape) (labels .scan) (some (labels .rawReverse))
    program (code .scan) t1 (t1 3) state.1 none
  have h2 := MachineDrain.drainTrace (4 : Tape) (labels .rawReverse) (some (labels .computedReverse))
    program (code .rawReverse) t2 (t2 4) state.1 none
  have h3 := MachineDrain.drainTrace (6 : Tape) (labels .computedReverse) (some (labels .relation))
    program (code .computedReverse) t3 (t3 6) state.1 none
  have h4 := MachineDrain.drainTrace (7 : Tape) (labels .relation) exit
    program (code .relation) t4 (t4 7) state.1 none
  simp only [Function.update_eq_self] at h0 h1 h2 h3 h4
  simp only [t1, t2, t3, t4, Function.update_of_ne (by decide : (3 : Tape) ≠ 2),
    Function.update_of_ne (by decide : (4 : Tape) ≠ 3),
    Function.update_of_ne (by decide : (4 : Tape) ≠ 2),
    Function.update_of_ne (by decide : (6 : Tape) ≠ 4),
    Function.update_of_ne (by decide : (6 : Tape) ≠ 3),
    Function.update_of_ne (by decide : (6 : Tape) ≠ 2),
    Function.update_of_ne (by decide : (7 : Tape) ≠ 6),
    Function.update_of_ne (by decide : (7 : Tape) ≠ 4),
    Function.update_of_ne (by decide : (7 : Tape) ≠ 3),
    Function.update_of_ne (by decide : (7 : Tape) ≠ 2)] at h1 h2 h3 h4
  exact join_trace_inline_MachineRegularOriginalClean (join_trace_inline_MachineRegularOriginalClean (join_trace_inline_MachineRegularOriginalClean (join_trace_inline_MachineRegularOriginalClean h0 h1) h2) h3) h4

structure Input (t : Table) (e : Fin t.darts) (base : Tape → List Bool) : Prop
    extends MachineRegularOriginalRow.Input t e base where
  queryEmpty : base 2 = []
  scanEmpty : base 3 = []

theorem cleanup_frame (q : Nat) (t : Table) (e : Fin t.darts)
    (base : Tape → List Bool) (input : Input t e base) :
    cleanupTapes (MachineRegularOriginalRow.finalTapes q t e base) =
      Function.update base 9
        (base 9 ++ encodeWords (MachineRegularOriginalRow.emittedWords q t e)) := by
  funext k
  fin_cases k <;>
    simp [cleanupTapes, MachineRegularOriginalRow.finalTapes,
      MachineRegularOriginalRow.affined, MachineRegularOriginalRow.copied,
      MachineRegularOriginalRow.looked, MachineAffineLookup.finalTapes,
      MachinePreservingLookup.finalTapes, MachineLookup.tapes,
      MachineRegularOriginalRow.lookupTape, input.queryEmpty, input.scanEmpty,
      input.reverseEmpty, input.computedEmpty, input.relationEmpty]

theorem suffix_length_le (t : Table) (e : Fin t.darts) :
    (encodeWords (MachineRegularOriginalRow.suffixWords t e)).length ≤ (tableBits t).length := by
  have hlength : (tableBits t).length =
      (encodeWords (MachineRegularOriginalRow.prefixWords t e)).length +
        ((encodeWords (rowWords t.rows[e])).length +
          (encodeWords (MachineRegularOriginalRow.suffixWords t e)).length) := by
    rw [tableBits, MachineRegularOriginalRow.tableWords_at_row,
      encodeWords_append, encodeWords_append, List.length_append, List.length_append]
  omega

theorem cleanupSteps_eq (q : Nat) (t : Table) (e : Fin t.darts)
    (base : Tape → List Bool) (input : Input t e base) :
    cleanupSteps (MachineRegularOriginalRow.finalTapes q t e base) =
      (encodeWords (MachineRegularOriginalRow.suffixWords t e)).length +
        (q + 2) * t.rows[e].reverseIndex.val + q +
          (encodeWords (relationWords t.rows[e].relation)).length + 8 := by
  have hquery : MachineRegularOriginalRow.finalTapes q t e base 2 = encodeWord 0 := by
    simp only [MachineRegularOriginalRow.finalTapes,
      Function.update_of_ne (by decide : (2 : Tape) ≠ 9), MachineRegularOriginalRow.affined,
      Function.update_of_ne (by decide : (2 : Tape) ≠ 6), MachineRegularOriginalRow.copied,
      Function.update_of_ne (by decide : (2 : Tape) ≠ 7),
      Function.update_of_ne (by decide : (2 : Tape) ≠ 3)]
    change MachineLookup.tapes (2 : Tape) 3 4 base
      (encodeWord 0 ++ base 2)
      (encodeWords ((GraphTables.tableWords t).drop (4098 * e.val + 3 + 1)) ++ base 3)
      (encodeWord t.rows[e].reverseIndex.val ++ base 4) 2 = _
    rw [MachineLookup.tapes_index _ _ _ (by decide) (by decide), input.queryEmpty,
      List.append_nil]
  have hscan : MachineRegularOriginalRow.finalTapes q t e base 3 =
      encodeWords (MachineRegularOriginalRow.suffixWords t e) := by
    simp only [MachineRegularOriginalRow.finalTapes,
      Function.update_of_ne (by decide : (3 : Tape) ≠ 9), MachineRegularOriginalRow.affined,
      Function.update_of_ne (by decide : (3 : Tape) ≠ 6), MachineRegularOriginalRow.copied,
      Function.update_of_ne (by decide : (3 : Tape) ≠ 7), Function.update_self,
      input.scanEmpty, List.append_nil]
  have hraw : MachineRegularOriginalRow.finalTapes q t e base 4 =
      encodeWord t.rows[e].reverseIndex.val := by
    simp only [MachineRegularOriginalRow.finalTapes,
      Function.update_of_ne (by decide : (4 : Tape) ≠ 9), MachineRegularOriginalRow.affined,
      Function.update_of_ne (by decide : (4 : Tape) ≠ 6), MachineRegularOriginalRow.copied_reverse,
      input.reverseEmpty, List.append_nil]
  have hcomputed : MachineRegularOriginalRow.finalTapes q t e base 6 =
      encodeWord ((q + 1) * t.rows[e].reverseIndex.val + q) := by
    simp only [MachineRegularOriginalRow.finalTapes,
      Function.update_of_ne (by decide : (6 : Tape) ≠ 9), MachineRegularOriginalRow.affined_reverse,
      input.computedEmpty, List.append_nil]
  have hrelation : MachineRegularOriginalRow.finalTapes q t e base 7 =
      encodeWords (relationWords t.rows[e].relation) := by
    simp only [MachineRegularOriginalRow.finalTapes,
      Function.update_of_ne (by decide : (7 : Tape) ≠ 9), MachineRegularOriginalRow.affined_relation,
      input.relationEmpty, List.append_nil]
  simp only [cleanupSteps, hquery, hscan, hraw, hcomputed, hrelation, encodeWord_length]
  ring

theorem cleanupSteps_le (q : Nat) (t : Table) (e : Fin t.darts)
    (base : Tape → List Bool) (input : Input t e base) :
    cleanupSteps (MachineRegularOriginalRow.finalTapes q t e base) ≤
      (q + 3) * (tableBits t).length + q + 8200 := by
  rw [cleanupSteps_eq q t e base input]
  have hsuffix := suffix_length_le t e
  have htable := tableWords_length_le_bits t
  have hr : t.rows[e].reverseIndex.val ≤ (tableBits t).length := by
    have hrlt := t.rows[e].reverseIndex.isLt
    omega
  have hmul := Nat.mul_le_mul_left (q + 2) hr
  have hrel := relationBits_length_le t.rows[e].relation
  nlinarith

def steps (q : Nat) (t : Table) (e : Fin t.darts) (base : Tape → List Bool) : Nat :=
  MachineRegularOriginalRow.steps q t e base +
    cleanupSteps (MachineRegularOriginalRow.finalTapes q t e base)

def timeBound (q inputLength outputLength : Nat) : Nat :=
  (5 * q + 20) * inputLength + 2 * outputLength + 5 * q + 40995

theorem steps_le (q : Nat) (t : Table) (e : Fin t.darts)
    (base : Tape → List Bool) (input : Input t e base) :
    steps q t e base ≤ timeBound q (tableBits t).length (base 9).length := by
  have hrow := MachineRegularOriginalRow.steps_le q t e base
  have hclean := cleanupSteps_le q t e base input
  unfold steps timeBound MachineRegularOriginalRow.timeBound at *
  nlinarith

theorem traceAt {σ Λ : Type} (q : Nat) (labels : Label → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt Alphabet Λ (State σ))
    (code : ∀ l, program (labels l) = instruction q labels exit l)
    (t : Table) (e : Fin t.darts) (base : Tape → List Bool) (input : Input t e base)
    (ambient : σ) (register : Option Bool) :
    (advance (TM2.step program))^[steps q t e base]
      (some ⟨some (labels (.row (.lookup .seed))),
        ((ambient, MachineRegularOriginalRow.zeroBuffer), register), base⟩) =
      some ⟨exit, ((ambient, MachineRegularOriginalRow.zeroBuffer), none),
        Function.update base 9
          (base 9 ++ encodeWords (MachineRegularOriginalRow.emittedWords q t e))⟩ := by
  have row := MachineRegularOriginalRow.traceAt q (fun l => labels (.row l))
    (some (labels .query)) program (fun l => code (.row l)) t e base input.toInput ambient register
  have clean := cleanupTraceAt q labels exit program code
    (MachineRegularOriginalRow.finalTapes q t e base)
    ((ambient, MachineRegularOriginalRow.zeroBuffer), none)
  rw [cleanup_frame q t e base input] at clean
  exact join_trace_inline_MachineRegularOriginalClean row clean

def machine (q : Nat) : FinTM2 where
  K := Tape
  k₀ := 1
  k₁ := 9
  Γ := Alphabet
  Λ := Label
  main := .row (.lookup .seed)
  σ := State Unit
  initialState := (((), MachineRegularOriginalRow.zeroBuffer), none)
  m := instruction q id none

def machineInTime (H : BaseTable) (t : Table) (e : Fin t.darts)
    (base : Tape → List Bool) (input : Input t e base) (register : Option Bool) :
    StateTransition.EvalsToInTime (machine internalDegree).step
      ⟨some (.row (.lookup .seed)), (((), MachineRegularOriginalRow.zeroBuffer), register), base⟩
      (some ⟨none, (((), MachineRegularOriginalRow.zeroBuffer), none),
        Function.update base (9 : Tape)
          (base (9 : Tape) ++ encodeWords (rowWords (MachineRegularOriginalRow.inheritedRow H t e)))⟩)
      (timeBound internalDegree (tableBits t).length (base (9 : Tape)).length) where
  steps := steps internalDegree t e base
  evals_in_steps := by
    change (advance (TM2.step (instruction internalDegree id none)))^[_] _ = _
    rw [MachineRegularOriginalRow.inheritedRow_words]
    exact traceAt internalDegree id none (instruction internalDegree id none)
      (fun _ => rfl) t e base input () register
  steps_le_m := steps_le internalDegree t e base input

end DFVSGames.Foundations.Complexity.MachineRegularOriginalClean
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